Abstract

We present the derivation of the dyadic Green’s function for the aplanatic solid immersion lens based microscopy system. The presented dyadic Green’s function is general and is applicable at non-aplanatic points as well in the object plane. Thus, the electromagnetic wave formulation is used to describe the optical system without paraxial assumptions. Various important and useful properties of SIL based microscopy system are also presented. The effect of the numerical aperture of the objective on the peak intensities, resolutions and the depth of field are also reported. Some interesting longitudinal effects are also reported.

© 2011 OSA

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2011 (2)

2010 (5)

2009 (4)

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(5), 1036–1041 (2009).
[CrossRef]

S. H. Goh, C. J. R. Sheppard, A. C. T. Quah, C. M. Chua, L. S. Koh, and J. C. H. Phang, “Design considerations for refractive solid immersion lens: application to subsurface integrated circuit fault localization using laser induced techniques,” Rev. Sci. Instrum. 80(1), 013703 (2009).
[CrossRef] [PubMed]

Y. J. Yoon, W. C. Kim, N. C. Park, K. S. Park, and Y. P. Park, “Feasibility study of the application of radially polarized illumination to solid immersion lens-based near-field optics,” Opt. Lett. 34(13), 1961–1963 (2009).
[CrossRef] [PubMed]

T. Hakkarainen, T. Setälä, and A. T. Friberg, “Subwavelength electromagnetic near-field imaging of point dipole with metamaterial nanoslab,” J. Opt. Soc. Am. A 26(10), 2226–2234 (2009).
[CrossRef] [PubMed]

2008 (3)

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2(5), 311–314 (2008).
[CrossRef]

S. B. Ippolito, P. Song, D. L. Miles, and J. D. Sylvestri, “Angular spectrum tailoring in solid immersion microscopy for circuit analysis,” Appl. Phys. Lett. 92(10), 101109 (2008).
[CrossRef]

A. Nickolas 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(8), 758–768 (2008).
[CrossRef]

2007 (3)

C. A. Michaels, “Mid-infrared imaging with a solid immersion lens and broadband laser source,” Appl. Phys. Lett. 90(12), 121131 (2007).
[CrossRef]

E. Ramsay, K. A. Serrels, M. J. Thomson, A. J. Waddie, M. R. Taghizadeh, R. J. Warburton, and D. T. Reid, “Three-dimensional nanoscale subsurface optical imaging of silicon circuits,” Appl. Phys. Lett. 90(13), 131101 (2007).
[CrossRef]

J. Zhang, C. W. See, and M. G. Somekh, “Imaging performance of widefield solid immersion lens microscopy,” Appl. Opt. 46(20), 4202–4208 (2007).
[CrossRef] [PubMed]

2006 (3)

2005 (2)

A. K. Zvezdin and V. I. Belotelov, “Electrodynamic Green-function technique for investigating the magneto-optics of low-dimensional systems and nanostructures,” J. Opt. Soc. Am. B 22(1), 228–239 (2005).
[CrossRef]

S. B. Ippolito, B. B. Goldberg, and M. S. Unlu, “Theoretical analysis of numerical aperture increasing lens microscopy,” J. Appl. Phys. 97(5), 053105 (2005).
[CrossRef]

2004 (2)

2002 (1)

2001 (2)

L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191(3-6), 161–172 (2001).
[CrossRef]

S. B. Ippolito, B. B. Goldberg, and M. S. Unlu, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. 78(26), 4071–4073 (2001).
[CrossRef]

2000 (1)

Q. Wu, L. P. Ghislain, and V. B. Elings, “Imaging with solid immersion lenses, spatial resolution, and applications,” Proc. IEEE 88(9), 1491–1498 (2000).
[CrossRef]

1999 (2)

O. Keller, “Attached and radiated electromagnetic fields of an electric point dipole,” J. Opt. Soc. Am. B 16(5), 835–847 (1999).
[CrossRef]

T. Setälä, M. Kaivola, and A. T. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(1), 1200–1206 (1999).
[CrossRef]

1994 (1)

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, and G. S. Kino, “Near-field optical data storage using solid immersion lens,” Appl. Phys. Lett. 65(4), 388–390 (1994).
[CrossRef]

Altmeyer, S.

Behringer, E. R.

A. Nickolas 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(8), 758–768 (2008).
[CrossRef]

Belotelov, V. I.

Brunner, R.

Burkhardt, M.

Chen, J. B.

Chen, X. D.

Cheng, J. X.

Choudhury, A.

Chua, C. M.

S. H. Goh, C. J. R. Sheppard, A. C. T. Quah, C. M. Chua, L. S. Koh, and J. C. H. Phang, “Design considerations for refractive solid immersion lens: application to subsurface integrated circuit fault localization using laser induced techniques,” Rev. Sci. Instrum. 80(1), 013703 (2009).
[CrossRef] [PubMed]

Elings, V. B.

Q. Wu, L. P. Ghislain, and V. B. Elings, “Imaging with solid immersion lenses, spatial resolution, and applications,” Proc. IEEE 88(9), 1491–1498 (2000).
[CrossRef]

Ferstl, M.

Frank, J.

Friberg, A. T.

Ghislain, L. P.

Q. Wu, L. P. Ghislain, and V. B. Elings, “Imaging with solid immersion lenses, spatial resolution, and applications,” Proc. IEEE 88(9), 1491–1498 (2000).
[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(5), 1036–1041 (2009).
[CrossRef]

S. H. Goh, C. J. R. Sheppard, A. C. T. Quah, C. M. Chua, L. S. Koh, and J. C. H. Phang, “Design considerations for refractive solid immersion lens: application to subsurface integrated circuit fault localization using laser induced techniques,” Rev. Sci. Instrum. 80(1), 013703 (2009).
[CrossRef] [PubMed]

Goldberg, B. B.

A. Nickolas 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(8), 758–768 (2008).
[CrossRef]

S. B. Ippolito, B. B. Goldberg, and M. S. Unlu, “Theoretical analysis of numerical aperture increasing lens microscopy,” J. Appl. Phys. 97(5), 053105 (2005).
[CrossRef]

S. B. Ippolito, B. B. Goldberg, and M. S. Unlu, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. 78(26), 4071–4073 (2001).
[CrossRef]

Guo, H. M.

Hakkarainen, T.

Helseth, L. E.

L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191(3-6), 161–172 (2001).
[CrossRef]

Hohng, S.

Ippolito, S. B.

A. Nickolas 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(8), 758–768 (2008).
[CrossRef]

S. B. Ippolito, P. Song, D. L. Miles, and J. D. Sylvestri, “Angular spectrum tailoring in solid immersion microscopy for circuit analysis,” Appl. Phys. Lett. 92(10), 101109 (2008).
[CrossRef]

S. B. Ippolito, B. B. Goldberg, and M. S. Unlu, “Theoretical analysis of numerical aperture increasing lens microscopy,” J. Appl. Phys. 97(5), 053105 (2005).
[CrossRef]

S. B. Ippolito, B. B. Goldberg, and M. S. Unlu, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. 78(26), 4071–4073 (2001).
[CrossRef]

Jouravlev, M. V.

Kaivola, M.

T. Setälä, M. Kaivola, and A. T. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(1), 1200–1206 (1999).
[CrossRef]

Keller, O.

Kim, J. H.

Kim, K. S.

Kim, W. C.

Kim, Y.

Kino, G. S.

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, and G. S. Kino, “Near-field optical data storage using solid immersion lens,” Appl. Phys. Lett. 65(4), 388–390 (1994).
[CrossRef]

Koh, L. S.

S. H. Goh, C. J. R. Sheppard, A. C. T. Quah, C. M. Chua, L. S. Koh, and J. C. H. Phang, “Design considerations for refractive solid immersion lens: application to subsurface integrated circuit fault localization using laser induced techniques,” Rev. Sci. Instrum. 80(1), 013703 (2009).
[CrossRef] [PubMed]

Köklü, F. H.

Lajunen, H.

Lee, G. C. F.

Lee, J.

Liang, Z. C.

Lim, K. M.

Mamin, H. J.

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, and G. S. Kino, “Near-field optical data storage using solid immersion lens,” Appl. Phys. Lett. 65(4), 388–390 (1994).
[CrossRef]

Martinsson, P.

Mason, D. R.

Michaels, C. A.

C. A. Michaels, “Mid-infrared imaging with a solid immersion lens and broadband laser source,” Appl. Phys. Lett. 90(12), 121131 (2007).
[CrossRef]

Miles, D. L.

S. B. Ippolito, P. Song, D. L. Miles, and J. D. Sylvestri, “Angular spectrum tailoring in solid immersion microscopy for circuit analysis,” Appl. Phys. Lett. 92(10), 101109 (2008).
[CrossRef]

Milster, T. D.

Nickolas Vamivakas, A.

A. Nickolas 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(8), 758–768 (2008).
[CrossRef]

Park, K. S.

Park, N. C.

Park, Y. P.

Pesch, A.

Phang, J. C. H.

K. M. Lim, G. C. F. Lee, C. J. R. Sheppard, J. C. H. Phang, C. L. Wong, and X. D. Chen, “Effect of polarization on a solid immersion lens of arbitrary thickness,” J. Opt. Soc. Am. A 28(5), 903–911 (2011).
[CrossRef] [PubMed]

S. H. Goh, C. J. R. Sheppard, A. C. T. Quah, C. M. Chua, L. S. Koh, and J. C. H. Phang, “Design considerations for refractive solid immersion lens: application to subsurface integrated circuit fault localization using laser induced techniques,” Rev. Sci. Instrum. 80(1), 013703 (2009).
[CrossRef] [PubMed]

Pitter, M. C.

Quah, A. C. T.

S. H. Goh, C. J. R. Sheppard, A. C. T. Quah, C. M. Chua, L. S. Koh, and J. C. H. Phang, “Design considerations for refractive solid immersion lens: application to subsurface integrated circuit fault localization using laser induced techniques,” Rev. Sci. Instrum. 80(1), 013703 (2009).
[CrossRef] [PubMed]

Ramsay, E.

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2(5), 311–314 (2008).
[CrossRef]

E. Ramsay, K. A. Serrels, M. J. Thomson, A. J. Waddie, M. R. Taghizadeh, R. J. Warburton, and D. T. Reid, “Three-dimensional nanoscale subsurface optical imaging of silicon circuits,” Appl. Phys. Lett. 90(13), 131101 (2007).
[CrossRef]

Reid, D. T.

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2(5), 311–314 (2008).
[CrossRef]

E. Ramsay, K. A. Serrels, M. J. Thomson, A. J. Waddie, M. R. Taghizadeh, R. J. Warburton, and D. T. Reid, “Three-dimensional nanoscale subsurface optical imaging of silicon circuits,” Appl. Phys. Lett. 90(13), 131101 (2007).
[CrossRef]

Rugar, D.

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, and G. S. Kino, “Near-field optical data storage using solid immersion lens,” Appl. Phys. Lett. 65(4), 388–390 (1994).
[CrossRef]

Sandfuchs, O.

See, C. W.

Serrels, K. A.

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2(5), 311–314 (2008).
[CrossRef]

E. Ramsay, K. A. Serrels, M. J. Thomson, A. J. Waddie, M. R. Taghizadeh, R. J. Warburton, and D. T. Reid, “Three-dimensional nanoscale subsurface optical imaging of silicon circuits,” Appl. Phys. Lett. 90(13), 131101 (2007).
[CrossRef]

Setälä, T.

T. Hakkarainen, T. Setälä, and A. T. Friberg, “Subwavelength electromagnetic near-field imaging of point dipole with metamaterial nanoslab,” J. Opt. Soc. Am. A 26(10), 2226–2234 (2009).
[CrossRef] [PubMed]

T. Setälä, M. Kaivola, and A. T. Friberg, “Decomposition of the point-dipole field into homogeneous and evanescent parts,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59(1), 1200–1206 (1999).
[CrossRef]

Sheppard, C. J. R.

K. M. Lim, G. C. F. Lee, C. J. R. Sheppard, J. C. H. Phang, C. L. Wong, and X. D. Chen, “Effect of polarization on a solid immersion lens of arbitrary thickness,” J. Opt. Soc. Am. A 28(5), 903–911 (2011).
[CrossRef] [PubMed]

S. H. Goh, C. J. R. Sheppard, A. C. T. Quah, C. M. Chua, L. S. Koh, and J. C. H. Phang, “Design considerations for refractive solid immersion lens: application to subsurface integrated circuit fault localization using laser induced techniques,” Rev. Sci. Instrum. 80(1), 013703 (2009).
[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(5), 1036–1041 (2009).
[CrossRef]

C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43(22), 4322–4327 (2004).
[CrossRef] [PubMed]

Somekh, M. G.

Song, P.

S. B. Ippolito, P. Song, D. L. Miles, and J. D. Sylvestri, “Angular spectrum tailoring in solid immersion microscopy for circuit analysis,” Appl. Phys. Lett. 92(10), 101109 (2008).
[CrossRef]

Studenmund, W. R.

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, and G. S. Kino, “Near-field optical data storage using solid immersion lens,” Appl. Phys. Lett. 65(4), 388–390 (1994).
[CrossRef]

Swan, A. K.

A. Nickolas 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(8), 758–768 (2008).
[CrossRef]

Sylvestri, J. D.

S. B. Ippolito, P. Song, D. L. Miles, and J. D. Sylvestri, “Angular spectrum tailoring in solid immersion microscopy for circuit analysis,” Appl. Phys. Lett. 92(10), 101109 (2008).
[CrossRef]

Taghizadeh, M. R.

E. Ramsay, K. A. Serrels, M. J. Thomson, A. J. Waddie, M. R. Taghizadeh, R. J. Warburton, and D. T. Reid, “Three-dimensional nanoscale subsurface optical imaging of silicon circuits,” Appl. Phys. Lett. 90(13), 131101 (2007).
[CrossRef]

Terris, B. D.

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, and G. S. Kino, “Near-field optical data storage using solid immersion lens,” Appl. Phys. Lett. 65(4), 388–390 (1994).
[CrossRef]

Thomson, M. J.

E. Ramsay, K. A. Serrels, M. J. Thomson, A. J. Waddie, M. R. Taghizadeh, R. J. Warburton, and D. T. Reid, “Three-dimensional nanoscale subsurface optical imaging of silicon circuits,” Appl. Phys. Lett. 90(13), 131101 (2007).
[CrossRef]

Unlu, M. S.

S. B. Ippolito, B. B. Goldberg, and M. S. Unlu, “Theoretical analysis of numerical aperture increasing lens microscopy,” J. Appl. Phys. 97(5), 053105 (2005).
[CrossRef]

S. B. Ippolito, B. B. Goldberg, and M. S. Unlu, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. 78(26), 4071–4073 (2001).
[CrossRef]

Unlü, M. S.

Ünlü, M. S.

A. Nickolas 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(8), 758–768 (2008).
[CrossRef]

Waddie, A. J.

E. Ramsay, K. A. Serrels, M. J. Thomson, A. J. Waddie, M. R. Taghizadeh, R. J. Warburton, and D. T. Reid, “Three-dimensional nanoscale subsurface optical imaging of silicon circuits,” Appl. Phys. Lett. 90(13), 131101 (2007).
[CrossRef]

Wang, L.

Warburton, R. J.

K. A. Serrels, E. Ramsay, R. J. Warburton, and D. T. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nat. Photonics 2(5), 311–314 (2008).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

The setup of the SIL based microscopy system. (a) Diagram showing various interfaces and the path of a ray travelling through these interfaces. Various regions, coordinate systems, and important angles are also indicated. The horizontal axis shown above is the longitudinal axis z ^ . (b) A practical representation of the SIL-based microscopy system. (c) The non-SIL based system obtained by substituting n S I L = n o b j , which is used in all the numerical experiments presented in sections 5-8. (d) Illustration of the angles γ S I L and γ o b j A used in Eqs. (3) and (4).

Fig. 2
Fig. 2

Basic characteristics of the dyadic Green’s function of SIL based microscopy system for N A max = 0.2857 . (a) The intensity along the x and y axes in the CCD region corresponding to a x ^ directed dipole at the aplanatic point, for the case x C C D = 0 , ρ C C D = y C C D while the case y C C D = 0 , ρ C C D = x C C D (b) The intensity along the z axis in the CCD region corresponding to a x ^ directed dipole at the aplanatic point. (c) The intensity along the x axis in the CCD region corresponding to a z ^ directed dipole at the aplanatic point.

Fig. 3
Fig. 3

Comparison of the dyadic Green’s function of SIL based microscopy system with non-SIL based microscopy system for N A = 0.2857 . (a) The normalized intensity along the x axis in the CCD region corresponding to a x ^ directed dipole at the aplanatic point. (b) The normalized intensity along the z axis in the CCD region corresponding to a x ^ directed dipole at the aplanatic point. (c) The normalized intensity along the x axis in the CCD region corresponding to a z ^ directed dipole at the aplanatic point.

Fig. 4
Fig. 4

Comparison of the shift in image of a x ^ directed dipole in SIL and non-SIL based microscopy systems for N A = 0.2857 . (a) For dipole location r S I L = ( 2 λ , 0 , 0 ) , the intensity along the x axis in the CCD region. (b) For dipole location r S I L = ( 0 , 0 , 2 λ ) , the intensity along the z axis in the CCD region.

Fig. 5
Fig. 5

Visual demonstration of resolution limit according to the Rayleigh Criterion for two x ^ directed dipoles along x axis. (a) For the SIL based system, Δ x = 0.245 λ satisfies the Rayleigh criterion and the dipoles with Δ x = 0.245 λ can be indeed resolved by the SIL based system, (b) The non-SIL based system cannot resolve the sources with Δ x = 0.245 λ . (c) For the non-SIL based system, Δ x = 2.874 λ satisfies the Rayleigh criterion and the dipoles with Δ x = 2.874 λ can be resolved by the non-SIL based system.

Fig. 6
Fig. 6

Effect of numerical aperture on the point spread function. (a) The normalized intensity along the x axis in the CCD region for various values of numerical aperture. (b) The normalized intensity along the z axis in the CCD region for various values of numerical aperture. (c) The peak intensities of SIL based and non-SIL based microscopy systems for various values of numerical apertures. (d) The full width at half maximum of the point spreads of SIL based and non-SIL based microscopy systems for various values of numerical apertures.

Fig. 7
Fig. 7

Effect of numerical aperture of the objective along the longitudinal axis. We consider six positions z S I L = { 0 , 0.2 , 0.4 , 0.6 , 0.8 , 1 } λ , x S I L = x S I L = 0 . (a) N A max (b) N A = 0.2 (c) N A = 0.1 .

Fig. 8
Fig. 8

Effect of numerical aperture of the objective along the longitudinal axis. (a) The peak points in the image (CCD) region vs. the actual source locations in the object (SIL) region. The faint gray line shows the expected peak points. (b) The peak intensities in the image (CCD) plane corresponding to the actual locations in the object (SIL) region. For both (a) and (b), the non-SIL based microscopy system uses N A = 0.2857 . M S I L l o n from Eq. (24) is used for SIL based microscopy system and M = n C C D M 2 / n o b j is used for non-SIL based microscopy system.

Equations (24)

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E S I L ( r S I L , r S I L ) ω 2 μ 0 ( θ ^ S I L θ ^ S I L + ϕ ^ S I L ϕ ^ S I L ) exp ( i k S I L r S I L ) 4 π r S I L exp ( i k S I L r S I L ) . p ( r S I L )
E o b j ( r o b j A , r S I L ) ω 2 μ 0 ( t p A θ ^ o b j A θ ^ S I L A + t s A ϕ ^ o b j A ϕ ^ S I L A ) exp ( i k o b j r o b j A ) 4 π r o b j A exp ( i k S I L r S I L ) . p ( r S I L )
t s A = 2 n S I L cos γ S I L A n S I L cos γ S I L A + n o b j cos γ o b j A exp ( i k S I L r S I L A i k o b j r o b j A ) r o b j A r S I L A
t p A = 2 n S I L cos γ S I L A n o b j cos γ S I L A + n S I L cos γ o b j A exp ( i k S I L r S I L A i k o b j r o b j A ) r o b j A r S I L A
t s A = 2 n S I L cos θ o b j A n S I L cos θ o b j A + n o b j cos θ S I L A n S I L n o b j
t p A = 2 n S I L cos θ o b j A n o b j cos θ o b j A + n S I L cos θ S I L A n S I L n o b j
E C C D ( θ C C D B , ϕ C C D B ) ( ( E o b j . s ^ o b j ) s ^ C C D + ( E o b j . p ^ o b j ) p ^ C C D ) n o b j cos θ C C D B n C C D cos θ o b j A
E C C D ( θ C C D B , ϕ C C D B ) = ω 2 μ 0 exp ( i k o b j f o b j ) 4 π f o b j exp ( i k S I L r S I L ) × n o b j cos θ C C D B n C C D cos θ o b j A ( t p A θ ^ C C D B θ ^ S I L A + t s A ϕ ^ S I L A ϕ ^ S I L A ) p ( r S I L ) = ω 2 μ 0 exp ( i k o b j f o b j ) 8 π f o b j exp ( i k S I L r S I L ) × n o b j cos θ C C D B n C C D cos θ o b j A [ E ¯ x E ¯ y E ¯ z ] p ( r S I L )
E ¯ x = [ ( t s A + t p A cos θ C C D B cos θ S I L A ) ( t s A t p A cos θ C C D B cos θ S I L A ) cos 2 ϕ S I L A ( t s A t p A cos θ C C D B cos θ S I L A ) sin 2 ϕ S I L A 2 t p A sin θ C C D B cos θ S I L A cos ϕ S I L A ] E ¯ y = [ ( t s A t p A cos θ C C D B cos θ S I L A ) sin 2 ϕ S I L A ( t s A + t p A cos θ C C D B cos θ S I L A ) + ( t s A t p A cos θ C C D B cos θ S I L A ) cos 2 ϕ S I L A 2 t p A sin θ C C D B cos θ S I L A sin ϕ S I L A ] E ¯ z = 2 t p A sin θ S I L A [ cos θ C C D B cos ϕ S I L A cos θ C C D B sin ϕ S I L A sin θ C C D B ]
E C C D ( r C C D ) = i k C C D f C C D exp ( i k C C D f C C D ) 2 π                     Ω C C D E C C D ( θ C C D B , ϕ C C D B ) exp ( i k C C D r C C D ) sin θ C C D B   d θ C C D B d ϕ C C D B
E C C D ( r C C D ) = ω 2 μ 0 G P S F p ( r S I L )
G P S F = α [ I 0 + I 21 I 22 2 i I 11 I 22 I 0 I 21 2 i I 12 0 0 0 ]
α = i k C C D 8 π f o b j f C C D ( n o b j n C C D ) 1 2 exp ( i ( k o b j f o b j + k C C D f C C D ) ) I 0 = 0 θ max sin θ o b j A cos θ o b j A ( t s A + t p A cos θ S I L A ) J 0 ( ρ ) exp ( i z ) d θ o b j A I 11 = 0 θ max sin θ o b j A cos θ o b j A ( t p A sin θ S I L A ) J 1 ( ρ ) exp ( i z ) cos ψ d θ o b j A I 12 = 0 θ max sin θ o b j A cos θ o b j A ( t p A sin θ S I L A ) J 1 ( ρ ) exp ( i z ) sin ψ d θ o b j A I 21 = 0 θ max sin θ o b j A cos θ o b j A ( t s A t p A cos θ S I L A ) J 2 ( ρ ) exp ( i z ) cos 2 ψ d θ o b j A I 22 = 0 θ max sin θ o b j A cos θ o b j A ( t s A t p A cos θ S I L A ) J 2 ( ρ ) exp ( i z ) sin 2 ψ d θ o b j A
ρ = x 2 + y 2 ; ψ = tan 1 y x ; x = ( k C C D sin θ C C D B x C C D B + k S I L sin θ S I L A x S I L ) y = ( k C C D sin θ C C D B y C C D B + k S I L sin θ S I L A y S I L ) z = k C C D cos θ C C D B z C C D B k S I L cos θ S I L A z S I L
N A max = ( n o b j ) 2 n S I L
t = t s = t p = 2 n S I L n S I L + n o b j n S I L n o b j
x = x ˜ θ o b j ;  where  x ˜ = k o b j M ( x C C D + M ( n S I L n o b j ) 2 x S I L ) y = y ˜ θ o b j ;  where  y ˜ = k o b j M ( y C C D + M ( n S I L n o b j ) 2 y S I L ) z = z ^ + θ o b j 2 z ˜ ;                    where  z ^ = ( k C C D z C C D k S I L z S I L )                   and  z ˜ = 1 2 ( k C C D z C C D ( f o b j f C C D ) 2 k S I L z S I L ( n S I L n o b j ) 2 ) ρ = θ o b j ρ ˜ ;  where  ρ ˜ = x ˜ 2 + y ˜ 2 ψ = tan 1 y ˜ x ˜ ;
I 0 t 0 θ max θ o b j ( 1 1 4 θ o b j 2 ) ( 2 1 2 ( n S I L n o b j ) 2 θ o b j 2 ) J 0 ( ρ ) d θ o b j 2 t 0 θ max θ o b j J 0 ( ρ ˜ θ o b j ) d θ o b j 2 t θ max 2 J 1 ( ρ ˜ θ max ) ( ρ ˜ θ max ) , ,
G P S F = α 2 t θ max 2 J 1 ( ρ ˜ θ max ) ( ρ ˜ θ max ) [ 1 0 0 0 1 0 0 0 0 ]
M S I L l a t = M ( n S I L n o b j ) 2
I 0 t exp ( i z ^ ) 0 θ max θ o b j ( 1 1 4 θ o b j 2 ) ( 2 1 2 ( n S I L n o b j ) 2 θ o b j 2 ) exp ( i z ˜ θ o b j 2 ) d θ o b j 2 t exp ( i z ^ ) 0 θ max θ o b j exp ( i z ˜ θ o b j 2 ) d θ o b j = t θ max 2 sin ( z ˜ θ max 2 2 ) ( z ˜ θ max 2 2 ) exp ( i z ˜ θ max 2 2 + i z ^ )
G P S F = α t θ max 2 sin ( z ˜ θ max 2 2 ) ( z ˜ θ max 2 2 ) exp ( i z ˜ θ max 2 2 + i z ^ ) [ 1 0 0 0 1 0 0 0 0 ]
z C C D = n S I L 1 n C C D ( f C C D f o b j ) 2 ( n S I L n o b j ) 2 z S I L = M ( n S I L n o b j ) 3 z S I L
M S I L l o n = ( n S I L n o b j ) 3 M

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