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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 8, Iss. 7 — Aug. 1, 2013
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A complete and computationally efficient numerical model of aplanatic solid immersion lens scanning microscope

Rui Chen, Krishna Agarwal, Colin J. R. Sheppard, Jacob C. H. Phang, and Xudong Chen  »View Author Affiliations


Optics Express, Vol. 21, Issue 12, pp. 14316-14330 (2013)
http://dx.doi.org/10.1364/OE.21.014316


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Abstract

This paper presents a computational model for modeling an aplanatic solid immersion lens scanning microscope. The scanning microscope model consists of three subsystems, each of which can be computed as a separate system, connected to the preceding or succeeding subsystem through the input/output only. Numerical techniques are used to enhance the computational efficiency of each subsystem. A distinct merit of the proposed model is that it can be used to simulate imaging results for diverse setups of the scanning microscope, like various polarizations, numerical aperture, and different detector pinhole sizes. It allows the study and analysis of both theoretical aspects like achievable resolution, and practical aspects like expected images for different object patterns and experimental setups. Further, due to its computational efficiency, diverse large scale structures can be easily simulated in scanning microscope and good experimental approaches determined before indulging into the time consuming and costly process of experimentation.

© 2013 OSA

1. Introduction

The need for high resolution imaging necessitates the use of high numerical aperture (NA) optical systems. Solid immersion lens (SIL) microscopy provides higher spatial resolution, improved light collection efficiency, and the capability of subsurface imaging [1

1. S. M. Mansfield and G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615–2616 (1990) [CrossRef] .

3

3. S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. 78, 4071–4073 (2001) [CrossRef] .

]. Experimental implementations of this technique as a wide-field microscope [2

2. Q. Wu, L. P. Ghislain, and V. B. Elings, “Imaging with solid immersion lenses, spatial resolution, and applications,” Proc. IEEE 88, 1491–1498 (2000) [CrossRef] .

, 4

4. F. H. Köklü, J. I. Quesnel, A. N. Vamivakas, S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, “Widefield subsurface microscopy of integrated circuits,” Opt. Express 16, 9501–9506 (2008) [CrossRef] [PubMed] .

] and a scanning microscope [5

5. Semicaps, “Optical fault localization system,” http://www.semicaps.com/innovations.htm (2011).

9

9. F. H. Köklü and M. S. Ünlü, “Subsurface microscopy of interconnect layers of an integrated circuit,” Opt. Lett. 35, 184–186 (2010) [CrossRef] [PubMed] .

] have been reported in the field of imaging solid state devices. For the SIL microscope, only the focusing of incident light through SIL has been theoretically investigated in detail [3

3. S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. 78, 4071–4073 (2001) [CrossRef] .

, 8

8. A. N. Vamivakas, R. D. Younger, B. B. Goldberg, A. K. Swan, M. S. Ünlü, E. R. Behringer, and S. B. Ippolito, “A case study for optics: The solid immersion microscope,” Am. J. Phys. 76, 758–768 (2008) [CrossRef] .

, 10

10. S. H. Goh and C. J. R. Sheppard, “High aperture focusing through a spherical interface: Application to refractive solid immersion lens (RSIL) for subsurface imaging,” Opt. Commun. 282, 1036–1041 (2009) [CrossRef] .

12

12. 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) [CrossRef] .

]. However, the remaining optical path of the microscope also has a significant influence on the resolution and image quality, as studied for aplanatic solid immersion lens (ASIL) wide-field microscope in [13

13. 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) [CrossRef] .

]. It indicates the importance of such a complete study and modelling of ASIL scanning microscope (ASIL-SM) as well. A computational model allows a controlled study of the impact of each individual or a set of parameters, which is often difficult to achieve in experiments.

A paraxial scanning microscope without SIL was investigated in [14

14. C. J. R. Sheppard and A. Choudhury, “Image formation in scanning microscope,” Opt. Acta 24, 1051–1073 (1977) [CrossRef] .

16

16. C. J. R. Sheppard and T. Wilson, “The theory of the direct-view confocal microscope,” J. Microsc.-Oxf. 124, 107–117 (1981) [CrossRef] .

]. An immersion type confocal scanning light microscope (CSLM) with numerical aperture NA = 1.3 was investigated experimentally in [17

17. G. J. Brakenhoff, P. Blom, and P. Barends, “Confocal scanning light-microscopy with high aperture immersion lenses,” J. Microsc.-Oxf. 117, 219–232 (1979) [CrossRef] .

]. Furthermore, [18

18. T. Wilson, R. Juskaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. 141, 298–313 (1997) [CrossRef] .

, 19

19. P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998) [CrossRef] .

] considered image formation of dielectric point scatterers in conventional and confocal microscopes with high NA. In addition, a numerical method, viz., finite-difference time-domain (FDTD) was used for modeling the optical microscope [20

20. P. Török, P. R. T. Munro, and E. E. Kriezis, “High numerical aperture vectorial imaging in coherent optical microscopes,” Opt. Express 16, 507–523 (2008) [CrossRef] [PubMed] .

, 21

21. R. L. Coe and E. J. Seibel, “Computational modeling of optical projection tomographic microscopy using the finite difference time domain method,” J. Opt. Soc. Am. A 29, 2696–2707 (2012) [CrossRef] .

]. A four-step process, i.e., illumination, scattering, resampling, and image formation, was designed to treat the numerical model of the optical microscope. Due to the huge computational load of FDTD, examples of only small scatterers were given for a scanning microscope [20

20. P. Török, P. R. T. Munro, and E. E. Kriezis, “High numerical aperture vectorial imaging in coherent optical microscopes,” Opt. Express 16, 507–523 (2008) [CrossRef] [PubMed] .

] and some simplifications were assumed in order to deal with larger object structures [21

21. R. L. Coe and E. J. Seibel, “Computational modeling of optical projection tomographic microscopy using the finite difference time domain method,” J. Opt. Soc. Am. A 29, 2696–2707 (2012) [CrossRef] .

].

The problem of developing a computation efficient model for ASIL-SM is very challenging. The case of ASIL microscope is very different from other free space or oil-immersion microscopes [22

22. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14, 11277–11291 (2006) [CrossRef] [PubMed] .

]. It is not only a high NA system (higher than oil immersion lens for silicon substrate), it also involves refraction along a spherical interface which is offset from the focal point of the objective. Thus, the integrals are quite complicated and a minor approximation may lead to significant error in computation. Further, the computation of the vector diffraction integrals in ASIL microscope [11

11. K. M. Lim, G. C. F. Lee, C. J. R. Sheppard, J. C. H. Phang, C. L. Wong, and X. Chen, “Effect of polarization on a solid immersion lens of arbitrary thickness,” J. Opt. Soc. Am. A 28, 903–911 (2011) [CrossRef] .

, 23

23. L. Hu, R. Chen, K. Agarwal, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Dyadic Green’s function for aplanatic solid immersion lens based sub-surface microscopy,” Opt. Express 19, 19280–19295 (2011) [CrossRef] [PubMed] .

] suffers from a big computational challenge due to the highly oscillatory behavior of the integrands. In the meanwhile, the computational requirements of a scanning system is several times more than its wide-field counterpart for any microscopy system, depending upon the scanning resolution. Thus, the development of a numerical model that is fast as well as accurate is of prime importance and significant challenge for ASIL-SM. Addressing this challenge is the main motivation of this paper.

In addition to the above, we also seek to address the properties and engineering parameters of ASIL-SM. We report the theoretical resolution capability for small scatterers. We also study the effect of polarization, NA, and detector pinhole size on the resolution and imaging characteristics.

While using the basic theoretical premise developed in [13

13. 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) [CrossRef] .

], we propose a complete and computationally efficient model of ASIL-SM with finite-sized detector. The highlights of our model are as follows:
  • This model is a complete model in the theoretical sense, since it considers the entire process from focusing of the incident light to the image generation with minimal and practical assumptions.
  • Our model has been made computationally efficient by the use of the conjugate gradient fast Fourier transform (CG-FFT) [24

    24. Y. Zhong and X. Chen, “An FFT twofold subspace-based optimization method for solving electromagnetic inverse scattering problems,” IEEE Trans. Antennas Propag. 59, 914–927 (2011) [CrossRef] .

    ] in the second subsystem, chirp z transform (CZT) in the first subsystem and CZT with fast convolution (CZT-FC) in the third subsystem.
  • The modular nature of the model (Fig. 1) makes it easy to simulate and engineer several system parameters, such as the polarization of the illumination, the object structures in the focal plane, or the pinhole size.
  • In our knowledge, this paper is the first paper demonstrating the imaging properties of the ASIL-SM microscope as a complete system. Besides studying the theoretical resolution using small (point-like) scatterers, we also demonstrate the capability of simulating images of large and complicated object structures. Interesting and practical object patterns are used for investigation. Especially, practically important examples of USAF target and an object structure with various materials used in silicon fabrication industry are also illustrated.
  • The effect of polarization, NA, and detector pinhole sizes are studied as important system design parameters. Such a study is of practical importance for predicting and understanding the microscopy results better.

Fig. 1 Block diagram of the computational model.

The structure of this paper is as follows. The proposed model of ASIL-SM is presented in Section 2. It is followed in Section 3 by numerical simulations and analysis for small scatterers and large objects. Finally, we conclude this paper in Section 4.

2. Computational model of ASIL scanning microscope with finite sized detector

2.1. Background

The diagram of ASIL-SM is shown in Fig. 2. We assume that the radius R of the ASIL is far larger than the wavelength λ in air, and that the refractive index of ASIL nsil is same as that of the substrate. The NA of the ASIL is NAsil = (nsil/nobj)2NAobj, where NAobj is the NA of the objective and nobj is the refractive index of material between ASIL and objective. Further, fobj and fccd are the focal lengths of the objective lens and the detector lens respectively, and RPH denotes the radius of pinhole. With reference to Fig. 1, the first subsystem comprises of the computation of the electric fields in the focal region formed due to the focusing of the incident beam (coherent, collimated, and of a certain polarization). In terms of microscope components, it involves the objective lens and ASIL. The second subsystem comprises of the electromagnetic interaction of the focal fields with the object structure, which results in the induction of secondary sources. It should include multiple scattering effect between different portions of the discretized object structure. More details about the second subsystem can be found in [13

13. 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) [CrossRef] .

, 24

24. Y. Zhong and X. Chen, “An FFT twofold subspace-based optimization method for solving electromagnetic inverse scattering problems,” IEEE Trans. Antennas Propag. 59, 914–927 (2011) [CrossRef] .

]. In this paper, we refrain from going into detail about the subsystem 2, and concentrate more on the subsystems 1 and 3. The third subsystem comprises of the radiation of electric fields from the induced sources, which pass through the ASIL, objective, detector lens and the pinhole, in order to compute a net intensity value corresponding to the focal point in the ASIL region.

Fig. 2 Diagrammatic description of ASIL scanning microscope. The refractive index of ASIL is the same as that of the substrate where the object structures are present.

The basic computational models of subsystems 1 and 3 are the same as in [13

13. 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) [CrossRef] .

]. It is notable that our model of subsystem 1 is in line with the extensive work done on modeling the focusing of incident light. Thus, like other works on focusing, our model of subsystem 1 also includes the evaluation of the diffraction integrals [10

10. S. H. Goh and C. J. R. Sheppard, “High aperture focusing through a spherical interface: Application to refractive solid immersion lens (RSIL) for subsurface imaging,” Opt. Commun. 282, 1036–1041 (2009) [CrossRef] .

, 11

11. K. M. Lim, G. C. F. Lee, C. J. R. Sheppard, J. C. H. Phang, C. L. Wong, and X. Chen, “Effect of polarization on a solid immersion lens of arbitrary thickness,” J. Opt. Soc. Am. A 28, 903–911 (2011) [CrossRef] .

]. Another approach is to use a vectorial spherical harmonics theory for focusing [12

12. 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) [CrossRef] .

]. Both direct evaluation of the diffraction integral as well as the use of vectorial spherical harmonics theory suffer from a big computational challenge due to highly oscillatory behavior of the integrands. Further, even for the subsystem 3, the computation of the dyadic Green’s function (DGF) also involves the computation of diffraction integrals with highly oscillatory integrands [23

23. L. Hu, R. Chen, K. Agarwal, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Dyadic Green’s function for aplanatic solid immersion lens based sub-surface microscopy,” Opt. Express 19, 19280–19295 (2011) [CrossRef] [PubMed] .

, 25

25. R. Chen, K. Agarwal, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Resolution of aplanatic solid immersion lens based microscopy,” J. Opt. Soc. Am. A 29, 1059–1070 (2012) [CrossRef] .

]. The computational load for subsystem 3 is significantly larger than subsystem 1 since the DGF should be computed for each pair of points in the object and image regions.

Our model uses an alternative and relatively less-used approach of the Fourier transform (FT) to evaluate the diffraction integrals of subsystems 1 and 3. Some relevant works on using FT in diffraction integrals for optical setups different from ASIL-SM appear in [22

22. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14, 11277–11291 (2006) [CrossRef] [PubMed] .

, 26

26. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,”J. Opt. Soc. Am. 54, 240–242 (1964) [CrossRef] .

29

29. J. Lin, O. G. Rodriguez-Herrera, F. Kenny, D. Lara, and J. C. Dainty, “Fast vectorial calculation of the volumetric focused field distribution by using a three-dimensional fourier transform,” Opt. Express 20, 1060–1069 (2012) [CrossRef] [PubMed] .

]. However, none of these works can be applied directly for the case ASIL-SM. This is because these works do not deal with the presence of a spherical interface such as the ASIL behind the objective lens. The problem is further complicated by the fact that the focal point of the objective is not coincident with the center of the ASIL, the refractive index mismatch is higher, and the numerical aperture of ASIL-SM is large (non-paraxial).

While the use of FT helps in dealing with the oscillatory integrand issue of the diffraction integral and reducing the computational complexity, we augment the computation speed further by using the CZT [22

22. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14, 11277–11291 (2006) [CrossRef] [PubMed] .

, 30

30. L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Acoust. Speech 17, 86–92 (1969).

]. Finally, we use the convolution theorem [31

31. A. Oppenheim, R. Schafer, and J. Buck, Discrete-Time Signal Processing, 2nd ed (Prentice Hall, 1999).

] to further speed up the computation of the DGF. The details are provided in sections 2.2 to 2.5.

2.2. Focusing of incident light through ASIL

The optical diagram of the first subsystem is given in Fig. 3. A non-paraxial monochromatic wave Einc(θ)=[Eincx,Eincy,Eincz]T is incident on the objective lens with NAobj=nobjsinθobjmax, where sinθobjmax is the maximum value of θobj. After refraction at the objective lens and the spherical interface of the ASIL, the local polarization components immediately inside the ASIL surface Esil=[Esilx,Esily,Esilz]T for the ray along the angular direction (θobj, ϕobj) can be described by the following equation [13

13. 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) [CrossRef] .

]:
Esil=(n0nobjcosθobj)1/2fobjrOAexp(ikobjfobjikobjrOA)TEinc(θ)
(1)
where the matrix, T, is written as
T=12[(ts+tpcosθsil)(tstpcosθsil)cos2ϕobj(tstpcosθsil)sin2ϕobj2tpsinθsilcosϕobj(tstpcosθsil)sin2ϕobj(ts+tpcosθsil)+(tstpcosθsil)cos2ϕobj2tpsinθsilsinϕobj2tpsinθsilcosϕobj2tpsinθsilsinϕobj2tpcosθsil].
(2)
The factor (n0nobjcosθobj)1/2 is the apodization function to account for the sine condition and energy conservation. ts and tp are transmission coefficients at the interface of the ASIL when the wave is traveling from the objective to the ASIL. The term of fobjrOAexp(ikobjfobjikobjrOA) denotes the change in the magnitude and phase due to the wave propagating from the Gaussian reference sphere (GRS) to the outer surface of the ASIL. It is worth mentioning that Eq. (1) is more accurate for ASIL case [13

13. 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) [CrossRef] .

], whereas [11

11. K. M. Lim, G. C. F. Lee, C. J. R. Sheppard, J. C. H. Phang, C. L. Wong, and X. Chen, “Effect of polarization on a solid immersion lens of arbitrary thickness,” J. Opt. Soc. Am. A 28, 903–911 (2011) [CrossRef] .

] modelled the focusing effect for an SIL of arbitrary thickness.

Fig. 3 Equivalent representation of subsystem I. The details can be found in [13].

Taking the point F in Fig. 3 as the reference point, we can express the electric field at a point P(xsil, ysil, zsil) in the focal region of ASIL by the following expression [13

13. 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) [CrossRef] .

]:
E(xsil,ysil,zsil)=ΩirFAexp(iksilrFA)2πkzsilEsilexp(ikxsilxsil+ikysilysil+ikzsilzsil)dkxsildkysil,
(3)
where Ω denotes the integration domain (kxsil2+kysil2)1/2ksilsinθsilmax. Considering the relationship nobjrOA = nsilrFA, we can rewrite Eq. (3) as
E(xsil,ysil,zsil)=αΩEfft(kxsil,kysil)exp(ikxsilxsil+ikysilysil)dkxsildkysil
(4)
where Efft=(cosθobj)1/2exp(ikzsilzsil)cosθsilTEinc(θ) and α=ifobj2πksilnobjnsil(n0nobj)1/2exp(ikobjfobj). E(xsil, ysil, zsil) can be understood as the two dimensional inverse Fourier transform (2D-IFT) of Efft(kxsil, kysil), except for the constant factor α. After discretization of Eq. (4), we have
E(p,q,zsil)=αm=M/2M/21n=N/2N/21Efft(m,n)exp[2πi(pmM+qnN)]ΔkxsilΔkysil;(M/2pM/21),(N/2qN/21),
(5)
where M=2sinθsilmaxksil/Δkxsil and N=2sinθsilmaxksil/Δkysil are the numbers of samples in the x and y directions in both spatial and spatial frequency domains, respectively. Therefore, the electric field in the focal region of the ASIL using FFT is given by
E(p,q,zsil)=αIFFT[Efft(m,n)]ΔkxsilΔkysil.
(6)
For the FFT algorithm, the spatial domain (xsil, ysil) has the same sample size (M and N) as the spatial frequency domain (kxsil, kysil) respectively. However, in the spatial domain, the region of interest is near and around the focal point only, which is only a small portion compared with the required sample size of the Fourier frequency domain for capturing the highly oscillatory behavior of the integrand. In order to sufficiently resolve the details of the region of interest, zero-padding is often used to increase the output sampling size, and thus the resolution of the spatial domain is determined by
Δxsil=2π/(MfftΔkxsil),Δysil=2π/(NfftΔkysil),
(7)
where Mfft and Nfft is the sample size after zero-padding. It subsequently increases the size of the input matrix and the computation time.

Compared to FFT, CZT allows us to avoid the use of the zero-padding, and select only the region of interest as the output space. Due to this, the number of computations is reduced significantly and the CZT scheme is much faster than FFT. For CZT, we choose the same spatial frequency domain as FFT and the spatial domain as xsil ∈ [x1, x2] and ysil ∈ [y1, y2], then Eq. (5) is given by [30

30. L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Acoust. Speech 17, 86–92 (1969).

, 31

31. A. Oppenheim, R. Schafer, and J. Buck, Discrete-Time Signal Processing, 2nd ed (Prentice Hall, 1999).

]
E(p,q)=αm=M/2M/21n=N/2N/21Efft(m,n)AxmAynWxpmWxqnΔkxsilΔkysil;(P/2pP/2),(Q/2qQ/2),
(8)
where Ax = exp(−iΔkxsilx1), Ay = exp(−iΔkysily1), Wx = exp[iΔkxsil(x2x1)/P] and Wy = exp[iΔkysil (y2y1)/Q]. The flexibility of the CZT allows the choice of an arbitrarily shaped and sized output region, and expresses the CZT as a convolution, permitting the use of the well-known convolution algorithm. For the details, please refer to [30

30. L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Acoust. Speech 17, 86–92 (1969).

, 31

31. A. Oppenheim, R. Schafer, and J. Buck, Discrete-Time Signal Processing, 2nd ed (Prentice Hall, 1999).

].

2.3. Fast computation of DGF using CZT

The DGF for ASIL microscope has been presented recently in [23

23. L. Hu, R. Chen, K. Agarwal, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Dyadic Green’s function for aplanatic solid immersion lens based sub-surface microscopy,” Opt. Express 19, 19280–19295 (2011) [CrossRef] [PubMed] .

]. By means of the DGF, we can obtain the intensity distribution in the CCD/detector region corresponding to an induced current distribution in the focal plane of ASIL. In this section, the fast calculation of DGF for the imaging of scattered light using CZT is presented. The optical diagram is shown in Fig. 4. For a current dipole Il(rsil) located at rsil(xsil, ysil, zsil) close to the aplanatic point of ASIL, Osil, the electric field immediately after the interface of the Gaussian reference sphere representing the CCD lens is given by [23

23. L. Hu, R. Chen, K. Agarwal, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Dyadic Green’s function for aplanatic solid immersion lens based sub-surface microscopy,” Opt. Express 19, 19280–19295 (2011) [CrossRef] [PubMed] .

]
Eccd=iωμexp(ikobjfobj)8πfobjexp(iksilrsil)(nobjcosθccdnccdcosθobj)12[Gx,Gy,Gz]Il(rsil)
(9)
where ksil = [kxsil, kysil, kzsil]T denotes the propagation vector of light inside the ASIL and
Gx=[(ts+tpcosθccdcosθsil)(tstpcosθccdcosθsil)cos2ϕobj(tstpcosθccdcosθsil)sin2ϕobj2tpsinθccdcosθsilcosϕobj],
(10a)
Gy=[(tstpcosθccdcosθsil)sin2ϕobj(ts+tpcosθccdcosθsil)+(tstpcosθccdcosθsil)cos2ϕobj2tpsinθccdcosθsilsinϕobj],
(10b)
Gz=2tpsinθsil[cosθccdcosϕobjcosθccdsinϕobjsinθccd].
(10c)
ts and tp are redefined transmission coefficients at the ASIL interface for a wave travelling from the ASIL to the objective [23

23. L. Hu, R. Chen, K. Agarwal, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Dyadic Green’s function for aplanatic solid immersion lens based sub-surface microscopy,” Opt. Express 19, 19280–19295 (2011) [CrossRef] [PubMed] .

], which are different from the transmission coefficients in Eq. (2). The far-field Eccd can now be introduced into the angular spectrum representation and thus the electric field at a point rccd(xccd, yccd, zccd) in the CCD region is given by [23

23. L. Hu, R. Chen, K. Agarwal, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Dyadic Green’s function for aplanatic solid immersion lens based sub-surface microscopy,” Opt. Express 19, 19280–19295 (2011) [CrossRef] [PubMed] .

]
E(rccd)=iωμG¯¯PSFIl(rsil).
(11)
In the present paper, the DGF is written differently from that in [23

23. L. Hu, R. Chen, K. Agarwal, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Dyadic Green’s function for aplanatic solid immersion lens based sub-surface microscopy,” Opt. Express 19, 19280–19295 (2011) [CrossRef] [PubMed] .

], such that it is amenable to a 2D-IFT format (similar to Eq. (4))
G¯¯PSF=βΩccdg¯¯exp[ikxccd(xccd+Mlatxsil)+kyccd(yccd+Mlatysil)]dkxccddkyccd,
(12a)
g¯¯=[Gx,Gy,Gz]cosθccd(cosθccdcosθobj)1/2exp(ikzccdzccdikzsilzsil),
(12b)
where β=i16π2kccdfccdfobj(nobjnccd)1/2exp(ikccdfccd+ikobjfobj) is a constant. Ωccd is the integration domain (kxccd2+kyccd2)1/2kccdsinθccdmax, and θccdmax is the maximum value of θccd. We also note that kxsil= −kxccd Mlat and kysil = −kyccd Mlat, where Mlat=(nsilnobj)2nobjfccdnccdfobj is transverse magnification of ASIL. Based on the similarity of Eq. (12a) with Eq. (4), we can follow the processes in Section 2.2 to compute the DGF efficiently and accurately using CZT.

Fig. 4 Equivalent representation of subsystem III. The details can be found in [13].

While the adaptation of the fast techniques for focusing and propagation may appear to be similar to each other, we highlight that the adaptation of the propagation integral in subsystem 3 is more involved and requires care in computing the Fourier terms of each component of the DGF. We note that while the remaining procedure of applying CZT is similar to section 2.2, the modification of the DGF into the form of Eq. (12a) is crucial for being able to apply CZT.

2.4. Image formation using convolution theorem

Based on the analysis of Sections 2.2 and 2.3, the first and third subsystem can be evaluated using CZT instead of direct integration. We assume that an object structure is present in the focal plane of ASIL, as shown in Fig. 5(b). After we obtain the electric field distribution in the focal plane, shown in Fig. 5(a), the induced current distribution Il(xsil, ysil) in the focal plane of ASIL can be obtained by MoM [13

13. 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) [CrossRef] .

]. The electric field in the focal plane of CCD region is then given by
E(xccd,yccd)=iωμG¯¯PSF(xsil+xccd/Mlat,ysil+yccd/Mlat)Il(xsil,ysil),
(13)
where
G¯¯PSF=βMlat2g¯¯exp[ikxsil(xsil+xccdMlat)ikysil(ysil+yccdMlat)]dkxsildkysil
(14)
which is just a modified expression of Eq. (12a). For Eq. (13), the numerical implementation is straightforward. In the domain where the object structure is located, an equidistant sampling xsil = mΔxsil and ysil = nΔysil with m = −M/2,...,M/2 and n = −N/2,...,N/2, is made (or simply reused from the discretization in subsystem 1). The sampling in the focal plane of detector region is xccd = pΔxccd and yccd = qΔyccd with p = −M/2,...,M/2 and q = −N/2,...,N/2, where we choose Δxccd = MlatΔxsil, Δyccd = MlatΔysil, such that the grid points in the detector region are image conjugates of the grid points in the object region.

Fig. 5 The description of the location of object structure when it is scanned relative to the optical microscope. (a) The focal spot and (b) the object domain in the focal plane of ASIL. The location of the object domain when (c) the first pixel (1,1) and (d) the (m,n) pixel are scanned, i.e,. the corresponding pixel is at the center of focal spot. The red arrows in (c) and (d) denote the scanning directions.

Taking the element Gxx of DGF and Ilx(xsil, ysil) for example, we can express the electric field produced by these two elements in the discrete form
Exx(p,q)=iωμmnIlx(m,n)Gxx(p+m,q+n),
(15)
which is the two-dimensional discrete convolution of matrices Ilx(−p, −q) and Gxx(p, q), which can be expressed by
Exx(p,q)=iωμ(Ilx(p,q)*Gxx(p,q)).
(16)
where the symbol ‘*’ denotes the convolution operator. According to the convolution theorem [31

31. A. Oppenheim, R. Schafer, and J. Buck, Discrete-Time Signal Processing, 2nd ed (Prentice Hall, 1999).

], FFT and IFFT can be used to evaluate this equation:
Exx(p,q)=iωμIFFT{FFT[Ilx(p,q)]FFT[Gxx(p,q)]},
(17)
Similarly, other elements of DGF can also be evaluated using the fast convolution theorem and then we can obtain the image of object structure by considering all of the elements of DGF.

2.5. Scanning system implementation

For the scanning system, the object structure is assumed to be scanned relative to the optical system, and the total intensity of the light passing through a finite sized detector pinhole is collected at the detector. Noting the effects of employing a finite sized detector, the detected signal can be written as the integration of the intensity (the modulus square of the electric field vector) over the detector area S (i.e., the pinhole size),
I=S|E(xccd,yccd)|2dS=0RPH02π|E(xccd,yccd)|2drccddϕ.
(18)

After discretization, the equidistant sampling of the object in the focal plane of ASIL are xsil = mΔxsil and ysil = nΔysil with m ∈ [−M/2, M/2] and n ∈ [−N/2, N/2], as shown in Fig. 5(b). When the object is scanned laterally as shown in Figs. 5(c) and 5(d), the detector pinhole is fixed at the focal point of the detector lens and the object is shifted in the focal plane of ASIL. The scanning action makes each pixel of the object locate at the center of focal spot in turn. When the (m, n)th pixel is moved to the focal point of ASIL, the induced current distribution inside the focal spot in Fig. 5(d) produces an electric field in the detector plane. Therefore, the signal detected on the (m, n)th pixel in the detector region, I(m, n), determined by Eq. (18), corresponds to the (m, n)th pixel in the object region. After scanning is done, I(m, n), a matrix with the same dimension as discretization M × N, is the image of the original object structure using ASIL-SM with the pinhole of radius, RPH.

2.6. Example of enhancement in computational efficiency

In order to understand the accuracy and computational efficiency of the proposed model, we take an example in which we compare the direct integral approach (DI) used in [13

13. 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) [CrossRef] .

]. The setup of the numerical example is similar to the setup used in the numerical examples in section 3. In Table 1, we show the relative error and the time taken by the direct integral approach (DI), the FFT-approach (FFT), and the proposed CZT approach (CZT) using a personal computer with an Intel(R) Core2 Duo 3.16GHz processor and 3GB of RAM. The output spatial domain in the focal plane is [−1.0λ, 1.0λ] with 101 × 101 pixels. We also compare the effect of the value of M, N. It is seen that even with very large value of M, N, M = N = 800, the computation using CZT is about 36 times lesser than DI. Further, for an error of less than 10−2, M = N = 100 is sufficient and the speed up is by a factor of 2000. We have used M = N = 200 for both subsytem 1 and subsystem 3 in the remaining part of the paper. We also summarize the computation efficiency for the whole system in Table 2 using the same computer. It is seen that the computation of a few hours in DI approach get reduced to a few seconds using the proposed approach.

Table 1. Comparison of calculation times and relative error for DI, FFT and CZT methods

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Table 2. Comparison between computational time using direct integral (DI) in [13] and CZT methods presented in this paper for each scan point of the ASIL-SM.

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3. Numerical simulations and analysis

In all the numerical simulations below, we consider the following settings. The SIL is made of silicon with a refractive index of nsil = 3.5 and the operating wavelength is 1340 nm in free space, both of which are appropriate for subsurface imaging of silicon chips. The radius of the SIL is R = 1.5 mm. The incident light has unity magnitude. The focal length of the objective and detector lens are fobj = 0.01 m and fccd = 0.1 m, respectively. The objective and detector media are free space, i.e., nobj = 1 and nccd = 1, respectively. Thus, the lateral magnification of the microscope is Mlat = 122.5. The discretization (scanning pixel size) is Δxsil = Δysil = 0.02λ for both small scatterer and large object examples. The semiconductor industry typically uses planar resolution target chips (like Metrochip [32] and MRS-5 [33]) for characterizing the resolution of the microscopes used in silicon failure analysis. Thus, we consider planar object structures only and thickness of the plane is taken as 0.02λ.

In our numerical examples, we have considered two polarizations, x-polarization and circular polarization. Further, for small scatterers, we consider two values of NA, NAsil = 2.4 and 3.3. Throughout the numerical examples, we have considered various pinhole sizes like 1 μm, 25 μm, and 100 μm. We highlight that pinhole radius of 1 μm and 100μm correspond to the confocal and wide-field cases respectively. In our opinion, polarization and detector pinhole radius are two important system parameters that can be used ASIL-SM engineering. We make a note that pinhole radius of 1 μm reduces the signal strength considerably, but this effect can be practically dealt with by using high sensitivity and low noise InGaAs detector.

3.1. Imaging of small scatterers

For the example of small scatterers [13

13. 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) [CrossRef] .

], we show the normalized intensity distribution of the image of the small scatterer for NAsil = 3.3 and 2.4 in Figs. 6 and 7 respectively. The comparison for pinhole radius of RPH = 1μm and 100μm and different polarizations are included. The paraxial approximation [34

34. T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy, vol. 1 (London: Academic Press, 1984).

], referred to as PA, is also provided.

Fig. 6 Normalized intensity distribution of the image of a small scatterer along a lateral axis for different pinhole radii for NAsil = 3.3, (a) RPH = 1μm and (b) RPH = 100μm, using x-polarized illumination (two curves: ysil = 0 and xsil = 0) and circularly polarized illumination, respectively. PA denotes paraxial approximation.
Fig. 7 The caption is the same as that of Fig. 6 except for NAsil = 2.4.

For x-polarized illumination, it is seen that when high NA (NAsil = 3.3, Fig. 6) is employed, the image asymmetry of a small scatterer is much larger than that using low NA (NAsil = 2.4, Fig. 7). It is also seen that the curve corresponding to symmetric paraxial theory is quite close to the curve corresponding to xsil = 0 (see Fig. 6 for clear illustration of this fact). The curve corresponding to circularly polarized illumination, as expected, lies in between the curves corresponding to xsil = 0 and ysil = 0 of x- polarized illumination, as shown in Figs. 6 and 7, since it is a superposition of x- and y- polarized illuminations. Further, it is observed that the bigger pinhole radius results in more pronounced sidelobes as seen in Figs. 6(b) and 7(b).

Fig. 8 shows the resolution (FWHM) of imaging a small scatterer as a function of radius of pinhole, RPH, for different values of NAsil using x-polarized and circularly polarized illumination, respectively. It is seen that the resolution of ASIL-SM gradually deteriorates with the increase of pinhole size and then remains at almost a constant value when the pinhole is big enough to capture most of the signal produced by the small scatterers. Although the results shown here correspond to substantially large NA, such observations were reported for the paraxial microscope also (non-SIL type) in [19

19. P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998) [CrossRef] .

, 34

34. T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy, vol. 1 (London: Academic Press, 1984).

].

Fig. 8 Resolution of imaging a small scatterer as a function of radius of pinhole, RPH, for different NAsil using x-polarized illumination ((a) ysil = 0 and (b) xsil = 0) and (c) circularly polarized illumination, respectively.

A quantitative comparison of the resolution of ASIL confocal microscope (RCM - FWHM for RPH = 1μm), ASIL wide field microscope (RWFM - FWHM for RPH = 100μm), and the resolution predicted by subsystem 1 (FWHM of focal spot) is provided in Table 3. For x-polarized illumination, it is found that the ratio of RCM and RWFM along y-axis is close to 1.4, as shown in Table 3 (red color numbers). This is similar to the observation made in [14

14. C. J. R. Sheppard and A. Choudhury, “Image formation in scanning microscope,” Opt. Acta 24, 1051–1073 (1977) [CrossRef] .

,34

34. T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy, vol. 1 (London: Academic Press, 1984).

] for a paraxial non-SIL microscope that the theoretically expected improved resolution for confocal microscope is about 1.4 times better than wide-field microscope for a point object. Further, the RWFM agrees with the prediction obtained using the FWHM of the focal spot for x- polarization along the y-axis (xsil = 0). However, these observations do not hold for the x-polarized wave along the x-axis (ysil = 0), as well as for the circularly polarized illumination.

Table 3. Comparison of RWMF, RCM and FWHM of subsystem 1.(Unit:λ)

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3.2. Imaging of large object structures

The proposed model is capable of simulating and analyzing the images of large object structures. Several examples are presented in this section to illustrate this: an annular ring pattern (Fig. 9(a)), a ‘08’ digital pattern (Fig. 9(b)), a USAF resolution target pattern (Fig. 9(c)) and a pattern with materials similar to integrated circuits (called IC pattern for simplicity) (Fig. 9(d)). In the first three examples, the refractive index of the structures is 1.5, which corresponds to silica (silicon dioxide) in silicon substrate. The last example closely mimics the patterns and materials used in integrated circuits. It considers features with different materials, viz., silicon nitride (n = 2.0), silica (n = 1.5), cobalt silicide (n = 1.3), and gold (n = 0.41 + i9.11), all refractive indices at 1340 nm [35

35. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972) [CrossRef] .

].

Fig. 9 Refractive index (magnitude) distribution of the cross section of the object structures, (a) an annular ring, (b) a ‘08’ digital pattern, (c) a designed three-bar resolving power test target and (d) the IC pattern.

The annular ring pattern and the ‘08’ digital pattern are discussed first. The inner and outer radii of the annular ring are Ri = 0.12λ and Ro = 0.24λ, respectively. The ‘08’ digital pattern, is sized such that each line has a width of t = 0.1λ and the distance between any two adjacent parallel lines is d = 0.2λ, as shown in Fig. 9(b). Fig. 10 shows the simulated images for the annular ring and the ‘08’ digital pattern in the top and bottom rows respectively.

Fig. 10 Image of the annular ring (the top row) and ‘08’ digital pattern (the bottom row) using proposed model with NAsil = 3.3 and different pinhole radius, RPH = 1μm (a,d,g and j), RPH = 25μm (b,e,h and k), and RPH = 100μm (c,f,i and l). The first three columns are for x- polarized incidence while the last three columns for the circular polarized incidence. The horizontal and vertical coordinates are [−0.7, 0.7] and [−0.55, 0.55] for xsil(λ) and ysil(λ) in the bottom row, respectively. Both of coordinates are [−0.3, 0.3] in the top row.

For x-polarized illumination and RPH = 1μm, 25μm, we can see only two symmetric spots away from the optical axis in the x-direction in Figs. 10(a) and 10(b) for the annular ring pattern and only the vertical lines in Figs. 10(g) and 10(h) for the ‘08’ digital pattern. For x-polarized illumination and RPH = 100μm in Figs. 10(c) and 10(i), the image quality of both patterns deteriorates even further. Thus, it can be concluded that x- polarized illumination (and likewise other linear polarizations) is not suitable for imaging such structures, irrespective of the size of pinhole. Comparatively, circular polarized illumination results in better images. It is observed that small pinhole radius results in reasonable imaging (see Figs. 10(d),10(e), 10(j) and 10(k)). However, there are some artifacts in the form of a small dot in the centre (annular pattern) or in the form of the lines in the middle (’08’ pattern). For pinhole radius RPH = 25μm, we see a bleaching effect for both the patterns (Figs. 10(e) and 10(k)). The images are of poor quality for high values of pinhole radius (Figs. 10(f) and 10(l)).

Fig. 11 shows the simulated images for the x-polarized and circular polarized illuminations. It is found that we can resolve the target element of line width 0.10 λ for circularly polarized illumination (Fig. 11(e)), and 0.12 λ for x-polarized illumination (Fig. 11(b)), using a pinhole radius of 25μm. On the other hand, for the pinhole radius of 1μm, the images are not good for either circular or x-polarized illumination. For the circular polarized illumination, both the horizontal and vertical bar patterns have artifacts. For the x-polarized illumination, the vertical bar patterns have artifacts while the horizontal bar patterns have very low intensity.

Fig. 11 The caption is the same as Fig. 10 except that the three-bars pattern is imaged. The horizontal and vertical coordinates are [−2.55, 2.55] and [−2.10, 2.10] for xsil(λ) and ysil(λ), respectively.

Fig. 12 The caption is the same as Fig. 10 except that the IC pattern is imaged with the pinhole size RPH = 1μm, 15μm and 25μm. Both the horizontal and vertical coordinates are [−1.60, 1.60] for xsil(λ) and ysil(λ). The last three columns create images using a base 10 logarithmic scale for intensity.

This example raises a concern about the visibility and fidelity of the features in the microscope image in presence of noise, especially for the case of small pinhole like 1μm, where the measured intensity is very low and the signal-to-noise ratio (SNR) is very poor. In order to study the effect of noise on the image quality (when viewed in logarithmic scale), we add successively increasing noise to the detected intensity (circular polarization, 1μm pinhole radius) and plot the logarithmic image in Fig. 13. We see that even in the presence of Gaussian noise with SNR = 20dB, as shown in Fig. 13, all the features are clearly visible.

Fig. 13 The logarithmic image of the IC pattern in the presence of Gaussian noise with (a) no noise, (b) SNR = 40dB, (c) SNR = 30dB and (d) SNR = 20dB, using circular polarization and 1μm pinhole radius.

4. Conclusion

This paper presents a complete and computationally efficient model of ASIL-SM and provides the ability to analyze image formation in a more systematic way than can be performed experimentally. For example, imaging different object structures would each require their own fabrication, whereas the presented model can predict results by simply changing a few parameters and then if needed, some structures can be checked by experiment [36

36. K. Agarwal, R. Chen, L. S. Koh, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Experimental validation of the computational model of aplanatic solid immersion lens scanning microscope,” presented at Focus on microscopy 2013, Maastricht, The Netherlands, 24–27 Mar. 2013

].

The computational efficiency is derived by using fast algorithms like CG-FFT and CZT-FC for each subsystem in the computational model. This is an important contribution to the research and development in ASIL based microscopy technology since modeling and simulating ASIL microscopy system is very challenging due to spherical refraction interface, highly oscillatory diffraction integrals, and pressing requirement of reducing the scanning resolution. It is found that the presented model significantly reduces the computational time, especially for the large object structures from few hours to a few seconds. We highlight that the model has been experimentally tested and simulated results match the experimental results very well. Comparison of the simulation model with experimental results was reported in [36

36. K. Agarwal, R. Chen, L. S. Koh, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Experimental validation of the computational model of aplanatic solid immersion lens scanning microscope,” presented at Focus on microscopy 2013, Maastricht, The Netherlands, 24–27 Mar. 2013

].

Further, engineering parameters of ASIL-SM, like the polarization, NA, and the detector pinhole size have been identified. Their effects on resolution and imaging quality have been studied. We hope that this study provides a preliminary analysis of suitable imaging setup for ASIL-SM and incites ASIL-SM system engineering using these parameters. We have already begun working on designing more complicated polarizations and pupil filter designs for achieving better resolution.

Acknowledgments

This work was supported by the Singapore Ministry of Education (MOE) grant under Project No. MOE2009-T2-2-086.

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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) [CrossRef] .

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P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972) [CrossRef] .

36.

K. Agarwal, R. Chen, L. S. Koh, C. J. R. Sheppard, J. C. H. Phang, and X. Chen, “Experimental validation of the computational model of aplanatic solid immersion lens scanning microscope,” presented at Focus on microscopy 2013, Maastricht, The Netherlands, 24–27 Mar. 2013

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(180.5810) Microscopy : Scanning microscopy
(260.2110) Physical optics : Electromagnetic optics
(110.1758) Imaging systems : Computational imaging
(290.5855) Scattering : Scattering, polarization

ToC Category:
Microscopy

History
Original Manuscript: February 8, 2013
Revised Manuscript: April 17, 2013
Manuscript Accepted: May 28, 2013
Published: June 10, 2013

Virtual Issues
Vol. 8, Iss. 7 Virtual Journal for Biomedical Optics

Citation
Rui Chen, Krishna Agarwal, Colin J. R. Sheppard, Jacob C. H. Phang, and Xudong Chen, "A complete and computationally efficient numerical model of aplanatic solid immersion lens scanning microscope," Opt. Express 21, 14316-14330 (2013)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-12-14316


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References

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