Abstract

The characterization of imaging methods as three-dimensional (3D) linear filtering operations provides a useful way to compare the 3D performance of optical surface topography measuring instruments, such as coherence scanning interferometry, confocal and structured light microscopy. In this way, the imaging system is defined in terms of the point spread function in the space domain or equivalently by the transfer function in the spatial frequency domain. The derivation of these characteristics usually involves making the Born approximation, which is strictly only applicable to weakly scattering objects; however, for the case of surface scattering, the system is linear if multiple scattering is assumed to be negligible and the Kirchhoff approximation is assumed. A difference between the filter characteristics derived in each case is found. However this paper discusses these differences and explains the equivalence of the two approaches when applied to a weakly scattering object.

© 2013 Optical Society of America

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References

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  1. B. S. Lee and T. C. Strand, “Profilometry with a coherence scanning microscope,” Appl. Opt. 29, 3784–3788 (1990).
    [CrossRef]
  2. L. Deck and P. De Groot, “High-speed noncontact profiler based on scanning white-light interferometry,” Appl. Opt. 33, 7334–7338 (1994).
    [CrossRef]
  3. R. K. Leach, Optical Measurement of Surface Topography (Springer, 2011), 187–208.
  4. G. S. Kino and S. S. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).
    [CrossRef]
  5. F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
    [CrossRef]
  6. C. L. Giusca and R. K. Leach, “Calibration of the scales of areal surface topography measuring instruments: Part 2—Amplification coefficient, linearity and squareness,” Meas. Sci. Technol. 23, 065005 (2012).
    [CrossRef]
  7. J. N. Petzing, J. M. Coupland, and R. K. Leach, “The measurement of rough surface topography using coherence scanning interferometry,” in Good Practice Guide No. 116, (National Physical Laboratory, 2010), pp. 92–112.
  8. W. Hillmann, “Surface profiles obtained by means of optical methods—are they true representations of the real surface,” CIRP Ann. 39, 581–583 (1990).
    [CrossRef]
  9. K. Palodhi, J. M. Coupland, and R. K. Leach, “A linear model of fringe generation and analysis in coherence scanning interferometry,” presented at the ASPE Summer Topical Meeting on Precision Interferometric Metrology, Asheville, North Carolina, 23–25 June2010.
  10. K. Palodhi, J. M. Coupland, and R. K. Leach, “Determination of the point spread function of a coherence scanning interferometer,” in Proceedings of the 13th International Conference on Metrology and Properties of Engineering Surfaces (NPL, 2011), pp. 199–202.
  11. R. Mandal, K. Palodhi, J. M. Coupland, R. K. Leach, and D. Mansfield, “Application of linear systems theory to characterize coherence scanning interferometry,” Proc. SPIE 8430, 84300T (2012).
    [CrossRef]
  12. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  13. R. Dandliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323–328 (1970).
    [CrossRef]
  14. N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
    [CrossRef]
  15. C. J. R. Sheppard and C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
    [CrossRef]
  16. A. F. Fercher, H. Bartelt, H. Becker, and E. Wiltschko, “Image formation by inversion of scattered field data: experiments and computational simulation,” Appl. Opt. 18, 2427–2439 (1979).
    [CrossRef]
  17. J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19, 07010 (2008).
    [CrossRef]
  18. P. D. Ruiz, J. M. Huntley, and J. M. Coupland, “Depth-resolved imaging and displacement measurement techniques viewed as linear filtering operations,” Exp. Mech. 13, 453–465 (2010).
    [CrossRef]
  19. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999), 695–734.
  20. P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987), 17–33.
  21. R. J. Wombellan and J. A. DeSanto, “Reconstruction of rough-surface profiles with the Kirchhoff approximation,” J. Opt. Soc. Am. A 8, 1892–1897 (1991).
    [CrossRef]
  22. E. I. Thorsosand and D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 1, S165–S190 (1991).
    [CrossRef]
  23. C. J. Raymond, “Milestones and future directions in applications of optical scatterometry,” Proc. SPIE CR72, 147–177 (1999).
    [CrossRef]
  24. C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
    [CrossRef]
  25. C. J. R. Sheppard, “Imaging of random surfaces and inverse scattering in the Kirchhoff approximation, waves in random media,” Waves Random Media 8, 53–66 (1998).
  26. J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, 1978), p. 56.

2012

C. L. Giusca and R. K. Leach, “Calibration of the scales of areal surface topography measuring instruments: Part 2—Amplification coefficient, linearity and squareness,” Meas. Sci. Technol. 23, 065005 (2012).
[CrossRef]

R. Mandal, K. Palodhi, J. M. Coupland, R. K. Leach, and D. Mansfield, “Application of linear systems theory to characterize coherence scanning interferometry,” Proc. SPIE 8430, 84300T (2012).
[CrossRef]

2010

P. D. Ruiz, J. M. Huntley, and J. M. Coupland, “Depth-resolved imaging and displacement measurement techniques viewed as linear filtering operations,” Exp. Mech. 13, 453–465 (2010).
[CrossRef]

2008

J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19, 07010 (2008).
[CrossRef]

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

1999

C. J. Raymond, “Milestones and future directions in applications of optical scatterometry,” Proc. SPIE CR72, 147–177 (1999).
[CrossRef]

1998

C. J. R. Sheppard, “Imaging of random surfaces and inverse scattering in the Kirchhoff approximation, waves in random media,” Waves Random Media 8, 53–66 (1998).

1994

1992

C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

1991

R. J. Wombellan and J. A. DeSanto, “Reconstruction of rough-surface profiles with the Kirchhoff approximation,” J. Opt. Soc. Am. A 8, 1892–1897 (1991).
[CrossRef]

E. I. Thorsosand and D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 1, S165–S190 (1991).
[CrossRef]

1990

C. J. R. Sheppard and C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
[CrossRef]

W. Hillmann, “Surface profiles obtained by means of optical methods—are they true representations of the real surface,” CIRP Ann. 39, 581–583 (1990).
[CrossRef]

G. S. Kino and S. S. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).
[CrossRef]

B. S. Lee and T. C. Strand, “Profilometry with a coherence scanning microscope,” Appl. Opt. 29, 3784–3788 (1990).
[CrossRef]

1985

1979

1970

R. Dandliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323–328 (1970).
[CrossRef]

1969

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Bartelt, H.

Becker, H.

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987), 17–33.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999), 695–734.

Chim, S. S.

Cogswell, C. J.

C. J. R. Sheppard and C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
[CrossRef]

Coupland, J. M.

R. Mandal, K. Palodhi, J. M. Coupland, R. K. Leach, and D. Mansfield, “Application of linear systems theory to characterize coherence scanning interferometry,” Proc. SPIE 8430, 84300T (2012).
[CrossRef]

P. D. Ruiz, J. M. Huntley, and J. M. Coupland, “Depth-resolved imaging and displacement measurement techniques viewed as linear filtering operations,” Exp. Mech. 13, 453–465 (2010).
[CrossRef]

J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19, 07010 (2008).
[CrossRef]

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

J. N. Petzing, J. M. Coupland, and R. K. Leach, “The measurement of rough surface topography using coherence scanning interferometry,” in Good Practice Guide No. 116, (National Physical Laboratory, 2010), pp. 92–112.

K. Palodhi, J. M. Coupland, and R. K. Leach, “A linear model of fringe generation and analysis in coherence scanning interferometry,” presented at the ASPE Summer Topical Meeting on Precision Interferometric Metrology, Asheville, North Carolina, 23–25 June2010.

K. Palodhi, J. M. Coupland, and R. K. Leach, “Determination of the point spread function of a coherence scanning interferometer,” in Proceedings of the 13th International Conference on Metrology and Properties of Engineering Surfaces (NPL, 2011), pp. 199–202.

Dandliker, R.

R. Dandliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323–328 (1970).
[CrossRef]

De Groot, P.

Deck, L.

DeSanto, J. A.

Fercher, A. F.

Gao, F.

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, 1978), p. 56.

Giusca, C. L.

C. L. Giusca and R. K. Leach, “Calibration of the scales of areal surface topography measuring instruments: Part 2—Amplification coefficient, linearity and squareness,” Meas. Sci. Technol. 23, 065005 (2012).
[CrossRef]

Gu, M.

C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

Hillmann, W.

W. Hillmann, “Surface profiles obtained by means of optical methods—are they true representations of the real surface,” CIRP Ann. 39, 581–583 (1990).
[CrossRef]

Huntley, J. M.

P. D. Ruiz, J. M. Huntley, and J. M. Coupland, “Depth-resolved imaging and displacement measurement techniques viewed as linear filtering operations,” Exp. Mech. 13, 453–465 (2010).
[CrossRef]

Jackson, D. R.

E. I. Thorsosand and D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 1, S165–S190 (1991).
[CrossRef]

Kino, G. S.

Leach, R. K.

R. Mandal, K. Palodhi, J. M. Coupland, R. K. Leach, and D. Mansfield, “Application of linear systems theory to characterize coherence scanning interferometry,” Proc. SPIE 8430, 84300T (2012).
[CrossRef]

C. L. Giusca and R. K. Leach, “Calibration of the scales of areal surface topography measuring instruments: Part 2—Amplification coefficient, linearity and squareness,” Meas. Sci. Technol. 23, 065005 (2012).
[CrossRef]

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

J. N. Petzing, J. M. Coupland, and R. K. Leach, “The measurement of rough surface topography using coherence scanning interferometry,” in Good Practice Guide No. 116, (National Physical Laboratory, 2010), pp. 92–112.

K. Palodhi, J. M. Coupland, and R. K. Leach, “A linear model of fringe generation and analysis in coherence scanning interferometry,” presented at the ASPE Summer Topical Meeting on Precision Interferometric Metrology, Asheville, North Carolina, 23–25 June2010.

K. Palodhi, J. M. Coupland, and R. K. Leach, “Determination of the point spread function of a coherence scanning interferometer,” in Proceedings of the 13th International Conference on Metrology and Properties of Engineering Surfaces (NPL, 2011), pp. 199–202.

R. K. Leach, Optical Measurement of Surface Topography (Springer, 2011), 187–208.

Lee, B. S.

Lobera, J.

J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19, 07010 (2008).
[CrossRef]

Mandal, R.

R. Mandal, K. Palodhi, J. M. Coupland, R. K. Leach, and D. Mansfield, “Application of linear systems theory to characterize coherence scanning interferometry,” Proc. SPIE 8430, 84300T (2012).
[CrossRef]

Mansfield, D.

R. Mandal, K. Palodhi, J. M. Coupland, R. K. Leach, and D. Mansfield, “Application of linear systems theory to characterize coherence scanning interferometry,” Proc. SPIE 8430, 84300T (2012).
[CrossRef]

Palodhi, K.

R. Mandal, K. Palodhi, J. M. Coupland, R. K. Leach, and D. Mansfield, “Application of linear systems theory to characterize coherence scanning interferometry,” Proc. SPIE 8430, 84300T (2012).
[CrossRef]

K. Palodhi, J. M. Coupland, and R. K. Leach, “Determination of the point spread function of a coherence scanning interferometer,” in Proceedings of the 13th International Conference on Metrology and Properties of Engineering Surfaces (NPL, 2011), pp. 199–202.

K. Palodhi, J. M. Coupland, and R. K. Leach, “A linear model of fringe generation and analysis in coherence scanning interferometry,” presented at the ASPE Summer Topical Meeting on Precision Interferometric Metrology, Asheville, North Carolina, 23–25 June2010.

Petzing, J.

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

Petzing, J. N.

J. N. Petzing, J. M. Coupland, and R. K. Leach, “The measurement of rough surface topography using coherence scanning interferometry,” in Good Practice Guide No. 116, (National Physical Laboratory, 2010), pp. 92–112.

Raymond, C. J.

C. J. Raymond, “Milestones and future directions in applications of optical scatterometry,” Proc. SPIE CR72, 147–177 (1999).
[CrossRef]

Ruiz, P. D.

P. D. Ruiz, J. M. Huntley, and J. M. Coupland, “Depth-resolved imaging and displacement measurement techniques viewed as linear filtering operations,” Exp. Mech. 13, 453–465 (2010).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, “Imaging of random surfaces and inverse scattering in the Kirchhoff approximation, waves in random media,” Waves Random Media 8, 53–66 (1998).

C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

C. J. R. Sheppard and C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
[CrossRef]

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987), 17–33.

Strand, T. C.

Streibl, N.

Thorsosand, E. I.

E. I. Thorsosand and D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 1, S165–S190 (1991).
[CrossRef]

Weiss, K.

R. Dandliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323–328 (1970).
[CrossRef]

Wiltschko, E.

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999), 695–734.

Wombellan, R. J.

Appl. Opt.

CIRP Ann.

W. Hillmann, “Surface profiles obtained by means of optical methods—are they true representations of the real surface,” CIRP Ann. 39, 581–583 (1990).
[CrossRef]

Exp. Mech.

P. D. Ruiz, J. M. Huntley, and J. M. Coupland, “Depth-resolved imaging and displacement measurement techniques viewed as linear filtering operations,” Exp. Mech. 13, 453–465 (2010).
[CrossRef]

J. Microsc.

C. J. R. Sheppard and M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

C. J. R. Sheppard and C. J. Cogswell, “Three-dimensional image formation in confocal microscopy,” J. Microsc. 159, 179–194 (1990).
[CrossRef]

J. Opt. Soc. Am. A

Meas. Sci. Technol.

F. Gao, R. K. Leach, J. Petzing, and J. M. Coupland, “Surface measurement errors using commercial scanning white light interferometers,” Meas. Sci. Technol. 19, 015303 (2008).
[CrossRef]

C. L. Giusca and R. K. Leach, “Calibration of the scales of areal surface topography measuring instruments: Part 2—Amplification coefficient, linearity and squareness,” Meas. Sci. Technol. 23, 065005 (2012).
[CrossRef]

J. M. Coupland and J. Lobera, “Holography, tomography and 3D microscopy as linear filtering operations,” Meas. Sci. Technol. 19, 07010 (2008).
[CrossRef]

Opt. Commun.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

R. Dandliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1, 323–328 (1970).
[CrossRef]

Proc. SPIE

R. Mandal, K. Palodhi, J. M. Coupland, R. K. Leach, and D. Mansfield, “Application of linear systems theory to characterize coherence scanning interferometry,” Proc. SPIE 8430, 84300T (2012).
[CrossRef]

C. J. Raymond, “Milestones and future directions in applications of optical scatterometry,” Proc. SPIE CR72, 147–177 (1999).
[CrossRef]

Waves Random Media

E. I. Thorsosand and D. R. Jackson, “Studies of scattering theory using numerical methods,” Waves Random Media 1, S165–S190 (1991).
[CrossRef]

C. J. R. Sheppard, “Imaging of random surfaces and inverse scattering in the Kirchhoff approximation, waves in random media,” Waves Random Media 8, 53–66 (1998).

Other

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, 1978), p. 56.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999), 695–734.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987), 17–33.

R. K. Leach, Optical Measurement of Surface Topography (Springer, 2011), 187–208.

J. N. Petzing, J. M. Coupland, and R. K. Leach, “The measurement of rough surface topography using coherence scanning interferometry,” in Good Practice Guide No. 116, (National Physical Laboratory, 2010), pp. 92–112.

K. Palodhi, J. M. Coupland, and R. K. Leach, “A linear model of fringe generation and analysis in coherence scanning interferometry,” presented at the ASPE Summer Topical Meeting on Precision Interferometric Metrology, Asheville, North Carolina, 23–25 June2010.

K. Palodhi, J. M. Coupland, and R. K. Leach, “Determination of the point spread function of a coherence scanning interferometer,” in Proceedings of the 13th International Conference on Metrology and Properties of Engineering Surfaces (NPL, 2011), pp. 199–202.

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Figures (5)

Fig. 1.
Fig. 1.

Scattering from a 3D object.

Fig. 2.
Fig. 2.

Surface scattering to a distant boundary.

Fig. 3.
Fig. 3.

Principle of stationary phase.

Fig. 4.
Fig. 4.

Superposition of the reference and scattered fields in a Mirau objective.

Fig. 5.
Fig. 5.

Scattered field at spherical boundary surface.

Equations (52)

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

OB(r)=HB(rr)ΔB(r)d3r,
HB(r)=GNA2(r,k0)k02S(k0)dk0.
GNA(r,k0)=j4πk0δ(|k|k0)step(k·o^k01NA2)ej2πk·rd3k,
O˜B(k)=Δ˜B(k)H˜B(k),
H˜B(k)=G˜NA(k,k0)G˜NA(kk,k0)d3kk02S(k0)dk0,
G˜NA(k,k0)=j4πk0δ(|k|k0)step(k·o^k01NA2).
Es(r)=k02G(rr)ΔB(r)(Es(r)+Er(r))d3r,
Es(r)=k02VG(rr)ΔB(r)Et(r)d3r,
Es(r)=V(G(rr)2Et(r)Et(r)2G(rr))d3r,
Es(r)=S(G(rr)Et(r)nEt(r)G(rr)n)ds,
Et(r)=(1+R)e2πjkr·r,
Et(r)n=2πjkr·n^S(1R)e2πjkr·r,
G(rbr)e2πjk0|rb|4π|rb|e2πjk0r·rb|rb|.
G(rbr)n=2πjG(rbr)k0rb|rb|·n^S.
Es(rb)=je2πjk0|rb|2|rb|Se2πj(k0r·rb|rb|kr·r)[R(k0rb|rb|kr)+(k0rb|rb|+kr)]·n^Sds.
A(r)=W(rx,ry)δ(rzs(rx,ry)),
Es(rb)=je2πjk0|rb|2|rb|e2πj(k0rb|rb|kr)·r[R(k0rb|rb|kr)+(k0rb|rb|+kr)]·n^SA(r)n^S·zd3r.
Em(r)=Σ[G*(rrb)Es(rb)nEs(rb)G*(rrb)n]ds.
Em(r)=k02Σ1|rb|2e2πj(k0rb|rb|kr)·r[R(k0rb|rb|kr)+(k0rb|rb|+kr)]·n^S×A(r)n^S·zd3re2πjk0rrb|rb|ds.
Em(r)=k021|rb|2e2πj(k0rb|rb|kr)·r[R(k0rb|rb|kr)+(k0rb|rb|+kr)]·n^SA(r)n^S·zd3re2πjk0rrb|rb|δ(|rb|r0)d3rb.
Em(r)=12k0[e2πj(kkr)·r[R(kkr)+(k+kr)]·n^SA(r)n^S.zd3r]δ(|k|k0)e2πjk·rd3k.
Em(r)=R2k0e2πj(kkr)·r(|kkr|2(kkr)·z)A(r)d3rδ(|k|k0)e2πjk·rd3k.
Em(r)=4πjRe2πj(kkr)·r(|kkr|22(kkr)·z)A(r)d3rG˜ideal(k)e2πjk·rd3k.
Em(r)=4πjRe2πj(kkr)·r(|kkr|22(kkr)·z)A(r)d3rG˜NA(k,k0)e2πjk·rd3k.
I(r)=|Em(r)Er(r)|2=|Er(r)|2+|Em(r)|2Em(r)*Er(r)Em(r)Er(r)*.
O(r)=Em(r)Er(r)*.
OF(r)=e2πj(kkr)·r(|kkr|22(kkr)·z)ΔF(r)d3rG˜NA(k,k0)e2πj(kkr)·rd3k,
ΔF(r)=4πjRA(r)=4πjRW(rx,ry)δ(rzs(rx,ry)).
OF(r)=ΔF(r)e2πjk·rd3r(|k|22k.z)G˜NA(k+kr,k0)e2πjk·rd3k
O˜F(k)=Δ˜F(k)(|k|22k·z)G˜NA(k+kr,k0).
O˜F(k)=Δ˜F(k)H˜F(k),
H˜F(k)=(|k|22k·z)G˜NA(kr,k0)G˜NA(kkr,k0)d3krS(k0)dk0.
H˜B(k)=G˜NA(k,k0)G˜NA(kk,k0)d3kk02S(k0)dk0.
ΔB(r)=4π2(1n2(r)),
ΔF(r)=4πjRW(rx,ry)δ(rzs(rx,ry)).
ΔF(r)4πj(1n1+n)δ(rzs(rx,ry)).
H˜F(k)G˜NA(kr,k0)G˜NA(kkr,k0)d3krk0S(k0)dk0.
HF(r)GNA2(r,k0)k0S(k0)dk0.
Es(rb)=U(r)G(rbr)d3r,
G(rbr)=e2πjk0|rbr|4π|rbr|,
G(rbr)e2πjk0|rb|4π|rb|e2πjk0r·rb|rb|,
Es(rb)=e2πjk0|rb|4π|rb|U(r)e2πjk0r·rb|rb|d3r.
Em(r)=Σ[G*(rrb)ES(rb)nES(rb)G*(rrb)n]ds,
G*(rrb)n=2πjk0G*(rrb),
ES(rb)n=2πjk0ES(rb).
Em(r)=4πjk0ΣG*(rrb)ES(rb)ds=jk04πΣ[U(r)e2πjk0r·rb|rb|d3r]1|rb|2e2πjk0r·rb|rb|ds.
Em(r)=jk04π[U(r)e2πjk0r·rb|rb|d3r]1|rb|2δ(|rb|r0)e2πjk0r·rb|rb|d3rb.
Em(r)=j4πk0[U(r)e2πjk·rd3r]δ(|k|k0)e2πjk·rd3k.
E˜m(k)=U˜(k)G˜ideal(k),
G˜ideal(k)=j4πk0δ(|k|k0).
G˜NA(k,k0)=j4πk0δ(|k|k0)step(k·o^k01NA2),
GNA(r,k0)=j4πk0δ(|k|k0)step(k.o^k01NA2)ej2πk.rd3k.

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