## A complete and computationally efficient numerical model of aplanatic solid immersion lens scanning microscope |

Optics Express, Vol. 21, Issue 12, pp. 14316-14330 (2013)

http://dx.doi.org/10.1364/OE.21.014316

Acrobat PDF (1469 KB)

### 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

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

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

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. 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] .

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

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] .

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. 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. 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. 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] .

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] .

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

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

*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] .

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. 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] .

*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. 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] .

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. 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] .

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

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. 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] .

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] .

- 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] in the second subsystem, chirp
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] .*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.

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

### 2.1. Background

*R*of the ASIL is far larger than the wavelength

*λ*in air, and that the refractive index of ASIL

*n*is same as that of the substrate. The NA of the ASIL is

_{sil}*NA*= (

_{sil}*n*/

_{sil}*n*)

_{obj}^{2}

*NA*, where

_{obj}*NA*is the NA of the objective and

_{obj}*n*is the refractive index of material between ASIL and objective. Further,

_{obj}*f*and

_{obj}*f*are the focal lengths of the objective lens and the detector lens respectively, and

_{ccd}*R*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

_{PH}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. 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] .

**29**, 2350–2359 (2012) [CrossRef] .

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. 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] .

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] .

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. 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] .

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

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

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] .

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

### 2.2. Focusing of incident light through ASIL

*θ*. After refraction at the objective lens and the spherical interface of the ASIL, the local polarization components immediately inside the ASIL surface

_{obj}*θ*,

_{obj}*ϕ*) can be described by the following equation [13

_{obj}**29**, 2350–2359 (2012) [CrossRef] .

**T**, is written as

*t*and

_{s}*t*are transmission coefficients at the interface of the ASIL when the wave is traveling from the objective to the ASIL. The term of

_{p}**29**, 2350–2359 (2012) [CrossRef] .

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] .

*F*in Fig. 3 as the reference point, we can express the electric field at a point

*P*(

*x*,

_{sil}*y*,

_{sil}*z*) in the focal region of ASIL by the following expression [13

_{sil}**29**, 2350–2359 (2012) [CrossRef] .

*n*=

_{obj}r_{OA}*n*, we can rewrite Eq. (3) as where

_{sil}r_{FA}**E**(

*x*,

_{sil}*y*,

_{sil}*z*) can be understood as the two dimensional inverse Fourier transform (2D-IFT) of

_{sil}*E*(

_{fft}*k*

_{xsil},

*k*

_{ysil}), except for the constant factor

*α*. After discretization of Eq. (4), we have

*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 For the FFT algorithm, the spatial domain (

*x*,

_{sil}*y*) has the same sample size (

_{sil}*M*and

*N*) as the spatial frequency domain (

*k*

_{xsil},

*k*

_{ysil}) 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 where

*M*and

_{fft}*N*is the sample size after zero-padding. It subsequently increases the size of the input matrix and the computation time.

_{fft}*x*∈ [

_{sil}*x*

_{1},

*x*

_{2}] and

*y*∈ [

_{sil}*y*

_{1},

*y*

_{2}], then Eq. (5) is given by [30, 31]

*A*= exp(−

_{x}*i*Δ

*k*

_{xsil}

*x*

_{1}),

*A*= exp(−

_{y}*i*Δ

*k*

_{ysil}

*y*

_{1}),

*W*= exp[

_{x}*i*Δ

*k*

_{xsil}(

*x*

_{2}−

*x*

_{1})/

*P*] and

*W*= exp[

_{y}*i*Δ

*k*

_{ysil}(

*y*

_{2}−

*y*

_{1})/

*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, 31].

### 2.3. Fast computation of DGF using CZT

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] .

**Il**(

**r**

*) located at*

_{sil}**r**

*(*

_{sil}*x*,

_{sil}*y*,

_{sil}*z*) close to the aplanatic point of ASIL,

_{sil}*O*, the electric field immediately after the interface of the Gaussian reference sphere representing the CCD lens is given by [23

_{sil}**19**, 19280–19295 (2011) [CrossRef] [PubMed] .

**k**

*= [*

_{sil}*k*

_{xsil},

*k*

_{ysil},

*k*

_{zsil}]

*denotes the propagation vector of light inside the ASIL and*

^{T}*t*and

_{s}*t*are redefined transmission coefficients at the ASIL interface for a wave travelling from the ASIL to the objective [23

_{p}**19**, 19280–19295 (2011) [CrossRef] [PubMed] .

**E**

*can now be introduced into the angular spectrum representation and thus the electric field at a point*

_{ccd}**r**

*(*

_{ccd}*x*,

_{ccd}*y*,

_{ccd}*z*) in the CCD region is given by [23

_{ccd}**19**, 19280–19295 (2011) [CrossRef] [PubMed] .

**19**, 19280–19295 (2011) [CrossRef] [PubMed] .

*is the integration domain*

_{ccd}*θ*. We also note that

_{ccd}*k*

_{xsil}= −

*k*

_{xccd}

*M*and

_{lat}*k*

_{ysil}= −

*k*

_{yccd}

*M*, where

_{lat}### 2.4. Image formation using convolution theorem

**Il**(

*x*,

_{sil}*y*) in the focal plane of ASIL can be obtained by MoM [13

_{sil}**29**, 2350–2359 (2012) [CrossRef] .

*x*=

_{sil}*m*Δ

*x*and

_{sil}*y*=

_{sil}*n*Δ

*y*with

_{sil}*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

*x*=

_{ccd}*p*Δ

*x*and

_{ccd}*y*=

_{ccd}*q*Δ

*y*with

_{ccd}*p*= −

*M*/2,...,

*M*/2 and

*q*= −

*N*/2,...,

*N*/2, where we choose Δ

*x*=

_{ccd}*M*Δ

_{lat}*x*, Δ

_{sil}*y*=

_{ccd}*M*Δ

_{lat}*y*, such that the grid points in the detector region are image conjugates of the grid points in the object region.

_{sil}*G*of DGF and

_{xx}*Il*(

_{x}*x*,

_{sil}*y*) for example, we can express the electric field produced by these two elements in the discrete form which is the two-dimensional discrete convolution of matrices

_{sil}*Il*(−

_{x}*p*, −

*q*) and

*G*(

_{xx}*p*,

*q*), which can be expressed by where the symbol ‘*’ denotes the convolution operator. According to the convolution theorem [31], FFT and IFFT can be used to evaluate this equation: 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

*S*(

*i.e.*, the pinhole size),

*x*=

_{sil}*m*Δ

*x*and

_{sil}*y*=

_{sil}*n*Δ

*y*with

_{sil}*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,

*R*.

_{PH}### 2.6. Example of enhancement in computational efficiency

**29**, 2350–2359 (2012) [CrossRef] .

*λ*, 1.0

*λ*] 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.

## 3. Numerical simulations and analysis

*n*= 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

_{sil}*R*= 1.5 mm. The incident light has unity magnitude. The focal length of the objective and detector lens are

*f*= 0.01 m and

_{obj}*f*= 0.1 m, respectively. The objective and detector media are free space, i.e.,

_{ccd}*n*= 1 and

_{obj}*n*= 1, respectively. Thus, the lateral magnification of the microscope is

_{ccd}*M*= 122.5. The discretization (scanning pixel size) is Δ

_{lat}*x*= Δ

_{sil}*y*= 0.02

_{sil}*λ*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 0.02

*λ*.

*x*-polarization and circular polarization. Further, for small scatterers, we consider two values of NA,

*NA*= 2.4 and 3.3. Throughout the numerical examples, we have considered various pinhole sizes like 1

_{sil}*μm*, 25

*μm*, and 100

*μm*. We highlight that pinhole radius of 1

*μm*and 100

*μm*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

**29**, 2350–2359 (2012) [CrossRef] .

*NA*= 3.3 and 2.4 in Figs. 6 and 7 respectively. The comparison for pinhole radius of

_{sil}*R*= 1

_{PH}*μm*and 100

*μm*and different polarizations are included. The paraxial approximation [34], referred to as PA, is also provided.

*x*-polarized illumination, it is seen that when high NA (

*NA*= 3.3, Fig. 6) is employed, the image asymmetry of a small scatterer is much larger than that using low NA (

_{sil}*NA*= 2.4, Fig. 7). It is also seen that the curve corresponding to symmetric paraxial theory is quite close to the curve corresponding to

_{sil}*x*= 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

_{sil}*x*= 0 and

_{sil}*y*= 0 of

_{sil}*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).

*R*, for different values of

_{PH}*NA*using

_{sil}*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] .

*R*= 1

_{PH}*μ*m), ASIL wide field microscope (RWFM - FWHM for

*R*= 100

_{PH}*μ*m), 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] .

*x*- polarization along the

*y*-axis (

*x*= 0). However, these observations do not hold for the

_{sil}*x*-polarized wave along the

*x*-axis (

*y*= 0), as well as for the circularly polarized illumination.

_{sil}### 3.2. Imaging of large object structures

*n*= 2.0), silica (

*n*= 1.5), cobalt silicide (

*n*= 1.3), and gold (

*n*= 0.41 +

*i*9.11), 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] .

*R*= 0.12

_{i}*λ*and

*R*= 0.24

_{o}*λ*, respectively. The ‘08’ digital pattern, is sized such that each line has a width of

*t*= 0.1

*λ*and the distance between any two adjacent parallel lines is

*d*= 0.2

*λ*, 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.

*x*-polarized illumination and

*R*= 1

_{PH}*μ*m, 25

*μ*m, 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

*R*= 100

_{PH}*μ*m 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

*R*= 25

_{PH}*μm*, 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)).

*t*

_{1}= 0.2

*λ*to

*t*

_{9}= 0.04

*λ*at a step 0.02

*λ*. In addition to the aforementioned nine groups, four squares corresponding to the pattern size of the smallest four groups are also included.

*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 25

*μm*. On the other hand, for the pinhole radius of 1

*μm*, 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.

*λ*× 0.02

*λ*. The other structures are silica, silicon nitride, and cobalt silicide from outer to inner as shown in Fig. 9(d). The thickness of each line is 0.06

*λ*and the distance between the lines is 0.20

*λ*. Fig. 12 shows the simulated images of the IC pattern for the

*x*-polarized and circular polarized illuminations. As expected, irrespective of the size of pinhole, the gold stubs have very strong intensity (Figs. 12(a)–12(c) and Figs. 12(g)–12(i)) due to the large refractive index (magnitude) of gold. However, the horizontal gold stubs are most clearly imaged structures for

*x*- polarized incidence as shown in Figs. 12(a)–12(c). This is different from dielectric structures where vertical features were more prominent in Figs. 10 and 11. We note that while other features are difficult to visualize in the images in (Figs. 12(a)–12(c) and Figs. 12(g)–12(i)), they can be easily visualized by using a log scale plot of the intensity, as shown in (Figs. 12(d)–12(f) and Figs. 12(j)–12(l)). In the log scale, we note that all the features are visible. The artifacts similar to those noted in the previous examples are also clearly visible. Further, we see that circular polarization with pinhole radius of 15

*μm*gives the best image.

*μm*, 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, 1

*μm*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.

## 4. Conclusion

## Acknowledgments

## References and links

1. | S. M. Mansfield and G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. |

2. | Q. Wu, L. P. Ghislain, and V. B. Elings, “Imaging with solid immersion lenses, spatial resolution, and applications,” Proc. IEEE |

3. | S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. |

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 |

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

6. | L. P. Ghislain and V. B. Elings, “Near-field scanning solid immersion microscope,” Appl. Phys. Lett. |

7. | D. A. Fletcher, K. B. Crozier, C. F. Quate, G. S. Kino, K. E. Goodson, D. Simanovskii, and D. V. Palanker, “Near-field infrared imaging with a microfabricated solid immersion lens,” Appl. Phys. Lett. |

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. |

9. | F. H. Köklü and M. S. Ünlü, “Subsurface microscopy of interconnect layers of an integrated circuit,” Opt. Lett. |

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. |

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 |

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 |

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 |

14. | C. J. R. Sheppard and A. Choudhury, “Image formation in scanning microscope,” Opt. Acta |

15. | C. J. R. Sheppard and T. Wilson, “Image formation in scanning microscopes with partially coherent source and detector,” Opt. Acta |

16. | C. J. R. Sheppard and T. Wilson, “The theory of the direct-view confocal microscope,” J. Microsc.-Oxf. |

17. | G. J. Brakenhoff, P. Blom, and P. Barends, “Confocal scanning light-microscopy with high aperture immersion lenses,” J. Microsc.-Oxf. |

18. | T. Wilson, R. Juskaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Opt. Commun. |

19. | P. Török, P. D. Higdon, and T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. |

20. | P. Török, P. R. T. Munro, and E. E. Kriezis, “High numerical aperture vectorial imaging in coherent optical microscopes,” Opt. Express |

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 |

22. | M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express |

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 |

24. | Y. Zhong and X. Chen, “An FFT twofold subspace-based optimization method for solving electromagnetic inverse scattering problems,” IEEE Trans. Antennas Propag. |

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 |

26. | C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,”J. Opt. Soc. Am. |

27. | J. Lin, X. C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodriguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffracted field,” Opt. Lett. |

28. | C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik |

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 |

30. | L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Acoust. Speech |

31. | A. Oppenheim, R. Schafer, and J. Buck, |

32. | http://www.tedpella.com/metrochip_html/metrochip-calibration-target.htm. |

33. | http://www.2spi.com/catalog/magnifiers/magnification-standard-geller-MRS-5.php. |

34. | T. Wilson and C. Sheppard, |

35. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

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

- S. M. Mansfield and G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett.57, 2615–2616 (1990). [CrossRef]
- Q. Wu, L. P. Ghislain, and V. B. Elings, “Imaging with solid immersion lenses, spatial resolution, and applications,” Proc. IEEE88, 1491–1498 (2000). [CrossRef]
- S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett.78, 4071–4073 (2001). [CrossRef]
- 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. Express16, 9501–9506 (2008). [CrossRef] [PubMed]
- Semicaps, “Optical fault localization system,” http://www.semicaps.com/innovations.htm (2011).
- L. P. Ghislain and V. B. Elings, “Near-field scanning solid immersion microscope,” Appl. Phys. Lett.72, 2779–2781 (1998). [CrossRef]
- D. A. Fletcher, K. B. Crozier, C. F. Quate, G. S. Kino, K. E. Goodson, D. Simanovskii, and D. V. Palanker, “Near-field infrared imaging with a microfabricated solid immersion lens,” Appl. Phys. Lett.77, 2109–2111 (2000). [CrossRef]
- 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]
- F. H. Köklü and M. S. Ünlü, “Subsurface microscopy of interconnect layers of an integrated circuit,” Opt. Lett.35, 184–186 (2010). [CrossRef] [PubMed]
- 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]
- 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. A28, 903–911 (2011). [CrossRef]
- 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. A29, 32–43 (2012). [CrossRef]
- 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. A29, 2350–2359 (2012). [CrossRef]
- C. J. R. Sheppard and A. Choudhury, “Image formation in scanning microscope,” Opt. Acta24, 1051–1073 (1977). [CrossRef]
- C. J. R. Sheppard and T. Wilson, “Image formation in scanning microscopes with partially coherent source and detector,” Opt. Acta25, 315–325 (1978). [CrossRef]
- C. J. R. Sheppard and T. Wilson, “The theory of the direct-view confocal microscope,” J. Microsc.-Oxf.124, 107–117 (1981). [CrossRef]
- 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]
- 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]
- 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]
- P. Török, P. R. T. Munro, and E. E. Kriezis, “High numerical aperture vectorial imaging in coherent optical microscopes,” Opt. Express16, 507–523 (2008). [CrossRef] [PubMed]
- 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. A29, 2696–2707 (2012). [CrossRef]
- M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express14, 11277–11291 (2006). [CrossRef] [PubMed]
- 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. Express19, 19280–19295 (2011). [CrossRef] [PubMed]
- 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]
- 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. A29, 1059–1070 (2012). [CrossRef]
- C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,”J. Opt. Soc. Am.54, 240–242 (1964). [CrossRef]
- J. Lin, X. C. Yuan, S. S. Kou, C. J. R. Sheppard, O. G. Rodriguez-Herrera, and J. C. Dainty, “Direct calculation of a three-dimensional diffracted field,” Opt. Lett.36, 1341–1343 (2011). [CrossRef] [PubMed]
- C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik107, 79–87 (1997).
- 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. Express20, 1060–1069 (2012). [CrossRef] [PubMed]
- L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Acoust. Speech17, 86–92 (1969).
- A. Oppenheim, R. Schafer, and J. Buck, Discrete-Time Signal Processing, 2nd ed (Prentice Hall, 1999).
- http://www.tedpella.com/metrochip_html/metrochip-calibration-target.htm .
- http://www.2spi.com/catalog/magnifiers/magnification-standard-geller-MRS-5.php .
- T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy, vol. 1 (London: Academic Press, 1984).
- P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972). [CrossRef]
- 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

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