Micro-optical components for depth-controlled Bessel beams

Bessel beams have an extended focal zone and self-healing properties, which can be advantagous in many different fields, e.g., in material processing or in life science applications. This work describes the conceptualization, design, fabrication, and characterization of several micro-optical components to generate and control Bessel beams. Basic considerations including axicons with rounded tips, limitations regarding the aperture, and the evaluation of Bessel beams using asymmetric illumination sources were confirmed with beam propagation methods and measurements. Based on this knowledge different micro-optical components were developed, including an adaptive liquid crystal ring aperture array to segment the extended focal zone of Bessel beams for depth-control and several transmissive aspherical lens arrays: a collimation lens for diverging light sources, two different axicons to generate a Bessel beam, and a lensacon, a combination of a lens and an axicon, to reduce the number of optical surfaces. Finally, a compact micro-optical system with minimal alignment and assembly errors was demonstrated by successfully combining the developed micro-optical components.

Angelina Müller received her Bachelor’s and Master’s degree in Microsystems Engineering from the University of Freiburg. In May 2013 she joined the Laboratory for Microactuators of Prof. Dr. Ulrike Wallrabe at the IMTEK – Department of Microsystems Engineering. Her research focuses on the development of different micro-optical components and systems including the fabrication of aspherical micro-lenses and liquid crystal based devices.

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Angelina Müller

Micro-optical components fordepth-controlled Bessel beams


Angelina Müller

IMTEK, University of Freiburg

Microactuators – Design and Technology Vol. 17


Dissertation zur Erlangung des Doktorgradesder Technischen Fakultät

der Albert-Ludwigs-Universität Freiburg im Breisgau

Angelina MüllerFebruary 2019

Dean: Prof. Dr.Hannah Bast

Referees: Prof. Dr.Ulrike WallrabeProf. Dr.Ulrich Theodor Schwarz

Date of Submission: 05. February 2019

Date of Disputation: 05. April 2019

Microactuators – Design and Technologyedited by Prof. Dr. Ulrike Wallrabe

Volume 17

Angelina Müller


Angelina Müller

Laboratory for Microactuators

IMTEK – Department of Microsystems Engineering

University of Freiburg

Freiburg im Breisgau, Germany

ISSN: 2567-0921


Angelina Müller

AbstractNew, non-invasive, highly compact, and integrated optical systems are needed, forexample, for optogenetic stimulation. This thesis describes the conceptulization, de-sign, fabrication, and characterization of several micro-optical components to gener-ate and control Bessel beams. Their self-reconstructing properties promise enhancedtissue penetration and advanced depth-control.Necessary basic analytical considerations included axicons with rounded tips, limi-tations regarding the aperture width to control a Bessel beam along its propagationdirection, and the evaluation of Bessel beams using asymmetric illumination sources.All considerations were confirmed with beam propagation methods and measure-ments. Although the rounding of the axicon tip generates a sharp focal point, theinterfering waves converge to the intensity profile of an ideal axicon. An easy touse expression was derived for the shortest possible Bessel beam without diffractioneffects, showing a dependency of the aperture width on its radial position. Thisuniversally valid expression can be used for other applications where Bessel beamsneed to be tailored. The use of asymmetric and astigmatic illumination sources,i.e., edge emitting laser diodes, with rounded tip axicons showed a transition of asymmetric Bessel beam into a bow-tie shaped pattern. For characteristic propertieslike intensity, core radius, and ellipticity, two characteristic regions were determined:an approximately rotationally symmetric central part (slow-axis) transitioning intoa asymmetric bow-tie shaped intensity distribution (fast-axis).Micro-optical components were developed based on this knowledge. The first com-ponent was an adaptive liquid crystal ring aperture array to segment the extendedfocal zone of Bessel beams for depth-control. The fabrication process based onwafer-level clean room technologies allowed an efficient production of different de-signs. Their experimental characterization demonstrated a clear segmentation ofBessel beams and good optical quality with only a small wavefront error betweenλ/5 and λ/2.The second category of components were different transmissive aspherical lens ar-rays, where four different lens types were produced: a collimation lens for diverginglight sources, two different axicons to generate a Bessel beam, and a lensacon, acombination of a lens and an axicon, to reduce the number of optical surfaces. Thefabrication of the lenses was realized by a further developed rapid-prototyping pro-cess for lens molds based on surface deformations due to thermal expansion of asilicone. The needed initial cavity was simulated using precise shape optimization

i

with finite element methods. Material parameters showed a critical influence onthe result. Therefore, a novel, more precise method to determine an exact valuefor the Poisson’s ratio and the linear coefficient of thermal expansion of the usedsilicone was developed. The characterization of the different lens types showed ahigh, state-of-the-art optical quality with wavefront errors between λ/7 and λ/2.Finally, a compact micro-optical system with minimal alignment and assembly errorswas demonstrated by successfully combining the previous micro-optical componentsinto an integrated, fully functional system. Thereby, a proof-of-concept for suchsystems with relevance far beyond optogenetic stimulation was achieved.

ii

ZusammenfassungNeue, nicht-invasive, hoch kompakte und integrierte optische Systeme werden bei-spielsweise für optogenetische Stimulation benötigt. Diese Arbeit beschreibt Konzep-tion, Entwurf, Herstellung und Charakterisierung verschiedener mikrooptischer Kom-ponenten zur Erzeugung und Kontrolle von Besselstrahlen. Deren selbstrekonstru-ierende Eigenschaften versprechen verbesserte Penetration in Gewebe sowie erwei-terte Tiefenkontrolle.Notwendige grundlegende analytische Betrachtungen umfassten Axicons mit abge-rundeten Spitzen, Begrenzungen bei der Blendenöffnung zur Kontrolle von Bessel-strahlen entlang der Ausbreitungsrichtung sowie die Evaluierung von Besselstrahlenbei asymmetrischen Lichtquellen. Alle Betrachtungen wurden durch Strahlausbrei-tungsmethoden und Messungen bestätigt. Obwohl die Verrundung der Axiconspitzeeinen scharfen Fokuspunkt generiert, konvergieren die interferierenden Wellen zueinem Intensitätsprofil eines idealen Axicons. Um einen möglichst kurzen Bessel-strahl ohne Beugungseffekte zu erzeugen, wurde eine einfach zu benutzende Glei-chung hergeleitet, die eine Abhängigkeit der Aperturbreite mit dem Radius her-stellt. Diese Gleichung ist allgemeingültig und kann auch für andere Anwendun-gen von maßgeschneiderten Besselstrahlen verwendet werden. Die Verwendung vonasymmetrischen und astigmatischen Lichtquellen, zum Beispiel kantenemittierendeLaserdioden, zusammen mit Axicons mit verrundeter Spitze zeigte den Übergangeines symmetrischen Besselstrahls zu einem schmetterlingsförmigen Muster. Fürcharakteristische Eigenschaften wie Intensität, Radius und Elliptizität wurden zweicharakteristische Regionen bestimmt: ein näherungsweise rotationssymmetrischerMittelteil (slow-axis), der in eine asymmetrische schmetterlingsförmige Intensitäts-verteilung (fast-axis) übergeht.Basierend auf diesen Erkenntnissen wurden mikrooptische Komponenten entwick-elt. Die erste Komponente war ein adaptives Flüssigkristall Ringblendenarray, umdie lange Fokuszone des Besselstrahls in kleinere Teilsegmente unterteilen zu kön-nen und somit den Strahl zu kontrollieren. Der Fabrikationsprozess basierend aufStandard wafer-level Reinraumtechnologien erlaubte eine effiziente Herstellung un-terschiedlicher Entwürfe. Deren Charakterisierung zeigte, dass der Besselstrahl inunterschiedliche Segmente zerteilt werden kann und nur geringe Wellenfrontfehlerzwischen λ/5 und λ/2 aufweist.Die zweite Komponentenkategorie bestand aus transmissiven asphärischen Linsenar-rays, wobei vier unterschiedliche Linsentypen hergestellt wurden: eine Kollimation-

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slinse für divergente Strahlquellen, zwei verschiedene Axicons zur Erzeugung vonBesselstrahlen sowie ein Lensacon, eine Kombination einer Linse mit einem Axicon,um die Anzahl der optischen Oberflächen zu reduzieren. Der Herstellungsprozessder Linsen erfolgte durch die Weiterentwicklung einer Rapid-Prototyping Technolo-gie für Linsengussformen basierend auf einer Oberflächendeformation durch thermi-sche Ausdehnung eines Silikons. Die dafür benötigte Urform wurde durch präziseFormoptimierungen mittels Finiter Elemente Methode bestimmt. Die Materialpa-rameter zeigten hierbei einen kritischen Einfluss auf das Ergebnis. Zu diesem Zweckwurde eine neuartige und genauere Methode zur exakten Bestimmung der Poisson-zahl und des linearen thermischen Ausdehnungskoeffizienten des Silikons entwickelt.Die Charakterisierung der verschiedenen Linsentypen zeigte eine hohe optische Qua-lität auf dem Stand der Technik, wobei der Wellenfrontfehler zwischen λ/7 und λ/2variierte.Abschließend wurde ein kompaktes mikrooptisches System mit minimalen Ausrich-tungs - und Fertigungsfehlern entwickelt, bei dem die zuvor entwickelten Komponen-ten erfolgreich in ein voll funktionierendes integriertes System kombiniert wurden.Damit wurde ein Machbarkeitsnachweis für solche weit über optogenetische Stimu-lation hinaus relevante Systeme erbracht.

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Nomenclature

Greek Symbols

Symbol Description

α Conical angle of a Bessel beamαcritical Critical conical angleαCTE Linear coefficient of thermal expansionαCTE Coefficient of thermal expansion including εp and ∆T∆αcone Angular shiftβ Opening angle of the bow-tie shaped intensity patternχ Rotational viscosityε Ellipticity of the intensity in the first maximum Imax,1ε‖ Parallel permittivityε⊥ Perpendicular permittivityε0 Vacuum permittivityεalign Permittivity of the alignment layerεlc Permittivity of the liquid crystal∆ε Dielectric anisotropyη Kinematic viscosityγ Axicon apex angleιi Coefficient describing the deviations from a spherical surfaceκ Dual spatial base vectorΛ Angle between molecule axis and the directorλ Wavelengthν Poisson’s ratioΦ Wavefront phaseφ Azimuthal componentΠ Power of the illumination beamΨ Electric field distributionψ Amplitude of the electric field distributionρ Mass densityσ StressσHn Measurement uncertaintyθ Opening angleθd Diffraction opening angleθdestructive Opening angle that causes destructive interference

v

Symbol Description

θfast Opening angle of the fast axisθm Effective opening angleθs Angle of the substrate modeθslow Opening angle of the slow axis∆θs Opening angle of the substrate modeε Strainεp Polymerization shrinkageε Contribution from εp and ∆Tϕ Angle of the lateral displacement, angular offset∆ϕ Opening angle of the lateral displacementζ Correction factor

Roman Symbols

Symbol Description

A Areaa AstigmatismB AbsorbermatrixBn Bernstein polynomialbη,n Bernstein basis polynomialC CapacityCη Coefficient of the Bernstein polynomialCHn Factor relating the probability distributionsc Curvature of the surfaceDi,j Linear elastic tensord Aperture widthdalign Alignment layer thicknessdlc Liquid crystal cell thicknessdmin Optimal aperture widthds Laser diode substrate thicknessd Scale-invariant aperture widthdmin Scale-invariant optimal aperture widthE Young’s modulusF Elastic energy densityfc Focal lengthffit Even polynomial of 14th degreefHn,T Interpolation functionftarget Target function, deflection of the expanding polymerfobj Objective function∆ftarget Deviation of S(x,y) from ftargetgcritical Critical parameter for destructive interference

vi

Symbol Description

H Height or cavity depthh(r) Surface profileha Position of the substrate mode in the plane of the axiconhs Position of the substrate mode in the plane of the lens∆h Surface deformation from pseudo-shrinkageI IntensityIcore Central intensity of the Bessel beamIint Integrated intensityImax Maximal intensityImax,1 Maximal intensity of the first ring of the Bessel beamJ0 0-th order Bessel functionJm m-th order Bessel functionj1,0 First zero point of the 0-th order Bessel functionK1 Elastic constant: splayK2 Elastic constant: twistK3 Elastic constant: bendkc Conical constantk Wavenumberkr Radial wave vectorkz Axial wave vectorl Length of the light pathla Distance between lens and axiconls Length of the light path of the substrate mode after the axicon∆lcone Shift of the propagation length∆ls Length difference of the main beam and the substrate moden Director of the liquid crystaln Refractive indexne Extraordinary refractive indexno Ordinary refractive index∆n Difference in refractive index, optical anisotropyP Propagatorr Radial componentr1 Lower interval boundary, rounding radiusr2 Upper interval boundaryrcore Radius of the first Bessel beam minimum, central core radiusrc Radius of curvaturerfast Maximal radius of the fast axisrlens Focal length of the lens at the axicon tiprmax Maximal radiusrobstacle Radius of an obstaclerslow Maximal radius of the slow axisr Scale-invariant radius∆rmin Shift of the first Bessel beam minimum

vii

Symbol Description

S Order parameter, degree of orientational orderS(x,y) Surface deviationsrms Rms value of the surface deviationT Temperaturet Timetoff Switch off timeton Switch on time∆T Temperature differenceVthreshold Threshold voltageW Gaussian probabilityx, y, z Cartesian coordinatesz0 Focal length of the lens at the axicon tipzend End position of the segmented Bessel beam on the optical axiszm Modulation period, m ∈ Zzmax Maximal propagation length of a Bessel beamzmax,fast Maximal propagation length of the fast axis on the optical axiszmax,slow Maximal propagation length of the slow axis on the optical axiszobstacle Position of an obstacle on the optical axis, propagation lengthzs Position of the substrate mode on the optical axiszstart Start position of the segmented Bessel beam on the optical axisz Scale-invariant position on the optical axis∆z Length of the segmented Bessel beam∆zC Length of the segmented Bessel beam in the classical limit∆zF Length of the segmented Bessel beam in the Fraunhofer limit∆zmin Shortest segmented Bessel beam without diffraction∆zs Length of the substrate mode on the optical axis∆zC Scale-invariant ∆zC∆zF Scale-invariant ∆zF∆zmin Scale-invariant ∆zmin

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Abbreviations

Abbreviation Description

2D Two-dimensional3D Three-dimensional5CB 4-Cyano-4’-pentylbiphenylBPM Beam propagation methodCNC Computerized Numerical ControlDPSS Diode pumped solid stateFEM Finite element methodFFT Fast Fourier transformHeNe Helium-neonITO Indium tin oxideLD Laser diodeLED Light emitting diodeMEMS Micro electro mechanical systemsND Neutral densityNOA Norland Optical AdhesivePCB Printed circuit boardPDMS PolydimethylsiloxanePEDOT Poly(3,4-ethylenedioxythiophene)PET Polyethylene terephthalatePV Peak-to-valleyrms Root mean squareUV Ultraviolet

ix

Contents

Abstract i

Zusammenfassung iii

Nomenclature v

1 Introduction 11.1 Project environment . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Outline and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Measurement methods 72.1 Optical surface measurements . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Surface data evaluation . . . . . . . . . . . . . . . . . . . . . . 72.2 Optical characterization method . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Optical measurement setup . . . . . . . . . . . . . . . . . . . 92.2.2 Beam propagation method . . . . . . . . . . . . . . . . . . . . 102.2.3 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Bessel beams 133.1 Introduction to Bessel beams . . . . . . . . . . . . . . . . . . . . . . 133.2 General description of Bessel beams . . . . . . . . . . . . . . . . . . . 143.3 Bessel beams generated from axicons . . . . . . . . . . . . . . . . . . 15

3.3.1 Analytical considerations . . . . . . . . . . . . . . . . . . . . . 153.3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Segmented Bessel beams . . . . . . . . . . . . . . . . . . . . . . . . . 213.4.1 Analytical considerations . . . . . . . . . . . . . . . . . . . . . 213.4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Bessel beams generated from edge emitting laser diodes . . . . . . . . 283.5.1 Analytical considerations . . . . . . . . . . . . . . . . . . . . . 283.5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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Contents

4 Adaptive ring aperture 434.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Overview of different concepts . . . . . . . . . . . . . . . . . . . . . . 44

4.2.1 Selection of the method . . . . . . . . . . . . . . . . . . . . . 454.3 Liquid crystal technology . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3.1 Twisted-nematic liquid crystal device . . . . . . . . . . . . . . 474.4 Fabrication and assembly . . . . . . . . . . . . . . . . . . . . . . . . . 494.5 Chip design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.6 Optical characterization . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.6.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 Transmissive aspherical optics 635.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2 Overview of different fabrication technologies . . . . . . . . . . . . . . 635.3 Basic fabrication method of the transmissive aspherical optics . . . . 655.4 Analytical considerations . . . . . . . . . . . . . . . . . . . . . . . . . 665.5 Simulation and shape optimization . . . . . . . . . . . . . . . . . . . 67

5.5.1 Evaluation of the simulation model . . . . . . . . . . . . . . . 695.6 Material parameters of PDMS . . . . . . . . . . . . . . . . . . . . . . 705.7 Fabrication process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.7.1 Fabrication limits . . . . . . . . . . . . . . . . . . . . . . . . . 745.8 Lens designs and fabrication . . . . . . . . . . . . . . . . . . . . . . . 775.9 Experimental characterization . . . . . . . . . . . . . . . . . . . . . . 80

5.9.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.9.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6 Compact micro-optical system 996.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.2 Fabrication and assembly . . . . . . . . . . . . . . . . . . . . . . . . . 996.3 Experimental characterization . . . . . . . . . . . . . . . . . . . . . . 102

6.3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7 Conclusion and Outlook 1077.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

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Contents

A Appendix 113A.1 Appendix: adaptive ring aperture . . . . . . . . . . . . . . . . . . . . 113

A.1.1 Electric characterization . . . . . . . . . . . . . . . . . . . . . 113A.2 Appendix: transmissive aspherical optics . . . . . . . . . . . . . . . . 115A.3 Appendix: Poisson’s ratio and coefficient of thermal expansion . . . . 118

A.3.1 FEM simulations . . . . . . . . . . . . . . . . . . . . . . . . . 118A.3.2 Experimental procedure . . . . . . . . . . . . . . . . . . . . . 118A.3.3 Data evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 121A.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121A.3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Acknowledgments 127

Author’s Bibliography 129

References 131

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1 IntroductionThe idea to use and manipulate light has been fascinating mankind since thousandsof years. One of the oldest remaining glass fabrication recipes dates back over 2700years [1]. Since then, optical technology revolutionized and can be found nearlyeverywhere in our modern lives. Both macro- and micro-optics are widely used intelecommunication systems, consumer-electronics, cameras, but also in analyticalsciences and biomedical applications. Miniaturization and system integration ofmicro-optical components is essential to reduce cost, robustness, and stability ofoptical systems [1].Another application field in life sciences are miniaturized microscopes for advancedimaging applications in freely behaving animals. Recently, its outstanding scientificvalue was recognized by being named the Method of the Year 2018 by Nature Meth-ods [2].Optogenetics has been a steadily expanding research field over the last severalyears [3]. The technique involves genetically altered proteins (opsins) to modifyphotosensitive cell membranes, which can then be specifically addressed using lightof a specific wavelength (e.g., channelrhodopsin 2 responds to blue light) [3,4]. Thekey advantage of this technology compared to, e.g., electrical stimulation is theability to target only specific neurons, with a high level of spatial selectivity. Themethod is used to stimulate specific brain regions, and motor or cardiac musclefunctions in new types of neural prosthetics [4–6]. The most common way to in-troduce light into the brain is by inserting a rigid optical fiber into the region ofinterest. Also, most micro-fabricated approaches rely on rigid needle-type devicespenetrating the tissue. Therefore, new, non-invasive, highly compact, and integratedoptical stimulation systems are needed to avoid penetration of tissue, also far beyondoptogenetics.

1.1 Project environmentThe main motivation for this work originates from a research project within theCluster of Excellence BrainLinks-BrainTools. The scientific challenge of this clus-ter was to improve the interaction of technical systems with the brain. Within theproject, the goal was to develop a miniaturized stimulation device for optogeneticstimulation of small rodent brains without invasive penetration of brain tissue. The

1

1 Introduction

Laser diode array

Collimation lens array

Adaptive ring aperture

Axicon array

Depth-controlled Bessel beam

z

Brain tissue

Figure 1.1 Schematic of the miniaturized stimulation device. The indicated area (dashedline) shows the micro-optical components developed in this thesis.

device should be able to optically stimulate neurons with high spatio-temporal res-olution and should simultaneouly be able to perform multi-focal stimulation. ABessel beam should be used to avoid physical penetration of the brain, using itsunique features: the extended focal zone and self-healing properties, which can beused for enhanced penetration of scattering tissue [7].

The miniaturized stimulation device consists of an array of edge emitting blue laserdiodes, an aspherical lens array for collimation, an adaptive ring aperture to controlthe beam along the optical axis (depth-control), and a conically shaped lens array(axicon array) to generate Bessel beams, as illustrated in Figure 1.1.

In cooperation with the Laboratory for Power Electronics of Prof. Oliver Ambacherthe tool should be developed and then tested in vitro and in vivo with the othercooperation partners: Prof. Ulrich Egert from the Laboratory for Biomicrotechnol-ogy, Prof. Marlene Bartos from the Institute for Physiology, Systemic and CellularNeurophysiology, and Prof. Ilka Diester from the Optophysiology Laboratory, De-partment of Biology III.

The project-relevant specifications were set to be: a 3× 3 array with a pitch of1.2 mm, resulting in nine 1 mm diameter elements. Each element should be individ-ually addressable, adaptive and should fit to a rat brain. The light source should beable to activate the light-sensitive protein channelrhodopsin 2. Its activation band-with is between 420 nm and 550 nm, with a maximum activation at 470 nm [4].

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1.2 Objectives

1.2 ObjectivesThe primary objectives of this thesis were the concept, design, fabrication and char-acterization of the adaptive ring aperture and various transmissive optical compo-nents, i.e., all components in the indicated area of Figure 1.1. Accordingly, theobjectives can be summarized as:

• Basic theoretical and analytical considerations of Bessel beams regarding theuse of axicons with rounded tips, limitations of depth-controlled Bessel beams,and the evaluation of Bessel beams using asymmetric illumination sources forcustomized collimation lenses.

• Concept, design, fabrication, and characterization of an adaptive ring aperturearray using wafer-level clean room fabrication techniques for depth-control.

• Design, fabrication, and characterization of different transmissive asphericallens arrays made with a rapid-prototype process, including a collimation lensarray, two axicon arrays, and a lensacon array, which is a combination of alens and an axicon array.

• Additionally, a compact micro-optical system was developed to further reducethe number of optical surfaces with minimal alignment and assembly errors.This integration was achieved by combining the adaptive ring aperture arraywith a lensacon array into a single system.

The goal of this thesis was the design, fabrication, and characterizationof micro-optical components to generate depth-controlled Bessel beams.

3

1 Introduction

1.3 Outline and scopeFor clarity, this work is divided into four main parts, as illustrated in more detail inFigure 1.2. In each part an overview of the literature and a general description withanalytical considerations is given, and experimental results are discussed. A list oforiginal publications, some of which include results shown in this work, is found inthe Author’s Bibliography on page 129.

In Chapter 2, Measurement methods, the methods used for experimental char-acterization are described. As these measurement methods appear in modified con-figurations throughout this thesis, their basic function and corresponding data eval-uations are introduced in this chapter. Furthermore, beam propagation methods(BPM) are presented to evaluate the propagation of the beam along the opticalaxis. A separate methods section reappears in each following chapter, specifyingthe actual used components, applied BPM simulations, and data analysis.

The first main part of this work is described in Chapter 3, Bessel beams. Ageneral description of Bessel beams is discussed, followed by three applications:Bessel beams generated from axicons, segmented Bessel beams to control the beam,and Bessel beams generated from edge emitting laser diodes. Each applicationincludes an analytical model, BPM simulations, and experimental verification. Theother parts in this work are based on the knowledge gained from this chapter.

The second main part is Chapter 4, Adaptive ring apertures. It contains adiscussion of different technologies to realize such a device, including the selectedtwisted nematic liquid crystal cell. The fabrication and assembly process is detailed,and different designs are presented. Electrical and optical characterizations weredone to evaluate the performance of the components.

Chapter 5, Transmissive aspherical optics, describes the third main part. Therapid-prototype fabrication method for micro-lenses based on thermal expansionof a silicone is presented. Analytical considerations are explained, and a shape-optimized finite element simulations is introduced. The fabrication process is dis-cussed, including its limitations. Four different lens types were designed, fabricated,and experimentally characterized. In this context, the material properties of thesilicone PDMS became important. Whereas the Young’s modulus has been inten-sively studied, the Poisson’s ratio had been lacking. Therefore, a new method todetermine the Poisson’s ratio and the coefficient of thermal expansion was developedwithin this work, which is described in detail in the Appendix.

The last main part, Chapter 6, Compact micro-optical system, combines theprevious chapters. A prototype of the compact system was demonstrated to generatedepth-controlled Bessel beams using only the components developed within thisthesis integrated into one system.

4

1.3 Outline and scope

Chapter 3Bessel beams

Chapter 4Adaptive ring

aperture

Chapter 5Transmissive

aspherical optics*

Chapter 6Compact micro-optical system

z

Lase

r diod

e arra

y

Collim

ation

lens

array

Adapti

ve rin

g ape

rture

Axicon

array

Figure 1.2 Schematic overview of the thesis structure with indications of each chapter.*New method for the Poisson’s ratio in the Appendix.

5

2 Measurement methodsWithin this thesis, different experimental characterizations were performed using op-tical surface measurements and a variable optical measurement setup. As these mea-surement methods appear in modified configurations throughout this thesis, theirbasic function and corresponding data evaluations are introduced in this chapter, in-cluding beam propagation simulations to evaluate the analytical considerations andmeasurements. In each following chapter, a short methods section then describesthe actual used components, beam propagation simulations, and data analysis.

2.1 Optical surface measurementsA chromatic confocal distance sensor from Polytec with different optical pens wasused to measure the surface topography depending on the measurement range andthe lateral resolution, see Table 2.1 [8]. The measurement principle is based on anachromatised white light source, by which the distance can be calculated dependingon the back reflected spectrum of wavelength. The setup is illustrated in Figure 2.1.

2.1.1 Surface data evaluationThe surface measurements of various lens arrays that will be discussed in Chapter 5needed to be evaluated regarding their rotational symmetry, deviation from the in-tended shape, and type of aberrations. Therefore, the following evaluation methodsare discussed.

Surface deviation: This evaluation serves to illustrate all surface aberrations orirregularities in the surface that will interfere with the wavefront. Therefore, thesurface deviations from a rotationally symmetric surface were found by fitting themeasured data to an even polynomial of 14th degree fFit(x,y), with additional offsetand tilt compensation. The original measurement data was subtracted, and the rootmean square (rms) value srms of the surface deviation S(x,y) was determined.

Deviation from the target function: The fabrication process to produce the lensarrays will have some surface deviations from the intended finite element simulatedlens profiles. Consequently, the fitted data fFit(x,y) with offset and tilt compensationwas transformed into radial coordinates fFit(r), with r =

√x2 + y2 to compare the

7

2 Measurement methods

2 cm

6 cm

Figure 2.1 Left: Photograph of the profilometer setup with xy-stage and confocal sensor(CL2). Right: Close-up view of the confocal sensor with a lens array in a custmizedmounting.

produced lenses to the intended lens shape found in the rotationally symmetric finiteelement simulation (target function ftarget). As an approximate value of the totalprocess deviation its rms value was determined.

Zernike polynomials: The surface deviations S(x,y) can be expanded by Zernikepolynomials, which are a common description for optical applications [1, 9]. Thesurface deviations were determined using the first eleven Zernike polynomials, whichare orthogonal on the unit disk. The “fringe” convention was used to calculate theZernike coefficients with a Matlab code (ZernikeCalc-function) [10]. The diameterwas normalized to obtain a fit over the unit disk.

Radius of curvature: The radius of curvature of a lens can be determined fromthe 3-D data using the deviation from the target fFit function by:

rc =

∣∣∣∣∣∣∣∣(1 + ∂fFit(r)

∂r

) 32

∂2fFit(r)∂r2

∣∣∣∣∣∣∣∣ . (2.1)

The focal length is the inverse of the radius of curvature.

8

2.2 Optical characterization method

Table 2.1 Properties of the confocal distance sensor with different optical pens [8].

Confocal pen CL2 CL4

Range (µm) 400 4000Max. target tilt () ± 28 ± 21Spot diameter (µm) 4 12.3Lateral resolution (µm) 1.7 4.6Axial resolution (nm) 17 110Axial resolution 10× averaging (nm) 2.7 37Precision (nm) 55 300

2.2 Optical characterization methodThe experimental characterization of the interference pattern from Bessel beamsis one of the essential methods in this thesis. Therefore, a modular optical mea-surement setup with different light sources and magnifications was used to measurethe beam profile along the propagation direction. Additionally, beam propagationsimulations were used to compare and evaluate the measurements. For both mea-surements and simulations, various data evaluation algorithms were developed de-pending on the desired information.

2.2.1 Optical measurement setupThe modular optical measurement setup is illustrated in Fig. 2.2. The movable linearstage has a range of 670 mm. The cameras from IDS (IDS-UI-1490-SE-M-GL andIDS-UI-1482-LE-M) have a pixel pitch of 1.67 µm and 2.2 µm, respectively. Theycan be fixed to the linear stage using a rail to additionally vary the measurementrange. The intensity patterns for smaller apex angles were measured with the useof microscope objectives (magnification factors 4, 10 or 20). The used light sourceswere a helium-neon (HeNe) laser (λ = 632.8 nm) with a mono-mode fiber anda diode pumped solid state (DPSS) laser (λ = 473 nm) with a mono-mode fiberattached to it. The laser light was then collimated using a lens with focal length fcthat differs depending on the application. A neutral density (ND) filter, apertures,and other optical elements can be integrated in the setup. Two different commercialaxicons were used within this thesis. They are made from UV-fused silica with ananti-reflection coating and apex angles of γ = 178 ± 0.5 and γ = 170 ± 0.1.The measurement was automatized using LabView and is based on the programof Brunne [11]. The exposure times of the cameras were found using an iterativeprocess, such that the maximal pixel intensity was set to a value of 250 in 8-bitgray scale to avoid overexposure. The maximal exposure time divided by the realexposure time results in the scaling factor with which the intensity can be multiplied.

9

2 Measurement methods

Light source

Collimation lens Axicon

Obstacle (optional)

yx

z

fc ND filter and other optional optical elements

Fiber

Movable range

Camera

Figure 2.2 Schematic illustration of the experimental setup. The illumination sourcewith a mono-mode fiber is collimated using a lens with focal length fc.

2.2.2 Beam propagation methodA numerical beam propagation method (BPM) based on fast Fourier transformations(FFT) was used to simulate the beam propagation and thereby verifies the analyticalconsiderations and measurements within this thesis. The basic concept was adaptedfrom the rotationally symmetric case by Siegman [12]. The propagation of an initialwave Ψinit up to a certain point z in the image plane is given by

Ψ(z) = F−1(P (r,z)F(ψinit)), (2.2)

with propagatorP (r,z) = e−iz

√k2−(κ2

x+κ2y), (2.3)

where k = 2π/λ is the wavenumber, κ is the dual spatial base vector of the xy-plane.Assuming a thin lens, the initial overall electrical field behind the axicon at zeroz-position can be described by

Ψinit(r,z = 0) = ψ0 e−ik∆nh(r), (2.4)

where ψ0 is the amplitude, ∆n is the difference in refractive index of the lens andthe surrounding media, and h(r) is the surface profile, which varies depending onthe application, e.g., a rounded axicon profile.

The method needed to be performed in two steps to verify the self-reconstructionproperties of Bessel beams with simulations. First, the wave function was propa-gated to the desired z-position, i.e., the plane of the absorber. Second, the wave wasmultiplied with an absorber matrix B that represents the shading by the absorberand is then further propagated. The modified wavefront can be propagated to theimage plane as described in Equation 2.2:

Ψ(zimage) = F−1(P (zimage − zabsorb)F

(B F−1 (P (zabsorb − zinit)F(Ψinit))

)). (2.5)

10

2.2 Optical characterization method

2.2.3 Data analysisThe measured and simulated intensity profiles needed to be analyzed by extractingdifferent properties of the Bessel beam: the maximum intensity of the central core,the width of the core radius, and the overall profile along the optical axis. For thispurpose, various evaluation algorithms were developed using Matlab.All developed algorithms were designed to analyze individually taken pictures inthe xy-plane. Therefore, the optical center of gravity was found to compensate anypossible unwanted movement by the motion of the camera and was then furtheranalyzed from there on.

Reconstruction of the image plane along the optical axis: The image planewas reconstructed along the optical axis taking the optical center of gravity as thecentral intensity Icore of each plane. The intensity was logarithmically scaled to il-lustrate the side lobes.

Intensity profile evaluation: The central intensity of each plane was analyzedby taking the optical central intensity Icore of each plane from the center of gravityas a function of distance z. It was normalized depending on the application.

Polar basket evaluation: After finding the optical center of gravity, each imageplane was divided in equidistant polar coordinate areas of 11.25. There, the follow-ing characteristic parameters were found by first fitting a polynomial of 4th-orderinto the data around the minima and maxima and then fitting a Bessel functionof 0th-order in each polar interval: the core intensity Icore, the radius of the firstminimum rcore, and the intensity of the first maximum Imax,1, all of them as a func-tion of the distance z. The intensities were normalized to the total integrated initialintensity distribution after the collimation lens.

11

3 Bessel beamsIn this chapter, a general description of Bessel beams is given, and afterwards threedifferent applications are discussed in detail. First, the generation of Bessel beamsusing non-ideal rounded axicons is discussed, followed by a new concept to seg-ment Bessel beams for depth-control, and lastly, the generation of Bessel beamswith asymmetric and astigmatic illumination sources. For each case, an analyticalconsideration is presented, followed by simulations using beam propagation meth-ods (BPM), and subsequently experimental verifications. All presented analyticalconsiderations and results were published in Optics Express [13, 14]1,2.

3.1 Introduction to Bessel beamsDiffraction-free light beams were first described by Durnin et al. more than 30 yearsago [15, 16]. The transverse amplitude profile of such beams can be described bya Bessel function giving the Bessel beam its name [17]. Since then, more types ofpropagation invariant beams have been introduced, including vortex beams, Mathieubeams, or higher order Bessel beams [18–21]. The ideal non-diffracting beam hasan interference pattern produced by the superposition of plane waves whose relativephase difference remains unchanged in free propagation [21]. In experiments, onlyan approximation of the beam profile can be obtained, which is often called pseudo-or quasi-nondiffractive [22–24].The unique feature of Bessel beams, including the extended focal zone and theself-reconstruction of these beams, was investigated in detail by Bouchal et al. [25].There are general analytical studies comparing Bessel beams to Gaussian beams and

1 AM: Conceived, performed, and evaluated all experiments and simulations, prepared the graphs,and wrote the paper.MCW: Helped conceiving the analytical considerations, supervised the data analysis, helpedinterpreting the results, and reviewed the manuscript.UTS: Provided laser diodes, helped interpreting the results, and reviewed the manuscript.MR, KH, OA: Provided laser diodes and reviewed the manuscript.UW: Proposed and initiated this research, supervised the work, and reviewed the manuscript.

2 AM: Conceived, performed, and evaluated all experiments and simulations, prepared the graphs,and wrote the paper.MCW: Helped conceiving the analytical considerations, supervised the data analysis, helpedinterpreting the results, and reviewed the manuscript.UW: Proposed and initiated this research, supervised the work, and reviewed the manuscript.

13

3 Bessel beams

the generation of Bessel beams with Gaussian intensity distributions [22,23,26,27].

For Bessel beams, there are many different applications including non-linear op-tics [27–30] or material processing [31–35], e.g., to polymerize complex shapes usingtwo-photon polymerization [36]. Another application field is optical imaging, whereBessel beams are used to enhance optical coherence tomography [37–40], opticaltweezers [41], or light sheet microscopy [42].The generation of Bessel beams is as diverse as their application fields. They can begenerated using an aperture and lenses [15,23] or by spatial light modulators, whichare also used to replace other optical elements such as gratings [43,44]. One commonway to generate a Bessel beam is the use of a conical lens, a so-called axicon. Itwas first descriebed by McLeod in 1954 [45]. Since then, axicons have often beenanalyzed and applied [13, 27, 45–49], as well as their reflective variants, the axiconmirrors [50–52].

3.2 General description of Bessel beamsBessel beams are one class of non-diffracting solutions of the Helmholtz equation[15, 16]. The electric field distribution of an ideal Bessel beam can be generallydescribed by

Ψm(r, φ, z) = ψ0 eikzz e±imφ Jm(krr), (3.1)

where r, φ and z are the radial, azimuthal and longitudinal components, respectively.Jm is the mth-order Bessel function and kr = k sin (φ), kz = k cos (φ) the radialand axial wavevectors, where k =

√k2

r + k2z = 2π/λ is the wavenumber, with

wavelength λ [1, 27].

In this thesis the main focus is on the zeroth-order Bessel function of first kind fromEquation 3.1, which describes a Bessel beam with a non-diffracting bright core:

Ψ0(r, z) = ψ0 eikzz J0(krr). (3.2)

Using this solution, the characteristic ring pattern does not change over its propa-gation length (z-direction), and the intensity I ∝ |Ψ|2 obeys:

I(x,y,z ≥ 0) = I(x,y), (3.3)

resulting in an unaltered intensity profile along the propagation direction, whichgives the beam its propagation-invariant or diffraction-free feature.In 1987, Durnin proposed to think of Bessel beams as a superposition of plane wavewith same amplitude, traveling at the same angle α = arctan (kr/kz), relative to thepropagation axis, with different azimuthal angles φ between 0 and 2π [16]. Thatmeans in the angular spectrum the Bessel beam can be represented as a ring (in

14

3.3 Bessel beams generated from axicons

k-space). This ring can be Fourier transformed, using Fourier optics, resulting in aBessel beam. The main characteristic parameter of the Bessel beam is the centralcore radius:

rcore = j1,0

kr= 2.405 λ

2πn sin (α) , (3.4)

where j1,0 ≈ 2.405 is the first zero point of the 0th Bessel function, n the refractiveindex, and α the opening angle of the conical wavefront. Further, any positionalong the propagation length z corresponds to a radial position r in the initial beamprofile:

z = r

tan (α) . (3.5)

Another feature is the self-reconstruction ability of Bessel beams. If a small obstacleof arbitrary shape, with radius robstacle, is placed in the optical path of the Besselbeam, the distance after which the beam is reconstructed zobstacle is given by:

zobstacle = robstacle

tan (α) . (3.6)

3.3 Bessel beams generated from axiconsIn contrast to an ideal, infinitely extended Bessel beam, a real generated Besselbeam, e.g., by an axicon, is only an approximation and, as mentioned above, oftenreferred to as a quasi-Bessel beam [49,53,54]. In this thesis, a Bessel beam refers toa quasi-Bessel beam for the sake of simplicity.

3.3.1 Analytical considerationsThe generation of a Bessel beam using an axicon is illustrated in Figure 3.1. Theintensity has an amplitude I0 ∝ |Ψ0|2 (from Equation 3.2), which varies slowly anddepends on the intensity distribution of the illuminating beam:

I(r, z) = I0 J20 (krr). (3.7)

When illuminated with a plane wave of uniform intensity, the axicon leads to alinear intensity rise along the optical axis z [43]. However, when illuminated by aGaussian beam at zmax/2, the exponential drop in the initial intensity profile takesover with a decrease in intensity [17]. The obtained intensity distribution is then

I(r, z) = Πkrz

rmaxzmaxe−

2z2z2max J2

0 (krr), (3.8)

15

3 Bessel beams

γ

Gaussian beam

Axicon

zmax

α

z0

rmax

Position in z-direction

Inte

nsity

A

A’AA’

Figure 3.1 Schematic illustration of a Bessel beam generated by an axicon using a Gauss-ian illumination profile. The set of plane waves interfere and generate the characteristicring pattern (cross-section AA’). The finite propagation distance is zmax.

where Π is the power of the illumination beam [49,55]. Further, the finite propaga-tion distance zmax with finite radius, rmax, defined as 1/e2 of the maximum intensity,similar to Equation 3.5:

zmax = rmax

tan (α) . (3.9)

They are related by the cone angle

α = (naxicon − nair)(π − γ

2

), (3.10)

where γ is the apex angle of the axicon, and n is the refractive index of the axiconand the surrounding media (air), respectively.

An ideal axicon has a sharp tip without any rounding. In reality Bessel beams gen-erated from axicons show a rounding of the tip due to fabrication limitations. Thisrounding appears as a small lens which modulates the beam intensity periodically,as described in [13, 49]. If two wavefronts, assuming a plane wavefront with phaseΦ1 = kz from the axicon tip and a conical wavefront with phase Φ2 = kz/ cos(α),interfere constructively, their phase has to be a multiple of 2π:

|Φ2 − Φ1| = 2πm, m ∈ Z.

By substituting Φ1 and Φ2, the modulation period for small angles in propagationdirection is approximately

zm ' 2λmα2 . (3.11)

16


Another intensity variation resulting from the rounded axicon tip can be observedin the first minimum of the Bessel beam. The interference of the rather plane wavefrom the rounded axicon tip and the conical wave causes a shift and modulationof the first minimum. The total electric field distribution is a combination of theelectric field distribution from the axicon, Ψaxicon, and the rounding, which can berepresented by a lens Ψlens:

Ψtotal = Ψaxicon + Ψlens

= ψ0,axicon (z) e ikzz kr∆rJ ′0(j1,0) + ψ0,lens(z) eikz

= ψ0,lens eikz

(1 + ψ0,axicon

ψ0,lensei(kz−k)z (kr∆r J ′0(j1,0))

), (3.12)

where ∆r = r − rcore, and J ′0(j1,0) is the derivative of the 0th Bessel function at itsfirst minimum. The total intensity distribution I ∝ |Ψ|2 becomes

Itotal ∝ 1 + 2ψ0,axicon

ψ0,lenskr ∆rJ ′0(j1,0) cos2((kz − k) z)

+(ψ0,axicon

ψ0,lenskr ∆rJ ′0(j1,0)

)2

, (3.13)

and the shift of the minimum ∆rmin is then

∆rmin = −ψ0,axicon

ψ0,lens

cos ((kz − k)z)krJ ′0(j1,0) . (3.14)

The resulting modulation period in the propagation direction zm is the same asdescribed in Equation 3.11. The amplitude of this oscillation, i.e., the amount ofvariation of the first zero point is: (ψ0, axicon/ψ0, lens)(α/kJ ′0(j1,0)).Substituting ∆rmin back into the total intensity (Equation 3.13), the amplitude onlyvanishes at discrete values where cos ((kz − k)z) = ±1, provided that the lineariza-tion is consistent.

The radius of curvature rc and the focal length z0 = rc/(n − 1) of the lens at theaxicon tip (Figure 3.2) influence the beam. Ideally, the Bessel beam should startslightly behind the tip, as soon as two conical wavefronts interfere with each other.In case of the rounded tip, a Bessel beam can be generated only beyond the focallength of the lens z0, assuming a continuous, sufficiently smooth transition from therounding into the conical axicon slope. This effect is further called the “lens effect”.

3.3.2 MethodsOptical measurement setup: The analytical considerations above were evaluatedusing the setup described in Section 2.2.1. A commercial axicon with apex angle

17

3 Bessel beams

Φ1

Φ2

α

zz0

rlens

Figure 3.2 Schematic of the rounded axicon with two resulting wave fronts. The planewave phase φ1 and the conical wave phase φ2 with indications of the ideal axicon and thefocal length z0.

178, the camera with 2.2 µm pixel pitch, and the DPSS laser with wavelength473 nm were used. A Gaussian intensity profile with 1/e2 diameter of 3 mm wasgenerated using a aspherical collimation lens (17.5 mm focal length) to illuminatethe axicon.

BPM simulations: The beam propagation method (BPM) described in Sec-tion 2.2.2 was used to simulate the propagation of a Bessel beam generated from arounded axicon. The surface profile h(r) was determined by measuring the surface ofthe actual used commercial axicon using a white light interferometer. A polynomialof 3rd order was fitted to the measured data:

h(r) = 6

√√√√ar2 + br4 + c6 + r6

tan6(γ/2)) . (3.15)

The parameters a, b, and c describe the deviation from an ideal axicon and γ theapex angle. An additional radial weighting was introduced ((100 µm)2 + r2)−1/2 toweigh the radial sections equally and at the same time cut off the weighting factorto avoid excessive noise from the few data points with diverging weight at the center.

Data evaluation: The polar basked evaluation described in Section 2.2.3 was usedto evaluate the data. The intensities are normalized to the total integrated intensityof the initial intensity distribution measured after the collimation lens.

3.3.3 ResultsThe surface profile of the axicon was measured to determine the rounding of the tip.The apex angle was found to be (178.553 ±8×10−3), which is in the same range asthe manufacturer’s tolerance. The rounding of the axicon tip corresponds to a smalllens with radius of curvature rc = (48.84 ± 0.085) mm and radius rlens = (852.5 ±

18


0 50 100 150 200 250 300 350 4000.00.10.20.30.40.50.60.70.80.91.0

z0 zmaxNor

mal

ized

core

inte

nsity

/a.u

.

Position in z-direction / mm

MeasurementSimulationIdeal axicon

50 100 150 200 250 300 350 400152025303540455055606570758085

z0 zmax

Cor

era

dius

/µm



Figure 3.3 Normalized core intensity (left) and core radius (right) along the optical axisof an ideal axicon compared to an axicon with rounded tip with BPM simulation andmeasurement.

0.04) µm. This small lens will generate a focal point at z0 = (105.3 ± 0.2) mm,only beyond which a Bessel beam can be detected.

Figure 3.3 illustrates the comparison of a Bessel beam from an ideal simulated axiconto a measured and simulated axicon with rounded tip. In Figure 3.3 (left), the nor-malized intensity in the central core Imax is plotted as a function of the propagationdirection z. The “lens effect” can be clearly seen at z0 = 105 mm. The oscillationsmentioned in Section 3.3 converge for both simulations and measurements to theintensity of the ideal axicon, with a period described in Equation 3.11. In Figure 3.3(right) the central core radius is illustrated along the propagation direction. Thecore radius, i.e., the radius at the first minimum of the Bessel beam for an idealaxicon is rcore,ideal = 22.35 µm.The spatial intensity pattern was obtained by reconstructing the lateral cross-sections,as shown in Figure 3.4 (left). The intensity is logarithmically scaled to illustratethe side lobes of the Bessel beam on the left side. The measurement has a longervisible core after zmax, resulting from differences in the illumination compared tothe perfect simulation. Figure 3.4 (right) presents two characteristic cross-sectionsthat are normalized to the brightest pixel. The symmetric interference pattern isfound in both positions. The oscillations from the intensities in Figure 3.3, can beobserved at the different intensity values of the individual rings (beam profile andcross-section) in Figure 3.4.

Self-reconstruction was evaluated using a laser-structured absorber made from blackcardboard with a radius robstacle = (375 ± 2) µm at the center of the beam. Us-ing Equation 3.6, the distance at which the beam is geometrically reconstructed iszobstacle = 46.31 mm. Figure 3.5 shows the self-reconstruction of the intensity inthe central maximum at two positions: A1 = 125 mm and A2 = zmax. Comparingthe intensity profile of the un-obstructed beam (no absorber), the intensity after

19

3 Bessel beams

xz

A1 A2

yx

A2A1

100µm

Mea

sure

.S

imul

atio

n

A1A2 50 mm50 µm

A1A2

xz

yx

Figure 3.4 Left: Simulation (top) and measurement (bottom) of the Bessel beam profilesalong the optical axis. To illustrate the side lobes the intensities were logarithmicallyscaled. Right: Simulation and measurement at different positions of the transverse plane:(A1) z = 125 mm and (A2) zmax. The images are each normalized to the brightest pixel.

0 50 100 150 200 250 300 350 400 4500.000.050.100.150.200.250.300.350.400.450.50

Ideal axiconNo absorberPosition A1Position A2

Nor

mal

ized

core

inte

nsity

/a.u

.

Position in z-direction / mm0 50 100 150 200 250 300 350 400 450

0.000.050.100.150.200.250.300.350.400.450.50

Nor

mal

ized

core

inte

nsity

/a.u

.


No absorberPosition A1Position A2

Figure 3.5 Normalized core intensity of the simulated (left) and measured (right) beamprofile along the optical axis with an absorber placed at different positions on the opticalaxis (A1 = 125 mm and A2 = zmax). The shaded regions indicate the distance behind theabsorbers at which the beam is geometrically reconstructed.

the self-reconstruction of the Bessel beam is in good agreement with the theoreticalvalue (shaded area). The slight variation between measurements and simulations isa result of a stronger diffraction at the edge of the laser-structured absorber used inthe measurement, whereas in the simulation a perfect disk was assumed.

20

3.4 Segmented Bessel beams

Axicon

Ring aperture

Plane wave zstart zend

r

z

d SegmentedBessel beam

∆z

αγ

Figure 3.6 Schematic drawing of the segmented Bessel beam region ∆z, which was gen-erated using an axicon with apex angle γ and a ring aperture with aperture size d.

3.4 Segmented Bessel beamsA segmented Bessel beam is generated by blocking parts of a Bessel beam using, e.g.,a ring aperture, as illustrated in Figure 3.6. The main advantage of these segmentedBessel beams compared to the change of a conical wavefront, e.g., by changing theaxicon angle, is that the length and position of the Bessel beam can be varied bythe radius of the ring aperture without changing the core radius, which is calleddepth-controlled in this thesis. A “scanning” or depth-controlled Bessel beam canbe achieved with an adjustable ring aperture, as illustrated in Figure 3.7.

3.4.1 Analytical considerationsThe ring aperture in front of the axicon limits the Bessel beam region, and a smaller,segmented beam region ∆z at position zstart until zend is generated. The ring apertureis approximated to be a slit with width d that is much narrower than its radius r(d r). The length of the resulting Bessel region can be derived from Equation 3.5.There are two cases which were considered:

• The classical limit, where only pure geometric considerations are taken intoaccount, is marked with a subscript C:

∆zC = d

tan (α) . (3.16)

• The diffraction at an aperture using Fraunhofer approximation is marked witha subscript F. Considering the Fraunhofer diffraction as an approximation inthe far field, the intensity with diffraction opening angle θd is then [56]:

I(θd) = I0sinc2(d

2kr). (3.17)

21

3 Bessel beams

Taking the conical wavefront angle α together with the diffraction openingangle θd into account, a new angle is defined α′ = α + θd. The diffractionopening angle is then:

θd = α′ − α= arctan ( r

z + ∆z )− arctan (rz

)

' r∆zz2

11 + r

z2 . (3.18)

Equation 3.18 holds only in the limit ∆z z, i.e., in the limit of smalldiffraction angles

θd α. (3.19)

Further substituting z from Equation 3.5, the diffraction angle θd becomes:

θd = ∆z tan2 (α)r

11 + tan2 (α) . (3.20)

This diffraction angle can be substituted back into Equation 3.17 and solved forthe first zero point of the intensity distribution. The length of the segmentedBessel beam in the Fraunhofer limit is then:

∆zF = 2λrd tan2 (α) (1 + tan2 (α)). (3.21)

A correction factor ζ is introduced to compare the analytic approximation withsimulations and measurements in which the 1/e2 intensity of the Bessel beamis defined as the width. The length of the Bessel beam within the Fraunhoferlimit is then:

∆zF '2λrζ

d tan2 (α) . (3.22)

The subleading term 1 + tan2 (α) is negligible as it will only introduce smalldeviations.

The classical approximation is expected to be valid if its beam length is much longerthan the pure diffraction result of the Fraunhofer limit, i.e., if zC zF, and viceversa:

d

tan (α) 2λrζ

d tan2 (α) ,

d2 2λrζtan (α) . (3.23)

22


axicon

ring aperture

plane wave zstart zend

r

z

d segmentedBessel beam

∆z

4 mm

10 mm

20 µm

rz

Ring aperture

∆z

∆z

∆z

Beam profile

(d1)

(d2)

(d3)

αγ

Figure 3.7 Simulated intensity profiles of the segmented Bessel beams (logarithmicallyscaled) with axicon angle 170 and wavelength 632.8 nm. The optimal apertures dminused at different radial positions r: r1 = 1 mm, d1,min = 197.85 µm, r2 = 2 mm, d2,min =281.58 µm and r3 = 3 mm, d3,min = 344.90 µm.

To be able to compare different axicon angles and wavelengths, the aperture andradius is normalized with the wavelength and the axicon angle. The aperture widthand radius are defined to be:

d := d

λand r := r

λ tan (α) . (3.24)

The scale invariant z-position is defined as:

z := z tan (α)λ

, (3.25)

reducing Equation 3.16 and Equation 3.22 to:

∆zC = d and ∆zF = 2 rζd. (3.26)

The condition from Equation 3.19 expressed in scale invariant terms in the Fraun-hofer limit is 2/d α.

The smallest segmented Bessel beam region ∆zmin is expected to be somewhere nearthe intersection of the classical and the Fraunhofer predictions, see Figure 3.8, near∆zC = ∆zF, where

dmin =√

2rζ = ∆zmin. (3.27)

23

3 Bessel beams

0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

1600∆zC

∆zF

∆z

d

dmin

Figure 3.8 Normalized segmented Bessel beam width ∆z as a function of the aperturesizes d. The intersection of the Fraunhofer and classical regime is marked with dmin. Atthis position the smallest segmented Bessel beam width ∆zmin is expected.

3.4.2 MethodsOptical measurement setup: The experimental setup is similar to the one de-scribed in Section 2.2.1. An additional ring aperture was introduced directly behindthe axicon tip. The dependence of the minimal segmented Bessel beam region wasverified using different wavelengths and axicon angles. The light sources were aDPSS laser with wavelength 473 nm and a HeNe laser with wavelength 632.8 nm.Commercial axicons with apex angles 178 and 170 were used. The initial beam ra-dius of 10 mm, generated using an aspheric collimation lens (100 mm focal length),was chosen to obtain an approximately plane wave. Both cameras were used, de-pending on the wavelength and axicon angle. For the ring aperture, a high resolutionphoto plot (25 000 dpi) on a polyethylene terephthalate (PET) film with a thick-ness of 180 µm and an accuracy of ±1 µm was used. The different aperture radiito generate different segmented Bessel beam regions were defined in the center ofeach aperture: 100 µm, 200 µm, 400 µm and 800 µm. The small deviation fromthe conical wavefront that changes the effective width of the aperture by a factor ofcos(α) was neglected.

BPM simulations: The segmented Bessel beams were simulated with a surfaceprofile h(r) = ((180 − γ)/2)r. Similar to the self-reconstruction simulation fromSection 2.2.2. An absorber matrix B, which represents the desired ring aperture,was multiplied to the initial conical profile and propagated to the desired imageplane.

Data analysis: The reconstruction of the image plane along the optical axis fromSection 2.2.3 was used. The beam was normalized by the initial Bessel beam withouta ring aperture and a correction factor of 1.098, coming from surface reflections onboth sides of the PET film. The same procedure was used for the BPM simulations.

24


0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

1600∆zF

∆zC

∆zSim, 178°, λ = 473 nm

∆zSim, 178°, λ = 632 nm

∆zSim, 170°, λ = 473 nm

∆zSim, 170°, λ = 632 nm

∆z

d

Figure 3.9 Comparison of simulations and analytical estimates as a function of the aper-tures sizes d at a fixed scale invariant radius r ≈ 79200. Different axicon apex angles 178and 170 and wavelengths λ = 632.8 nm and λ = 473 nm were simulated.

The length of the segmented Bessel beam was determined where the maximumintensity drops to 1/e2, starting from the expected center of the beam.

3.4.3 ResultsThe different aperture angles and wavelengths with constant radius r are illustratedin Figure 3.9. The chosen axicon angles, which differ by a factor of 5, and wave-lengths from blue to red cover a wide range of parameters to verify that the ap-proximation is generally valid. It can be observed that the simulations are in goodagreement with the analytical approximation and that the scaling is suitable, as allsimulations predict approximately the same beam length in the scaled coordinates.The deviations using small aperture sizes d in the Fraunhofer limit appear due tothe violation of the condition from Equation 3.19. The correction factor of Equa-tion 3.22 that accounts for the 1/e2 intensity drop is ζ = 0.70.

The normalized central maximum of the Bessel beam with axicon apex angle 170,radius r = 2 mm and the wavelength λ = 632.8 nm is compared in simulationsand measurements in Figure 3.10. The intensity loss in the measurement originatesfrom surface reflections on the ring aperture substrate (PET film) and was takeninto account with a factor of 1.098. Both measurements and simulations showedthe quadratic sinc function profile in the Fraunhofer limit (at d = 100 µm), whichappears increasingly asymmetric with decreasing aperture width (Figure 3.10). Itis expected to originate from the projection of the Fraunhofer intensity profile ontothe optical axis with varying angle α′, where the linear approximation in the laststep of Equation 3.18 is not valid anymore, if the diffraction angles are as smallas assumed. Another reason could be the circular geometry, which was neglectedin the analysis and becomes relevant in this case, especially for larger diffraction

25

3 Bessel beams

35 40 45 50 55 60 650.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6


∆z

35 40 45 50 55 60 650.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6


∆z

35 40 45 50 55 60 650.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

∆z

MeasurementSimulation

Position in z-direction / mm30 µm

35 65 35 65Position in z-direction / mm35 65

(d1) (d2) (d3)

Position in z-direction / mm Position in z-direction / mm

Nor

mal

ized

inte

nsity

/ a.

u.

Figure 3.10 Normalized core intensity around the segmented Bessel beam region ∆zfor simulations and measurements using different aperture sizes d with wavelength λ =632.8 nm and an axicon apex angle of γ = 170. The shaded areas indicate the analyticalestimate for ∆z. The actual measured aperture widths are: d1 = (101.9 ± 0.7) µm,d2 = (199.6 ± 1.1) µm, and d3 = (801.0 ± 0.6) µm. On the bottom of each graph the cor-responding simulated intensity profiles along the propagation length with logarithmicallyscaled side lobes is shown.

angles, as illustrated in Figure 3.10 (top left). The diffraction pattern inside andoutside the circle, or before and behind the illuminated region on the optical axis willdiffer. In the other regime towards the classical limit, a unit step intensity profilecan be observed as expected. The oscillations of the intensity in measurements andsimulations arise from the interference of both edges. This could be confirmed withthe oscillation period. The maximal mismatch of this modulation in the intensity ishalf a period.

Further, the quantitative comparison between the analytical estimate with simu-lations and measurements were done by two different combinations: λ = 473 nm,γ = 178 and λ = 632.8 nm, γ = 170, which is illustrated in Figure 3.11. Theerror is defined to be half of an oscillation period from Figure 3.10. The measuredring apertures have a manufacturing deviation of ±1 µm and were neglected. Theresults are in good agreement within the error.

The minimal segmented Bessel beam region ∆zmin and the corresponding ring aper-ture size dmin were derived in Equation 3.27 and are illustrated in Figure 3.12. Theminimum simulated Bessel beam length ∆zmin was found by determining the opti-mal aperture dmin using simulations with finer iterations steps around the theoreticalminimum. From there the corresponding Bessel beam length ∆zmin was obtained.

It can be observed that the optimal simulated aperture width dmin is larger thanthe estimated optimal aperture, which likely results from edge effects. However, the

26


500 1000 1500 2000 2500 30000

500100015002000250030003500400045005000

∆zF

∆zC

∆zSim

∆zMeasurement

∆z

d

γ = 178°= 473 nm

r = 5 mm

0 200 400 600 800 1000 12000

200400600800

100012001400160018002000 ∆zF

∆zC

∆zSim

∆zMeasurement

∆z

d

γ = 170°= 632 nm

r = 2 mm

Figure 3.11 Comparison of analytical approximations, simulations and measurements ofthe beam length ∆z as a function of different aperture size d for two different configurationsof the axicon apex angles, radius, and wavelength.

0.0 3.0x104 6.0x104 9.0x104 1.2x1050

100

200

300

400

500

Simulation ∆zmin

Analytical estimate dmin = ∆zmin

Simulation dmin

r

d min

0

100

200

300

400

500

∆z m

in

Figure 3.12 Optimal aperture width dmin (orange) and the minimal segmented Besselbeam region ∆zmin (cyan) simulated and analytically estimated as a function of differentradii r.

trend shows the same square root behavior, scaled by a factor of 1.155. This effectis also seen in Figure 3.11, where the minimum of ∆z is shifted towards slightlylarger values d, while the minimum value of ∆z is approximately the one of theintersection.

27

3 Bessel beams

3.5 Bessel beams generated from edge emitting laserdiodes

As mentioned in Section 1.1, blue edge emitting gallium-nitride based laser diodeswere chosen as illumination source for the miniaturized stimulation tool and werethus investigated within this thesis. Their far-field intensity profile has an ellipticalshape, astigmatism and substrate modes [57]. Therefore, different properties regard-ing opening angle, maximal intensity along the optical axis, effects of astigmatism,and effects of substrate modes and far-field ripples are investigated.

3.5.1 Analytical considerationsThe elliptic beam profile of the laser diode (LD) with 1/e2 half opening angles inthe far field is approximately:

I(r,φ) = I0 e−2θ2

(cos2(φ)θ2slow

+ sin2(φ)θ2fast

), (3.28)

where φ is the azimuthal angle and θslow and θfast are the half opening angles of theshorter (“slow”) and longer (“fast”) axis, respectively.After collimation with a perfect lens, for sufficiently small opening angles the inten-sity profile is:

I(r,φ) ≈ I0 e−2r2

(cos2(φ)r2slow

+ sin2(φ)r2fast

), (3.29)

where rslow = fc tan θslow, rfast = fc tan θfast and fc the focal length.

Similar to Equation 3.5, two different propagation lengths can be defined

zslow = rslow/ tan (α) and zfast = rfast/ tan (α). (3.30)

These propagation lengths distinguish two different regions in the generated Besselbeam (Figure 3.13): First, the slow axis region with z rslow/ tan (α). Here, aBessel beam is expected as it corresponds to the approximately rotationally sym-metric central region of the initial profile. Second the fast axis region, starting fromz = rslow/ tan (α) to z = rfast/ tan (α), corresponding to the initial profile where theazimuthal asymmetry of the intensity distribution on the corresponding radius is nomore rotationally symmetric, and a bow-tie shaped intensity pattern similar to aMathieu beam is observed [7, 58,59].

28

3.5 Bessel beams generated from edge emitting laser diodes

z

2 mm

20 µm

β

Fast axis region

Slow axis region

Axicon

Illumination beam rfast

rslow

Figure 3.13 Schematic of the beam propagation using an asymmetric beam profile andan axicon. The transition between a symmetric Bessel beam and a bow-tie shaped patternwith opening angle β depends on the radial position.

Opening angle

The opening angle of the bow-tie shaped intensity pattern, as illustrated in Fig-ure 3.13, can be derived by geometric considerations. The light from a point onthe fast axis φ = π/2 and a point slightly off axis φ = (π/2− β/2) with the sameradius on the optical axis propagates from z = 0 to z = r/ tan (α) and forms theinner most part of the bow-tie shaped pattern near the axis. That assuming the az-imuthal distribution is approximately the same as the corresponding distribution ofthe incident beam. The opening angle β, defined until the 1/e2 intensity drops, canbe derived by applying the condition I(r,π/2 − β/2)/I(r,π/2) = 1/e2, substitutingr = z tan (α) and solving for β:

β = 2 arcsin 1z tan (α)

√r−2

slow − r−2fast

. (3.31)

In Figure 3.14 (left), β is illustrated as a function of the ratio of the propagationlength z and the maximal propagation length in the fast axis zfast. Equation 3.31 isonly valid for sufficiently large values of z. The circular pattern only starts to openup around z ∼ 1/

(tan (α)

√r−2

slow − r−2fast

). Eventually, if the value of z is large, the

outer parts of the fast axis have a similar effect as a slit aperture, and the bow-tieshaped pattern will turn into a narrow line.

29

3 Bessel beams

0.0 0.2 0.4 0.6 0.8 1.0

rslow/rfast = 0.1 rslow/rfast = 0.4rslow/rfast = 0.2 rslow/rfast = 0.5rslow/rfast = 0.3 rslow/rfast = 0.6

1/8 π1/4 π

1/2 π

2/3 π

π

Ope

ning

angl

eβ

/rad

z/zfast

0.0 0.2 0.4 0.6 0.8 1.00.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

z Im

ax/z

slow

rslow / rfast

Figure 3.14 Left: Opening angle β dependent on the z-position (z/zfast) for differentvalues rslow/rfast. Right: The position of maximal intensity (normalized to the fast axiszImax/zfast) depending on the 1/e2 radii rslow/rfast.

Maximal intensity along the optical axis

For a symmetric Gaussian intensity distribution, the maximum intensity Imax ofa Bessel beam is at zmax/2. For asymmetric illumination the central maximum isnot necessarily at the same position. The intensity in the central maximum at adistinguished position z, Imax(z), is proportional to the integrated intensity Iint ona circle in the transverse plane at corresponding radius r:

Iint(r) =∫ 2πr

0Idl

= r∫ 2π

0e−2r2

((1

r2slowcos2(φ)

)+(

1r2fast

sin2(φ)))dφ. (3.32)

Substituting r = z tan (α) and simplifying the expression, Imax is proportional to:

Imax(z) ∝ 2πz tan (α)e−(z tan (α))2

(1

r2slow+ 1r2fast

)×

J0

((z tan (α))2

(1r2

fast− 1r2

slow

)). (3.33)

In Figure 3.14 (right), the numerical value for the ratio of the z-position with max-imum intensity zImax to the length of the slow axis range zslow as a function of theratio of the initial beam radii rslow/rfast is illustrated. If the initial intensity profileis symmetric (rslow = rfast) the maximum intensity is at position zImax = zslow/2. Ifthe initial intensity profile is asymmetric the position of maximal intensity is shiftedup to zImax = 0.89 zslow.

30


Effects of astigmatism

Edge emitting laser diodes show an astigmatism in the range of several micrometers(3−10 µm) [1,60]. The effect of this astigmatism is an asymmetric phase difference,which destroys interference and thus the generation of Bessel beams. Therefore, itis essential to find the condition under which the asymmetric intensity distributionwill suppress the interference effect from the astigmatism.An astigmatism in laser diodes is a shift a between the origin of the wavefront in slowand fast axis, see Figure 3.15. When using a classical aspherical collimation lens, onlyone axis gets collimated, while the other axis still has a curved wavefront, causinga shift in the propagation length ∆lastigm(r). This implies that the light arriving ata point z on the central axis comes from two different radii on the transverse plane.Depending on the azimuthal angle φ, a difference in the conical angle ∆α and in thelength of the light path, ∆lcone, can be found. Destructive interference will appearat a length ∆l = λ/2. The length of the light path relates to a propagation lengthby the conical angle l = z/ cos (α). Consequently, a variation in the length ∆lconerelates to a variation in the conical angle ∆α by linear approximation:

∆lcone = z ∆α sin (α)cos2(α) . (3.34)

The total shift is then the combination of the geometric and astigmatic length shift:

∆ltot = ∆lcone + ∆lastigm

= z ∆α sin (α)cos2(α) + 1

2ar2

fc. (3.35)

A shift a of the origin, after the wavefront is collimated with focal length fc, resultsin an angular shift ∆α = (r a)/f 2

c at a radial position r. Equation 3.35 can besimplified by substituting r = fc tan (θ):

∆ltot = a

(12 −

1cos (α)

)tan2(θ). (3.36)

Assuming only two wave fronts exist, one from the fast and one from the slow axis,the condition for destructive interference (∆ltot = λ/2) is:

θdestructive = arctan√√√√ λ cos (α)a(cos (α)− 2)

. (3.37)

The relative amplitude of the interference can be estimated by the ratio of initialintensity distributions of slow and fast axis:

Iinterference

Itotal≈ Islow

Ifast. (3.38)

31

3 Bessel beams

Substituting the Gaussian distribution of Equation 3.28, the ratio is:

Iθ|φ=0

Iθ|φ=π2

= e−2θ2

(1

θ2slow− 1θ2fast

)= e

−2 θ2θ2m . (3.39)

The effective opening angle which describes the faster decline of the slow axis com-pared to the fast axis is:

θm =(√

θ−2slow − θ−2

fast

)−1. (3.40)

In order to have a relative amplitude of the interference effect less than 1/e2, theconstraint

θm ≤ θdestructive (3.41)

needs to be satisfied.The critical angle αcritical can then be found by substituting Equation 3.37 andEquation 3.40 in Equation 3.41:

tan2 (θm) ≤(

λ cos (α)a(cos(α)− 2)

), (3.42)

and solving for α:

α ≤ αcritical = arccos(

2g2 + g

), (3.43)

where g = (2a tan2 (θm))/λ is the critical parameter only dependent on the prop-erties of the laser diode. For angles smaller than αcritical, or critical parametersg < gcritical = (2 cos(α)/(2 − cos(α)), interference will be highly suppressed due tothe difference in the initial intensity profiles (see Figure 3.16 (left)). The maximumrealizable refractive axicon angle with a typical refractive index of n = 1.5 can gen-erate at most an angle α = 30. The smallest relevant critical parameter is theng = 1.53 (see Figure 3.16 (left)). Different astigmatism values a are indicated in Fig-ure 3.16 (left) with the effective half opening angle θm = 166.2mrad and λ = 436nm,which will be relevant in Section 3.5.3.

In Figure 3.16 (right), the normalized central intensity (normalized to an inten-sity profile without astigmatism) for symmetric (θm = 0) and asymmetric (θm =166.2 mrad) illumination sources with same astigmatism are illustrated using anideal axicon with apex angle 178 and wavelength 436 nm. The symmetric illumina-tion sources have a perfect destructive interference at the center for certain valuesof z. In asymmetric sources the interference is suppressed with smaller values ofa. The suppression is smaller than expected, as in reality there are not only twowave fronts from the fast and slow axis, but a continuous wavefront with smoothlyvarying phase and intensity.

32


z0

ll+∆lcone

∆α

r

∆r

αa

∆lastigm

Laser diode

Collimationlens

Axicon

z

Figure 3.15 Schematic of a wavefront from a laser diode in slow and fast axis withastigmatism a. The astigmatic wavefront is collimated only in one direction generating anangular shift ∆α.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Astigmatism a = 2 µmAstigmatism a = 5 µmAstigmatism a = 10 µm

1/6 π

0

1/8 π

1/2 π

αcr

itica

l

g = (2a tan2(θm))/λ

1/4 π

0 100 200 300 400 500 600 700 8000.0

0.2

0.4

0.6

0.8

1.0

AsymSym

AsymSymAsymSymAsymSym

20 µm

2 µm zmax, mean

Nor

mal

ized

cent

erin

tens

ity/a

.u.


5 µm

10 µm

Figure 3.16 Left: The critical angle αcritical as a function of the parameter g. The shadedarea indicates the physical limitation of an axicon with maximal wavefront angle α = 30and refractive index n = 1.5. The additional indications are calculated examples for dif-ferent astigmatism values with mean half opening angle θm = 166.2mrad and λ = 436 nm.Right: Normalized core intensity of an ideal axicon with 178 apex angle simulated withsymmetric (sym) and asymmetric (asym) illumination and different astigmatism values a.

33

3 Bessel beams

Effects of substrate modes and far-field ripples in laser diodes

The substrate and waveguide material of the used edge emitting laser diodes is gal-lium nitride. As a result, the waveguide mode may partially leak into the substrate,generating substrate modes that can be observed in the initial far-field intensitydistribution at an angle θs. Its opening angle ∆θs can be approximated with thesubstrate thickness ds (see Figure 3.17) [57]:

∆θs ∼λ

ds. (3.44)

It is not collimated with the main Gaussian intensity distribution because its origin isslightly displaced compared to the laser mode. The transverse offset is approximatelyds/2, with an angle relative to the main beam:

ϕ ' ds

2fc. (3.45)

The lateral displacement of the substrate mode gives an opening angle ∆ϕ behindthe collimation lens. In addition to the substrate modes, additional narrow peaksmay arise in the far-field, which are caused by light scattering from geometric ripplesbetween the waveguide and the cladding interface, which is typical for these laserdiodes. The position of the substrate mode on the optical axis zs can be calculatedusing the position of the substrate mode in the plane of the aspherical lens, hs =fc tan (θs), the angle ϕ, and the position in the plane of the axicon ha:

zs = ha

tan (α + ϕ) = (hs − la tan (ϕ))tan (α + ϕ) , (3.46)

where ∆ha = ∆hs+la tan (ϕ), and la is the distance between the lens and the axicon.The length over which the substrate mode appears comes from the opening of thesubstrate mode ∆hs and its defocusing ∆ϕ:

∆zs ∼∆ha

tan (α + ϕ) + h− la tan (ϕ)sin2(α + ϕ) ∆ϕ. (3.47)

The defocusing, ripples and substrate modes will run into each other such that allcan be considered for ∆hs. As the substrate mode is coherent with the laser mode,it can interfere with the main beam. To estimate this effect, the length difference ofthe corresponding optical paths (from the original beam l and the substrate models) has to be determined, neglecting any fixed phase difference between the modes:

∆l = l − ls = z

cos (α) −√h2

a + z2. (3.48)

The condition for constructive interference is ∆l = mλ, m ∈ Z:z(α + ϕ)2

2 − h2az

= mλ. (3.49)

34


Laser diode

Aspherical lens Axicon

ds

θsϕ

hala

Opening angle ∆ϕ∆hs

hs

∆ha

∆zs

zs

∆θs

fc

as

ls

Figure 3.17 Schematic influence of the substrate modes in edge emitting laser diodes.The displacement of the substrate mode will result in a widening in the z-direction ∆zs.

For typical laser diodes and not extremely small axicon angles α, m will appear inthe order of at least a few hundred, e.g., m = 150 for λ = 436 nm and γ = 178(ha = 4.7 mm, α + ϕ = 0.46). A very rapidly varying interference pattern in thez-direction is generated, where the maxima might not even be distinguishable. Foreven larger m, i.e., not small angles α, the path difference will be greater than thecoherence length of the laser diode so that no interference effects will be visible,e.g., m = 448 (∆l = 195 µm) for γ = 160. Moreover, the ripples running into thesubstrate mode may also cause decoherence so that there will be no interference atall.

3.5.2 Methods

Optical measurement setup: The measurement setup from Section 2.2.1 wasused with an edge emitting laser diode (λ = 463 nm) as illumination source. Anaspheric collimation lens with focal length 17.5 mm for collimation, a commercialaxicon with apex angle 178, and a camera with pixel pitch of 2.2 µm were used.The properties of the laser diode are summarized in Table 3.1. The far-field intensityprofile of the laser diode is shown in Figure 3.18. It was driven using a pulse lengthof 1 µs and a duty cycle of 1% with driving current 150 mA, slightly above thethreshold.

BPM simulation: The asymmetric intensity distribution from Equation 3.29 wasused to simulate a collimated laser diode. The axicon with rounded tip from Sec-tion 3.3 (Equation 3.15) was used as surface profile h(r). Additionally, the self-reconstruction of the Bessel beam was simulated, as described in Section 2.2.2.

Data analysis: For the evaluation of the asymmetric beam profile, the polar basketevaluation described in Section 2.2.3 was used. The intensities are normalized tothe total integrated intensity of the initial intensity distribution measured after thecollimation lens. The asymmetric behavior of the beam profile was investigated bythe following expression for the ellipticity of the intensity in the first maximum Imax,1

35

3 Bessel beams

Table 3.1 Characteristic properties of the used edge emitting laser diode.

Parameter Symbol Value

Wavelength λ 436 nmRefractive index naxicon 1.4668Core radius rcore 20.48 µmOpening angle, slow axis θslow (143.8± 0.02) mradOpening angle, fast axis θfast (286.4± 0.03) mradOpening angle substrate mode θsubstrate 271.6 mradBessel beam start position z0 (104.63± 0.2) mmBessel beam end position, slow axis zmax,slow (310.8± 0.03) mmBessel beam end position, fast axis zmax,fast (632.4± 0.07) mm

1 mm

Fast axis

Slow axis

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.40.0

0.2

0.4

0.6

0.8

1.0

θmax, slow

θmax, fast

Slow axis FitFast axis Fit

Nor

mal

ized

inte

nsity

/a.u

.

Radial position / rad

θmax,fast

θmax, slow

Figure 3.18 Left: Far-field intensity profile of the laser diode with indication for 1/e2

intensity drop. Right: Measured intensity profile of the fast and slow axis with markingsat characteristic points. The shaded areas indicate the rounding of the axicon tip.

36


yx

100µm

B1 B3 B4B2

yx

xz

50 mm

yz

50 µm

B1

B2

Slow axis

Fast axis

B3 B4

B2 B3 B4

B1

Sim

ulat

ion

Mea

sure

.

B1 B3 B4B2

Figure 3.19 Left: Simulations of the beam profiles along the optical axis in the planeof the slow and fast axis. To illustrate the side lobes, the intensities were logarithmicallyscaled. Right: Simulation (top) and measurement (bottom) at different positions of thetransverse plane: (B1) z = 125 mm, (B2) zmax,slow, (B3)

√zmax,slow zmax,fast and (B4)

zmax,fast. The images are each normalized to the brightest pixel.

as a measure for the transition from a circular profile to a line-shaped profile:

ε = (Ifast,1 − Islow,1)

(Ifast,1 + Islow,1), (3.50)

where ε = 0 corresponds to a perfectly symmetric beam and ε = 1 to an asymmet-ric beam with vanishing intensity in the slow axis direction. This parameter wasobtained experimentally by fitting a sinusoidal curve to the values Imax,1(∆φ).

3.5.3 ResultsIn Figure 3.19 the reconstructed intensity pattern along the propagation directionfrom simulations in slow and fast axis is illustrated. In Figure 3.19 (right) thelateral cross-sections at four different positions are illustrated in simulations (top)and measurements (bottom). The characteristic positions are (B1) z = 125 mmbehind z0, (B2) at the end of the slow axis region zmax,slow, (B3) an intermediateposition √zmax,slow zmax,fast, and (B4) at the end of the fast axis region zmax,fast. Theintensity pattern starts symmetrical in the slow axis region. Beginning from theend of the slow axis region zmax,slow an asymmetry can be observed. Moving furtheroutside, the asymmetric pattern transforms into a more line shaped profile. Thedeviations in the experiments starting from position B3, come from imperfection,e.g., substrate modes and ripples further outside where the profile is less smooth,which can be seen near θmax,slow in Figure 3.18.

The normalized core intensity Icore comparing simulation and measurement is illus-trated in Figure 3.20. The “lens effect” and the characteristic oscillation explainedin Section 3.3 can be observed. An additional deviation is marked with ∆zs, whichresults from the variations in the intensity profile including substrate modes andripples. This position agrees well with the angle ϕ ∼3.8 mrad that can be measured

37

3 Bessel beams

0 100 200 300 400 500 6000.0

0.1

0.2

0.3

0.4

zmax, fastzmax, slow

z0


Nor

mal

ized

core

inte

nsity

/a.u

.


∆zs

Figure 3.20 Normalized core intensity of the simulated and measured beam profile alongthe optical axis. The shaded area ∆zs indicates the influence of the substrate mode.

from the tilt of the substrate mode in the collimated beam. The resulting width∆z ∼ 95 mm, where the substrate mode will appear is consistent with the actualsubstrate thickness ds ∼96 µm.

The behavior of the core radius (rcore) and the ellipticity as a function of the z-position is illustrated in Figure 3.21. The core radius approaches the value of theideal axicon rideal = 20.48 µm after the “lens effect” (z0). The ellipticity ε fromEquation (3.50) was introduced to study the transition from a circle (ε = 0) towardsa line profile (ε = 1). For the laser diode, ε starts rising shortly after z0 as theintensity on the slow axis decreases already at that point, while the intensity on thefast axis is still approximately constant. The deviation between measurement andsimulation can be explained by variations in the initial intensity profile and a totalloss of contrast due to misalignment, reflections and scattering. This effect can alsobe observed in the cross-sectional images (B3) and (B4) in Figure 3.19.

Self-reconstruction of the asymmetric beam was investigated using the same setupand absorber described in Section 3.3. In Figure 3.22 the simulated (left) and mea-sured (right) intensity profiles along the optical axis are shown. To illustrate the self-reconstruction in the different characteristic regions, the positions of the absorberwere chosen to be near the characteristic points: B1 = 125 mm, B2 ≈ zmax,slow,B3 ≈ √zmax,slow zmax,fast, and B4 ≈ zmax,fast. As already observed in Section 3.3,the intensity after the self-reconstruction of the beam is in good agreement with theintensity profile of the un-obstructed beam and the theoretical values. This resultcan be observed in all positions throughout the propagation direction, except forthe absorber at the end of the fast axis region. In this case, the intensity is alreadyof the order of noise and distortions, which can be seen in Figure 3.19 (right).

38


100 150 200 250 300 350 4001416182022242628303234363840

MeasurmentSimulation

Ideal radius

z0

Cor

era

dius

/µm


zmax,slow0.0

0.2

0.4

0.6

0.8

1.0

Ellip

ticity

Figure 3.21 Measurement and simulation of the core radius and ellipticity in the intensityof the first maximum, see Equation (3.50), with indications of the characteristic positionson the optical axis.

0 100 200 300 400 500 6000.00

0.05

0.10

0.15

0.20

0.25

0.30IdealaxiconNo absorberPosition C1Position C2Position C3Position C4

Nor

mal

ized

core

inte

nsity

/a.u

.

Position in z-direction / mm0 100 200 300 400 500 600

0.00

0.05

0.10

0.15

0.20

0.25

0.30No absorberPosition C1Position C2Position C3Position C4

Nor

mal

ized

core

inte

nsity

/a.u

.


Figure 3.22 Normalized core intensity of the simulated (left) and measured (right) beamprofile along the optical axis (z-direction) with an absorber of 375 ± 2 µm radius placedat different positions on the optical axis. The positions are B1 = 125 mm, B2 = 276 mm,B3 = 382 mm and B4 = 546 mm. The shaded regions indicate the distance behind theabsorbers after which the beam is geometrically reconstructed.

39

3 Bessel beams

3.6 ConclusionIn this chapter, a short overview of Bessel beams, which are one class of quasipropagation invariant beams, was given. It included the characteristic propertiessuch as the extended focal zone with its bright core radius and self-reconstruction.Afterwards, the generation of Bessel beams using a non-ideal axicon waspresented and the analytical considerations were evaluated using BPM simulationsand measurements. It was demonstrated that it is possible to simulate a Besselbeam generated from a rounded axicon using a Gaussian illumination profile.The rounding at the tip of the axicon corresponds to a small lens with focal lengthz0 = (105.3 ± 0.2) mm. This lens generates a high intensity peak in the spatialintensity profile, which was refereed to as “lens effect”. Only beyond this point, aBessel beam is generated, as verified in simulations and measurements. The derivedoscillations converge to the intensity of an ideal axicon-generated Bessel beam. Fromthe cross-sectional images at different positions, a good agreement between simula-tions and measurements was found, likewise in the self-reconstruction experiments.Further, segmented Bessel beams were investigated, which can be generatedusing ring apertures placed in a conical wavefront. A simple analytical estimatein the classical and Fraunhofer limits was developed to determine segmented Besselbeams choosing appropriate dimensionless parameters which were independent of thewavelength and the axicon angle. An easy to use expression for the shortest possiblebeam segment and corresponding optimal aperture was found, assuming that theshortest Bessel region occurs near the intersection of these two limits. To verifythe considerations made, BPM simulations and measurements were performed byplacing a ring aperture behind an axicon that is illuminated with an approximateplane wave. The small increase in the optimal aperture width by approximately15.5 % can be explained with the hard aperture and also explains the oscillationswhich appear in the simulated intensity profile. Additionally, different wavelengths,axicon angles, and aperture sizes at different radial positions confirm the analyticalestimates to be generally valid. Therefore, this expression can be used for otherapplications where Bessel beams need to be tailored.The last part of this chapter describes Bessel beams generated from asym-metric and astigmatic laser diodes. Analytical considerations regarding theopening angle, the maximal intensity along the optical axis, effects of astigmatismand effects of substrate modes and far-field ripples were investigated and evaluatedin BPM simulations and measurements. The use of a non-perfect, asymmetric edgeemitting laser diode and a refractive axicon with rounded tip showed a transitionof a symmetric Bessel beam into a bow-tie shaped pattern. Analytical considera-tions to estimate the opening angle of the bow-tie shaped pattern were done, andthe maximum of the intensity for the asymmetric beam was found. Astigmatismwas investigated, which may destroy the formation of a Bessel beam. This effect is

40

3.6 Conclusion

highly suppressed in realistic asymmetric light sources with realistic values of theastigmatism, as observed for the used edge emitting laser diode.Substrate modes, which are also characteristic for the these laser diodes, have adifferent opening angle, are not collimated and interfere with the main beam. Com-paring characteristic properties such as the intensity, core radius and the ellipticityalong the optical axis, two characteristic regions were determined: the approxi-mately isotropic central part of the intensity distribution, which creates a Besselbeam in the slow-axis region that turns smoothly into an asymmetric bow-tie shapedbeam cross-section in the fast axis region, which is illuminated by the outer, highlyanisotropic part of the intensity profile. Consequently, a collimation lens with shortfocal length using only the slow-axis region would be beneficial to avoid the resultingimperfections from the fast-axis described above.The measurement results agree with the BPM simulations, the main difference beingsome variations in the intensity of the central maximum. Reason for the deviationscan be explained by substrate modes and the assumption of a perfect asymmetricGaussian profile in the simulation. The existence of the central maximum wasrobust and not disturbed by most imperfections, which was additionally verified byself-reconstruction experiments in all regions.

41

4 Adaptive ring apertureIn the following chapter, the adaptive ring aperture developed to generate depth-controlled Bessel beams is described. The requirements for such an aperture and anoverview of different concepts with the selected method are discussed. The fabri-cation process based on standard micro-fabrication techniques is explained, includ-ing the assembly and packaging technique to obtain a functional device. Differentaperture designs are displayed, which were chosen to evaluate the results gained inSection 3.4. Further, optical characterizations of the developed device are presented.A brief electrical characterization can be found in the Appendix in Section A.1.1.

4.1 RequirementsAs described in Section 1.1, the design of an adaptive ring aperture was one of themain goals of this thesis. The requirements were as follows:It should be an adaptive 3×3 array with discretized, individually addressable elec-trodes, as illustrated in Figure 4.1. Each element should have a fixed diameter of1 mm, which limits the maximum number of individually addressable ring electrodesto four (Section 3.4, Equation 3.27). The device needed to be transmissive, compact,and it should be fabricated using mainly standard clean room processes.

Electrode R1

1 mm

Electrode R2Electrode R3Electrode R4E2E1 E3

E6E5E4

E7 E8 E9

Figure 4.1 Schematic of the 3×3 ring aperture array. Each element consists of up to 4individually addressable electrodes (R1-R4).

43

4 Adaptive ring aperture

Figure 4.2 Left: Cross-section of an electrochromic device. Right: Images of an elec-trochromic device (electrochromic-iris) in the four individual states [71]. Images reprintedwith permission from IOP Publishing.

4.2 Overview of different conceptsIn the literature, there are many different compact apertures which are continu-ously movable with various concepts of actuation. Different approaches includemechanical apertures [61–63], optofluidic apertures using electrowetting [64–68] orpiezo-actuation [69], electrochromic apertures [70,71], liquid crystal elastomer aper-tures [72], and liquid crystal apertures [73]. Here, different concepts are presentedand discussed that are suitable to build a device with the requirements mentionedabove.

Electrochromic devices

Electrochromism uses electro-chromic materials whose absorption characteristics canchange by an electric field (redox reaction) or with temperature [74]. Electrochromicdevices consist of an electrochemical cell sandwiched between glass or flexible sub-strates, each with transparent electrodes and thin electrochromic materials, e.g.,poly(3,4-ethylenedioxythiophene) PEDOT and an additional electrolyte, as illus-trated in Figure 4.2 [71]. They operate as a rechargeable electrochemical cell and canmaintain their redox state for a given potential, which is called memory effect [71,75].These devices have a thin and compact design and low power consumption at verysmall voltages (∼ 2 V) [70, 71]. New materials to enhance the absorption contrastare still under development. The switching time is in the range of seconds [71,76].

Electrowetting devices

Electrowetting devices can manipulate small amounts of liquids on surfaces using anelectrical potential [77]. There exist several electrowetting devices in the literaturewith a movable aperture (often called optofluidic iris) based on electrowetting ondielectrics [65–67, 78]. These concepts use a thin insulation layer to protect theelectrode from chemical reactions with the conducting liquid. But none of them

44

4.2 Overview of different concepts

have discretized, individually switchable ring apertures. One way to produce suchan aperture could be the use of electrowetting displays [79–81]. The drawback isthat the opaque medium in each pixel can be shifted to the side, e.g., a reservoiron the side, which would make the design bigger. Another way would be to use theprinciple of the variable slit aperture proposed by Schuhladen et al. and design theelectrodes in a ring-shaped pattern [68], or the proposed optical switch by Lee etal., further devloped to a discretized ring aperture [82].

Liquid crystal devices

Liquid crystals are widely used in various electro-optical devices. The most commonapplications being liquid crystal displays, which have been used since the 70s [83].They are commerically produced and have been characterized extensively. Someexamples include digital watches, mobile phones and liquid crystal displays in flat-screens [74,84,85]. In optics there are several applications using liquid crystal spatiallight modulators [86]. They can be reflective or transmissive, modulating either thephase or the amplitude in an array of small individually adressable pixels [87, 88],making them a versatile tool for complex beam generation [89–92]. The workingprinciple of a twisted nematic liquid crystal cell is discussed in Section 4.3.1.

4.2.1 Selection of the methodIn this thesis, the liquid crystal device concept was chosen. It has a straightforwardfabrication technique and is widely used in literature. Nematic liquid crystals arewell characterized and easy to handle. The fabrication can be done in the cleanroom with standard MEMS wafer-level processes.In contrast, electrowetting is insensitive to polarization. It has a thin design, andthe switching time is in the millisecond range, but there is no discretized apertureas desired. Hence, this would lead to a completely new process development, whichwas not intended within this thesis. The electrochromic concept has a compactdesign with very low voltages but the switching time of several seconds and therather limited works on suitable high contrast materials in the desired wavelengthdisqualify this concept.To evaluate the liquid crystal device concept for a ring aperture, a process sequencewas developed, and a prototype was fabricated. This prototype was presented in aconference paper [93], it will not be discussed further in this thesis.

45


4.3 Liquid crystal technologyLiquid crystals possess both properties of liquids and crystals. They have a crystal-like order but can flow like liquids [94]. It is an intermedite state (mesophase), whichhas an orientation of one or two dimensions. The nematic liquid crystal with itsdirector is illustrated in Figure 4.3. The director n is a unit vector which is definedalong the “long axis” of the rod-like structure [94]. The order parameter S describesthe degree of orientational order:

S = 12(3〈cos2(Λ)〉 − 1

), (4.1)

where Λ is the angle between the axis of the molecule and the director n (Figure 4.3).The brackets 〈〉 denote the average over all molecules. A perfect crystal has Λ = 0,resulting in an order parameter S = 1. In a liquid, the average cosine-square termis zero, resulting in an order parameter of S = 0. A liquid crystal is in the range of0.3 < S < 0.9 [95]. The anisotropic molecular structure of nematic liquid crystalscauses an optical anisotropy which results in two refractive indices: an extraordinaryrefractive index, ne, when the light is polarized in direction of the director, andan ordinary refractive index, no, perpendicular to it. The optical anisotropy orbirefringence of the liquid crystal is

∆n = ne − no. (4.2)

If the temperature increases the extraordinary refractive index decreases as theordinary refractive index increases until the clear point temperature is reached, andthe liquid crystal changes from its nematic state to an isotropic state.Similar to the optical anisotropy, the nematic liquid crystal has a dieletric anisotropyresulting in

∆ε = ε‖ − ε⊥, (4.3)

where ε‖ and ε⊥ are parallel and perpendicular to the director.By applying an electric field the liquid crystal molecules are slightly displaced, and anelectric dipole is induced. The static deformation pattern of liquid crystals dependson the balance between elastic and electrostatic (or magnetic) forces [94]. The elasticconstants are the splay (K1), twist (K2) and bend (K3) described by Frank [96]. Thefree energy expression yields the elastic energy density in the Frank-Oseen form:

F = 12(K1(∇n)2 +K2(n∇× n)2 +K3(n×∇× n)2

). (4.4)

46

4.3 Liquid crystal technology

nΛ

Figure 4.3 Nematic liquid crystal with director n, which indicates the average direction.

4.3.1 Twisted-nematic liquid crystal deviceOne of the major modes how liquid crystal devices are used is as twisted-nematicliquid crystal cells [88, 97–99]. Their working principle in normally black mode isillustrated in Figure 4.4. Two alignment layers in orthogonal orientation align theliquid crystals (in a twisted way). When polarized light comes in, the polarizationdirection is rotated by 90, according to the twisted liquid crystals. In addition,two parallel polarization filters are added. The first linear polarizer allows onlythe parallel polarized part of the light to propagate through the device. In the un-actuated state of the cell the light is rotated by 90 and is then blocked by the secondlinear polarizer. When a voltage is applied, the liquid crystals align themselves alongthe applied field, and their birefringent effect is suppressed. The initially parallelpolarized light can now travel through the second polarizer. To avoid the productionof elliptically polarized light, the Mauguin condition for a 90° twisted nematic liquidcrystal cell needs to be fulfilled:

∆ndlc λ

2 , (4.5)

where dlc is the thickness of the cell, and λ the wavelength [94]. The thresholdvoltage for a twisted nematic cell, at which the liquid crystal starts aligning itselfalong the electric field is [97,98]:

Vthreshold = π

√1

ε0∆ε

(K1 + K3 − 2K2

4

). (4.6)

If the electric field is removed, the elastic forces restore the twist, and the light isblocked. The time to switch between the actuaded (on) state and the un-actuated(off) state can be determined using the rotational viscosity (rotation of the moleculearound an axis perpendicular to the director) χ = ηρ [99] by

ton ∝ χd2lc

∆εE−π2(K1+K3−2K24 ) , (4.7)

toff ∝ χd2lc

π2(K1+K3−2K24 ) . (4.8)

47


Alignment layer

Polarizer

V

Electrode

Off On

Liquid crystal dlc

Figure 4.4 Twisted nematic liquid crystal cell operation in normally black mode. Left:Un-actuated state (Off), the incoming light is rotated by 90 and blocked by the secondlinear polarizer. Right: Actuated state (On), the liquid crystals align themselves alongthe applied electric field, and the incoming light can pass through the device.

Material properties of the nematic liquid crystal

In this thesis, the commonly used nematic liquid crystal 4-cyano-4-pentylbiphenylfrom Sigma Aldrich was used, better known under the name 5CB, with a rod-likestructure (∼ 2 nm long and ∼ 0.5 nm wide) [95]. By applying an electric field,the liquid crystal molecules are slightly displaced, and an electric dipole is induced(Figure 4.4 (right)). The temperature range for the nematic state of CB5 is between23 C and 35 C [95]. The material properties of 5CB are summarized in Table 4.1.

Table 4.1 Properties of used liquid crystal 5CB [94,95,99].

Parameter Symbol Value

Extraordinary refractive index ne 1.735Ordinary refractive index no 1.531Optical anisotropy ∆n 0.204Parallel permittivity ε‖ 22Perpendicular permittivity ε⊥ 5.9Dielectric anisotropy ∆ε 16.1Frank elastic constant: splay K1 ∼7.2× 10−12NFrank elastic constant: twist K2 ∼4.3× 10−12 NFrank elastic constant: bend K3 ∼10× 10−12 NKinematic viscosity η 2.4× 10−5 m2

sMass density ρ 1008 kg

m3

48

4.4 Fabrication and assembly

4.4 Fabrication and assemblyThe fabrication process is based on wafer-level micro-fabrication techniques summa-rized in Figure 4.5. The substrate is a 700 µm thick borosilicate glass with a 100 nmthick indium tin oxide (ITO) layer and a sheet resistance of 20 Ω

sq. purchased fromPräzisions Glas. The fabrication was done as follows:

1. Contact pads: The chromium/gold (Cr 50 nm/Au 150 nm) metallization wasrealized using physical vapor deposition with a lift-off process (AZ 5214E).

2. Transparent electrodes: A second lithography step with a standard positiveresist (AZ1518) was performed, followed by wet etching of ITO.

3. Alignment layer structuring: A hybrid polymer (Ormocomp/Ormothin, 1:3)was spin-coated and lithographically structured.

4. Rubbing: The orientation of the alignment layer was realized by rubbing amicrofiber cloth along the wafer (parallel to the flat).

5. Spacing and cavity: A dry film resist, Ordyl SY330, with a thickness of 30 µmwas laminated and structured by lithography.

Contact pads and electrode layerThe chromium-gold (Cr/Au) layer was fabricated by a lift-off process using AZ 5214Eand the physical vapor deposition technique evaporation. A 50 nm chromium layerwas used as adhesion layer underneath the 150 nm thick gold layer. They weremeasured to be (170.96 ± 7.08) nm thick using an mechanical profilometer (KLATencor P11). The variation results from a poor accuracy in the control of thedeposition rate.Subsequently, the transparent electrode (ITO) layer was structured by a secondlithography step, which was followed by a wet etch of ITO with 6 M hydrochloricacid solution with additional 0.2 M iron(III) chloride at room temperature. The etchtime of 10-15 min was controlled measuring the sheet resistance. The ITO layer wasmeasured to be (101.10 ± 8.22) nm, which agreed well with the value provided bythe company.

Alignment layerThe alignment layer was realized using a UV-curable hybrid polymer, Ormocompfrom Micro Resist Technology. An additional thinner (Ormothin) was applied ina 1:3 ratio to get the desired thicknesses of 350 nm [100]. The two componentswere mixed thoroughly and filtered (filter size: 2 µm). It was spin-coated at roomtemperature with 6000 rpm for 30 s, which resulted in a thickness of (348.19 ±

49


Chrom / Gold

Alignment layer

1.

5.

3.

2.

4.

Rubbing

ITO etching

Ordyl

Borosilicate glass

ITO

Figure 4.5 Fabrication steps of the liquid crystal ring aperture device:1. Cr/Au metallization by vapor deposition and a lift-off process.2. Electrode structuring by a lithography step and wet etching.3. Alignment layer with a hybrid polymer by spin-coating and lithographic structuring.4. Rubbing of the alignment layer with a microfiber cloth.5. Lamination of dry-film-resist and lithographic structuring.

16.44) nm. After a soft bake at 80 C for 2 min, UV-exposure was performed withthe mask aligner (MA/BA6, Süss Microtec) using proximity exposure for 100 s, witha subsequent post-exposure bake at 130 C for 10 min. The substrates were thendeveloped with OrmoDev at room temperature for 2 min, followed by a final hardbake at 150 C for 3 h.

Cavity and spacingOrdyl SY330 from Elga Europe is a permanent dry film resist with a thickness 30 µm.It was used as a cavity for the liquid crystal and as a spacer for the device. It waslaminated using an office laminator (Mylam 12) with a temperature of approximately110 C [101]. Afterwards, the resist was exposed using the above mentioned maskaligner with proximity exposure for 16 s, followed by a post-exposure bake for 3minat 80 C. After the development a hard bake at 150 C for 1 h was done. Thethickness of the Ordyl layer was determined to be (21.86± 0.65) µm, which resultsmost likely from material shrinkage.

AssemblyAfter the fabrication steps, the wafer was diced into single chips, and the assemblywas done. The wafer included two different chip types: the bottom part of theliquid crystal cell (bottom chip) containing the structured electrodes with 7 differentdesigns (A1-A7) and the Ordyl cavity. The top part (counter chip) contained thecounter electrode. Both parts are illustrated in Figure 4.7.

50

4.5 Chip design

Counter chip

Bottom chip

Liquid crystal

Linear polarizer

Wire bonding

PCB

Wire bond loop

Figure 4.6 Assembly steps of the liquid crystal ring aperture device. Left: The bottomchip is placed into the PCB and electrically connected. Right: The counter chip is gluedon top, the liquid crystal is filled via capillary forces, and the linear polarizers are placedparallel to each other on both sides.

The bottom chip was placed into a specially designed printed circuit board (PCB),for electrical connection, illustrated in Figure 4.6. It consist of two layers (doublesided copper layers) and a milled pocket to align and fix the bottom chip in itsdesired position and to be able to transmit light (Figure 4.8). The chip was thenconnected with the PCB by wire bonding using aluminum wires (Delvotek wirebonder). To ensure the connection of the counter chip, several wire bond loops wereplaced on the PCB at the necessary position (Figure 4.6). The assembly of theliquid crystal cell was done manually using a die bonder. The smaller counter chip(Figure 4.7) was aligned 90° offset to the already connected chip and glued on topusing a UV-curable adhesive. For this purpose, few small droplets of the adhesivewere dispensed on the Ordyl cavity to minimize the thickness of the adhesive layerand make sure that no adhesive flows inside the cavity.Alignment marks on both chip sides assure that the top and bottom alignment layersare 90 shifted to each other. After the assembly, the liquid crystal was filled intothe cavity by capillary forces, and the two linear polarizers from Edmund Optics(cut into the right shape) were added in parallel orientation to each other on bothsides, as illustrated in Figure 4.6. The finalized device is depicted in Figure 4.8.

4.5 Chip designIn Section 3.4, the basic concept of segmented Bessel beams was presented to gener-ate depth-controlled Bessel beams and the optimal aperture width for the shortestpossible Bessel beam segment (without diffraction) was determined. The knowledgegained in the previous section was used to design different ring apertures, as illus-trated in Figure 4.9. The dimensions of all seven designed ring apertures is presentedin Table 4.2.The bottom chip (A1-7), consisted of the the structured electrodes and the Ordylcavity. Its dimensions were 1.2 mm × 1.2 mm (Figure 4.7 (left)). The counter chip

51


0.5 mm

Ordyl

Contact pads

ITO electrodes

Chip outline

Figure 4.7 Left: Bottom chip A3 with the ITO electrode array, metal contact pads andthe Ordyl cavity. Right: Counter chip. Both chips have alignment marks for the assembly.

2 mm

1 mm

Figure 4.8 PCB with the filled and connected liquid crystal cell. The chip and the PCBwere connected via wire bonding. Inset: Close-up view of the ITO electrodes.

Table 4.2 Designed adaptive ring apertures with the diameter(∅) of each element, number(#) of rings, the aperture width, and the spacing between the single eletrodes.

Design ∅ element / mm # of rings Aperture width /µm Spacing /µm

A1 1 3 100− 150 50A2 1 4 50− 100 50A3 1 4 100− 140 20A4, 5 1 7 36− 65 30A6 4 10 50− 355 50A7 4 12 50− 344 20

52

4.5 Chip design

1 mm

Figure 4.9 Ring aperture array designs. Top from left to right: The three ring aperturearrays A1, A2 and A3. Bottom from left to right: The single ring apertures A4 and A5designed for an 140 axicon, and A6 and A7 designed for a 170 axicon.

with the big counter electrode and dimensions of 1.2 mm × 0.5 mm was designedsmaller to be able to contact the bottom chip.The first three designed chips A1-A3 were ring aperture arrays with each 3×3 el-ements and a 1 mm diameter of each element, as illustrated in Figure 4.9 (top).The aperture width was not optimized for one specific axicon angle so that differentaxicon angles can be evaluated.The last four designed chips A4-A7 were additionally produced to prove differentconcepts regarding spacing, axicon angle and symmetry of the desired device (Fig-ure 4.9 (bottom)). Designs A4 and A5 have a diameter of 1 mm with optimalapertures width dmin designed for an axicon with 140 apex angle, one symmetricand one asymmetric. Designs A6 and A7 have a diameter of 4 mm with optimalapertures width dmin designed for an axicon with 170 apex angle. The optimal aper-ture width dmin was determined using Equation 3.27 from Section 3.4 with 473 nmwavelength. A summary of each aperture width for all designs can be found inTable A.1 in the Appendix.

53


4.6 Optical characterization

The optical performance of the ring aperture devices was evaluated using chip A1and chip A2 for the array designs and chip A6 for the single design.

4.6.1 Methods

Optical measurement setup: The experimental setup is similar to the one de-scribed in Section 2.2.1. The fabricated and assembled ring apertures were intro-duced directly in front of the commercial axicon with 170 apex angle. The DPSSlaser with 473 nm wavelength was collimated using an aspherical collimation lenswith 100 mm focal length to obtain an approximately plane wave. An additionalstatic aperture was placed in front of the ring aperture device to block any straylights. Dependent on the element size, the static aperture had a diameter of 1 mmfor the array or a diameter of 4 mm for the single element. The camera with 1.67 µmpixel pitch and a microscope objective with magnification factor 4 were used. Thering apertures were driven with a sinusoidal voltage of 6 Vrms and 500 Hz.

BPM simulations: BPM simulations were used for two reasons: First, to confirmthat the measurement results agree with the analytical considerations from Sec-tion 3.4, and second to estimate the wavefront error generated from the adaptivering apertures. For this purpose, a plane wave with an ideal axicon with apex an-gle 170 was used to simulate the propagation of the segmented Bessel beams with1 mm static aperture. The adaptive ring aperture was realized by a matrix similarto the absorber matrix B described in Section 2.2.2 representing the desired ringaperture. This matrix was multiplied to the initial conical profile and propagatedto the desired image plane.

An astigmatic term was added to estimate the magnitude of the wavefront error,such that the initial overall electric field extends to:

Ψinit(r,z = 0) = ψ0 e−ik(−∆nh(r)+mλr2

r20

), (4.9)

where ψ0 is the amplitude, ∆n the difference in refractive index of the lens and thesurrounding media, m is the factor of the wavefront error, r0 is the radius of thestatic aperture, and the ideal axicon slope h(r) = ((180 − γ)/2)r.

Data analysis: The reconstruction of the image plane along the optical axis andthe intensity profile evaluation from Section 2.2.3 were used. The intensity profilesalong the optical axis were normalized by subtracting the intensity profile of theun-actuated liquid crystal ring aperture.

54


200 µm

Figure 4.10 Photograph of the individually actuated electrodes of design A2, element E5at 6 Vrms: top left R1, top right R2, bottom left R3, bottom right R4.

4.6.2 ResultsWavefront errorAn example of the individually actuated electrodes is shown in Figure 4.10 for designA2. The pictures show that the alignment layers, which should be orthogonal toeach other, have some imperfections and defects. Those defects lead to wavefronterrors and were tried to be measured using a Shack-Hartmann wavefront sensor.Unfortunately, two different sensor systems achieved no reliable result. Therefore,BPM simulations were performed to determine the magnitude of the wavefront error,which included an additional astigmatic term in the initial wavefront. In Figure 4.11(left) the ideal Bessel beam and Bessel beams with different astigmatic peak-to-valley wavefront errors are illustrated for an axicon with 170 apex angle. It can beobserved that the wavefront error scales with the intensity and Bessel beam range.If a static aperture is introduced, as illustrated in Figure 4.11 (right), an additionalaperture effect can be observed. It consists of a characteristic cut-off profile withoscillations coming from the edge of the aperture.In Section 3.5 the behavior of astigmatism in the central intensity along the prop-agation direction was already introduced and showed that destructive interferenceappears at certain z values (Figure 4.11). From this knowledge, it can be assumedthat all wavefront errors that show no destructive interference within the Besselbeam range zmax have a rather small influence on the Bessel beam. For the designarrays A1, A2, and single element A6, that means all wavefront errors below 2/3λ.

Design A2The simulation and measurement results for design A2 of the whole Bessel beamwithout the ring aperture and the corresponding segmented Bessel beams createdwith the ring aperture are illustrated in Figure 4.12. The measured cross-sectionalimages in Figure 4.12 (right) were added to show the quality of the Bessel beamin the center of each segmented region. They clearly show the characteristic ring

55


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.00.10.20.30.40.50.60.70.80.91.01.11.2

λ/22/3λλ2λ

Cor

ein

tens

ity/a

.u.


Ideal Bessel beamλ/16λ/8λ/4λ/3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.00.10.20.30.40.50.60.70.80.91.01.11.2

λ/22/3λλ2λ

Cor

ein

tens

ity/a

.u.


Ideal Bessel beamBessel beam with static apertureλ/16λ/8λ/4λ/3

zmax

Figure 4.11 Core intensities along the propagation direction for an ideal Bessel beam anddifferent astigmatic wavefront errors. Left: Without an additional static aperture. Right:Using a static aperture with 1 mm diameter and indication for a corresponding zmax.

pattern of the Bessel beam with some wavefront errors resulting from defects in thealignment layer, which can be found in Figure 4.10. Additionally, it can be observedthat the measured beam is widened compared to the simulations, which originateslikely from smoother edges of the fabricated apertures. Up to some small variations,the measurements agree well with the simulations.For a more quantitative result the core intensity along the propagation direction insimulations (left) and measurements (right) are illustrated in Figure 4.13. The sim-ulation was done with a static aperture without wavefront errors. As suspected, thesegmented beams from the simulations follow the Bessel beam without an aperturevery closely in their corresponding regions. The measurement results were evaluatedat three different element positions to have some redundancy. Here, clear deviationsfrom the theoretical aperture width can be observed in R3 and R4, which can beexplained with a wavefront error increase with increasing radius.To estimate the magnitude of the wavefront errors, additional simulations with vari-ous wavefront errors were performed and then compared with the measurements. Anexample is illustrated in Figure 4.14 for element E9. A simulated wavefront error ofλ/5 was added and shows a good agreement with the measurement. The wavefronterrors of the other two elements E1 and E5 were determined to be approximatelyλ/4 and λ/2, respectively.

Design A1A second array design was measured and compared with BPM simulations to evalu-ate the findings from design A2. The simulation and measurement results for designA1 of the whole Bessel beam without the ring aperture and the segmented Besselbeams created with the ring aperture are illustrated in Figure 4.15. The measuredcross-sectional images were added to show the quality of the Bessel beam in thecenter of each region. Up to some variations, the measurements agree well with the

56


2 mm

20 µm

10µm

(c)

(b)

(c)

(d)(d)

(b)

(a)(a)

R1 R2 R3 R4

(e) (e)

R1

R2

R3

R4

2 mm

1 mm 10µm

R2

R3

R1

R1 R2 R3 R4

R4

xy

zy

zy

xy

Simulation MeasurementIncidentbeam

Figure 4.12 Left: Simulated incident beam and apertures. Center: Simulated (left) andmeasured (right) intensity profile of the Bessel beams (logarithmically scaled) with andwithout the ring apertures using design A2 element E9. The shaded areas indicate thepurely geometrical Bessel beam regions. Right: Measured cross-sectional areas of theBessel beam each normalized to the brightest pixel.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.00.10.20.30.40.50.60.70.80.91.01.11.2

Nor

mal

ized

core

inte

nsity

/a.u

.


Bessel beamwithout ring aperture

R4

R3

R2

R1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.00.10.20.30.40.50.60.70.80.91.01.11.2


R4

R3

R2

Nor

mal

ized

core

inte

nsity

/a.u

.


E1E5E9

R1

Figure 4.13 Normalized core intensity along the propagation direction of the ring apertureusing design A2. Left: Simulation of the Bessel beam with and without the ring aperturesand additional static aperture. Right: Measurement at three different element positionsE1, E5 and E9. The shaded areas indicate the theoretical Bessel beam region for eachelectrode.

57


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.00.10.20.30.40.50.60.70.80.91.01.11.2

Nor

mal

ized

core

inte

nsity

/a.u

.


Simulation

R1

R2

R3

R4

Figure 4.14 Normalized core intensity along the propagation direction of the measuredring aperture using design A2, element E9 with a simulated wavefront error of λ/5 anda static aperture. The shaded areas indicate the theoretical Bessel beam region for eachelectrode.

simulations. The cross-sectional images clearly show the characteristic ring patternof the Bessel beam with some small wavefront errors.The normalized core intensity along the propagation direction in simulation (left)and measurement (right) is illustrated in Figure 4.16. Again, the simulations consid-ered a static aperture without wavefront error. In the measurements the deviationfrom the beam without an aperture are much higher than before due to more de-fects in the alignment layer. The measurement result for aperture R3 deviate fromtheoretical region for all three elements indicating that the wavefront error increaseswith increasing ring radius.The magnitude estimate of the wavefront error for element E5 is illustrated in Fig-ure 4.17. The simulated wavefront error is 0.6λ. The wavefront error of element E1and E2 are 0.65λ and 0.6λ, respectively.

Design A6The single element design A6 was additionally measured to verify the optimal aper-ture of the segmented Bessel beams as derived in Section 3.4. The normalized coreintensity along the propagation direction is illustrated in Figure 4.18 in simulations(left) and measurement (right). The theoretical Bessel beam regions agree well withthe measurements. Especially at smaller radii where the optimal ring aperturesare small no widening due to diffraction is visible. Starting from electrode R7, awidening can be observed. Thus, compared to the smaller array designs with 1 mmdiameter a higher wavefront error of 0.75λ was estimated in design A6 with 4 mmdiameter. This error is higher than the 2/3λ mentioned before and has thereforea higher influence of the Bessel beam. An indication for this is the decrease of theintensity after ring electrode R6, followed by an increase in ring electrode R9, whichsuggests a high asymmetric wavefront error.

58


1 mm

R2

R3

R12 mm

20 µm

10µm

(c)

(b)

(c)

(d)

(b)

(a)

R1 R2 R3

R2

R3

R1 2 mm

10µm

(d)

(a)

R1 R2 R3

xy

zy

xy

zy

Simulation MeasurementIncidentbeam

Figure 4.15 Left: Simulated incident beam and apertures. Center: Simulated (left) andmeasured (right) intensity profile of the Bessel beams (logarithmically scaled) with andwithout the ring apertures using design A1 element E5. The shaded areas indicate thepurely geometrical Bessel beam regions. Right: Measured cross-sectional areas of theBessel beam each normalized to the brightest pixel.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.00.10.20.30.40.50.60.70.80.91.01.11.2

Nor

mal

ized

core

inte

nsity

/a.u

.



R3

R2

R1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.00.10.20.30.40.50.60.70.80.91.01.11.2

Bessel beam without ring aperture

E1E2E5

Nor

mal

ized

core

inte

nsity

/a.u

.


R1

R2

R3

Figure 4.16 Normalized core intensity along the propagation direction of the adaptivering aperture using design A1. Left: Simulation of the Bessel beam with and without thering apertures and additional static aperture. Right: Measurement using three differentelement positions E1, E2, and E5.The shaded areas indicate the theoretical Bessel beamregion for each electrode.

59


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.00.10.20.30.40.50.60.70.80.91.0

Nor

mal

ized

core

inte

nsity

/a.u

.


Simulation

R1

R2

R3

Figure 4.17 Normalized core intensity along the propagation direction of the adaptivering aperture using design A1 element E5 with a simulated wavefront error of 0.6λ. Theshaded areas indicate the theoretical Bessel beam region for each electrode.

0 5 10 15 20 25 30 35 40 45 500.00.10.20.30.40.50.60.70.80.91.01.1 Bessel beam

without ring aperture

Nor

mai

zed

core

inte

nsity

/a.u

.


R10R9

R8

R7

R6R5

R4

R3R2

R1

0 5 10 15 20 25 30 35 40 45 500.00.10.20.30.40.50.60.70.80.91.01.11.2

Bessel beam without ring apertureSimulation

R10

Nor

mal

ized

core

inte

nsity

/a.u

.


R1R2

R3

R4

R5

R6

R8

R7R9

Figure 4.18 Normalized core intensity along the propagation direction of the measuredring aperture design A6. Left: Simulation of the Bessel beam with and without thering apertures and additional static aperture. Right: Measurement and the simulatedwavefront error 0.75λ. The shaded areas indicate the theoretical Bessel beam region foreach electrode.

60

4.7 Conclusion

4.7 ConclusionIn this chapter, the miniaturized adaptive ring aperture to generate depth-controlledBessel beams was introduced. The requirements of the ring aperture and an overviewof different concepts was given. The basic principle of liquid crystal technology wasexplained and the properties of the used twisted nematic liquid crystal cells weredescribed.Further, the fabrication process using mainly wafer-level standard clean room pro-cesses was explained, and the assembly process was presented. Seven different de-signs were produced, including three arrays for the miniaturization device and foursingle element designs to verify the findings from Section 3.4.Two different array designs were experimentally characterized: design A1 and A2.Both agreed well with the theoretical region up to some small variations comingfrom wavefront errors due to defects in the alignment layer. The wavefront errorswere estimated using BPM simulations with an astigmatic wavefront which is ratheran estimate of the magnitude. The values varied between λ/5 and 0.65λ, which hasan influence on the beam profile but does not account for single imperfections in thealignment layers that were observed in the cross-section images.The single element design A6 was evaluated to verify the analytical considerationsmade in Section 3.4. The fabricated optimal aperture radii showed no widening dueto diffraction for the electrodes close to the center. But with increasing electroderadius the wavefront error gets larger and an estimate of approximately 0.75λ wasdetermined.

61

5 Transmissive aspherical opticsThis chapter describes the developed transmissive aspherical optics, ranging fromthe optical requirements to an overview of different fabrication technologies and abasic description of the used method. Simulations and shape optimization are intro-duced, and the material properties of the used silicone are discussed. The processtechnology, design and fabrication are described. The final, ready-to-use lens arraysare characterized using optical surface profilometry and an optical measurementsetup.

5.1 RequirementsThe miniaturized transmissive aspherical optics should consist of a 3 × 3 array,where each element has a fixed diameter of 1 mm and a pitch of 1.2 mm. Thefabrication process should be based on a rapid-prototype process, first developedby Brunne [11] for reflective elements, i.e., mirrors. Further, the lens performanceshould have a root mean square (rms) wavefront error smaller than λ/16, with highrotational symmetry. Finally, besides the two required lens types collimation lensand axicon an additional axicon and lensacon were designed:

• An aspherical collimation lens, which is designed for blue edge emitting laserdiode arrays with 450 nm wavelength.

• Two static axicon arrays with 174 and 170 apex angles.

• A lensacon, which is a combination of a lens and an axicon to reduce thenumber of surfaces.

5.2 Overview of different fabrication technologiesThe fabrication technologies for optical systems are limitless. There are differentoptical implementations, such as refraction, reflection, diffraction, or combinationsof them using static or adaptive lenses, depending on the application [1].In this thesis, the focus was on transmissive static optics fabrication technologies.The main commerical fabrication methods for aspheres are [102,103]:

63

5 Transmissive aspherical optics

• Precision glass molding, where a glass is heated until it becomes soft and ispressed into an aspherical mold.

• Precision polishing, where glass is ground and polished to a certain accuracy.

• Diamond turning, which is the same process as precision polishing with evensmaller tool size and thus higher accuracy.

• Polymer molding, where a glass lens is pressed into a diamond-turned moldwith a polymer, which then gets cured by UV-light.

• Injection molding, where molten liquid polymers are injected into asphericmolds.

The individual fabrication technologies vary in their precision, cost and the choiceof material, which depends on the potential application.Aside from the classical manufacturing methods, other fabrication techniques to gen-erate microlenses and microlens arrays exist, which also depend on their materials,fabrication methods and applications. A short summary is given below [104,105]:Electrical methods for microlenses include liquid crystal lenses. By applying anelectrical field the refractive index changes, and the focal length can be tuned [106].Mechanical methods use hot embossing [107], ink jet printed microlenses [104, 108]or liquid microlenses with different actuation principles, e.g., piezoelectric [109] ac-tuation or electrowetting [110, 111]. Chemical methods use ion exchange [112], wetetching [113] or self-assembly of a microlens arrays. Optical methods use photother-mal techniques in photo-sensitive glass [114], thermal reflow of a photoresist [115],gray-tone lithography [116] or direct writing techniques. The direct writing tech-niques can be further split into laser direct writing, electron beam writing, focusedion beam writing and laser ablation [117]. Another method is the fabrication ofmicrolenses formed by volume change of materials, for example a tunable lens madeout of two substrate materials with a high difference in the coefficient of thermalexpansion, which has been shown recently by Hu et al. [118].

Axicons and lensaconsAs mentioned in Section 3.1, axicons are conically shaped lenses which can be usedto generate Bessel beams, first described by McLeod [45]. Some of their appli-cations include large telescopes [46], material machining [35, 119, 120], or medicalapplications, such as optical coherence tomography [38,121,122], biophotonics [123]or optical tweezers [124]. As the applications, axicon fabrication methods have beenextensively discussed in literature. Commercial axicons are produced using classi-cal manufacturing methods, such as precision polishing or diamond turning fromsolid materials, e.g., fused silica [103, 125, 126]. Other fabrication methods include

64

5.3 Basic fabrication method of the transmissive aspherical optics

electron beam lithography [124], laser direct writing [127, 128] or femtosecond laserablation with additional laser polishing [129]. Tunable axicons can be produced,e.g., by fluidic axicons [130], liquid crystal axicons [126, 131, 132] or liquid crystalspatial light modulators [44].

Besides the classical conical axicon, other axicon types exist, including logarithmicaxicons [126,133,134], axilenses [135] or lensacons [136].A lensacon is a combination of a lens and an axicon, which was first described byKorolkov et al. in 1984 [136]. They can be produced by diffractive optical elements,holographic optical elements, doublet lenses [137,138] or liquid crystal devices [126].

In this thesis, a lensacon was designed on only one optical surface, which is discussedin more detail in Section 5.8.

5.3 Basic fabrication method of the transmissiveaspherical optics

The fabrication method is based on the surface deformation due to thermal expan-sion of a soft polymer, polydimethylsiloxane (PDMS), in a cavity, first described byBrunne et al. [50, 139]. PDMS is filled into a conically shaped silicon cavity, closedusing a glass slide, and heated. During heating, the still liquid PDMS expands andflows out of the cavity at the edges. The temperature is kept constant until thepolymer is completely cured. Afterwards, the glass slide is removed, the setup iscooled down, the polymer shrinks, which will be further called pseudo-shrinkage,and the surface deforms. This surface deformation is generally dependent on thedepth and shape of the initial cavity, the polymerization shrinkage, the coefficient ofthermal expansion and the Poisson’s ratio. To guarantee that the expanded liquidPDMS can flow out properly, an additional spacer is necessary.

The initial method by Brunne et al. was developed for thermally actuated reflectiveaxicon arrays with varying diameter of 0.6−1 mm and a maximum axicon apexangle of up to γ = 178.2 [11].In this thesis, four different static transmissive lens types were designed and pro-duced. Two of the designed lens types were axicons with a maximum desired apexangle of up to γ = 170. Compared to the rather shallow axicons produced byBrunne et al., molds deeper by a factor of 5.6 were used in this thesis, resulting inthe need of a completely new finite element simulation to predict the initial cavity.New shape optimization methods, non-linearities, and better predictions of the ma-terial parameters were applied. Additionally, a transfer from the reflective designto a transmissive design needed to be developed, which is discussed in detail inSection 5.7.

65


5.4 Analytical considerationsThe deformation in an isotropic material can be expressed using Hooke’s law [140]:

εi = D−1i,j σj, (5.1)

where ε is the strain, and σ is the stress. The linear elasticity tensor with Young’smodulus E and Poisson’s ratio ν is:

Di,j = E

(1 + ν)(1− 2 ν)

×

1− ν ν ν 0 0 0ν 1− ν ν 0 0 0ν ν 1− ν 0 0 00 0 0 1−2ν

2 0 00 0 0 0 1−2ν

2 00 0 0 0 0 1−2ν

2

. (5.2)

The deformation of PDMS due to thermal expansion has no skew deformations, sothat the shear components in Hooke’s law are negligible, and the strain is reducedto:

~ε = 1E

1 −ν −ν−ν 1 −ν−ν −ν 1

~σ +

111

(αCTE + εp

∆T

)∆T, (5.3)

where αCTE is the linear coefficient of thermal expansion, ∆T the temperature dif-ference, and ε the contribution from polymerization shrinkage εp and temperaturevariations.For infinitesimally shallow cavities, the strain can be obtained analytically usingHooke’s law from Equation 5.3 and subsequently the surface deformation ∆h(r)caused by the PDMS pseudo-shrinkage.The boundary conditions with fixed bottom and side walls (εx=y = 0) and a freetop surface (σz = 0) yield an in-plane stress σx, which can be substituted into thestrain:

εz =( 2 ν

1− ν + 1)(

αCTE + εp

∆T

)∆T. (5.4)

The surface deformation ∆h(r) generated by the pseudo-shrinkage of the PDMS ina cavity with depth H(r) is then:

∆h(r) = εzH(r). (5.5)

66

5.5 Simulation and shape optimization

m

Silicon

PDMS

Figure 5.1 Left: Simulated geometry. A silicon substrate with a funnel-shaped cavityfilled with PDMS. Center: User-controlled mesh. Right: Displacement field of the surfacedeformation resulting from the pseudo-shrinkage for an axicon with 170 apex angle curedat 120 C.

5.5 Simulation and shape optimizationA finite element method (FEM) was used to simulate the initial cavity that is neces-sary to form the desired surface deformation due to the thermal expansion of PDMSand the resulting pseudo-shrinkage. For this purpose, a 2-D axisymmetric structuralmechanics module with solid mechanics, a linear elastic material model and thermalexpansion were implemented in COMSOL Multiphysics.The geometry is illustrated in Figure 5.1 (left). It consists of a funnel-shaped siliconmold with PDMS inside. The mold structure is defined as a parametric curve with aBernstein polynomial B(r) of 5th order, as the Weierstrass approximation theoremstates that all continuous functions in an defined interval [r1, r2] can be approximatedas closely as desired by a polynomial function [141].

The radius of the funnel-shaped mold was fixed at 500 µm by the requirements inSection 5.1. The spacer was set to be 20 µm, and the maximum mold depth waslimited by the thickness of the silicon substrate of 2000 µm. A user-controlled meshwas applied, which was defined extra fine at the necessary surfaces and otherwisekept coarse to decrease calculation time, as illustrated in Figure 5.1 (center). Theboundary conditions were set as described in the analytical considerations: a fixedconstraint at the bottom and sidewalls of the substrate and a free boundary on thetop surface. Geometric non-linearities were taken into account to achieve a preciseresult, which is discussed in more detail in Section 5.5.1. In Figure 5.1 (right) thesurface deformation due to the PDMS pseudo-shrinkage is illustrated for an axiconwith 170 apex angle and a curing temperature of 120 C.

For a better parameter estimation of the predicted mold shape, the optimizationmodule was additionally implemented. The basic optimization objective function

67


using a BOBYQA (Bound Optimization BY Quadratic Approximation) solving al-gorithm was adapted from Binal Bruno and further optimized using two inequalityconstraints. The optimization used the least-square method and searched a mini-mum of the objective function. In this case the function was described by:

fobj(r) = 2πr∫ r2

r1|(h(r)− ftarget(r))|2 dr, (5.6)

where h(r) is the ideal lens shape, either a collimation lens, an axicon, or a lensacon(from Section 5.8) and ftarget(r) the deflection of the expanding polymer. The 2πrin front of the integral was used as a weighting for the area taken into account. Thefirst pointwise inequality constraint was added to make sure that the slope at thedeepest point of the mold was not steeper than 20 to be able to measure the laterfabricated mold with the confocal distance sensor described in Section 2.1. Thus,the lower bound corresponding to the deepest point of the axicon should fulfill:

δrB(r) ≥ 0.35 rad. (5.7)The second pointwise inequality constraint was used to ensure that the slope of theresulting Bessel beam z(r) with corresponding radius r rises monotonously. Thisis necessary to avoid that light coming from a higher radial position can interferewith previous parts of the Bessel beam. The slope can be determined by usingEquation 3.5 and the slope of the surface deviation (δrftarget(r)) as conical angle:

δrz(r) = δr

(r

tan (α)

)

≈ δr

(r

δrftarget(r)

)

≈ 1δrftarget(r)2 − δ

2rftarget(r) > 0, (5.8)

where δr is the short version of δδr.

The last constraint was the deviation of the FEM simulated target function ftarget(r)and the ideal lens profile h(r), which should result in a corresponding wavefront errorof λ/16:

(h(r)− ftarget(r)) 6λ

16∆n (5.9)

where ∆n is the difference in refractive index of the lens material and the surroundingmedium. As the pseudo-shrinkage will cause a rounding at the center of the moldmaking the constraint insoluble, a smaller interval [r1, r2] was defined in which thisconstraint should be valid. The upper interval boundary r2 = 450 µm was fixed

68

5.5 Simulation and shape optimization

0 50 100 150 200 250 300 350 400 450 500-45-40-35-30-25-20-15-10-50

r2

Surfa

cede

form

atio

n/µ

m

Radial position / µm

Ideal 170° axiconTarget function ftarget

r1

0 50 100 150 200 2500.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Dev

iatio

nfro

mid

eala

xico

n/µ

m

Rounding radius / µm

174° axicon170° axicon160° axicon

Figure 5.2 Left: Simulated surface deformation due to the pseudo-shrinkage of an axiconwith 170 apex angle at 120 C. The desired target function ftarget is defined in the interval[r1 = 175 µm, r2 = 450 µm]. Right: Different rounding radii (r1) with the correspondingaveraged surface deviation in the defined interval of three different axicon apex angleswith fixed upper boundary r2 = 450 µm.

at 90% of the maximum radius to avoid edge effects. The lower interval boundaryr1 defines the rounding radius of the axicon tip. An example of a defined interval[r1 = 175 µm, r2 = 450 µm] is illustrated in Figure 5.2 (left) with an ideal 170apex angle axicon (dotted curve) and the optimized surface deformation ftarget(r)with rounding (blue curve). The constrain is determined by taking the averageddeviation. An example of this deviation with various rounding radii for differentaxicon angles is illustrated in Figure 5.2 (right). A higher axicon angle (160) wasadditionally simulated to illustrate the precision limitations of this process.The ideal surface profile h(r) for each lens type is further discussed in Section 5.8.

5.5.1 Evaluation of the simulation modelThe analytical considerations from Section 5.4 were compared to a simplified sim-ulation model with constant slope, as illustrated in the inset of Figure 5.3. Thecavity depth was calculated using Equation 5.5. Two different simulations were per-formed: a geometric linear simulation and a simulation which included geometricnon-linearities (non-linear simulation). In structural mechanics, geometrically linearsimulations use the equations of equilibrium in the undeformed state and are notupdated with deformation. Usually, the deformations are so small that the deviationfrom the original geometry is not distinguishable. If this is not the case, geometricnon-linearities have to be included.

The simulated surface deformation due to the pseudo-shrinkage of an axicon with178 apex angle at 120 C with linear and non-linear simulations is illustrated inFigure 5.3. The linear and non-linear simulations were studied by comparing the

69


0 50 100 150 200 250 300 350 400 450 500-9-8-7-6-5-4-3-2-10

Analytical valueLinear simulationNon-linear simulation

Surfa

cede

form

atio

n/µ

m

Radial position / µm

Silicon

PDMS

Figure 5.3 Simulated surface deformation due to the pseudo-shrinkage of an axiconwith 178 apex angle at 120 C with linear and non-linear simulations. Inset: Simplifiedsimulation model.

170 171 172 173 174 175 176 177 178 179 1800369

12151821242730

Linear simulationNon-linear simulation

Solp

ede

viat

ion

/%

Axicon apex angle / °170 171 172 173 174 175 176 177 178 179 180

0.00.51.01.52.02.53.03.54.04.55.05.5

Slop

ede

viat

ion

/%

Axicon apex angle / °

Figure 5.4 Left: Linear and non-linear deviations in the slope of the surface deformationat the center (r = 250 µm) to the analytical values in dependence of the axicon apexangle. Right: Slope deviation between linear and non-linear simulations.

slope of the surface deviation at the center (r = 250 µm) with the analytical value(left) and to each other (right), as illustrated in Figure 5.3. Starting from very smallaxicon apex angles, the non-linear simulation shows less deviation in the slope thanthe linear simulation. In this thesis, axicons with 174 and 170 apex angle weredesigned and fabricated. Hence, geometric non-linearities were included for bothcases as they provide a more precise result.

5.6 Material parameters of PDMSPDMS is a soft polymer which is widely used as a MEMS material for its thermal,mechanical, and optical behavior, e.g., for adaptive lenses, lab-on-a-chip devices,and many more [50,109,142–146].

70

5.7 Fabrication process

The Poisson’s ratio of PDMS is a crucial variable for the prediction of the moldgeometry genetrated by the pseudo-shrinkage due to thermal expansion. In the lit-erature several different values for the Poisson’s ratio were determined or estimated,all ranging between 0.45 and 0.5 [50, 109, 146–154]. While this appears to be onlya variation of 10 %, for the application of this thesis, the difference in the surfacedeformation will change by 30 %.Consequently, within this thesis a new method was developed to determine an accu-rate value for the Poisson’s ratio and the coefficient of thermal expansion of Sylgard184 and Sylgard 182 from Dow Chemical. It has been published in Soft Matter [155].A detailed description of the developed method can be found in the Appendix inSection A.3.It was found that the Poisson’s ratio for Sylgard 184 is ν = 0.4950± 0.0010 and forSylgard 182 ν = 0.4974 ± 0.0006. Compared to the literature, the values are onemagnitude more precise [50, 109, 146, 147, 149, 152, 154]. The coefficient of thermalexpansion was observed to have a linear decrease with increasing temperature. ForSylgard 184 an extrapolated value of (248.06 ± 6.91) ppm/C at 120 C was foundand for Sylgard 182 (240.69± 4.44) ppm/C.

The value for the Young’s modulus of PDMS is temperature dependent but has arather small impact on the results [143,154]. This was verified by a FEM simulations,where a change in the Young’s modulus by 30% caused a surface deformation dueto the pseudo-shrinkage by only 18 h.

5.7 Fabrication processThe individual fabrication steps are summarized in Figure 5.5. The fabrication wasdone as follows:

1. Substrate preparation: According to the FEM simulation results, a funnel-shaped mold (initial cavity) was laser-structured into a 2000 µm thick siliconsubstrate and was cleaned afterwards.

2. Mold preparation: Uncured PDMS was filled into the silicon mold, degassed,sealed with a silane-coated glass slide and cured within a mechanical pressat elevated temperature. The cured mold was removed from the press, and,while the PDMS cooled down, a surface deformation formed due to the pseudo-shrinkage in the material.

3. Lens preparation: The formed surface deformations were used as lens molds.UV-curable adhesive was placed on top of the molds, and a glass slide waspressed against it, before curing it with UV-light. After the adhesive wascompletely cured, the lens array was removed from the mold, which was pos-sible because the adhesive has a better adhesion to glass than to PDMS.

71


Silicon

Laser structuring

Glass slide

PDMS

NOA 61

Glass slide

Lens array

Heat and pressure

Lens master mold

UV exposure

1.

2.

3.

4.

5.

6.

7.

8.

Figure 5.5 Fabrication process of the transmissive aspherical optics. A funnel-shapedcavity was laser-structured into a thick silicon substrate (1 and 2). The uncured liquidPDMS was filled into the cavity (3), sealed with a glass slide (4) and heated under pressure.The still liquid polymer expanded due to the elevated temperature and was pressed outof the mold. After polymerization, the device was cooled down, and the cured polymershrank. Small surface deformations formed which were used as lens molds (5). A UV-curable adhesive was dispensed on top of the lens mold (6), and a glass slide was pressedon top (7). The device was then cured with UV-exposure, and the glass slide with thecured UV-adhesive was removed (8).

72


2 mm

1 mm

Figure 5.6 Cleaned silicon substrate (174 axicon) with laser-structured initial cavities.Inset: Close-up view of the laser-structured array.

Substrate preparationA 2 mm thick, single side polished <100> silicon wafer was diced into 10× 10 mm2

pieces as substrate material. The cavities were structured into the substrate usinga marking laser (Trumpf TruMark 6330: 355 nm pulsed nanosecond laser with 2 Woptical output power and a focused laser beam of ∼ 15 µm). The lens arrays areillustrated in Figure 5.6. As mentioned before, an additional spacer with a thicknessof 20 µm was included to ensure the flow of PDMS out of the cavity. The residualsleft by the laser process were cleaned using 30 % KOH at 50 C for 5 min. Thisshort cleaning process has only a small influence with a etch depth of approximately1 µm. The verification of the shape is discussed in Section 5.8.

Lens mold preparationSylgard 182 was chosen for the fabrication of the different lens types. It has a longerpot life of 8 h compared to the commonly used Sylgard 184. That was necessaryto degas the deep silicon cavities properly. It was prepared as recommended in thedata sheet [156]. The liquid two-part silicone elastomer kit consists of a pre-polymerbase (part A) and a cross-linking curing agent (part B), which need to be mixedthoroughly in a 10:1 ratio. After mixing, the PDMS was degassed in a vacuumdesiccator at a pressure below 30 mbar for at least 15 min until the mixture showedno more air bubbles. After pouring the liquid PDMS into the cavities and degassingfor one hour, a glass slide was arranged on top, and everything was placed directlyinto a mechanical press which was preheated to the curing temperature. The glassslide was prior treated with a silane coating to prevent sticking of PDMS. A forceof at least 1 kN and a curing temperature of 120 C were applied for 40 min toprevent motion or deformation of the glass slide and to complete the curing process.While the still liquid PDMS expanded, the excess material was pressed out of thesubstrate via the small spacers.After curing, the pressure was released, and the glass slide was removed, such that

73


3 mm

1 mm

Figure 5.7 Different lens arrays from left to right: Collimation lens array, 174 axiconarray, 170 axicon array and lensacon array. Inset: Close-up view of the collimation lensarray.

the substrate was able to cool down to room temperature. The material shrank,and a small surface deformation was formed providing the required lens mold.

Lens array preparation

The produced lens molds served as the master mold for the lens arrays, whichconsisted of a plasma-cleaned glass substrate with 12 × 12 mm2 dimensions and atransparent UV-curable adhesive, NOA61 (Norland Adhesive), as lens material [157].Small droplets of the adhesive were placed on top of the molds avoiding air bubbles.A milled alignment structure was used to center the glass slide to the PDMS mold,which was then pressed on top. To avoid extensive heating, which would lead toan expansion of the PDMS, a LED with 385 nm wavelength was used to cure theadhesive. A photograph of the finished lens arrays is shown in Figure 5.7.

5.7.1 Fabrication limits

Surface roughness

The surface roughness of the silicon cavity has an impact on the pseudo-shrinkageof the PDMS. Therefore, an estimate of the influence from the surface roughnesswas done, as illustrated in Figure 5.8 (left). Different quadratic silicon particles fordifferent cavity depths were simulated, and the deviation from the surface deforma-tion without a particle (mold deviation) was found, as shown in Figure 5.8 (right).For small particle sizes, the deviation is below the required deviation corresponds toa wavefront error (λ/16) and thus negligible. The step profile for the laser processwas set to have 20 µm steps, as it only has a small mold deviation of 0.9%.

74


0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

Max. allowedmold deviation

Mol

dde

viat

ion

/nm

Particle size / µm

0.1 mm0.2 mm0.3 mm0.4 mm0.5 mm0.6 mm0.7 mm0.8 mm1 mmSilicon

PDMS

Silicon particle

Figure 5.8 Left: FEM simulation model with a silicon particle. Right: Difference in thesurface deformation due to pseudo-shrinkage at different cavity depths. Everything belowthe blue curve is smaller than the required wavefront error and has a negligible influenceon the surface deformation.

Aspect ratio

From the funnel-shaped structure in the shape optimization (Section 5.5), the ques-tion arose how deep the cavity has to be until the maximum surface deformation isreached. For this purpose, a second simplified FEM simulation was done, similar tothe FEM simulation in Figure 5.8 (left), where the cavity depth H and radius r werevaried. The normalized surface deformation reaches a saturation at an aspect ratio(H/r) of approximately 4, exemplarily shown for five different radii in Figure 5.9(left) and in more detail as a contour plot in Figure 5.9 (right). For this process itcan be concluded that for the collimation lens and the 170 axicon, the saturationis reached at a radius smaller than 100 µm, and for the lensacon at 200 µm. As aconsequence, the initial cavity can be fabricated with smaller depth. In this thesis,this result was not further investigated as the deep cavities could not be verified bythe confocal setup, but can be considered for future process improvements.

Heat transfer

To guarantee that the drainage of the still liquid PDMS out of the cavity is ensured,the heat transfer within the system was investigated. For this purpose a FEMsimulation was performed, as illustrated in Figure 5.10 (left). A temperature of120 C was applied at the outer surface of the top and bottom glass slides. 90 %of the temperature is reached in the center of gravity of the PDMS mold after 3.1 s(Figure 5.10 (right)). This value is much smaller than the curing time of 35 min,thus ensuring proper drainage of the still liquid material.

75


0 1 2 3 4 5 6 7 8 9 10

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00 r = 100 µmr = 200 µmr = 300 µmr = 400 µmr = 500 µm

∆h

/rad

ius

Aspect ratio100 200 300 400 500

0123456789

10

Aspe

ctra

tioRadius / µm

-0.325

-0.275

-0.225

-0.175

-0.125

-0.075

-0.025

∆h / r

Figure 5.9 Left: Normalized surface deformation ∆h/r reaches a saturation with increas-ing aspect ratio. Right: Simulated contour plot of the normalized surface deformation∆h/r depending on the aspect ratio.

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 302030405060708090

100110120

Tem

pera

ture

/°C

Time / s

°C

mm0 2 4

0

1

2

3

4mm

Glass slide

Glass slide

Silicon

PDMS

Figure 5.10 Left: Simulated heat transfer in the setup with silicon substrate, PDMSand a glass slide on top and bottom at 90 % of the final temperature. Right: Simulatedtemperature profile over time. 90 % of the final temperature is reach after t = 3.1 s.

76

5.8 Lens designs and fabrication

5.8 Lens designs and fabricationAn aspheric surface can be described by a sagitta profile h(r) as

h(r) = cr2

(1 +√

1− (1 + kc)r2c2+

N∑i=1

ιiri, (5.10)

where r is the radius, c the curvature of the surface, kc the conic constant, and ιithe coefficients that describe the deviation from a spherical surface.All lens types were designed plano-convex with a glass substrate and UV-curableadhesive (NOA61) as lens material. The design wavelength and refractive index ofeach lens were set to be 450 nm and nNOA61 = 1.573, respectively, and the maximumradius was rmax = 500 µm.

Collimation lensThe collimation lens was designed with an effective focal length of fdesign = 5 mmand radius rmax corresponding to an opening angle of θdesigned = 99.66 mrad, whichis smaller than the slow-axis from Section 3.5. The optical simulation tool ZEMAXwas used to design the collimation lens. An even asphere with conic constant kc = 0was simulated, and the optimized surface profile was found to be:

hcollimation(r) = ι2r2 + ι4r

4 + ι6r6, (5.11)

where ι2 = 1.74×10−4 µm−1, ι4 = −7.80×10−12 µm−3, and ι6 = 6.16×10−19 µm−5.

AxiconsTwo different axicon lenses were designed: a 170 and a 174 apex angle axicon.The surface profile in both cases was

haxicon,γi(r) = ((180 − γi)/2)r, (5.12)

where γ1 = 170 and γ2 = 174.

The rounding of the axicon tip has a strong influence on the generated Bessel beam,as mentioned in Section 3.3. Therefore, the rounding radius r1 defined in Section 5.5should be minimal and still fulfill the constraint of Equation 5.9. For both axiconlenses the deviation from the ideal shape depending on the rounding radius r1 isillustrated in Figure 5.11. The lower interval boundary was determined to be 130 µm(0.26 rmax) for the 174 axicon, and 175 µm (0.35 rmax) for the 170 apex angleaxicon. The upper interval boundary was fixed at 450 µm (0.9 rmax) for bothaxicons to avoid edge effects.

77


0 50 100 150 200 2500.000.050.100.150.200.250.300.350.400.450.50

Max. alloweddeviation

λ

Dev

iatio

nfro

mid

eala

xico

n/µ

m

Rounding radius / µm

174° axicon170° axicon

Figure 5.11 Different FEM simulated rounding radii r1, with corresponding deviationfrom the ideal shape of the two designed axicons. The upper interval value r2 was fixed at450 µm. The shaded area fulfills the maximum allowed deviation of the surface deviationthat corresponds to the λ/16 constraint.

Lensacon

The design of a lensacon was realized by combining the 5 mm collimation lens andthe 170 axicon into a single plano-convex lens to reduce the number of opticalsurfaces and thereby reduce the components necessary for the stimulation devicementioned in Section 1.1.

Figure 5.12 illustrates the lensacon design in ZEMAX. The desired lensacon surface(S2) was designed with the help of an additional conical surface (S3) and a circularobstruction (S1) to neglect the rounding radius (r1 = 175 µm) in the center. Theconical surface represents an ideal 170 axicon with profile haxicon,170(r) = 0.087 r.The axicon surface (S3) subtracts the conical component of the lensacon surface(S2), which can then be optimized for a 5 mm collimation lens.The lensacon surface was determined with an odd sphere and conic constant kc = 0to be:

hlensacon(r) = ι2r2 + ι3r

3 + ι4r4 + ι5r

5 + ι6r6 + ι7r

7, (5.13)

where ι2 = 1.74 × 10−3 µm−1, ι3 = −1.29 × 10−7 µm−2, ι4 = 5.60 × 10−8 µm−3,ι5 = −1.33× 10−10 µm−4, ι6 = 1.62× 10−13 µm−5, and ι7 = −7.97× 10−17 µm−6.

The spot diagram with all three surfaces (S1-S3) shows that the lensacon is colli-mated and within the diffraction limit. In the cosine space (angular spectrum) therms radius is 6.77 × 10−4 rad (airy radius is 1.37 × 10−3 rad). The spot diagramwithout the additional axicon (S3) shows the characteristic ring pattern in the cosinespace, and the concial angle is α = 0.05 rad.

78

5.8 Lens designs and fabrication

S1 S3S2

Figure 5.12 Lensacon design configuration using ZEMAX with circular obstruction (S1),lensacon surface (S2), and the ideal 170 axicon surface (S3).

Lens fabrication

The designed surface profiles (h(r)) of the individual lens types were implementedin the FEM simulation from Section 5.5 and optimized using the material propertiessummarized in Table 5.1. The resulting initial cavities were described using Bern-stein polynomials. A definition of the polynomials and corresponding coefficientsfor each lens are summarized in the Appendix in Table A.2.

The obtained initial cavities were transformed into step profiles of 20 µm deep stepsusing Matlab, as illustrated for all four lens types in Figure 5.13, and manuallytransferred to the laser software.

The depth of the individual lasered steps needed to be controlled, as the opticaloutput power of the laser varies. Therefore, a test sample was laser-structured withdifferent parameters, including the velocity, the repetition times, and the hatch dis-tance. The test sample was then measured using the confocal profilometer, analyzed,and the parameters which should result in a depth of 20 µm were chosen. Addi-tionally, the surface roughness generated by the laser process was measured to berrms = 2.54 µm, which is below the critical surface roughness found in Section 5.7.1.Further, the substrates were aligned such that the cavities were laser-structured inthe center and produced as described in Section 5.7.

The cavity depth of the silicon substrate and the mold depth (lens height) for allfour lens types are summarized in Table 5.2.

Table 5.1 Material properties of PDMS and silicon used in the FEM simulation.

Material PDMS Silicon

Density 1040 kg/m3 2329 kg/m3

Poisson’s ratio 0.4954 0.28Coeff. of thermal expansion 229 ppm/C 2.6 ppm/CYoung’s modulus 1.76 MPa 170 GPa

79


0 100 200 300 400 500 600 700 800 900-1800-1600-1400-1200-1000-800-600-400-200

0

174° axicon170° axiconCollimation lensLensacon

Cav

ityde

pth

/µm

Radius / µm

Figure 5.13 Different step profiles from the initial cavity predicted from the FEM simu-lations of four different lens types.

Table 5.2 Cavity depth and mold depth of the four lens types.

Lens type Cavity depth /µm Mold depth /µm

Collimation lens 1120 45.24174 apex angle axicon 480 24.47170 apex angle axicon 1120 34.22Lensacon 1800 73.91

Verification of the shape optimization

The intended simulated shape of the silicon cavity was verified in measurementsfor a 174 axicon. The laser structured silicon cavity was cleaned with KOH, asdescribed in Section 5.7, and measured using the confocal profilometer mentioned inSection 2.1. Figure 5.14 (left) compares the measured silicon mold to the simulatedshape. The deviation between both is illustrated in Figure 5.14 (right). The rmsdeviation was 7.73 %. The other lens types were not verified as it was not possibleto measure their deep profile with the confocal profilometer.

5.9 Experimental characterization

The fabricated lens arrays were characterized using optical surface measurementsand an optical measurement setup. Both measurement methods are described inmore detail in Chapter 2. For each lens type, three arrays were measured at threedifferent positions.

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0 50 100 150 200 250 300 350 400 450 500-500

-400

-300

-200

-100

0

Mol

dde

pth

/µm

Radius / µm

SimulationMeasurement

0 50 100 150 200 250 300 350 400 450 500-15

-10

-5

0

5

10

15

20

25

Dev

iatio

n/µ

m

Radius / µm

Figure 5.14 Left: Comparison of the simulated and measured silicon substrate for anaxicon with 174 apex angle. Right: Deviation between measurement and simulation.

5.9.1 MethodsOptical surface measurements: The surface topology of the fabricated lens ar-rays was measured using a chromatic confocal distance sensor with confocal penCL2, described in Section 2.1.

Optical measurement setup: The produced lens types were further evaluatedusing the setup described in Section 2.2.2. The DPSS laser with 473 nm wavelengthwas collimated using a lens with 100 mm focal length to obtain an approximatelyplane wave and was then blocked by a static aperture with 1 mm diameter to ensuremeasuring only one element. The camera with 1.67 µm pixel pitch and a microscopeobjective with magnification factor 4 were used. For the lensacon array, the lightsource was placed at a distance of approximately 5 mm, depending on the actualfabricated element.

BPM simulations: The ideal surface profiles h(r), partially designed with theoptical ray-tracing tool ZEMAX from Section 5.8, were analyzed and comparedto the measurements using BPM simulations. Therefore, a plane wave with thecorresponding ideal surface profile h(r) was simulated, similar to the one describedin Section 2.2.2.

Data evaluation: The reconstruction of the image plane along the optical axis,the intensity profile evaluation, and the polarbasket evaluation from Section 2.2.3were used to determine the characteristic beam properties.

5.9.2 ResultsThe evaluation of the results for each lens type follow the same pattern. First,the surface profile was analyzed, as described in more detail in Section 2.1.1. Thatincludes the evaluation of surface deviations from a rotationally symmetric wavefront

81


Piston X Tilt Defocus Astigmatism

X Coma Spherical abberation Trefoil I

Z 00 Z 1

-1 Z 11 Z 2

0 Z 2-2Z 2

2

Z 3 1 Z 3

-1 Z 40 Z 3

-3Z 33

Y Coma Trefoil II

Y Tilt 0° 45°

Figure 5.15 First eleven Zernike polynomials with “fringe” convention, generated withdedicated Matlab code [10].

(rms value srms), the deviation from the intended shape (deviation of the targetfunction ∆ftarget), and aberrations of the surface deviation (srms) using the firsteleven Zernike polynomials (Figure 5.15). All references to the wavefront error in λare referred to the wavelength used in the optical measurement setup λ = 473 nm.A wavefront error of λ/16 corresponds to a surface deviation of 51.6 nm.Afterwards, the lens arrays were measured with the optical measurement setup, andthe relevant properties were discussed. To show the repeatability of this process, twoadditional arrays for each lens type were fabricated and the surfaces were measuredand analyzed in the same manner. The results are summarized in Table A.3-A.6 inthe Appendix (Section A.2).

Collimation lensSurface deviation: The surface profile of a molded collimation array is depictedin Figure 5.16 (left), with indications of the individual lens positions within thearray (right). The surface profile was evaluated, and the back focal length wasdetermined from the radius of curvature as described in Section 2.1.1. The resultsfor three different positions are summarized in Table 5.3 with mean values andstandard deviations. The peak-to-valley value was determined in addition to therms surface deviation to show the maximal variations in the single lens positions.The mean value of the rms surface deviation for array Cl-A1 for three positions wassrms = (0.153 ± 0.008) µm. The results for the two additional arrays Cl-A2 andCl-A3 are summarized in Table A.3 in the Appendix. The mean surface deviationwere determined to be srms = (0.154 ± 0.020) µm and srms = (0.347 ± 0.047) µm,respectively. Array CLA3 has a rather poor optical quality compared to the othertwo arrays and is further excluded from the comparison.

82


1 mm

P2P1 P3

P6P5P4

P7 P8 P9

xy

Figure 5.16 Left: Measured three dimensional surface data of a fabricated collimationlens array (Cl-A1). Right: Array with indications of lens positions.

Table 5.3 Surface evaluation of the collimation array Cl-A1 at three different positionswith the overall rms value srms, peak-to-valley (PV) value, the rms deviation of the targetfunction (∆ ftarget), and the focal length with mean value (Mean) and standard deviation(SD).

Lens position srms /µm PV /µm ∆ ftarget / % Focal length / mm

P1 0.154 1.105 6.04 5.33P5 0.160 1.116 4.17 5.23P9 0.144 1.138 5.16 5.36

Mean 0.153 1.120 5.13 5.31SD ±0.008 ±0.017 ±0.93 ±0.07

Deviation from the target function: The deviation of the measured surfacedata from the intended shape is illustrated along the lens radius in Figure 5.17 (left)for array Cl-A1. Deviations from the target function are visible in all three lenspositions. Lens position P1 and P5 follow the same profile, which is probably a resultof the lasered cavity. Lens P9 has a different deviation in the center, which mightbe caused by a bigger particle in the cavity. However, starting from r ≈ 200 µm,the same profile as the other two lenses can be observed with a mean deviation of(5.13 ± 0.93) %.

Zernike polynomials: An overview of the most dominant aberrations extractedfrom the surface deviations of array Cl-A1 for all three positions can be found inFigure 5.17 (right), including defocus, astigmatism, coma, spherical aberration andtrefoil. As the surface deviations were used to determine the Zernike polynomials,aberrations less than the corresponding required wavefront error λ/16 should beexpected (at best zero). Thus, the shaded area indicates negligible surface errorscorresponding to the required λ/16 wavefront error. The surface deviations from

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0 50 100 150 200 250 300 350 400 4502.02.53.03.54.04.55.05.56.06.57.07.5 P1

P5P9

Dev

iatio

nfro

mf ta

rget

/%

Radius / µm Defocu

s

0°Asti

gm

45° Asti

gm

XCom

a

YCom

a

Spheri

cal

Trefoil

I

Trefoil

II-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Zern

ike

coef

f./µ

m

P1P5P9

Figure 5.17 Left: Deviation of the measured surface data from the simulated target func-tion for array Cl-A1 at three different positions. Right: Zernike coefficients extracted fromthe surface data of the same array representing the individual contributing aberrations.The aberrations within the shaded area are negligible.

the individual positions show that the lenses have astigmatism, coma, and trefoil inthe range of approximately λ/4. Possible error sources for those aberrations couldbe in the laser-structured silicon cavity, the pseudo-shrinkage, and smaller particles,which will cause a non-symmetric lens profile and therefore aberrations.

Optical characterization method: The collimation lens array Cl-A1 was furthercharacterized using the optical measurement setup. The normalized intensity alongthe propagation direction in measurements and simulation is shown in Figure 5.18.The determined focal lengths in the BPM simulation and measurements were theback focal lengths. Thus, a small shift between the designed focal length fdesign =5 mm (effective focal length) and the simulation (back focal length) 5.1 mm can beobserved. The focal length determined from the surface data of 5.3 mm was alsofound in the optical measurements, which results in a shift from the focal length byapproximately 6 %.

Short conclusion: The collimation lens arrays showed a good optical quality,in two of the three fabricated arrays. Averaging array Cl-A1 and Cl-A2 over allmeasured lens positions a rms surface deviation of srms = (0.154 ± 0.013) µmwas achieved, which corresponds to λ/3 surface error or 0.19λ wavefront error.The peak-to-valley surface error was averaged to be (1.100 ± 0.087) µm. Thesevalues are comparable to state-of-the-art micro-lenses, which vary in their rms waveaberration approximately between λ/12 and λ/2 [104, 158]. The focal length variesby (6.2 ± 1.6) %, which is in the same range as Thorlabs’ manucfacturing tolerancefor the focal length of aspheric condenser lenses of ±8% [159]. The deviation fromthe intended shape varied between −8.3 % and 5.9 %.

84


4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 6.4 6.60.00.10.20.30.40.50.60.70.80.91.0 P1

P5P9Simulation

Nor

mal

ized

inte

nsity

/a.u

.


fdesign

Figure 5.18 Normalized intensity profile along the propagation direction for array Cl-A1at three different positions with BPM simulation.

Axicon with 174 apex angleSurface deviation: The surface profile of the 174 axicon array is illustrated inFigure 5.19. The evaluation of the surface profile was performed in the same wayas the collimation array. The characteristic properties are summarized in Table 5.4.The mean rms surface deviation for the three measured positions of array Ax3-A1 was srms = (0.113 ± 0.013) µm. For array Ax3-A2 and Ax3-A3 they weresrms = (0.107 ± 0.006) µm and srms = (0.149 ± 0.029) µm, respectively (Table A.4in the Appendix).

Deviation from the target function: The deviation of the measured surface datafrom the target function is shown for Ax3-A1 in Figure 5.20 (left). The deviations inlens position P1 and P9 show the same behavior, whereas position P5 has a strongdeviation at the tip, which might be explained by trapped particles in the center ofthe silicon cavity.

Zernike polynomials: The Zernike coefficients extracted from the surface datafor all three positions are illustrated in Figure 5.20 (right). Here, the astigmatism isthe dominating aberration for lens position P1 and P5. The other aberrations arenegligible.

The surface data was further evaluated by determining the slope of the axicons in theinterval [r1 = 130 µm, r2 = 450 µm] to compare the lenses to the ideal lens shape.From the slope of the axicons the corresponding apex angle, Bessel beam range(zmax) and core radius (rcore) were determined and are summarized in Table 5.5.The apex angle of the axicon, as well as the characteristic properties of the Besselbeam, agree well with values from an ideal 174 axicon.

85


Figure 5.19 Measured three dimensional surface data of a 174 axicon array (Ax3-A1).

Table 5.4 Surface evaluation of the 174 axicon array Ax3-A1 at three different positionswith the rms value srms, peak-to-valley (PV) value, and the rms deviation of the targetfunction (∆ ftarget) with mean value (Mean) and standard deviation (SD).

Lens position srms /µm PV /µm ∆ ftarget / %

P1 0.099 0.591 4.16P5 0.118 0.790 6.38P9 0.123 0.841 2.11

Mean 0.113 0.741 4.22SD ±0.013 ±0.132 ±2.06

0 50 100 150 200 250 300 350 400 450-11-10-9-8-7-6-5-4-3-2-10

P1P5P9D

evia

tion

from

f targ

et/%

Radius / µm Defocu

s

0°Asti

gm

45° Asti

gm

XCom

a

YCom

a

Spheri

cal

Trefoil

I

Trefoil

II-0.12-0.10-0.08-0.06-0.04-0.020.000.020.040.06

Zern

ike

coef

f./µ

m

P1P5P9

Figure 5.20 Left: Deviation of the measured surface data from the simulated targetfunction for array Ax3-A1 at three different measured positions. Right: Zernike coefficientsextracted from the surface data representing the individual contributing aberrations. Theaberrations within the shaded area are negligible.

86


Table 5.5 Characteristic properties of the 174 axicon array Ax3-A1 at three differentpositions with mean value (Mean) and standard deviation (SD).

Lens position Apex angle / rcore /µm zmax / mm

Ideal axicon 174.00 6.06 15.07

P1 173.60 5.69 14.14P5 173.59 5.68 14.10P9 173.68 5.76 14.32

Mean 173.63 5.71 14.19SD ±0.05 ±0.05 ±0.11

zmax

P5

P9

2 mm

P1

10µm

(b)

(c)

(a)

zx

(b)

(c)

(a) 50 µm0.5 zmax

xy

Figure 5.21 Measured intensity profile of the Bessel beam (logarithmically scaled) forthree different positions with cross-sections at zmax/2. The mean Bessel beam range zmaxfrom Table 5.5 was used.

Optical characterization method: Similar to the collimation lens array, the qual-ity of the generated Bessel beam was evaluated using the optical measurement setupand array Ax3-A1. The measured Bessel beams along the propagation direction isillustrated in Figure 5.21 for three different positions in the array. Cross-sectionalimages were added to show the quality of the Bessel beam at zmax/2. A typical Besselbeam can be observed for all three positions with small wavefront errors for P1 andP5 (visible at the first ring) resulting from the astigmatism found in Figure 5.20(left).The intensity profile and the core radius of the Bessel beam along the propagationdirection for three different positions are illustrated in Figure 5.22. As expected, a“lens effect” can be observed. The focal length of 5.8 mm corresponds to a roundingradius of 175 µm, which is 45 µm larger than intended. The core radius of all threepositions followed the ideal core radius and show that typical Bessel beam weregenerated. The characteristic properties determined from the surface data from

87


0 2 4 6 8 10 12 14 16 1802468

1012141618

P1P5P9Simulation

Nor

mal

ized

core

inte

nsity

/a.u

.


zmax

5 6 7 8 9 10 11 12 13 14 154.55.05.56.06.57.07.58.08.59.0

Ideal rcore

Mean rcore


P1P5P9

Cor

era

dius

/µm

zmax

Figure 5.22 Left: Measured and simulated normalized core intensity of the profile alongthe propagation direction. Right: Core radius along the propagation direction. The meanBessel beam range zmax from Table 5.5 was used.

Table 5.5 show a good agreement with the propagation results in Figure 5.22.

Short conclusion: The 174 axicon lens arrays showed a very good optical qualityfor all three arrays (Ax3-A1-A3 each at three positions). The total mean surfacedeviation of all three arrays was found to be srms = (0.123 ± 0.025) µm thatcorresponds to a surface error of 0.26λ or 0.15λ wavefront error. Further, for allthree arrays the axicon angle varied by (−0.48 ± 0.094) from the designed apexangle, and the core radius varied by (7.34 ± 0.08)%. The surface error and theangle variation are in the same range as Altechna’s manucfacturing tolerance forplano-convex axicons with a surface irregularity (rms) of below 0.158 µm and anglevariations of ±(0.02 − 0.5) [125]. The deviation from the intended shape variedbetween −6.1 % and 5.3 %.Additionally, the optical measurements showed the characteristic beam propertiesof the Bessel beam throughout the propagation length, which additionally indicatesa high optical quality and the required high rotational symmetry.

Axicon with 170 apex angleSurface deviation: The surface profile of the 170 axicon array is illustrated inFigure 5.23. The evaluated surface data is summarized in Table 5.6, with a meanrms surface deviation of all three positions for array Ax5-A1 of srms = (0.123 ±0.014) µm. The two additional arrays Ax5-A2 and Ax5-A3 showed mean rms surfacedeviation of srms = (0.213 ± 0.068) µm and srms = (0.174 ± 0.018) µm , respectively(Table A.5 in the Appendix).

Deviation from the target function: The deviation of the measured surfacedata from the target function is illustrated in Figure 5.24 (left). All three lensesfollow the same profile with different values, but compared to the previous lens

88


Figure 5.23 Measured three dimensional surface data of a 170 array (Ax5-A1).



P1 0.122 0.595 12.19P5 0.109 0.959 13.34P9 0.138 0.979 7.12

Mean 0.123 0.844 10.23SD ±0.014 ±0.216 ±4.39

types with much higher values. As mentioned before, the reason is most likely thedeviation from the intended shape of the silicon cavity, which is then transferredinto the PDMS mold and consequently into the lens array. The difference, comparedto the previous lens types, is a much higher aspect ratio in the initial cavity towardssmaller radii (Figure 5.13).

Zernike polynomials: The Zernike coefficients extracted from the surface data forall three positions are illustrated in Figure 5.24 (right). The dominant aberrationsin the range between λ/6 and λ/2 are mainly astigmatism and trefoil.

Similar to the other axicon arrays, the intended shape needed to be determined byfitting the slope of the axicons in the interval [ r1 = 175 µm, r2 = 450 µm], found inSection 5.8. Once more, the characteristic properties: apex angle of the axicon, coreradius, and Bessel beam range were determined from the slope and are summarizedin Table 5.7. Likewise, the characteristic properties agree with an ideal 170 axicon.

Optical characterization method: The axicon array Ax5-A1 was evaluated withthe optical measurement setup. The generated Bessel beam along the propagationdirection is illustrated in Figure 5.25 for three different positions. The cross-sectional

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Figure 5.24 Left: Deviation of the measured surface data from the simulated targetfunction for three different measured positions of array Ax5-A1. Right: Zernike coefficientsextracted from the surface data representing the individual contributing aberrations. Theaberrations within the shaded area are negligible.

Table 5.7 Characteristic properties of the 170 axicon array Ax5-A1 at three differentpositions with mean value (Mean) and standard deviation (SD).

Lens position Apex angle / rcore /µm zmax / mm

Ideal axicon 170.00 3.64 9.03

P1 170.48 3.82 9.49P5 170.56 3.86 9.58P9 170.22 3.72 9.24

Mean 170.42 3.80 9.44SD ±0.02 ±0.07 ±0.18

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Figure 5.25 Measured intensity profile of the Bessel beam (logarithmically scaled) forthree different positions with cross-sections at zmax/2. The mean Bessel beam range zmaxfrom Table 5.7 was used.

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Figure 5.26 Left: Measured and simulated normalized core intensity of the profile alongthe propagation direction. Right: Core radius along the propagation direction. The meanBessel beam range zmax from Table 5.7 was used.

images at zmax/2 show that the Bessel beam has higher aberrations, which wasalready expected from the surface data (Figure 5.24 (right)). Nevertheless, a typicalBessel beam can be observed for all three positions.The intensity profile and the core radius along the propagation direction of theBessel beam in Figure 5.26 confirmed the Bessel beam profile. The “lens effect” in theintensity profile had a focal length of 4.4 mm, which corresponds to a rounding radiusof 220 µm. It can be observed that the Bessel beam ends before the determinedzmax most probably because the static aperture must have been smaller than 1 mm.The core radius agrees with the ideal axicon radius with some small variations.

Short conclusion: The 170 axicon lens arrays showed a good optical quality for allmeasured arrays (Ax5-A1-A3 each at three positions), where the total mean surfacedeviation of all three arrays was found to be srms = (0.174 ± 0.019) µm. This value

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Figure 5.27 Measured three dimensional surface data of a lensacon array (La-A1).

corresponds to a 0.37λ surface error or 0.21λ wavefront error. The axicon angle forall three arrays varied by (−0.35 ± 0.17) from the designed apex angle and thecore radius by (−3.45±0.07)%. The deviation from the intended shape was between3.8 % and 13.2 %. Similar to the 174 axicon the optical measurements showed thecharacteristic beam properties of the Bessel beam throughout the propagation lengthwith some visible wavefront errors.

LensaconSurface deviation: The surface profile of the lensacon array La-A1 is shown inFigure 5.27, and the evaluated surface data is summarized in Table 5.8. A mean rmssurface deviation of srms = (0.227±0.022) µm was determined for all three positions.For the lensacon array similar values of srms = (0.238 ± 0.017) µm and srms =(0.205± 0.057) µm) were determined, respectively (Table A.6 in the Appendix).

Deviation from the target function: The deviation of the measured surfacedata from the target function is illustrated in Figure 5.28 (left). As before, all threelenses show the same deviation behavior. Here, the highest deviations occur mostprobably according to the much steeper and stepper initial cavity (Figure 5.13).

Zernike polynomials: The Zernike coefficients extracted from the surface data aredepicted in Figure 5.28 (right) for all three positions. Compared to the previous lensarrays, much higher aberrations can be observed for both astigmatisms directionsbetween λ/3 and 0.7λ.

Optical characterization method: The lensacon array La-A1 was further char-acterized using the optical measurement setup. The reconstructed measured Besselbeam along the propagation direction is illustrated in Figure 5.29 for three differentpositions. The cross-sectional images are shown for two different positions. The firstposition is close to the “lens effect” at z = 2.4 mm generated by the rounding at the

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P3 0.205 1.289 5.11P7 0.249 1.390 5.48P9 0.228 1.259 5.22

Mean 0.227 1.313 5.27SD ±0.022 ±0.069 ±0.19

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Figure 5.28 Left: Deviation of the measured surface data from the simulated targetfunction for array La-A1 at three different measured positions. Right: Zernike coefficientsextracted from the surface data representing the individual contributing aberrations. Theaberrations within the shaded area are negligible.

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P7

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Figure 5.29 Measured intensity profile of the Bessel beam (logarithmically scaled) forthree different positions with cross-sections at 2.4 mm and zmax.

tip. Here, a clear, ring-shaped profile with only some wavefront aberrations can beobserved. The second position is at the maximal Bessel range zmax = 3.34 mm. Asthe lensacon is a combination of a lens and an axicon, the chracteristic properties ofa Bessel beam need to include the collimation lens. Therefore, the maximal Besselrange is determined by zmax = rmax/(α+(rmax/fc)), with cone angle α [160,161]. Thecross-sectional profile shows a wavefront with large aberrations, which was expectedfrom Figure 5.28 (right).

The intensity profile and the core radius along the propagation direction of the Besselbeam are illustrated in Figure 5.30. The normalized core intensity shows a rapiddecrease shortly after the “lens effect”, which is characteristic for lensacons [138,161,162]. Lens P9 follows the simulation in good agreement, whereas lenses P3 and P7decrease much faster. The core radius for all three positions show a much larger coreradius than the theoretical rcore = (0.38λ)/(α + (r/fc)) = 1.20 µm. This error wascaused by limitations in precise positioning of the distance from the light source.

Short conclusion: The lensacon arrays were produced to reduce the number ofoptical surfaces. All three arrays (La-A1-A3 each at three positions) showed an ac-ceptable optical quality with an average surface deviation srms = (0.224± 0.035) µmthat corresponds to 0.47λ surface error or 0.27λ wavefront error. The deviation fromthe intended shape was found to be between −6.4 % and 15.6 % resulting from thestepper initial cavity. The intensity and radial profile displayed a Bessel beam withvarious wavefront errors and a much larger core radius than expected. This couldbe explained by a positioning error of the light source to the lensacon array. Still,the lensacon can be used if a reduction of optical surfaces is needed.

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5.10 Conclusion

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Figure 5.30 Left: Measured and simulated normalized core intensity of the profile alongthe propagation direction. Right: Core radius along the propagation direction for threedifferent lens positions.

5.10 Conclusion

In this chapter, the transmissive aspherical optics were introduced. The require-ments and an overview of different concepts were presented. The method based ona rapid-prototype process was explained, and the simulation with shape optimiza-tion needed for this purpose was introduced. The critical material parameters ofthe used PDMS necessary for the simulation were specified. The lens fabricationprocess was explained and some limitations, including surface roughness and aspectratio of the initial silicon cavity as well as the heat transfer within the system wereanalyzed.Four different lens types were evaluated by their surface quality as summarizedfor the surface deviation in Figure 5.31 and corresponding wavefront errors in Fig-ure 5.32. The surface rms values show that different lens arrays could be producedwith a surface deviation between λ/4 and λ/2, depending on the lens type. Thatcorresponds to a wavefront error between λ/7 and 0.14λ (λ = 473 nm). The peak-to-valley value was additionally determined to illustrate the maximum surface de-viation. It ranges between λ and 1.7λ.Although, this is not the intended high precision rms wavefront error of λ/16 asrequired in Section 5.1, the produced lens arrays have a comparable surface qual-ity to other state-of-the-art lenses, which typically ranges between λ/12 and λ/2(λ = 473 nm) [104, 125, 158]. A decrease in optical quality was observed withincreasing lens height and cavity steepness, which indicates that the fabricationprocess gets less precise with increasing steepenss in the initial cavity.Within one array, all four lens types follow the same deviation profile from theintended shape (∆ftarget) up to some smaller variations. The highest deviationwas found in the lensacon array, where the deviations varied from −6.5% up to

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15.6%. The smallest variations were found in the 174 axicon (with shallower pro-file). Compared to the collimation lens and the 174 axicon arrays, the other twolens types showed higher deviations from the intended shape, concluding that deeperand steeper cavity profiles result in higher deviations in the fabricated silicon cav-ity from the intended (FEM simulated) cavity shape. This deviation will then betransferred into the PDMS mold and from there to the microlens arrays.The initial silicon shape could only be verified for the 174 axicon. The other lenstypes were too deep to be measured using the confocal profilometer setup. Otherdeviations in the profiles resulted most probably from process variations, includingparticles, temperature variations and errors during the pseudo-shrinkage. The mea-surements showed that the lens arrays have aberrations caused by the fabricationprocess that will cause non-symmetric lens profiles. Notably, the majority of theaberrations for each lens type showed a aberrations in the 0- or y-direction. Thiscould result from distortions in the laser.Additional measurements using an optical measurement setup confirmed the resultsfrom the surface data and provided supplementary information on the lens quality.The two fabricated axicons showed surface errors and angle variations in the samerange as Altechna’s manucfacturing tolerances for plano-convex axicons [125]. Ad-ditionally, the optical measurements showed the characteristic beam properties ofthe Bessel beam throughout the propagation length, which indicates a high opticalquality and the required high rotational symmetry.Other sources of error were the Poisson’s ratio and the coefficient of thermal expan-sion which were used to simulate the intended lens shape. Those simulated materialparameters vary from the experimentally found values by 0.4 % for the Poisson’sratio and 5 % for the thermal expansion, which changes the pseudo-shrinkage andthe generated surface deformation by 8.4 %. The temperature variation within thelaboratory, from 23 C up to 28 C, during the production of the lens was an addi-tional source of error. This temperature variation results in a surface deformationchange by 5 %. Still, it was possible to successfully fabricate and evaluate each lenstype.

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5.10 Conclusion

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Figure 5.32 From the surface data determined rms (left) and peak-to-valley (PV) (right)wavefront error of each array average over three positions (λ = 473 nm) with an additionalmean value.

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6 Compact micro-optical systemThis chapter describes the compact micro-optical system combining the miniatur-ized adaptive ring aperture array with the lensacon array to reduce the number ofcomponents. For this purpose, an adapted fabrication and assembly process wasdeveloped. The produced system was experimentally characterized, and the resultsare discussed.

6.1 ConceptThe micro-optical device for optogenetic stimulation introduced in Chapter 1 hasthree micro-optical components (collimation lens array, ring aperture array andaxicon array), which need to be aligned precisely to the laser diode arrays and toeach other. Therefore, the concept of the compact micro-optical system was toreduce the number of components and thereby minimize alignment and assemblyerrors.The first step to reduce the number of components and thereby the number ofoptical surfaces is already discussed in Chapter 5: the combination of a lens andan axicon into a lensacon. The next step is the integration of the miniaturized ringaperture array with the lensacon array into one compact micro-optical system. Forthis reason, the lensacon array was fabricated directly on top of the adaptive ringaperture array, as illustrated in Fig. 6.1.

6.2 Fabrication and assemblyThe individual fabrication and assembly steps are summarized in Figure 6.2. Thefabrication was done as follows:

1. Ring aperture preparation: The adaptive ring aperture array was fabricated,as described in Section 4.4.

2. Lens mold preparation: The lens mold was fabricated, as detailed in Sec-tion 5.7.

3. Combination of the two micro-optical components: The adaptive ring apertureand the lens mold were precisely aligned and assembly using a flip-chip bonder.

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6 Compact micro-optical system

3 mm

Figure 6.1 Photograph of the compact micro-optical system with ring aperture array andthe lensacon array on the back side of the bottom chip.

Ring aperture preparationAfter the wafer-level fabrication of the adaptive ring aperture, as described in Sec-tion 4.4, the bottom and counter chip were aligned to each other and glued togetherwith UV-curable adhesive. For functional testing of the system, the liquid crystalwas filled into the cavity via capillary forces and was visually checked.

Lens mold preparationThe lensacon mold was fabricated as detailed in Section 5.7. The substrate prepa-ration and the lensacon mold preparation, where the surface deformation is formeddue to pseudo-shrinkage of PDMS, were the same. The lensacon mold was thentransferred into the clean room, where the array preparation on the adaptive ringaperture was performed.

Combination of the two micro-optical componentsThe next step was the combination of the lensacon mold with the adaptive ringaperture at the back side of the bottom chip, as illustrated in Figure 6.2. A mis-alignment of the two components would result in an asymmetric beam profile. Toavoid this, the lensacon array needed to be aligned accurately to the ring aperturearray. Therefore, a flip-chip bonder (Fineplacer lambda, Finetech) was used. First,the backside of the bottom chip was aligned to the lensacon molds (Figure 6.2 (2)).Then, the UV-curable adhesive was placed on top of the molds, and the ring aper-ture array was pressed onto the molds with a force of 100 mN. The adhesive wascured with a portable UV-LED (Opsytec, λ = 365 nm) for 5 min, after which thepressure was released. As the LED was manually positioned close to the adhesive, anexact exposure distance could not be guaranteed, resulting in a variation of the light

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6.2 Fabrication and assembly

1.

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PCB

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Figure 6.2 Fabrication process of the compact micro-optical system. The counter andbottom chips of the ring aperture array were aligned, glued together, and the liquid crystalwas filled in (1). UV-curable adhesive was applied on top of the lens molds, and the liquidcrystal device was aligned (2) and pressed on top of the lens mold (3). After curing, thelens mold was removed (4) and placed into the PCB, where the electrical connection wasdone via wire bonding (5).

intensity. To overcome this problem, the device was placed under the mask alignerdescribed in Section 4.4, where it was additionally exposed using flood-exposure10× 5 s with 30 s breaks to make sure the UV-adhesive was completely cured with-out generating too much heat (otherwise the PDMS would expand). Finally, thelensacon mold was removed and the device could be assembled further.Several wire bond loops were placed on the PCB at the necessary position to en-sure the connection of the counter chip. Then, the combined system was carefullyplaced into the PCB and pressed against the wire. Afterwards, the bottom chip waselectrically connected to the PCB by wire bonding.

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6.3 Experimental characterizationThe experimental characterization was performed with the same optical surface mea-surement methods and optical characterization method as described in Section 5.9.2.

6.3.1 MethodsOptical surface measurements: The surface topology of the fabricated lens ar-rays was measured using a chromatic confocal distance sensor with objective CL2,described in Section 2.1.

Optical measurement setup: The produced compact micro-optical system wasfurther evaluated using the setup described in Section 2.2.1. The DPSS laser with473 nm wavelength was placed at a distance of approximately 5 mm, dependingon the actual fabricated element. The camera with 1.67 µm pixel pitch and amicroscope objective with magnification factor 4 were used. The ring apertureswere driven with a sinusoidal voltage of 6 Vrms and 500 Hz.

Data evaluation: The reconstruction of the image plane along the optical axis, aswell as the intensity profile evaluation from Section 2.2.3 were used.

6.3.2 ResultsThe presented results were generated with the adaptive ring aperture array designA1 and element E6.

Surface deviation: The evaluated surface data showed a rms surface deviation ofsrms = 0.405 µm and a peak-to-valley value of 1.756 µm.

Deviation from the target function: The deviation of the measured surfacedata from the target function ftarget is illustrated along the radius in Figure 6.3(left). The deviation along the radius was found to be smaller than the deviationsin the lensacon array from Section 5.9.2.

Zernike polynomials: Compared to the aberration of a lensacon in Section 5.9.2,the Zernike coefficients here showed a three times higher astigmatism in the rangeof approximately 2λ, as illustrated in Figure 6.3. The reason for such a high astig-matism could be caused by a deviation in the pseudo-shrinkage.

Optical characterization method: The compact micro-optical system was mea-sured with the optical measurement setup. The generated Bessel beam along thepropagation direction is illustrated in Figure 6.4, including the un-actuated ringaperture (top), the actuated ring apertures R1 (center) and R2 (bottom). Thecontrast between un-actuated state and actuated state was approximately twelve.

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Figure 6.3 Left: Deviation of the measured surface data from the simulated target func-tion. Right: Zernike coefficients extracted from the surface data representing the individ-ual contributing aberrations. The aberrations within the shaded area are negligible.

The outer ring aperture electrode R3 was not actuated as the lensacon array wasshape optimized until r = 450 µm (to avoid edge effects), which is in the middle ofR3. Consequently, the surface profile of the lensacon had high deviations from thetarget function at those outer radial positions and would not create a Bessel beam.The cross-sectional images show that the Bessel beam has high aberrations, whichwas already indicated by the surface data (Figure 6.3 (right)). The first actuated ringaperture R1 only includes the rounded tip and the mentioned “lens effect”, whichwill not produce a Bessel beam. The second actuated ring aperture R2 generates aBessel beam, which can be observed along the intensity profile. The high aberrationsfrom the lensacon and the additional wavefront error induced by the ring aperture(probably in the order of λ/2) will result in rather high aberrations, as observed inthe cross-section in Figure 6.4 (c).For a more quantitative result, the normalized intensity profile along the propagationdirection of the compact micro-optical system is illustrated in Figure 6.5. As alreadyseen in Figure 6.4, the intensity profiles of the actuated ring aperture showed abroader intensity profile than expected from theoretical positions. This broadeningoriginates from the deviations in the surface profile, which could additionally resultin a difference in conical angle and thus in different theoretical positions.

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Figure 6.4 Measured intensity profile of the Bessel beam (logarithmically scaled) withcross-sections at (a) zmax/2 = 1.67 mm, (b) z = 1.4 mm and (c) z = 2.4 mm, and addi-tional indications of the theoretical segmented Bessel beams regions. Top: un-actuadedring aperture; center and bottom: actuated ring aperture R1 and R2, respectively.

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Figure 6.5 Measured normalized core intensity along the propagation direction. Theshaded areas indicate the theoretical positions of each corresponding ring electrode.

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6.4 Conclusion

6.4 ConclusionThe realization of a compact micro-optical system was demonstrated by integratinga miniaturized ring aperture array and a rapid-prototype produced lensacon arrayto generate depth-control Bessel beams.Both micro-optical components were first designed and fabricated individually withinthis thesis and were then combined with an adapted fabrication and assembly pro-cess. Compared to three micro-optical components that needed assembly and align-ment to each other, here only one micro-optical component is produced with onenecessary assembly and alignment step.To eliminate the need for a customized mounting, the lensacon was fabricated di-rectly onto the ring aperture. It was successfully shown that the produced systemis able to generate depth-controlled Bessel beams.The optical performance of the prototype could still be improved by minimizing thehigh aberrations of the lensacon and wavefront errors, induced by imperfections ofthe alignment layer in the ring aperture array that was discussed in Section 4.6.2.Further, the use of different lensacon properties, e.g., a different axicon angle wouldbe beneficial for this system. The proof-of-concept was successfully demonstratedby combining both components into an integrated fully functional system.

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7 Conclusion and Outlook

7.1 ConclusionThe goal of this thesis was the design, fabrication and characterization of micro-optical components to generate depth-controlled Bessel beams. These micro-opticalcomponents were part of a project to develop a miniaturized device for optogeneticstimulation of small rodent brains. Therefore, different micro-optical componentswith requirements (3 × 3 arrays with 1 mm element diameter) necessary for thestimulation device were developed: an adaptive ring aperture array and various as-pherical lens arrays, including a collimation lens, two axicons with different angles,and a lensacon, which is a combination of a lens and an axicon.

Prerequisite for the development of the micro-optical components were basic ana-lytical considerations of Bessel beams. Of particular interest were the use of axiconswith rounded tips, limitations regarding the aperture width to control a Bessel beamalong its propagation direction, and the evaluation of Bessel beams using asymmet-ric illumination sources. All considerations were confirmed with beam propagationmethods. More importantly, it was also possible to confirm these findings with mea-surements.The rounding of the axicon tip affects the Bessel beam and generates a sharp focalpoint only after which a Bessel beam can be observed. The derived oscillations thatresult from the constructive interference of the conical wavefront from the axiconand the plane wavefront of the rounded tip converge to the intensity profile of anideal axicon.The limitations of depth-controlled Bessel beams were analytically estimated in theclassical and Fraunhofer limits with respect to the shortest possible beam segment(without broadening by diffraction) and the corresponding optimal aperture. Aneasy to use expression was found, assuming that the shortest Bessel region occursnear the intersection of these two limits. This expression is universally valid andcan be used for other applications where Bessel beam need to be tailored.Analytical considerations for asymmetric and astigmatic illumination sources wereinvestigated regarding opening angle, maximal intensity along the optical axis, ef-fects of astigmatism, and effects of substrate modes and far-field ripples. The use ofedge emitting laser diodes with an axicon with a rounded tip showed a transition ofa symmetric Bessel beam into a bow-tie shaped pattern. As a result, an estimate of

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the opening angle and the maximum of the intensity for the asymmetric beam werefound. It appears that the astigmatism was highly suppressed in the actual edgeemitting laser diodes with realistic astigmatism values. Further, the characteristicsubstrate modes for these laser diodes have a different angle, are not collimated,and interfere with the main beam. Comparing the characteristic properties like in-tensity, core radius, and ellipticity along the optical axis, two characteristic regionswere determined. The slow-axis region creates a Bessel beam by the approximatelyrotationally symmetric central part of the intensity distribution. From there, theBessel beam turns smoothly into the fast-axis, where an asymmetric bow-tie shapedintensity distribution is found. Consequently, the collimation lens was designed witha short focal length to use only the rotationally symmetric slow-axis region.

Micro-optical components were then developed based on the knowledge gained fromthese analytical considerations. The first micro-optical component, an adaptive ringaperture array for depth-control of a Bessel beam, was conceptualized, designed,fabricated, and characterized using liquid crystal technology. Twisted nematic liq-uid crystal cells in different designs were fabricated based on standard wafer-levelclean room technologies and were then experimentally evaluated. The array designsshowed small wavefront errors that were approximated to be between λ/5 and 0.65λ(λ = 473 nm), resulting from imperfections in the alignment layer. However, a clearsegmentation of the Bessel beam with a good optical quality was found. The char-acterization of the single element design showed a higher wavefront error of 0.75λ,which can be explained by a wavefront error increase that is proportional to theradius.

The second type of micro-optical components, various transmissive aspherical lensarrays, were designed, fabricated, and characterized. A rapid-prototype processto fabricate transmissive aspherical lens arrays based on surface deformation dueto thermal expansion of PDMS was developed further. The fabrication methodhad been developed previously for reflective axicon mirrors with only small conicalangles. Within this thesis, a substantial enhancement was achieved by using thefabrication process for transmissive aspherical lens arrays with much steeper anglesand deeper mold depths. For this purpose, a new method to determine an exact valuefor the Poisson’s ratio and the coefficient of thermal expansion of the used PDMSwas developed. Compared to the literature, values one magnitude more precisewere obtained. FEM shape optimization simulations improved the predictions ofthe initial cavity.

Four different 3× 3 micro-lens array types (the required collimation lens and axiconas well as an additional axicon and a lensacon) with a 1 mm lens diameter weredesigned, fabricated, and characterized. All lens arrays showed a wavefront errorbetween λ/7 and λ/2, which is comparable to state-of-the-art micro-lenses (typical

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7.2 Outlook

values λ/12 − λ/2 with λ = 473 nm). Notably, the generated self-interfering beamprofiles indicated a high optical quality and the required high rotational symmetry.The aspheric collimation lens arrays with a design focal length of 5 mm, showed onlysmall aberrations, a surface error of (0.154 ± 0.013) µm, and an error in focal lengthby (6.2 ± 1.6)%. The 174 and 170 apex angle axicon arrays showed a small astig-matism and surface errors of (0.123 ± 0.025) µm and (0.174 ± 0.019) µm, respec-tively. Characteristic beam properties of the Bessel beam were observed throughoutthe propagation length. Ultimately, a lensacon array was fabricated. Lensaconswere designed as a combination of 5 mm focal length collimation lenses and 170axicons to reduce the number of optical surfaces. They showed astigmatism in bothdirections, but the surface errors of (0.224 ± 0.035) µm proved that this developedrapid-prototype process can be used to fabricate highly aspherical lens shapes thatare able to generate Bessel beams.

Finally, a compact micro-optical system with minimal alignment and assembly errorswas produced to further reduce the number of surfaces by combining the adaptivering aperture array with the transmissive aspherical optics into a single device. Theproof-of-concept to assemble such micro-optical components into an integrated, fullyfunctional system was successfully demonstrated.

Besides the here addressed applications in optogenetic stimulation, these micro-optical components can be used in many other fields, such as lithography or lightsheet microscopy. The adaptive ring aperture can be implemented in other opticalsystems where a beam tailoring is necessary, e.g., in consumer electronics (iris incameras). The rapid-prototype fabrication process for the transmissive asphericaloptics can be used for free-form shapes, e.g., logarithmic axicons, elliptic axicons ortransmissive phase plates.

7.2 Outlook

The two different fabrication processes introduced for the micro-components in thisthesis open up space for further improvements.

The adaptive ring aperture could be improved optimizing the alignment layer, whichwould result in a smaller wavefront error. This could include the use of other mate-rials and fabrication techniques forming the alignment layer. One example would bethe photoalignment technique, where a photo-active layer is patterned by polarizedlight. Another enhancement could be done by also patterning the counter electrodeto address more different electrode configurations.

109


The process parameters of the transmissive aspherical optics, which still showedsome variations, could be improved using other techniques to determine the initialmold shape in the silicon. Smaller laser steps or other fabrication technologies toshape the initial mold for the pseudo-shrinkage could advance the process. Limita-tions of the process could be explored by producing other free-form shapes such aselliptic axicons.

The compact micro-optical system could benefit from different lensacon parameters(axicon angle and focal length for which the lensacon is designed). An additionalimprovement would be to increase the lens dimensions so that all electrodes of thering aperture can be actuated. Improving the single components would be advan-tageous for the device and could reduce wavefront errors even more.

First in vitro tests on rat brain slices have been performed and have shown thatBessel beams generated from axicons can penetrate the brain tissue slightly deepercompared to Gaussian beams. This route needs to be investigated further, e.g.,by comparing different regions of interests of the intensity distributions at variousdepths.

110

A Appendix

A.1 Appendix: adaptive ring apertureThe exact aperture width of each design is summarized in Table A.1.

A.1.1 Electric characterizationThe actuation of the ring aperture was done by applying a sinusoidal voltage signalbetween 3 Vrms and 7 Vrms with a frequency of 500 Hz. The threshold voltage wascalculated to be 0.72 V.The capacity of the liquid crystal can be determined by the equivalent circuit illus-trated in Figure A.1 [98]. The capacities Clc = ε0εlcAlc

dlcand Calign = ε0εalignAalign

dalignare

the liquid crystal and alignment layer capacities, respectively. The total capacityfor the equivalent circuit was determined to be C = 8.67 pF. To get an estimatefor the capacity, electrode R3 of design A2 was determined experimentally using afrequency sweep from 100 Hz to 10 kHz. It was determined to be C = 7.73 pF,which lies in the same range as the calculated total capacity.The respondse time was calculated using Equation 4.7 and Equation 4.8, mentionedin section 4.3.1 to be in the millisecond range (∼ 0.22 s). The experimental de-termination showed that the “switching on” time is in the same order, while the“switch off” time takes several second. This “switch off” time is most likely domi-nated by imperfections in the alignment layer rather than the electrical propertiesof the device.

dlc

dalign

Liquid crystalAlignment layer

ITO

RITO

RITO

Ralign

Rlc

RalignCalign

Calign

Clc VV

Figure A.1 Left: Schematic of the liquid crystal cell. Right: Equivalent circuit of theliquid crystal cell with resistance RITO,align,andlC, and capacity Calign,lC the capacities.

113

A Appendix

Table A.1 Different ring aperture designs with outer radius (rout), inner radius (rin),aperture width (daperture), and spacing between each electrode.

Design rout /µm rin /µm daperture /µm Spacing /µm

A1 500 400 100 50350 200 150 50150 0 150 0

A2 500 400 100 50350 250 100 50200 100 100 5050 0 50 0

A3 500 400 100 20380 280 100 20260 160 100 20140 0 140 0

A4, A5 500 435 65 30405 347 58 30317 265 52 30235 191 44 30161 124 37 3094 66 28 3036 0 36 0

A6 2000 1645 355 501595 1380 215 501330 1134 196 501084 908 176 50858 702 156 50652 517 135 50467 354 113 50304 214 90 50164 100 64 5050 0 50 50

A7 2000 1656 344 201636 1418 218 201398 1198 200 201178 994 184 20974 808 166 20788 640 148 20620 489 131 20469 356 113 20336 242 94 20222 146 76 20126 70 56 2050 0 50 0

114

A.2 Appendix: transmissive aspherical optics

A.2 Appendix: transmissive aspherical opticsBernstein polynomialsThe Bernstein polynomials are defined as:

Bn(r) =n∑η=0

Cηbη,n(r), (A.1)

where bη,n is the Bernstein basis polynomial of degree n:

bη,n(r) = n!η! (n− η!)r

η(1− r)n−η, (A.2)

The coefficients Cη of the simulated Bernstein polynomials are summarized in Ta-ble A.2.

Table A.2 Coefficients Cn of the Bernstein polynomials for each lens type.

Lens type C0 C1 C2 C3 C4 C5 C6

Collimation lens 1 5 3.3 4.3 2.1 0.0148 -Axicon 174 0.476 2.465 0.339 2.441 0.052 1× 10−6 -Axicon 170 1.0267 3.0997 0.2727 3.9993 −0.2042 0.0054 -Lensacon 1.6612 5.9866 23.8560 10.3536 7.6910 0.0576 0.0001

Additional resultsThe additionally measured lens arrays at three different positions are summarizedin Table A.3-A.6. The surface evaluation included the rms value surface devia-tion (srms), peak-to-valley (PV) value, and the rms deviation of the intended shape(∆ ftarget) with mean value (Mean) and standard deviation (SD). For the two axi-cons additional values for the apex angle (Apex), the core radius (rcore), and theBessel beam range (zmax) were evaluated.

115

A Appendix

Table A.3 Surface evaluation of the collimation arrays Cl-A2 and Cl-A3 at three differentpositions with mean values (Mean) and standard deviation (SD).

Lens position srms /µm PV /µm ∆ ftarget / % Focal length / mm

CLe2 P1 0.144 0.930 6.98 4.77CLe2 P4 0.177 1.134 6.69 4.67CLe2 P7 0.141 1.175 8.11 4.62

Mean 0.154 1.079 7.26 4.69SD ±0.020 ±0.131 ±0.75 ±0.08

CLe3 P1 0.307 2.008 3.91 5.18CLe3 P4 0.336 2.213 7.98 5.85CLe3 P7 0.398 2.582 3.12 5.13

Mean 0.347 2.268 5.00 5.38SD ±0.047 ±0.291 ±2.61 ±0.40

Table A.4 Surface evaluation and characteristic properties of the 174 axicon arraysAx3-A2 and Ax3-A3 at three different positions with mean values (Mean) and standarddeviation (SD).

Lens pos. srms/µm PV/µm ∆ftarget/% Apex/ rcore/µm zmax/µm

AX3e3 P1 0.113 0.785 5.27 173.45 5.56 13.82AX3e3 P4 0.102 0.906 6.32 173.41 5.52 13.72AX3e3 P8 0.106 0.700 5.20 173.49 5.59 13.88

Mean 0.107 0.797 5.60 173.45 5.56 13.81SD ±0.006 ±0.104 ±0.63 ±0.04 ±0.03 ±0.09

AX3e4 P2 0.166 1.051 3.63 173.41 5.53 13.73AX3e4 P5 0.165 1.095 4.54 173.50 5.60 13.91AX3e4 P8 0.115 0.823 3.91 173.57 5.66 14.08

Mean 0.149 0.990 4.03 173.50 5.60 13.91SD ±0.029 ±0.146 ±0.46 ±0.08 ±0.07 ±0.17

116

A.2 Appendix: transmissive aspherical optics

Table A.5 Surface evaluation and characteristic properties of the 170 axicon arraysAx5-A2 and Ax5-A3 at three different positions with mean values (Mean) and standarddeviation (SD).

Lens pos. srms/µm PV/µm ∆ftarget/% Apex/ rcore/µm zmax/µm

AX5e2 P1 0.155 0.983 11.85 170.39 3.79 9.41AX5e2 P5 0.197 1.308 10.43 170.46 3.82 9.48AX5e2 P9 0.288 1.575 9.68 170.54 3.85 9.55

Mean 0.213 1.289 10.65 170.46 3.82 9.48SD ±0.068 ±0.296 ±1.10 ±0.07 ±0.03 ±0.07

AX5e5 P2 0.193 1.428 4.76 170.14 3.69 9.17AX5e5 P5 0.171 1.383 3.97 170.12 3.68 9.15AX5e5 P8 0.156 0.968 7.48 170.23 3.72 9.25

Mean 0.174 1.259 5.40 170.17 3.70 9.19SD ±0.018 ±0.254 ±1.86 ±0.05 ±0.02 ±0.05

Table A.6 Surface evaluation and characteristic properties of the lensacon arrays La-A2and La-A3 at three different positions with mean values (Mean) and standard deviation(SD).


LAe3 P1 0.258 1.383 8.18LAe3 P3 0.228 1.397 7.51LAe3 P9 0.229 1.422 6.74

Mean 0.238 1.401 7.48SD ±0.017 ±0.020 ±0.72

LAe4 P2 0.271 1.373 7.16LAe4 P4 0.180 0.984 8.02LAe4 P9 0.165 0.924 7.22

Mean 0.205 1.094 7.47SD ±0.057 ±0.243 ±0.48

117

A Appendix

A.3 Appendix: Poisson’s ratio and coefficient ofthermal expansion

A new method to determine an accurate value for the Poisson’s ratio and the coef-ficient of the rapid prototyping process from Chapter 5 was found. The Poisson’sratio and the coefficient of thermal expansion were determined by a set of definedmolds measuring the resulting surface deformation at different temperatures. Inthis case, cylindrical cavities were used to have a reliable fabrication, avoid edgeeffects, and allow for an efficient rational simulation. The content presented in thisappendix has been published in Soft Matter [155]1.

A.3.1 FEM simulationsA FEM simulation similar to the one described in Section 5.5 was performed. Thegeometry and mesh with corner refinement at the edge of the cavity are illustratedin Figure A.2. Any possible mesh dependency at the edge of the cavity was ruled outby simulating different mesh sizes, chamfers, and fillets with different radii, whichresulted in a rapid convergence with decreasing mesh size. The boundary conditionsincluded a roller constraint at the bottom and a free boundary on the sidewalls andthe top surface. As the depth of the cavities is varied, from shallow to deep, theresulting surface deformation becomes relatively large. Therefore, geometric non-linearities were taken into account, as explained in Section 5.5.1. An interpolationfHn,T (ν, αCTE) over the Poisson’s ratio and the coefficient of thermal expansionwas done using four nested parametric sweeps (different cavity depths Hn, curingtemperatures T , Poisson’s ratios ν, and linear coefficient of thermal expansionsαCTE) for each cavity depth and temperature.

A.3.2 Experimental procedureTwo commonly used PDMS types were investigated: Sylgard 184 and Sylgard 182,which differ in their viscosity, shore hardness, and curing time, as summarized in Ta-ble A.7. The two component silicone elastomer kits were prepared as recommendedin the data sheets in a 10:1 ratio [156, 163]. The detailed procedure is described inSection 5.7.The aluminum cavities were fabricated by precise CNC milling, as depicted in Fig-ure A.3, top. The substrate has 16 cavities. Four cavities have either 0.5 mm,

1 AM: Conceived, performed, and evaluated all experiments and simulations, prepared the graphs,and wrote the paper.MCW: Conceived the basic idea, supervised the data analysis, helped interpreting the results,and reviewed the manuscript.UW: Proposed and initiated this research, supervised the work, and reviewed the manuscript.

118

A.3 Appendix: Poisson’s ratio and coefficient of thermal expansion

Aluminum

PDMS

Figure A.2 Left: Cross section of the FEM model using COMSOL Multiphysics. Right:Close up view of the mesh in the region near the cavity (green box in the left image) [155].

Table A.7 Characteristic properties of Sylgard 184 and Sylgard 182 [156,163].

Properties Sylgard 184 Sylgard 182

Viscosity (Mixed) 5100 cP 4575 cPThermal Conductivity 0.27 W/mK 0.16 W/mKPot Life (Working time at 25 C) 1.5 h 8 hCuring time at 25 C 48 h 336 hHeat curing time at 125 C 20 min 30 minDurometer Shore A 43 51Linear coefficient of thermal expansion 340 ppm/C 325 ppm/C

0.75 mm, 1 mm or 1.5 mm depth, with a constant diameter of 3 mm, and a spacerwith thickness of 0.5 mm.A silane-coated glass slide was carefully arranged on top of the aluminum cavities,filled with the liquid PDMS, and then directly placed into a mechanical press, whichwas preheated at different temperatures, as summarized in Table A.8. A force of atleast 1 kN was applied to prevent motion or deformation of the glass slide. Whilethe still liquid PDMS expanded, the excess material was pressed out of the substratevia the spacers.

After curing, the pressure was released, and the glass slide was removed. Thesubstrate cooled down to room temperature, and the pseudo-shrinkage formed smallsurface deviations, which were measured using the optical profilometer described inSection 2.1. The minimum of each surface deformation was determined manually,after leveling the surface profile, using the measured vertical positions between thecavities, as shown in Figure A.3. Guided by the curing time, the temperatureprotocol was found experimentally, and is summarized in Table A.8. Measurementswere performed on all four different cavity depths, with at least four cavities, eachat six different temperatures.

119

A Appendix

0 2 4 6 8 10 12 14 16 18 20-180

-160

-140

-120

-100

-80

-60

-40

-20

0

∆h/µ

m

Position in x-direction / mm

0.5 mm0.75 mm1 mm1.5 mm

3 mmH∆h

Height H 0.5 mm

Figure A.3 Left: Schematic of the milled aluminum substrate. Top: Cross-section of thealuminium substrate with four different cavity depths. Bottom: Measured profile of thegenerated surface deformation using Sylgard 184 at 120 C.

Table A.8 Curing protocol of Sylgard 184 and Sylgard 182 with start temperature (Tstart),ramp time (tramp), end temperature (Tend), and the curing time (tcuring) applied afterwards.

PDMS Tstart / C tramp / h Tend / C tcuring / h

Sylgard 184 50 1 60 665 0.3 75 2.580 0.1 90 1105 0 105 0.5120 0 120 0.3135 0 135 0.2

Sylgard 182 50 2 60 11.265 0.75 75 4.780 0.25 90 295 0.1 105 1120 0 120 0.6135 0 135 0.4

120


A.3.3 Data evaluationThe maximum surface deformation ∆h, measured at different cavity depths hadto be related to two quantities, ν and αCTE, to relate the measurements to thesimulation. Equating the measured ∆h to the interpolation fHn,T (ν,αCTE) left oneunknown degree of freedom, i.e., with a line in the space (αCTE, ν) for each cavitydepth Hn. The intersection of these lines of different cavities provided the predictedvalues of ν and αCTE. A Gaussian probability distribution W of the surface defor-mation ∆h for each cavity depth Hn at a constant temperature T was assumed toaccount for the measurement uncertainty:

WHn = 1σ2Hn

e−(−fHn,T (ν,αCTE)−∆hHn )2

(σHn )2 CHn , (A.3)

with the corresponding measurement uncertainty σHn , which was found statisticallyfrom the four cavities at each depth. The factor C co-variantly relates the probabilitydistribution in the space of ∆h to a probability distribution in the space of ν andαCTE:

CH =

√√√√(∂fHn,T (ν, αCTE)∂ν

)2

+(∂fHn,T (ν, αCTE)

∂αCTE

)2

. (A.4)

In Figure A.4 (left), the distributions for different depths are exemplarily shown for acuring temperature of 60 C. In Figure A.4 (right), the distributions were multipliedto get a total probability distribution of the overall measurement result for ν andαCTE. The Poisson’s ratio and coefficient of thermal expansion for one temperaturecorrelated and had dependent uncertainties. One variable was integrated to obtainindependent uncertainty values.

A.3.4 ResultsThe surface deformations for both materials at different cavity depths and temper-atures are summarized in Figure A.5. The probability distributions for differentcavity depths cross approximately at a single point, as illustrated in Figure A.4,which validates the measurement method.As already discussed in Section 5.7.1, the heat transfer was simulated to ensure thedrainage of the liquid PDMS (Figure A.6) using the 1.5 mm deep mold. The tem-perature in the center of the PDMS reaches 90 % of the final temperature in 3.97 sfor Sylgard 184 and 6.98 s for Sylgard 182, which is in both cases well below thecuring time.To evaluate the strain distribution (relative volume change) within the cavities, amost extreme case of the strain at a cavity depth of 1.5 mm and 135 C curingtemperature was simulated. The maximum strain was found to be 10.83 %. Only3.8 % of the overall volume change had a strain of more than 5 %. Consequently,

121

A Appendix

αCTE / (ppm/ °C)

ν

αCTE / (ppm / °C)ν

Figure A.4 Data evaluation in (αCTE, ν) space using Mathematica. Left: Probabilitydistribution for different cavity depths. Right: 1 σ probability region for different cavitiesand the 1 and 2 σ regions of the multiplied probability distributions of all different cavitydepths.

60 75 90 105 120 1350

20406080

100120140160180200

∆h/µ

m

Curing temperature / °C

0.5 mm S182 1.5 mm S1820.5 mm S184 1.5 mm S1840.75 mm S1820.75 mm S1841 mm S1821 mm S184

Figure A.5 Measured surface deformations ∆h at different curing temperatures anddifferent cavity depths for both Sylgard 182 (S182) and Sylgard 184 (S184). The errorbars indicate the sample (N > 4) over at least four cavities.

122


°C

0 5 10 15 20 25 30 35 402030405060708090

100110120

Tem

pera

ture

/°C

Time / s

Sylgard 182Sylgard 184

Figure A.6 Left: FEM simulation model of the complete setup using Sylgard 184 at90 % of the final temperature. Right: Simulated heat transfer for both PDMS types withindications at 90 % of the final temperature.

50 60 70 80 90 100 110 120 130 1400.4900.4910.4920.4930.4940.4950.4960.4970.4980.4990.5000.501

Sylgard 182 Sylgard 184Sylgard 182 mean Sylgard 184 mean

Pois

son'

sra

tio


Figure A.7 Poisson’s ratio, for Sylgard 182 and Sylgard 184 at six different temperatureswith mean value. The shaded areas show the error of the mean values.

it can be concluded that the result is dominated by small strains, and the possiblenon-linear deformation near the edge is negligible.In Figure A.7, the Poisson’s ratio for Sylgard 182 and Sylgard 184 at six differenttemperatures are illustrated. The error of the mean values is indicated with shadedareas. For both materials an approximately constant Poisson’s ratio can be found.Averaging the overall curing temperatures, the Poisson’s ratio for Sylgard 184 isν = 0.4950± 0.0010, and for Sylgard 182 it is ν = 0.4974± 0.0006.Additionally, the coefficient of thermal expansion from Equation 5.4, αCTE = αCTE+ε

∆T , was fitted to the measurement results, which included a contribution fromthe polymerisation shrinkage εp and the temperature ramp. It was found thatε is in the order of the polymerisation shrinkage around 0.1 %. This error isclearly smaller than the measurement uncertainty and thus negligible. An ap-proximately linear decrease in the coefficient of thermal expansion with increasingcuring temperature can be observed in Figure A.8. For Sylgard 184, the coeffi-

123

A Appendix

50 60 70 80 90 100 110 120 130 140220

230

240

250

260

270

280

290

300

slopeS182 = (-0.51 0.05) ppm/°C2, R2 = 0.966slopeS184 = (-0.65 0.07) ppm/°C2, R2 = 0.959

Sylgard 182Fit Sylgard 182Sylgard 184Fit Sylgard 184

Coe

ff.th

erm

al e

xpan

sion

/pp

m/°

C


Figure A.8 Coefficient of thermal expansion for Sylgard 182 and Sylgard 184 at sixdifferent temperatures with linear fits and slope.

cient of thermal expansion was found to be approximately 9 % less than the valuegiven in the data sheet 340 ppm/C [163]. It was extrapolated at 25 C to be(309.63 ± 6.91) ppm/C. The decreasing coefficient (slope) is indicated in Fig-ure A.8 and is (−0.65 ± 0.07) ppm/C2. For Sylgard 182, the extrapolated valueat 25 C was found to be (289.55 ± 4.44) ppm/C, which is approximately 11 %less than the value in the data sheet 325 ppm/C [156]. The slope was found to be(−0.51± 0.05) ppm/C2.

A.3.5 ConclusionsA new method to accurately determine the Poisson’s ratio using the thermal ex-pansion properties of PDMS, and an optical surface scanning method was found.The Poisson’s ratio for Sylgard 184 is ν = 0.4950 ± 0.0010, and for Sylgard 182 itis ν = 0.4974± 0.0006. In addition to the Poisson’s ratio, the coefficient of thermalexpansion was found. Whereas the Poisson’s ratio remains constant, a linear de-crease of the coefficient of thermal with increasing temperature was observed. ForSylgard 184 the slope was found to be (−0.65± 0.07) ppm/C2 and for Sylgard 182it is (−0.51± 0.05) ppm/C2.

124

AcknowledgmentsIt is my great pleasure to thank all persons whose contribution made this work pos-sible and those who assisted me during the past years:

I thank Prof. Ulrike Wallrabe for accepting and supervising this thesis. Sheinitiated the project and gave me the freedom and trust to pursue all the challengingtasks. The scientific and strategic advice over the years and the direct contributionsto this work are highly valued.

I also want to thank Prof. Ulrich Schwarz for accepting to co-referee this thesisand for the interesting discussions we had over the years.

Dr. Matthias Wapler has been my adviser and office mate ever since I started atthe lab and has always pushed me with new and exiting ideas and ways to tacklea problem. He contributed to many aspects of this work, from the overall scientificperspective to the most detailed discussions, for which I am very grateful.

I thank all members of the Laboratory for Microactuators for their support andhelp completing this thesis. Especially, Dr. Jens Brunne who helped me in thebeginning of my PhD thesis and Dr. Moritz Stürmer, my long-time office matewho supported me in many topics. I want to thank: Dr. Ali Moazenzadeh,for the nice discussions we had about my work during the years and Dr. MikelGorostiaga who gave me helpful tips with the CNC-mill. I thank my colleagues:Dr. Roland Lausecker, Dr. Fralett Suárez Sandoval, Binal Bruno, FlorianLemke, Hitesh Gowda and Bibhu Kar, the other staff members, especiallyKatja Ickler, and the students at the Laboratory for Microactuators.

I would like to thank the RSC and ESC at IMTEK for their support and helpfuldiscussions. Further, I thank our project partners, especially Prof. Oliver Am-bacher, Markus Reisacher, Dr. Samora Okujeni, and Prof. Ulrich Egert,and whoever I shared my thoughts with or who I may have forgotten.

I wish to thank Thorbjörn Jörger for his long-time friendship, advice and supportsince the first semester. His help in any electrical engineering related question ismuch appreciated.

They were not immediately involved in this work, but I thank my Mädels-groupand all of my friends, who supported me in many different ways.

127

A Appendix

I am most thankful to my family, my parents and my sister, who made it possiblefor me to come so far and have always supported me in every step of the way.

Finally, I thank Dr. Andreas Weltin for all imaginable help over the years duringmy PhD in the most diverse topics. In the many discussions we had, he helped meto improve my skills by challenging me to be more critical and explore a topic fromdifferent angles. I am very grateful for his diligent proof-reading of this work.

Funding for parts of this work by the German Research Foundation (DFG) within theBrainLinks-BrainTools Cluster of Excellence (EXC 1086) is gratefully acknowledged.

128

Author’s Bibliography

Journal articles

[A1] A. Müller, M.C. Wapler, U.T. Schwarz, M. Reisacher, K. Holc, O.Ambacher and U. Wallrabe, “Quasi-Bessel beams from asymmetric andastigmatic illumination sources”, Optics Express, vol. 24, no. 15, pp.17433–17452, 2016, DOI: 10.1364/OE.24.017433.

[A2] A. Müller, M.C. Wapler and U. Wallrabe, “Segmented Bessel beams”,Optics Express, vol. 25, no. 19, pp. 22640–22647, 2017,DOI: 10.1364/OE.25.022640.

[A3] A. Müller, M.C. Wapler and U. Wallrabe, “A quick and accuratemethod to determine the Poisson’s ratio and the coefficient of thermalexpansion of PDMS”, Soft Matter, 2019, vol. 15, pp. 779–784DOI: 10.1039/C8SM02105H.

[A4] S. Bär, T. Oerther, M. Weigel, A. Müller, P. Hucker, J.G. Korvink, C.Ko, M.C. Wapler, and J. Leupold, “On the Application of BalancedSteady-State Free Precession to MR Microscopy”, Magnetic ResonanceMaterials in Physics, Biology and Medicine, 2019,DOI: 10.1007/s10334-019-00736-4.

Own conference contributions

[A5] A. Müller, M.C. Wapler and U. Wallrabe, Depth controlled Besselbeam, International Conference on Optical MEMS and Nanophotonics(OMN), Singapore, Singapore, pp. 243–244, 2016,DOI: 10.1109/OMN.2016.7565918

[A6] A. Müller, M.C. Wapler, M. Reisacher, K. Holc, O. Ambacher and U.Wallrabe, Bessel beams for depth-controlled quasi-noninvasiveoptogenetic stimulation, 3rd International Workshop on Technologiesfor Optogenetics (OPTOGEN), Freiburg, Germany, 2016.

129

https://doi.org/10.1364/OE.24.017433

https://doi.org/10.1364/OE.25.022640

https://doi.org/10.1039/C8SM02105H

https://doi.org/10.1007/s10334-019-00736-4

https://doi.org/10.1109/OMN.2016.7565918

A Appendix

[A7] A. Müller, M.C. Wapler, M. Reisacher, O. Ambacher and U. Wallrabe,Tiefenkontrollierbare Bessel-Strahlen für quasi-nichtinvasiveoptogenetische Stimulation, 118. Jahrestagung der DeutschenGesellschaft für angewandte Optik, Dresden, Germany, 2017.

[A8] A. Müller, M.C. Wapler, P. Vaity, M. Reisacher, O. Ambacher, S.Okujeni, U. Egert, M. Bartos, I. Diester, and U. Wallrabe,Non-diffracting light beams for optogenetics, BLBT InternationalConference, Freiburg, Germany, 2017.

[A9] A. Müller, M.C. Wapler and U. Wallrabe, Steuerbare Ringblenden fürsegmentierte Besselstrahlen, MikroSystemTechnik Kongress (MSTKongress), München, Germany, 2017.

Conference contributions as co-author

[A10] K. Holc, A. Jakob, T. Weig, K. Köhler, U.T. Schwarz, A. Müller, M.Pauls, M. Wapler, U. Wallrabe and O. Ambacher, New tools foroptogenetics: nitride laser diodes combined with axicons fornon-invasive neuronal stimulation, Proceedings of the InternationalWorkshop on Nitride Semiconductors, Wroclaw, Poland, 2014.

[A11] U. Wallrabe, A. Müller, M. Reisacher, O. Ambacher, K. Holc, and M.C.Wapler, Controlling Bessel beams for optophysiology, Dreiländertagung,Biomedizinische Technik BMT, Basel, Swiss, 2016.

[A12] Sebastien Bär, Thomas Oerther, Angelina Müller, Matthias Weigel,Matthias Wapler and Jochen Leupold, Ingredients for balanced SSFPMicroimaging, International Society for Magnetic Resonance inMedicine (ISMRM), Honolulu, Hawaii, USA, 2017.

[A13] U. Wallrabe, J. Brunne, A. Müller, F. Lemke, B. Bruno, M. Stürmerand M.C. Wapler, Nicht-sphärische adaptive Optik für dieLebenswissenschaften, 118. Jahrestagung der Deutschen Gesellschaftfür angewandte Optik, Dresden, Germany, 2017.

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References[1] S. Sinzinger and J. Jahns, Microoptics. Weinheim: Wiley-VCH, 1999.

[2] “Method of the Year 2018: Imaging in freely behaving animals,” NatureMethods, vol. 16, no. 1, 2019.

[3] W. Deng, E. M. Goldys, M. M. Farnham, and P. M. Pilowsky,“Optogenetics, the intersection between physics and neuroscience: lightstimulation of neurons in physiological conditions,” American Journal ofPhysiology-Regulatory, Integrative and Comparative Physiology, vol. 307,no. 11, pp. R1292–R1302, 2014.

[4] F. Zhang, V. Gradinaru, A. R. Adamantidis, R. Durand, R. D. Airan,L. de Lecea, and K. Deisseroth, “Optogenetic interrogation of neural circuits:technology for probing mammalian brain structures,” Nature Protocols,vol. 5, no. 3, pp. 439–456, 2010.

[5] M. T. Alt, E. Fiedler, L. Rudmann, J. S. Ordonez, P. Ruther, andT. Stieglitz, “Let There Be Light-Optoprobes for Neural Implants,”Proceedings of the IEEE, vol. 105, no. 1, pp. 101–138, 2017.

[6] J. Williams, A. Watson, A. L. Vazquez, and A. B. Schwartz, “Viral-mediatedoptogenetic stimulation of peripheral motor nerves in non-human primates,”bioRxiv, 2018.

[7] F. O. Fahrbach and A. Rohrbach, “Propagation stability ofself-reconstructing Bessel beams enables contrast-enhanced imaging in thickmedia,” Nature Communications, vol. 3, p. 632, 2012.

[8] Polytec, “TopCCS PRIMA: Datasheet.” URL:https : //www.polytec.com/fileadmin/d/Oberfl%C3%A4chenmesstechnik/OM_DS_TopSens−Series_E_42364.pdf, accessed on 07/19/2018.

[9] V. Lakshminarayanan and A. Fleck, “Zernike polynomials: A guide,”Journal of Modern Optics, vol. 58, no. 7, pp. 545–561, 2011.

[10] R. Gray and J. Howard, “ZernikeCalc v1.1.”URL:https : //de.mathworks.com/matlabcentral/fileexchange/33330−zernikecalc,accessed on 10/30/2018.

131

https://www.polytec.com/fileadmin/d/Oberfl%C3%A4chenmesstechnik/OM_DS_TopSens-Series_E_42364.pdf

https://www.polytec.com/fileadmin/d/Oberfl%C3%A4chenmesstechnik/OM_DS_TopSens-Series_E_42364.pdf

https://de.mathworks.com/matlabcentral/fileexchange/33330-zernikecalc

References

[11] J. Brunne, Adaptive Axiconspiegel. PhD thesis, University of Freiburg,Freiburg, 2014.

[12] A. E. Siegman, “Quasi fast Hankel transform,” Optics Letters, vol. 1, no. 1,pp. 13–15, 1977.

[13] A. Müller, M. C. Wapler, U. T. Schwarz, M. Reisacher, K. Holc,O. Ambacher, and U. Wallrabe, “Quasi-Bessel beams from asymmetric andastigmatic illumination sources,” Optics Express, vol. 24, no. 15,pp. 17433–17452, 2016.

[14] A. Müller, M. C. Wapler, and U. Wallrabe, “Segmented Bessel beams,”Optics Express, vol. 25, no. 19, pp. 22640–22647, 2017.

[15] J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” PhysicalReview Letters, vol. 58, no. 15, pp. 1499–1501, 1987.

[16] J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,”Journal of the Optical Society of America A, vol. 4, no. 4, pp. 651–654, 1987.

[17] J. Durnin, J. J. Miceli, J. H. Eberly, and J. J. Miceli, “Comparison of Besseland Gaussian beams,” Optics Letters, vol. 13, no. 2, pp. 79–80, 1988.

[18] S. Ruschin, “Modified Bessel nondiffracting beams,” Journal of the OpticalSociety of America A, vol. 11, no. 12, pp. 3224–3228, 1994.

[19] S. Chávez-Cerda, G. Mcdonald, and G. New, “Nondiffracting beams:travelling, standing, rotating and spiral waves,” Optics Communications,vol. 123, no. 1-3, pp. 225–233, 1996.

[20] J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use ofan axicon,” Optics Communications, vol. 177, no. 1-6, pp. 297–301, 2000.

[21] Z. Bouchal, “Nondiffracting optical beams: Physical properties, experiments,and applications,” Czechoslovak Journal of Physics, vol. 53, no. 7,pp. 537–578, 2003.

[22] F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” OpticsCommunications, vol. 64, no. 6, pp. 491–495, 1987.

[23] P. L. Overfelt and C. S. Kenney, “Comparison of the propagationcharacteristics of Bessel, Bessel–Gauss, and Gaussian beams diffracted by acircular aperture,” Journal of the Optical Society of America A, vol. 8, no. 5,pp. 732–745, 1991.

132

References

[24] R. Grunwald, S. Huferath, M. Bock, U. Neumann, and S. Langer, “Angulartolerance of Shack-Hartmann wavefront sensors with microaxicons.,” OpticsLetters, vol. 32, no. 11, pp. 1533–1535, 2007.

[25] Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distortednondiffracting beam,” Optics Communications, vol. 151, no. 4-6,pp. 207–211, 1998.

[26] D. Ding and X. Liu, “Approximate description for Bessel, Bessel–Gauss, andGaussian beams with finite aperture,” Journal of the Optical Society ofAmerica A, vol. 16, no. 6, pp. 1286–1293, 1999.

[27] D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,”Contemporary Physics, vol. 46, no. 1, pp. 15–28, 2005.

[28] T. Wulle and S. Herminghaus, “Nonlinear Optics of Bessel Beams,” PhysicalReview Letters, vol. 70, no. 10, pp. 1401–1404, 1993.

[29] L. Niggl and M. Maier, “Efficient conical emission of stimulated RamanStokes light generated by a Bessel pump beam,” Optics Letters, vol. 22,no. 12, pp. 910–912, 1997.

[30] S. Sogomonian, U. T. Schwarz, and M. Maier, “Phase-front transformation ofa first-order Bessel beam in Raman-resonant four-wave mixing,” Journal ofthe Optical Society of America B, vol. 18, no. 4, pp. 497–504, 2001.

[31] F. Courvoisier, M. K. Bhuyan, P. A. Lacourt, M. Jacquot, L. Furfaro, andJ. M. Dudley, “Surface nanoprocessing with nondiffracting femtosecond,”Optics Letters, vol. 34, no. 20, pp. 3163–3165, 2009.

[32] M. Duocastella and C. B. Arnold, “Bessel and annular beams for materialsprocessing,” Laser and Photonics Reviews, vol. 6, no. 5, pp. 607–621, 2012.

[33] J. Dudutis, P. Gečys, and G. Račiukaitis, “Non-ideal axicon-generated Besselbeam application for intra-volume glass modification,” Optics Express,vol. 24, no. 25, pp. 28433–28443, 2016.

[34] L. Rapp, R. Meyer, L. Furfaro, C. Billet, R. Giust, and F. Courvoisier, “Highspeed cleaving of crystals with ultrafast Bessel beams,” Optics Express,vol. 25, no. 8, p. 9312, 2017.

[35] J. Dudutis, R. Stonys, G. Račiukaitis, and P. Gečys, “Aberration-controlledBessel beam processing of glass,” Optics Express, vol. 26, no. 3,pp. 3627–3637, 2018.

133

References

[36] B. Bhuian, R. Winfield, S. O’Brien, and G. Crean, “Pattern generation usingaxicon lens beam shaping in two-photon polymerisation,” Applied SurfaceScience, vol. 254, no. 4, pp. 841–844, 2007.

[37] Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolutionoptical coherence tomography over a large depth range with an axicon lens,”Optics Letters, vol. 27, no. 4, pp. 243–245, 2002.

[38] K.-S. Lee and J. P. Rolland, “Bessel beam spectral-domain high-resolutionoptical coherence tomography with micro-optic axicon providing extendedfocusing range,” Optics Letters, vol. 33, no. 15, pp. 1696–1698, 2008.

[39] N. Weber, D. Spether, A. Seifert, and H. Zappe, “Highly compact imagingusing Bessel beams generated by ultraminiaturized multi-micro-axiconsystems,” Journal of the Optical Society of America A, vol. 29, no. 5,pp. 808–816, 2012.

[40] A. Curatolo, P. R. T. Munro, D. Lorenser, P. Sreekumar, C. C. Singe, B. F.Kennedy, and D. D. Sampson, “Quantifying the influence of Bessel beams onimage quality in optical coherence tomography,” Scientific Reports, vol. 6,p. 23483, 2016.

[41] V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia,“Simultaneous micromanipulation in multiple planes using aself-reconstructing light beam,” Nature, vol. 419, pp. 145–147, 2002.

[42] F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy withself-reconstructing beams,” Nature Photonics, vol. 4, no. 11, pp. 780–785,2010.

[43] L. Niggl, T. Lanzl, and M. Maier, “Properties of Bessel beams generated byperiodic gratings of circular symmetry,” Journal of the Optical Society ofAmerica A, vol. 14, no. 1, pp. 27–33, 1997.

[44] N. Chattrapiban, E. A. Rogers, D. Cofield, W. T. Hill, III, and R. Roy,“Generation of nondiffracting Bessel beams by use of a spatial lightmodulator,” Optics Letters, vol. 28, no. 22, pp. 2183–2185, 2003.

[45] J. H. McLeod, “The Axicon: A New Type of Optical Element,” Journal ofthe Optical Society of America, vol. 44, no. 8, pp. 592–597, 1954.

[46] J. H. McLeod, “Axicons and Their Uses,” Journal of the Optical Society ofAmerica, vol. 50, no. 2, pp. 166–168, 1960.

134

References

[47] V. V. Korobkin, L. Polonski, V. P. Poponin, and L. N. Pyatnitski, “Focusingof Gaussian and super-Gaussian laser beams by axicons to obtain continuouslaser sparks,” Soviet Journal of Quantum Electronics, vol. 16, no. 2,pp. 178–182, 1986.

[48] R. M. Herman and T. A. Wiggins, “Production and Uses of DiffractionlessBeams,” Journal of the Optical Society of America A, vol. 8, no. 6,pp. 932–942, 1991.

[49] O. Brzobohatý, T. Cizmár, and P. Zemánek, “High quality quasi-Besselbeam generated by round-tip axicon,” Optics Express, vol. 16, no. 17,pp. 12688–12700, 2008.

[50] J. Brunne and U. Wallrabe, “Tunable MEMS axicon mirror arrays.,” OpticsLetters, vol. 38, no. 11, pp. 1939–1941, 2013.

[51] J. Brunne, M. C. Wapler, and U. Wallrabe, “Fast and robust piezoelectricaxicon mirror,” Optics Letters, vol. 39, no. 15, pp. 4631–4634, 2014.

[52] X. Yu, A. Todi, and H. Tang, “Bessel beam generation using a segmenteddeformable mirror,” Applied Optics, vol. 57, no. 16, pp. 4677–4682, 2018.

[53] R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. Nibbering, T. Elsaesser,V. Kebbel, H. J. Hartmann, and W. Jüptner, “Generation of femtosecondBessel beams with microaxicon arrays.,” Optics Letters, vol. 25, no. 13,pp. 981–983, 2000.

[54] P. Wu, C. Sui, and W. Huang, “Theoretical analysis of a quasi-Bessel beamfor laser ablation,” Photonics Research, vol. 2, no. 3, pp. 82–86, 2014.

[55] T. Cizmár, Optical traps generated by non-traditional beams. PhD thesis,Masaryk University, Brno, 2006.

[56] J. W. Goodman, Introduction to Fourier Optics. New York: McGraw-Hill,second ed., 1996.

[57] K. Holc, A. Jakob, T. Weig, K. Köhler, O. Ambacher, and U. T. Schwarz,“Influence of surface roughness on the optical mode profile in GaN-basedviolet ridge waveguide laser diodes,” Proceedings of SPIE, vol. 9002,p. 90020I, 2014.

[58] J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda,“Alternative formulation for invariant optical fields: Mathieu beams,” OpticsLetters, vol. 25, no. 20, pp. 1493–1495, 2000.

135

References

[59] J. Gutiérrez-Vega, M. Iturbe-Castillo, G. Ramírez, E. Tepichín,R. Rodrıíguez-Dagnino, S. Chávez-Cerda, and G. New, “Experimentaldemonstration of optical Mathieu beams,” Optics Communications, vol. 195,no. 1-4, pp. 35–40, 2001.

[60] H. Sun, “Measurement of laser diode astigmatism,” Optical Engineering,vol. 36, no. 4, pp. 1082–1087, 1997.

[61] R. R. Syms, H. Zou, J. Stagg, and H. Veladi, “Sliding-blade MEMS iris andvariable optical attenuator,” Journal of Micromechanics andMicroengineering, vol. 14, no. 12, pp. 1700–1710, 2004.

[62] C.-H. Kim, K.-D. Jung, and W. Kim, “A Wafer-Level Micro MechanicalGlobal Shutter for a Micro Camera,” 2009 IEEE 22nd InternationalConference on Micro Electro Mechanical Systems, pp. 156–159, 2009.

[63] G. Zhou, H. Yu, Y. Du, and F. S. Chau,“Microelectromechanical-systems-driven two-layer rotary-blade-basedadjustable iris diaphragm,” Optics Letters, vol. 37, no. 10, pp. 1745–1747,2012.

[64] C. G. Tsai and J. A. Yeh, “Circular dielectric liquid iris,” Optics Letters,vol. 35, no. 14, pp. 2484–2486, 2010.

[65] P. Müller, R. Feuerstein, and H. Zappe, “Integrated optofluidic iris,” Journalof Microelectromechanical Systems, vol. 21, no. 5, pp. 1156–1164, 2012.

[66] L. Li, C. Liu, H. Ren, and Q.-H. Wang, “Adaptive liquid iris based onelectrowetting.,” Optics Letters, vol. 38, no. 13, pp. 2336–2338, 2013.

[67] M. Xu, H. Ren, and Y.-H. Lin, “Electrically actuated liquid iris,” OpticsLetters, vol. 40, no. 5, pp. 831–834, 2015.

[68] S. Schuhladen, K. Banerjee, M. Stürmer, P. Müller, U. Wallrabe, andH. Zappe, “Variable optofluidic slit aperture,” Light: Science & Applications,vol. 5, no. 1, p. e16005, 2016.

[69] J. Draheim, Minimalistic adaptive lenses. PhD thesis, University of Freiburg,2011.

[70] J. H. Kang, S. M. Paek, S. J. Hwang, and J. H. Choy, “Optical irisapplication of electrochromic thin films,” Electrochemistry Communications,vol. 10, no. 11, pp. 1785–1787, 2008.

136

References

[71] T. Deutschmann and E. Oesterschulze, “Integrated electrochromic iris devicefor low power and space-limited applications,” Journal of Optics, vol. 16,no. 7, pp. 1–5, 2014.

[72] S. Schuhladen, F. Preller, R. Rix, S. Petsch, R. Zentel, and H. Zappe,“Iris-Like Tunable Aperture Employing Liquid-Crystal Elastomers,”Advanced Materials, vol. 26, no. 42, pp. 7247–7251, 2014.

[73] D. Wang, C. Liu, L. Li, X. Zhou, and Q.-H. Wang, “Adjustable liquidaperture to eliminate undesirable light in holographic projection,” OpticsExpress, vol. 24, no. 3, pp. 2098–2105, 2016.

[74] S. Mias and H. Camon, “A review of active optical devices: I. Amplitudemodulation,” Journal of Micromechanics and Microengineering, vol. 18,no. 8, p. 083001, 2008.

[75] R. J. Mortimer, “Electrochromic materials,” Annual Review of MaterialsResearch, vol. 41, no. 1, pp. 241–268, 2011.

[76] D. Zhou, D. Xie, X. Xia, X. Wang, C. Gu, and J. Tu, “All-solid-stateelectrochromic devices based on WO3NiO films: material developments andfuture applications,” Science China Chemistry, vol. 60, no. 1, pp. 3–12, 2017.

[77] F. Mugele and J.-C. Baret, “Electrowetting: from basics to applications,”Journal of Physics: Condensed Matter, vol. 17, no. 28, pp. R705–R774, 2005.

[78] S. Rong, Y. Li, and L. Li, “Electrowetting optical switch with circulartunable aperture,” Optical Engineering, vol. 57, no. 8, p. 087102, 2018.

[79] R. Shamai, D. Andelman, B. Berge, and R. Hayes, “Water, electricity, andbetween... On electrowetting and its applications,” Soft Matter, vol. 4,pp. 38–45, 2007.

[80] H.-C. Chiang, Y.-H. Tsai, Y.-J. Yan, T.-W. Huang, and O.-Y. Mang, “Oildefect detection of electrowetting display,” Proceedings of SPIE, OpticalManufacturing and Testing XI, vol. 9575, p. 957514, 2015.

[81] Y. Xie, M. Sun, M. Jin, G. Zhou, and L. Shui, “Two-phase microfluidic flowmodeling in an electrowetting display microwell,” The European PhysicalJournal E, vol. 39, no. 16, pp. 1–5, 2016.

[82] J. Lee, Y. Park, and S. K. Chung, “Switchable liquid shutter for security anddesign of mobile electronic devices,” 2018 IEEE Micro Electro MechanicalSystems (MEMS), pp. 14–16, 2018.

137

References

[83] G. H. Heilmeier, “Liquid-Crystal Display Devices,” Scientific American,vol. 222, no. 4, pp. 100–107, 1970.

[84] G. W. Taylor, “Introduction to liquid crystals,” Ferroelectrics, vol. 73, no. 1,pp. 265–265, 1987.

[85] E. R. Waclawik, M. J. Ford, P. S. Hale, J. G. Shapter, and N. H. Voelcker,“Liquid-Crystal Displays: Fabrication and Measurement of a TwistedNematic Liquid-Crystal Cell,” vol. 81, no. 6, 2004.

[86] S. Ahderom, M. Raisi, K. Lo, K. Alameh, and R. Mavaddat, “Applications ofliquid crystal spatial light modulators in optical communications,” 2002IEEE 5th International Conference on High Speed Networks and MultimediaCommunication, pp. 239–242, 2002.

[87] G. D. Love, “Wave-front correction and production of Zernike modes with aliquid-crystal spatial light modulator,” Applied Optics, vol. 36, no. 7, p. 1517,1997.

[88] D. Meschede, Optics, Light and Lasers. Weinheim: Wiley-VCH, 2004.

[89] J. Leach, G. M. Gibson, M. J. Padgett, E. Esposito, G. McConnell, A. J.Wright, and J. M. Girkin, “Generation of achromatic Bessel beams using acompensated spatial light modulator,” Optics Express, vol. 14, no. 12,pp. 5581–5587, 2006.

[90] V. Nikolenko, B. O. Watson, R. Araya, A. Woodruff, D. S. Peterka, andR. Yuste, “SLM microscopy: scanless two-photon imaging andphotostimulation using spatial light modulators,” Frontiers in NeuralCircuits, vol. 2, no. 5, pp. 1–14, 2008.

[91] J. L. Martinez, E. J. Fernandez, P. M. Prieto, and P. Artal, “Chromaticaberration control with liquid crystal spatial phase modulators,” OpticsExpress, vol. 25, no. 9, pp. 9793–9801, 2017.

[92] Z. Yao, L. Jiang, X. Li, A. Wang, Z. Wang, M. Li, and Y. Lu,“Non-diffraction-length, tunable, Bessel-like beams generation by spatiallyshaping a femtosecond laser beam for high-aspect-ratio micro-hole drilling,”Optics Express, vol. 26, no. 17, pp. 21960–21967, 2018.

[93] A. Müller, M. C. Wapler, and U. Wallrabe, “Depth-controlled bessel beams,”Proceedings International Conference on Optical Mems and Nanophotonics(OMN), 2016.

[94] B. Bahadur, ed., Liquid Crystals-Applications and Uses, vol. 1. Singapore:World Scientific, 1990.

138

References

[95] H. Ren and S.-T. Wu, eds., Introduction to Adaptive Lenses. Weinheim:Wiley VCH, 2012.

[96] F. C. Frank, “I. Liquid crystals. On the theory of liquid crystals,”Discussions of the Faraday Society, vol. 25, pp. 19–28, 1958.

[97] M. Schadt and W. Helfrich, “Voltage-dependent optical activity of a twistednematic liquid crystal,” Applied Physics Letters, vol. 18, no. 4, pp. 127–128,1971.

[98] M. Schadt, “Liquid crystal materials and liquid crystal displays,” AnnualReview of Materials Science, vol. 27, pp. 305–379, 1997.

[99] G. W. Gray and S. M. Kelly, “Liquid crystals for twisted nematic displaydevices,” Journal of Materials Chemistry, vol. 9, no. 9, pp. 2037–2050, 1999.

[100] Micro Resist Technology, “Ormocomp Processing Guidelines.” URL:https : //www.microresist.de/sites/default/files/download/flyer_hybrid_polymers_July2015_0.pdf, accessed on 06/10/2016.

[101] Elga Europe, “Ordyl SY330 Product Data Sheet.” URL:http : //www.elgaeurope.it/user/download_ctg.aspx?TIPO = F&FILE =OBJ00078.PDF&NOME = Product + Data + Sheet_SY300.pdf, accessed on08/15/2016.

[102] Edmund Optics GmbH, “Manufacturing methods of aspheres.” URL:https://www.edmundoptics.com/resources/application-notes/optics/all-about-aspheric-lenses/, accessed on 30/10/2018.

[103] Thorlabs GmbH, “Technical Data Sheet Axicons.”URL:https : //www.thorlabs.com/catalogpages/V21/753.PDF, accessed on30/11/2018.

[104] H. Ottevaere, R. Cox, H. P. Herzig, T. Miyashita, K. Naessens,M. Taghizadeh, R. Völkel, H. J. Woo, and H. Thienpont, “Comparing glassand plastic refractive microlenses fabricated with different technologies,”Journal of Optics A: Pure and Applied Optics, vol. 8, no. 7, pp. 407–429,2006.

[105] T. Hou, C. Zheng, S. Bai, Q. Ma, D. Bridges, A. Hu, and W. W. Duley,“Fabrication, characterization, and applications of microlenses,” AppliedOptics, vol. 54, no. 24, pp. 7366–7376, 2015.

[106] H.-C. C. Lin, M.-S. S. Chen, and Y.-H. H. Lin, “A review of electricallytunable focusing liquid crystal lenses,” Transactions on Electrical andElectronic Materials, vol. 12, no. 6, pp. 234–240, 2011.

139

https://www.microresist.de/sites/default/files/download/flyer_hybrid_polymers_July2015_0.pdf

https://www.microresist.de/sites/default/files/download/flyer_hybrid_polymers_July2015_0.pdf

http://www.elgaeurope.it/user/download_ctg.aspx?TIPO=F&FILE=OBJ00078.PDF&NOME=Product+Data+Sheet_SY300.pdf

http://www.elgaeurope.it/user/download_ctg.aspx?TIPO=F&FILE=OBJ00078.PDF&NOME=Product+Data+Sheet_SY300.pdf

https://www.edmundoptics.com/resources/application-notes/optics/all-about-aspheric-lenses/

https://www.edmundoptics.com/resources/application-notes/optics/all-about-aspheric-lenses/

https://www.thorlabs.com/catalogpages/V21/753.PDF

References

[107] X. J. Shen, L. W. Pan, and L. Lin, “Microplastic embossing process:Experimental and theoretical characterizations,” Sensors and Actuators, A:Physical, vol. 97, no. 98, pp. 428–433, 2002.

[108] B. P. Keyworth, D. J. Corazza, J. N. McMullin, and L. Mabbott,“Single-step fabrication of refractive microlens arrays,” Applied Optics,vol. 36, no. 10, pp. 2198–2202, 1997.

[109] F. Schneider, J. Draheim, C. Müller, and U. Wallrabe, “Optimization of anadaptive PDMS-membrane lens with an integrated actuator,” Sensors andActuators A: Physical, vol. 154, no. 2, pp. 316–321, 2009.

[110] F. Krogmann, W. Mönch, and H. Zappe, “A MEMS-based variablemicro-lens system,” Journal of Optics A: Pure and Applied Optics, vol. 8,no. 7, 2006.

[111] D. Kopp and H. Zappe, “Tubular astigmatism-tunable fluidic lens,” OpticsLetters, vol. 41, no. 12, p. 2735, 2016.

[112] J. Baehr and K.-H. Brenner, “Applications and potential of the maskstructured ion exchange technique (MSI) in micro-optics,” in Gradient Index,Miniature, and Diffractive Optical Systems III, p. 121, 2003.

[113] G. Du, Q. Yang, F. Chen, H. Liu, Z. Deng, H. Bian, S. He, J. Si, X. Meng,and X. Hou, “Direct fabrication of seamless roller molds with gapless andshaped-controlled concave microlens arrays,” Optics Letters, vol. 37, no. 21,pp. 4404–4406, 2012.

[114] N. F. Borrelli, D. L. Morse, R. H. Bellman, and W. L. Morgan, “Photolytictechnique for producing microlenses in photosensitive glass,” Applied Optics,vol. 24, no. 16, p. 2520, 1985.

[115] R. Kirchner and H. Schift, “Thermal reflow of polymers for innovative andsmart 3D structures: A review,” Materials Science in SemiconductorProcessing, 2018.

[116] J. Yao, Z. Cui, F. Gao, Y. Zhang, Y. Guo, C. Du, H. Zeng, and C. Qiu,“Refractive micro lens array made of dichromate gelatin with gray-tonephotolithography,” Microelectronic Engineering, vol. 57-58, pp. 729–735,2001.

[117] S. Mihailov and S. Lazare, “Fabrication of refractive microlens arrays byexcimer laser ablation of amorphous Teflon,” Applied Optics, vol. 32, no. 31,pp. 6211–6218, 1993.

140

References

[118] Y. Hu, Y. Xiong, X. Chen, H. Bai, Y. Tian, and G. Liu, “Controllable longfocal length microlens based on thermal expansion,” Applied Optics, vol. 57,no. 15, 2018.

[119] M. K. Bhuyan, O. Jedrkiewicz, V. Sabonis, M. Mikutis, S. Recchia, A. Aprea,M. Bollani, and P. D. Trapani, “High-speed laser-assisted cutting of strongtransparent materials using picosecond Bessel beams,” Applied Physics A:Materials Science and Processing, vol. 120, no. 2, pp. 443–446, 2015.

[120] J. Dudutis, R. Stonys, G. Račiukaitis, and P. Gečys, “Glass dicing withelliptical Bessel beam,” Optics & Laser Technology, vol. 111, pp. 331–337,2019.

[121] R. A. Leitgeb, M. Villiger, A. H. Bachmann, L. Steinmann, and T. Lasser,“Extended focus depth for Fourier domain optical coherence microscopy,”Optics Letters, vol. 31, no. 16, p. 2450, 2006.

[122] P. Zhang, D. J. Wahl, J. Mocci, S. K. Manna, R. K. Meleppat, S. Bonora,M. V. Sarunic, E. N. Pugh, and R. J. Zawadzki, “Adaptive optics withcombined optical coherence tomography and scanning laser ophthalmoscopyfor in vivo mouse retina imaging,” in Ophthalmic Technologies XXVIII,vol. 10474, p. 79, SPIE, 2018.

[123] X. Tsampoula, V. Garćs-Chávez, M. Comrie, D. J. Stevenson, B. Agate,C. T. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellulartransfection using a nondiffracting light beam,” Applied Physics Letters,vol. 91, no. 2007, pp. 5–8, 2007.

[124] W. C. Cheong, B. P. Ahluwalia, X. C. Yuan, L. S. Zhang, H. Wang, H. B.Niu, and X. Peng, “Fabrication of efficient microaxicon by directelectron-beam lithography for long nondiffracting distance of Bessel beamsfor optical manipulation,” Applied Physics Letters, vol. 87, no. 2, pp. 10–13,2005.

[125] ALTECHNA, “Product description of plano-convex-axicons.” URL:https://www.altechna.com/products/plano-convex-axicons/, accessed on30/10/2018.

[126] J. Algorri, V. Urruchi, B. García-Cámara, and J. Sánchez-Pena, “LiquidCrystal Lensacons, Logarithmic and Linear Axicons,” Materials, vol. 7, no. 4,pp. 2593–2604, 2014.

[127] X. F. Lin, Q. D. Chen, L. G. Niu, T. Jiang, W. Q. Wang, and H. B. Sun,“Mask-free production of integratable monolithic micro logarithmic axiconlenses,” Journal of Lightwave Technology, vol. 28, no. 8, pp. 1256–1260, 2010.

141

https://www.altechna.com/products/plano-convex-axicons/

References

[128] A. Žukauskas, M. Malinauskas, C. Reinhardt, B. N. Chichkov, andR. Gadonas, “Closely packed hexagonal conical microlens array fabricated bydirect laser photopolymerization,” Applied Optics, vol. 51, no. 21, p. 4995,2012.

[129] S. Schwarz, S. Rung, C. Esen, and R. Hellmann, “Fabrication of ahigh-quality axicon by femtosecond laser ablation and CO2 laser polishingfor quasi-Bessel beam generation,” Optics Express, vol. 26, no. 18,pp. 23287–23294, 2018.

[130] G. Milne, G. D. M. Jeffries, and D. T. Chiu, “Tunable generation of Besselbeams with a fluidic axicon,” Applied Physics Letters, vol. 92, no. 26,p. 261101, 2008.

[131] A. K. Kirby, P. J. Hands, and G. D. Love, “Liquid crystal multi-mode lensesand axicons based on electronic phase shift control,” Optics Express, vol. 15,no. 21, pp. 13496–13501, 2007.

[132] J. F. Algorri, V. Urruchi, N. Bennis, and J. M. Sánchez-Pena, “Modal liquidcrystal microaxicon array,” Optics Letters, vol. 39, no. 12, p. 3476, 2014.

[133] Z. Jaroszewicz, J. Sochacki, A. Kolodziejczyk, and L. R. Staronski,“Apodized annular-aperture logarithmic axicon: smoothness and uniformityof intensity distributions,” Optics Letters, vol. 18, no. 22, p. 1893, 1993.

[134] Z. Cao, K. Wang, and Q. Wu, “Logarithmic axicon characterized byscanning optical probe system,” Optics Letters, vol. 38, no. 10, p. 1603, 2013.

[135] N. Davidson, A. A. Friesem, and E. Hasman, “Holographic axilens: highresolution and long focal depth,” Optics Letters, vol. 16, no. 7, pp. 523–525,1991.

[136] V. P. Koronkevich, I. A. Mikhaltsova, E. G. Churin, and Y. I. Yurlov,“Lensacon,” Applied Optics, vol. 34, no. 25, pp. 5761–5772, 1995.

[137] C. Parigger, Y. Tang, D. H. Plemmons, and J. W. L. Lewis, “Sphericalaberration effects in lens–axicon doublets: theoretical study,” Applied Optics,vol. 36, no. 31, pp. 8214–8221, 1997.

[138] A. Burvall, K. Kołacz, Z. Jaroszewicz, and A. T. Friberg, “Simple lensaxicon,” Applied Optics, vol. 43, no. 25, p. 4838, 2004.

[139] U. Wallrabe, J. Brunne, M. Pauls, and R. Grunwald, “Optisches element undherstellung desselben.” DE102012219655A1, Deutschland, 2012.

142

References

[140] L. D. Landau and E. M. Lifschitz, Theory of Elasticity, vol. 7. Oxford:Butterworth-Heinemann, 1999.

[141] E. Hewitt, “Certain generalizations of the Weierstrass approximationtheorem,” Duke Mathematical Journal, vol. 14, no. 2, pp. 419–427, 1947.

[142] J. M. Ng, I. Gitlin, A. D. Stroock, and G. M. Whitesides, “Components forintegrated poly(dimethylsiloxane) microfluidic systems,” Electrophoresis,vol. 23, no. 20, pp. 3461–3473, 2002.

[143] F. Schneider, T. Fellner, J. Wilde, and U. Wallrabe, “Mechanical propertiesof silicones for MEMS,” Journal of Micromechanics and Microengineering,vol. 18, no. 6, pp. 1–9, 2008.

[144] G. Ouyang, K. Wang, L. Henriksen, M. Akram, and X. Chen, “A noveltunable grating fabricated with viscoelastic polymer (PDMS) and conductivepolymer (PEDOT),” Sensors and Actuators A: Physical, vol. 158, no. 2,pp. 313–319, 2010.

[145] J. Friend and L. Yeo, “Fabrication of microfluidic devices usingpolydimethylsiloxane,” Biomicrofluidics, vol. 4, no. 2, pp. 1–5, 2010.

[146] P. Du, I. K. Lin, H. Lu, and X. Zhang, “Extension of the beam theory forpolymer bio-transducers with low aspect ratios and viscoelasticcharacteristics,” Journal of Micromechanics and Microengineering, vol. 20,no. 9, pp. 1–13, 2010.

[147] B. Wang and S. Krause, “Properties of Dimethylsiloxane Microphases inPhase-Separated Dimethylsiloxane Block Copolymers,” Macromolecules,vol. 20, no. 9, pp. 2201–2208, 1987.

[148] K. Hosokawa, K. Hanada, and R. Maeda, “A polydimethylsiloxane (PDMS)deformable diffraction grating for monitoring of local pressure in microfluidicdevices,” Journal of Micromechanics and Microengineering, vol. 12, no. 1,pp. 1–6, 2002.

[149] P. A. Roman, Polydimethylsiloxane tensile mechanical properties andmembrane deflection theory Properties and Membrane Deflection Theory.PhD thesis, Oregon Health & Science University, 2004.

[150] V. Studer, G. Hang, A. Pandolfi, M. Ortiz, W. F. Anderson, and S. R.Quake, “Scaling properties of a low-actuation pressure microfluidic valve,”Journal of Applied Physics, vol. 95, no. 1, pp. 393–398, 2004.

143

References

[151] F. M. Sasoglu, A. J. Bohl, and B. E. Layton, “Design and microfabricationof a high-aspect-ratio PDMS microbeam array for parallel nanonewton forcemeasurement and protein printing,” Journal of Micromechanics andMicroengineering, vol. 17, pp. 623–632, 2007.

[152] D. W. Inglis, “A method for reducing pressure-induced deformation insilicone microfluidics,” Biomicrofluidics, vol. 4, no. 2, pp. 1–8, 2010.

[153] R. H. Pritchard, P. Lava, D. Debruyne, and E. M. Terentjev, “Precisedetermination of the Poisson ratio in soft materials with 2D digital imagecorrelation,” Soft Matter, vol. 9, no. 26, pp. 6011–6198, 2013.

[154] I. D. Johnston, D. K. McCluskey, C. K. L. Tan, and M. C. Tracey,“Mechanical characterization of bulk Sylgard 184 for microfluidics andmicroengineering,” Journal of Micromechanics and Microengineering, vol. 24,no. 3, pp. 1–7, 2014.

[155] A. Müller, M. C. Wapler, and U. Wallrabe, “A quick and accurate method todetermine the poisson’s ratio and the coefficient of thermal expansion ofpdms,” Soft Matter, vol. 15, pp. 779–784, 2019.

[156] Dow Chemical Company, “Technical Data Sheet Sylgard 182.” URL:https://consumer.dow.com/content/dam/dcc/documents/en-us/productdatasheet/11/11-12/11-1251-01-sylgard-182-silicone-elastomer.pdf,accessed on 09/10/2018.

[157] Norland Products, “NAO 61.”URL:https://www.norlandprod.com/adhesives/noa%2061.html, accessed on05/07/2017.

[158] F. Krogmann, W. Mönch, and H. Zappe, “Electrowetting for tunablemicrooptics,” Journal of Microelectromechanical Systems, vol. 17, no. 6,pp. 1501–1512, 2008.

[159] Thorlabs GmbH, “Aspheric Condenser Lenses.” URL:https : //www.thorlabs.com/newgrouppage9.cfm?objectgroup_id = 3835,accessed on 30/11/2018.

[160] S. N. Khonina, N. L. Kazanskii, A. V. Ustinov, and S. G. Volotovskii, “Thelensacon: nonparaxial effects,” Journal of Optical Technology, vol. 78, no. 11,p. 724, 2011.

[161] N. L. Kazanskiy and S. N. Khonina, “Nonparaxial effects in lensacon opticalsystems,” Optoelectronics, Instrumentation and Data Processing, vol. 53,no. 5, pp. 484–493, 2017.

144

https://consumer.dow.com/content/dam/dcc/documents/en-us/productdatasheet/11/11-12/11-1251-01-sylgard-182-silicone-elastomer.pdf

https://consumer.dow.com/content/dam/dcc/documents/en-us/productdatasheet/11/11-12/11-1251-01-sylgard-182-silicone-elastomer.pdf

https://www.norlandprod.com/adhesives/noa%2061.html

https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=3835

References

[162] A. V. Ustinov, “Action of a generalized parabolic (aspheric) lens in differentoptical models,” no. April 2017, p. 103420I, 2017.

[163] Dow Chemical Company, “Technical Data Sheet Sylgard 184.” URL:https://consumer.dow.com/content/dam/dcc/documents/en-us/productdatasheet/11/11-31/11-3184-sylgard-184-elastomer.pdf, accessed on09/10/2018.

145

https://consumer.dow.com/content/dam/dcc/documents/en-us/productdatasheet/11/11-31/11-3184-sylgard-184-elastomer.pdf

https://consumer.dow.com/content/dam/dcc/documents/en-us/productdatasheet/11/11-31/11-3184-sylgard-184-elastomer.pdf

Bessel beams have an extended focal zone and self-healing properties, which can be advantagous in many different fields, e.g., in material processing or in life science applications. This work describes the conceptualization, design, fabrication, and characterization of several micro-optical components to generate and control Bessel beams. Basic considerations including axicons with rounded tips, limitations regarding the aperture, and the evaluation of Bessel beams using asymmetric illumination sources were confirmed with beam propagation methods and measurements. Based on this knowledge different micro-optical components were developed, including an adaptive liquid crystal ring aperture array to segment the extended focal zone of Bessel beams for depth-control and several transmissive aspherical lens arrays: a collimation lens for diverging light sources, two different axicons to generate a Bessel beam, and a lensacon, a combination of a lens and an axicon, to reduce the number of optical surfaces. Finally, a compact micro-optical system with minimal alignment and assembly errors was demonstrated by successfully combining the developed micro-optical components.

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Angelina Müller



Angelina Müller

IMTEK, University of Freiburg

Microactuators – Design and Technology Vol. 17

Documents

Micro-optical components for depth-controlled Bessel beams