Abstract

We harness coherent optical processing to simultaneously sense the angle of arrival and the frequency of radio waves. Signals captured by a distributed antenna array are up-converted to optical domain using electro-optic modulators coupled to individual antennas. Employing a common laser source to feed all the modulators ensures spatially coherent up-conversion of radio-frequency (RF) waves to optical beams carried by optical fibers. Fiber-length dispersion extends the spatial aperture of the distributed antenna array into the temporal dimension. The interference of beams emanating from the fibers is captured by a CCD and used to computationally reconstruct RF waves in k-space.

© 2017 Optical Society of America

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References

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2016 (1)

D. Novak, R. B. Waterhouse, A. Nirmalathas, C. Lim, P. A. Gamage, T. R. Clark, M. L. Dennis, and J. A. Nanzer, “Radio-Over-Fiber Technologies for Emerging Wireless Systems,” IEEE J. Quantum Electron. 52(1), 0600311 (2016).
[Crossref]

2015 (1)

J. A. Nanzer, A. Wichman, J. Klamkin, T. P. Mckenna, and T. R. Clark., “Millimeter-Wave Photonics for Communications and Phased Arrays,” Fiber Integr. Opt. 34(4), 91–106 (2015).
[Crossref]

2014 (1)

Y. Kim, H. Ji, J. Lee, Y. H. Nam, B. L. Ng, I. Tzanidis, Y. Li, and J. Zhang, “Full dimension mimo (FD-MIMO): the next evolution of MIMO in LTE systems,” IEEE Wirel. Commun. 21(2), 26–33 (2014).
[Crossref]

2013 (2)

M. Bajpai, P. Gupta, P. Munshi, V. Titarenko, and P. J. Withers, “A Graphical Processing Unit–Based Parallel Implementation of Multiplicative Algebraic Reconstruction Technique Algorithm for Limited View Tomography,” Res. Nondestruct. Eval. 24(4), 211–222 (2013).
[Crossref]

J. Capmany, J. Mora, I. Gasulla, J. Sancho, J. Lloret, and S. Sales, “Microwave Photonic Signal Processing,” J. Lightwave Technol. 31(4), 571–586 (2013).
[Crossref]

2012 (1)

J. Murakowski, G. J. Schneider, and D. W. Prather, “Passive millimeter-wave holography enabled by optical up-conversion,” Rf Millim.- Wave Photonics II 8259, 825903 (2012).

2009 (2)

J. Yao, “Microwave Photonics,” J. Lightwave Technol. 27(3), 314–335 (2009).
[Crossref]

C. Byrne, “Block-iterative algorithms,” Int. Trans. Oper. Res. 16(4), 427–463 (2009).
[Crossref]

2008 (1)

F. Bucholtz, V. J. Urick, M. Godinez, and K. J. Williams, “Graphical approach for evaluating performance limitations in externally modulated analog photonic links,” IEEE Trans. Microw. Theory Tech. 56(1), 242–247 (2008).
[Crossref]

2007 (3)

2006 (3)

R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microw. Theory Tech. 54(2), 832–846 (2006).
[Crossref]

A. J. Seeds and K. J. Williams, “Microwave photonics,” J. Lightwave Technol. 24(12), 4628–4641 (2006).
[Crossref]

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[Crossref]

1999 (2)

P. M. Blanchard, A. H. Greenaway, A. R. Harvey, and K. Webster, “Coherent optical beam forming with passive millimeter-wave arrays,” J. Lightwave Technol. 17(3), 418–425 (1999).
[Crossref]

D. Mishra, K. Muralidhar, and P. Munshi, “A robust mart algorithm for tomographic applications,” Numer. Heat Transf. Part B-Fundam. 35(4), 485–506 (1999).
[Crossref]

1998 (1)

C. L. Byrne, “Accelerating the EMML algorithm and related iterative algorithms by rescaled block-iterative methods,” IEEE Trans. Image Process. 7(1), 100–109 (1998).
[Crossref] [PubMed]

1997 (1)

P. M. V. Subbarao, P. Munshi, and K. Muralidhar, “Performance of iterative tomographic algorithms applied to non-destructive evaluation with limited data,” NDT Int. 30(6), 359–370 (1997).
[Crossref]

1993 (1)

S. Kaczmarz, “Approximate Solution of Systems of Linear-Equations (reprinted from Bulletin-Int-Acad-Polonaise-Sci, Lett a, Pg 355-357, 1937),” Int. J. Control 57(6), 1269–1271 (1993).
[Crossref]

1992 (1)

M. Reis and N. Roberty, “Maximum-Entropy Algorithms for Image-Reconstruction from Projections,” Inverse Probl. 8(4), 623–644 (1992).
[Crossref]

1984 (1)

A. H. Andersen and A. C. Kak, “Simultaneous Algebraic Reconstruction Technique (SART): a Superior Implementation of the Art Algorithm,” Ultrason. Imaging 6(1), 81–94 (1984).
[Crossref] [PubMed]

1983 (1)

Y. Censor, “Finite Series-Expansion Reconstruction Methods,” Proc. IEEE 71(3), 409–419 (1983).
[Crossref]

1977 (1)

J. G. Colsher, “Iterative three-dimensional image reconstruction from tomographic projections,” Comput. Graph. Image Process. 6(6), 513–537 (1977).
[Crossref]

1974 (1)

R. Gordon and G. Herman, “3-Dimensional Reconstruction from Projections - Review of Algorithms,” Int. Rev. Cytol.- Surv. Cell Biol. 38, 111–151 (1974).

Ackerman, E. I.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[Crossref]

Andersen, A. H.

A. H. Andersen and A. C. Kak, “Simultaneous Algebraic Reconstruction Technique (SART): a Superior Implementation of the Art Algorithm,” Ultrason. Imaging 6(1), 81–94 (1984).
[Crossref] [PubMed]

Bajpai, M.

M. Bajpai, P. Gupta, P. Munshi, V. Titarenko, and P. J. Withers, “A Graphical Processing Unit–Based Parallel Implementation of Multiplicative Algebraic Reconstruction Technique Algorithm for Limited View Tomography,” Res. Nondestruct. Eval. 24(4), 211–222 (2013).
[Crossref]

Betts, G. E.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[Crossref]

Biswas, I.

C. A. Schuetz, R. D. Martin, I. Biswas, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Sparse aperture millimeter-wave imaging using optical detection and correlation techniques,” in Proceedings of SPIE - The International Society for Optical Engineering, 6548 (SPIE, 2007).
[Crossref]

Blanchard, P. M.

Bucholtz, F.

F. Bucholtz, V. J. Urick, M. Godinez, and K. J. Williams, “Graphical approach for evaluating performance limitations in externally modulated analog photonic links,” IEEE Trans. Microw. Theory Tech. 56(1), 242–247 (2008).
[Crossref]

Byrne, C.

C. Byrne, “Block-iterative algorithms,” Int. Trans. Oper. Res. 16(4), 427–463 (2009).
[Crossref]

Byrne, C. L.

C. L. Byrne, “Accelerating the EMML algorithm and related iterative algorithms by rescaled block-iterative methods,” IEEE Trans. Image Process. 7(1), 100–109 (1998).
[Crossref] [PubMed]

Capmany, J.

Censor, Y.

Y. Censor, “Finite Series-Expansion Reconstruction Methods,” Proc. IEEE 71(3), 409–419 (1983).
[Crossref]

Chang, G.-K.

Clark, T. R.

D. Novak, R. B. Waterhouse, A. Nirmalathas, C. Lim, P. A. Gamage, T. R. Clark, M. L. Dennis, and J. A. Nanzer, “Radio-Over-Fiber Technologies for Emerging Wireless Systems,” IEEE J. Quantum Electron. 52(1), 0600311 (2016).
[Crossref]

J. A. Nanzer, A. Wichman, J. Klamkin, T. P. Mckenna, and T. R. Clark., “Millimeter-Wave Photonics for Communications and Phased Arrays,” Fiber Integr. Opt. 34(4), 91–106 (2015).
[Crossref]

Colsher, J. G.

J. G. Colsher, “Iterative three-dimensional image reconstruction from tomographic projections,” Comput. Graph. Image Process. 6(6), 513–537 (1977).
[Crossref]

Cox, C. H.

C. H. Cox, E. I. Ackerman, G. E. Betts, and J. L. Prince, “Limits on the performance of RF-over-fiber links and their impact on device design,” IEEE Trans. Microw. Theory Tech. 54(2), 906–920 (2006).
[Crossref]

Dennis, M. L.

D. Novak, R. B. Waterhouse, A. Nirmalathas, C. Lim, P. A. Gamage, T. R. Clark, M. L. Dennis, and J. A. Nanzer, “Radio-Over-Fiber Technologies for Emerging Wireless Systems,” IEEE J. Quantum Electron. 52(1), 0600311 (2016).
[Crossref]

Ellinas, G.

Gamage, P. A.

D. Novak, R. B. Waterhouse, A. Nirmalathas, C. Lim, P. A. Gamage, T. R. Clark, M. L. Dennis, and J. A. Nanzer, “Radio-Over-Fiber Technologies for Emerging Wireless Systems,” IEEE J. Quantum Electron. 52(1), 0600311 (2016).
[Crossref]

Gasulla, I.

George, J.

Ghosh, A.

F. W. Vook, A. Ghosh, and T. A. Thomas, “MIMO and beamforming solutions for 5G technology,” in 2014 IEEE MTT-S International Microwave Symposium (IMS2014) (2014), pp. 1–4.
[Crossref]

Godinez, M.

F. Bucholtz, V. J. Urick, M. Godinez, and K. J. Williams, “Graphical approach for evaluating performance limitations in externally modulated analog photonic links,” IEEE Trans. Microw. Theory Tech. 56(1), 242–247 (2008).
[Crossref]

Gordon, R.

R. Gordon and G. Herman, “3-Dimensional Reconstruction from Projections - Review of Algorithms,” Int. Rev. Cytol.- Surv. Cell Biol. 38, 111–151 (1974).

Greenaway, A. H.

Gupta, P.

M. Bajpai, P. Gupta, P. Munshi, V. Titarenko, and P. J. Withers, “A Graphical Processing Unit–Based Parallel Implementation of Multiplicative Algebraic Reconstruction Technique Algorithm for Limited View Tomography,” Res. Nondestruct. Eval. 24(4), 211–222 (2013).
[Crossref]

Harvey, A. R.

Herman, G.

R. Gordon and G. Herman, “3-Dimensional Reconstruction from Projections - Review of Algorithms,” Int. Rev. Cytol.- Surv. Cell Biol. 38, 111–151 (1974).

Ji, H.

Y. Kim, H. Ji, J. Lee, Y. H. Nam, B. L. Ng, I. Tzanidis, Y. Li, and J. Zhang, “Full dimension mimo (FD-MIMO): the next evolution of MIMO in LTE systems,” IEEE Wirel. Commun. 21(2), 26–33 (2014).
[Crossref]

Jia, Z.

Kaczmarz, S.

S. Kaczmarz, “Approximate Solution of Systems of Linear-Equations (reprinted from Bulletin-Int-Acad-Polonaise-Sci, Lett a, Pg 355-357, 1937),” Int. J. Control 57(6), 1269–1271 (1993).
[Crossref]

Kak, A. C.

A. H. Andersen and A. C. Kak, “Simultaneous Algebraic Reconstruction Technique (SART): a Superior Implementation of the Art Algorithm,” Ultrason. Imaging 6(1), 81–94 (1984).
[Crossref] [PubMed]

Kim, Y.

Y. Kim, H. Ji, J. Lee, Y. H. Nam, B. L. Ng, I. Tzanidis, Y. Li, and J. Zhang, “Full dimension mimo (FD-MIMO): the next evolution of MIMO in LTE systems,” IEEE Wirel. Commun. 21(2), 26–33 (2014).
[Crossref]

Klamkin, J.

J. A. Nanzer, A. Wichman, J. Klamkin, T. P. Mckenna, and T. R. Clark., “Millimeter-Wave Photonics for Communications and Phased Arrays,” Fiber Integr. Opt. 34(4), 91–106 (2015).
[Crossref]

Kobyakov, A.

Lee, J.

Y. Kim, H. Ji, J. Lee, Y. H. Nam, B. L. Ng, I. Tzanidis, Y. Li, and J. Zhang, “Full dimension mimo (FD-MIMO): the next evolution of MIMO in LTE systems,” IEEE Wirel. Commun. 21(2), 26–33 (2014).
[Crossref]

Li, Y.

Y. Kim, H. Ji, J. Lee, Y. H. Nam, B. L. Ng, I. Tzanidis, Y. Li, and J. Zhang, “Full dimension mimo (FD-MIMO): the next evolution of MIMO in LTE systems,” IEEE Wirel. Commun. 21(2), 26–33 (2014).
[Crossref]

Lim, C.

D. Novak, R. B. Waterhouse, A. Nirmalathas, C. Lim, P. A. Gamage, T. R. Clark, M. L. Dennis, and J. A. Nanzer, “Radio-Over-Fiber Technologies for Emerging Wireless Systems,” IEEE J. Quantum Electron. 52(1), 0600311 (2016).
[Crossref]

Lloret, J.

Martin, R. D.

C. A. Schuetz, R. D. Martin, I. Biswas, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Sparse aperture millimeter-wave imaging using optical detection and correlation techniques,” in Proceedings of SPIE - The International Society for Optical Engineering, 6548 (SPIE, 2007).
[Crossref]

Mckenna, T. P.

J. A. Nanzer, A. Wichman, J. Klamkin, T. P. Mckenna, and T. R. Clark., “Millimeter-Wave Photonics for Communications and Phased Arrays,” Fiber Integr. Opt. 34(4), 91–106 (2015).
[Crossref]

Minasian, R. A.

R. A. Minasian, “Photonic signal processing of microwave signals,” IEEE Trans. Microw. Theory Tech. 54(2), 832–846 (2006).
[Crossref]

Mirotznik, M. S.

C. A. Schuetz, R. D. Martin, I. Biswas, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Sparse aperture millimeter-wave imaging using optical detection and correlation techniques,” in Proceedings of SPIE - The International Society for Optical Engineering, 6548 (SPIE, 2007).
[Crossref]

Mishra, D.

D. Mishra, K. Muralidhar, and P. Munshi, “A robust mart algorithm for tomographic applications,” Numer. Heat Transf. Part B-Fundam. 35(4), 485–506 (1999).
[Crossref]

Mora, J.

Munshi, P.

M. Bajpai, P. Gupta, P. Munshi, V. Titarenko, and P. J. Withers, “A Graphical Processing Unit–Based Parallel Implementation of Multiplicative Algebraic Reconstruction Technique Algorithm for Limited View Tomography,” Res. Nondestruct. Eval. 24(4), 211–222 (2013).
[Crossref]

D. Mishra, K. Muralidhar, and P. Munshi, “A robust mart algorithm for tomographic applications,” Numer. Heat Transf. Part B-Fundam. 35(4), 485–506 (1999).
[Crossref]

P. M. V. Subbarao, P. Munshi, and K. Muralidhar, “Performance of iterative tomographic algorithms applied to non-destructive evaluation with limited data,” NDT Int. 30(6), 359–370 (1997).
[Crossref]

Murakowski, J.

J. Murakowski, G. J. Schneider, and D. W. Prather, “Passive millimeter-wave holography enabled by optical up-conversion,” Rf Millim.- Wave Photonics II 8259, 825903 (2012).

C. A. Schuetz, R. D. Martin, I. Biswas, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Sparse aperture millimeter-wave imaging using optical detection and correlation techniques,” in Proceedings of SPIE - The International Society for Optical Engineering, 6548 (SPIE, 2007).
[Crossref]

Muralidhar, K.

D. Mishra, K. Muralidhar, and P. Munshi, “A robust mart algorithm for tomographic applications,” Numer. Heat Transf. Part B-Fundam. 35(4), 485–506 (1999).
[Crossref]

P. M. V. Subbarao, P. Munshi, and K. Muralidhar, “Performance of iterative tomographic algorithms applied to non-destructive evaluation with limited data,” NDT Int. 30(6), 359–370 (1997).
[Crossref]

Nam, Y. H.

Y. Kim, H. Ji, J. Lee, Y. H. Nam, B. L. Ng, I. Tzanidis, Y. Li, and J. Zhang, “Full dimension mimo (FD-MIMO): the next evolution of MIMO in LTE systems,” IEEE Wirel. Commun. 21(2), 26–33 (2014).
[Crossref]

Nanzer, J. A.

D. Novak, R. B. Waterhouse, A. Nirmalathas, C. Lim, P. A. Gamage, T. R. Clark, M. L. Dennis, and J. A. Nanzer, “Radio-Over-Fiber Technologies for Emerging Wireless Systems,” IEEE J. Quantum Electron. 52(1), 0600311 (2016).
[Crossref]

J. A. Nanzer, A. Wichman, J. Klamkin, T. P. Mckenna, and T. R. Clark., “Millimeter-Wave Photonics for Communications and Phased Arrays,” Fiber Integr. Opt. 34(4), 91–106 (2015).
[Crossref]

Ng, B. L.

Y. Kim, H. Ji, J. Lee, Y. H. Nam, B. L. Ng, I. Tzanidis, Y. Li, and J. Zhang, “Full dimension mimo (FD-MIMO): the next evolution of MIMO in LTE systems,” IEEE Wirel. Commun. 21(2), 26–33 (2014).
[Crossref]

Nirmalathas, A.

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Supplementary Material (7)

NameDescription
» Visualization 1: MP4 (1599 KB)      RF sources in k-space.
» Visualization 2: MP4 (194 KB)      k-space mapping for array of antennas coupled to fibers with identical lengths.
» Visualization 3: MP4 (622 KB)      Effect of changing fiber lengths on weight map.
» Visualization 4: MP4 (642 KB)      k-space mapping for array of antennas coupled to fibers with lengths increasing linearly from the top to the bottom fiber.
» Visualization 5: MP4 (624 KB)      Effect of randomizing fiber lengths on the weight map.
» Visualization 6: MP4 (474 KB)      k-space mapping for array of antennas coupled to fibers with randomly varied lengths.
» Visualization 7: MP4 (8585 KB)      Weight maps measured experimentally.

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Figures (11)

Fig. 1
Fig. 1 Conceptual diagram of a k-space tomograph.

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Fig. 2
Fig. 2 Notional configuration of the system used for numerical simulations of a k-space tomograph; inset shows an example of a weight map an corresponding to one of the photo-detectors in the detector array; the labeled equi-frequency contours in the weight map provide scale in the k-space.

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Fig. 3
Fig. 3 Comparison of a distribution of sources in k-space and their tomographic reconstruction from Eq. (1). (a), (b), (c), (d) show scenes set up with various numbers of emitters distributed throughout k-space whereas (a’), (b’), (c’), (d’) show the corresponding reconstructions. These scenes contain 23, 43, 83, and 163 sources, respectively.

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Fig. 4
Fig. 4 Linear k-space tomograph constructed at the University of Delaware. Inset shows a simplified schematic diagram.

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Fig. 5
Fig. 5 Schematic diagram of an 8-channel k-space tomograph. EDFA: erbium-doped fiber amplifier. PBS: polarizing beam splitter. BS: beam splitter. LNA: low-noise amplifier.

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Fig. 6
Fig. 6 Experimental testing of k-space tomography with a system of Figs. 4 and 5. (a) Weight map corresponding to one of the photodetectors in the tomograph of Figs. 4 and 5. (b) Distribution of light among photo-detectors in the array with a single RF plane wave at normal incidence and 16.7 GHz frequency. (c)-(f) Reconstruction of k-space distributions for various source distributions. The plus ‘ + ’ signs indicate the nominal positions and frequencies of sources as set up in the experiment. Sizes of the ‘ + ’ signs in (f) indicate the relative powers of the sources.

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Fig. 7
Fig. 7 RF sources in k-space. Single-frame of an animation, Visualization 1, showing five transmitters, four frequency-hopping and one frequency-swept. Crowded frequency spectrum (see Visualization 1) shown in lower left pane, and the frequency variation makes it difficult to identify the sources. Discerning the angle of arrival, the azimuth and elevation in the upper left pane, allows for easy identification of the positions of the sources. K-space tomography provides full 3D spatial-spectral location of all transmitters, right pane.

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Fig. 8
Fig. 8 k-space mapping for array of antennas coupled to fibers with identical lengths. Left side corresponds to a two-dimensional k-space weight map. On the right, antennas are coupled to optical fibers that form an array. A lens images light emanating from the fiber array onto a line of photo-detectors. Different photo-detectors receive different contributions from RF sources represented as points in k-space, see Visualization 2. The brighter the regions in k-space, the greater the contribution from the RF sources with the corresponding k-vectors to the signal detected at the photo-detector position of the bright star on the right.

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Fig. 9
Fig. 9 Effect of changing fiber lengths on weight map. As the lengths of the fibers at the top of the array increase relative to the fibers at the bottom, the bright region in the k-space weight map deforms from a straight line to a hyperbola (see Visualization 3). For fibers with fixed lengths increasing linearly from the top to the bottom fiber, the bright region of high contribution in the weight map on the left shifts as the position of the photo-detector changes on the right (see Visualization 4). The bright curve retains the shape of a hyperbola.

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Fig. 10
Fig. 10 Effect of randomizing fiber lengths on the weight map. The fibers change from equal lengths to a random length distribution (Visualization 5). As a result, the well-organized weight map containing a single straight bright line, and fainter parallel traces induced by antenna side-lobes, becomes discombobulated. Multiple bright regions scattered randomly throughout the k-space provide diverse contribution to the detection. For a fixed (pseudo‑)random distribution of fiber lengths (Visualization 6), each position of the photodetector (represented by the bright star on the right) corresponds to a different random distribution of contributions from the k-space as represented by the weight maps on the left. With sufficiently diverse collection of weight maps, the sources in k-space may be reconstructed computationally.

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Fig. 11
Fig. 11 Weight maps measured experimentally. Left pane shows weight maps for different pixels of the photo-detector array, see Visualization 7. The animation of Visualization 7 is analogous to that of Visualization 6 obtained using computer simulations. As the photo-detector index is incremented from 1 to 174, the weight map changes as shown in the video of Visualization 7. Right pane shows the contribution of a single k-vector, indicated on the weight map with a white circle at frequency 16.7 GHz and 0.0-degrees incidence angle, to the different pixels of the photo-detector array.

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Equations (14)

Equations on this page are rendered with MathJax. Learn more.

P n = ∑ k a nk S k = a n ⋅S,
C n = 1 N ∑ m=0 M−1 B m e i( ωt+ θ nm + φ m ) +c.c.
φ m = ∑ k=0 K−1 S ˜ k cos[ Ω k (t+ t m )+ ϕ km ],
C n LSB = i e iωt 2 N ∑ mk B m S ˜ k e i[ θ nm − ϕ km − Ω k (t+ t m )] +c.c.
〈 | C n LSB | 2 〉= 1 2N ∑ k | ∑ m S ˜ k B m e i( θ nm − ϕ km − Ω k t m ) | 2 .
〈 | C LSB ( x ) | 2 〉= 1 2N ∑ k | ∑ m S ˜ k a m ( x ) B m e −i( ϕ km + Ω k t m ) | 2 ,
〈 | C LSB ( x ) | 2 〉= 1 2N ∑ k | S ˜ k | 2 | ∑ m a m ( x ) B m e −i( K k ⋅ X m + Ω k t m ) | 2 .
P n = ∫ P/D area 〈 | C LSB ( x ) | 2 〉 d 2 x=〈 | C LSB ( x n ) | 2 〉A= = A 2N ∑ k | S ˜ k | 2 | ∑ m a m ( x n ) B m e −i( K k ⋅ X m + Ω k t m ) | 2 ,
P n = ∑ k a nk S k ,
a nk = A 2N | ∑ m a m ( x n ) B m e −i( K k ⋅ X m + Ω k t m ) | 2 .
a m ( x n )= 1 z−i z 0 exp[ 2πi λ | x n − x m | 2 2( z−i z 0 ) ],
w 0 2 = z 0 λ π .
a m ( x n )= 1 z−i z 0 exp[ 2πi λ | x n | 2 2( z−i z 0 ) ]exp[ − 2πi λ x n ⋅ x m ( z−i z 0 ) ],
P n = 1 2N A z 2 exp[ − 1 2 | 2π λ w 0 x n z | 2 ]× × ∑ k | S ˜ k | 2 | ∑ m B m exp[ −i( 2π λ x m ⋅ x n z + K k ⋅ X m + Ω k t m ) ] | 2 .

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