Abstract

Coherent vortex beams have shown great potential in many applications including information transmission under non-ideal conditions, as information can be encoded in the orbital angular momentum. However, inhomogeneity of atmosphere tends to scramble the vortex structure and give rise to speckle. It is therefore of great interest to reconstruct the topological charge of a vortex beam after it propagates through a scattering medium. Here, we propose a feasible solution for this. The proposed method measures holographically the scattered field and reconstructs the spiral phase from it by taking advantage of both the deterministic nature and the ergodicity of the scattering process. Our preliminary experiments show promising results and suggest that the proposed method can have great potential in information transmission under non-ideal conditions.

© 2021 Optical Society of America

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References

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2020 (2)

Y. Li, L. Wang, L. Gong, and Q. Wang, “Speckle characteristics of vortex beams scattered from rough targets in turbulent atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 257, 107342 (2020).
[Crossref]

X.-B. Hu, M.-X. Dong, Z.-H. Zhu, W. Gao, and C. Rosales-Guzmán, “Does the structure of light influence the speckle size?” Sci. Rep. 10, 199 (2020).
[Crossref]

2019 (1)

Z. Yu, M. Xia, H. Li, T. Zhong, F. Zhao, H. Deng, Z. Li, D. Li, D. Wang, and P. Lai, “Implementation of digital optical phase conjugation with embedded calibration and phase rectification,” Sci. Rep. 9, 1537 (2019).
[Crossref]

2018 (3)

M. P. Lavery, “Vortex instability in turbulent free-space propagation,” New J. Phys. 20, 043023 (2018).
[Crossref]

X. Qiu, F. Li, W. Zhang, Z. Zhu, and L. Chen, “Spiral phase contrast imaging in nonlinear optics: seeing phase objects using invisible illumination,” Optica 5, 208–212 (2018).
[Crossref]

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light Sci. Appl. 7, 17146 (2018).
[Crossref]

2017 (5)

2016 (6)

E. Edrei and G. Scarcelli, “Optical imaging through dynamic turbid media using the Fourier-domain shower-curtain effect,” Optica 3, 71–74 (2016).
[Crossref]

R. Liu, F. Wang, D. Chen, Y. Wang, Y. Zhou, H. Gao, P. Zhang, and F. Li, “Measuring mode indices of a partially coherent vortex beam with Hanbury Brown and Twiss type experiment,” Appl. Phys. Lett. 108, 051107 (2016).
[Crossref]

A. Badon, D. Li, G. Lerosey, A. C. Boccara, M. Fink, and A. Aubry, “Smart optical coherence tomography for ultra-deep imaging through highly scattering media,” Sci. Adv. 2, e1600370 (2016).
[Crossref]

R. V. Vinu and S. R. Kumar, “Determining helicity and topological structure of coherent vortex beam from laser speckle,” Appl. Phys. Lett. 109, 111108 (2016).
[Crossref]

S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photon. Res. 4, B1–B4 (2016).
[Crossref]

A. Trichili, A. B. Salem, A. Dudley, M. Zghal, and A. Forbes, “Encoding information using Laguerre Gaussian modes over free space turbulence media,” Opt. Lett. 41, 3086–3089 (2016).
[Crossref]

2015 (1)

G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107, 021104 (2015).
[Crossref]

2014 (2)

2012 (1)

2011 (1)

2010 (1)

J. Ng, Z. Lin, and C. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104, 103601 (2010).
[Crossref]

2009 (2)

2008 (2)

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
[Crossref]

G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25, 225–230 (2008).
[Crossref]

2007 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]

2004 (4)

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref]

1999 (1)

1995 (1)

I. Basistiy, M. Soskin, and M. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[Crossref]

1994 (1)

W. I. Beavers, D. E. Dudgeon, J. W. Beletic, and M. T. Lane, “Speckle imaging through the atmosphere,” Lincoln Lab. J. 2, 207–228 (1994).

1992 (1)

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref]

1976 (1)

1970 (1)

A. Aabeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

’t Hooft, G. W.

Aabeyrie, A.

A. Aabeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Ahmed, N.

Allen, L.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref]

Almeida, S.

Anwar, A.

G. R. Salla, C. Perumangattu, S. Prabhakar, A. Anwar, and R. P. Singh, “Recovering the vorticity of a light beam after scattering,” Appl. Phys. Lett. 107, 021104 (2015).
[Crossref]

Aubry, A.

A. Badon, D. Li, G. Lerosey, A. C. Boccara, M. Fink, and A. Aubry, “Smart optical coherence tomography for ultra-deep imaging through highly scattering media,” Sci. Adv. 2, e1600370 (2016).
[Crossref]

Badon, A.

A. Badon, D. Li, G. Lerosey, A. C. Boccara, M. Fink, and A. Aubry, “Smart optical coherence tomography for ultra-deep imaging through highly scattering media,” Sci. Adv. 2, e1600370 (2016).
[Crossref]

Balthazar, W.

Bao, C.

Barnett, S. M.

Basistiy, I.

I. Basistiy, M. Soskin, and M. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[Crossref]

Beavers, W. I.

W. I. Beavers, D. E. Dudgeon, J. W. Beletic, and M. T. Lane, “Speckle imaging through the atmosphere,” Lincoln Lab. J. 2, 207–228 (1994).

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref]

Beletic, J. W.

W. I. Beavers, D. E. Dudgeon, J. W. Beletic, and M. T. Lane, “Speckle imaging through the atmosphere,” Lincoln Lab. J. 2, 207–228 (1994).

Boccara, A. C.

A. Badon, D. Li, G. Lerosey, A. C. Boccara, M. Fink, and A. Aubry, “Smart optical coherence tomography for ultra-deep imaging through highly scattering media,” Sci. Adv. 2, e1600370 (2016).
[Crossref]

Boyd, R. W.

Cai, Y.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110, 181104 (2017).
[Crossref]

Chan, C.

J. Ng, Z. Lin, and C. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104, 103601 (2010).
[Crossref]

Chen, D.

R. Liu, F. Wang, D. Chen, Y. Wang, Y. Zhou, H. Gao, P. Zhang, and F. Li, “Measuring mode indices of a partially coherent vortex beam with Hanbury Brown and Twiss type experiment,” Appl. Phys. Lett. 108, 051107 (2016).
[Crossref]

Chen, J.-L.

T. Durt, D. Kaszlikowski, J.-L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A 69, 032313 (2004).
[Crossref]

Chen, L.

Courtial, J.

Cui, Y.

Y. Ma, G. Rui, B. Gu, and Y. Cui, “Trapping and manipulation of nanoparticles using multifocal optical vortex metalens,” Sci. Rep. 7, 14611 (2017).
[Crossref]

Da Silva, L.

Deng, H.

Z. Yu, M. Xia, H. Li, T. Zhong, F. Zhao, H. Deng, Z. Li, D. Li, D. Wang, and P. Lai, “Implementation of digital optical phase conjugation with embedded calibration and phase rectification,” Sci. Rep. 9, 1537 (2019).
[Crossref]

Dolinar, S.

Dong, M.-X.

X.-B. Hu, M.-X. Dong, Z.-H. Zhu, W. Gao, and C. Rosales-Guzmán, “Does the structure of light influence the speckle size?” Sci. Rep. 10, 199 (2020).
[Crossref]

Dudgeon, D. E.

W. I. Beavers, D. E. Dudgeon, J. W. Beletic, and M. T. Lane, “Speckle imaging through the atmosphere,” Lincoln Lab. J. 2, 207–228 (1994).

Dudley, A.

Durt, T.

T. Durt, D. Kaszlikowski, J.-L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A 69, 032313 (2004).
[Crossref]

Edrei, E.

Eliel, E.

Erhard, M.

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light Sci. Appl. 7, 17146 (2018).
[Crossref]

Erkmen, B. I.

Feld, M. S.

Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
[Crossref]

Fickler, R.

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light Sci. Appl. 7, 17146 (2018).
[Crossref]

Fink, M.

A. Badon, D. Li, G. Lerosey, A. C. Boccara, M. Fink, and A. Aubry, “Smart optical coherence tomography for ultra-deep imaging through highly scattering media,” Sci. Adv. 2, e1600370 (2016).
[Crossref]

Forbes, A.

Franke-Arnold, S.

Fu, S.

Gao, C.

Gao, H.

R. Liu, F. Wang, D. Chen, Y. Wang, Y. Zhou, H. Gao, P. Zhang, and F. Li, “Measuring mode indices of a partially coherent vortex beam with Hanbury Brown and Twiss type experiment,” Appl. Phys. Lett. 108, 051107 (2016).
[Crossref]

Gao, W.

X.-B. Hu, M.-X. Dong, Z.-H. Zhu, W. Gao, and C. Rosales-Guzmán, “Does the structure of light influence the speckle size?” Sci. Rep. 10, 199 (2020).
[Crossref]

Gbur, G.

Gibson, G.

Gong, L.

Y. Li, L. Wang, L. Gong, and Q. Wang, “Speckle characteristics of vortex beams scattered from rough targets in turbulent atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 257, 107342 (2020).
[Crossref]

Goodman, J. W.

J. W. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Am. 66, 1145–1150 (1976).
[Crossref]

J. W. Goodman, Speckle Phenomena In Optics: Theory And Applications (Roberts & Company, 2007).

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref]

Gu, B.

Y. Ma, G. Rui, B. Gu, and Y. Cui, “Trapping and manipulation of nanoparticles using multifocal optical vortex metalens,” Sci. Rep. 7, 14611 (2017).
[Crossref]

Hu, X.-B.

X.-B. Hu, M.-X. Dong, Z.-H. Zhu, W. Gao, and C. Rosales-Guzmán, “Does the structure of light influence the speckle size?” Sci. Rep. 10, 199 (2020).
[Crossref]

Huang, H.

Huguenin, J.

Jüptner, W.

Kaszlikowski, D.

T. Durt, D. Kaszlikowski, J.-L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A 69, 032313 (2004).
[Crossref]

Kloosterboer, J.

Krenn, M.

M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, “Twisted photons: new quantum perspectives in high dimensions,” Light Sci. Appl. 7, 17146 (2018).
[Crossref]

Kumar, S. R.

R. V. Vinu and S. R. Kumar, “Determining helicity and topological structure of coherent vortex beam from laser speckle,” Appl. Phys. Lett. 109, 111108 (2016).
[Crossref]

Kwek, L. C.

T. Durt, D. Kaszlikowski, J.-L. Chen, and L. C. Kwek, “Security of quantum key distributions with entangled qudits,” Phys. Rev. A 69, 032313 (2004).
[Crossref]

Lai, P.

Z. Yu, M. Xia, H. Li, T. Zhong, F. Zhao, H. Deng, Z. Li, D. Li, D. Wang, and P. Lai, “Implementation of digital optical phase conjugation with embedded calibration and phase rectification,” Sci. Rep. 9, 1537 (2019).
[Crossref]

Lane, M. T.

W. I. Beavers, D. E. Dudgeon, J. W. Beletic, and M. T. Lane, “Speckle imaging through the atmosphere,” Lincoln Lab. J. 2, 207–228 (1994).

Lavery, M. P.

Leach, J.

Lemos, M.

Lerosey, G.

A. Badon, D. Li, G. Lerosey, A. C. Boccara, M. Fink, and A. Aubry, “Smart optical coherence tomography for ultra-deep imaging through highly scattering media,” Sci. Adv. 2, e1600370 (2016).
[Crossref]

Li, D.

Z. Yu, M. Xia, H. Li, T. Zhong, F. Zhao, H. Deng, Z. Li, D. Li, D. Wang, and P. Lai, “Implementation of digital optical phase conjugation with embedded calibration and phase rectification,” Sci. Rep. 9, 1537 (2019).
[Crossref]

G. Li, W. Yang, D. Li, and G. Situ, “Cyphertext-only attack on the double random-phase encryption: experimental demonstration,” Opt. Express 25, 8690–8697 (2017).
[Crossref]

A. Badon, D. Li, G. Lerosey, A. C. Boccara, M. Fink, and A. Aubry, “Smart optical coherence tomography for ultra-deep imaging through highly scattering media,” Sci. Adv. 2, e1600370 (2016).
[Crossref]

Li, F.

X. Qiu, F. Li, W. Zhang, Z. Zhu, and L. Chen, “Spiral phase contrast imaging in nonlinear optics: seeing phase objects using invisible illumination,” Optica 5, 208–212 (2018).
[Crossref]

R. Liu, F. Wang, D. Chen, Y. Wang, Y. Zhou, H. Gao, P. Zhang, and F. Li, “Measuring mode indices of a partially coherent vortex beam with Hanbury Brown and Twiss type experiment,” Appl. Phys. Lett. 108, 051107 (2016).
[Crossref]

Li, G.

Li, H.

Z. Yu, M. Xia, H. Li, T. Zhong, F. Zhao, H. Deng, Z. Li, D. Li, D. Wang, and P. Lai, “Implementation of digital optical phase conjugation with embedded calibration and phase rectification,” Sci. Rep. 9, 1537 (2019).
[Crossref]

Li, Y.

Y. Li, L. Wang, L. Gong, and Q. Wang, “Speckle characteristics of vortex beams scattered from rough targets in turbulent atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 257, 107342 (2020).
[Crossref]

Li, Z.

Z. Yu, M. Xia, H. Li, T. Zhong, F. Zhao, H. Deng, Z. Li, D. Li, D. Wang, and P. Lai, “Implementation of digital optical phase conjugation with embedded calibration and phase rectification,” Sci. Rep. 9, 1537 (2019).
[Crossref]

Lin, Z.

J. Ng, Z. Lin, and C. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104, 103601 (2010).
[Crossref]

Liu, J.-P.

T.-C. Poon and J.-P. Liu, Introduction to Modern Digital Holography: with MATLAB (Cambridge University, 2014).

Liu, L.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110, 181104 (2017).
[Crossref]

Liu, R.

R. Liu, F. Wang, D. Chen, Y. Wang, Y. Zhou, H. Gao, P. Zhang, and F. Li, “Measuring mode indices of a partially coherent vortex beam with Hanbury Brown and Twiss type experiment,” Appl. Phys. Lett. 108, 051107 (2016).
[Crossref]

Liu, X.

X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110, 181104 (2017).
[Crossref]

Ma, Y.

Y. Ma, G. Rui, B. Gu, and Y. Cui, “Trapping and manipulation of nanoparticles using multifocal optical vortex metalens,” Sci. Rep. 7, 14611 (2017).
[Crossref]

Maleev, I.

D. Palacios, I. Maleev, A. Marathay, and G. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref]

Malik, M.

Marathay, A.

D. Palacios, I. Maleev, A. Marathay, and G. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref]

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Molina-Terriza, G.

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X. Liu, X. Peng, L. Liu, G. Wu, C. Zhao, F. Wang, and Y. Cai, “Self-reconstruction of the degree of coherence of a partially coherent vortex beam obstructed by an opaque obstacle,” Appl. Phys. Lett. 110, 181104 (2017).
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[Crossref]

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Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “Optical phase conjugation for turbidity suppression in biological samples,” Nat. Photonics 2, 110–115 (2008).
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Nat. Phys. (1)

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Opt. Express (3)

Opt. Lett. (3)

Optica (2)

Photon. Res. (2)

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[Crossref]

Phys. Rev. Lett. (3)

J. Ng, Z. Lin, and C. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104, 103601 (2010).
[Crossref]

D. Palacios, I. Maleev, A. Marathay, and G. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
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Z. Yu, M. Xia, H. Li, T. Zhong, F. Zhao, H. Deng, Z. Li, D. Li, D. Wang, and P. Lai, “Implementation of digital optical phase conjugation with embedded calibration and phase rectification,” Sci. Rep. 9, 1537 (2019).
[Crossref]

X.-B. Hu, M.-X. Dong, Z.-H. Zhu, W. Gao, and C. Rosales-Guzmán, “Does the structure of light influence the speckle size?” Sci. Rep. 10, 199 (2020).
[Crossref]

Y. Ma, G. Rui, B. Gu, and Y. Cui, “Trapping and manipulation of nanoparticles using multifocal optical vortex metalens,” Sci. Rep. 7, 14611 (2017).
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Figures (5)

Fig. 1.
Fig. 1. Conceptual representation of the generation of speckle patterns by vortex beams. A plane wave illuminates the spiral phase plate and then travels forward a distance of ${z_1}$. After passing through the diffuser, the scrambled field propagates forward a distance of ${z_2}$, where a speckle pattern can be observed.
Fig. 2.
Fig. 2. Optical setup. A plane wave enters a Mach–Zehnder interferometer formed by a PBS, beam splitters BS1 and BS2, and mirror M1. An sCMOS camera is used to record the interferometric fringe. P1, P2, P3, P4, and P5 are linear polarizers. A1 and A2 are apertures. In our case, the scattering medium is a ground glass DG20-600 from Thorlabs.
Fig. 3.
Fig. 3. Vortex recovery algorithm. (a), (d) Original pattern displayed on the SLM, and (b), (e) are speckle patterns captured by the camera. (c), (f) Amplitude of the sub-images’ Fourier transform. (g) Noise-suppressed intermediate result. (h) Final reconstruction of vortex structure.
Fig. 4.
Fig. 4. Experimental results. These results correspond to topological charges equal to 2, 3, and 5 from top to bottom. The first two columns represent the original vortex patterns and speckle patterns captured by camera, while the last two columns show the intermediate results and final reconstructions.
Fig. 5.
Fig. 5. Experimental results with a DG20-220 ground glass. From left to right, the corresponding topological charge is equal to 1, 3, 5, and 8.

Equations (8)

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ψ z 1 ( x 1 , y 1 ) = P { ψ 0 ( x 0 , y 0 ) ; z 1 } = d x 0 d y 0 ψ 0 ( x 0 , y 0 ) × exp { j π λ z 1 [ ( x 1 x 0 ) 2 + ( y 1 y 0 ) 2 ] } ,
ψ z 1 ( x 1 , y 1 ) = ψ z 1 ( x 1 , y 1 ) ϕ s ( x 1 , y 1 ) .
I ( x 2 , y 2 ) = | P { ψ z 1 ( x 1 , y 1 ) ; z 2 } | 2 .
ψ 0 ( x 0 , y 0 ) = P { P { ψ z 2 ( x 2 , y 2 ) ; z 2 } ϕ s ( x 1 , y 1 ) ; z 1 } = P { ψ z 1 ( x 1 , y 1 ) ; z 1 } ,
ϕ s ( x 1 , y 1 ) = P { ψ z 2 r ( x 2 , y 2 ) ; z 2 } P { ψ r ( x 0 , y 0 ) ; z 1 } = ψ z 1 r ( x 1 , y 1 ) ψ z 1 r ( x 1 , y 1 ) ,
ψ 0 ( x 0 , y 0 ) = P { ψ z 1 ( x 1 , y 1 ) ψ z 1 r ( x 1 , y 1 ) P { ψ r ( x 0 , y 0 ) ; z 1 } ; z 1 } .
U v ( x i , y i ) = ϕ i ψ z 1 ( x i + x r λ z , y i + y r λ z ) ,
U r ( x i , y i ) = ϕ i ψ z 1 r ( x i + x r λ z , y i + y r λ z ) ,

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