Abstract

The effect of atmosphere turbulence on light’s spatial structure compromises the information capacity of photons carrying the Orbital Angular Momentum (OAM) in free-space optical (FSO) communications. In this paper, we study two aberration correction methods to mitigate this effect. The first one is the Shack-Hartmann wavefront correction method, which is based on the Zernike polynomials, and the second is a phase correction method specific to OAM states. Our numerical results show that the phase correction method for OAM states outperforms the Shark-Hartmann wavefront correction method, although both methods improve significantly purity of a single OAM state and the channel capacities of FSO communication link. At the same time, our experimental results show that the values of participation functions go down at the phase correction method for OAM states, i.e., the correction method ameliorates effectively the bad effect of atmosphere turbulence.

© 2011 OSA

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References

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  1. L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. M. Gruneisen, W. Miller, R. Dymale, and A. Sweiti, “Holographic generation of complex fields with spatial light modulators: application to quantum key distribution,” Appl. Opt. 47, A32–A42 (2008).
    [CrossRef] [PubMed]
  6. J. Garcia-Escartin and P. Chamorro-Posada, “Quantum multiplexing with the orbital angular momentum of light,” Phys. Rev. A 78, 062320 (2008).
    [CrossRef]
  7. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  9. J. Anguita, M. Neifeld, and B. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47, 2414–2429 (2008).
    [CrossRef] [PubMed]
  10. G. Tyler and R. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34, 142–144 (2009).
    [CrossRef] [PubMed]
  11. Y. X. Zhang and J. Chang, “Effects of turbulent aberrations on probability distribution of orbital angular momentum for optical communication,” Chin. Phys. Lett. 26, 074220 (2009).
    [CrossRef]
  12. C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9, 94 (2007).
    [CrossRef]
  13. S. Zhao, J. Leach, and B. Zheng, “Correction Effect of Shark-Hartmann Algorithm on Turbulence Aberrations for free space Optical Communications Using Orbital Angular Momentum,” in 12th IEEE International Conference on Communication Technology (ICCT), pp. 580–583 (2010).
    [CrossRef]
  14. L. Andrews and R. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005).
    [CrossRef]
  15. R. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (1978).
    [CrossRef]
  16. J. Strasburg and W. Harper, “Impact of atmospheric turbulence on beam propagation,” Proc. SPIE 5413, 93 (2004).
    [CrossRef]
  17. R. Frehlich, “Simulation of laser propagation in a turbulent atmosphere,” Appl. Opt. 39, 393–397 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. A. Jesacher, A. Schwaighofer, S. Frhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express 15, 5801–5808 (2007).
    [CrossRef] [PubMed]
  22. R. Bowman, “Aberration correction for spatial light modulators,” Master’s thesis (Churchill College, 2007).
  23. P. Vontobel, A. Kavcic, D. Arnold, and H. Loeliger, “A generalization of the Blahut–Arimoto algorithm to finite-state channels,” IEEE Trans. Inf. Theory 54, 1887–1918 (2008).
    [CrossRef]
  24. J. R. Fienup and J. J. Miller, “Aberration correction by maximizing generalized sharpness metrics,” J. Opt. Soc. Am. A 20, 609–620 (2003).
    [CrossRef]

2011

2010

I. Djordjevic and M. Arabaci, “LDPC-coded orbital angular momentum (OAM) modulation for free-space optical communication,” Opt. Express 18, 24722–24728 (2010).
[CrossRef] [PubMed]

S. Zhao, J. Leach, and B. Zheng, “Correction Effect of Shark-Hartmann Algorithm on Turbulence Aberrations for free space Optical Communications Using Orbital Angular Momentum,” in 12th IEEE International Conference on Communication Technology (ICCT), pp. 580–583 (2010).
[CrossRef]

2009

Y. X. Zhang and J. Chang, “Effects of turbulent aberrations on probability distribution of orbital angular momentum for optical communication,” Chin. Phys. Lett. 26, 074220 (2009).
[CrossRef]

G. Tyler and R. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34, 142–144 (2009).
[CrossRef] [PubMed]

2008

M. Gruneisen, W. Miller, R. Dymale, and A. Sweiti, “Holographic generation of complex fields with spatial light modulators: application to quantum key distribution,” Appl. Opt. 47, A32–A42 (2008).
[CrossRef] [PubMed]

J. Anguita, M. Neifeld, and B. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47, 2414–2429 (2008).
[CrossRef] [PubMed]

J. Garcia-Escartin and P. Chamorro-Posada, “Quantum multiplexing with the orbital angular momentum of light,” Phys. Rev. A 78, 062320 (2008).
[CrossRef]

M. Mahdieh, “Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor,” Opt. Commun. 281, 3395–3402 (2008).
[CrossRef]

P. Vontobel, A. Kavcic, D. Arnold, and H. Loeliger, “A generalization of the Blahut–Arimoto algorithm to finite-state channels,” IEEE Trans. Inf. Theory 54, 1887–1918 (2008).
[CrossRef]

2007

2005

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett 94, 153901 (2005).
[CrossRef] [PubMed]

2004

2003

2001

B. Platt, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17, 573–577 (2001).

2000

1996

1992

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

1978

R. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (1978).
[CrossRef]

Ahmed, N.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “Demonstration of 12.8-bit/s/Hz spectral efficiency using 16-QAM signals over multiple orbital-angular-momentum modes,” in 37th European Conference and Exhibition on Optical Communication, paper We.10.P1.76 (2011).

Allen, L.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Andrews, L.

L. Andrews and R. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005).
[CrossRef]

Andrews, R.

C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9, 94 (2007).
[CrossRef]

Anguita, J.

Arabaci, M.

Arnold, D.

P. Vontobel, A. Kavcic, D. Arnold, and H. Loeliger, “A generalization of the Blahut–Arimoto algorithm to finite-state channels,” IEEE Trans. Inf. Theory 54, 1887–1918 (2008).
[CrossRef]

Barnett, S.

Beijersbergen, M.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Bernet, S.

Birnbaum, K.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “Demonstration of 12.8-bit/s/Hz spectral efficiency using 16-QAM signals over multiple orbital-angular-momentum modes,” in 37th European Conference and Exhibition on Optical Communication, paper We.10.P1.76 (2011).

Bowman, R.

R. Bowman, “Aberration correction for spatial light modulators,” Master’s thesis (Churchill College, 2007).

Boyd, R.

Chamorro-Posada, P.

J. Garcia-Escartin and P. Chamorro-Posada, “Quantum multiplexing with the orbital angular momentum of light,” Phys. Rev. A 78, 062320 (2008).
[CrossRef]

Chang, J.

Y. X. Zhang and J. Chang, “Effects of turbulent aberrations on probability distribution of orbital angular momentum for optical communication,” Chin. Phys. Lett. 26, 074220 (2009).
[CrossRef]

Choi, J.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “Demonstration of 12.8-bit/s/Hz spectral efficiency using 16-QAM signals over multiple orbital-angular-momentum modes,” in 37th European Conference and Exhibition on Optical Communication, paper We.10.P1.76 (2011).

Courtial, J.

Djordjevic, I.

Dolinar, S.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “Demonstration of 12.8-bit/s/Hz spectral efficiency using 16-QAM signals over multiple orbital-angular-momentum modes,” in 37th European Conference and Exhibition on Optical Communication, paper We.10.P1.76 (2011).

Dymale, R.

Erkmen, B.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “Demonstration of 12.8-bit/s/Hz spectral efficiency using 16-QAM signals over multiple orbital-angular-momentum modes,” in 37th European Conference and Exhibition on Optical Communication, paper We.10.P1.76 (2011).

Fazal, I.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “Demonstration of 12.8-bit/s/Hz spectral efficiency using 16-QAM signals over multiple orbital-angular-momentum modes,” in 37th European Conference and Exhibition on Optical Communication, paper We.10.P1.76 (2011).

Fienup, J. R.

Franke-Arnold, S.

Frehlich, R.

Frhapter, S.

Garcia-Escartin, J.

J. Garcia-Escartin and P. Chamorro-Posada, “Quantum multiplexing with the orbital angular momentum of light,” Phys. Rev. A 78, 062320 (2008).
[CrossRef]

Gibson, G.

Gopaul, C.

C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9, 94 (2007).
[CrossRef]

Gruneisen, M.

Harper, W.

J. Strasburg and W. Harper, “Impact of atmospheric turbulence on beam propagation,” Proc. SPIE 5413, 93 (2004).
[CrossRef]

Hill, R.

R. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (1978).
[CrossRef]

Jesacher, A.

Kavcic, A.

P. Vontobel, A. Kavcic, D. Arnold, and H. Loeliger, “A generalization of the Blahut–Arimoto algorithm to finite-state channels,” IEEE Trans. Inf. Theory 54, 1887–1918 (2008).
[CrossRef]

Leach, J.

S. Zhao, J. Leach, and B. Zheng, “Correction Effect of Shark-Hartmann Algorithm on Turbulence Aberrations for free space Optical Communications Using Orbital Angular Momentum,” in 12th IEEE International Conference on Communication Technology (ICCT), pp. 580–583 (2010).
[CrossRef]

Loeliger, H.

P. Vontobel, A. Kavcic, D. Arnold, and H. Loeliger, “A generalization of the Blahut–Arimoto algorithm to finite-state channels,” IEEE Trans. Inf. Theory 54, 1887–1918 (2008).
[CrossRef]

Mahdieh, M.

M. Mahdieh, “Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor,” Opt. Commun. 281, 3395–3402 (2008).
[CrossRef]

Maurer, C.

Miller, J. J.

Miller, W.

Neifeld, M.

Padgett, M.

Pas’ko, V.

Paterson, C.

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett 94, 153901 (2005).
[CrossRef] [PubMed]

Phillips, R.

L. Andrews and R. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005).
[CrossRef]

Platt, B.

B. Platt, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17, 573–577 (2001).

Ritsch-Marte, M.

Schwaighofer, A.

Shamee, B.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “Demonstration of 12.8-bit/s/Hz spectral efficiency using 16-QAM signals over multiple orbital-angular-momentum modes,” in 37th European Conference and Exhibition on Optical Communication, paper We.10.P1.76 (2011).

Spreeuw, R.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Strasburg, J.

J. Strasburg and W. Harper, “Impact of atmospheric turbulence on beam propagation,” Proc. SPIE 5413, 93 (2004).
[CrossRef]

Sweiti, A.

Tur, M.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “Demonstration of 12.8-bit/s/Hz spectral efficiency using 16-QAM signals over multiple orbital-angular-momentum modes,” in 37th European Conference and Exhibition on Optical Communication, paper We.10.P1.76 (2011).

Tyler, G.

Vasic, B.

Vasnetsov, M.

Voitsekhovich, V.

Vontobel, P.

P. Vontobel, A. Kavcic, D. Arnold, and H. Loeliger, “A generalization of the Blahut–Arimoto algorithm to finite-state channels,” IEEE Trans. Inf. Theory 54, 1887–1918 (2008).
[CrossRef]

Wang, J.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “Demonstration of 12.8-bit/s/Hz spectral efficiency using 16-QAM signals over multiple orbital-angular-momentum modes,” in 37th European Conference and Exhibition on Optical Communication, paper We.10.P1.76 (2011).

Willner, A.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “Demonstration of 12.8-bit/s/Hz spectral efficiency using 16-QAM signals over multiple orbital-angular-momentum modes,” in 37th European Conference and Exhibition on Optical Communication, paper We.10.P1.76 (2011).

Woerdman, J.

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Yan, Y.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “Demonstration of 12.8-bit/s/Hz spectral efficiency using 16-QAM signals over multiple orbital-angular-momentum modes,” in 37th European Conference and Exhibition on Optical Communication, paper We.10.P1.76 (2011).

Yang, J.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “Demonstration of 12.8-bit/s/Hz spectral efficiency using 16-QAM signals over multiple orbital-angular-momentum modes,” in 37th European Conference and Exhibition on Optical Communication, paper We.10.P1.76 (2011).

Zhang, Y. X.

Y. X. Zhang and J. Chang, “Effects of turbulent aberrations on probability distribution of orbital angular momentum for optical communication,” Chin. Phys. Lett. 26, 074220 (2009).
[CrossRef]

Zhao, S.

S. Zhao, J. Leach, and B. Zheng, “Correction Effect of Shark-Hartmann Algorithm on Turbulence Aberrations for free space Optical Communications Using Orbital Angular Momentum,” in 12th IEEE International Conference on Communication Technology (ICCT), pp. 580–583 (2010).
[CrossRef]

Zheng, B.

S. Zhao, J. Leach, and B. Zheng, “Correction Effect of Shark-Hartmann Algorithm on Turbulence Aberrations for free space Optical Communications Using Orbital Angular Momentum,” in 12th IEEE International Conference on Communication Technology (ICCT), pp. 580–583 (2010).
[CrossRef]

12th IEEE International Conference on Communication Technology (ICCT)

S. Zhao, J. Leach, and B. Zheng, “Correction Effect of Shark-Hartmann Algorithm on Turbulence Aberrations for free space Optical Communications Using Orbital Angular Momentum,” in 12th IEEE International Conference on Communication Technology (ICCT), pp. 580–583 (2010).
[CrossRef]

Appl. Opt.

Chin. Phys. Lett.

Y. X. Zhang and J. Chang, “Effects of turbulent aberrations on probability distribution of orbital angular momentum for optical communication,” Chin. Phys. Lett. 26, 074220 (2009).
[CrossRef]

IEEE Trans. Inf. Theory

P. Vontobel, A. Kavcic, D. Arnold, and H. Loeliger, “A generalization of the Blahut–Arimoto algorithm to finite-state channels,” IEEE Trans. Inf. Theory 54, 1887–1918 (2008).
[CrossRef]

J. Fluid Mech.

R. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (1978).
[CrossRef]

J. Opt. Soc. Am. A

J. Refract. Surg.

B. Platt, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. 17, 573–577 (2001).

New J. Phys.

C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9, 94 (2007).
[CrossRef]

Opt. Commun.

M. Mahdieh, “Numerical approach to laser beam propagation through turbulent atmosphere and evaluation of beam quality factor,” Opt. Commun. 281, 3395–3402 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

L. Allen, M. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

J. Garcia-Escartin and P. Chamorro-Posada, “Quantum multiplexing with the orbital angular momentum of light,” Phys. Rev. A 78, 062320 (2008).
[CrossRef]

Phys. Rev. Lett

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett 94, 153901 (2005).
[CrossRef] [PubMed]

Proc. SPIE

J. Strasburg and W. Harper, “Impact of atmospheric turbulence on beam propagation,” Proc. SPIE 5413, 93 (2004).
[CrossRef]

Other

R. Bowman, “Aberration correction for spatial light modulators,” Master’s thesis (Churchill College, 2007).

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, B. Shamee, A. Willner, K. Birnbaum, J. Choi, B. Erkmen, S. Dolinar, and M. Tur, “Demonstration of 12.8-bit/s/Hz spectral efficiency using 16-QAM signals over multiple orbital-angular-momentum modes,” in 37th European Conference and Exhibition on Optical Communication, paper We.10.P1.76 (2011).

L. Andrews and R. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

The aberration caused by atmosphere turbulence and the mitigation effect of the aberration correction methods. (a) propagation through turbulent atmosphere without aberration correction (b) propagation through turbulent atmosphere with aberration correction.

Fig. 2
Fig. 2

Decomposition of the beam after passing through atmospheric turbulence with and without a correction. (a) original decomposition (b) without a correction (c) corrected by the Shark-Hartmann wavefront correction method (d) corrected by the phase correction method for OAM states.

Fig. 3
Fig. 3

The effect of turbulence on the propagating OAM quantum states as functions of C n 2, where C n 2 varies from 1 × 10−14m−2/3 to 1 × 10−11m−2/3, representing the strength of turbulent aberration changing from weak to strong. The distance of propagating is 100m, the outer scale is 50m, and the inner scale is 0.0002m. The simulation grid comprises 128 × 128 elements, and the grid spacing size is 0.0003m.

Fig. 4
Fig. 4

Comparison of the channel capacity for a communication link employing OAM states of single photon through atmospheric turbulence, corrected by Shark-Hartmann wavefront correction method and the phase correction method for OAM states.

Fig. 5
Fig. 5

The sketch of the experimental setup.

Fig. 6
Fig. 6

The variance of the participation functions for the reference spot with the change of the index of refraction by the phase correction method for OAM states.

Equations (15)

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

Φ n ( k x , k y ) = 0.033 C n 2 [ 1 + 1.802 k x 2 + k y 2 k l 2 0.254 [ k x 2 + k y 2 k l 2 ] 7 / 12 ] × exp [ k x 2 + k y 2 k l 2 ] [ k x 2 + k y 2 + 1 L 0 2 ] 11 / 6 ,
Φ ( k x , k y ) = 2 π k 0 2 Δ Z Φ n ( k x , k y ) ,
φ ( x , y ) = F F T ( C σ ( k x , k y ) ) ,
σ 2 ( k x , k y ) = ( 2 π N Δ x ) 2 Φ ( k x , k y ) ,
φ ( x , y ) = k = 0 N a k Z k ( x , y ) .
φ x ( x , y ) = k = 1 N a k Z k x ( x , y ) ,
φ y ( x , y ) = k = 0 N a k Z k y ( x , y ) .
( G x ( 1 ) G y ( 1 ) G x ( m ) G y ( m ) ) = ( D 1 x ( 1 ) D n x ( 1 ) D 1 y ( 1 ) D n y ( 1 ) D 1 x ( m ) D n x ( m ) D 1 y ( m ) D n y ( m ) ) ( a 1 a n ) .
G = D A ,
A = D 1 G .
A ( k x , k y ) exp [ i Φ ( k x , k y ) ] = F { exp [ i φ ( x , y ) ] } .
Ψ ( r , θ , z ) = p l a p , l ( z ) L G p , l ( r , θ ) .
P ( l ) = p | a p , l ( z ) | 2 ,
a p , l ( z ) = L G p , l ( r , θ ) | Ψ ( r , θ , z ) .
P = ( i , j N I i , j ) 2 i , j N I i , j 2 ,

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