Abstract

In this paper, we consider computational super-resolution inverse diffraction phase retrieval. The optical setup is lensless, with a spatial light modulator for aperture phase coding. The paper is focused on experimental tests of the super-resolution sparse phase amplitude retrieval algorithm. We start from simulations and proceed to physical experiments. Both simulation tests and experiments demonstrate good-quality imaging for super-resolution with a factor of 4 and a serious advantage over diffraction-limited resolution as defined by Abbe’s criterion.

© 2017 Optical Society of America

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2017 (1)

Q. Lian, B. Shi, and S. Chen, “Transfer orthogonal sparsifying transform learning for phase retrieval,” Digit Signal Process. 62, 11–25 (2017).
[Crossref]

2016 (3)

2015 (6)

E. J. Candès, X. Lib, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).
[Crossref]

C. Guo, S. Liu, and J. T. Sheridan, “Iterative phase retrieval algorithms. I: optimization,” Appl. Opt 54, 4698–4708 (2015).
[Crossref]

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory, 61, 1985–2007 (2015).
[Crossref]

E. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).

K. Guo, S. Dong, P. Nanda, and G. Zheng, “Optimization of sampling pattern and the design of Fourier ptychographic illuminator,” Opt. Express 23, 6171–6180 (2015).
[Crossref]

2013 (2)

A. Greenbaum, W. Luo, B. Khademhosseinieh, T. W. Su, A. F. Coskun, and A. Ozcan, “Increased space-bandwidth product in pixel super-resolved lensfree on-chip microscopy,” Sci. Rep. 3, 1717 (2013).
[Crossref]

V. Katkovnik and J. Astola, “Sparse ptychographical coherent diffractive imaging from noisy measurements,” J. Opt. Soc. Am. A 30, 367–379 (2013).
[Crossref]

2012 (4)

V. Katkovnik and J. Astola, “High-accuracy wavefield reconstruction: decoupled inverse imaging with sparse modeling of phase and amplitude,” J. Opt. Soc. Am. A 29, 44–54 (2012).
[Crossref]

A. Fannjiang, “Absolute uniqueness of phase retrieval with random illumination,” Inverse Probl. 28, 075008 (2012).
[Crossref]

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
[Crossref]

M. Agour, P. Almoro, and C. Falldorf, “Investigation of smooth wave fronts using SLM-based phase retrieval and a phase diffuser,” J. Eur. Opt. Soc. Rapid. Publ. 7, 12046 (2012).
[Crossref]

2011 (3)

P. Almoro, G. Pedrini, P. N. Gundu, W. Osten, and S. G. Hanson, “Enhanced wavefront reconstruction by random phase modulation with a phase diffuser,” Opt. Lasers Eng. 49, 252–257 (2011).
[Crossref]

N. Verrier and M. Atlan, “Off-axis digital hologram reconstruction: some practical considerations,” Appl. Opt. 50, H136–H146 (2011).
[Crossref]

W. Bishara, U. Sikora, O. Mudanyali, T. W. Su, O. Yaglidere, S. Luckhart, and A. Ozcan, “Holographic pixel super-resolution in portable lensless on-chip microscopy using a fiber-optic array,” Lab Chip 11, 1276–1279 (2011).
[Crossref]

2010 (2)

A. F. Doval and C. Trillo, “Dimensionless formulation of the convolution and angular spectrum reconstruction methods in digital holography,” Proc. SPIE 7387, 73870U (2010).
[Crossref]

L. Camacho, V. Micy, Z. Zalevsky, and J. Garcha, “Quantitative phase microscopy using defocussing by means of a spatial light modulator,” Opt. Express 18, 6755–6766 (2010).
[Crossref]

2009 (3)

2007 (3)

J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[Crossref]

P. Almoro, G. Pedrini, and W. Osten, “Aperture synthesis in phase retrieval using a volume-speckle field,” Opt. Lett. 32, 733–735 (2007).
[Crossref]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref]

2006 (2)

2005 (1)

1982 (1)

1980 (1)

W. O. Saxton, “Correction of artefacts in linear and nonlinear high resolution electron micrographs,” J. Microsc. Spectrosc. Electron. 5, 665–674 (1980).

1973 (1)

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations,” J. Phys. D 6, 2200–2216 (1973).
[Crossref]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Agour, M.

M. Agour, P. Almoro, and C. Falldorf, “Investigation of smooth wave fronts using SLM-based phase retrieval and a phase diffuser,” J. Eur. Opt. Soc. Rapid. Publ. 7, 12046 (2012).
[Crossref]

Almoro, P.

Ambs, P.

Astola, J.

Atlan, M.

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698–709 (2007).
[Crossref]

Bishara, W.

W. Bishara, U. Sikora, O. Mudanyali, T. W. Su, O. Yaglidere, S. Luckhart, and A. Ozcan, “Holographic pixel super-resolution in portable lensless on-chip microscopy using a fiber-optic array,” Lab Chip 11, 1276–1279 (2011).
[Crossref]

Camacho, L.

Candès, E.

E. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).

Candès, E. J.

E. J. Candès, X. Lib, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).
[Crossref]

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory, 61, 1985–2007 (2015).
[Crossref]

Chapman, H. N.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Chen, S.

Q. Lian, B. Shi, and S. Chen, “Transfer orthogonal sparsifying transform learning for phase retrieval,” Digit Signal Process. 62, 11–25 (2017).
[Crossref]

Cohen, O.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Colicchio, B.

Coskun, A. F.

A. Greenbaum, W. Luo, B. Khademhosseinieh, T. W. Su, A. F. Coskun, and A. Ozcan, “Increased space-bandwidth product in pixel super-resolved lensfree on-chip microscopy,” Sci. Rep. 3, 1717 (2013).
[Crossref]

Dabov, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref]

Danielyan, A.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
[Crossref]

Dieterlen, A.

Dong, S.

Doval, A. F.

A. F. Doval and C. Trillo, “Dimensionless formulation of the convolution and angular spectrum reconstruction methods in digital holography,” Proc. SPIE 7387, 73870U (2010).
[Crossref]

Egiazarian, K.

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
[Crossref]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref]

Eldar, Y. C.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17, 23920–23946 (2009).
[Crossref]

Falldorf, C.

M. Agour, P. Almoro, and C. Falldorf, “Investigation of smooth wave fronts using SLM-based phase retrieval and a phase diffuser,” J. Eur. Opt. Soc. Rapid. Publ. 7, 12046 (2012).
[Crossref]

Fannjiang, A.

A. Fannjiang, “Absolute uniqueness of phase retrieval with random illumination,” Inverse Probl. 28, 075008 (2012).
[Crossref]

Fienup, J. R.

Foi, A.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref]

Garcha, J.

Gazit, S.

Gemayel, P.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

Greenbaum, A.

A. Greenbaum, W. Luo, B. Khademhosseinieh, T. W. Su, A. F. Coskun, and A. Ozcan, “Increased space-bandwidth product in pixel super-resolved lensfree on-chip microscopy,” Sci. Rep. 3, 1717 (2013).
[Crossref]

Gundu, P. N.

P. Almoro, G. Pedrini, P. N. Gundu, W. Osten, and S. G. Hanson, “Enhanced wavefront reconstruction by random phase modulation with a phase diffuser,” Opt. Lasers Eng. 49, 252–257 (2011).
[Crossref]

Guo, C.

C. Guo, S. Liu, and J. T. Sheridan, “Iterative phase retrieval algorithms. I: optimization,” Appl. Opt 54, 4698–4708 (2015).
[Crossref]

Guo, K.

Hanson, S.

Hanson, S. G.

P. Almoro, G. Pedrini, P. N. Gundu, W. Osten, and S. G. Hanson, “Enhanced wavefront reconstruction by random phase modulation with a phase diffuser,” Opt. Lasers Eng. 49, 252–257 (2011).
[Crossref]

Im, H.

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681 (2016).
[Crossref]

Iwamoto, Y.

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681 (2016).
[Crossref]

Jeong, S.

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681 (2016).
[Crossref]

Katkovnik, V.

V. Katkovnik and J. Astola, “Sparse ptychographical coherent diffractive imaging from noisy measurements,” J. Opt. Soc. Am. A 30, 367–379 (2013).
[Crossref]

V. Katkovnik and J. Astola, “High-accuracy wavefield reconstruction: decoupled inverse imaging with sparse modeling of phase and amplitude,” J. Opt. Soc. Am. A 29, 44–54 (2012).
[Crossref]

A. Danielyan, V. Katkovnik, and K. Egiazarian, “BM3D frames and variational image deblurring,” IEEE Trans. Image Process. 21, 1715–1728 (2012).
[Crossref]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref]

Khademhosseinieh, B.

A. Greenbaum, W. Luo, B. Khademhosseinieh, T. W. Su, A. F. Coskun, and A. Ozcan, “Increased space-bandwidth product in pixel super-resolved lensfree on-chip microscopy,” Sci. Rep. 3, 1717 (2013).
[Crossref]

Kohler, C.

Kress, B.

B. Kress and P. Meyrueis, Applied Digital Optics: from Micro-Optics to Nanooptics (Wiley, 2009).

Lee, H.

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681 (2016).
[Crossref]

Li, X.

E. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via Wirtinger flow: theory and algorithms,” IEEE Trans. Inf. Theory, 61, 1985–2007 (2015).
[Crossref]

Lian, Q.

Q. Lian, B. Shi, and S. Chen, “Transfer orthogonal sparsifying transform learning for phase retrieval,” Digit Signal Process. 62, 11–25 (2017).
[Crossref]

Lib, X.

E. J. Candès, X. Lib, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).
[Crossref]

Liu, S.

C. Guo, S. Liu, and J. T. Sheridan, “Iterative phase retrieval algorithms. I: optimization,” Appl. Opt 54, 4698–4708 (2015).
[Crossref]

Luckhart, S.

W. Bishara, U. Sikora, O. Mudanyali, T. W. Su, O. Yaglidere, S. Luckhart, and A. Ozcan, “Holographic pixel super-resolution in portable lensless on-chip microscopy using a fiber-optic array,” Lab Chip 11, 1276–1279 (2011).
[Crossref]

Luo, W.

A. Greenbaum, W. Luo, B. Khademhosseinieh, T. W. Su, A. F. Coskun, and A. Ozcan, “Increased space-bandwidth product in pixel super-resolved lensfree on-chip microscopy,” Sci. Rep. 3, 1717 (2013).
[Crossref]

Maallo, A. M. S.

Meyrueis, P.

B. Kress and P. Meyrueis, Applied Digital Optics: from Micro-Optics to Nanooptics (Wiley, 2009).

Miao, J.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Micy, V.

Misell, D. L.

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations,” J. Phys. D 6, 2200–2216 (1973).
[Crossref]

Mitsuo, T.

Mudanyali, O.

W. Bishara, U. Sikora, O. Mudanyali, T. W. Su, O. Yaglidere, S. Luckhart, and A. Ozcan, “Holographic pixel super-resolution in portable lensless on-chip microscopy using a fiber-optic array,” Lab Chip 11, 1276–1279 (2011).
[Crossref]

Nanda, P.

Osten, W.

Ozcan, A.

A. Greenbaum, W. Luo, B. Khademhosseinieh, T. W. Su, A. F. Coskun, and A. Ozcan, “Increased space-bandwidth product in pixel super-resolved lensfree on-chip microscopy,” Sci. Rep. 3, 1717 (2013).
[Crossref]

W. Bishara, U. Sikora, O. Mudanyali, T. W. Su, O. Yaglidere, S. Luckhart, and A. Ozcan, “Holographic pixel super-resolution in portable lensless on-chip microscopy using a fiber-optic array,” Lab Chip 11, 1276–1279 (2011).
[Crossref]

Pathania, D.

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681 (2016).
[Crossref]

Pedrini, G.

Pivovarov, M.

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681 (2016).
[Crossref]

Quang Duc, D. P.

Satoshi, H.

Saxton, W. O.

W. O. Saxton, “Correction of artefacts in linear and nonlinear high resolution electron micrographs,” J. Microsc. Spectrosc. Electron. 5, 665–674 (1980).

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Segev, M.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17, 23920–23946 (2009).
[Crossref]

Serrano-Garcia David, I.

Shechtman, Y.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Sheridan, J. T.

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J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681 (2016).
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Appl. Comput. Harmon. Anal. (2)

E. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).

E. J. Candès, X. Lib, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Anal. 39, 277–299 (2015).
[Crossref]

Appl. Opt (1)

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[Crossref]

Appl. Opt. (6)

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J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681 (2016).
[Crossref]

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Matlab Demo code, http://dx.doi.org/10.6084/m9.figshare.4833932.

Supplementary Material (1)

NameDescription
» Code 1       Matlab Demo code

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Figures (15)

Fig. 1.
Fig. 1.

Flow chart of the SR-SPAR algorithm.

Fig. 2.
Fig. 2.

Optical setup. λ4, retardation plate; P, polarizer; L1,L2, lenses; BS, beamsplitter; SLM, spatial light modulator; CMOS, registration camera.

Fig. 3.
Fig. 3.

Simulation—“cameraman” test image, pixel-wise resolution: rS=1, ΔAbbe/Δc=4.8, RMSEphase=0.00253.

Fig. 4.
Fig. 4.

Simulation—logo “TUT” test image, pixel-wise resolution: rS=1, ΔAbbe/Δc=4.8, RMSEphase=0.000852.

Fig. 5.
Fig. 5.

Simulation—“cameraman” test image, sub-pixel resolution: rS=4, ΔAbbe/Δc=38.4, RMSEphase=0.00228.

Fig. 6.
Fig. 6.

Simulation—logo “TUT” test image, sub-pixel resolution: rS=4, ΔAbbe/Δc=38.4, RMSEphase=0.000592.

Fig. 7.
Fig. 7.

Experiment—phase object “cameraman,” pixel-wise resolution: rS=1, ΔAbbe/Δc=4.8.

Fig. 8.
Fig. 8.

Experiment—phase object logo “TUT,” pixel-wise resolution: rS=1, ΔAbbe/Δc=4.8.

Fig. 9.
Fig. 9.

Experiment—phase object “cameraman,” sub-pixel resolution: rS=4, ΔAbbe/Δc=38.4.

Fig. 10.
Fig. 10.

Experiment—phase object logo “TUT,” sub-pixel resolution: rS=4, ΔAbbe/Δc=38.4.

Fig. 11.
Fig. 11.

Phase RMSE dependence on the number of observations L.

Fig. 12.
Fig. 12.

Phase RMSE dependence on the standard deviation σ of zero-mean Gaussian random phase masks in Eq. (11).

Fig. 13.
Fig. 13.

Phase RMSE dependence on SLM superpixel size M.

Fig. 14.
Fig. 14.

Phase RMSE dependence on iteration number T.

Fig. 15.
Fig. 15.

Simulation. USAF test chart phase image with different propagation distances, sub-pixel resolution, and rS=4. Top row, phase images—on the left and right are reconstructions for z=21.55  mm and z=1  mm, respectively, while the original phase is in the middle. Bottom row—longitudinal cross sections: left for areas 2–4 of the USAF test chart (horizontal cross section of vertical lines) and right for areas 3–5 of the USAF test chart (vertical cross section of horizontal lines). The arrows (red) on the original phase image show the locations of the cross sections.

Equations (15)

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ys=|Ps{uo}|2,s=1,,L,
us=Ps{uo}=P{Msuo}.
ys=|P{Msuo}|2,s=1,,L.
zs=G{|us|2},s=1,,L,
us(x,y,z)=F1[H(fx,fy,z)·F[us(x,y,0)]],
H(fx,fy,z)={exp[i2πλz1λ2(fx2+fy2)],fx2+fy21λ2,0otherwise,
p(zs[l]=k)=exp(ys[l]χ)(ys[l]χ)kk!,
φ^o=BM3Dphase(φo,thφ),
B^o=BM3Dampl(Bo,thB).
ϕn,m(s)=ϵn,m(s),
Ms(n,m)=exp(jϕn,m(s)·σ),
zzmax=NΔc2/λ,
L=s=1Lk,l[|us[k,l]|2χzs[k,l]log(|us[k,l]|2χ)].
bs=|vs|+|vs|2+4z˜sγ1(1+γ1χ)2(1+γ1χ).
ΔAbbe=λNA2zλNΔS,

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