Focusing and imaging in microsphere-based microscopy · PDF fileFocusing and imaging in microsphere-based microscopy ... Arfken and H. J. Weber, Mathematical Methods for Physicists,

Focusing and imaging inmicrosphere-based microscopy

Thanh Xuan Hoang,1,∗

Yubo Duan,1 Xudong Chen,2

and George Barbastathis1,3

1Singapore–MIT Alliance for Research and Technology (SMART) Centre, Singapore 138602,Singapore

2Department of Electrical and Computer Engineering, National University of Singapore,Singapore 117576, Singapore

3Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge,Massachusetts 02139, USA∗[email protected]

Abstract: Microsphere-based microscopy systems have garnered lots ofrecent interest, mainly due to their capacity in focusing light and imagingbeyond the diffraction limit. In this paper, we present theoretical foundationsfor studying the optical performance of such systems by developing a com-plete theoretical model encompassing the aspects of illumination, sampleinteraction and imaging/collection. Using this model, we show that surfacewaves play a significant role in focusing and imaging with the microsphere.We also show that by designing a radially polarized convergent beam, wecan focus to a spot smaller than the diffraction limit. By exploiting surfacewaves, we are able to resolve two dipoles spaced 98 nm apart in simulationusing light at a wavelength of 402.292 nm. Using our model, we alsoexplore the effect of beam geometry and polarization on optical resolutionand focal spot size, showing that both geometry and polarization greatlyaffect the shape of the spot.

© 2015 Optical Society of America

OCIS codes: (180.0180) Microscopy; (180.4243) Near-field microscopy; (220.3630) Lenses;(350.3950) Micro-optics; (350.5730) Resolution.

References and links1. S. H. Goh, C. J. R Sheppard, A. C. T. Quah, C. M. Chua, L. S. Koh, and J. C. H. Phang, “Design considerations for

refractive solid immersion lens: application to subsurface integrated circuit fault localization using laser inducedtechniques,” Rev. Sci. Instrum. 80, 013703 (2009).

2. J. P. Hadden, J. P. Harrison, A. C. Stanley-Clarke, L. Marseglia, Y-L. D. Ho, B. R. Patton, J. L. O′Brien, and J.G. Rarity, “Strongly enhanced photon collection from diamond defect centres under micro-fabricated integratedsolid immersion lenses,” Appl. Phys. Lett. , 97, 241901 (2010).

3. T. X. Hoang, “Focusing light through spherical interface for subsurface microscopy,” Ph.D. dissertation (2014).4. T. X. Hoang, R. Chen, K. Agarwal, C. J. R. Sheppard, and X. Chen, “Imaging with annular focusing through a

dielectric interface,” in Proceedings of Focus On Microscopy 2014, Sydney, Australia 2014 April 13-16.5. D. R. Mason, M. V. Jouravlev, and K. S. Kim, “Enhanced resolution beyond the Abbe diffraction limit with

wavelength-scale solid immersion lenses,” Opt. Lett. 35, 2007–2009 (2010).6. D. A. Fletcher, K. E. Goodson, and G. S. Kino, “Focusing in microlenses close to a wavelength in diameter,”

Opt. Lett. 26, 399–401 (2001).7. J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I. C. Hwang, L. J.

Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-Field Focusing and Magnification through Self- AssembledNanoscale Spherical Lenses,” Nature (London) 460, 498–501 (2009).

#233013 - $15.00 USD Received 22 Jan 2015; revised 5 Apr 2015; accepted 16 Apr 2015; published 1 May 2015 © 2015 OSA 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.012337 | OPTICS EXPRESS 12337

8. D. Wildanger, B. R. Patton, H. Schill, L. Marseglia, J. P. Hadden, S. Knauer, A. Schnle, J. G. Rarity, J. L. O’Brien,S. W. Hell, and J. M. Smith, “Solid Immersion Facilitates Fluorescence Microscopy with Nanometer Resolutionand Sub-Angstrom emitter localization,” Adv. Mater 24, OP309–OP313 (2012).

9. Y. Yan, L. Li, C. Feng, W. Guo, S. Lee, and M. Hong, “Microsphere-Coupled Scanning Laser ConfocalNanoscope for Sub-Diffraction-Limited Imaging at 25 nm Lateral Resolution in the Visible Spectrum,” ACSNano, 8, 1809–1816 (2014).

10. V. M. Sundaram and S. B. Wen, “Analysis of deep sub-micron resolution in microsphere based imaging” Appl.Phys. Lett. 105, 204102 (2014).

11. A. Darafsheh, C. Guardiola, A. Palovcak, J. C. Finlay, and A. Carabe, “Optical super-resolution imaging byhigh-index microspheres embedded in elastomers,” Opt. Lett. 40, 5-8 (2015).

12. S. Lee, L. Li, and Z. Wang, “Optical resonances in microsphere photonic nanojets,” J. Opt. 16, 015704 (2014).13. Z. Wang, W. Guo, L. Li, B. Lukyanchuk, A. Khan, Z. Liu, Z. Chen, and M. Hong, “Optical virtual imaging at

50nm lateral resolution with a white-light nanoscope,” Nat. Commun. 2, 218 (2011).14. R. Ye, Y.-H. Ye, H. F. Ma, L. Cao, J. Ma, F. Wyrowski, R. Shi, and J.-Y. Zhang, “Experimental imaging properties

of immersion microscale spherical lenses,” Sci. Rep. 4, 1-5 (2014).15. Y. Duan, G. Barbastathis, and B. Zhang, “Classical imaging theory of a microlens with super-resolution,” Opt.

Lett. 38, 2988–2990 (2013).16. T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Multipole and plane wave expansions of diverging and converging

fields,” Opt. Express 22, 8949–8961 (2014).17. R. Chen, K. Agarwal, Y. Zhong, C. J.R. Sheppard, J. C.H. Phang, and X. Chen, “Complete modeling of subsurface

microscopy system based on aplanatic solid immersion lens,” J. Opt. Soc. Am. A 29, 2350–2359 (2012).18. A. J. Devaney and E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,”

J. Math. Phys. 15, 234–244 (1974).19. S. Orlov, U. Peschel, T. Bauer, and P. Banzer, “Analytical expansion of highly focused vector beams into vector

spherical harmonics and its application to Mie scattering,” Phys. Rev. A 85, 063825 (2012).20. J. D. Jackson, Classical electrodynamics (Wiley, 1998).21. G. Gouesbet and G. Grehan, Generalised Lorenz-Mie Theories (Springer-Verlag, 2011).22. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Fifth ed. (Academic Press, 2000).23. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Pt.2 (McGraw-Hill, 1953).24. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an

aplanatic system,” Proc. Roy. Soc. London Ser. A 253, 358–379 (1959).25. T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Rigorous analytical modeling of high-aperture focusing through

a spherical interface,” J. Opt. Soc. Am. A 30, 1426–1440 (2013).26. T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Multipole theory for tight focusing of polarized light, including

radially polarized and other special cases,” J. Opt. Soc. Am. A 29, 32–43 (2012).27. T. X. Hoang, X. Chen, and C. J. R. Sheppard, “Interpretation of the scattering mechanism for particles in a

focused beam,” Phys. Rev. A 86, 033817 (2012).28. W. C. Chew and Y. M. Wang, “Efficient Ways to Compute the Vector Addition Theorem,” J. Elect. Wave. Appl.

7, 651–665 (1993).29. M. Born and W. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 2005).30. J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, 2008).31. W. T. Grandy, Scattering of Waves from Large Spheres (Cambridge University Press, 2000).32. S. Quabis, R. Dorn, M. Eberler, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).33. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91,

233901 (2003).34. M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-

free microcavity biosensor platform,” Nature Nanotech. 9, 933–939 (2014).35. J. T. Robinson, L. Chen, and M. Lipson, “On-chip gas detection in silicon optical microcavities,” Opt. Express

16, 4296–4301 (2008).36. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt.

Soc. Am. A 10, 343–352 (1993).37. A. B. Pluchino, “Surface waves and the radiative properties of micron-sized particles,” Appl. Opt. 20, 2986–2992

(1981).38. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J.

Math. Phys. 10, 82–124 (1969).39. A. Devilez, N. Bonod, J. Wenger, D. Gerard, B. Stout, H. Rigneault, and E. Popov, “Three-dimensional subwave-

length confinement of light with dielectric microspheres,” Opt. Express 17, 2089–2094 (2009).40. A. Heifetz, S.-C. Kong, A. V. Sahakian, A. Taflove, and V. Backman, “Photonic Nanojets,” J. Comput. Theor.

Nanosci. 6, 1979–1992 (2009).41. A. Devilez, B. Stout, N. Bonod, E. Popov, “Spectral analysis of three-dimensional photonic jets,” œ16, 14200

14212 (2008).42. F. J. Duarte, “Multiple-prism grating solid-state dye laser oscillator: optimized architecture,” Appl. Opt. 38,


6347–6349 (1999).

1. Introduction

Optical imaging faces a fundamental limit because of diffraction, and thus sub-wavelengthimaging has been a topic of great interest for the last several decades. Among well-establishedtechniques, solid immersion has been known for improving spatial resolution [1] and increasingoptical collection efficiency [2]. A bulk solid immersion lens (SIL) with a radius of 1.5 mm wasused to resolve features with a resolution of 110 nm using 1.342 µm wavelength laser illumina-tion [3, 4].

Instead of using the bulk SIL, a micro-SIL was predicted to focus light to a tighter spot [5, 6]and has been reported to be able to image features beyond the diffraction limit in free space[5, 7]. Furthermore, stimulated emission depletion (STED) microscopy at visible wavelengthsusing a micro-SIL was able to reach a resolution of 2.4±0.3 nm, 1.8 times finer than withoutthe micro-SIL [8]. This ability of the micro-SIL is simply attributed to near-field focusing andmagnification effects [7].

Instead of using a micro-SIL, which is a truncated microsphere, a complete-microsphere-based visible light confocal microscope was developed, achieving a resolution of 25 nm [9].The microsphere-based microscope is currently an active topic in both theory [10] and exper-iment [11]. In theory, researchers have posited many different explanations for the ability ofthe microsphere-based microscope to achieve resolution beyond the diffraction limit; the sub-wavelength focusing ability was explained as being similar to superoscillatory lenses in [9],while photonic nanojet formation and optical resonance were proposed as the cause in [12].Meanwhile, [13] explained the phenomenon using virtual imaging and ray optics, while [14]provided an explanation using near-field optics and geometrical optics. However, super oscil-lations occur in a focusing system and not an imaging system—the high side-lobes commonlyassociated with super oscillations would deteriorate image quality in the latter. Hence, superoscillations alone would not be a complete explanation for super-resolved imaging. Likewise,photonic nanojets are also a focusing phenomenon, and may not be applicable to describingimaging, as was pointed out in [15]. Lastly, explanations based on ray optics and virtual imag-ing are not appropriate for a microsphere since ray optics has been known to be not applicablefor near-field interactions.

Based on vectorial electromagnetic analysis, Duan et al. in [15] show that the microsphere-based microscope cannot resolve two points separated less than 100 nm apart and that the bestresolution is obtained when optical resonance occurs inside the microsphere. However, sincetheir simulations do not agree with previous results, the authors acquiesce that their theoreticaltreatment may be incomplete. In this paper, we propose using multipole and plane wave ex-pansions [16] to rigorously model the entire microsphere-based microscope system, includingboth the focusing (illumination) and imaging sub-systems. Our theoretical model can be usedfor imaging a designed feature which mimicked a real object and hence provides an insightfulunderstanding about the microsphere-based microscopy.

2. Theory

A complete model of a microscope requires a full understanding of three processes: illuminat-ing the specimen with incident light, interaction of the specimen with the incident light, andcollection of scattered light from the specimen [17]. In general, all three processes contributeto the resolution attainable by the microscope, and hence it is necessary to study all three as acomplete system. Interaction between the incident light and the specimen can be solved numeri-cally as was presented in [17]. Hence, we will focus on illumination and collection in this paper.


First, let us rigorously derive the field at the output of a conventional imaging system; this resultwill serve as a foundation for the more complex derivation with regards to a microsphere-basedmicroscopy.

2.1. Imaging in a conventional microscope

Figure 1a) shows a simplified diagram of a conventional infinity-corrected imaging system.The source object O, fully contained inside sphere S, radiates electromagnetic energy in alldirections. The portion of this radiation that falls within a cone with half-angle αm < π/2 iscaptured by the objective lens L1 and collimated into a beam, which is then focused by tubelens L2 onto a CCD detector to generate the image. For this derivation, we will assume thatthe source is emanating fully coherent light at a wavelength of λ , and we will model this as acollection of electromagnetic multipole fields. Each of these electromagnetic multipole fieldscan be treated separately as shown in Fig. 1b), and we will derive its output by modeling eachlens as a section of a Gaussian reference sphere (GRS).

First, we adopt terminologies from Devaney and Wolf [18] and expand the source radiation

field as the following sum of electromagnetic multipole fields N(1)lm and M(1)

lm :

E(r) =∞

∑l=1

l

∑m=−l

[pmElN

(1)lm (r)+ pm

MlM(1)lm (r)], (1)

where pmEl and pm

Ml are multipole expansion coefficients [18, 19] (are also called as multipolemoments in [18, 20] and are related to beam shape coefficients in [21]), These coefficients aredetermined from the source distribution or boundary conditions. The electromagnetic multipole

fields are defined in terms of the conventional spherical Hankel function of the first kind h(1)l (kr)and the scalar spherical harmonics Y m

l (θ ,φ) as follows:

N(1)lm (r) = ∇×∇× [rh(1)l (kr)Y m

l (θ ,φ)]

= rl(l +1)

rh(1)l (kr)Y m

l (θ ,φ)+ θ[

1r

ddr

(rh(1)l (kr))∂

∂θY m

l (θ ,φ)]

+ φim

sinθ1r

ddr

(rh(1)l (kr))Y ml (θ ,φ),

M(1)lm (r) = ik∇× [rh(1)l (kr)Y m

l (θ ,φ)]

=−kh(1)l (kr)

[θ

msinθ

Y ml (θ ,φ)+ iφ

∂∂θ

Y ml (θ ,φ)

], (2)

where r, θ , and φ are the spherical unit vectors in spherical coordinates with the origin atthe source, i is the imaginary unit, k = 2π/λ is the wave number. Since there exist different

Fig. 1. Imaging with Conventional Lenses: a) Physical Setup and b) System Modeling.


definitions of the scalar spherical harmonics [22, 23], we adopt the definition of Arfken andWeber [22] and explicitly write it here for convenience:

Y ml (θ ,φ) = (−1)m

[2l +1

4π(l −m)!(l +m)!

] 12

Pml (cosθ)exp(imφ)

= clmPml (cosθ)exp(imφ), (3)

where Pml (cosθ) is the associated Legendre polynomial of degree l and order m and is defined

as follows:

Pml (x) =

12l l!

(1− x2)m/2 dl+m

dxl+m (x2 −1)l . (4)

One can notice that we omit the Condon-Shortley phase (−1)m in the definition of the asso-ciated Legendre polynomial but include it in the definition of the scalar spherical harmonics.Consequently, our definition of the scalar spherical harmonics is the same as that in [21, 20].For convenience, we can express Eq. (1) in the form of E(r) = rEr + θEθ + φEφ [16, 3] where

Er = k∞

∑l=1

l

∑m=−l

l(l +1)2l +1

clm

(pm

El [h(1)l−1(kr)+h(1)l+1(kr)]Pm

l (cosθ))

exp(imφ),

Eθ = k∞

∑l=1

l

∑m=−l

l(l +1)2l +1

clm

(pm

El

[h(1)l−1(kr)

l− h(1)l+1(kr)

l +1

]dPm

l (cosθ)dθ

−m2l +1

l(l +1)pm

Mlh(1)l (kr)

Pml (cosθ)

sinθ

)exp(imφ),

Eφ = ik∞

∑l=1

l

∑m=−l

l(l +1)2l +1

clm

(mpm

El

[h(1)l−1(kr)

l− h(1)l+1(kr)

l +1

]Pm

l (cosθ)sinθ

− 2l +1l(l +1)

pmMlh

(1)l (kr)

dPml (cosθ)

dθ

)exp(imφ). (5)

It is reasonable to assume in practice that the source radiates a limited number of electromag-netic multipole fields, and hence the infinite summations in Eq. (5) can be terminated at l = L.

Now the general expression in Eq. (5) can be simplified using a far-field approximationassuming we are operating in that regime. It is worth noting that Eq. (5) is valid everywhereoutside the source and in general the electromagnetic wave associated with Eq. (5) does notbehave as a spherical wave near the source. For this paper, we define the far-field regime to beone where the radiating field can be accurately approximated by spherical waves, i.e. when theradial coordinate r satisfies the following inequality:

kr � L(L+1)/2 (6)

Note that in most conventional microscopes, the objective is placed far enough from the object(i.e. the focal length f1 is long enough) that it is located in the far-field regime, and thus afar-field approximation would be valid. Assuming Eq. (6) holds, for all l ≤ L we can make thefollowing approximation:

h(1)l (kr)≈ (−i)l+1 eikr

kr(7)

which in turn allows us to write

h(1)l−1(kr)+h(1)l+1(kr)≈ 0 and l(l+1)2l+1

[h(1)l−1(kr)

l − h(1)l+1(kr)l+1

]≈ (−i)l eikr

kr . (8)


Substituting Eqs. (7) and (8) into Eq. (5), we obtain

Er ≈ 0, Eθ ≈ eikr

r

L

∑l=1

l

∑m=−l

Kml (θ)exp(imφ), Eφ ≈ eikr

r

L

∑l=1

l

∑m=−l

Gml (θ)exp(imφ), (9)

where

Kml (θ) = (−i)lclm

[pm

EldPm

l (cosθ)dθ

+ impmMl

Pml (cosθ)

sinθ

],

Gml (θ) = (−i)l−1clm

[mpm

ElPm

l (cosθ)sinθ

+ ipmMl

dPml (cosθ)

dθ

].

Equation (9) represents the fact that an electromagnetic wave behaves as a spherical wave at aposition far away from the source.

Since the objective lens L1 is located in the far-field, we can use ray optics to solve forlight refracting through it [24]. The vectorial coordinate system, which we will use, is given inFig. 1(b). At the Gaussian reference sphere GRS1, the incident vectorial electric field Eα1 α1 +

Eβ1β1 can be associated with a ray traveling from the centre of the GRS1. This ray is then

refracted and becomes a collimated ray, which is associated with an electric field Eρ ρ +Eϕ ϕ .We can show that

Eα1 = Eθ ≈ eik f1

f1

L

∑l=1

l

∑m=−l

Kml (α1)exp(imβ1), (10)

Eβ1= Eφ ≈ eik f1

f1

L

∑l=1

l

∑m=−l

Gml (α1)exp(imβ1), (11)

Eρ =Eα1√cosα1

, Eϕ =Eβ1√cosα1

. (12)

The collimated beam is then focused by the tube lens L2 which, in this paper, is a conventionallens with a long focal length f . The focused ray is associated with an electric field Eα α +Eβ βwhere

sinα =f1f

sinα1, β = β1, (13)

Eα = Eρ√

cosα = Eα1

√cosαcosα1

, Eβ = Eϕ√

cosα = Eβ1

√cosαcosα1

. (14)

The focal field at the focus of lens L2 can be estimated using the well-known angular spectrumrespresentation:

E(x,y,z) = − ik f eik f

2π

∫ αm

0sinαdα

∫ 2π

0dβ

⎡⎣Eα

⎛⎝cosβ cosα

sinβ cosα−sinα

⎞⎠

+Eβ

⎛⎝−sinβ

cosβ0

⎞⎠⎤⎦eik[ρ sinα cos(β−φ)+zcosα ]. (15)

Substituting Eq. (14) into Eq. (15), we can write the focal field as

E(x,y,z) =L

∑l=1

l

∑m=−l

⎛⎝ Im−1

l (r)+ Im+1l (r)

i(Im−1l (r)− Im+1

l (r))−2Im

l (r)

⎞⎠ , (16)


where

Im+1l (r) =− ik f eik f

2im+1ei(m+1)φ

∫ αm

0Jm+1(kρ sinα)[Um

l (α)cosα + iV ml (α)]eikzcosα sinα dα,

Im−1l (r) =− ik f eik f

2im−1ei(m−1)φ

∫ αm

0Jm−1(kρ sinα)[Um

l (α)cosα − iV ml (α)]eikzcosα sinα dα,

Iml (r) =− ik f eik f

2imeimφ

∫ αm

0Jm(kρ sinα)Um

l (α)sinαeikzcosα sinα dα,

Uml (α) =

eik f1

f1Km

l (α1)

√cosαcosα1

, V ml (α) =

eik f1

f1Gm

l (α1)

√cosαcosα1

.

Equation (16) gives the vector field impinging on the CCD. When the object is small comparedto the wavelength, the source can be modeled as a single dipole (i.e. l = 1). In this case, theformula in Eq. (16) is equivalent to the dyadic Green function of the imaging system. However,if the object is big compared to the wavelength, the radiation field will contain many multipolefields. In fact, Eq. (16) expresses the focal field as a sum of diffraction integrals applied to theconstituent multipole fields from the source [25].

2.2. Illumination and imaging in a microsphere-based microscope

Fig. 2. Microsphere based Microscopy.

Now let us consider a modification to the conventional microscope—inserting a microspherebetween the object and the objective lens and adding a beam splitter for illumination, as shownin Fig. 2. We will refer to this new optical recipe as the microsphere-based microscope. In-cident light reflects off the beam splitter and illuminates the specimen after passing throughthe objective lens and microsphere. This light then interacts with the specimen, which in turngenerates some outgoing radiation, represented as a collection of dipole fields by the numericalmethod [17]. This outgoing light passes through the microsphere and is then imaged by the twolenses, similar to the conventional microscope. Since our derivation for a conventional micro-scope is general enough for any coherent source located inside some finitely large sphere S, amodel for the imaging aspect of the microscope can be derived by constructing a multipole-based expression for the field generated by a dipole behind the microsphere. As was mentionedbefore, understanding the microscope requires understanding the illumination aspect as well,so we will first start with a derivation for the focusing field before presenting the results for theimaging aspect.

2.2.1. Illumination through a microsphere

A converging wave from a positive lens is incident on a microsphere, as shown in Fig. 3(a),where k is the wave vector, αm is the half-angle of the incident cone of light, R is the radius


Fig. 3. Illumination with a) An incident converging wave and b) An incident plane wave.

of the microsphere centered at point O, ρ is the radial axis of a cylindrical coordinate systemcentered at O, z is the longitudinal axis as well as optical axis, F and f are the focus and focallength of the lens, respectively, and d is the distance between F and O.

It is a nontrivial task to determine the multipole expansion coefficients for a converging wave.Given an incident plane wave approaching the lens (which we will assume to be aplanatic forthis derivation), we can express the electric field on the GRS as EGRS = Eα α +Eβ β and themultipole coefficients of the focused beam can be estimated by matching EGRS with Eq. (5) asshown in [3, 25, 26, 27]:

pmEl =− il f eik f clm

l(l +1)

∫ 2π

0

∫ α1m

0

(dPm

l (cosα)

dαEα − im

Pml (cosα)

sinαEβ

)e−imβ sinα dα dβ ,

pmMl =

il f eik f clm

l(l +1)

∫ 2π

0

∫ α1m

0

(dPm

l (cosα)

dαEβ + im

Pml (cosα)

sinαEα

)e−imβ sinα dα dβ . (17)

We can then substitute these coefficients into a modified version of Eq. (5) (where instancesof h(1), the spherical Hankel function of the first kind, are replaced with the spherical Hankelfunction of the second kind, h(2)) to obtain an expression for the vectorial field of a convergingwave. Note that this expression would be based on a spherical coordinate system centered onF in Fig. 3(a). In practice, the focus of the objective lens and the center of the sphere are notcoincident (i.e. d > 0), and hence the coordinate system for this converging wave field wouldnot be centered on the microsphere. Thus, to use previous results for scattering and focusingby a microsphere, we would need to translate the coordinate system for the converging wave’svectorial field so that the origin is at the center of the microsphere. We do this by employing thetranslational addition theorem [28] and obtain the following multipole expansion coefficients[3]:

pm′El′ =

∞

∑l=1

l

∑m=−l

[Alml′m′ pm

El + iBlml′m′ pm

Ml ], pm′Ml′ =

∞

∑l=1

l

∑m=−l

[Alml′m′ pm

Ml − iBlml′m′ pm

El ] (18)

where Alml′m′ and Blm

l′m′ are the translational coefficients derived in [28]. One can find details forderiving Eq. (18) in Appendix B of [3]. These translational coefficients can be ignored whenl′ is sufficiently large (i.e. l′ > kd + 1.8η 3

√kd for a desired accuracy of 10−η ). The multipole

expansion coefficients in Eq. (18) can be substituted into a modified version of Eq. (5) (where

instances of h(1)l are replaced with the spherical Bessel function jl) to obtain an expression forthe translated vectorial field.

Using this translated field, we can now derive expressions for multipole expansion coeffi-cients corresponding to the vectorial field both inside and outside the microsphere using the


boundary condition [3]. Inside the microsphere, we obtain the following multipole expansioncoefficients:

pmEl = (ε/εs)cl′ p

m′El′ pm

Ml = (k/ks)dl′ pm′Ml′ (19)

which can then be substituted into a modified version of Eq. (5) (where instances of h(1) arereplaced with the spherical Bessel function of the first kind and instances of k are replaced withks). In Eq. (19), ε and εs are the permittivity of the surrounding material and the microsphererespectively, k and ks are the wave numbers for the surrounding material and the microsphererespectively, and the internal Mie scattering coefficients are given by cl′ and dl′ . For the scatte-ring field, we have the following expression for the multipole expansion coefficients

pmEl = al′ p

m′El′ pm

Ml = bl′ pm′Ml′ (20)

which can then be substituted into the unmodified Eq. (5). al′ and bl′ are the external Mie scat-tering coefficients. We followed the derivation procedure in [29, 30] and derived our scatteringcoefficients in [3, 27], we list them here for convenience:

al =εεs

ksk Jl(kR)J′l (ksR)− J′l (kR)Jl(ksR)

H(1)′l (kR)Jl(ksR)− ε

εs

ksk H(1)

l (kR)J′l (ksR), bl =

μμs

ksk Jl(kR)J′l (ksR)− J′l (kR)Jl(ksR)

H(1)′l (kR)Jl(ksR)− μ

μs

ksk H(1)

l (kR)J′l (ksR),

cl =i ks

k

H(1)′l (kR)Jl(ksR)− ε

εs

ksk H(1)

l (kR)J′l (ksR), dl =

i ksk

H(1)′l (kR)Jl(ksR)− μ

μs

ksk H(1)

l (kR)J′l (ksR),

where Jl(x) = x jl(x) is Riccati-Bessel function; H(1)l (x) = xh(1)l (x) and H(2)

l (x) = xh(2)l (x) areRiccati-Hankel functions.

For completeness, we would like to include results for a plane wave (shown in Fig. 3(b))for comparison. An incident linearly polarized plane wave Einc = xE0eikr cosθ can be expressedin terms of the electromagnetic multipole fields with the following multipole expansion coeffi-cients:

pmEl =

E0

kil+1

√π(2l +1)l(l +1)

(δ 1m −δ−1

m ), pmMl =

E0

kil

√π(2l +1)l(l +1)

(δ 1m +δ−1

m ). (21)

where δ ba is the Kronecker delta function (i.e. δ b

a = 1 ⇐⇒ a = b). These are analogous tothe coefficients in Eq. (18), and the internal and scattering fields can be obtained in a fashionsimilar to the converging wave case. Likewise, for a circularly polarized plane wave Einc =(x+ iy) E0√

2eikr cosθ , the multipole expansion coefficients are:

pmEl =

√2E0

kil+1

√π(2l +1)l(l +1)

δ 1m, pm

Ml =

√2E0

kil

√π(2l +1)l(l +1)

δ 1m. (22)

2.2.2. Far-field imaging through a microsphere

As mentioned before, the light radiating back from an illuminated object can be modeled bya number of discrete dipoles, and thus now we will present a rigorous expression for the elec-tromagnetic field at the CCD due to an axis-aligned dipole (along x, y or z) placed behind themicrosphere, i.e. we derive the dyadic Green function for the collection subsystem. The outputfield due to an arbitrary dipole can then be calculated using the dyadic Green function with theprojected length of the dipole moment along the three axes.


Consider a dipole located next to but outside of a microsphere; it will radiate what we willcall an incident field, and the microsphere interacts with this field, creating a scattering field.What arrives at lens L1 is a sum of these two fields, and we can derive multipole coefficientsfor the total field resulting from a dipole along each of the basic directions x, y and z. Thederivation, not explicitly shown here, is similar to the previous section—derive an expressionfor the incident field centered on the microsphere and apply the boundary condition [3]. Theresulting coefficients are as follows:

1. A Hertzian electric dipole with the current dipole moment Il pointing in the x direction:

pmEl =

ωμIl

2√

6π

(al [A

1,1lm −A1,−1

lm ]+ [C1,1lm −C1,−1

lm ]),

pmMl =−i

ωμIl

2√

6π

(bl [B

1,1lm −B1,−1

lm ]+ [D1,1lm −D1,−1

lm ]), (23)

where the translational coefficients (A, B, C and D) are calculated via the method in [28].

2. A Hertzian electric dipole with the current dipole moment Il pointing in the y direction:

pmEl =

iωμIl

2√

6π

(al [A

1,1lm +A1,−1

lm ]+ [C1,1lm +C1,−1

lm ]),

pmMl =

ωμIl

2√

6π

(bl [B

1,1lm +B1,−1

lm ]+ [D1,1lm +D1,−1

lm ]). (24)

3. A Hertzian electric dipole with the current dipole moment Il pointing in the z direction:

pmEl =

ωμIl

2√

3π

(alA

1,0lm +C1,0

lm

), pm

Ml =−iωμIl

2√

3π

(blB

1,0lm +D1,0

lm

). (25)

Fig. 4. Far field approximation for expressing the field in O coordinate system.

We note that Eqs. (23) , (24), and (25) are the multipole expansion coefficients of the corre-sponding total fields expressed in a coordinate system where the origin is at the center (O′) ofthe microsphere, as shown in Fig. 4. However, from the point of view of the far field, the dipoleappears to be at position O due to the presence of the microsphere. Thus, for aberration-freeimaging, the lens L1 has to be positioned such that O is at the front focal plane of the lens, notO′. To properly analyze this situation, we would have to shift the coordinate system’s origin


from O′ to O, i.e. substituting Eqs. (23) , (24), and (25) into Eq. (18). Alternatively, we canapproximate the multipole coeficients in the coordinate system O by employing the fact that inpractical cases, the distance between O and O′ is much less than the focal length of the lensL1 (d f1). This means that the electromagnetic wave on the GRS can be approximated as aspherical wave in both O′ and O-coordinate systems. At a far-field position P (r,θ ,φ ), we canapproximate θ ≈ θ ′, φ = φ ′, r = r′+d cosα1. For wave vector k1 (k,α1,β1) at P, we have k = k′,α1 ≈ α ′, β1 = β ′, and eikr′/r′ ≈ e−ikd cosα1eikr/r. Using these approximations and Eq. (9), wecan express the multipole coefficients pm

El and pmMl representing the field in the O-coordinate

system in terms of those in the O′-coordinate system (pmEl and pm

El from Eqs. (23) , (24), and(25)):

pmEl = e−ikd cosα1 pm

El pmMl = e−ikd cosα1 pm

Ml (26)

We can then substitute into Eq. (16) to obtain the desired dyadic Green function.

3. Discussion

3.1. Resonance in the microsphere

When light is scattered by a sphere, the total energy lost from the incident wave can be deter-mined by using the optical cross-section theorem [29], which relates the rate at which energyis lost from the incident field to the amplitude of the scattered field in the direction of inci-dence (the forward direction). The mathematical formula of the theorem expresses the extinc-tion cross-section in terms of the forward scattering amplitude, and it was first derived by van deHulst. The extinction cross section for scattering by a microsphere contains a series of spikes asa function of wavelength [31]. It has been shown that these spikes manifest optical resonances[31], and they have been investigated and exploited in numerous applications, such as biosens-ing [34] and environmental monitoring [35]. Each of the spikes corresponds to one scatteringcoefficient (al or bl), i.e. the optical resonance is due to one particular partial wave (Nm

l or Mml )

temporarily trapped inside the microsphere. More intuitively, these resonances can be shown tobe analogous to quantum-mechanical shape resonances in which the electromagnetic energy istemporarily trapped near the surface of the microsphere in a dielectric potential well [36].

Let us consider the rays associated with a so-called impact parameter D traveling close tothe edge of the microsphere, as shown in Fig. 3(b) [31]. These rays tunnel through the centrifu-gal barrier to the surface of the microsphere and are then totally internally reflected multipletimes inside the near-surface annulus (D/n ≤ r ≤ R), each time only losing a small amount ofenergy [31]. They are usually refered to as surface waves, which in turn account for the opticalresonances [37]. Resonance occurs when the ray associated with one of the partial waves, aftertraveling around the microsphere, is in phase with itself when it has traveled to the point whereit originally entered the microsphere—there is a standing wave along the surface of the micro-sphere. Nussenzveig defines the multipole fields associated with these rays as being in the edgedomain; these fields are associated with multipole orders l− < l < l+, where l± ≈ kR±η0

3√

kRand η0 is a constant of order unity (empirically, η0 > 3) [38]. Since optical resonance is com-pletely due to surface waves, we will consider only edge domain multipole fields in our analysis.

Let us consider a microsphere with radius R = 2.37µm and a refractive index ns = 1.46 tomatch the parameters in [13, 15]. With these parameters and a choice of η0 = 4, λ = 400nmwe obtain:

kR ≈ 37, l− ≈ 25, l+ ≈ 49. (27)

There are many multipole orders in the edge domain for this case, and we will study in partic-ular orders l = 40 and l = 43. We plot the Mie scattering coefficients c40, d40, c43, and d43 as a


function of the wavelength λ for 0.35µm < λ < 0.5µm in Fig. 5(a). We observe from the plotthat there are several visible peaks for each scattering coefficient (in fact, there are an infinitenumber of such peaks over the electromagnetic spectrum); the width of the peaks decreases aswavelength increases. The largest peaks for each of the scattering coefficients are essentiallyspikes. We do not consider wavelengths shorter than 0.35 µm because the peaks in the scatte-ring coefficients start widening so much that the maxima are too low, resulting in negligibleresonance. We also do not consider wavelengths longer than 0.5 µm because the peaks are sonarrow that they are not applicable in practice.

In Fig. 5(b), we show the magnitude of the Mie scattering coefficients for 1 ≤ l ≤ 50 atλ = 401.6345nm (the location of the middle d43 peak in Fig. 5(a)) as well as just |dl | atλ = 403.07nm (the off-resonant case). We observe that the resonant scattering coefficientd43 = −21.3i is much higher in magnitude than the other coefficients. This high value of thescattering coefficient represents the constructive interference of the partial wave l = 43 insidethe microsphere. It has been known that Mie scattering does not explain the effect of the multi-ple reflections inside the microsphere [27].

To further our understanding of the multiple reflections occurring inside the microsphere, itis necessary to adopt the Debye series, since standard Mie theory considers the entire processas a whole [25, 31]. The Debye series expresses each of the Mie scattering coefficients (al , bl ,cl , dl) in terms of an infinite summation. Each term represents one particular scattering eventoccurring at the microsphere’s surface. In the case of resonance, we need to include severaldozen terms in order for the Debye series to converge to the Mie scattering coefficient. Foroff-resonance, we only need to include several terms for convergence [31]. The constructivecontribution of many internal reflections physically explain the high magnitude of the resonantscattering coefficient, d43 in this particular example.

3.2. Microsphere focusing of a converging wave

Using our derivation for illumination through a microsphere, we will now show by example thata microsphere can focus light from a converging wave into a tight three-dimensional spot; suchfocal spots are important to many applications, e.g. [39]. We consider a converging wave, theso-called axial dipole wave [3], formed by an aplanatic lens focusing a radially polarized wave.Substituting the description of this radially polarized wave (Eα = sinα , Eβ = 0) into Eq. (17)yields the following multipole expansion coefficients:

pmEl =− il f eik f [π(2l +1)]

12

l(l +1)δ 0

m

∫ αm

0

dPl(cosα)

dαsin2 α dα, pm

Ml = 0. (28)

0.3 0.35 0.4 0.45 0.50

100

200

300

λ (μm)

|cl|,|

d l|

c

43

d43

c40

d40

10 20 30 40 500

5

10

15

20

25

Multipole order l

Mag

nitu

de

|al| for resonance

|bl| for resonance

|cl| for resonance

|dl| for resonance

|dl| for off−resonance

Resonant mode: l=43

a) Mie coefficients with l = 40 and l = 43. b) The first 50 Mie coefficients.

Fig. 5. a) Scattering coefficients as a function of the wavelength λ : the peaks correspondto the resonant wavelengths and b) Scattering coefficients with a resonant wavelength ofλ = 401.6345nm and an off-resonant wavelength of λ = 403.07nm.


−2 0 2 4 6−3

−2

−1

0

1

2

3

z(μm)

x(μm

)

Electric Intensity

2

4

6

8

10

12

14

16

x 104

−2 0 2 4 6−3

−2

−1

0

1

2

3

z(μm)

x(μm

)

Electric Intensity

1

2

3

4

5

Fig. 6. Electric intensity in linear (left) and logarithmic (right) scales with λ =401.6345nm. Light propagates from the left to the right sides.

Using the derived results in Section 2.2.1 with parameters λ = 401.6345nm, R = 2.37µm,d = 3R, and αm = arcsin(R/d), we can compute the electric intensity of the illuminated micro-sphere, shown in Fig. 6. It should be noted that λ = 401.6345nm is the resonant wavelength forthe TE mode (dl pm

MlMml with l = 43) as indicated in Fig.5. However, a converging beam with

radial polarization is completely described by the TM modes, i.e. pmMl = 0 which explains that

we do not observe any resonance in Fig. 6.Figure 6 shows that the field is highly confined to the surface, especially along the optical

axis. The focal spot has a FWHM of 234 nm (≈ λ/1.72) as indicated by the green curve inFig. 7(a). It has been known that focusing a radially polarized beam with a high numericalaperture (NA) aplanatic lens results in a tighter focal spot than that of focusing a linearly polar-ized beam [32, 33]. But a radially polarized beam with a low NA may result in a focal spot ofa donut-shaped spot. In our case, the incident beam is a low NA (= 1/3) but the focus is tight.This tight spot is attributed to the microsphere which guides light to converge to the focus witha semi-angle of 90o as observed in Fig. 6, i.e. the effective NA is 1. To make the focal spoteven tighter, we can increase the contribution of the surface wave to the focal spot by blockingthe center portion of the incident beam (i.e. block all of the light within a cone with semi-angleα0). Using this annular focusing technique, we can reduce the FWHM to 171 nm, representedby the blue curve in Fig. 7(a).

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

x(μm)

Nor

mal

ized

Inte

nsity

d=3R, αm

=0.34,αo=0

d=3R,αm

=0.34,αo=0.3

d=7R, Resonance, αo=0

d=7R, Off−resonance, αo=0

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

x(μm)

Nor

mal

ized

Inte

nsity

CP: ResonanceCP: Off−resonanceLP

x: Resonance

LPy: Resonance

a) A converging wave. b) A plane wave.

Fig. 7. Electric intensity at focus along the transversal direction with a) λ = 401.6345nm(off-resonance) and λ = 439.44nm (resonance). b) λ = 403.07nm (off-resonance) and λ =445.5127nm (resonance). CP and LP refer to circularly polarized and linearly polarizedlight, respectively.

An alternative approach would be to move the microsphere toward the lens, i.e. increasingd. At d = 7R, we obtain a FWHM of 174 nm as represented by the pink curve in Fig. 7(a). Wecan further tighten the spot transversally by tuning the wavelength so that resonance occurs.Figure 8 shows two such wavelengths: 397.44 nm and 439.44 nm. Resonance is characterized


by a radial mode number p (p peaks along the radial direction) and an angular mode number l.We excite the microsphere with (l = 43, p = 2) and (l = 43, p = 1) in Fig. 8(a) and Fig. 8(b),respectively. The resulting transversal cross section for the 439.44 nm case is represented bythe red curve in Fig. 7(a), with a FWHM of 138 nm. The transversal cross section for the397.44 nm is nearly identical and thus we have omitted it from Fig. 7(a). One important propertyof focusing a radially polarized beam is that the side-lobes of the electric intensity distributionare relatively low compared to those of linear and circular polarizations. The low side-lobes areimportant in imaging applications since high side-lobes may cause distortion and poor contrast[15].

−5−4−3−2 −1 0 1 2 3 4 5−5−4−3−2−1

012345

z(μm)

x(μm

)

Electric Intensity in Logarithm Scale

−1

0

1

2

3

4

−5−4−3−2−1 0 1 2 3 4 5−5−4−3−2−1

012345

z(μm)

x(μm

)

Electric Intensity in Logarithm Scale

−2

0

2

4

6

a) λ = 397.44nm (|c43|= 23.6364). b) λ = 439.44nm (|c43|= 330.6155).

Fig. 8. Resonance with a converging wave illumination and d = 7R.

We note that this FWHM is close to λ/(2ns), where ns is the index of refraction for theinterior of the microsphere, although it is not directly dependent on the wavelength; λ/(2ns) is136 nm and 150.5 nm for λ = 397.44nm and λ = 439.44nm, respectively, although the FWHMremained almost the same at 138 nm in both cases. In fact, the FWHM was smaller than λ/(2ns)in the λ = 439.44nm case. In this resonant case, the effective NA of the incident beam isapproximately equal to ns which is attributed to the dominant contribution of the surface wave.

3.3. Convergent versus planar incident field

For scanning or confocal microsphere-based microscopy, the illumination usually involves aconverging beam, although analysis of such systems have sometimes been conducted using aplane wave illumination model, e.g. in [9]. However, the beam geometry and polarization playa significant role in determining the shape of the focal spot, as we will now discuss.

For λ = 439.44nm, an incident converging wave with radial polarization results in a tightspot as observed in Fig. 8(b). However, if we switch to a circularly-polarized plane wave atthe same wavelength, the result is a wider donut-shaped spot, as demonstrated by the double-lobed pattern in Fig. 9(a). To obtain a tight spot on the microsphere’s surface with circular orlinear polarization, we may choose to excite a TE mode in the microsphere instead of the TMmode. For example, if we excite the microsphere at a wavelength of 445.5127 nm, this willinduce resonance for the mode l = 43, p = 1 (|d43| = 280.258), as shown in Fig. 9(b). Crosssections of the electric intensities at the focus along the x direction are shown in Fig. 7(b) forincident circularly polarized plane waves as well as plane waves linearly polarized in the xand y directions. The FWHM along the x direction for circular polarization is 124.8 nm, while


for linear polarization, the FWHMs are 172.2 nm and 98.3 nm for polarization along x and y,respectively. The asymmetry in the FWHMs for linear polarization is similar to the asymmetrywhen focusing linearly polarized light using an aplanatic lens [24].

To study the focusing property of the microsphere off-resonance, we choose the nonresonantwavelength of λ = 403.07nm. Figure 9c) shows a photonic jet formed after the microspherefor the case of off-resonance [40]. The electric intensity along the transversal direction at themaximum intensity of the photonic jet is represented by the solid pink curve in Fig.7(b), and ithas a FWHM of 268.4 nm (≈ λ/1.5). Using the Debye series, we can show that the formationof the photonic jet is mainly due to light refraction at the front and back interfaces, i.e. wecan ignore the contribution of the surface wave to the formation. In [41], the strong focusingability of the microsphere is explained as a consequence of the interaction between the incidentwave and the microsphere which enhances the high spatial frequency components in the angularspectrum content of the scattering field.

−3 −2 −1 0 1 2 3 4−3

−2

−1

0

1

2

3

z(μm)

x(μm

)

Electric Intensity

2

4

6

8

x 104

−3 −2 −1 0 1 2 3 4−3

−2

−1

0

1

2

3

z(μm)

x(μm

)Electric Intensity

0.5

1

1.5

2

x 105

−3 −2 −1 0 1 2 3 4−3

−2

−1

0

1

2

3

z(μm)

x(μm

)

Electric Intensity

100

200

300

a) λ = 439.44nm. b) λ = 445.51nm. c) λ = 403.07nm.

Fig. 9. A plane wave illumination with a circular polarization.

Looking at Fig. 7(b), it is obvious that even with an incident plane wave, the shape of thefocal spot depends greatly on the incoming polarization. A comparison with the convergentwave results in Fig. 7(a) also shows that geometry plays an important role as well. Based onthese observations, it would appear that Yan et al.’s superoscillatory explanation for their 25 nmresolution is incomplete [9], since an analysis of plane wave illumination was used to explainresults from an experiment that uses convergent wave illumination. It is not clear what state ofpolarization the aforementioned convergent wave is in, but if it were radially polarized, then itwould form a tight focal spot near the surface of the microsphere without significant side lobes,as shown in Fig. 7(a).

3.4. Far-field imaging

Now that we have examined the illumination aspect of the microsphere-based microscope withsimulation examples, let us also do the same for the imaging/collection aspect as well. However,recall earlier that we assumed that the objective lens operates in the far-field regime, i.e. Eq. (6)holds. Thus, before we present imaging simulation results, we would first like to present somenumerical examples supporting this far-field assumption.

Since any object can be modeled as a sum of dipoles, we will examine the case of a singledipole placed at a distance d from the center of the microsphere with specific wavelengths; anyconclusions can be extended to general objects because we are in effect investigating the dyadicGreen function. Let us investigate the magnitude of the multipole expansion coefficients. Sincewe are only interested in objects outside the microsphere (i.e. d ≥ R), the truncation multipolenumber L will only depend on the translational distance d.

Figure 10 shows the magnitudes of the first 45 multipole expansion coefficients for a x-dipoleon the surface of the microsphere (d = R and R = 2.37µm) at two different wavelengths—


10 20 30 40 450

0.01

0.02

0.03

0.04

Multipole Order l

|PE

l1

|, |P

Ml

1|

|p

El1 |

|pMl1 |

10 20 30 40 450

0.5

1

1.5

Multipole Order l

|PE

l1

|, |P

Ml

1|

|p

El1 |

|pMl1 |

|d43

|=280.258

a) Multipole coefficients with off-resonance. b) Multipole coefficients with resonance.

Fig. 10. Multipole coefficients with a) λ = 450nm and b) λ = 445.5127nm.

450 nm for an off-resonant case and 445.5127 nm for a resonant case. As evident from themagnitude plots, we can accurately model the total field even if we ignore all coefficients withl > 40 for the off-resonant case and l > 45 for the resonant case. This is due to the fact thatkd ≈ 33 for both cases, and if we apply the rule of thumb from Section 2.2.1, we get thatL = kd +3(kd)(1/3) ≈ 43.

In practice, a microscope usually uses an objective lens with a focal length in millimeters.For example, the Olympus MDPlan 80×/0.9 objective from [13] has an effective focal lengthof f1 = 2.25mm, and thus k f1 ≈ 3× 104 for both wavelengths. This is more than an orderof magnitude greater than L(L+ 1)/2 ≈ 946, and hence the far-field condition in Eq. (6) issatisfied, an assumption we made for our theoretical model.

a) No sphere. b) Sphere in off-resonance. c) Sphere in resonance.

Fig. 11. Images of two dipoles spaced 109 nm apart with a) λ = 445.5127nm, b) λ =450nm, and c) λ = 445.5127nm.

Now that we have validated the far-field condition, we will discuss the imaging results shownin Fig. 11 for a pair of mutually incoherent dipoles 109 nm apart in three different situations—a) without a microsphere, b) behind a microsphere in off-resonance, and c) behind a micro-sphere in resonance. Imaging without a microsphere, as shown in Fig. 11a), demonstrates thata conventional microscope is unable to resolve the two dipoles due to the diffraction limit. Al-though using a microsphere without resonance imparts magnification on the sample, as shownin Fig. 11(b), we still cannot resolve the two dipoles. It is only with microsphere in resonance(i.e. at λ = 445.5127nm) that we can resolve the two, as seen in Fig. 11(c). The resonant con-dition requires a laser with a high monochromaticity. We adopt Johnson’s derivation in [36]and derive an analytical formula for estimating the laser linewidth Δλ for efficiently exciting aresonant TE mode as follows:

Δλ =−2λ0

(n2s −1)k0R

(H(1)

l (k0R)− Jl(k0R))2 . (29)


In our case, we substitute ns = 1.46, λ0 = 0.4455127µm, R = 2.37µm, k0 ×R = (2π/λ0)×(2.37) = 33.4247, and l = 43 into Eq. (29) and obtain Δλ ≈ 6.5×10−4 nm. This accuracy canbe experimentally produced –for an example– by using optimized solid-state multiple-prismgrating laser oscillators which is shown to produce a laser with a laser linewidth Δλ ≈ 4×10−4

nm (Δ f = 350MHz) [42].We also note that the microsphere in resonance transfers energy to the far-field region more

efficiently. The maximum intensity in Fig.11(c) is much larger than that in Fig.11(b). This isdue to the evanescent wave on the microsphere being converted into a propagating wave moreefficiently. This wave conversion has been previously exploited in microdroplet lasing, in whichthe microsphere’s surface plays the role of resonance cavity mirrors in a standard laser [31].

Obviously, resolution and conversion efficiency depend on the resonant wavelength, i.e. theresonant mode. For example, at a resonant wavelength of λ = 402.2920nm, we can resolve twodipoles placed 98 nm apart. Furthermore, the resonant scattering coefficient |d48| = 628.6162is slightly more than twice that of the resonant scattering coefficient from the previous case,making the maximum intensity more than 5 times higher compared to that in Fig. 11(c).

It should be noted that the microsphere induces a magnification of 2.93 in the off-resonantcase and a magnification of 1.83 in the resonant case. Ray optics gives an approximate magni-fication of ns/(2− ns) ≈ 2.70 [13], quite close to that of the non-resonant case. This accurateapproximation is due to the fact that refraction plays a dominant role in off-resonance, whilethat is not the case in resonance. However, even with a higher magnification of 2.93, the off-resonant case does not yield a higher resolution than that of the resonant case [15]. Given theseresults, it is evident that coupling from the evanescent field onto a propagating field plays apivotal role in resolution enhancement with microsphere-based microscopy.

3.5. Future directions

As mentioned before, a complete model of the microsphere-based microscope comprises threecomponents: focusing of the incident light, interaction of the incident light with the sample,and imaging of the scattered light. As a next step, we would like to apply the rigorous focus-ing and imaging models developed in this paper along with the numerical model for sample-illumination interaction in [17] to properly simulate a microsphere-microscopy system imagingspecific specimens in various modalities, such as wide-field, scanning and confocal. In doing so,we would be able to validate our theory and provide a physically rigorous explanation for pre-viously obtained experimental results, such as those from [9, 13]. We would also be able to useour complete model to further explore the resolution limits of microsphere-based microscopyin a principled fashion.

Acknowledgments

We thank Dr. Zhengyun Zhang for proofreading our manuscript. This research was supportedby the National Research Foundation Singapore through the Singapore-MIT Alliance for Re-search and Technology’s Center for Environmental Sensing and Modeling interdisciplinary re-search program. This research was also supported by the National Research Foundation, PrimeMinisters Office, Singapore under its Competitive Research Programme (CRP Award No. NRF-CRP10-2012-04).


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Focusing and imaging in microsphere-based microscopy · PDF fileFocusing and imaging in microsphere-based microscopy ... Arfken and H. J. Weber, Mathematical Methods for Physicists,