Abstract

In water-window soft x-ray microscopy the studied object is typically larger than the depth of focus and the sample illumination is often partially coherent. This blurs out-of-focus features and may introduce considerable fringing. Understanding the influence of these phenomena on the image formation is therefore important when interpreting experimental data. Here we present a wave-propagation model operating in 3D for simulating the image formation of thick objects in partially coherent soft x-ray microscopes. The model is compared with present simulation methods as well as with experiments. The results show that our model predicts the image formation of transmission soft x-ray microscopes more accurately than previous models.

© 2014 Optical Society of America

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References

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2012 (3)

H. M. Hertz, O. von Hofsten, M. Bertilson, U. Vogt, A. Holmberg, J. Reinspach, D. Martz, M. Selin, A. E. Christakou, J. Jerlström-Hultqvist, and S. Svärd, “Laboratory cryo soft X-ray microscopy,” J. Struct. Biol. 177(2), 267–272 (2012).
[Crossref] [PubMed]

S. Rehbein, P. Guttmann, S. Werner, and G. Schneider, “Characterization of the resolving power and contrast transfer function of a transmission X-ray microscope with partially coherent illumination,” Opt. Express 20(6), 5830–5839 (2012).
[Crossref] [PubMed]

J. Oton, C. O. Sorzano, E. Pereiro, J. Cuenca-Alba, R. Navarro, J. M. Carazo, and R. Marabini, “Image formation in cellular X-ray microscopy,” J. Struct. Biol. 178(1), 29–37 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (2)

A. Sakdinawat and D. Attwood, “Nanoscale X-ray imaging,” Nat. Photonics 4(12), 840–848 (2010).
[Crossref]

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[Crossref] [PubMed]

2009 (2)

M. Uchida, G. McDermott, M. Wetzler, M. A. Le Gros, M. Myllys, C. Knoechel, A. E. Barron, and C. A. Larabell, “Soft X-ray tomography of phenotypic switching and the cellular response to antifungal peptoids in Candida albicans,” Proc. Natl. Acad. Sci. U.S.A. 106(46), 19375–19380 (2009).
[Crossref] [PubMed]

S. Trattner, M. Feigin, H. Greenspan, and N. Sochen, “Validity criterion for the Born approximation convergence in microscopy imaging,” J. Opt. Soc. Am. A 26(5), 1147–1156 (2009).
[Crossref] [PubMed]

2007 (2)

P. A. C. Takman, H. Stollberg, G. A. Johansson, A. Holmberg, M. Lindblom, and H. M. Hertz, “High-resolution compact X-ray microscopy,” J. Microsc. 226(2), 175–181 (2007).
[Crossref] [PubMed]

O. von Hofsten, P. A. Takman, and U. Vogt, “Simulation of partially coherent image formation in a compact soft X-ray microscope,” Ultramicroscopy 107(8), 604–609 (2007).
[Crossref] [PubMed]

2002 (1)

1998 (1)

G. Schneider, “Cryo X-ray microscopy with high spatial resolution in amplitude and phase contrast,” Ultramicroscopy 75(2), 85–104 (1998).
[Crossref] [PubMed]

1995 (1)

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, “Application of the Parabolic Wave-Equation to X-Ray-Diffraction Optics,” Opt. Commun. 118(5-6), 619–636 (1995).
[Crossref]

1991 (1)

1985 (1)

Attwood, D.

A. Sakdinawat and D. Attwood, “Nanoscale X-ray imaging,” Nat. Photonics 4(12), 840–848 (2010).
[Crossref]

Barron, A. E.

M. Uchida, G. McDermott, M. Wetzler, M. A. Le Gros, M. Myllys, C. Knoechel, A. E. Barron, and C. A. Larabell, “Soft X-ray tomography of phenotypic switching and the cellular response to antifungal peptoids in Candida albicans,” Proc. Natl. Acad. Sci. U.S.A. 106(46), 19375–19380 (2009).
[Crossref] [PubMed]

Bertilson, M.

H. M. Hertz, O. von Hofsten, M. Bertilson, U. Vogt, A. Holmberg, J. Reinspach, D. Martz, M. Selin, A. E. Christakou, J. Jerlström-Hultqvist, and S. Svärd, “Laboratory cryo soft X-ray microscopy,” J. Struct. Biol. 177(2), 267–272 (2012).
[Crossref] [PubMed]

M. Bertilson, O. von Hofsten, H. M. Hertz, and U. Vogt, “Numerical model for tomographic image formation in transmission x-ray microscopy,” Opt. Express 19(12), 11578–11583 (2011).
[Crossref] [PubMed]

Carazo, J. M.

J. Oton, C. O. Sorzano, E. Pereiro, J. Cuenca-Alba, R. Navarro, J. M. Carazo, and R. Marabini, “Image formation in cellular X-ray microscopy,” J. Struct. Biol. 178(1), 29–37 (2012).
[Crossref] [PubMed]

Christakou, A. E.

H. M. Hertz, O. von Hofsten, M. Bertilson, U. Vogt, A. Holmberg, J. Reinspach, D. Martz, M. Selin, A. E. Christakou, J. Jerlström-Hultqvist, and S. Svärd, “Laboratory cryo soft X-ray microscopy,” J. Struct. Biol. 177(2), 267–272 (2012).
[Crossref] [PubMed]

Cuenca-Alba, J.

J. Oton, C. O. Sorzano, E. Pereiro, J. Cuenca-Alba, R. Navarro, J. M. Carazo, and R. Marabini, “Image formation in cellular X-ray microscopy,” J. Struct. Biol. 178(1), 29–37 (2012).
[Crossref] [PubMed]

Feigin, M.

Greenspan, H.

Guttmann, P.

S. Rehbein, P. Guttmann, S. Werner, and G. Schneider, “Characterization of the resolving power and contrast transfer function of a transmission X-ray microscope with partially coherent illumination,” Opt. Express 20(6), 5830–5839 (2012).
[Crossref] [PubMed]

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[Crossref] [PubMed]

Hadley, G. R.

Heim, S.

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[Crossref] [PubMed]

Hertz, H. M.

H. M. Hertz, O. von Hofsten, M. Bertilson, U. Vogt, A. Holmberg, J. Reinspach, D. Martz, M. Selin, A. E. Christakou, J. Jerlström-Hultqvist, and S. Svärd, “Laboratory cryo soft X-ray microscopy,” J. Struct. Biol. 177(2), 267–272 (2012).
[Crossref] [PubMed]

M. Bertilson, O. von Hofsten, H. M. Hertz, and U. Vogt, “Numerical model for tomographic image formation in transmission x-ray microscopy,” Opt. Express 19(12), 11578–11583 (2011).
[Crossref] [PubMed]

P. A. C. Takman, H. Stollberg, G. A. Johansson, A. Holmberg, M. Lindblom, and H. M. Hertz, “High-resolution compact X-ray microscopy,” J. Microsc. 226(2), 175–181 (2007).
[Crossref] [PubMed]

Heymann, J. B.

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[Crossref] [PubMed]

Holmberg, A.

H. M. Hertz, O. von Hofsten, M. Bertilson, U. Vogt, A. Holmberg, J. Reinspach, D. Martz, M. Selin, A. E. Christakou, J. Jerlström-Hultqvist, and S. Svärd, “Laboratory cryo soft X-ray microscopy,” J. Struct. Biol. 177(2), 267–272 (2012).
[Crossref] [PubMed]

P. A. C. Takman, H. Stollberg, G. A. Johansson, A. Holmberg, M. Lindblom, and H. M. Hertz, “High-resolution compact X-ray microscopy,” J. Microsc. 226(2), 175–181 (2007).
[Crossref] [PubMed]

Jerlström-Hultqvist, J.

H. M. Hertz, O. von Hofsten, M. Bertilson, U. Vogt, A. Holmberg, J. Reinspach, D. Martz, M. Selin, A. E. Christakou, J. Jerlström-Hultqvist, and S. Svärd, “Laboratory cryo soft X-ray microscopy,” J. Struct. Biol. 177(2), 267–272 (2012).
[Crossref] [PubMed]

Johansson, G. A.

P. A. C. Takman, H. Stollberg, G. A. Johansson, A. Holmberg, M. Lindblom, and H. M. Hertz, “High-resolution compact X-ray microscopy,” J. Microsc. 226(2), 175–181 (2007).
[Crossref] [PubMed]

Knoechel, C.

M. Uchida, G. McDermott, M. Wetzler, M. A. Le Gros, M. Myllys, C. Knoechel, A. E. Barron, and C. A. Larabell, “Soft X-ray tomography of phenotypic switching and the cellular response to antifungal peptoids in Candida albicans,” Proc. Natl. Acad. Sci. U.S.A. 106(46), 19375–19380 (2009).
[Crossref] [PubMed]

Kopylov, Y. V.

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, “Application of the Parabolic Wave-Equation to X-Ray-Diffraction Optics,” Opt. Commun. 118(5-6), 619–636 (1995).
[Crossref]

Kurokhtin, A. N.

Larabell, C. A.

M. Uchida, G. McDermott, M. Wetzler, M. A. Le Gros, M. Myllys, C. Knoechel, A. E. Barron, and C. A. Larabell, “Soft X-ray tomography of phenotypic switching and the cellular response to antifungal peptoids in Candida albicans,” Proc. Natl. Acad. Sci. U.S.A. 106(46), 19375–19380 (2009).
[Crossref] [PubMed]

Le Gros, M. A.

M. Uchida, G. McDermott, M. Wetzler, M. A. Le Gros, M. Myllys, C. Knoechel, A. E. Barron, and C. A. Larabell, “Soft X-ray tomography of phenotypic switching and the cellular response to antifungal peptoids in Candida albicans,” Proc. Natl. Acad. Sci. U.S.A. 106(46), 19375–19380 (2009).
[Crossref] [PubMed]

Lindblom, M.

P. A. C. Takman, H. Stollberg, G. A. Johansson, A. Holmberg, M. Lindblom, and H. M. Hertz, “High-resolution compact X-ray microscopy,” J. Microsc. 226(2), 175–181 (2007).
[Crossref] [PubMed]

Marabini, R.

J. Oton, C. O. Sorzano, E. Pereiro, J. Cuenca-Alba, R. Navarro, J. M. Carazo, and R. Marabini, “Image formation in cellular X-ray microscopy,” J. Struct. Biol. 178(1), 29–37 (2012).
[Crossref] [PubMed]

Martz, D.

H. M. Hertz, O. von Hofsten, M. Bertilson, U. Vogt, A. Holmberg, J. Reinspach, D. Martz, M. Selin, A. E. Christakou, J. Jerlström-Hultqvist, and S. Svärd, “Laboratory cryo soft X-ray microscopy,” J. Struct. Biol. 177(2), 267–272 (2012).
[Crossref] [PubMed]

McDermott, G.

M. Uchida, G. McDermott, M. Wetzler, M. A. Le Gros, M. Myllys, C. Knoechel, A. E. Barron, and C. A. Larabell, “Soft X-ray tomography of phenotypic switching and the cellular response to antifungal peptoids in Candida albicans,” Proc. Natl. Acad. Sci. U.S.A. 106(46), 19375–19380 (2009).
[Crossref] [PubMed]

McNally, J. G.

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[Crossref] [PubMed]

Mueller, F.

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[Crossref] [PubMed]

Müller, W. G.

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[Crossref] [PubMed]

Myllys, M.

M. Uchida, G. McDermott, M. Wetzler, M. A. Le Gros, M. Myllys, C. Knoechel, A. E. Barron, and C. A. Larabell, “Soft X-ray tomography of phenotypic switching and the cellular response to antifungal peptoids in Candida albicans,” Proc. Natl. Acad. Sci. U.S.A. 106(46), 19375–19380 (2009).
[Crossref] [PubMed]

Nagashima, K.

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[Crossref] [PubMed]

Navarro, R.

J. Oton, C. O. Sorzano, E. Pereiro, J. Cuenca-Alba, R. Navarro, J. M. Carazo, and R. Marabini, “Image formation in cellular X-ray microscopy,” J. Struct. Biol. 178(1), 29–37 (2012).
[Crossref] [PubMed]

Oton, J.

J. Oton, C. O. Sorzano, E. Pereiro, J. Cuenca-Alba, R. Navarro, J. M. Carazo, and R. Marabini, “Image formation in cellular X-ray microscopy,” J. Struct. Biol. 178(1), 29–37 (2012).
[Crossref] [PubMed]

Pereiro, E.

J. Oton, C. O. Sorzano, E. Pereiro, J. Cuenca-Alba, R. Navarro, J. M. Carazo, and R. Marabini, “Image formation in cellular X-ray microscopy,” J. Struct. Biol. 178(1), 29–37 (2012).
[Crossref] [PubMed]

Popov, A. V.

A. N. Kurokhtin and A. V. Popov, “Simulation of high-resolution x-ray zone plates,” J. Opt. Soc. Am. A 19(2), 315–324 (2002).
[Crossref] [PubMed]

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, “Application of the Parabolic Wave-Equation to X-Ray-Diffraction Optics,” Opt. Commun. 118(5-6), 619–636 (1995).
[Crossref]

Rehbein, S.

S. Rehbein, P. Guttmann, S. Werner, and G. Schneider, “Characterization of the resolving power and contrast transfer function of a transmission X-ray microscope with partially coherent illumination,” Opt. Express 20(6), 5830–5839 (2012).
[Crossref] [PubMed]

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[Crossref] [PubMed]

Reinspach, J.

H. M. Hertz, O. von Hofsten, M. Bertilson, U. Vogt, A. Holmberg, J. Reinspach, D. Martz, M. Selin, A. E. Christakou, J. Jerlström-Hultqvist, and S. Svärd, “Laboratory cryo soft X-ray microscopy,” J. Struct. Biol. 177(2), 267–272 (2012).
[Crossref] [PubMed]

Sakdinawat, A.

A. Sakdinawat and D. Attwood, “Nanoscale X-ray imaging,” Nat. Photonics 4(12), 840–848 (2010).
[Crossref]

Schneider, G.

S. Rehbein, P. Guttmann, S. Werner, and G. Schneider, “Characterization of the resolving power and contrast transfer function of a transmission X-ray microscope with partially coherent illumination,” Opt. Express 20(6), 5830–5839 (2012).
[Crossref] [PubMed]

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[Crossref] [PubMed]

G. Schneider, “Cryo X-ray microscopy with high spatial resolution in amplitude and phase contrast,” Ultramicroscopy 75(2), 85–104 (1998).
[Crossref] [PubMed]

Selin, M.

H. M. Hertz, O. von Hofsten, M. Bertilson, U. Vogt, A. Holmberg, J. Reinspach, D. Martz, M. Selin, A. E. Christakou, J. Jerlström-Hultqvist, and S. Svärd, “Laboratory cryo soft X-ray microscopy,” J. Struct. Biol. 177(2), 267–272 (2012).
[Crossref] [PubMed]

Sochen, N.

Sorzano, C. O.

J. Oton, C. O. Sorzano, E. Pereiro, J. Cuenca-Alba, R. Navarro, J. M. Carazo, and R. Marabini, “Image formation in cellular X-ray microscopy,” J. Struct. Biol. 178(1), 29–37 (2012).
[Crossref] [PubMed]

Stollberg, H.

P. A. C. Takman, H. Stollberg, G. A. Johansson, A. Holmberg, M. Lindblom, and H. M. Hertz, “High-resolution compact X-ray microscopy,” J. Microsc. 226(2), 175–181 (2007).
[Crossref] [PubMed]

Streibl, N.

Svärd, S.

H. M. Hertz, O. von Hofsten, M. Bertilson, U. Vogt, A. Holmberg, J. Reinspach, D. Martz, M. Selin, A. E. Christakou, J. Jerlström-Hultqvist, and S. Svärd, “Laboratory cryo soft X-ray microscopy,” J. Struct. Biol. 177(2), 267–272 (2012).
[Crossref] [PubMed]

Takman, P. A.

O. von Hofsten, P. A. Takman, and U. Vogt, “Simulation of partially coherent image formation in a compact soft X-ray microscope,” Ultramicroscopy 107(8), 604–609 (2007).
[Crossref] [PubMed]

Takman, P. A. C.

P. A. C. Takman, H. Stollberg, G. A. Johansson, A. Holmberg, M. Lindblom, and H. M. Hertz, “High-resolution compact X-ray microscopy,” J. Microsc. 226(2), 175–181 (2007).
[Crossref] [PubMed]

Trattner, S.

Uchida, M.

M. Uchida, G. McDermott, M. Wetzler, M. A. Le Gros, M. Myllys, C. Knoechel, A. E. Barron, and C. A. Larabell, “Soft X-ray tomography of phenotypic switching and the cellular response to antifungal peptoids in Candida albicans,” Proc. Natl. Acad. Sci. U.S.A. 106(46), 19375–19380 (2009).
[Crossref] [PubMed]

Vinogradov, A. V.

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, “Application of the Parabolic Wave-Equation to X-Ray-Diffraction Optics,” Opt. Commun. 118(5-6), 619–636 (1995).
[Crossref]

Vogt, U.

H. M. Hertz, O. von Hofsten, M. Bertilson, U. Vogt, A. Holmberg, J. Reinspach, D. Martz, M. Selin, A. E. Christakou, J. Jerlström-Hultqvist, and S. Svärd, “Laboratory cryo soft X-ray microscopy,” J. Struct. Biol. 177(2), 267–272 (2012).
[Crossref] [PubMed]

M. Bertilson, O. von Hofsten, H. M. Hertz, and U. Vogt, “Numerical model for tomographic image formation in transmission x-ray microscopy,” Opt. Express 19(12), 11578–11583 (2011).
[Crossref] [PubMed]

O. von Hofsten, P. A. Takman, and U. Vogt, “Simulation of partially coherent image formation in a compact soft X-ray microscope,” Ultramicroscopy 107(8), 604–609 (2007).
[Crossref] [PubMed]

von Hofsten, O.

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Supplementary Material (1)

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Figures (5)

Fig. 1
Fig. 1 Schematic representation of the simulation model. The illumination is decomposed to plane waves that are individually defined by their wave vector k. In the wave propagation step the electric field Ek is propagated forward in the z-direction for each incident plane wave Ek(za). In the imaging step the output Ek(zb) from the wave propagation step is imaged coherently using fourier optics methods to the detector plane. The final partially coherent image is formed by taking the sum over all coherent images weighted with the illumination intensity.
Fig. 2
Fig. 2 Comparison of computational models. (a) shows the original 2D phantom of [15] with mylar discs, (b) shows the extended 2D phantom with cylinders, and (c) shows the full 3D phantom with mylar spheres. (d) depicts the result of a defocus series when the algorithm of [15] is applied to the original 2D phantom. (e) shows the defocus-dependent intensity at x = 0 when the present 3D model is applied to the extended 2D phantom. In (f) the same calculation is performed for the full 3D object, indicating significant differences compared to the 2D cases. For comparison, the method of [16] is applied to the full 3D phantom and plotted at x = 0 in (g). Finally, in (h) and (j) the four methods are compared quantitatively. The magenta lines in (h) shows how the intensity varies in the center of a mylar feature as a function of defocus, while the blue, green and red in (i) are y-profiles at different defocus positions. “Normalized intensity” here refers to the ratio between the image intensity with and without object in the calculation volume.
Fig. 3
Fig. 3 Measured brightness after the capillary condenser and the central stop at the XRM at HZB. The areas outside of the dashed line were outside the field of view during the measurement and their contents have been extrapolated. θx and θy are the angles the illumination makes with the optical axis in the x- and y-direction.
Fig. 4
Fig. 4 (a) Experimentally acquired images on two overlapping diatom frustules for defocus position: −5, 0 and +5 µm. The red rectangles indicate the pore which is studied in greater detail in Fig. 5. (b) Renderings of a phantom designed to have similar coarse and fine structure and material composition as part of the frustule-pair used in the experiments. (c) Images from simulations of the image formation using our 3D wave-propagation model.
Fig. 5
Fig. 5 Focus series of a pore in the diatom frustule from the experimental data (a) and our 3D wave-propagation method’s simulation of the same pore (b). (c) shows the simulated focus series of the pore using the PSF-enhanced method and (d) shows it using ideal projections. Note that the defocus is relative to the diatom center and not the pore, which is why the more focused images are found at the more negative defocus positions.

Equations (14)

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I D ( Mx,My )= 1 M 2 I( x,y, z a )×exp[ z a z b μ(x,y,z)dz ],
I D (Mx,My)= I zi (Mx,My) z a z b ( μ(x,y,z) I zi (Mx,My) e z a z μ(x,y,ξ)dξ ) x,y h(x,y,D(z)) dz,
E k (r,t)= E 0 k exp( j( k 0 zωt) ) u k (r),
u z = j 2 k 0 ( x,y 2 + k 0 2 χ )u,
u n+1 = u n + 1 6 ( k 1 +2 k 2 +2 k 3 + k 4 )+TB C n ,
k 1 =f( z n , u n ) k 3 =f( z n + Δz 2 , u n + Δz 2 k 2 ) k 2 =f( z n + Δz 2 , u n + Δz 2 k 1 ) k 4 =f( z n +Δz, u n +Δz k 3 ) ,
E D,k (Mx,My)= E 0 k K k (Mx,My)= E 0 k 1 M 1 { { u k ( z b ) }×A },
A( ν x , ν y )={ exp( jπλ( ν x 2 + ν y 2 ) f 1 +Δ f 1 ) , ν x 2 + ν y 2 N A zp /λ 0 , ν x 2 + ν y 2 >N A zp /λ ,
I D = k I 0 k | K k | 2 ,
J( P 1 , P 2 )= E( P 1 ,t)E ( P 2 ,t) ,
μ( P 1 , P 2 )= J( P 1 , P 2 ) I( P 1 ) I( P 2 ) ,
J( Q 1 , Q 2 )= A A J( P 1 , P 2 )K( P 1 , Q 1 )K ( P 2 , Q 2 ) d P 1 d P 2 ,
I(Q)= A A μ( P 1 , P 2 )[ I( P 1 ) K( P 1 ,Q) ] [ I( P 2 ) K( P 2 ,Q) ] d P 1 d P 2 .
I(Q)= k h μ( P k , P h )( E D,k (Q)× E D,h (Q) ) .

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