Abstract

The applicability of optical scanning holography (OSH) to the field of microscopic imaging for biological applications is assessed. A generalized mathematical description of OSH that takes into account polarization effects, high numerical apertures, and generalized illumination wave fronts is presented. This description is used to show that the proposed single-beam scanning technique relaxes the restrictions under which OSH functions correctly compared with the conventional double-beam scanning method. It is also shown that, although in general OSH is restricted to thin samples, this condition can be relaxed in nonrefracting fluorescence samples, which are of importance in biological microscopy.

© 2002 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]

2001

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

2000

1999

J. White, L. Johannes, F. Mallard, A. Girod, S. Grill, S. Reinsch, P. Keller, B. Tzschaschel, A. Echard, B. Goud, E. H. K. Stelzer, “Rab6 coordinates a novel golgi to ER retrograde transport pathway in live cells,” J. Cell Biol. 147, 743–759 (1999).
[CrossRef] [PubMed]

Y. Takaki, H. Ohzu, “Fast numerical reconstruction technique for high-resolution hybrid holographic microscopy,” Appl. Opt. 38, 2204–2211 (1999).
[CrossRef]

C. J. R. Sheppard, “Vectors and Fourier transforms in optics,” Optik 110, 157–160 (1999).

Y. Takaki, H. Kawai, H. Ohzu, “Hybrid holographic microscopy free of conjugate and zero-order images,” Appl. Opt. 38, 4990–4996 (1999).
[CrossRef]

1998

1997

1995

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. Shinoda, Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1344 (1995).
[CrossRef]

1992

1985

1966

L. O. Heflinger, R. F. Wuerker, R. E. Brooks, “Holographic interferometry,” J. Appl. Phys. 37, 642–649 (1966).
[CrossRef]

R. F. VanLigten, H. Osterberg, “Holographic microscopy,” Nature 211, 282–283 (1966).
[CrossRef]

Agard, D. A.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Brooks, R. E.

L. O. Heflinger, R. F. Wuerker, R. E. Brooks, “Holographic interferometry,” J. Appl. Phys. 37, 642–649 (1966).
[CrossRef]

Collot, L.

Doh, K. B.

T.-C. Poon, K. B. Doh, B. Schilling, K. Shinoda, Y. Suzuki, M. H. Wu, “Holographic three-dimensional display using an electron-beam-addressed spatial light modulator,” Opt. Rev. 4, 567–571 (1997).
[CrossRef]

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. Shinoda, Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1344 (1995).
[CrossRef]

Duncan, B. D.

Echard, A.

J. White, L. Johannes, F. Mallard, A. Girod, S. Grill, S. Reinsch, P. Keller, B. Tzschaschel, A. Echard, B. Goud, E. H. K. Stelzer, “Rab6 coordinates a novel golgi to ER retrograde transport pathway in live cells,” J. Cell Biol. 147, 743–759 (1999).
[CrossRef] [PubMed]

Ellenberg, J.

J. Lippincott-Schwartz, J. F. Presley, K. J. M. Zaal, K. Hirschberg, C. D. Miller, J. Ellenberg, “Monitoring the dynamics and mobility of membrane proteins tagged with green fluorescent protein,” in Methods in Cell Biology, K. F. Sullivan, S. A. Kay, eds. (Academic, San Diego, 1999), Vol. 58, pp. 261–281.
[PubMed]

Girod, A.

J. White, L. Johannes, F. Mallard, A. Girod, S. Grill, S. Reinsch, P. Keller, B. Tzschaschel, A. Echard, B. Goud, E. H. K. Stelzer, “Rab6 coordinates a novel golgi to ER retrograde transport pathway in live cells,” J. Cell Biol. 147, 743–759 (1999).
[CrossRef] [PubMed]

Goud, B.

J. White, L. Johannes, F. Mallard, A. Girod, S. Grill, S. Reinsch, P. Keller, B. Tzschaschel, A. Echard, B. Goud, E. H. K. Stelzer, “Rab6 coordinates a novel golgi to ER retrograde transport pathway in live cells,” J. Cell Biol. 147, 743–759 (1999).
[CrossRef] [PubMed]

Grill, S.

J. White, L. Johannes, F. Mallard, A. Girod, S. Grill, S. Reinsch, P. Keller, B. Tzschaschel, A. Echard, B. Goud, E. H. K. Stelzer, “Rab6 coordinates a novel golgi to ER retrograde transport pathway in live cells,” J. Cell Biol. 147, 743–759 (1999).
[CrossRef] [PubMed]

Gross, M.

Gustafsson, M. G. L.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Hanser, B.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Don Mills, Ontario, 1987), pp. 593–610.

Heflinger, L. O.

L. O. Heflinger, R. F. Wuerker, R. E. Brooks, “Holographic interferometry,” J. Appl. Phys. 37, 642–649 (1966).
[CrossRef]

Hirschberg, K.

J. Lippincott-Schwartz, J. F. Presley, K. J. M. Zaal, K. Hirschberg, C. D. Miller, J. Ellenberg, “Monitoring the dynamics and mobility of membrane proteins tagged with green fluorescent protein,” in Methods in Cell Biology, K. F. Sullivan, S. A. Kay, eds. (Academic, San Diego, 1999), Vol. 58, pp. 261–281.
[PubMed]

Indebetouw, G.

Johannes, L.

J. White, L. Johannes, F. Mallard, A. Girod, S. Grill, S. Reinsch, P. Keller, B. Tzschaschel, A. Echard, B. Goud, E. H. K. Stelzer, “Rab6 coordinates a novel golgi to ER retrograde transport pathway in live cells,” J. Cell Biol. 147, 743–759 (1999).
[CrossRef] [PubMed]

Jonkman, J.

J. Jonkman, E. H. K. Stelzer, “Resolution and contrast in confocal and two-photon microscopy,” in Confocal and Two-photon Microscopy: Foundations, Applications, and Advances, A. Diaspro, ed. (Wiley–Liss, New York, 2001), pp. 101–125.

Kam, Z.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Kawai, H.

Keller, P.

J. White, L. Johannes, F. Mallard, A. Girod, S. Grill, S. Reinsch, P. Keller, B. Tzschaschel, A. Echard, B. Goud, E. H. K. Stelzer, “Rab6 coordinates a novel golgi to ER retrograde transport pathway in live cells,” J. Cell Biol. 147, 743–759 (1999).
[CrossRef] [PubMed]

Kim, E.-S.

Kim, S.-G.

Kim, T.

Klysubun, P.

G. Indebetouw, P. Klysubun, T. Kim, T.-C. Poon, “Imaging properties of scanning holographic microscopy,” J. Opt. Soc. Am. A 17, 380–390 (2000).
[CrossRef]

P. Klysubun, G. Indebetouw, “A posteriori processing of spatiotemporal digital microholograms,” J. Opt. Soc. Am. A18, 326–331 (2001).
[CrossRef]

Le Clerc, F.

Lee, B.

Lippincott-Schwartz, J.

J. Lippincott-Schwartz, J. F. Presley, K. J. M. Zaal, K. Hirschberg, C. D. Miller, J. Ellenberg, “Monitoring the dynamics and mobility of membrane proteins tagged with green fluorescent protein,” in Methods in Cell Biology, K. F. Sullivan, S. A. Kay, eds. (Academic, San Diego, 1999), Vol. 58, pp. 261–281.
[PubMed]

Mallard, F.

J. White, L. Johannes, F. Mallard, A. Girod, S. Grill, S. Reinsch, P. Keller, B. Tzschaschel, A. Echard, B. Goud, E. H. K. Stelzer, “Rab6 coordinates a novel golgi to ER retrograde transport pathway in live cells,” J. Cell Biol. 147, 743–759 (1999).
[CrossRef] [PubMed]

Massig, J. H.

Miller, C. D.

J. Lippincott-Schwartz, J. F. Presley, K. J. M. Zaal, K. Hirschberg, C. D. Miller, J. Ellenberg, “Monitoring the dynamics and mobility of membrane proteins tagged with green fluorescent protein,” in Methods in Cell Biology, K. F. Sullivan, S. A. Kay, eds. (Academic, San Diego, 1999), Vol. 58, pp. 261–281.
[PubMed]

Ohzu, H.

Osterberg, H.

R. F. VanLigten, H. Osterberg, “Holographic microscopy,” Nature 211, 282–283 (1966).
[CrossRef]

Pedrini, G.

Poon, T.-C.

G. Indebetouw, P. Klysubun, T. Kim, T.-C. Poon, “Imaging properties of scanning holographic microscopy,” J. Opt. Soc. Am. A 17, 380–390 (2000).
[CrossRef]

T.-C. Poon, T. Kim, G. Indebetouw, B. W. Schilling, M. H. Wu, K. Shinoda, Y. Suzuki, “Twin-image elimination experiments for three-dimensional images in optical scanning holography,” Opt. Lett. 25, 215–217 (2000).
[CrossRef]

G. Indebetouw, T. Kim, T.-C. Poon, B. W. Schilling, “Three-dimensional location of fluorescent inhomogeneities in turbid media by scanning heterodyne holography,” Opt. Lett. 23, 135–137 (1998).
[CrossRef]

B. W. Schilling, T.-C. Poon, G. Indebetouw, B. Storrie, K. Shinoda, Y. Suzuki, M. H. Wu, “Three-dimensional holographic fluorescence microscopy,” Opt. Lett. 22, 1506–1508 (1997).
[CrossRef]

T.-C. Poon, K. B. Doh, B. Schilling, K. Shinoda, Y. Suzuki, M. H. Wu, “Holographic three-dimensional display using an electron-beam-addressed spatial light modulator,” Opt. Rev. 4, 567–571 (1997).
[CrossRef]

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. Shinoda, Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1344 (1995).
[CrossRef]

B. D. Duncan, T.-C. Poon, “Gaussian beam analysis of optical scanning holography,” J. Opt. Soc. Am. A 9, 229–236 (1992).
[CrossRef]

T.-C. Poon, “Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis,” J. Opt. Soc. Am. A 2, 521–527 (1985).
[CrossRef]

Presley, J. F.

J. Lippincott-Schwartz, J. F. Presley, K. J. M. Zaal, K. Hirschberg, C. D. Miller, J. Ellenberg, “Monitoring the dynamics and mobility of membrane proteins tagged with green fluorescent protein,” in Methods in Cell Biology, K. F. Sullivan, S. A. Kay, eds. (Academic, San Diego, 1999), Vol. 58, pp. 261–281.
[PubMed]

Reinsch, S.

J. White, L. Johannes, F. Mallard, A. Girod, S. Grill, S. Reinsch, P. Keller, B. Tzschaschel, A. Echard, B. Goud, E. H. K. Stelzer, “Rab6 coordinates a novel golgi to ER retrograde transport pathway in live cells,” J. Cell Biol. 147, 743–759 (1999).
[CrossRef] [PubMed]

Schedin, S.

Schilling, B.

T.-C. Poon, K. B. Doh, B. Schilling, K. Shinoda, Y. Suzuki, M. H. Wu, “Holographic three-dimensional display using an electron-beam-addressed spatial light modulator,” Opt. Rev. 4, 567–571 (1997).
[CrossRef]

Schilling, B. W.

Sedat, J. W.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Shaw, P. J.

P. J. Shaw, “Comparison of wide-field/deconvolution and confocal microscopy for 3D imaging,” in Handbook of Biological Confocal Microscopy, 2nd ed. J. B. Pawley, ed. (Plenum, New York, 1995), pp. 373–387.

Sheppard, C. J. R.

C. J. R. Sheppard, “Vectors and Fourier transforms in optics,” Optik 110, 157–160 (1999).

Shinoda, K.

T.-C. Poon, T. Kim, G. Indebetouw, B. W. Schilling, M. H. Wu, K. Shinoda, Y. Suzuki, “Twin-image elimination experiments for three-dimensional images in optical scanning holography,” Opt. Lett. 25, 215–217 (2000).
[CrossRef]

T.-C. Poon, K. B. Doh, B. Schilling, K. Shinoda, Y. Suzuki, M. H. Wu, “Holographic three-dimensional display using an electron-beam-addressed spatial light modulator,” Opt. Rev. 4, 567–571 (1997).
[CrossRef]

B. W. Schilling, T.-C. Poon, G. Indebetouw, B. Storrie, K. Shinoda, Y. Suzuki, M. H. Wu, “Three-dimensional holographic fluorescence microscopy,” Opt. Lett. 22, 1506–1508 (1997).
[CrossRef]

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. Shinoda, Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1344 (1995).
[CrossRef]

Stadelmaier, A.

Stelzer, E. H. K.

J. White, L. Johannes, F. Mallard, A. Girod, S. Grill, S. Reinsch, P. Keller, B. Tzschaschel, A. Echard, B. Goud, E. H. K. Stelzer, “Rab6 coordinates a novel golgi to ER retrograde transport pathway in live cells,” J. Cell Biol. 147, 743–759 (1999).
[CrossRef] [PubMed]

J. Jonkman, E. H. K. Stelzer, “Resolution and contrast in confocal and two-photon microscopy,” in Confocal and Two-photon Microscopy: Foundations, Applications, and Advances, A. Diaspro, ed. (Wiley–Liss, New York, 2001), pp. 101–125.

Storrie, B.

Suzuki, Y.

T.-C. Poon, T. Kim, G. Indebetouw, B. W. Schilling, M. H. Wu, K. Shinoda, Y. Suzuki, “Twin-image elimination experiments for three-dimensional images in optical scanning holography,” Opt. Lett. 25, 215–217 (2000).
[CrossRef]

T.-C. Poon, K. B. Doh, B. Schilling, K. Shinoda, Y. Suzuki, M. H. Wu, “Holographic three-dimensional display using an electron-beam-addressed spatial light modulator,” Opt. Rev. 4, 567–571 (1997).
[CrossRef]

B. W. Schilling, T.-C. Poon, G. Indebetouw, B. Storrie, K. Shinoda, Y. Suzuki, M. H. Wu, “Three-dimensional holographic fluorescence microscopy,” Opt. Lett. 22, 1506–1508 (1997).
[CrossRef]

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. Shinoda, Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1344 (1995).
[CrossRef]

Takaki, Y.

Tiziani, H. J.

Tzschaschel, B.

J. White, L. Johannes, F. Mallard, A. Girod, S. Grill, S. Reinsch, P. Keller, B. Tzschaschel, A. Echard, B. Goud, E. H. K. Stelzer, “Rab6 coordinates a novel golgi to ER retrograde transport pathway in live cells,” J. Cell Biol. 147, 743–759 (1999).
[CrossRef] [PubMed]

VanLigten, R. F.

R. F. VanLigten, H. Osterberg, “Holographic microscopy,” Nature 211, 282–283 (1966).
[CrossRef]

White, J.

J. White, L. Johannes, F. Mallard, A. Girod, S. Grill, S. Reinsch, P. Keller, B. Tzschaschel, A. Echard, B. Goud, E. H. K. Stelzer, “Rab6 coordinates a novel golgi to ER retrograde transport pathway in live cells,” J. Cell Biol. 147, 743–759 (1999).
[CrossRef] [PubMed]

Wu, M. H.

T.-C. Poon, T. Kim, G. Indebetouw, B. W. Schilling, M. H. Wu, K. Shinoda, Y. Suzuki, “Twin-image elimination experiments for three-dimensional images in optical scanning holography,” Opt. Lett. 25, 215–217 (2000).
[CrossRef]

T.-C. Poon, K. B. Doh, B. Schilling, K. Shinoda, Y. Suzuki, M. H. Wu, “Holographic three-dimensional display using an electron-beam-addressed spatial light modulator,” Opt. Rev. 4, 567–571 (1997).
[CrossRef]

B. W. Schilling, T.-C. Poon, G. Indebetouw, B. Storrie, K. Shinoda, Y. Suzuki, M. H. Wu, “Three-dimensional holographic fluorescence microscopy,” Opt. Lett. 22, 1506–1508 (1997).
[CrossRef]

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. Shinoda, Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1344 (1995).
[CrossRef]

Wuerker, R. F.

L. O. Heflinger, R. F. Wuerker, R. E. Brooks, “Holographic interferometry,” J. Appl. Phys. 37, 642–649 (1966).
[CrossRef]

Yamaguchi, I.

Zaal, K. J. M.

J. Lippincott-Schwartz, J. F. Presley, K. J. M. Zaal, K. Hirschberg, C. D. Miller, J. Ellenberg, “Monitoring the dynamics and mobility of membrane proteins tagged with green fluorescent protein,” in Methods in Cell Biology, K. F. Sullivan, S. A. Kay, eds. (Academic, San Diego, 1999), Vol. 58, pp. 261–281.
[PubMed]

Zhang, T.

Appl. Opt.

J. Appl. Phys.

L. O. Heflinger, R. F. Wuerker, R. E. Brooks, “Holographic interferometry,” J. Appl. Phys. 37, 642–649 (1966).
[CrossRef]

J. Cell Biol.

J. White, L. Johannes, F. Mallard, A. Girod, S. Grill, S. Reinsch, P. Keller, B. Tzschaschel, A. Echard, B. Goud, E. H. K. Stelzer, “Rab6 coordinates a novel golgi to ER retrograde transport pathway in live cells,” J. Cell Biol. 147, 743–759 (1999).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Nature

R. F. VanLigten, H. Osterberg, “Holographic microscopy,” Nature 211, 282–283 (1966).
[CrossRef]

Opt. Eng.

T.-C. Poon, K. B. Doh, B. W. Schilling, M. H. Wu, K. Shinoda, Y. Suzuki, “Three-dimensional microscopy by optical scanning holography,” Opt. Eng. 34, 1338–1344 (1995).
[CrossRef]

Opt. Lett.

Opt. Rev.

T.-C. Poon, K. B. Doh, B. Schilling, K. Shinoda, Y. Suzuki, M. H. Wu, “Holographic three-dimensional display using an electron-beam-addressed spatial light modulator,” Opt. Rev. 4, 567–571 (1997).
[CrossRef]

Optik

C. J. R. Sheppard, “Vectors and Fourier transforms in optics,” Optik 110, 157–160 (1999).

Proc. Natl. Acad. Sci. USA

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Other

P. Klysubun, G. Indebetouw, “A posteriori processing of spatiotemporal digital microholograms,” J. Opt. Soc. Am. A18, 326–331 (2001).
[CrossRef]

P. J. Shaw, “Comparison of wide-field/deconvolution and confocal microscopy for 3D imaging,” in Handbook of Biological Confocal Microscopy, 2nd ed. J. B. Pawley, ed. (Plenum, New York, 1995), pp. 373–387.

J. Lippincott-Schwartz, J. F. Presley, K. J. M. Zaal, K. Hirschberg, C. D. Miller, J. Ellenberg, “Monitoring the dynamics and mobility of membrane proteins tagged with green fluorescent protein,” in Methods in Cell Biology, K. F. Sullivan, S. A. Kay, eds. (Academic, San Diego, 1999), Vol. 58, pp. 261–281.
[PubMed]

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Don Mills, Ontario, 1987), pp. 593–610.

J. Jonkman, E. H. K. Stelzer, “Resolution and contrast in confocal and two-photon microscopy,” in Confocal and Two-photon Microscopy: Foundations, Applications, and Advances, A. Diaspro, ed. (Wiley–Liss, New York, 2001), pp. 101–125.

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Figures (4)

Fig. 1
Fig. 1

OSH signal generation. An illumination pattern, created by the interference of coherent beams, is scanned across a point object. The signal at the detector is proportional to the intensity of the illumination pattern at the point object, which varies as the illumination is scanned. The resulting detector signal for a line scan is shown; a 2D raster scan of the point object produces a map of the illumination pattern. For spatially extended objects, the detected signal is the sum of the signals from all of the points in the object.

Fig. 2
Fig. 2

Important wave fronts in OSH. Eo and Er are incident on the sample; Eobj exits the sample, and E˜obj is incident on the mask, M. The optics before the sample are not shown.

Fig. 3
Fig. 3

Schematic of scanning configurations for OSH. For single-beam scanning, Mi1 is used to scan the beam and Mi2 is a fixed mirror. For double-beam scanning, Mi2 is scanned and Mi1 is fixed. The details of the scan optics are not shown. A combination of a plane reference wave and a diverging object wave are shown here; however, the analysis in the text is not restricted to this configuration. BS, beam splitter; L, lens; OL, objective lens; Mi, mirror; S, sample; M, mask; D, detector.

Fig. 4
Fig. 4

Sample thickness effects. (a) In a thick sample, the light is refracted and/or absorbed as it passes through the sample. For points at some depth into the sample, the illumination pattern is dependent on the properties of the sample itself. (b) For a thin sample this is not a problem, because the illumination is not significantly altered before it exits the sample. (c) For a nonscattering fluorescent sample (as discussed in Subsection 5.B), the illumination pattern is not significantly affected by the sample, even when the sample is thick.

Tables (1)

Tables Icon

Table 1 Summary of the Different OSH Modes

Equations (44)

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H(r)=fsample(r)fenc(r),
Irec(r)=H(r)Rf(r),
Rf,ideal(r)=FT-1[1/FT(fenc(r))],
Irec(r)=fsample(r)fenc(r)Rf,ideal(r)=fsample(r)δ(r)=fsample(r).
Eill(r)=Eo(r)+exp(iϕ)Er(r).
Eill(r, rs)=Eo(r-rs)+exp(iϕ)Er(r-rs),
Eobj(r, rs)=T(r, t)Eill(r, rs).
T(r, t)=Tins(r)exp[iΨ(r, t)],
S(rs)=M(ρ)E˜obj(ρ, rs)·E˜obj*(ρ, rs)d2ρ,
S(rs)=M(ρ)·exp(-2πiρ·r)Eobj(r, rs)d2r·exp(2πiρ·r)Eobj*(r, rs)d2rd2ρ.
S(rs)=exp[-2πiρ·(r-r)]M(ρ)d2ρEobj(r, rs)·Eobj*(r, rs)d2rd2r=M˜(r-r)Eobj(r, rs)·Eobj*(r, rs)d2rd2r.
S(rs)=M˜(r-r)[T(r, t)Eill(r, rs)]·[T*(r, t)Eill*(r, rs)]d2rd2r.
S(rs)=M˜(r-r)T(r, t)T*(r, t)
·[Eo(r-rs)·Eo*(r-rs)+Er(r-rs)·Er*(r-rs)]d2rd2r+exp(iϕ)M˜(r-r)
T(r, t)T*(r, t)Er(r-rs)·Eo*(r-rs)d2rd2r+exp(-iϕ)M˜(r-r)T(r, t)T*(r, t)
·Er*(r-rs)·Eo(r-rs)d2rd2r.
S(rs)B(rs)+exp(iϕ)V(rs)+exp(-iϕ)H(rs),
H2(rs)=M˜(r-r)T(r, t)T*(r, t)Er*(r-rs)·Eo(r-rs)d2rd2r,
H2,i(rs)=δ(r-r)T(r, t)T*(r, t)
Er*(r-rs)·Eo(r-rs)d2rd2r
=|T(r, t)|2Er*(r-rs)·Eo(r-rs)d2r=|T(rs, t)|2[Er*(rs)·Eo(rs)].
|T(rs, t)|2=Tins(r)exp[iΨ(r, t)]Tins*(r)exp[-iΨ(r, t)]=|Tins(rs)|2,
H2,i(rs)=|T(rs)|2[Er*(rs)·Eo(rs)].
H2,i,pw(rs)=|T(rs)|2[Eˆr·Eo(rs)].
H2,i,pw,pol(rs)=|T(rs)|2Eo(rs).
H2,c(rs)=exp(-2πiρd·(r-r))T(r)T*(r)Er*(r-rs)·Eo(r-rs)d2rd2r=exp(2πiρd·r)T*(r)Er*(r-rs)d2r·exp(-2πiρd·r)T(r)Eo(r-rs)d2r={[exp(2πiρd·rs)T*(rs)]¯Er*(rs)}·{[exp(-2πiρd·rs)T(rs)]¯Eo(rs)},
H2,c,pw(rs)=Hc0·exp(-2πiρd·r)T(r)Eo(r-rs)d2r=[exp(-2πiρd·rs)T(rs)][Hc0·Eo(rs)],
H2,c,pw(rs)=T(rs)[Hc0·Eo(rs)].
Eill(r, rs)=Eo(r-rs)+exp(iϕ)Er(r).
H1(rs)=M˜(r-r)T(r, t)T*(r, t)Er*(r)·Eo(r-rs)d2rd2r,
H1,i(rs)=|T(r, t)|2Er*(r)·Eo(r-rs)d2r=(|T(rs)|2Er*(rs))˙Eo(rs),
H1,i,pol(rs)=(|T(rs)|2Er*(rs))[Eˆr·Eo(rs)].
H1,i,cal(rs)=Er*(rs)[Eˆr·Eo(rs)].
H1,i,pw,pol(rs)=|T(rs)|2[Eˆr·Eo(rs)].
H1,c(rs)=exp(-2πiρd·(r-r))T(r)T*(r)Er*(r)·Eo(r-rs)d2rd2r=Hc0·exp(-2πiρd·r)T(r)Eo(r-rs)d2r=[exp(-2πiρd·rs)T(rs)][Hc0·Eo(rs)],
H1,c(rs)=T(rs)[Hc0·Eo(rs)],
H1,c,cal(rs)=exp(-2πiρd·rs)[Hc0·Eo(rs)].
Eill(r, rs)=Eo(r, rs)+exp(iϕ)·Er(r, rs).
T(r)= δT(r, z)δz dz,
Eill(r, rs, z)=Eo(r-rs, z)+exp(iϕ)Er(r-rs, z).
Eill(r, rs, z)=Eo(r, rs, z)+exp(iϕ)Er(r, rs, z).
Iem(r, rs, z)=Tfl(r, z)|Eill(r, rs, z)|2,
S(rs)=Iem(r, rs, z)d2rdz=|Tfl(r, z)|2[|Eo(r-rs, z)|2+|Er(r-rs, z)|2+exp(iϕ)Er(r-rs, z)Eo*(r-rs, z)+exp(-iϕ)Er*(r-rs, z)Eo(r-rs, z)]d2rdz,
Hfl(rs)=|Tfl(r, z)|2Er*(r-rs, z)·Eo(r-rs, z)d2rdz=|Tfl(rs, z)|2[Er*(rs, z)·Eo(rs, z)]dz.

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