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

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [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

2012 (5)

2011 (4)

2010 (1)

2009 (1)

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]

2008 (3)

2006 (1)

2001 (1)

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

2000 (2)

Q. Wu, L. P. Ghislain, and V. B. Elings, “Imaging with solid immersion lenses, spatial resolution, and applications,” Proc. IEEE88, 1491–1498 (2000).
[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]

1998 (2)

L. P. Ghislain and V. B. Elings, “Near-field scanning solid immersion microscope,” Appl. Phys. Lett.72, 2779–2781 (1998).
[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]

1997 (2)

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]

C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik107, 79–87 (1997).

1990 (1)

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

1981 (1)

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

1979 (1)

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]

1978 (1)

C. J. R. Sheppard and T. Wilson, “Image formation in scanning microscopes with partially coherent source and detector,” Opt. Acta25, 315–325 (1978).
[CrossRef]

1977 (1)

C. J. R. Sheppard and A. Choudhury, “Image formation in scanning microscope,” Opt. Acta24, 1051–1073 (1977).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

1969 (1)

L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Acoust. Speech17, 86–92 (1969).

1964 (1)

Agarwal, K.

Barends, P.

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]

Behringer, E. R.

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]

Blom, P.

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]

Brakenhoff, G. J.

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]

Buck, J.

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

Chen, R.

Chen, X.

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]

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]

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]

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]

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]

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]

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

Choudhury, A.

C. J. R. Sheppard and A. Choudhury, “Image formation in scanning microscope,” Opt. Acta24, 1051–1073 (1977).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Coe, R. L.

Crozier, K. B.

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]

Dainty, J. C.

Elings, V. B.

Q. Wu, L. P. Ghislain, and V. B. Elings, “Imaging with solid immersion lenses, spatial resolution, and applications,” Proc. IEEE88, 1491–1498 (2000).
[CrossRef]

L. P. Ghislain and V. B. Elings, “Near-field scanning solid immersion microscope,” Appl. Phys. Lett.72, 2779–2781 (1998).
[CrossRef]

Fletcher, D. A.

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]

Ghislain, L. P.

Q. Wu, L. P. Ghislain, and V. B. Elings, “Imaging with solid immersion lenses, spatial resolution, and applications,” Proc. IEEE88, 1491–1498 (2000).
[CrossRef]

L. P. Ghislain and V. B. Elings, “Near-field scanning solid immersion microscope,” Appl. Phys. Lett.72, 2779–2781 (1998).
[CrossRef]

Goh, S. H.

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]

Goldberg, B. B.

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]

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]

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

Goodson, K. E.

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]

Higdon, P.

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]

Higdon, P. D.

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]

Hoang, T. X.

Hu, L.

Ippolito, S. B.

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]

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]

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

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Juskaitis, R.

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]

Kenny, F.

Kino, G. S.

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]

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

Koh, L. S.

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

Köklü, F. H.

Kou, S. S.

Kriezis, E. E.

Lara, D.

Larkin, K. G.

C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik107, 79–87 (1997).

Lasser, T.

Lee, G. C. F.

Leitgeb, R. A.

Leutenegger, M.

Lim, K. M.

Lin, J.

Mansfield, S. M.

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

McCutchen, C. W.

Munro, P. R. T.

Oppenheim, A.

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

Palanker, D. V.

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]

Phang, J. C. H.

Quate, C. F.

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]

Quesnel, J. I.

Rabiner, L.

L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Acoust. Speech17, 86–92 (1969).

Rader, C.

L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Acoust. Speech17, 86–92 (1969).

Rao, R.

Rodriguez-Herrera, O. G.

Schafer, R.

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).

Seibel, E. J.

Sheppard, C.

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

Sheppard, C. J. R.

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]

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]

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]

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]

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]

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]

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]

C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik107, 79–87 (1997).

C. J. R. Sheppard and T. Wilson, “The theory of the direct-view confocal microscope,” J. Microsc.-Oxf.124, 107–117 (1981).
[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 A. Choudhury, “Image formation in scanning microscope,” Opt. Acta24, 1051–1073 (1977).
[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

Simanovskii, D.

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]

Swan, A. K.

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]

Török, P.

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]

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]

Ünlü, M. S.

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

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, 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]

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

Vamivakas, A. N.

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Figures (13)

Fig. 1
Fig. 1

Block diagram of the computational model.

Fig. 2
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.

Fig. 3
Fig. 3

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

Fig. 4
Fig. 4

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

Fig. 5
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.

Fig. 6
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
Fig. 7

The caption is the same as that of Fig. 6 except for NAsil = 2.4.

Fig. 8
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.

Fig. 9
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.

Fig. 10
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.

Fig. 11
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
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.

Fig. 13
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.

Tables (3)

Tables Icon

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

Tables Icon

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.

Tables Icon

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

Equations (21)

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E sil = ( n 0 n obj cos θ obj ) 1 / 2 f obj r OA exp ( i k obj f obj i k obj r O A ) T E inc ( θ )
T = 1 2 [ ( t s + t p cos θ sil ) ( t s t p cos θ sil ) cos 2 ϕ obj ( t s t p cos θ sil ) sin 2 ϕ obj 2 t p sin θ sil cos ϕ obj ( t s t p cos θ sil ) sin 2 ϕ obj ( t s + t p cos θ sil ) + ( t s t p cos θ sil ) cos 2 ϕ obj 2 t p sin θ sil sin ϕ obj 2 t p sin θ sil cos ϕ obj 2 t p sin θ sil sin ϕ obj 2 t p cos θ sil ] .
E ( x sil , y sil , z sil ) = Ω i r F A exp ( i k sil r F A ) 2 π k z sil E sil exp ( i k x sil x sil + i k y sil y sil + i k z sil z sil ) d k x sil d k y sil ,
E ( x sil , y sil , z sil ) = α Ω E f f t ( k x sil , k y sil ) exp ( i k x sil x sil + i k y sil y sil ) d k x sil d k y sil
E ( p , q , z sil ) = α m = M / 2 M / 2 1 n = N / 2 N / 2 1 E f f t ( m , n ) exp [ 2 π i ( p m M + q n N ) ] Δ k x sil Δ k y sil ; ( M / 2 p M / 2 1 ) , ( N / 2 q N / 2 1 ) ,
E ( p , q , z sil ) = α IFFT [ E f f t ( m , n ) ] Δ k x sil Δ k y sil .
Δ x sil = 2 π / ( M f f t Δ k x sil ) , Δ y sil = 2 π / ( N f f t Δ k y sil ) ,
E ( p , q ) = α m = M / 2 M / 2 1 n = N / 2 N / 2 1 E f f t ( m , n ) A x m A y n W x p m W x q n Δ k x sil Δ k y sil ; ( P / 2 p P / 2 ) , ( Q / 2 q Q / 2 ) ,
E c c d = i ω μ exp ( i k obj f obj ) 8 π f obj exp ( i k sil r sil ) ( n obj cos θ c c d n c c d cos θ obj ) 1 2 [ G x , G y , G z ] Il ( r sil )
G x = [ ( t s + t p cos θ c c d cos θ sil ) ( t s t p cos θ c c d cos θ s i l ) cos 2 ϕ obj ( t s t p cos θ c c d cos θ sil ) sin 2 ϕ obj 2 t p sin θ c c d cos θ sil cos ϕ obj ] ,
G y = [ ( t s t p cos θ c c d cos θ sil ) sin 2 ϕ obj ( t s + t p cos θ c c d cos θ sil ) + ( t s t p cos θ c c d cos θ sil ) cos 2 ϕ obj 2 t p sin θ c c d cos θ sil sin ϕ obj ] ,
G z = 2 t p sin θ sil [ cos θ c c d cos ϕ o b j cos θ c c d sin ϕ o b j sin θ c c d ] .
E ( r c c d ) = i ω μ G ¯ ¯ P S F Il ( r sil ) .
G ¯ ¯ P S F = β Ω c c d g ¯ ¯ exp [ i k x c c d ( x c c d + M l a t x sil ) + k y c c d ( y c c d + M lat y sil ) ] d k x c c d d k y c c d ,
g ¯ ¯ = [ G x , G y , G z ] cos θ c c d ( cos θ c c d cos θ obj ) 1 / 2 exp ( i k z c c d z c c d i k z sil z sil ) ,
E ( x c c d , y c c d ) = i ω μ G ¯ ¯ P S F ( x sil + x c c d / M lat , y sil + y ccd / M lat ) I l ( x sil , y sil ) ,
G ¯ ¯ P S F = β M lat 2 g ¯ ¯ exp [ i k x sil ( x sil + x c c d M lat ) i k y sil ( y sil + y c c d M lat ) ] d k x sil d k y sil
E x x ( p , q ) = i ω μ m n I l x ( m , n ) G x x ( p + m , q + n ) ,
E x x ( p , q ) = i ω μ ( I l x ( p , q ) * G x x ( p , q ) ) .
E x x ( p , q ) = i ω μ IFFT { FFT [ I l x ( p , q ) ] FFT [ G x x ( p , q ) ] } ,
I = S | E ( x c c d , y c c d ) | 2 d S = 0 R P H 0 2 π | E ( x c c d , y c c d ) | 2 d r c c d d ϕ .

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