Abstract

We present new methods for optimizing the spatial bandwidth capacity in off-axis holography using spatial multiplexing. We use optimal spatial multiplexing of off-axis holograms to fill the entire spatial frequency domain, including the space previously occupied by the intensity of the sample. Our approach enables spatial digital compression of eight off-axis holograms into a single real-valued multiplexed hologram, having the same number of pixels as each of the input holograms, but still allowing their full reconstruction without resolution or magnification loss in the reconstructed complex wave fronts. This new method allows 33% improvement in usage of the spatial bandwidth capacity compared to the best available off-axis holography real-value multiplexing method. Since the output multiplexed hologram contains only real values, it can be used for rapid display of eight wave front reconstructions at once, which is useful for real-time visualization, when the hologram display device is slower than the acquiring camera. We further generalize this technique to digital multiplexing of 16 real-valued holograms into a single complex-valued hologram by simple arithmetic operations in the hologram domain. Then, the extraction of the 16 wave fronts includes a single 2-D discrete Fourier transform to access the spatial frequency domain, allowing fast reconstruction, which is useful for real-time processing of off-axis holograms, with improved processing rate compared to current hologram processing algorithms. These new approaches allow the full reconstruction of all compressed data without loss of resolution or magnification, even though the samples are dense such that their frequency content employs the entire range. Both multiplexing architectures are then demonstrated for experimentally-acquired off-axis holograms for quantitative phase imaging of biological cells.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article

Corrections

Gili Dardikman, Nir A. Turko, Noa Nativ, Simcha K. Mirsky, and Natan T. Shaked, "Optimal spatial bandwidth capacity in multiplexed off-axis holography for rapid quantitative phase reconstruction and visualization: erratum," Opt. Express 26, 20848-20848 (2018)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-26-16-20848

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References

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    [PubMed]
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    [PubMed]

2018 (1)

A. V. Zea, J. F. Barrera, and R. Torroba, “Cross-talk free selective reconstruction of individual objects from multiplexed optical field data,” Opt. Lasers Eng. 100, 90–97 (2018).

2017 (3)

Z. Zhong, H. Bai, M. Shan, Y. Zhang, and L. Guo, “Fast phase retrieval in slightly off-axis holography,” Opt. Lasers Eng. 97, 9–18 (2017).

C. Rosales-Guzmán, N. Bhebhe, N. Mahonisi, and A. Forbes, “Multiplexing 200 spatial modes with a single hologram,” J. Opt. 19(11), 113501 (2017).

N. A. Turko and N. T. Shaked, “Simultaneous two-wavelength phase unwrapping using an external module for multiplexing off-axis holography,” Opt. Lett. 42(1), 73–76 (2017).
[PubMed]

2016 (2)

2015 (4)

T. Tahara, R. Mori, Y. Arai, and Y. Takaki, “Four-step phase-shifting digital holography simultaneously sensing dual-wavelength information using a monochromatic image sensor,” J. Opt. 17, 125707 (2015).

S. Karepov, N. T. Shaked, and T. Ellenbogen, “Off-axis interferometer with adjustable fringe contrast based on polarization encoding,” Opt. Lett. 40(10), 2273–2276 (2015).
[PubMed]

P. Girshovitz, I. Frenklach, and N. T. Shaked, “Broadband quantitative phase microscopy with extended field of view using off-axis interferometric multiplexing,” J. Biomed. Opt. 20(11), 111217 (2015).
[PubMed]

P. Girshovitz and N. T. Shaked, “Fast phase processing in off-axis holography using multiplexing with complex encoding and live-cell fluctuation map calculation in real-time,” Opt. Express 23(7), 8773–8787 (2015).
[PubMed]

2014 (3)

2013 (3)

Y. Wu, Y. Yang, H. Zhai, Z. Ma, L. Deng, and Q. Ge, “Single-exposure approach for expanding the sampled area of a dynamic process by digital holography with combined multiplexing,” J. Opt. 15(8), 085402 (2013).

P. Girshovitz and N. T. Shaked, “Compact and portable low-coherence interferometer with off-axis geometry for quantitative phase microscopy and nanoscopy,” Opt. Express 21(5), 5701–5714 (2013).
[PubMed]

T. Tahara, Y. Awatsuji, K. Nishio, S. Ura, O. Matoba, and T. Kubota, “Space-bandwidth capacity-enhanced digital holography,” Appl. Phys. Express 6(2), 022502 (2013).

2012 (1)

2009 (2)

2008 (1)

2007 (1)

2006 (1)

2005 (1)

2002 (1)

1994 (1)

1965 (1)

Arai, Y.

T. Tahara, R. Mori, Y. Arai, and Y. Takaki, “Four-step phase-shifting digital holography simultaneously sensing dual-wavelength information using a monochromatic image sensor,” J. Opt. 17, 125707 (2015).

Awatsuji, Y.

T. Tahara, Y. Awatsuji, K. Nishio, S. Ura, O. Matoba, and T. Kubota, “Space-bandwidth capacity-enhanced digital holography,” Appl. Phys. Express 6(2), 022502 (2013).

Bai, H.

Z. Zhong, H. Bai, M. Shan, Y. Zhang, and L. Guo, “Fast phase retrieval in slightly off-axis holography,” Opt. Lasers Eng. 97, 9–18 (2017).

Barrera, J. F.

A. V. Zea, J. F. Barrera, and R. Torroba, “Cross-talk free selective reconstruction of individual objects from multiplexed optical field data,” Opt. Lasers Eng. 100, 90–97 (2018).

Bhebhe, N.

C. Rosales-Guzmán, N. Bhebhe, N. Mahonisi, and A. Forbes, “Multiplexing 200 spatial modes with a single hologram,” J. Opt. 19(11), 113501 (2017).

Burton, D. R.

Charrière, F.

Colomb, T.

Cuche, E.

Deng, L.

Y. Wu, Y. Yang, H. Zhai, Z. Ma, L. Deng, and Q. Ge, “Single-exposure approach for expanding the sampled area of a dynamic process by digital holography with combined multiplexing,” J. Opt. 15(8), 085402 (2013).

Depeursinge, C.

Dürr, F.

Ellenbogen, T.

Emery, Y.

Ferraro, P.

Finizio, A.

Forbes, A.

C. Rosales-Guzmán, N. Bhebhe, N. Mahonisi, and A. Forbes, “Multiplexing 200 spatial modes with a single hologram,” J. Opt. 19(11), 113501 (2017).

Frenklach, I.

P. Girshovitz, I. Frenklach, and N. T. Shaked, “Broadband quantitative phase microscopy with extended field of view using off-axis interferometric multiplexing,” J. Biomed. Opt. 20(11), 111217 (2015).
[PubMed]

I. Frenklach, P. Girshovitz, and N. T. Shaked, “Off-axis interferometric phase microscopy with tripled imaging area,” Opt. Lett. 39(6), 1525–1528 (2014).
[PubMed]

Gdeisat, M. A.

Ge, Q.

Y. Wu, Y. Yang, H. Zhai, Z. Ma, L. Deng, and Q. Ge, “Single-exposure approach for expanding the sampled area of a dynamic process by digital holography with combined multiplexing,” J. Opt. 15(8), 085402 (2013).

Ge, X. L.

Girshovitz, P.

Guo, C. S.

Guo, L.

Z. Zhong, H. Bai, M. Shan, Y. Zhang, and L. Guo, “Fast phase retrieval in slightly off-axis holography,” Opt. Lasers Eng. 97, 9–18 (2017).

Herráez, M. A.

Javidi, B.

Karepov, S.

Kubota, T.

T. Tahara, Y. Awatsuji, K. Nishio, S. Ura, O. Matoba, and T. Kubota, “Space-bandwidth capacity-enhanced digital holography,” Appl. Phys. Express 6(2), 022502 (2013).

Kühn, J.

Lalor, M. J.

Limberger, H. G.

Liu, X.

Lohmann, A. W.

Lu, Y. J.

Ma, Z.

Y. Wu, Y. Yang, H. Zhai, Z. Ma, L. Deng, and Q. Ge, “Single-exposure approach for expanding the sampled area of a dynamic process by digital holography with combined multiplexing,” J. Opt. 15(8), 085402 (2013).

Mahonisi, N.

C. Rosales-Guzmán, N. Bhebhe, N. Mahonisi, and A. Forbes, “Multiplexing 200 spatial modes with a single hologram,” J. Opt. 19(11), 113501 (2017).

Marquet, P.

Matoba, O.

T. Tahara, Y. Awatsuji, K. Nishio, S. Ura, O. Matoba, and T. Kubota, “Space-bandwidth capacity-enhanced digital holography,” Appl. Phys. Express 6(2), 022502 (2013).

Memmolo, P.

Miccio, L.

Montfort, F.

Mori, R.

T. Tahara, R. Mori, Y. Arai, and Y. Takaki, “Four-step phase-shifting digital holography simultaneously sensing dual-wavelength information using a monochromatic image sensor,” J. Opt. 17, 125707 (2015).

Mu, G.

Nishio, K.

T. Tahara, Y. Awatsuji, K. Nishio, S. Ura, O. Matoba, and T. Kubota, “Space-bandwidth capacity-enhanced digital holography,” Appl. Phys. Express 6(2), 022502 (2013).

Ohtsuka, Y.

Oka, K.

Paturzo, M.

Rinehart, M. T.

Roitshtain, D.

Rosales-Guzmán, C.

C. Rosales-Guzmán, N. Bhebhe, N. Mahonisi, and A. Forbes, “Multiplexing 200 spatial modes with a single hologram,” J. Opt. 19(11), 113501 (2017).

Salathé, R.-P.

Saleh, B. A. E.

B. A. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

Sha, B.

Shaked, N. T.

N. A. Turko and N. T. Shaked, “Simultaneous two-wavelength phase unwrapping using an external module for multiplexing off-axis holography,” Opt. Lett. 42(1), 73–76 (2017).
[PubMed]

D. Roitshtain, N. A. Turko, B. Javidi, and N. T. Shaked, “Flipping interferometry and its application for quantitative phase microscopy in a micro-channel,” Opt. Lett. 41(10), 2354–2357 (2016).
[PubMed]

S. Karepov, N. T. Shaked, and T. Ellenbogen, “Off-axis interferometer with adjustable fringe contrast based on polarization encoding,” Opt. Lett. 40(10), 2273–2276 (2015).
[PubMed]

P. Girshovitz and N. T. Shaked, “Fast phase processing in off-axis holography using multiplexing with complex encoding and live-cell fluctuation map calculation in real-time,” Opt. Express 23(7), 8773–8787 (2015).
[PubMed]

P. Girshovitz, I. Frenklach, and N. T. Shaked, “Broadband quantitative phase microscopy with extended field of view using off-axis interferometric multiplexing,” J. Biomed. Opt. 20(11), 111217 (2015).
[PubMed]

P. Girshovitz and N. T. Shaked, “Real-time quantitative phase reconstruction in off-axis digital holography using multiplexing,” Opt. Lett. 39(8), 2262–2265 (2014).
[PubMed]

I. Frenklach, P. Girshovitz, and N. T. Shaked, “Off-axis interferometric phase microscopy with tripled imaging area,” Opt. Lett. 39(6), 1525–1528 (2014).
[PubMed]

P. Girshovitz and N. T. Shaked, “Compact and portable low-coherence interferometer with off-axis geometry for quantitative phase microscopy and nanoscopy,” Opt. Express 21(5), 5701–5714 (2013).
[PubMed]

N. T. Shaked, “Quantitative phase microscopy of biological samples using a portable interferometer,” Opt. Lett. 37(11), 2016–2018 (2012).
[PubMed]

N. T. Shaked, Y. Zhu, M. T. Rinehart, and A. Wax, “Two-step-only phase-shifting interferometry with optimized detector bandwidth for microscopy of live cells,” Opt. Express 17(18), 15585–15591 (2009).
[PubMed]

Shan, M.

Z. Zhong, H. Bai, M. Shan, Y. Zhang, and L. Guo, “Fast phase retrieval in slightly off-axis holography,” Opt. Lasers Eng. 97, 9–18 (2017).

Tahara, T.

T. Tahara, R. Mori, Y. Arai, and Y. Takaki, “Four-step phase-shifting digital holography simultaneously sensing dual-wavelength information using a monochromatic image sensor,” J. Opt. 17, 125707 (2015).

T. Tahara, Y. Awatsuji, K. Nishio, S. Ura, O. Matoba, and T. Kubota, “Space-bandwidth capacity-enhanced digital holography,” Appl. Phys. Express 6(2), 022502 (2013).

Takaki, Y.

T. Tahara, R. Mori, Y. Arai, and Y. Takaki, “Four-step phase-shifting digital holography simultaneously sensing dual-wavelength information using a monochromatic image sensor,” J. Opt. 17, 125707 (2015).

Teich, M. C.

B. A. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

Torroba, R.

A. V. Zea, J. F. Barrera, and R. Torroba, “Cross-talk free selective reconstruction of individual objects from multiplexed optical field data,” Opt. Lasers Eng. 100, 90–97 (2018).

Tulino, A.

Turko, N. A.

Ura, S.

T. Tahara, Y. Awatsuji, K. Nishio, S. Ura, O. Matoba, and T. Kubota, “Space-bandwidth capacity-enhanced digital holography,” Appl. Phys. Express 6(2), 022502 (2013).

Wang, X.

Wax, A.

Wu, Y.

Y. Wu, Y. Yang, H. Zhai, Z. Ma, L. Deng, and Q. Ge, “Single-exposure approach for expanding the sampled area of a dynamic process by digital holography with combined multiplexing,” J. Opt. 15(8), 085402 (2013).

Xie, Y. Y.

Yang, Y.

Y. Wu, Y. Yang, H. Zhai, Z. Ma, L. Deng, and Q. Ge, “Single-exposure approach for expanding the sampled area of a dynamic process by digital holography with combined multiplexing,” J. Opt. 15(8), 085402 (2013).

Yue, Q. Y.

Zea, A. V.

A. V. Zea, J. F. Barrera, and R. Torroba, “Cross-talk free selective reconstruction of individual objects from multiplexed optical field data,” Opt. Lasers Eng. 100, 90–97 (2018).

Zhai, H.

Y. Wu, Y. Yang, H. Zhai, Z. Ma, L. Deng, and Q. Ge, “Single-exposure approach for expanding the sampled area of a dynamic process by digital holography with combined multiplexing,” J. Opt. 15(8), 085402 (2013).

X. Wang, H. Zhai, and G. Mu, “Pulsed digital holography system recording ultrafast process of the femtosecond order,” Opt. Lett. 31(11), 1636–1638 (2006).
[PubMed]

Zhang, Y.

Z. Zhong, H. Bai, M. Shan, Y. Zhang, and L. Guo, “Fast phase retrieval in slightly off-axis holography,” Opt. Lasers Eng. 97, 9–18 (2017).

Zhong, Z.

Z. Zhong, H. Bai, M. Shan, Y. Zhang, and L. Guo, “Fast phase retrieval in slightly off-axis holography,” Opt. Lasers Eng. 97, 9–18 (2017).

Zhu, Y.

Appl. Opt. (4)

Appl. Phys. Express (1)

T. Tahara, Y. Awatsuji, K. Nishio, S. Ura, O. Matoba, and T. Kubota, “Space-bandwidth capacity-enhanced digital holography,” Appl. Phys. Express 6(2), 022502 (2013).

Chin. Opt. Lett. (1)

J. Biomed. Opt. (1)

P. Girshovitz, I. Frenklach, and N. T. Shaked, “Broadband quantitative phase microscopy with extended field of view using off-axis interferometric multiplexing,” J. Biomed. Opt. 20(11), 111217 (2015).
[PubMed]

J. Opt. (3)

Y. Wu, Y. Yang, H. Zhai, Z. Ma, L. Deng, and Q. Ge, “Single-exposure approach for expanding the sampled area of a dynamic process by digital holography with combined multiplexing,” J. Opt. 15(8), 085402 (2013).

C. Rosales-Guzmán, N. Bhebhe, N. Mahonisi, and A. Forbes, “Multiplexing 200 spatial modes with a single hologram,” J. Opt. 19(11), 113501 (2017).

T. Tahara, R. Mori, Y. Arai, and Y. Takaki, “Four-step phase-shifting digital holography simultaneously sensing dual-wavelength information using a monochromatic image sensor,” J. Opt. 17, 125707 (2015).

Opt. Express (6)

Opt. Lasers Eng. (2)

Z. Zhong, H. Bai, M. Shan, Y. Zhang, and L. Guo, “Fast phase retrieval in slightly off-axis holography,” Opt. Lasers Eng. 97, 9–18 (2017).

A. V. Zea, J. F. Barrera, and R. Torroba, “Cross-talk free selective reconstruction of individual objects from multiplexed optical field data,” Opt. Lasers Eng. 100, 90–97 (2018).

Opt. Lett. (8)

M. Paturzo, P. Memmolo, L. Miccio, A. Finizio, P. Ferraro, A. Tulino, and B. Javidi, “Numerical multiplexing and demultiplexing of digital holographic information for remote reconstruction in amplitude and phase,” Opt. Lett. 33(22), 2629–2631 (2008).
[PubMed]

P. Girshovitz and N. T. Shaked, “Real-time quantitative phase reconstruction in off-axis digital holography using multiplexing,” Opt. Lett. 39(8), 2262–2265 (2014).
[PubMed]

D. Roitshtain, N. A. Turko, B. Javidi, and N. T. Shaked, “Flipping interferometry and its application for quantitative phase microscopy in a micro-channel,” Opt. Lett. 41(10), 2354–2357 (2016).
[PubMed]

S. Karepov, N. T. Shaked, and T. Ellenbogen, “Off-axis interferometer with adjustable fringe contrast based on polarization encoding,” Opt. Lett. 40(10), 2273–2276 (2015).
[PubMed]

N. T. Shaked, “Quantitative phase microscopy of biological samples using a portable interferometer,” Opt. Lett. 37(11), 2016–2018 (2012).
[PubMed]

N. A. Turko and N. T. Shaked, “Simultaneous two-wavelength phase unwrapping using an external module for multiplexing off-axis holography,” Opt. Lett. 42(1), 73–76 (2017).
[PubMed]

X. Wang, H. Zhai, and G. Mu, “Pulsed digital holography system recording ultrafast process of the femtosecond order,” Opt. Lett. 31(11), 1636–1638 (2006).
[PubMed]

I. Frenklach, P. Girshovitz, and N. T. Shaked, “Off-axis interferometric phase microscopy with tripled imaging area,” Opt. Lett. 39(6), 1525–1528 (2014).
[PubMed]

Other (3)

M. Rubin, G. Dardikman, S. K. Mirsky, N. A. Turko, and N. T. Shaked, “Six-pack off-axis holography,” Accepted to Opt. Lett. (2017).

C. M. Vest, Holographic Interferometry (Wiley, 1979).

B. A. E. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

Supplementary Material (3)

NameDescription
» Visualization 1       Dynamic off-axis hologram of cancer cells during flow
» Visualization 2       OPD maps of cancer cells during flow, reconstructed from 8-pack multiplexed holograms (using complex algorithm)
» Visualization 3       OPD maps of cancer cells during flow, reconstructed without multiplexing

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Figures (7)

Fig. 1
Fig. 1 Schematic illustrations of the holograms (left) and the coinciding spatial power spectra (right) for various off-axis hologram multiplexing architectures, including bandwidth calculations. (a) Conventional off-axis holography. (b) SPACE [25]. (c) Diagonal multiplexing of two channels [25]. (d) 6PH [16]. (e) 8PH, proposed in the paper. In the spectra images, DC denotes the auto-correlation terms, the numbered circles around it denote the CC terms, where the coinciding complex conjugate CC terms are denoted by a number and an asterisk. Note that the relative size of the CC and DC terms change when the total bandwidth changes, since the number of pixels remains constant.
Fig. 2
Fig. 2 Hermitian (conjugate) symmetry for even arrays in (a) 1-D (b) 1-D after fftshift (c) 2-D. (d) 2-D after fftshift. DC denotes the zero-frequency component, and N denotes the Niquist frequency component, both are real valued. Complex conjugation is denoted by an asterisk.
Fig. 3
Fig. 3 Quantitative phase imaging of cancer cells during flow. (a) Experimental setup for generating the regular holograms as the input for the digital multiplexing process; MO: microscope objective, RR: retro reflector, TL: tube lens, BS: beam splitter (b) Dynamic off-axis hologram (Visualization 1) (c) Left: multiplexed power spectrum, right: multiplexed 8PH hologram. (d) Eight reconstructed OPD profiles from a single multiplexed hologram (Visualization 2). (e) Eight OPD profiles reconstructed without the 8PH compression (Visualization 3), for comparison. White scale bars represent 5 μm on the sample. Colorbar represents OPD values in nm.
Fig. 4
Fig. 4 A schematic illustration of the SFD of various complex multiplexed holograms, matching the different stages of Algorithms G, H, I and J for fast hologram processing presented in this paper. (a) Step 1 in Algorithm G. (b, c) Multiplexed holograms 2 and 3, respectively, in step 2 of Algorithm G. (d) Final multiplexed hologram consisting of 12 wave fronts in Algorithm G, with overlaps of CC terms, but still allowing full reconstruction. (e) Step 1 in Algorithm H, prior to resampling four times in the row dimension. (f) Step 1 in Algorithm I. (g) Step 2 in Algorithm I. (h) Step 1 in Algorithm J, prior to resampling four times in the row dimension. j indicates the imaginary unit. Note that although the coinciding SFDs are shown, all operations are performed in the hologram domain, without the need to transform the holograms to the SFD first.
Fig. 5
Fig. 5 A schematic drawing of the modified Mach-Zehnder interferometer used for the experimental demonstrations for generating the regular DC-free holograms as the input for the digital multiplexing process. AOTF, acousto optical tunable filter; BS1, BS2, beam splitters; M1, M2, mirrors; RR1, RR2, retroreflectors; S, sample; MO1, MO2, microscope objectives; L1, L2, tube lenses. The two phase-shifted off-axis holograms from the two cameras are subtracted to obtain a DC-free off-axis hologram.
Fig. 6
Fig. 6 (a) Hologram from camera 1 (b) Hologram from camera 2 (c) Hologram obtained by subtracting the holograms from (a) and (b). (d-f) Coinciding SFD absolute values for (a-c). (g) Absolute value of multiplexed hologram in Algorithm G. (h) Coinciding SFD absolute value. (i) Absolute value of multiplexed resized hologram in Algorithm H. (j) Coinciding SFD absolute value. (k) Absolute value of multiplexed hologram in Algorithm I. (l) Coinciding SFD absolute value. (m) Absolute value of multiplexed resized hologram in Algorithm J. (n) Coinciding SFD absolute value.
Fig. 7
Fig. 7 Reconstruction results. (a) Standard reconstruction by Algorithm B [17]. (b‑d) Reconstruction using Algorithm G of holograms in locations 1,5,9, respectively (see upper quarter in Fig. 4(d)). (e-g) Reconstruction using Algorithm H of holograms in locations 1,2,3, respectively (see Fig. 4(e)). (h) Reconstruction using Algorithm I. (i) Reconstruction using Algorithm J. White scale bars represent 5 μm on the sample. Colorbar represents OPD values in nm.

Tables (2)

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Table 1 Comparison of various off-axis hologram multiplexing architectures. The best result in each column is highlighted.

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Table 2 A comparison between various phase reconstruction algorithms. The reconstruction rate is given in fps for various square input hologram sizes, after being averaged over 1200 reconstructions. The last column shows the MSE in rad2 for the 512 × 512 input, averaged over 9 simulated phase images. The best result in each column in highlighted.

Equations (8)

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| Es+Er | 2 = | Es | 2 + | Er | 2 +| Es || Er |exp(j(φs+φo))+| Es || Er |exp(j(φs+φo)),
φo= 2π λ [ xsin(θx)+ysin(θy) ],
FT{Imultiplexed}= k=1 N [ FT{ | Es,k | 2 }+ | Er,k | 2 δ(u,v) ] + k=1 N | Er,k |FT{ | Es,k |exp( jφs,k ) } *δ[ u 2π λ sin(θx,k),v 2π λ sin(θy,k) ] + k=1 N | Er,k |FT{ | Es,k |exp( jφs,k ) } *δ[ u+ 2π λ sin(θx,k),v+ 2π λ sin(θy,k) ],
u 0 = 2π λ sin(θx)θx= sin 1 ( λ 2π u 0 ), v 0 = 2π λ sin(θy)θy= sin 1 ( λ 2π v 0 ),
ωmax,u= 2π 2Δx = π Δx , ωmax,v= 2π 2Δy = π Δy .
CC2n1= 1 2 [ cc2n1+flip(cc2n)* ];CC2n= 1 2j [ cc2n1flip(cc2n)* ],
IM=I1exp( 3jπ 4yΔy )+I2exp( jπ 4yΔy )+I3exp( jπ 4yΔy )+I4exp( 3jπ 4yΔy ).
I=IM,1+IM,2exp( jπ 2xΔx )+IM,3exp( jπ xΔx ).

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