Abstract

The paper presents two novel, space-domain reconstruction algorithms for holographic tomography utilizing scanning of illumination and a fixed detector that is highly suitable for imaging of living biomedical specimens. The first proposed algorithm is an adaptation of the filtered backpropagation to the scanning illumination tomography. Its space-domain implementation enables avoiding the error-prone interpolation in the Fourier domain, which is a significant problem of the state-of-the-art tomographic algorithm. The second proposed algorithm is a modified version of the former, which ensures the spatially invariant reconstruction accuracy. The utility of the proposed algorithms is demonstrated with numerical simulations and experimental measurement of a cancer cell.

© 2016 Optical Society of America

Full Article  |  PDF Article
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2016 (2)

W. Krauze, P. Makowski, M. Kujawińska, and A. Kuś, “Generalized total variation iterative constraint strategy in limited angle optical diffraction tomography,” Opt. Express 24(5), 4924–4936 (2016).
[Crossref]

J. Kostencka and T. Kozacki, “Space-domain, filtered backpropagation algorithm for tomographic configuration with scanning of illumination,” Proc. SPIE 9890M, 9890 (2016).

2015 (5)

2014 (5)

2013 (2)

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marque, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7(2), 113–117 (2013).
[Crossref]

K. Falaggis, T. Kozacki, and M. Kujawinska, “Computation of highly off-axis diffracted fields using the band-limited angular spectrum method with suppressed Gibbs related artifacts,” Appl. Opt. 52(14), 3288–3297 (2013).
[Crossref] [PubMed]

2012 (1)

2009 (3)

2008 (1)

2007 (1)

2006 (2)

F. Charrière, N. Pavillon, T. Colomb, C. Depeursinge, T. J. Heger, E. A. D. Mitchell, P. Marquet, and B. Rappaz, “Living specimen tomography by digital holographic microscopy: morphometry of testate amoeba,” Opt. Express 14(16), 7005–7013 (2006).
[Crossref] [PubMed]

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11(3), 034005 (2006).
[Crossref] [PubMed]

1998 (1)

1993 (1)

1983 (1)

S. Pan and A. Kak, “A computational study of reconstruction algorithms for diffraction tomography: Interpolation versus filtered-backpropagation,” IEEE Trans. Acoust. Speech Signal Process. 31(5), 1262–1275 (1983).
[Crossref]

1982 (2)

1981 (1)

1975 (1)

A. J. Devaney, H. J. Liff, and S. Apsell, “Spectral representations for free space propagation of complex phase perturbations of optical fields,” Opt. Commun. 15(1), 1–5 (1975).
[Crossref]

1969 (1)

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

Apsell, S.

A. J. Devaney, H. J. Liff, and S. Apsell, “Spectral representations for free space propagation of complex phase perturbations of optical fields,” Opt. Commun. 15(1), 1–5 (1975).
[Crossref]

Badizadegan, K.

Baethge, G.

M. Dudek, M. Kujawińska, V. Parat, G. Baethge, A. Michalska, B. Dahmani, and H. Ottevaere, “Tomographic and numerical studies of polymer bridges between two optical fibers for telecommunication applications,” Opt. Eng. 53(1), 016113 (2014).
[Crossref]

Boss, D.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marque, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7(2), 113–117 (2013).
[Crossref]

Bredebusch, I.

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11(3), 034005 (2006).
[Crossref] [PubMed]

Brenner, K.-H.

Carl, D.

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11(3), 034005 (2006).
[Crossref] [PubMed]

Charrière, F.

Chen, B.

Choi, C.

Choi, W.

Choi, Y.

Colomb, T.

Cotte, Y.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marque, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7(2), 113–117 (2013).
[Crossref]

Dahmani, B.

M. Dudek, M. Kujawińska, V. Parat, G. Baethge, A. Michalska, B. Dahmani, and H. Ottevaere, “Tomographic and numerical studies of polymer bridges between two optical fibers for telecommunication applications,” Opt. Eng. 53(1), 016113 (2014).
[Crossref]

Dasari, R. R.

Depeursinge, C.

Devaney, A. J.

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4(4), 336–350 (1982).
[Crossref] [PubMed]

A. J. Devaney, “Inverse-scattering theory within the Rytov approximation,” Opt. Lett. 6(8), 374–376 (1981).
[Crossref] [PubMed]

A. J. Devaney, H. J. Liff, and S. Apsell, “Spectral representations for free space propagation of complex phase perturbations of optical fields,” Opt. Commun. 15(1), 1–5 (1975).
[Crossref]

Domschke, W.

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11(3), 034005 (2006).
[Crossref] [PubMed]

Dudek, M.

M. Dudek, M. Kujawińska, V. Parat, G. Baethge, A. Michalska, B. Dahmani, and H. Ottevaere, “Tomographic and numerical studies of polymer bridges between two optical fibers for telecommunication applications,” Opt. Eng. 53(1), 016113 (2014).
[Crossref]

A. Kuś, M. Dudek, B. Kemper, M. Kujawińska, and A. Vollmer, “Tomographic phase microscopy of living three-dimensional cell cultures,” J. Biomed. Opt. 19(4), 046009 (2014).
[Crossref] [PubMed]

J. Kostencka, T. Kozacki, M. Dudek, and M. Kujawińska, “Noise suppressed optical diffraction tomography with autofocus correction,” Opt. Express 22(5), 5731–5745 (2014).
[Crossref] [PubMed]

Falaggis, K.

Fang-Yen, C.

Feld, M. S.

Gilboa, B.

Górski, W.

Habaza, M.

Heger, T. J.

Heo, J. H.

Hosseini, P.

Ina, H.

Jang, S.

Jin, K. H.

Jourdain, P.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marque, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7(2), 113–117 (2013).
[Crossref]

Kak, A.

S. Pan and A. Kak, “A computational study of reconstruction algorithms for diffraction tomography: Interpolation versus filtered-backpropagation,” IEEE Trans. Acoust. Speech Signal Process. 31(5), 1262–1275 (1983).
[Crossref]

Kang, J. W.

Kao, C.-M.

E. Y. Sidky, C.-M. Kao, and X. Pan, “Accuracte image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. Opt. Soc. Am. 25, 1772–1782 (2009).

Kemper, B.

A. Kuś, M. Dudek, B. Kemper, M. Kujawińska, and A. Vollmer, “Tomographic phase microscopy of living three-dimensional cell cultures,” J. Biomed. Opt. 19(4), 046009 (2014).
[Crossref] [PubMed]

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11(3), 034005 (2006).
[Crossref] [PubMed]

Kim, K.

Kim, Y.

Kobayashi, S.

Kostencka, J.

J. Kostencka and T. Kozacki, “Space-domain, filtered backpropagation algorithm for tomographic configuration with scanning of illumination,” Proc. SPIE 9890M, 9890 (2016).

J. Kostencka and T. Kozacki, “Computational and experimental study on accuracy of off-axis reconstructions in optical diffraction tomography,” Opt. Eng. 54(2), 024107 (2015).
[Crossref]

J. Kostencka, T. Kozacki, A. Kuś, and M. Kujawińska, “Accurate approach to capillary-supported optical diffraction tomography,” Opt. Express 23(6), 7908–7923 (2015).
[Crossref] [PubMed]

J. Kostencka, T. Kozacki, M. Dudek, and M. Kujawińska, “Noise suppressed optical diffraction tomography with autofocus correction,” Opt. Express 22(5), 5731–5745 (2014).
[Crossref] [PubMed]

Kou, S. S.

Kozacki, T.

Krauze, W.

Kujawinska, M.

W. Krauze, P. Makowski, M. Kujawińska, and A. Kuś, “Generalized total variation iterative constraint strategy in limited angle optical diffraction tomography,” Opt. Express 24(5), 4924–4936 (2016).
[Crossref]

J. Kostencka, T. Kozacki, A. Kuś, and M. Kujawińska, “Accurate approach to capillary-supported optical diffraction tomography,” Opt. Express 23(6), 7908–7923 (2015).
[Crossref] [PubMed]

A. Kuś, W. Krauze, and M. Kujawinska, “Active limited-angle tomographic phase microscope,” J. Biomed. Opt. 20(11), 111216 (2015).
[Crossref] [PubMed]

A. Kuś, M. Dudek, B. Kemper, M. Kujawińska, and A. Vollmer, “Tomographic phase microscopy of living three-dimensional cell cultures,” J. Biomed. Opt. 19(4), 046009 (2014).
[Crossref] [PubMed]

M. Dudek, M. Kujawińska, V. Parat, G. Baethge, A. Michalska, B. Dahmani, and H. Ottevaere, “Tomographic and numerical studies of polymer bridges between two optical fibers for telecommunication applications,” Opt. Eng. 53(1), 016113 (2014).
[Crossref]

J. Kostencka, T. Kozacki, M. Dudek, and M. Kujawińska, “Noise suppressed optical diffraction tomography with autofocus correction,” Opt. Express 22(5), 5731–5745 (2014).
[Crossref] [PubMed]

K. Falaggis, T. Kozacki, and M. Kujawinska, “Computation of highly off-axis diffracted fields using the band-limited angular spectrum method with suppressed Gibbs related artifacts,” Appl. Opt. 52(14), 3288–3297 (2013).
[Crossref] [PubMed]

T. Kozacki, K. Falaggis, and M. Kujawińska, “Computation of diffracted fields for the case of high numerical aperture using the angular spectrum method,” Appl. Opt. 51(29), 7080–7088 (2012).
[Crossref] [PubMed]

Kus, A.

W. Krauze, P. Makowski, M. Kujawińska, and A. Kuś, “Generalized total variation iterative constraint strategy in limited angle optical diffraction tomography,” Opt. Express 24(5), 4924–4936 (2016).
[Crossref]

J. Kostencka, T. Kozacki, A. Kuś, and M. Kujawińska, “Accurate approach to capillary-supported optical diffraction tomography,” Opt. Express 23(6), 7908–7923 (2015).
[Crossref] [PubMed]

A. Kuś, W. Krauze, and M. Kujawinska, “Active limited-angle tomographic phase microscope,” J. Biomed. Opt. 20(11), 111216 (2015).
[Crossref] [PubMed]

A. Kuś, M. Dudek, B. Kemper, M. Kujawińska, and A. Vollmer, “Tomographic phase microscopy of living three-dimensional cell cultures,” J. Biomed. Opt. 19(4), 046009 (2014).
[Crossref] [PubMed]

Lee, K.

Lee, S.

Liff, H. J.

A. J. Devaney, H. J. Liff, and S. Apsell, “Spectral representations for free space propagation of complex phase perturbations of optical fields,” Opt. Commun. 15(1), 1–5 (1975).
[Crossref]

Lim, J.

Magistretti, P.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marque, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7(2), 113–117 (2013).
[Crossref]

Makowski, P.

Marque, P.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marque, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7(2), 113–117 (2013).
[Crossref]

Marquet, P.

Michalska, A.

M. Dudek, M. Kujawińska, V. Parat, G. Baethge, A. Michalska, B. Dahmani, and H. Ottevaere, “Tomographic and numerical studies of polymer bridges between two optical fibers for telecommunication applications,” Opt. Eng. 53(1), 016113 (2014).
[Crossref]

Mitchell, E. A. D.

Osten, W.

Ottevaere, H.

M. Dudek, M. Kujawińska, V. Parat, G. Baethge, A. Michalska, B. Dahmani, and H. Ottevaere, “Tomographic and numerical studies of polymer bridges between two optical fibers for telecommunication applications,” Opt. Eng. 53(1), 016113 (2014).
[Crossref]

Pan, S.

S. Pan and A. Kak, “A computational study of reconstruction algorithms for diffraction tomography: Interpolation versus filtered-backpropagation,” IEEE Trans. Acoust. Speech Signal Process. 31(5), 1262–1275 (1983).
[Crossref]

Pan, X.

E. Y. Sidky, C.-M. Kao, and X. Pan, “Accuracte image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. Opt. Soc. Am. 25, 1772–1782 (2009).

Parat, V.

M. Dudek, M. Kujawińska, V. Parat, G. Baethge, A. Michalska, B. Dahmani, and H. Ottevaere, “Tomographic and numerical studies of polymer bridges between two optical fibers for telecommunication applications,” Opt. Eng. 53(1), 016113 (2014).
[Crossref]

Park, H.

Park, Y.

Pavillon, N.

Rappaz, B.

Roichman, Y.

Schäfer, M.

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11(3), 034005 (2006).
[Crossref] [PubMed]

Schnekenburger, J.

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11(3), 034005 (2006).
[Crossref] [PubMed]

Shaked, N. T.

Sheppard, C. J. R.

Shim, H.

Shin, S.

Sidky, E. Y.

E. Y. Sidky, C.-M. Kao, and X. Pan, “Accuracte image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. Opt. Soc. Am. 25, 1772–1782 (2009).

Singer, W.

So, P. T. C.

Stamnes, J. J.

Sung, Y.

Takeda, M.

Toy, F.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marque, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7(2), 113–117 (2013).
[Crossref]

Vollmer, A.

A. Kuś, M. Dudek, B. Kemper, M. Kujawińska, and A. Vollmer, “Tomographic phase microscopy of living three-dimensional cell cultures,” J. Biomed. Opt. 19(4), 046009 (2014).
[Crossref] [PubMed]

von Bally, G.

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11(3), 034005 (2006).
[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]

Yaqoob, Z.

Ye, J. C.

Yoon, J.

Appl. Opt. (5)

IEEE Trans. Acoust. Speech Signal Process. (1)

S. Pan and A. Kak, “A computational study of reconstruction algorithms for diffraction tomography: Interpolation versus filtered-backpropagation,” IEEE Trans. Acoust. Speech Signal Process. 31(5), 1262–1275 (1983).
[Crossref]

J. Biomed. Opt. (3)

A. Kuś, M. Dudek, B. Kemper, M. Kujawińska, and A. Vollmer, “Tomographic phase microscopy of living three-dimensional cell cultures,” J. Biomed. Opt. 19(4), 046009 (2014).
[Crossref] [PubMed]

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11(3), 034005 (2006).
[Crossref] [PubMed]

A. Kuś, W. Krauze, and M. Kujawinska, “Active limited-angle tomographic phase microscope,” J. Biomed. Opt. 20(11), 111216 (2015).
[Crossref] [PubMed]

J. Opt. Soc. Am. (2)

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982).
[Crossref]

E. Y. Sidky, C.-M. Kao, and X. Pan, “Accuracte image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. Opt. Soc. Am. 25, 1772–1782 (2009).

Nat. Photonics (1)

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marque, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7(2), 113–117 (2013).
[Crossref]

Opt. Commun. (2)

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

A. J. Devaney, H. J. Liff, and S. Apsell, “Spectral representations for free space propagation of complex phase perturbations of optical fields,” Opt. Commun. 15(1), 1–5 (1975).
[Crossref]

Opt. Eng. (2)

J. Kostencka and T. Kozacki, “Computational and experimental study on accuracy of off-axis reconstructions in optical diffraction tomography,” Opt. Eng. 54(2), 024107 (2015).
[Crossref]

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Opt. Express (7)

J. Lim, K. Lee, K. H. Jin, S. Shin, S. Lee, Y. Park, and J. C. Ye, “Comparative study of iterative reconstruction algorithms for missing cone problems in optical diffraction tomography,” Opt. Express 23(13), 16933–16948 (2015).
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[Crossref]

J. Kostencka, T. Kozacki, A. Kuś, and M. Kujawińska, “Accurate approach to capillary-supported optical diffraction tomography,” Opt. Express 23(6), 7908–7923 (2015).
[Crossref] [PubMed]

J. Kostencka, T. Kozacki, M. Dudek, and M. Kujawińska, “Noise suppressed optical diffraction tomography with autofocus correction,” Opt. Express 22(5), 5731–5745 (2014).
[Crossref] [PubMed]

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Opt. Lett. (5)

Proc. SPIE (1)

J. Kostencka and T. Kozacki, “Space-domain, filtered backpropagation algorithm for tomographic configuration with scanning of illumination,” Proc. SPIE 9890M, 9890 (2016).

Ultrason. Imaging (1)

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Supplementary Material (2)

NameDescription
» Visualization 1: AVI (4641 KB)      ISC-EDOF-FBPP reconstruction (xy-slices)
» Visualization 2: AVI (7744 KB)      xz reconstructions obtained with various methods

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Figures (15)

Fig. 1
Fig. 1 A scheme of ISC-HT; Ui – illuminating plane wave, Us – scattered wave.
Fig. 2
Fig. 2 Scheme of the direct inversion tomographic reconstruction algorithm for ISC-HT.
Fig. 3
Fig. 3 The results Π I S C - F B P P R obtained using (a,b) standard and (c,d) frequency-shifted Rytov propagation formulas for a cylinder sample of diameter 10µm and propagation direction tilted by 30° against the optical axis.
Fig. 4
Fig. 4 The backpropagated views obtained with standard (a,b) and the frequency-shifted (c,d) propagation formula.
Fig. 5
Fig. 5 Schematic illustration of the proposed ISC-FBPP.
Fig. 6
Fig. 6 Schematic illustration of the proposed ISC-EDOF-FBPP.
Fig. 7
Fig. 7 a) d) 3D distributions and b-c) e-f) cross-sections of tomographic transfer functions for the ISC-HT systems with: upper row – linear illumination scenario (the illumination vector ki rotates around y-axis in a range of ± 60°); lower row – conical illumination scenario (ki is inclined with respect to the optical (z) axis by 60°).
Fig. 8
Fig. 8 (a) Ideal reconstruction; (b-j) tomographic reconstructions of a cylindrical sample.
Fig. 9
Fig. 9 Central transverse cross-sections through the refractive index difference maps in Figs. 8(b)-8(j).
Fig. 10
Fig. 10 The reconstruction errors for various sample locations.
Fig. 11
Fig. 11 The ISC-HT measurement setup. Here, the vertical setup is demonstrated horizontally.
Fig. 12
Fig. 12 Transverse cross-sections of ISC-EDOF-FBPP reconstruction of the PaTu cell (See Visualization 1).
Fig. 13
Fig. 13 Axial cross-sections of the PaTu cell reconstructions obtained with DI, ISC-FBPP, ISC-EDOF-FBPP (see Visualization 2).
Fig. 14
Fig. 14 Fragments of the reconstructions of a PaTu cell obtained for different sample locations (zs); the RMSE error denotes discrepancy between the in-plane (zs = 0) and out-of-plane (|zs|>0) reconstructions.
Fig. 15
Fig. 15 The single Ewald sphere contribution: a) 3D view; b) cross-section at Kz = -k0n0cosαill.

Equations (19)

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φ ˜ s i ( k ) = n 0 k 0 / 2 k z O ˜ ( k k i ) ,
φ s i = i / ( k 0 n 0 ) u n w r a p [ ln ( U s / U i ) ] ,
O ( x , y , z ) = 1 n ( x , y , z ) 2 / n 0 2 .
Π O R C - F B P P R ( x , y ; z ) = e i n 0 k 0 z ( 2 π ) 2 φ ˜ s i ( k x , k y ) H ( k x , k y ; z ) exp { i ( k x x + k y y ) } d k x d k y ,
Π I S C - F B P P R ( x , y ; z ) = e i n 0 k i z z ( 2 π ) 2 φ ˜ s i ( k x ' , k y ' ) H ( k x ' , k y ' ; z ) exp { i [ k x ' x + k y ' y ] } d k x ' d k y ' .
Π I S C E D O F ( x , y ; z ) = e i n 0 k i z z ( 2 π ) 2 U ˜ s ( k x , k y ) H ( k x , k y ; z ) exp { i ( k x x + k y y ) } d k x d k y .
Π I S C E D O F ( x , y ; z ) = e i n 0 k i z z ( 2 π ) 2 U ˜ s i ( k x ' , k y ' ) H ( k x ' , k y ' ; z ) exp { i ( k x ' x + k y ' y ) } d k x ' d k y ' .
Π I S C E D O F R = i / ( k 0 n 0 ) u n w r a p [ ln ( Π I S C E D O F ) ] .
F T 1 { U ˜ s ( k x k i x , k y k i y ) } = U s ( x , y ) exp { i ( k i x x + k i y y ) } = U s i ( x , y )
F T 1 { φ ˜ s ( k x k i x , k y k i y ) } = φ s ( x , y ) exp { i ( k i x x + k i y y ) } φ s i ( x , y ) ,
O ˜ n o r m ( K ) = O ˜ ( K ) / c ( K ) .
R S M E = ( Δ n r e c Δ n i d e a l ) 2 / N ,
K yA = l 1 + l 2 = K St sin θ + sqrt ( R 2 K St 2 cos 2 θ ) ,
V A = δ K yA · δ k zA · δ R   / cos β A .
cos β A = K St cos θ / R ,
cos β A = 2 ( K y A 2 R 2 + K y A 2 K S t 2 + R 2 K S t 2 ) K y A 4 K S t 4 R 4 / 2 K y A R .
c ( K t , K z = k 0 n 0 cos α i l l ) = 2 K t R 2 ( K t 2 R 2 + K t 2 K S t 2 + R 2 K S t 2 ) K t 4 K S t 4 R 4 L 2 π K t
c = R L π 2 ( K t 2 R 2 + K t 2 k S t 2 + R 2 K S t 2 + K z 2 R 2 K t 2 K z 2 K S t 2 K z 2 ) K t 4 K S t 4 R 4 K z 4 ,
R 2 ( K t K S t ) 2 R cos α i l l > K z > R 2 ( K t + K S t ) 2 R cos α i l l .

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