Abstract

We study the resolution of a subsurface microscopy system based on the use of an aplanatic solid immersion lens. Resolution limits under various criteria are calculated theoretically as well as numerically. Images of combinations of dipoles of various orientations are considered. Both lateral and longitudinal resolutions are studied. The theoretical criteria are compared against the visually resolvable simulated images of the dipoles. The observations are explained explicitly through a detailed analysis of the dyadic Green’s function. A new resolution criterion is also proposed, which provides a very accurate estimate of the resolution limits.

© 2012 Optical Society of America

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References

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  1. S. M. Mansfield and G. S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57, 2615–2616 (1990).
    [CrossRef]
  2. 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]
  3. 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, 013703 (2009).
    [CrossRef]
  4. Y. J. Zhang, “Design of high-performance supersphere solid immersion lenses,” Appl. Opt. 45, 4540–4546 (2006).
    [CrossRef]
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    [CrossRef]
  6. 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]
  7. S. B. Ippolito, B. B. Goldberg, and M. S. Ünlü, “High spatial resolution subsurface microscopy,” Appl. Phys. Lett. 78, 4071–4073 (2001).
    [CrossRef]
  8. 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]
  9. L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001).
    [CrossRef]
  10. K. A. Serrels, E. Ramsay, and D. T. Reid, “70 nm resolution in subsurface optical imaging of silicon integrated-circuits using pupil-function engineering,” Appl. Phys. Lett. 94073113 (2009).
    [CrossRef]
  11. F. H. Köklü, and M. S. Ünlü, “Subsurface microscopy of interconnect layers of an integrated circuit,” Opt. Lett. 35, 184–186 (2010)
    [CrossRef]
  12. 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]
  13. L. Wang, M. C. Pitter, and M. G. Somekh, “Wide-field high-resolution solid immersion fluorescence microscopy applying an aplanatic solid immersion lens,” Appl. Opt. 49, 6160–6169 (2010).
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    [CrossRef]
  15. C. J. R. Sheppard, T. J. Connolly, J. Lee, and C. J. Cogswell, “Confocal imaging of a stratified medium,” Appl. Opt. 33, 631–640 (1994).
    [CrossRef]
  16. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
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    [CrossRef]
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  20. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2001).
  21. C. A. Balanis, Antenna Theory: Analysis and Design (Wiley, 2005).

2011

2010

2009

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, 013703 (2009).
[CrossRef]

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. A. Serrels, E. Ramsay, and D. T. Reid, “70 nm resolution in subsurface optical imaging of silicon integrated-circuits using pupil-function engineering,” Appl. Phys. Lett. 94073113 (2009).
[CrossRef]

2008

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]

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]

2006

2001

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

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

2000

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]

1994

1990

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

1978

Agarwal, K.

Balanis, C. A.

C. A. Balanis, Antenna Theory: Analysis and Design (Wiley, 2005).

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]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2001).

Chen, R.

Chen, X.

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, 013703 (2009).
[CrossRef]

Cogswell, C. J.

Connolly, T. J.

Corle, T. R.

T. R. Corle and G. S. Kino, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic, 1996).

Elings, V. B.

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]

Ghislain, L. P.

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]

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]

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, 013703 (2009).
[CrossRef]

Goldberg, B. B.

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]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

Helseth, L. E.

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

Hu, L.

Ippolito, S. B.

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]

Jouravlev, M. V.

Kim, K. S.

Kino, G. S.

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

T. R. Corle and G. S. Kino, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic, 1996).

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, 013703 (2009).
[CrossRef]

Köklü, F. H.

Kriezis, E. E.

Lee, G. C. F.

Lee, J.

Lim, K. M.

Mansfield, S. M.

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

Mason, D. R.

Munro, P. R. T.

Nayyar, V. P.

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

Phang, J. C. H.

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, 013703 (2009).
[CrossRef]

Ramsay, E.

K. A. Serrels, E. Ramsay, and D. T. Reid, “70 nm resolution in subsurface optical imaging of silicon integrated-circuits using pupil-function engineering,” Appl. Phys. Lett. 94073113 (2009).
[CrossRef]

Reid, D. T.

K. A. Serrels, E. Ramsay, and D. T. Reid, “70 nm resolution in subsurface optical imaging of silicon integrated-circuits using pupil-function engineering,” Appl. Phys. Lett. 94073113 (2009).
[CrossRef]

Serrels, K. A.

K. A. Serrels, E. Ramsay, and D. T. Reid, “70 nm resolution in subsurface optical imaging of silicon integrated-circuits using pupil-function engineering,” Appl. Phys. Lett. 94073113 (2009).
[CrossRef]

Sheppard, C. J. R.

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]

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]

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, 013703 (2009).
[CrossRef]

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, T. J. Connolly, J. Lee, and C. J. Cogswell, “Confocal imaging of a stratified medium,” Appl. Opt. 33, 631–640 (1994).
[CrossRef]

Somekh, M. G.

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.

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]

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]

Vamivakas, A. N.

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]

Verma, N. K.

Wang, L.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2001).

Wong, C. L.

Wu, Q.

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]

Younger, R. D.

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]

Zhang, Y. J.

Am. J. Phys.

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]

Appl. Opt.

Appl. Phys. Lett.

K. A. Serrels, E. Ramsay, and D. T. Reid, “70 nm resolution in subsurface optical imaging of silicon integrated-circuits using pupil-function engineering,” Appl. Phys. Lett. 94073113 (2009).
[CrossRef]

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

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

J. Opt. Soc. Am. A

Opt. Commun.

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

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]

Opt. Express

Opt. Lett.

Proc. IEEE

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]

Rev. Sci. Instrum.

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, 013703 (2009).
[CrossRef]

Other

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

T. R. Corle and G. S. Kino, Confocal Scanning Optical Microscopy and Related Imaging Systems (Academic, 1996).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2001).

C. A. Balanis, Antenna Theory: Analysis and Design (Wiley, 2005).

Supplementary Material (5)

» Media 1: MOV (699 KB)     
» Media 2: MOV (696 KB)     
» Media 3: MOV (705 KB)     
» Media 4: MOV (621 KB)     
» Media 5: MOV (649 KB)     

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Figures (11)

Fig. 1.
Fig. 1.

The setup of the SIL-based microscopy system and non-SIL-based system. (a) A practical representation of the SIL-based microscopy system. (b) The non-SIL-based system obtained by substituting nSIL=nobj, which is used in all the numerical experiments presented in Section 3. (c) 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^.

Fig. 2.
Fig. 2.

Definitions of parameters used for defining resolution in Figs. 35. (a) Sparrow criterion and Rayleigh criterion. (b) Modified Rayleigh criterion and Sparrow criterion.

Fig. 3.
Fig. 3.

One y^ and one x^ directed dipoles along x axis. (a) Plot of Imin/Imax for various values of NA for the SIL-based system and the non-SIL-based system. (b) Plot of Imin/max(In(r⃗SIL1),In(r⃗SIL2)) for various values of NA for the SIL-based system and the non-SIL-based system. (c)–(e) Visual demonstration of the resolution limit for the SIL-based system using NAmax. The sources cannot be resolved in (c), barely resolved in (d), and clearly resolved in (e). (f) Barely resolved dipoles for SIL-based system with NA=0.2. (g) Barely resolved dipoles for SIL-based system with NA=0.1. (h) Barely resolved dipoles for non-SIL-based system with NAmax.

Fig. 4.
Fig. 4.

Two y^ directed dipoles along x axis. (a) Plot of Imin/Imax for various values of NA for the SIL-based system and the non-SIL-based system. (b) Plot of Imin/max(In(r⃗SIL1),In(r⃗SIL2)) for various values of NA for the SIL-based system and the non-SIL-based system. (c)–(e) Visual demonstration of the resolution limit for the SIL-based system using NAmax. The sources cannot be resolved in (c), barely resolved in (d), and clearly resolved in (e). (f) Barely resolved dipoles for SIL-based system with NA=0.2. (g) Barely resolved dipoles for SIL-based system with NA=0.1. (h) Barely resolved dipoles for non-SIL-based system with NAmax.

Fig. 5.
Fig. 5.

Two x^ directed dipoles along x axis. (a) Plot of Imin/Imax for various values of NA for the SIL-based system and the non-SIL-based system. (b) Plot of Imin/max(In(r⃗SIL1),In(r⃗SIL2)) for various values of NA for the SIL-based system and the non-SIL-based system. (c)–(e) Visual demonstration of the resolution limit for the SIL-based system using NAmax. The sources cannot be resolved in (c), barely resolved in (d), and clearly resolved in (e). (f) Barely resolved dipoles for SIL-based system with NA=0.2. (g) Barely resolved dipoles for SIL-based system with NA=0.1. (h) Barely resolved dipoles for non-SIL-based system with NAmax.

Fig. 6.
Fig. 6.

Video clips for demonstrating lateral visual resolution for cases 1–3. (a) Clip 1 (Media 1) demonstrating visual resolution for case 1, i.e., one y^ and one x^ directed dipoles along the x axis (see Fig. 3 also). (b) Clip 2 (Media 2) demonstrating visual resolution for case 2, i.e., two y^ directed dipoles along the x axis (see Fig. 4 also). (c) Clip 3 (Media 3) demonstrating visual resolution for case 3, i.e., two x^ directed dipoles along the x axis (see Fig. 5 also).

Fig. 7.
Fig. 7.

The plot of angular ratio T for the three cases (a) |Tyx|, (b) |Tyy|, (c) |Txx|, and (d) |average(T;θobj)|. The plots are for the SIL-based system with NAmax.

Fig. 8.
Fig. 8.

Average values of the angular ratio T (averaged over the angular segments) for the three cases. (a) Case 1: one x^ and one y^ directed dipoles. (b) Case 2: two y^ directed dipoles (c) Case 3: two x^ directed dipoles.

Fig. 9.
Fig. 9.

Two x^ directed dipoles along z axis. (a) Plot of Imin/Imax for various values of NA for the SIL-based system and the non-SIL-based system. (b) Plot of Imin/max(In(r⃗SIL1),In(r⃗SIL2)) for various values of NA for the SIL-based system and the non-SIL-based system. (c)–(f) correspond to SIL-based system with NAmax; (g)–(h) correspond to NA=0.2; (i)–(j) correspond to NA=0.1. (c) The separation between the sources corresponds to the first minimum in (a) and (b); (d) the separation between the sources is slightly larger than Δz corresponding to the first minimum; (e) and (f) sources can be resolved for Δz0.475λ. (g) and (h) are analogous to (e) and (f), but for NA=0.2; (i)–(j) are analogous to (e)–(f), but for NA=0.1.

Fig. 10.
Fig. 10.

Video clips for demonstrating longitudinal visual resolution for NA=0.2 and NAmax. (a) Clip 4 (Media 4) demonstrates visual resolution for NA=0.2 (see Fig. 9 also). (b) Clip 5 (Media 5) demonstrates visual resolution for NAmax (see Fig. 9 also).

Fig. 11.
Fig. 11.

The plot of |E⃗(r⃗CCD=0⃗)/(ω2μα)| for various values of NAs. The minima coincide with the minima of Figs. 9(a) and 9(b).

Tables (2)

Tables Icon

Table 1. Notations and Explanation

Tables Icon

Table 2. Resolution Limit Based on Various Resolution Criteria

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

GPSF=α[I0+I21I222iI11I22I0I212iI12000],
α=ikCCD8πfobjfCCD(nobjnCCD)12exp[i(kobjfobj+kCCDfCCD)],I0=0θmaxsinθobjcosθobj(ts+tpcosθSIL)J0(ρ)exp(iz)dθobj,I11=0θmaxsinθobjcosθobj(tpsinθSIL)J1(ρ)exp(iz)cosψdθobj,I12=0θmaxsinθobjcosθobj(tpsinθSIL)J1(ρ)exp(iz)sinψdθobj,I21=0θmaxsinθobjcosθobj(tstpcosθSIL)J2(ρ)exp(iz)cos2ψdθobj,I22=0θmaxsinθobjcosθobj(tstpcosθSIL)J2(ρ)exp(iz)sin2ψdθobj,
ts=2nSILcosθobjnSILcosθobj+nobjcosθSILnSILnobj,
tp=2nSILcosθobjnobjcosθobj+nSILcosθSILnSILnobj,
ρ=x2+y2;ψ=tan-1yx;x=(kCCDsinθCCDxCCD+kSILsinθSILxSIL),y=(kCCDsinθCCDyCCD+kSILsinθSILySIL),z=kCCDcosθCCDzCCDkSILcosθSILzSIL.
NAmax=(nobj)2nSIL.
MSILlat=M(nSILnobj)2,
MSILlon=(nSILnobj)3M,
ρ1=|(kCCDsinθCCDxCCDkSILsinθSILΔx2)|,ρ2=|(kCCDsinθCCDxCCD+kSILsinθSILΔx2)|,ψ1=ψ2=0.
E⃗i=ω2μα0θmaxsinθobjcosθobj[hx(i)pxix^+hy(i)pyiy^]dθobj,
hx(i)(r⃗CCD)=(ts+tpcosθSIL)J0(ρi)+(tstpcosθSIL)J2(ρi),hy(i)(r⃗CCD)=(ts+tpcosθSIL)J0(ρi)(tstpcosθSIL)J2(ρi),
E⃗(r⃗CCD)=i=1,2E⃗i=ω2μα0θmaxsinθobjcosθobj[i=1,2hx(i)(r⃗CCD)pxix^+i=1,2hy(i)(r⃗CCD)pyiy^]dθobj.
r⃗CCD1:ρ1=0;ρ2=2χr⃗CCD2:ρ1=2χ;ρ2=0r⃗CCD3:ρ1=χ;ρ2=χ}whereχ=|kSILsinθSILΔx2|.
T(θobj,Δx)=max[i=1,2hx(i)(r⃗CCD3)pxi,i=1,2hy(i)(r⃗CCD3)pyi]max[max(i=1,2hx(i)(r⃗CCDj)pxi,i=1,2hy(i)(r⃗CCDj)pyi);j=1,2].
E(r⃗CCDj)=ω2μα0θmaxsinθobjcosθobj(ts+tpcosθSIL)[hx(2)(r⃗CCDj)x^+hy(1)(r⃗CCDj)y^]dθobj;j=1,2,3,
Tyx(θobj,Δx)=max[(ts+tpcosθSIL)J0(χ)±(tstpcosθSIL)J2(χ)](ts+tpcosθSIL).
E(r⃗CCDj)=i=1,2Ei=ω2μα0θmaxsinθobjcosθobj[i=1,2hy(i)(r⃗CCDj)y^]dθobj;j=1,2,3,
Tyy(θobj,Δx)=2[(ts+tpcosθSIL)J0(χ)(tstpcosθSIL)J2(χ)](ts+tpcosθSIL)[1+J0(2χ)](tstpcosθSIL)J2(2χ).
E(r⃗CCDj)=i=1,2Ei=ω2μα0θmaxsinθobjcosθobj[i=1,2hx(i)(r⃗CCDj)x^]dθobj;j=1,2,3,
Txx(θobj,Δx)=2[(ts+tpcosθSIL)J0(χ)+(tstpcosθSIL)J2(χ)](ts+tpcosθSIL)[1+J0(2χ)]+(tstpcosθSIL)J2(2χ).
ρ1=ρ2=0,z1=kCCDcosθCCDzCCD+kSILcosθSILΔz/2,z2=kCCDcosθCCDzCCDkSILcosθSILΔz/2.
E⃗(r⃗CCD)=x^ω2μα0θmaxsinθobjcosθobj(ts+tpcosθSIL)[exp(iz1)+exp(iz2)]dθobj=x^ω2μα0θmax2sinθobjcosθobj(ts+tpcosθSIL)exp(ikCCDcosθCCDzCCD)cos(kSILcosθSILΔz2)dθobj.
T(Δz,θobj)=exp(±ikCCDcosθCCDMSILlonΔz2).
h(Δz,θobj)=w(θobj,zCCD)cos(kSILcosθSILΔz2),w(θobj,zCCD)=2sinθobjcosθobj(ts+tpcosθSIL)exp(ikCCDcosθCCDzCCD).
E⃗(r⃗CCD=0⃗)=x^ω2μα0θmax2sinθobjcosθobj(ts+tpcosθSIL)cos(kSILcosθSILΔz2)dθobj.

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