Abstract

Unlike most optical coherence microscopy (OCM) systems, dynamic speckle-field interferometric microscopy (DSIM) achieves depth sectioning through the spatial-coherence gating effect. Under high numerical aperture (NA) speckle-field illumination, our previous experiments have demonstrated less than 1 μm depth resolution in reflection-mode DSIM, while doubling the diffraction limited resolution as under structured illumination. However, there has not been a physical model to rigorously describe the speckle imaging process, in particular explaining the sectioning effect under high illumination and imaging NA settings in DSIM. In this paper, we develop such a model based on the diffraction tomography theory and the speckle statistics. Using this model, we calculate the system response function, which is used to further obtain the depth resolution limit in reflection-mode DSIM. Theoretically calculated depth resolution limit is in an excellent agreement with experiment results. We envision that our physical model will not only help in understanding the imaging process in DSIM, but also enable better designing such systems for depth-resolved measurements in biological cells and tissues.

© 2017 Optical Society of America

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References

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2016 (3)

2015 (1)

N. D. Shemonski, F. A. South, Y. Z. Liu, S. G. Adie, P. S. Carney, and S. A. Boppart, “Computational high-resolution optical imaging of the living human retina,” Nat. Photonics 9(7), 440–443 (2015).
[Crossref] [PubMed]

2014 (5)

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat. Photonics 8(3), 256–263 (2014).
[Crossref]

B. Redding, Y. Bromberg, M. A. Choma, and H. Cao, “Full-field interferometric confocal microscopy using a VCSEL array,” Opt. Lett. 39(15), 4446–4449 (2014).
[Crossref] [PubMed]

R. Zhou, T. Kim, L. L. Goddard, and G. Popescu, “Inverse scattering solutions using low-coherence light,” Opt. Lett. 39(15), 4494–4497 (2014).
[Crossref] [PubMed]

Y. Choi, P. Hosseini, W. Choi, R. R. Dasari, P. T. C. So, and Z. Yaqoob, “Dynamic speckle illumination wide-field reflection phase microscopy,” Opt. Lett. 39(20), 6062–6065 (2014).
[Crossref] [PubMed]

B. Bhaduri, C. Edwards, H. Pham, R. Zhou, T. H. Nguyen, L. L. Goddard, and G. Popescu, “Diffraction phase microscopy: principles and applications in materials and life sciences,” Adv. Opt. Photonics 6(1), 57–119 (2014).
[Crossref]

2012 (1)

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, “Stain-free quantification of chromosomes in live cells using regularized tomographic phase microscopy,” PLoS One 7(11), e49502 (2012).
[Crossref] [PubMed]

2011 (1)

2009 (1)

2008 (1)

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

2007 (3)

H. U. Dodt, U. Leischner, A. Schierloh, N. Jährling, C. P. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4(4), 331–336 (2007).
[Crossref] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref] [PubMed]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

2006 (1)

P. J. Keller, F. Pampaloni, and E. H. K. Stelzer, “Life sciences require the third dimension,” Curr. Opin. Cell Biol. 18(1), 117–124 (2006).
[Crossref] [PubMed]

2005 (1)

J. A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2(12), 920–931 (2005).
[Crossref] [PubMed]

2004 (1)

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305(5686), 1007–1009 (2004).
[Crossref] [PubMed]

2003 (1)

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003).
[Crossref]

2000 (2)

P. T. C. So, C. Y. Dong, B. R. Masters, and K. M. Berland, “Two-photon excitation fluorescence microscopy,” Annu. Rev. Biomed. Eng. 2(1), 399–429 (2000).
[Crossref] [PubMed]

M. G. Somekh, C. W. See, and J. Goh, “Wide field amplitude and phase confocal microscope with speckle illumination,” Opt. Commun. 174(1-4), 75–80 (2000).
[Crossref]

1999 (1)

1997 (1)

1991 (3)

K. Creath, “Phase-measurement interferometry techniques for nondestructive testing,” Moire Techniques, Holographic Interferometry, Optical NDT, and Applications to Fluid Mechanics 1554, 701–707 (1991).

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

C. J. R. Sheppard, M. Gu, and X. Q. Mao, “Three-dimensional coherent transfer-function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81(5), 281–284 (1991).
[Crossref]

1990 (1)

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[Crossref] [PubMed]

1989 (2)

T. Wilson, “Optical sectioning in confocal fluorescent microscopes,” J. Microsc-Oxford 154(2), 143–156 (1989).
[Crossref]

D. A. Agard, Y. Hiraoka, P. Shaw, and J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

1988 (1)

M. Minsky, “Memoir on inventing the confocal scanning microscope,” Scanning 10(4), 128–138 (1988).
[Crossref]

1984 (1)

D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13(1), 191–219 (1984).
[Crossref] [PubMed]

1981 (1)

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1(4), 153–156 (1969).
[Crossref]

Adie, S. G.

N. D. Shemonski, F. A. South, Y. Z. Liu, S. G. Adie, P. S. Carney, and S. A. Boppart, “Computational high-resolution optical imaging of the living human retina,” Nat. Photonics 9(7), 440–443 (2015).
[Crossref] [PubMed]

Agard, D. A.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

D. A. Agard, Y. Hiraoka, P. Shaw, and J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13(1), 191–219 (1984).
[Crossref] [PubMed]

Babacan, S. D.

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat. Photonics 8(3), 256–263 (2014).
[Crossref]

Badizadegan, K.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref] [PubMed]

Becker, K.

H. U. Dodt, U. Leischner, A. Schierloh, N. Jährling, C. P. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4(4), 331–336 (2007).
[Crossref] [PubMed]

Berland, K. M.

P. T. C. So, C. Y. Dong, B. R. Masters, and K. M. Berland, “Two-photon excitation fluorescence microscopy,” Annu. Rev. Biomed. Eng. 2(1), 399–429 (2000).
[Crossref] [PubMed]

Bhaduri, B.

B. Bhaduri, C. Edwards, H. Pham, R. Zhou, T. H. Nguyen, L. L. Goddard, and G. Popescu, “Diffraction phase microscopy: principles and applications in materials and life sciences,” Adv. Opt. Photonics 6(1), 57–119 (2014).
[Crossref]

Boppart, S. A.

N. D. Shemonski, F. A. South, Y. Z. Liu, S. G. Adie, P. S. Carney, and S. A. Boppart, “Computational high-resolution optical imaging of the living human retina,” Nat. Photonics 9(7), 440–443 (2015).
[Crossref] [PubMed]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

Bromberg, Y.

Cande, W. Z.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Cao, H.

Carlton, P. M.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Carney, P. S.

N. D. Shemonski, F. A. South, Y. Z. Liu, S. G. Adie, P. S. Carney, and S. A. Boppart, “Computational high-resolution optical imaging of the living human retina,” Nat. Photonics 9(7), 440–443 (2015).
[Crossref] [PubMed]

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat. Photonics 8(3), 256–263 (2014).
[Crossref]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Choi, W.

Y. Choi, P. Hosseini, W. Choi, R. R. Dasari, P. T. C. So, and Z. Yaqoob, “Dynamic speckle illumination wide-field reflection phase microscopy,” Opt. Lett. 39(20), 6062–6065 (2014).
[Crossref] [PubMed]

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, “Stain-free quantification of chromosomes in live cells using regularized tomographic phase microscopy,” PLoS One 7(11), e49502 (2012).
[Crossref] [PubMed]

Y. Choi, T. D. Yang, K. J. Lee, and W. Choi, “Full-field and single-shot quantitative phase microscopy using dynamic speckle illumination,” Opt. Lett. 36(13), 2465–2467 (2011).
[Crossref] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref] [PubMed]

Choi, Y.

Choma, M. A.

Conchello, J. A.

J. A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2(12), 920–931 (2005).
[Crossref] [PubMed]

Creath, K.

K. Creath, “Phase-measurement interferometry techniques for nondestructive testing,” Moire Techniques, Holographic Interferometry, Optical NDT, and Applications to Fluid Mechanics 1554, 701–707 (1991).

Cuche, E.

Dasari, R. R.

Y. Choi, P. Hosseini, W. Choi, R. R. Dasari, P. T. C. So, and Z. Yaqoob, “Dynamic speckle illumination wide-field reflection phase microscopy,” Opt. Lett. 39(20), 6062–6065 (2014).
[Crossref] [PubMed]

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, “Stain-free quantification of chromosomes in live cells using regularized tomographic phase microscopy,” PLoS One 7(11), e49502 (2012).
[Crossref] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref] [PubMed]

Deininger, K.

H. U. Dodt, U. Leischner, A. Schierloh, N. Jährling, C. P. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4(4), 331–336 (2007).
[Crossref] [PubMed]

Del Bene, F.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305(5686), 1007–1009 (2004).
[Crossref] [PubMed]

Denk, W.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[Crossref] [PubMed]

Depeursinge, C.

Deussing, J. M.

H. U. Dodt, U. Leischner, A. Schierloh, N. Jährling, C. P. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4(4), 331–336 (2007).
[Crossref] [PubMed]

Diaspro, A.

Dodt, H. U.

H. U. Dodt, U. Leischner, A. Schierloh, N. Jährling, C. P. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4(4), 331–336 (2007).
[Crossref] [PubMed]

Dong, C. Y.

P. T. C. So, C. Y. Dong, B. R. Masters, and K. M. Berland, “Two-photon excitation fluorescence microscopy,” Annu. Rev. Biomed. Eng. 2(1), 399–429 (2000).
[Crossref] [PubMed]

Drexler, W.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003).
[Crossref]

Eder, M.

H. U. Dodt, U. Leischner, A. Schierloh, N. Jährling, C. P. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4(4), 331–336 (2007).
[Crossref] [PubMed]

Edwards, C.

B. Bhaduri, C. Edwards, H. Pham, R. Zhou, T. H. Nguyen, L. L. Goddard, and G. Popescu, “Diffraction phase microscopy: principles and applications in materials and life sciences,” Adv. Opt. Photonics 6(1), 57–119 (2014).
[Crossref]

Fang-Yen, C.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref] [PubMed]

Feld, M. S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref] [PubMed]

Fercher, A. F.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003).
[Crossref]

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Goddard, L. L.

T. Kim, R. J. Zhou, L. L. Goddard, and G. Popescu, “Solving inverse scattering problems in biological samples by quantitative phase imaging,” Laser Photonics Rev. 10(1), 13–39 (2016).
[Crossref]

R. Zhou, T. Kim, L. L. Goddard, and G. Popescu, “Inverse scattering solutions using low-coherence light,” Opt. Lett. 39(15), 4494–4497 (2014).
[Crossref] [PubMed]

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat. Photonics 8(3), 256–263 (2014).
[Crossref]

B. Bhaduri, C. Edwards, H. Pham, R. Zhou, T. H. Nguyen, L. L. Goddard, and G. Popescu, “Diffraction phase microscopy: principles and applications in materials and life sciences,” Adv. Opt. Photonics 6(1), 57–119 (2014).
[Crossref]

Goh, J.

M. G. Somekh, C. W. See, and J. Goh, “Wide field amplitude and phase confocal microscope with speckle illumination,” Opt. Commun. 174(1-4), 75–80 (2000).
[Crossref]

Golubovskaya, I. N.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Gu, M.

C. J. R. Sheppard, M. Gu, and X. Q. Mao, “Three-dimensional coherent transfer-function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81(5), 281–284 (1991).
[Crossref]

Gustafsson, M. G. L.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Hiraoka, Y.

D. A. Agard, Y. Hiraoka, P. Shaw, and J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

Hitzenberger, C. K.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003).
[Crossref]

Hosseini, P.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Huisken, J.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305(5686), 1007–1009 (2004).
[Crossref] [PubMed]

Jährling, N.

H. U. Dodt, U. Leischner, A. Schierloh, N. Jährling, C. P. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4(4), 331–336 (2007).
[Crossref] [PubMed]

Juskaitis, R.

Keller, P. J.

P. J. Keller, F. Pampaloni, and E. H. K. Stelzer, “Life sciences require the third dimension,” Curr. Opin. Cell Biol. 18(1), 117–124 (2006).
[Crossref] [PubMed]

Kim, T.

T. Kim, R. J. Zhou, L. L. Goddard, and G. Popescu, “Solving inverse scattering problems in biological samples by quantitative phase imaging,” Laser Photonics Rev. 10(1), 13–39 (2016).
[Crossref]

R. Zhou, T. Kim, L. L. Goddard, and G. Popescu, “Inverse scattering solutions using low-coherence light,” Opt. Lett. 39(15), 4494–4497 (2014).
[Crossref] [PubMed]

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat. Photonics 8(3), 256–263 (2014).
[Crossref]

Kim, Y. H.

Kou, S. S.

Kuang, C.

Lasser, T.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003).
[Crossref]

Lee, K. J.

Leischner, U.

H. U. Dodt, U. Leischner, A. Schierloh, N. Jährling, C. P. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4(4), 331–336 (2007).
[Crossref] [PubMed]

Lichtman, J. W.

J. A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2(12), 920–931 (2005).
[Crossref] [PubMed]

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Liu, Y. Z.

N. D. Shemonski, F. A. South, Y. Z. Liu, S. G. Adie, P. S. Carney, and S. A. Boppart, “Computational high-resolution optical imaging of the living human retina,” Nat. Photonics 9(7), 440–443 (2015).
[Crossref] [PubMed]

Lue, N.

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, “Stain-free quantification of chromosomes in live cells using regularized tomographic phase microscopy,” PLoS One 7(11), e49502 (2012).
[Crossref] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref] [PubMed]

Mao, X. Q.

C. J. R. Sheppard, M. Gu, and X. Q. Mao, “Three-dimensional coherent transfer-function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81(5), 281–284 (1991).
[Crossref]

Marks, D. L.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

Marquet, P.

Masters, B. R.

P. T. C. So, C. Y. Dong, B. R. Masters, and K. M. Berland, “Two-photon excitation fluorescence microscopy,” Annu. Rev. Biomed. Eng. 2(1), 399–429 (2000).
[Crossref] [PubMed]

Mauch, C. P.

H. U. Dodt, U. Leischner, A. Schierloh, N. Jährling, C. P. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4(4), 331–336 (2007).
[Crossref] [PubMed]

Minsky, M.

M. Minsky, “Memoir on inventing the confocal scanning microscope,” Scanning 10(4), 128–138 (1988).
[Crossref]

Mir, M.

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat. Photonics 8(3), 256–263 (2014).
[Crossref]

Nastasa, V.

Neil, M. A. A.

Nguyen, T. H.

B. Bhaduri, C. Edwards, H. Pham, R. Zhou, T. H. Nguyen, L. L. Goddard, and G. Popescu, “Diffraction phase microscopy: principles and applications in materials and life sciences,” Adv. Opt. Photonics 6(1), 57–119 (2014).
[Crossref]

Oh, S.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref] [PubMed]

Pampaloni, F.

P. J. Keller, F. Pampaloni, and E. H. K. Stelzer, “Life sciences require the third dimension,” Curr. Opin. Cell Biol. 18(1), 117–124 (2006).
[Crossref] [PubMed]

Peres, C.

Pham, H.

B. Bhaduri, C. Edwards, H. Pham, R. Zhou, T. H. Nguyen, L. L. Goddard, and G. Popescu, “Diffraction phase microscopy: principles and applications in materials and life sciences,” Adv. Opt. Photonics 6(1), 57–119 (2014).
[Crossref]

Popescu, G.

M. Shan, V. Nastasa, and G. Popescu, “Statistical dispersion relation for spatially broadband fields,” Opt. Lett. 41(11), 2490–2492 (2016).
[Crossref] [PubMed]

T. Kim, R. J. Zhou, L. L. Goddard, and G. Popescu, “Solving inverse scattering problems in biological samples by quantitative phase imaging,” Laser Photonics Rev. 10(1), 13–39 (2016).
[Crossref]

R. Zhou, T. Kim, L. L. Goddard, and G. Popescu, “Inverse scattering solutions using low-coherence light,” Opt. Lett. 39(15), 4494–4497 (2014).
[Crossref] [PubMed]

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat. Photonics 8(3), 256–263 (2014).
[Crossref]

B. Bhaduri, C. Edwards, H. Pham, R. Zhou, T. H. Nguyen, L. L. Goddard, and G. Popescu, “Diffraction phase microscopy: principles and applications in materials and life sciences,” Adv. Opt. Photonics 6(1), 57–119 (2014).
[Crossref]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Ralston, T. S.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

Redding, B.

Schierloh, A.

H. U. Dodt, U. Leischner, A. Schierloh, N. Jährling, C. P. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4(4), 331–336 (2007).
[Crossref] [PubMed]

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Sedat, J. W.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

D. A. Agard, Y. Hiraoka, P. Shaw, and J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

See, C. W.

M. G. Somekh, C. W. See, and J. Goh, “Wide field amplitude and phase confocal microscope with speckle illumination,” Opt. Commun. 174(1-4), 75–80 (2000).
[Crossref]

Shan, M.

Shao, L.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Shaw, P.

D. A. Agard, Y. Hiraoka, P. Shaw, and J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

Shemonski, N. D.

N. D. Shemonski, F. A. South, Y. Z. Liu, S. G. Adie, P. S. Carney, and S. A. Boppart, “Computational high-resolution optical imaging of the living human retina,” Nat. Photonics 9(7), 440–443 (2015).
[Crossref] [PubMed]

Sheppard, C. J. R.

S. S. Kou and C. J. R. Sheppard, “Image formation in holographic tomography: high-aperture imaging conditions,” Appl. Opt. 48(34), H168–H175 (2009).
[Crossref] [PubMed]

C. J. R. Sheppard, M. Gu, and X. Q. Mao, “Three-dimensional coherent transfer-function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81(5), 281–284 (1991).
[Crossref]

So, P. T. C.

Somekh, M. G.

M. G. Somekh, C. W. See, and J. Goh, “Wide field amplitude and phase confocal microscope with speckle illumination,” Opt. Commun. 174(1-4), 75–80 (2000).
[Crossref]

Sommargren, G. E.

South, F. A.

N. D. Shemonski, F. A. South, Y. Z. Liu, S. G. Adie, P. S. Carney, and S. A. Boppart, “Computational high-resolution optical imaging of the living human retina,” Nat. Photonics 9(7), 440–443 (2015).
[Crossref] [PubMed]

Stelzer, E. H. K.

P. J. Keller, F. Pampaloni, and E. H. K. Stelzer, “Life sciences require the third dimension,” Curr. Opin. Cell Biol. 18(1), 117–124 (2006).
[Crossref] [PubMed]

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305(5686), 1007–1009 (2004).
[Crossref] [PubMed]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Strickler, J. H.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[Crossref] [PubMed]

Sung, Y.

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, “Stain-free quantification of chromosomes in live cells using regularized tomographic phase microscopy,” PLoS One 7(11), e49502 (2012).
[Crossref] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Swoger, J.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305(5686), 1007–1009 (2004).
[Crossref] [PubMed]

Wang, C. J. R.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Webb, W. W.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[Crossref] [PubMed]

Wilson, T.

Wittbrodt, J.

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305(5686), 1007–1009 (2004).
[Crossref] [PubMed]

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1(4), 153–156 (1969).
[Crossref]

Yang, T. D.

Yaqoob, Z.

Zhou, R.

P. Hosseini, R. Zhou, Y. H. Kim, C. Peres, A. Diaspro, C. Kuang, Z. Yaqoob, and P. T. C. So, “Pushing phase and amplitude sensitivity limits in interferometric microscopy,” Opt. Lett. 41(7), 1656–1659 (2016).
[Crossref] [PubMed]

R. Zhou, T. Kim, L. L. Goddard, and G. Popescu, “Inverse scattering solutions using low-coherence light,” Opt. Lett. 39(15), 4494–4497 (2014).
[Crossref] [PubMed]

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat. Photonics 8(3), 256–263 (2014).
[Crossref]

B. Bhaduri, C. Edwards, H. Pham, R. Zhou, T. H. Nguyen, L. L. Goddard, and G. Popescu, “Diffraction phase microscopy: principles and applications in materials and life sciences,” Adv. Opt. Photonics 6(1), 57–119 (2014).
[Crossref]

Zhou, R. J.

T. Kim, R. J. Zhou, L. L. Goddard, and G. Popescu, “Solving inverse scattering problems in biological samples by quantitative phase imaging,” Laser Photonics Rev. 10(1), 13–39 (2016).
[Crossref]

Zieglgänsberger, W.

H. U. Dodt, U. Leischner, A. Schierloh, N. Jährling, C. P. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4(4), 331–336 (2007).
[Crossref] [PubMed]

Adv. Opt. Photonics (1)

B. Bhaduri, C. Edwards, H. Pham, R. Zhou, T. H. Nguyen, L. L. Goddard, and G. Popescu, “Diffraction phase microscopy: principles and applications in materials and life sciences,” Adv. Opt. Photonics 6(1), 57–119 (2014).
[Crossref]

Annu. Rev. Biomed. Eng. (1)

P. T. C. So, C. Y. Dong, B. R. Masters, and K. M. Berland, “Two-photon excitation fluorescence microscopy,” Annu. Rev. Biomed. Eng. 2(1), 399–429 (2000).
[Crossref] [PubMed]

Annu. Rev. Biophys. Bioeng. (1)

D. A. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13(1), 191–219 (1984).
[Crossref] [PubMed]

Appl. Opt. (3)

Biophys. J. (1)

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94(12), 4957–4970 (2008).
[Crossref] [PubMed]

Curr. Opin. Cell Biol. (1)

P. J. Keller, F. Pampaloni, and E. H. K. Stelzer, “Life sciences require the third dimension,” Curr. Opin. Cell Biol. 18(1), 117–124 (2006).
[Crossref] [PubMed]

J. Microsc-Oxford (1)

T. Wilson, “Optical sectioning in confocal fluorescent microscopes,” J. Microsc-Oxford 154(2), 143–156 (1989).
[Crossref]

Laser Photonics Rev. (1)

T. Kim, R. J. Zhou, L. L. Goddard, and G. Popescu, “Solving inverse scattering problems in biological samples by quantitative phase imaging,” Laser Photonics Rev. 10(1), 13–39 (2016).
[Crossref]

Methods Cell Biol. (1)

D. A. Agard, Y. Hiraoka, P. Shaw, and J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[Crossref] [PubMed]

Moire Techniques, Holographic Interferometry, Optical NDT, and Applications to Fluid Mechanics (1)

K. Creath, “Phase-measurement interferometry techniques for nondestructive testing,” Moire Techniques, Holographic Interferometry, Optical NDT, and Applications to Fluid Mechanics 1554, 701–707 (1991).

Nat. Methods (3)

J. A. Conchello and J. W. Lichtman, “Optical sectioning microscopy,” Nat. Methods 2(12), 920–931 (2005).
[Crossref] [PubMed]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref] [PubMed]

H. U. Dodt, U. Leischner, A. Schierloh, N. Jährling, C. P. Mauch, K. Deininger, J. M. Deussing, M. Eder, W. Zieglgänsberger, and K. Becker, “Ultramicroscopy: three-dimensional visualization of neuronal networks in the whole mouse brain,” Nat. Methods 4(4), 331–336 (2007).
[Crossref] [PubMed]

Nat. Photonics (2)

N. D. Shemonski, F. A. South, Y. Z. Liu, S. G. Adie, P. S. Carney, and S. A. Boppart, “Computational high-resolution optical imaging of the living human retina,” Nat. Photonics 9(7), 440–443 (2015).
[Crossref] [PubMed]

T. Kim, R. Zhou, M. Mir, S. D. Babacan, P. S. Carney, L. L. Goddard, and G. Popescu, “White-light diffraction tomography of unlabelled live cells,” Nat. Photonics 8(3), 256–263 (2014).
[Crossref]

Nat. Phys. (1)

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

Opt. Commun. (3)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1(4), 153–156 (1969).
[Crossref]

C. J. R. Sheppard, M. Gu, and X. Q. Mao, “Three-dimensional coherent transfer-function in a reflection-mode confocal scanning microscope,” Opt. Commun. 81(5), 281–284 (1991).
[Crossref]

M. G. Somekh, C. W. See, and J. Goh, “Wide field amplitude and phase confocal microscope with speckle illumination,” Opt. Commun. 174(1-4), 75–80 (2000).
[Crossref]

Opt. Lett. (7)

PLoS One (1)

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, “Stain-free quantification of chromosomes in live cells using regularized tomographic phase microscopy,” PLoS One 7(11), e49502 (2012).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003).
[Crossref]

Scanning (1)

M. Minsky, “Memoir on inventing the confocal scanning microscope,” Scanning 10(4), 128–138 (1988).
[Crossref]

Science (3)

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. K. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,” Science 305(5686), 1007–1009 (2004).
[Crossref] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990).
[Crossref] [PubMed]

Other (2)

M. Gu, Principles of Three Dimensional Imaging in Confocal Microscopes (World Scientific, Singapore; River Edge, NJ, 1996).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena (Springer, 1975), pp. 9–75.

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Figures (4)

Fig. 1
Fig. 1 Illustration of reflection-mode DSIM. (a) The system configuration of a reflection-mode DSIM based on a Linnik-type interferometer; (b) A description of the electromagnetic fields involved in the imaging system.
Fig. 2
Fig. 2 Illustration of a thin step phase object, defined by a rectangle function.
Fig. 3
Fig. 3 Axial response function with N A obj =1. The axial response function is obtained by calculating 2Re( Γ 12 ) at different defocus position z R .
Fig. 4
Fig. 4 Relationship between depth resolution δz (vertical axis) and objective numerical aperture N A obj (horizontal axis).

Equations (35)

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U i ( r )=S( k xi , k yi ) e i( k xi x+ k yi x+ k zi z ) ,
U r ( r d )=S( k xi , k yi ) e i( k xi x d + k yi y d + k yi z d ) .
2 U s ( r )+ β 2 U s ( r )= β o 2 χ( r )U( r ),
U bs ( k ,z; z R )= β 0 2 S( k xi , k yi ) e iq( z z R ) e i k zi z R 2q χ( k x k xi , k y k yi ,q k zi ).
U bs ( k d , z d =0; z R )= β o 2 S( k xi , k yi )P( k x , k y ) e i( q+ k zi ) z R 2q χ( k x k xi , k y k yi ,q k zi ),
χ( k x , k y , k z )=δ( k x )δ( k y )sinc( k z z 0 ).
U bs ( k x , k y ,z; z R )= β 0 2 S( k xi , k yi )P( k x , k y ) e i β 2 k x 2 k y 2 ( z z R ) e i k zi z R 2q × δ( k x k xi )δ( k y k yi )sinc[ ( β 2 k x 2 k y 2 + k zi ) z 0 ].
U bs ( x,y,z; z R )= β 0 2 S( k xi , k yi )P( k xi , k yi ) e i( k xi x+ k yi y ) e i β 2 k xi 2 k yi 2 ( z z R ) e i k zi z R 2 β 2 k xi 2 k yi 2 × sinc[ ( β 2 k xi 2 k yi 2 + k zi ) z 0 ].
U bs ( x,y,z; z R )= β 0 2 S( k xi , k yi )P( k xi , k yi ) e i( k xi x+ k yi y ) e i k zi ( z2 z R ) 2 k zi sinc( 2 k zi z 0 ).
U bs ( x,y,z )= β 0 2 S( k xi , k yi )P( k xi , k yi ) e i( k xi x+ k yi y ) e i k zi z e i k zi ( 2 z R ) 2 k zi .
U bs ( x d , y d , z d =0 )= β 0 2 S( k xi , k yi )P( k xi , k yi ) e i( k xi x+ k yi y ) e i k zi ( 2 z R ) 2 k zi .
S( k xi , k yi )=| S( k xi , k yi ) | e iφ( k xi , k yi ) = 1 N 2 A( k xi , k yi ) e iφ( k xi , k yi ) ,
2Re( Γ 12 )=2Re( U bs U r * )=2Re{ [ k xi , k yi U bs ( k xi , k yi ) ][ k xi ' , k yi ' U r * ( k xi ' , k yi ' ) ] }.
2Re( Γ 12 ) k xi , k yi { k xi ' , k yi ' 1 k zi A( k xi , k yi )A( k xi ' , k yi ' )cos[ ( k xi k xi ' )x+( k yi k yi ' )y+ k zi ( 2 z R )+Δφ ] } .
2Re( Γ 12 ) k xi , k yi k xi ' , k yi ' 1 k zi A( k xi , k yi )A( k xi ' , k yi ' ) cos[ ( k xi k xi ' )x+( k yi k yi ' )y+2 k zi z R ]cos( Δφ ) k xi , k yi k xi ' , k yi ' 1 k zi A( k xi , k yi )A( k xi ' , k yi ' ) sin[ ( k xi k xi ' )x+( k yi k yi ' )y+2 k zi z R ]sin( Δφ ).
2Re( Γ 12 ) k xi , k yi A 2 ( k xi , k yi )P( k xi , k yi ) cos( 2 k zi z R ) k zi M .
2Re( Γ 12 ) T 0 ( k xi , k yi ) P 2 ( k xi , k yi ) cos( 2 β 2 k xi 2 k yi 2 z R ) β 2 k xi 2 k yi 2 d k xi d k yi .
2Re( Γ 12 ) P( k r cosϕ, k r sinϕ ) cos( 2 β 2 k r 2 z R ) β 2 k r 2 d k r d ϕ .
2Re( Γ 12 ) 0 βsin θ max cos( 2 β 2 k r 2 z R ) β 2 k r 2 k r d k r .
2Re( Γ 12 ) βcos( θ max ) β cos( 2K z R ) dK
2Re( Γ 12 )β [ sinc( 2β z R /π ) cos( θ max )sinc( 2βcos( θ max ) z R /π ) ],
U fs ( k x , k y ,z; z R )= β 0 2 S( k xi , k yi )P( k x , k y ) e i β 2 k x 2 k y 2 ( z z R ) e i k zi z R 2q × δ( k x k xi )δ( k y k yi )sinc[ ( β 2 k x 2 k y 2 k zi ) z 0 ].
U fs ( x,y,z; z R )= β 0 2 S( k xi , k yi )P( k xi , k yi ) e i( k xi x+ k yi y ) e i β 2 k xi 2 k yi 2 ( z z R ) e i k zi z R 2 β 2 k xi 2 k yi 2 × sinc[ ( β 2 k xi 2 k yi 2 k zi ) z 0 ].
U fs ( x,y,z; z R )= β 0 2 S( k xi , k yi )P( k xi , k yi ) e i( k xi x+ k yi y ) e i k zi ( z z R ) e i k zi z R 2 k zi sinc( 0 ) = β 0 2 S( k xi , k yi )P( k xi , k yi ) e i( k xi x+ k yi y ) e i k zi z 2 k zi .
U fs ( x d , y d , z d =0; z R )= β 0 2 S( k xi , k yi )P( k xi , k yi ) e i( k xi x+ k yi y ) 2 k zi .
2 U s ( r )+ β 2 U s ( r )= β o 2 χ( r )U( r ),
( β 2 k 2 ) U s ( k x , k y , k z )= β 0 2 S( k xi , k yi )χ( k x , k y , k z ) k δ( k x k xi , k y k yi , k z k zi ) = β 0 2 S( k xi , k yi )χ( k x k xi , k y k yi , k z k zi ),
( β 2 k 2 ) U s ( k x , k y , k z )= β 0 2 S( k xi , k yi )χ( k x k xi , k y k yi , k z k zi ) e i( k z k zi ) z R .
U s ( k x , k y , k z )= β 0 2 S( k xi , k yi )χ( k x k xi , k y k yi , k z k zi ) e i( k z k zi ) z R k z 2 ( β 2 k x 2 k y 2 ) = β 0 2 S( k xi , k yi )χ( k x k xi , k y k yi , k z k zi ) e i( k z k zi ) z R 1 2q ( 1 k z +q 1 k z q ),
U s ( k x , k y ,z )= β 0 2 S( k xi , k yi )[ χ( k x k xi , k y k yi ,z z R ) e i k zi z ] z 1 2q ( sgn( z ) e iqz sgn( z ) e iqz ).
U bs ( k x , k y ,z )= β 0 2 S( k xi , k yi ) 2 [ χ( k x k xi , k y k yi ,z z R ) e i k zi z ] z e iqz q .
U fs ( k x , k y ,z )= β 0 2 S( k xi , k yi ) 2 [ χ( k x k xi , k y k yi ,z z R ) e i k zi z ] z e iqz q .
U bs ( k x , k y ,z )= β 0 2 S( k xi , k yi ) 2q + [ χ( k x k xi , k y k yi ,z' z R ) e i k zi z' ] e iq( zz' ) dz' = β 0 2 S( k xi , k yi ) e iqz 2q + χ( k x k xi , k y k yi ,z' z R ) e i( q+ k zi )z' dz' = β 0 2 S( k xi , k yi ) e iqz 2q + χ( k x k xi , k y k yi ,z z R ) e iWz dz.
U bs ( k x , k y ,z )= β 0 2 S( k xi , k yi ) e iqz e i( q+ k zi ) z R 2q χ( k x k xi , k y k yi ,q k zi ) = β 0 2 S( k xi , k yi ) e iq( z z R ) e i k zi z R 2q χ( k x k xi , k y k yi ,q k zi ).
U fs ( k x , k y ,z )= β 0 2 S( k xi , k yi ) e iqz e i( k zi q ) z R 2q χ( k x k xi , k y k yi ,q k zi ) = β 0 2 S( k xi , k yi ) e iq( z z R ) e i k zi z R 2q χ( k x k xi , k y k yi ,q k zi ).

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