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Structural and Multidisciplinary Optimization (2018) 58:885–901https://doi.org/10.1007/s00158-018-1932-4
RESEARCH PAPER
Topology optimization of multicomponent optomechanical systemsfor improved optical performance
S. Koppen1 ·M. van der Kolk1 · F. C. M. van Kempen2 · J. de Vreugd2 ·M. Langelaar1
Received: 11 October 2017 / Revised: 26 January 2018 / Accepted: 29 January 2018 / Published online: 23 March 2018© The Author(s) 2018. This article is an open access publication
AbstractThe stringent and conflicting requirements imposed on optomechanical instruments and the ever-increasing need forhigher resolution and quality imagery demands a tightly integrated system design approach. Our aim is to drive thethermomechanical design of multiple components through the optical performance of the complete system. To this end,we propose a new method combining structural-thermal-optical performance analysis and topology optimization whiletaking into account both component and system level constraints. A 2D two-mirror example demonstrates that the proposedapproach is able to improve the system’s spot size error by 95% compared to uncoupled system optimization while satisfyingequivalent constraints.
Keywords Topology optimization · Multidisciplinary design optimization · System optimization · Opticalinstrumentation · Optomechanics · Thermoelasticity · Structural-thermal-optical-performance analysis
1 Introduction
1.1 Optomechanical instruments
Optomechanical instruments control light by means ofoptics and generally have to meet very stringent opti-cal, mechanical and thermal requirements (e.g. Chin 1964;
� S. [email protected]
M. van der [email protected]
F. C. M. van [email protected]
J. de [email protected]
1 Faculty of Mechanical, Maritime & Materials Engineering(3ME), Department of Precision &Microsystems Engineering(PME), Delft University of Technology (TU Delft), Mekelweg2, 2628 CD Delft, Netherlands
2 Department of Optomechatronics, Netherlands Organisationfor Applied Scientific Research (TNO), Stieltjesweg 1, 2628CK Delft, Netherlands
Yoder and Vukobratovich 2015). They are found in high-tech applications in disciplines such as (nano)metrology,astronomy, life sciences, aerospace, lithography and com-munications. A cutting-edge example is the Sentinel 5 UV1space-based spectrometer, shown in Fig. 1. The challenge ofoptomechanics lies in maintaining the position and shape ofoptical elements such that the image quality is guaranteedunder all working environments (e.g. Giesen and Folgering2003; Meijer et al. 2008; Pijnenburg et al. 2012). Therefore,optomechanical design is a tightly integrated process involv-ing many disciplines such as thermal control, structuralmechanics, motion control and optics.
The optical performance generally encompasses thesystems’ image quality, optical resolution, and imageposition accuracy. Both image quality and optical resolutionare typically quantified by the spot size or image blurdiameter (e.g. Welsh 1991; Doyle et al. 2012). Thespot size of an aberration-free system is limited by thewavelength of light. This diffraction limit determines theminimum blur diameter that is achievable by an opticalsystem and hence provides a reference for image quality.The spot size mainly depends on the wavefront qualitydue to geometric aberrations, whereas the beam positionaccuracy depends on how accurate the optical componentsare positioned/oriented. These performance metrics can bedetermined using geometric ray tracing, which traces the
886 S. Koppen et al.
(a) Structural design of the Sentinel 5 UV1 spectrometer. The mirrorsare bolted to the housing, and the housing is semi-kinematically con-nected to the satellite frame via flexures in order to limit deformationsdue to imposed rigid body displacements of the frame.
(b) Optical design of the Sentinel 5 UV1 illustrating the mirror mountsand optical rays. Note the different mirror back layouts and consistentconnections to the frame. The spot position and size depends on themisalignments and surface deformations of each mirror and grating.
Fig. 1 The European Space Agency’s (ESA) Sentinel 5 shortwave UV1 band (270-300 nm) spectrometer system to be carried on the MetOp-SGsatellite
propagation of light rays through an optical system (e.g.Spencer andMurty 1962). For flat or spherical single-mirrorsystems the wavefront error scales linearly with the SurfaceForm Error (SFE) of the deformed surface and the pointingerror is directly related to the tilt of the surface (e.g. Genberg1984, 1999). The optical performance metric of interest thusdepends on application and system composition.
Many factors may contribute to the inability of an opticalsystem to produce a perfect image, including chromaticand geometrical aberrations, fabrication and alignmenterrors, (lack of) self-weight and environmental effects suchas temperature fluctuations. This work focuses on thereduction of optical performance errors of reflective opticalsystems induced by (quasi)-static thermal loads. Therefore,structural deformations and temperature differences ofthe frame should neither excessively distort the mountnor the optical surface. This implies that a mechanicallydisconnected frame and optical surface combination wouldbe optimal. However, the presence of thermal loads requiresmaterial to abduct the heat from the optical surfaces to theframe. In addition, the optical components also require astiff design, as the structure must constrain the componentssuch that they are not damaged or irreversibly movedafter exposure to external conditions such as vibrations,thermal shocks and gravity. To limit excitation from externalvibrations, the fundamental elastic eigenfrequency must behigher than a critical lower limit. Adding more practicalconstraints such as maximum mass and material usage,often linked to costs, the optomechanical mirror supportdesign clearly involves multiple conflicting structuralrequirements. To make well-founded and justified design
trade-offs, careful consideration of the thermomechanicaland dynamic performance of optical mounts is required.Optimization techniques can aid in this process.
1.2 Problem definition
The current typical design approach of optomechanicalinstruments is characterized by the optical discipline cre-ating a performance error budget that defines deformationlimits for each optical component to the structural disci-pline. In an iterative design process, the thermal disciplineprovides the temperature fields and gradients, after whichthe structural discipline aims to realize a design whichmeets the deformation limits during thermal and other envi-ronmental load cases. From this point of view, an opticalsystem functions as long as the components remain withinallowed tolerances of their nominal locations, orientationsand deformations. Thus, in the existing thermomechani-cal design process, the optical performance is not con-sidered directly. Instead, each component is designed andoptimized separately to meet a priori defined deformationlimits.
To expand the design space and enable further improve-ments of the optical performance we introduce:
1. system-level optical performance metrics that drive thethermomechanical design and optimization process,
2. simultaneous optimization of all components for thecombined metrics, and
3. system-level constraints that, where possible andapplicable, replace component-level constraints.
Topology optimization of multicomponent optomechanical systems for improved optical performance 887
1.3 Approach: simultaneousmulticomponentmultidisciplinary topology optimization
Model-based structural optimization techniques can aid infurther improving the performance of optical instruments.Topology optimization is a systematic, bottom-up structuraloptimization approach that provides maximum designfreedom without any prior knowledge of the design. Theprocedure optimizes the material layout within given designdomains in order to maximize a performance measure,while subjected to a given set of loads, boundary conditions,and constraints (e.g. Bendsøe and Sigmund 2003). Themethod, in combination with mathematical programming,has shown to be able to solve complex multidisciplinaryproblems with multiple active nonlinear response functionsand is capable of producing innovative solutions.
The mentioned design requirements have previouslybeen investigated in topology optimization. This includesthermal loads and prescribed temperature differences onstiffness problems (e.g. Rodrigues and Fernandes 1995;Li et al. 2001a; Gao and Zhang 2010; Deaton and Grandhi2013) and the application to the design of thermomechanicalcompliant mechanisms (e.g. Sigmund 2001; Ansola et al.2012). Recently, Zhu et al. (2016) presented a shape preser-ving design method for topology optimization. By sup-pressing the elastic strain energy in the local domain, theshape of the concerned domain can be effectively main-tained to satisfy the requirements. This is extended byLi et al. (2017) to achieve desired deformation behaviorwithin local structural domains by distinguishing and sup-pressing specific deformation in a certain direction. Thefundamental eigenfrequency as a response function has beenthoroughly investigated (e.g. Ma et al. 1995; Pedersen 2000;Du and Olhoff 2007; Tsai and Cheng 2013).
Previously, topology optimization has shown its bene-fits in the design of mirror mounts for optical performance.For example the minimization of deformations of opticalsurfaces under static (Park et al. 2005; Sahu et al. 2017)and thermal loads (Kim et al. 2005) and mass minimiza-tion while constraining deformations as well as the fun-damental eigenfrequency (Hu et al. 2017a). Furthermore,semi-kinematic flexible mirror mounts are topologicallyoptimized by, Hu et al. (2017b) who minimize surface formerrors subjected to both static and thermal loads constrainedby a minimum natural eigenfrequency, and Van der Kolket al. (2017), who achieve optimal damping characteristicsat source frequencies. System optical performance met-rics have not been included in the topology optimizationframework yet.
An integrated Multidisciplinary Design Optimization(MDO) approach is often applied to couple all involvedphysics and profit from the interactions between diffe-rent disciplines, resulting in superior designs. We apply an
integrated Structural-Thermal-Optical Performance (STOP)analysis and optimization procedure for this purpose toutilize simultaneous optimization of the optical, structuraland thermal design aspects (e.g. Johnston et al. 2004;Doyle et al. 2012; Kuisl et al. 2016). Prior work usingSTOP optimization couples various analysis tools to obtainthe optical performance. This shows great improvement,although the optimization often includes only a limitednumber of variables such as facesheet thickness and strutdiameters (e.g. Williams et al. 1999; Michels et al. 2005;Bonin and McMaster 2007). The results indicate that deviceperformance can profit from integration of optical knowl-edge, and that further improvement is accessible when alldesign parameters are considered. The STOP optimizationprocedure includes optical knowledge at the thermome-chanical design level, but the design freedom is not yetfully exploited.
Coupled multicomponent topology optimization can aidin exploiting the component interactions. Simultaneoustopology optimization of multiple components has mainlyfocused on layout design and combined topology and jointlocation optimization (e.g. Chickermane and Gea 1997; Liet al. 2001b; Zhu et al. 2009, 2015). Topology optimizationinvolving component interactions to improve a systemperformance has previously been investigated by Jin et al.(2016, 2017), with the focus on simultaneous optimizationof multiple coupled actuator mechanisms to minimize thecoupling interaction. However, this study was restricted to asingle physical discipline.
In this work, we combine the foregoing and additionallyapply multidisciplinary optimization to a component assem-bly and optimize for a system-level optical performancemetric. This approach will be referred to as integrated Sys-tem Design Optimization (SDO). To determine the systemoptical performance, this study uses a simplified versionof geometric ray tracing, the ray transfer matrix analy-sis. The method uses the paraxial approximation to cons-truct a linear operator that describes the behavior of an opti-cal system (e.g. Nazarathy et al. 1986; Smith 2007; Fischeret al. 2008).
The innovation point of the SDO method comparedto existing methods is threefold: it combines topologyoptimization and a full STOP analysis, uses system-leveloptical performance metrics to drive the thermomechanicaldesign of multiple components simultaneously, and usessystem-level constraints to replace multiple equivalentcomponent-level constraints.
We hypothesize that an integrated structural-thermal-optical thermomechanical design optimization proceduretaking into account all system components improves thesystem optical performance compared to individual com-ponent optimization, while subjected to equivalent designconstraints.
888 S. Koppen et al.
The formulation of the coupled thermomechanicaldiscretized equilibrium equations and modal analysis,topology optimization formulation and sensitivity analysisof a generalized response function are described inSection 2. The extension to a full STOP topologyoptimization approach is given in Section 3, which focuseson the optical performance prediction from finite elementanalysis results. The method is first validated on a single-component system, after which the hypothesis is testedby numerical optimization of a two-mirror example asdiscussed in Section 5. The results are followed by adiscussion, recommendations and conclusions presented inSections 6 and 7.
2 Formulation of coupled thermomechanicalanalysis framework for topologyoptimization
A schematic of a typical two-mirror opto-thermo-mechanical system is shown in Fig. 2, introducing thenecessary nomenclature for the mathematical model. Bothdomains consist of a linear-elastic homogeneous isotropicmaterial with conductivity ki , Young’s modulus Ei , Pois-son’s ratio νi , density ρi and Coefficient of ThermalExpansion (CTE) αi . A coordinate system is asserted forevery optical component to express the displacements ofthe optical surfaces. Additionally, each optical element(including space propagation in a medium with constantrefractive index) has a coordinate system to track the posi-tion and angle of rays with respect to the optical axis.The optical design includes the optical path lengths d,incident angles φ and initial ray position and angles. Theresponse of a system to a given load case depends onthe optical design and properties, as well as the compo-nent’s domain geometries, boundary conditions, structurallayouts and material properties. Any system responsefunction depends on the response of multiple compo-nents to a given load case, where the response of eachdomain directly depends on the layout of material, denotedby the design variables s, or indirectly through otherresponses.
The design variables following from density basedtopology optimization, one belonging to every finiteelement in the domain, are bounded by a lower and upperbound, i.e. 0 < s ≤ si ≤ 1 with i = 1, 2, ..., n, wheren is the number of variables. The underbound s has a verysmall value (to avoid numerical issues) denoting the absenceof material and, s = 1 providing the element with theassigned initial material properties. Each design variablecan have any intermediate value within the given bounds.The material properties are interpolated using a penalizationfunction as discussed in Section 2.1. Sections 2.2 to 2.4
Fig. 2 Overview and nomenclature of a typical optical system.The incoming converging wavefront is deflected by the mirrorsand focused onto a sensor. The design domain of each component�i(Ei, ki , νi , ρi , αi) is shown in grey, with nondesign domainillustrated in black. Prescribed displacements Ub and temperatures Td,respectively, represent the housing rigid body effects and temperatures.The beam imposes thermal loads Qc onto the mirror surfaces. Theposition and deformations of mirror i are locally described usingcoordinate systems Om,i . An incoming wavefront is described bymultiple rays, of which each rays’ position and the angle is expressedin local coordinate system Or,i with respect to the principal rayincoming on component i. Each mirror makes an angle φi with relationto the incoming principal ray. The initial optical path lengths ofprincipal ray i is given by di
discuss the thermomechanical and modal analysis, followedby the respective sensitivity analyses.
2.1 Penalization scheme
Intermediate density values are implicitly penalized to grad-ually force the design variables to approach their boundsand facilitate interpretation and improve manufacturability.Most material properties are a function of the design vari-ables, which varies depending on the applied penalizationscheme. Common interpolation functions for the Young’smodulus RE(s) and conductivity Rk(s), are the SIMP andRAMP functions (Bendsøe and Sigmund 1999).
Low-frequency eigenmodes localized in the void regionsof the structure should be avoided. Therefore, this studyapplies a continuously differentiable penalization schemefor RE(s) and Rk(s), as proposed by Zhu et al. (2009). Thefunction extends the SIMP interpolation with a linear term
Topology optimization of multicomponent optomechanical systems for improved optical performance 889
to avoid large ratios of Rρ
REfor design variables with near
zero densities, that is
RE = Rk(s) = qsp + (1 − q) s, (1)
where p is a penalization parameter and q is a weight factorcontrolling the relative influence. This study consistentlyuses a power p = 3 and weight q = 0.8. The densitiesare scaled linearly, thus the volume and mass of an elementdepends linearly on the accompanying design variable. Themethod can effectively penalize intermediate densities andavoid spurious eigenmodes for low-density elements whilekeeping the penalization factor relatively low.
The stress is given by
σ(s) = E(s) (εm + αT ) , (2)
where εm is the strain from applied forces, T theapplied temperature difference and E(s)αT the resultingthermal stress. The thermal stress is a nonlinear functionof the design variables via the penalization of the Young’smodulus, but the CTE itself is not directly penalized.
2.2 Thermomechanical equilibrium equations
The one-way coupled thermomechanical equilibrium equa-tions, neglecting heat transfer by radiation and convectionand discretized using finite elements, are given by[
K(s) −A(s)0 H(s)
] [UT
]=
[F(s)Q(s)
], (3)
where K(s) is the global stiffness matrix, H(s) the globalconductivity matrix and A(s) the global thermal expansionmatrix converting the temperatures to equivalent thermalloads (e.g. Cook 1981). These global matrices are a functionof the material layout. Depending on the considered loadingconditions, the forces F(s) and applied heat flux Q(s)can become design dependent. The material propertiesare assumed to be independent of strain and temperature,thus both linear geometric and linear material modelsare applied. Furthermore, the loads are assumed to beindependent of time, thus only steady state heat transfer and(quasi)-static forces apply.
Prescribed and unconstrained Degrees of Freedom(DOFs) sets are generally different for the mechanical andthermal problem. To express this in a convenient notation,one can separate the DOFs in terms of unconstrainedand prescribed displacement DOFs a and b, as well asunconstrained and prescribed temperature DOFs c and d.Each domain can be a function of externally applied forcesFa(s) and heat loads Qc(s), prescribed displacements Ub(s)and temperatures Td(s), unconstrained displacements Ua(s)and temperatures Tc(s) as well as the reaction forces Fb(s)and reaction heat loads Qd(s). For the sake of clarity we
abbreviate the multiplications and leave out the designvariable dependency, such that for instance,
KaU � KaaUa + KabUb. (4)
For a single domain one can now write the partitioneddiscretized heat transfer and elasticity equilibrium equationsas[
Kaa Kab
KTab Kbb
] [Ua
Ub
]=
[Fa
Fb
]+
[Aac Aad
Abc Abd
] [Tc
Td
][
Hcc Hcd
HTcd Hdd
] [Tc
Td
]=
[Qc
Qd
].
(5)
The equilibrium equations, (5), are solved to obtain bothtemperature and displacement fields.
2.3 Modal analysis andmean eigenvalue
The natural dynamic properties (neglecting damping) of astructure are found by solving the eigenvalue problem
(K − λiM) φi = 0 for i = 1, 2, .., n (6)
for each unknown eigenvalue λi and corresponding modeshape φi . Here i is the mode of interest and n the numberof DOFs. In this eigensystem the characteristic value equalsthe squared radial eigenfrequency, i.e. λi = ω2
i . All modesare mass normalized satisfying
φTi Mφj = δij for i, j = 1, 2, ..., n. (7)
The mean eigenvalue (Ma et al. 1995) is used as aresponse function to maximize or constrain the minimumfundamental frequency. The function combines the m lowesteigenfrequencies of the structure, calculated as
fω(s) =m∑
i=1
1
λi(s). (8)
This form ensures that the lowest resonant frequency is themain contributor and the influence of modes with a highereigenfrequency quickly decreases. This implants a simpleand effective natural eigenfrequency constraint and ensuresthe absence of mode switching.
2.4 Sensitivity analysis
For any response f (s), the gradient with respect to smust be determined to enable gradient-based optimization.The focus is first on the responses that depend on thethermomechanical analysis. Adjoint sensitivity analysis isefficient due to the large number of design variables, andtherefore we augment the response function as
f = f (s, Ua, Ub, Fa, Fb, Tc, Td, Qc, Qd)
−λTa (KaU − (Fa + AaT))
−λTc (HcT − Qc) ,
(9)
890 S. Koppen et al.
where λa and λc are the adjoint vectors related to themechanical and thermal equilibrium equations.
The total derivative with respect to the design variables,using the adjoint method to solve the gradient of theobjective function in relation to the variables, is given by
dfds = ∂f
∂s + ∂f∂Fa
∂Fa∂s + ∂f
∂Ub
∂Ub∂s + ∂f
∂Qc
∂Qc∂s + ∂f
∂Td
∂Td∂s
−λTa
(∂Ka∂s U + Kab
∂Ub∂s − ∂Fa
∂s − ∂Aa∂s T − Aad
∂Td∂s
)−λT
c
(∂Hc∂s T + Hcd
∂Td∂s − ∂Qc
∂s
)(10)
which holds if the Lagrange multipliers satisfy:
[Hcc −AT
ac0 Kaa
] [λc
λa
]= −
⎡⎢⎣
(∂f∂Tc
)T + Hcd
(∂f
∂Qd
)T
(∂f∂Ua
)T + Kab
(∂f∂Fb
)T
⎤⎥⎦ .
(11)
Here, loads and boundary conditions are assumed tobe design independent and the responses are assumed tosolely depend on the displacement field. Then, (10) can besimplified to
df
ds= −λT
a
(∂Ka
∂sU − ∂Aa
∂sT
)− λT
c∂Hc
∂sT, (12)
with the Lagrange multipliers determined by[
Hcc −ATac
0 Kaa
] [λc
λa
]=
[0
−(
∂f∂Ua
)T
]. (13)
The sensitivities of the mean eigenvalue, as described in (8),with respect to the design variables are
dfω
ds=
m∑i=1
− 1
λ2i
φTi
(∂K∂s
− λi
∂M∂s
)φi . (14)
The sensitivities of the global matrices can be derivedelement wise using direct differentiation. With this, thesensitivities have been determined up until the term ∂f
∂Ua,
which will be discussed in Section 3, after introducing theconsidered optimization problem and response functions.
3 Optical performancemetricsand sensitivities
This section describes various optical performance metricsrelevant for the analysis of reflective optical systems.First, the SFE response will be discussed, after whichthe analysis of average positional accuracy and spot sizeare explained. Most commonly the SFE is expressed bythe Root Mean Square Error (RMSE) of the deformedconfiguration compared to the undeformed or anotherpredefined configuration (Genberg 1984). For 3D unit disksurfaces the surface errors are often expressed in Zernike
polynomials, which are directly related to typical opticalaberrations. For diffraction limited flat or spherical single-mirror systems there exists a simple relation between theRMSE and the Strehl ratio. This is the peak aberrated imageintensity compared to the maximum attainable intensityusing an unaberrated system. The wavefront is proportionalto surface front error. Though, for complex mirrors or multi-mirror systems the WFE is not directly related to the SFEsand the image quality can only be determined by ray tracingtechniques.
To analyze a multi-component system without usingnumerical ray tracing, the deformed surfaces can beapproximated by a fit to directly obtain contributions to theoptical surface misalignments, i.e. rigid body movementsand SFEs. Next, the ray transfer matrix analysis can be usedto track the position and angle of a paraxial ray thougha multi-component system, leading to a measure for theaveraged positional accuracy. In order to track the rays, allsystem properties, i.e. optical paths lengths, incident anglesand specific properties of the optical components, as wellas component transfer functions and misalignments shouldbe known. Finally, the spot size is quantified by the meanaverage deviation of all rays with respect to the averagedspot position.
3.1 Surface form error
The RMSE is the standard deviation of the distance betweenthe deformed surface and the ideal surface and is a globalmeasure for the SFE, see Fig. 3. The deformed surface isconstructed from the out of plane displacements, stored in a
Fig. 3 Schematic representation of a generalized surface in unde-formed (blue) and deformed (green) configuration of mirror i. Outof plane node displacements are indicated by Uz,j for the deformedand by Uz,j for the fitted displacements, where j is the surface nodenumber. A least-squares polynomial fit through the deformed surface(purple) is used to extract the surface misalignments: the tangentialand out of plane misalignments, δx and δz as well as the rotationalmisalignment θy and curvature κ . The component specific propertiesand misalignments δ cause the input ray ri , with incident angle φi , tochange into ri+1
Topology optimization of multicomponent optomechanical systems for improved optical performance 891
vector Uz, which is fitted to a smooth surface represented bythe fitted out of plane displacements Uz using a linear leastsquares regression scheme (Lay 2006), such that
Uz = GUz, (15)
where G is the fit matrix. The fit matrix is constructed as
G = VY = V(
VT V)−1
VT , (16)
where V is the Vandermonde matrix and Y relates the outof plane displacements Uz directly to the coefficients ofthe least-squares solution, i.e. b = YUz. Vector b containsthe coefficients of the least-squares fit (size depends on theorder of the fitted polynomial) which will be used in Section3.2. The Vandermonde matrix consists of the terms of ageometric progression and evaluates a polynomial at a setof points, the surface nodal x-coordinates in this case. Thetangential displacements are assumed negligible comparedto the out of plane displacement, i.e. ||Ux|| � ||Uz||, whichensures V is constant during the optimization and the leastsquares fit is linear.
The fitted displacements are related to the coefficientsvia Uz = Vb. The nodal residuals of the linear least squaresfit, denoted by R, are calculated as
R = Uz − Uz = (I − G) Uz, (17)
where I is the identity matrix.The surface RMSE equals the absolute residual between
the deformed surface and the ideal surface. It is preferred touse the Mean Square Error (MSE) when using the responsein gradients based optimization to avoid division by zerowhen taking derivatives. The MSE is defined as
fMSE = MSE (Ua) = 1
n
n∑i=1
(Uz,i − Uz,i
)2 = 1
nRT R,
(18)
where n is the total number of nodes on the mirror face, Uz,i
is the observed and Uz,i the fitted out of plane displacementat nodal point i.
The sensitivities of the MSE in relation to the freedisplacements Ua, as required in (13), equal
∂fMSE
∂Ua= ∂fMSE
∂RJR(Ua) = 2
nRT (I − G) Lz, (19)
where JR(Ua) is the Jacobian matrix of R with respectto Ua and Lz selects the appropriate DOFs of the surfaceof interest from the vector of free displacements, i.e.Uz = LzUa.
3.2 Positional accuracy
Depending on the application, it is often essential thatthe image is kept within certain bounds (e.g. within theboundaries of a sensor). To determine the location of a
light ray on the image plane we use paraxial ray tracingof multiple rays. Considering a situation where all opticalcomponents are symmetric around the optical axis, thepositional error of ray j (i.e. the distance of ray j withrespect to the optical axis on the image plane), here denotedby εj , depends on the radial distance and angle of the raywith respect to the optical axis when entering the system,which will be denoted by vector r0. Furthermore, it dependson all misalignments δ1, ..., δN of all reflective optics (Nis the number of components) and system specific constantparameters p (initial optical path lengths d and angles ofincidence φ), thus εj
(r0,j , p, δ1, ..., δN
).
The lower order misalignments of a surface i in 2D,are the change in curvature κi , the axial displacement δz,i(despace), the rotational misalignment θy,i (tip/tilt), andthe radial displacement δx,i (decenter). The decenter isdirectly calculated from the tangential displacement of thesurface vertex. Other misalignments can be derived from thecoefficients of the surface fit bi , which is calculated by thelinear least square regression in (16) and shown in Fig. 3.
The radius of curvature Ri = 1κi
is assumed to beconstant (parabolic) over the surface for small angular
misalignments, that is(dzdx
)i� 1, and defined as the
reciprocal of the curvature κi , which equals κi ≈(d2zdx2
)i.
The misalignments are stored as δi = [Ri θy,i δz,i δx,i]T .Determination of the despace, tip/tilt and radius ofcurvature from the surface fit do not take into accountthe radial displacement distribution nor the average radialdisplacement (decenter) of the mirror surface.
3.2.1 Ray transfer matrix analysis
Ray transfer matrix analysis is used to determine the lightpath through a system based on paraxial approximationsby transforming the vector representing the ray, with theappropriate component transfer matrices, which depend onthe properties and misalignments of the component. Ageneral misaligned paraxial transformation for componenti, such as the mirror in Fig. 3, is denoted as[
ri+1
1
]=
[Mi Ei
0 1
] [ri
1
], (20)
where ri = [xidxdz i
]T , with xi and dxdz i
the distance andangle with respect to the optical axis i in the undeformedconfiguration. The ray vector after component i, ri+1, isa linear transformation to the incoming ray ri via thecomponent specific transfer matrix Mi , and the influence ofthe misalignments as contained in the misalignment matrixEi .
Using first-order optical canonical operator theory (e.g.Nazarathy et al. 1986) one can determine the influenceof the misalignments of an optical component δi on the
892 S. Koppen et al.
Table 1 General misaligned ray transfer matrices for generalized misaligned space propagation and mirror surfaces (Yuan et al. 2011). Here Ri
is the radius of curvature, δz,i and δx,i are despace and decenter misalignments, θy,i the tip/tilt contribution and di the optical path lengths of aprincipal ray i with angle of incidence φi on component i
position and angle of an incoming ray ri (Yuan et al. 2011).The generalized transfer matrices for the elements usedin this study (space propagation and reflective optics) areshown in Table 1. The space propagation transformationmatrix does not contain aberrations and changes ofrefraction index between optical components. The mirrortransformation matrix models reflective optics as perfectlyreflecting surfaces without scattering, transmission orabsorption.
The positional error εj after N sequential elements withaccompanying misalignments Ei ,1 considering an initial rayvector r0,j , equals the distance from the optical axis afterthe last component, that is εj = x
N. The output ray vector
equals
rN,j
= MSr0,j + ES, (21)
where the system transformation matrix and misalignmentvector are
MS =N−1∏i=0
MN−i
and ES =N−1∑i=1
Ei
⎛⎝N−i−1∏
j=0
MN−j
⎞⎠+E
N.
(22)
Thus, the positional error of each ray is a functionof all system and component optical properties andmisalignments.
3.2.2 Averaged positional error and sensitivities
The squared positional error of a single reflective optic,for the purpose of an uncoupled optimization, is solely afunction of the misalignments of that specific component,that is
fε,i = ε2i(Ua,i
) =⎛⎝ 1
m
m∑j=1
εj
(r0,j , δi
)⎞⎠
2
, (23)
where εi is the average positional error due to misalign-ments of component i and the number of rays in the system
1The number of optical elements in a system is generally larger thanthe number of optical components, since space propagation is alsoan optical element, i.e. N ≥ N . In most optomechanical applicationsN = 2N + 1.
m. For the same reason as in (18), the root is omitted.Statistically, it is unlikely that all positional error contribu-tions have the same direction and hence superposition of theerrors will lead to overdesign of the individual components,i.e. the errors are uncorrelated. Therefore, the positionalerrors of independent sources are combined via the RootSum Square (RSS).
The squared positional error taking into account allcomponents is defined as
fε = ε2(Ua,1, ..., Ua,N
) =⎛⎝ 1
m
m∑j=1
εj
(r0,j , δ1, ..., δN
)⎞⎠2
(24)
The sensitivities of the positional error contribution ofcomponent i in relation to the unconstrained displacementsare
∂fε,i
∂Ua,i= 2
m
m∑j=1
εj
(r0,j , δi
) ∂εj
∂Ua,i(25)
and the sensitivities of the positional error using anintegrated approach equals
∂fε
∂Ua,i= 2
m
m∑j=1
εj
(r0,j , δ1, ..., δN
) ∂εj
∂Ua,i. (26)
In both (24) and (26) the sensitivities of a single ray εj withrespect to Ua are
∂εj
∂Ua,i= ∂εj
∂δi
∂δi
∂Ua,i. (27)
Note that ∂fε
∂Ua,i
(Ua,1, ..., Ua,N
), thus the sensitivities of the
positional error with respect to the displacements dependsalso on the displacements of other components in thesystem. In contrary, the positional error due to componenti only depends on that components’ displacements, that is∂fε,i
∂Ua,i
(Ua,i
).
The sensitivities of the misalignments δi are defined as
∂δi
∂Ua,i=
[∂δ∗
i
∂bi
∂bi
∂Ua,i∂δx,i∂Ua,i
], (28)
Topology optimization of multicomponent optomechanical systems for improved optical performance 893
where δ∗i = [Ri θy,i δz,i]T . Corresponding sensitivity
matrix ∂δ∗i
∂biis a transformation scaling the fit coefficients to
the misalignments, this is defined as
∂δ∗i
∂bi
=
⎡⎢⎢⎣
∂Ri
∂bi,1
∂Ri
∂bi,2· · · ∂Ri
∂bi,k
∂θy,i∂bi,1
∂θy,i∂bi,2
· · · ∂θy,i∂b
i,k∂δz,i∂bi,1
∂δz,i∂bi,2
· · · ∂δz,i∂b
i,k
⎤⎥⎥⎦ , (29)
where k is the order of the fitted polynomial. For a thirdorder polynomial (parabola) k = 3 and
∂δ∗i
∂bi
=⎡⎣ 0 0 1
κi
0 1 01 0 0
⎤⎦ , (30)
where κi is considered constant over the surface. Theorder of the fit determines what type of aberrations canbe accounted for. The term ∂bi
∂Ua,i= YiLz,i can be derived
from (16) and ∂δx,i∂Ua,i
= Lx,i , where Lx,i picks the appropriateDOFs, i.e. δx,i = Lx,iUa,i .
3.3 Spot size and sensitivities
In order to measure the spot size, ray tracing is performedfor multiple rays with different initial distance and anglefrom the optical axis, see Fig. 2. The average resultingdeviation from the averaged positional error is a measure forthe spot size. The squared spot size due to the misalignmentsof component i is defined as
fε,i = ε2i(Ua,i
) = 1
m
m∑j=1
(εj − εi
)2, (31)
where εj and εi are calculated according to (21) and (23).For an integrated system optimization the spot size responseis calculated by
fε = ε2(Ua,i , ..., Ua,N
) = 1
m
m∑j=1
(εj − ε
)2, (32)
where ε is calculated with (24).The sensitivities of the MSE spot size due to the
misalignments of component i, with respect to theunconstrained displacements, as required to solve for theLagrange multipliers in (11), are
∂fε,i
Ua,i= 1
m
m∑j=1
(εj − εi
) (∂εj
∂Ua,i− ∂εi
∂Ua,i
), (33)
where ∂εi
∂Ua,iis defined in (25) and
∂εj
∂Ua,iis calculated using
(27). Similarly, the sensitivities of the MSE spot size takinginto account all components’ misalignments are
∂fε
Ua,i= 1
m
m∑j=1
(εj − ε
) (∂εj
∂Ua,i− ∂ε
∂Ua,i
), (34)
where ∂ε∂Ua,i
is defined in (26).
4 Numerical implementation
This section describes practical considerations of the imple-mentation. The initial conditions are set such that designsare initialized with a uniform density field that exactly sat-isfies the volume constraint. The optimization problem issolved using the Method of Moving Asymptotes (MMA)(Svanberg 1987). The optical performance measures are rel-atively sensitive to design changes. Therefore, the algorithmis set more conservative (the move limit move is set to 0.1)in order to avoid large jumps in the design space.
The optimization is subjected to termination criteriato avoid endless optimization and is considered to beterminated when the design variables and objective functionchange less than a threshold and all constraints are met, thatis⎡⎢⎢⎣
1n
√(s(k) − s(k−1)
)21n
√(f (s)(k) − f (s)(k−1)
)2gi(s)
⎤⎥⎥⎦ ≤
⎡⎣ εs
εf(s)εg(s)
⎤⎦ , (35)
where n is the number of design variables, k is theiteration number and i = 1, 2, ..., m with m the number ofconstraints.
In order to limit the design complexity and to avoidmesh dependency and checkerboard patterns, a generalmesh and element-type independent linear spatial filter isimplemented (Bruns and Tortorelli 2001). The simplestspatial filter, as used in the presented study, is the linearfilter with weights according to
wij ={
r − rij if rij ≤ r
0 if rij > r, (36)
where r is the radius of the filter, taken equal to the sizeof a single element. The radius rij is the distance betweenthe centroids of elements i and j and n is the number ofactive elements (equal to the number of design variables).The filtered variables are calculated as
ˆsi =∑n
j=1 wij sj∑nj=1 wij
, (37)
where s are the design variables updated by the optimizationalgorithm. The filter does not take account of the elementvolumes as only structured meshes are used.
To further stimulate fully black-and-white designs, thefiltered design variables are projected in the direction oftheir lower and upper bounds by a smooth Heavisideprojection function (e.g. Guest et al. 2004; Sigmund 2007;Xu et al. 2010). The projected design variables are penalized
894 S. Koppen et al.
using a projection parameter β around a threshold η, givenby
si =tanh (βη) + tanh
(β
( ˆsi − η))
tanh (βη) + tanh (β (1 − η)). (38)
All given values are constant during the optimization and nocontinuation strategy is used.
5 Results
This section discusses two case studies applying theforegoing theory and demonstrating the validity of theproposed method. First, the focus is on the optimizationof a single mirror mount to minimize SFEs. Next, theproposed method including the STOP analysis is tested on atwo-mirror case.
5.1 Single-component surface form errorminimization
The focus of this study is on the optimization of a flatmirror mount design subjected to a uniform temperatureincrease. The goal is to minimize the resulting spotsizedue do both boundary and loading conditions. For a single-mirror system the spotsize is directly related to the SFE ofthe mirror surface, hence minimizing the SFE will suffice.The aim is to show that the proposed method works well forsingle component problems under various conditions. Boththe boundary conditions and eigenfrequency constraint playa dominant role in the possibility to improve the opticalperformance. The study verifies the single-componenttopology optimization procedure and investigates theinfluence of the eigenfrequency constraint on the resultingtopology and performance.
5.1.1 Problem definition
The design domain �, as shown in Fig. 4, is subjected toan overall temperature increase T, causing the domainto expand and deform. As the temperature field is known,there is no need to solve the heat equation. It is assumedthe housing experiences the same temperature increase asthe considered design domain and that the design domain ismounted to a housing with different material properties. Theresulting change in expansion is introduced in the designdomain by application of known prescribed displacementson both fixtures.
The objective is to minimize the MSE of the opticalsurface � due to the specified thermal environment, whileconstrained by a minimum mean eigenfrequency and amaximum volume. Therefore, the mean eigenvalue (8) is
Fig. 4 Optical mount design domain � of width w, height h andthickness t with CTE α subjected to a uniform temperature increaseT. All non design space is indicated in black. Two regions withprescribed displacements are modelled as the interface with a rigidhousing structure, which has CTE α
2 , and thus known prescribeddisplacements Ub, which will induce SFE on surface � due to theboundary conditions and hence degrade the image quality of theincoming wavefront
adopted as a constraint such that the mean eigenvalue mustbe higher or equal than the minimum elastic mean eigen-frequency ω2
n. A parameter sweep is performed over a rangeof minimum eigenfrequency constraints to investigate theinfluence of the eigenfrequency constraint. The range spansfrom minimum eigenfrequency constraints where the con-straint is inactive up to values where the structure is unableto satisfy the constraint. Additionally, a constant volumeconstraint is added, in order to ensure a fair comparison.
The problem is formally defined as
mins
fMSE = MSE� (Ua) =(1n
RT R)
�
s.t. KU = AT(K − ω2
i M)φi = 0 for i = 1, 2, ..., n∑m
i=1ω2n
ω2i (s)
− 1 ≤ 0V�(s)
V− 1 ≤ 0
0 < s ≤ s ≤ s
(39)
In this study m = 6 for all cases, to prevent the occurrenceof mode switching.
5.1.2 Optimization results
Figure 5 shows the resulting topologies and accompanyingperformances obtained after optimization of (39) for a rangeof eleven different values for f
n(note that ω2
n = 4π2f 2n).
The required minimum eigenfrequency determines to whatdegree the optimizer is able to minimize the MSE. Caseswith a higher eigenfrequency constraint generally result in
Topology optimization of multicomponent optomechanical systems for improved optical performance 895
Fig. 5 Optimization results for (39) showing topologies (thermal deformations scaled by a factor 100) and RMS surface form error, in μmK−1,for a range of minimum eigenfrequency constraints f
n
higher SFE, as a compromise must be made. Note that abovea critical frequency the resulting designs do not satisfy theeigenfrequency constraint. The resulting RMS SFEs are gi-ven in μm K−1 because the RMS SFE scales linearly withrespect to both T and the CTE α, thus the imposed tempe-rature difference or CTE is irrelevant for the final topology.
The topology of the optimal design subjected to a meaneigenfrequency constraint of 500 Hz is shown in more detailin Fig. 6. Designs with a relatively low eigenfrequency cons-traint tend to possess compliant mechanism-like structuresin order to counteract surface deformations. In general,structures with a lower required dynamic stiffness have clearrotation points (shown in red) and thicker beams to supportthe mirror. The large amount of material underneath themirror both stiffens the structure and forces the flexure-likestructures, which are fixed to the frame (green), to bendoutwards effectively flattening the surface (blue).
5.2 Two-mirror system spotsize minimization
This example studies the topology optimization of a two-mirror system subjected to thermal loads from a light source,as used in for example high-power laser or EUV applica-tions. The study compares the following design approaches:
1. Uncoupled System (US) optimization, and2. Coupled System (CS) optimization, where the inte-
grated SDO approach is applied.
Both optimization procedures make use of a full STOPanalysis, though only the integrated approach considers the
component interactions and applies both system and compo-nent level constraints. Note that the uncoupled optimizationproblem is not a typical design approach used in practice,since it does consider system optical performances. The sep-arate components are however artificially decoupled with
Fig. 6 Detailed view of the resulting topology of optimization problem(39) with an eigenfrequency constraint of 500 Hz both in undeformedand deformed (scaled by a factor 4). The figure illustrates the deformedcompliant mechanism-like structure with rotation points indicated inred, mirror surface in blue and frame interface in green. Bottom figureshows the first modeshape of the structure. This is the bending mode,as expected
896 S. Koppen et al.
relation to the optical performance, in order to investigatethe difference with respect to the coupled SDO approach.
The optimization aims to minimize the spot size errordue to prescribed boundary conditions of the frame andthermal loads from the propagating beam, while boundedby a the position accuracy, mass, and eigenfrequencyconstraints. For multi-component systems both positionaccuracy and spotsize depend on the rigid body motionsand SFEs of all components involved. The main target isto investigate whether, and how, the components make useof the capability to interact and compensate for each othersoptical aberrations and how this may benefit the opticalperformance.
5.2.1 Problem definition
Consider the schematic structural-thermal-optical systemconsisting of two flat reflecting surfaces supported byoptical mounts with design domains as shown in Fig. 2.An incoming converging beam with a perfect wavefront isreflected by two mirrors before it is focused onto a sensorwith a theoretical spot size of zero (if it were unconstrainedby the diffraction limit).
The fundamental maximum resolution or minimum spotsize of any optical system is limited by the diffraction limit.The diffraction limit for the system as shown in Fig. 2,equals
D = 0.61λlNA
= 0.61λlnr sin θi
(40)
where D is the first minimum in the Airy disk, NA theNumerical Aperture of the system, λl the wavelength of thelight, nr the index of refraction of the medium and θi thehalf-angle of the incoming light. Assuming perfect vacuumconditions, i.e. nr = 1, sunlight of λl = 550 nm (center ofsunlight wavelength spectrum) the half-angle of the systemof 3.8◦ and a NA of 0.0665, the diffraction limit of thissystem equals 8.3μm.
The mesh is structured and consists of 10000 Quad4 isoparametric elements (4 Gauss points and bilinearshape functions) per domain. Each domain consists of Alu-minium, with Young’s modulus E = 70 GPa, Poisson ratio0.35, density ρ = 2700 kg/m3, coefficient of thermal con-ductivity k = 250 W m−1 K and CTE α = 25μm/mK−1.
Both mirrors are subjected to known rigid bodymovements from the housing, which is modelled as arigid interface. The interfaces of the first mirror mountare considered constant at 1 K difference with respect toambient conditions, whereas the interfaces of the secondmirror are constant at 0 K difference. The first mirror issubjected to a decenter rigid body effect of δx,1 = 200μmand a the same amount of despace misalignment. The leftside of the second mirror is moved out by 200μm and down
the same magnitude. The right side of the second mirrorinterface is also moved out by 200μm. This causes thesecond mirror to initially have a despace and tip/tilt error.
The heat load is modelled as a Gaussian profile over thesurface, i.e.
Qc,i (x) = Q0,i
σ√2π
e− 1
2
(x−μ
σ
)2, (41)
where Q0,i is the maximum amplitude in domain i in Watts,x is the location on the surface, with x = 0 m in the middleof the mirror surface, μ = 0 m and σ taken equal to 0.1m. The input heat loads are normalized in relation to themaximum value at x = μ. The first mirror is subjected to aGaussian heat profile with a maximum input of Q0,1 = 0.1W m−1. Assuming the first mirror absorbs 10% of the heatload, the second mirror is subjected to a heat load with a90% maximum amplitude in relation to the first mirror.
Each design domain is constrained by an individualeigenfrequency constraint, as well as a maximum allowableRMS SFE compared to a perfectly parabolic mirror.Therefore, the MSE SFE response (18), is adopted as aconstraint, taking into account proper scaling, i.e.
gMSE,i (Ua,i ) = log10
(1
nRT R
)�i
≤ − log10(MSE�i
)(42)
Additionally, the system is required to keep the positionof the image within a certain limit with respect to theoptical axis, therefore the positional error, (23) or (24),depending on the optimization problem, has to be adoptedas a constraint, such that
gε,i(Ua,i ) = log10(ε2i
)≤ − log10
(ε2i
)(43)
for an uncoupled optimization, and
gε(Ua,i , ..., Ua,N ) = log10(ε2
)≤ − log10
(ε2)
(44)
for the coupled case. For an uncoupled optimization, thesystem positional error budget must be split up into theindividual components, in this case such that the RSSvalue equals the total allowed system positional error, withequal weights per mirror, i.e. the positional error ε of thecoupled case equals the RSS of the positional tolerances ε1and ε2.
In the uncoupled optimization each mirror is subjectedto an individual maximum volume constraint as is typicalin industrial projects, however, when performing integratedSDO the full system is subjected to a maximum volume
Topology optimization of multicomponent optomechanical systems for improved optical performance 897
Table 2 Performance of the obtained optomechanical systems, and properties of their individual mirror mounts: RMS spot size, RMS positionalerror, volume, RMS SFE, mean eigenfrequency rigid body movements and mirror curvature. The values between parentheses indicate the valueat the first iteration
constraint equal to the sum of the volumes of the individualcomponents. In the example 50% volume is allowed. Thisallows the optimizer to move material between domainsfor the benefit of the system performance. The constrainttolerance values, redefined into more relevant measures(e.g. f
ninstead of ω2
n), are given in Table 2.The uncoupled optimization problem of the individual
components minimizes the spot size as a function of themisalignments of the component of interest while satisfyingthe volume, positional error contribution and RMS SFEconstraints. The US optimization is a combination of theoptimization of mirror one (U-M1), not taking into accountthe misalignments of the second mirror (U-M2) and viceversa. The problem is stated as
mins
fε,i = ε2i
(Ua,i
) = 1m
∑mj=1
(εj − εi
)2s.t.
[K −A0 H
]i
[UT
]i
=[
0Q
]i(
K − ω2i M
)φi = 0 for i = 1, 2, ..., n
gε,i (Ua,i ) = log10(ε2i
) + log10(ε2i
)≤ 0
gMSE,i (Ua,i )= log10(1n
RT R)
�i
+log10(MSE�i
)≤0
gω,i(Ua,i ) = ∑mj=1
ω2n(
ω2j
)i
− 1 ≤ 0
gV,i(Ua,i ) = V�i
V− 1 ≤ 0
0 < s ≤ s ≤ s
(45)
for i = 1, ...N , where N = 2 for this system.In the coupled case, the integrated SDO approach simul-
taneously optimizes all domains, thus objective function,
system positional error and volume constraint are a functionof the layout of all domains in the system, that is
mins
fε = ε2(Ua,1, Ua,2
) = 1m
∑mj=1
(εj − ε
)2s.t.
[K −A0 H
]i
[UT
]i
=[
0Q
]i(
K − ω2i M
)φi = 0 for i = 1, 2, ..., n
gε(Ua,1, Ua,2) = log10(ε2
) + log10(ε2) ≤ 0
gMSE,i (Ua,i )= log10(1n
RT R)
�i
+log10(MSE�i
)≤0
gω,i(Ua,i ) = ∑mj=1
ω2n(
ω2j
)i
− 1 ≤ 0
gV (Ua,1, Ua,2) = V�1+V�2V
− 1 ≤ 00 < s ≤ s ≤ s
(46)
for i = 1, 2.The lower bound on the design variables is s = 10−3. The
density filter radius equals the length of two finite elementsand the Heaviside projection parameters are set to β = 1.5and η = 0.45. The termination criteria are εs = 0.0015,εf(s) = 0.002, and εg(s) = 0.02.
5.2.2 Optimization results
Figure 8 shows the resulting RMS spot size diameter as afunction of iteration history for both optimization problems.The RMS spot size diameter versus the number of iterationsfor the coupled case using the SDO approach is illustratedin red. Whereas this approach requires only a singleobjective function, the uncoupled optimization consists of
898 S. Koppen et al.
two separated optimization problems (one for each opticalcomponent), of which the RMS spot size diameter versusiteration number are shown in green and purple. In orderto compare the system performances the RMS spot sizediameter as a function of iteration history of the uncoupledsystem is calculated afterwards as shown in blue. Note thatthe optimizer is unaware of the US performance duringthe optimization process. The resulting topologies for bothproblems are displayed in Fig. 7, accompanied by thefinal performances as well as the system and componentproperties and constraint tolerances shown in Table 2.
The two approaches result in some significant differ-ences, see Table 2 and Fig. 7. Whereas the curvature of thefirst mirror in both approaches is brought as close as pos-sible to zero, the second mirror in the coupled approach ismade even more concave than its original shape. Note thatthe spot size of US is not simply the average of both mirrors.The positional error constraint of U-M2 is active, althoughthe combined positional error of US is far below the allow-able error. Both systems have satisfied the allowable mass,however, the integrated SDO has transferred mass fromC-M2 to C-M1.
The optimization process has lowered the RMS spot sizediameter from 0.4mm in the first iteration to 106.5μmand 4.7μm for the uncoupled and coupled optimiza-tion approaches, respectively. Thus, the integrated SDOapproach used for the coupled case proves a ratio of increasein performance of 22.6 times compared to the uncoupledmethod, equivalent to an improvement of 95.6%. Note that
the integrated SDO approach is able to lower the RMS spotsize to below the diffraction limit of the system, that meansit is not possible to further improve the optical performancemeasure with a geometry-based performance metric.
6 Discussion and recommendations
Existing approaches for topology optimization of optome-chanical systems focus on the optimization of individualmirror mounts to minimize their surface deformations. Ourresearch extends this by
1. topology optimization of mirror mounts for thesystems’ optical performance expressed in terms ofspot size and position accuracy using a full structural-thermal-optical performance analysis procedure, and
2. simultaneous topology optimization of multiple mirrormounts to exploit the interaction of aberrations betweendifferent components due to thermal loads and framemovements and minimize a system optical performancewhile constrained by dynamic stiffness, weight andoptical performance measures, both on a componentand system level.
The results of the first case study, shown in Fig. 5,indicate there is a certain bandwidth of minimum eigen-frequency constraint values that influence the ability of theoptimizer to minimize the surface deformations. Thus, con-flicting requirement tolerance values should be thoroughly
(a) Final design after the uncoupled optimization, resulting in animproved but not diffraction-limited system.
(b) Final design after the integrated SDO, resulting in a diffraction-limited system.
Fig. 7 Final design after the optimization of the system in Fig. 2. Thesystem, topologies and deformations are not to scale (deformationsscaled by a factor 100). The uncoupled optimization attempts to obtain
two flat mirror surfaces, whereas the SDO is able to design both mir-rors with respect to each other and utilize the curvature of C-M2 tocorrect for the defocus error introduced by C-M1
Topology optimization of multicomponent optomechanical systems for improved optical performance 899
investigated, as their limits can highly influence the topo-logical layout and performance.
The results of the second study indicates that theuncoupled optimization aims to design two perfectly flatmirrors, whereas the layout of the second mirror in thecoupled optimization is such that its misalignments (mainlycurvature) effectively compensate for the misalignments ofthe first mirror, resulting in a spotsize improvement of over95%, without reduction of any other performance aspectconsidered in the optimization.
The activity of the positional error constraint in theoptimal solution of the uncoupled optimization (while thesystem easily remains within the accuracy limits) indicatesthe system is unnecessarily overconstrained, see Table 2.The SDO approach makes use of the enlarged feasibledomain, which is apparent from the mass transfer betweenthe mirrors in the optimal solution and the significantdifferences in topologies. Thus, the typical design approachunnecessarily overconstrains problems, whereas the SDOapproach enlarges the feasible domain and gives theoptimizer more freedom to minimize the objective whilestill satisfying all constraints.
During optimization the SFE constraints are notalways satisfied, which means that optical analysisis based on optical properties that do not accu-rately describe the deformed surface leading to inaccu-rate results. However, the converged optimal solutionsdo satisfy the SFE constraints and hence accuratelydescribe the system’s optical performance metrics. A moreenhanced surface error determination method may resultin faster convergence, and additionally a different optimalsolution.
One would expect the SDO problem to be more difficultto solve than the uncoupled problem, due to a larger designspace and potentially more complicated cost function.However, the results indicate the opposite as the requirednumber of iterations to solve the problem reduces. Theiteration history, Fig. 8, shows that the SDO approach is ableto drastically decrease the objective in the first number ofiterations, where the uncoupled optimization requires moreiterations.
The RMS spot size diameter of the uncoupled systemreaches the diffraction limit twice, halfway the iterationhistory, but the optimization continuous because componentresponses do not satisfy all constraints and terminationcriteria. On top of that, when the optimizer is able todecrease the objective function of the second mirror,the system spot size increases again. This indicatesthat the mirrors exactly counteract each others errorduring optimization, although the optimizer is unaware ofthis information. Thus, component interaction should beincluded to obtain a more optimal solution in terms ofsystem performance.
Fig. 8 RMS spot size diameter versus iteration history for theuncoupled system (US) as result of the individual contribution ofmirror mount one (U-M1) and mirror mount two (U-M2), as well asthe coupled system (CS). Note that the RMS spot size diameter of theUS is constructed afterwards
Thus, the case study demonstrates that an integratedstructural-thermal-optical design optimization proceduretaking into account all system components improvesthe system optical performance compared to individualcomponent optimization, while subjected to the same (orequivalent) design constraints.
The system designed using the SDO method keepsa considerable margin above the competitor and hencethe loads may increase considerably before the system’sspot size diameter reaches that of the uncoupled variant.Note that the individual components designed using theSDO method are only applicable to this specific systemconfiguration with these specific loads and boundaryconditions. In general, the SDO method will result indesigns that are more tailored to a specific case andgenerally are less robust with respect to other loadingconditions not considered in the optimization.
This study opens up further research directions, includ-ing:
– application towards various different load cases toverify and validate the general applicability of themethod,
– simultaneous optimization of the housing and opticalcomponents and simultaneous optimization when theirdomains are merged into a single mesh, in order togive the optimizer the freedom to relocate the boundarylocations,
– extension to multiple, and different types of compo-nents, e.g lenses, gratings, prisms and initially curvedmirrors,
900 S. Koppen et al.
– extension to 3D structures and consideration ofmanufacturability,
– including the uncertainty in both thermal and mechani-cal loading, i.e. robust design,
– extension to multi-material topology optimizationto achieve higher performances as there are morepossibilities to counter effect thermal expansion, createconductive isolation, as well as damp out externalvibrations (Van der Kolk et al. 2017), and
– enhancing thermal modeling and control by e.g.considering design-dependent heat loads affected bythe misalignments, including radiation influences, andsimultaneously optimizing locations and input of activethermal components (heaters/coolers).
It is expected that the more components the systemconsist of and the stronger their interaction is, the greater thebenefit of the SDO method will be. Allowing the optimizerto distribute unavoidable errors over multiple components inthe system, instead of letting the designer impose the errorbudgets on each component, enlarges the feasible domainand the potential for superior system designs.
7 Conclusions
The key to satisfy next generation optomechanical systemrequirements is to not distribute error budgets over compo-nents a priori, but to consider and optimize the system asa whole. This allows for focus where it matters, withoutoverconstraining the system unnecessarily. A structural-thermal-optical performance analysis is able to expose theperformance metrics that matter for optomechanical sys-tems without relying on intermediate derived performanceindicators. For a single component system or multicompo-nent uncoupled optimized system, only minimizing defor-mations (and nothing else) leads to better optical systems.However, there is additional room for improvement whenmulticomponent systems are optimized in a coupled fashionas this allows for error compensation between components.Since the feasible design space of the system level optimiza-tion completely encapsulates that of the individual compo-nent optimization, the globally optimal performance of thecoupled system is always better or equal to the uncoupledoptimization approach. This is also shown by the results ofthe numerical example; coupled optimization based on thefull structural-thermal-optical performance analysis is ableto reduce the spot size of a two-mirror system with 95%compared to uncoupled component optimization to belowthe system’s diffraction limit. The coupled analysis allowedthe two mirrors to compensate for each others errors, whichis a mechanism that would be otherwise invisible to theoptimizer. Despite the fact that real systems are more
complex than the simplified example considered in thisstudy, it shows that optomechanical designers should aimfor considering and integrating multiple components andphysics simultaneously in the design loop, and thus applythe SDO approach, when requirements seem irreconcilable.
Acknowledgements The authors thank R. Saathof and J. Day for theirrecommendations during the period of this research and K. Svanbergfor the use of the MATLAB implementation of the MMA optimizer(Svanberg 1987).
Open Access This article is distributed under the terms of theCreative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricteduse, distribution, and reproduction in any medium, provided you giveappropriate credit to the original author(s) and the source, provide alink to the Creative Commons license, and indicate if changes weremade.
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