Abstract

The propagation of a partially coherent rectangular flat-topped laser array in a turbulent atmosphere is studied. An analytical expression for the average intensity distribution at the receiving plane is obtained based on an extended Huygens–Fresnel principle. The effect of correlation length, intensity of turbulence, laser numbers, and beam orders on the beam quality in a target plane is studied by numerical examples.

© 2009 Optical Society of America

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2009

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248-256 (2009).
[CrossRef]

J. Marmo, H. Injeyan, H. Komine, S. McNaught, J. Machan, and J. Sollee, “Joint high power solid state laser program advancements at Northrop Grumman,” Proc. SPIE 7195, 719507 (2009).
[CrossRef]

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

P. Zhou, Z. Liu, X. Xu, and X. Chu, “Comparative study on the propagation performance of coherently combined and incoherently combined beams,” Opt. Commun. 282, 1640-1647(2009).
[CrossRef]

C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17, 1753-1765 (2009).
[CrossRef] [PubMed]

2008

E. Zhang, X. Ji, and B. Lü, “Propagation of the off-axis superposition of partially coherent beams through atmospheric turbulence,” Chin. Phys. B 18, 571-580 (2008).

X. Chu, Z. Liu, and Y. Wu, “Propagation of a general multi-Gaussian beam in turbulent atmosphere in a slant path,” J. Opt. Soc. Am. A 25, 74-49 (2008).
[CrossRef]

E. C. Cheung, J. G. Ho, G. D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wickham, “Diffractive-optics-based beam combination of a phase-locked fiber laser array,” Opt. Lett. 33, 354-356 (2008).
[CrossRef] [PubMed]

M. Fridman, V. Eckhouse, N. Davidson, and A. A. Friesem, “Simulataneous coherent and spectral addition of fiber lasers,” Opt. Lett. 33, 648-650 (2008).
[CrossRef] [PubMed]

X. Ji, E. Zhang, and B. Lü, “Superimposed partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. B 25, 825-833 (2008).
[CrossRef]

F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. 33, 1795-1797 (2008).
[CrossRef] [PubMed]

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16, 18437-18442 (2008).
[CrossRef] [PubMed]

Y. Cai, X. Lu, H. T. Eyyuboğlu, and Y. Baykal, “Paraxial propagation of a partially coherent flattened Gaussian beam through apertured ABCD optical systems,” Opt. Commun. 281, 3221-3229 (2008).
[CrossRef]

H. T. Eyyuboğlu, Y. Baykal, and Y. Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91, 265-271 (2008).
[CrossRef]

X. Xiao, X. Ji, and B. Lü, “The influence of turbulence on propagation properties of partially coherent sinh-Gaussian beams and their beam quality in the far field,” Opt. Laser Technol. 40, 129-136 (2008).
[CrossRef]

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46, 357-362 (2008).
[CrossRef]

X. Ji and Z. Pu, “Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence,” Appl. Phys. B 93, 915-923 (2008).
[CrossRef]

X. Ji and X. Li, “Influence of turbulence on the coherent and incoherent combinations of off-axis Gaussian beams,” Acta Phys. Sin. 57, 7674-7679 (2008).

J. E. Rothenberg, “Passive coherent phasing of fiber laser arrays,” Proc. SPIE 6873, 687315 (2008).
[CrossRef]

P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
[CrossRef]

2007

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13, 546-551 (2007).
[CrossRef]

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460-472 (2007).
[CrossRef]

L. Liu, M. A. Vorontsov, E. Polnau, T. Weyrauch, and L. A. Beresnev, “Adaptive phase-locked fiber array with wavefront phase tip-tilt compensation using piezoelectric fiber positioners,” Proc. SPIE 6708, 67080K (2007).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467-475 (2007).
[CrossRef]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

S. J. Augst, J. Ranka, T. Y. Fan, and A. Sanchez, “Beam combining of ytterbium fiber amplifiers,” J. Opt. Soc. Am. B 24, 1707-1715 (2007).
[CrossRef]

W. Liang, N. Satyan, F. Aflatouni, A. Yariv, A. Kewitsch, G. Rakuljic, and H. Hashemi, “Coherent beam combining with multilevel optical phase locked loops,” J. Opt. Soc. Am. B 24, 2930-2939 (2007).
[CrossRef]

2006

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14, 1353-1367(2006).
[CrossRef] [PubMed]

H. T. Eyyuboğlu, A. Arpali, and Y. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14, 4196-4207 (2006).
[CrossRef] [PubMed]

Y. Cai and S. He, “Partially coherent flattened Gaussian beam and its paraxial propagation properties,” J. Opt. Soc. Am. A 23, 2623-2628 (2006).
[CrossRef]

J. Anderegg, S. Brosnan, E. C. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A: Pure Appl. Opt. 8, 537-545(2006).
[CrossRef]

A. Desfarges-Berthelemot, V. Kermène, D. Sabourdy, J. Boullet, P. Roy, J. Lhermite, and A. Barthélémy, “Coherent combining of fibre lasers,” C. R. Physique 7, 244-253 (2006).
[CrossRef]

L. Michaille, C. R. Bennett, D. M. Taylor, and T. J. Shepherd, “Multi-core photonic crystal fibers for high power lasers and amplifiers,” Proc. SPIE 6102, 61020W (2006).
[CrossRef]

2005

C. J. Corcoran and F. Durville, “Experimental demonstration of a phase locked laser array using a self-Fourier cavity,” Appl. Phys. Lett. 86, 201118 (2005).
[CrossRef]

X. Ji and B. Lü, “Turbulence-induced quality degradation of partially coherent beams,” Opt. Commun. 251, 231-236(2005).
[CrossRef]

H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. 30, 1339-1341 (2005).
[CrossRef] [PubMed]

H. T. Eyyuboğlu, “Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere,” J. Opt. Soc. Am. A 22, 1527-1535 (2005).
[CrossRef]

2004

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A: Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

2003

2002

1994

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335-341 (1994).
[CrossRef]

1979

Aflatouni, F.

Alavinejad, M.

M. Alavinejad and B. Ghafary, “Turbulence-induced degradation properties of partially coherent flat-topped beams,” Opt. Lasers Eng. 46, 357-362 (2008).
[CrossRef]

Anderegg, J.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460-472 (2007).
[CrossRef]

J. Anderegg, S. Brosnan, E. C. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Arpali, A.

Asman, C. P.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460-472 (2007).
[CrossRef]

Augst, S. J.

Baker, J. T.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

Bammert, S.

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in CLEO/QELS/PHAST (Optical Society of America, 2007).

Barthélémy, A.

A. Desfarges-Berthelemot, V. Kermène, D. Sabourdy, J. Boullet, P. Roy, J. Lhermite, and A. Barthélémy, “Coherent combining of fibre lasers,” C. R. Physique 7, 244-253 (2006).
[CrossRef]

Baykal, Y.

H. T. Eyyuboğlu, Y. Baykal, and Y. Y. Cai, “Scintillations of laser array beams,” Appl. Phys. B 91, 265-271 (2008).
[CrossRef]

Y. Cai, X. Lu, H. T. Eyyuboğlu, and Y. Baykal, “Paraxial propagation of a partially coherent flattened Gaussian beam through apertured ABCD optical systems,” Opt. Commun. 281, 3221-3229 (2008).
[CrossRef]

Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467-475 (2007).
[CrossRef]

H. T. Eyyuboğlu, A. Arpali, and Y. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14, 4196-4207 (2006).
[CrossRef] [PubMed]

Bennai, B.

P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
[CrossRef]

Bennett, C. R.

L. Michaille, C. R. Bennett, D. M. Taylor, and T. J. Shepherd, “Multi-core photonic crystal fibers for high power lasers and amplifiers,” Proc. SPIE 6102, 61020W (2006).
[CrossRef]

Beresnev, L. A.

L. Liu, M. A. Vorontsov, E. Polnau, T. Weyrauch, and L. A. Beresnev, “Adaptive phase-locked fiber array with wavefront phase tip-tilt compensation using piezoelectric fiber positioners,” Proc. SPIE 6708, 67080K (2007).
[CrossRef]

Bochove, E. J.

Boullet, J.

A. Desfarges-Berthelemot, V. Kermène, D. Sabourdy, J. Boullet, P. Roy, J. Lhermite, and A. Barthélémy, “Coherent combining of fibre lasers,” C. R. Physique 7, 244-253 (2006).
[CrossRef]

Bourdon, P.

P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
[CrossRef]

Bronder, T. J.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

Brosnan, S.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460-472 (2007).
[CrossRef]

J. Anderegg, S. Brosnan, E. C. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Brosnan, S. J.

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G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460-472 (2007).
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Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13, 546-551 (2007).
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P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
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J. Marmo, H. Injeyan, H. Komine, S. McNaught, J. Machan, and J. Sollee, “Joint high power solid state laser program advancements at Northrop Grumman,” Proc. SPIE 7195, 719507 (2009).
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S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in CLEO/QELS/PHAST (Optical Society of America, 2007).

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A. Desfarges-Berthelemot, V. Kermène, D. Sabourdy, J. Boullet, P. Roy, J. Lhermite, and A. Barthélémy, “Coherent combining of fibre lasers,” C. R. Physique 7, 244-253 (2006).
[CrossRef]

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B. Li and B. Lü, “Characterization of off-axis superposition of partially coherent beams,” J. Opt. A: Pure Appl. Opt. 5, 303-307 (2003).
[CrossRef]

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X. Ji and X. Li, “Influence of turbulence on the coherent and incoherent combinations of off-axis Gaussian beams,” Acta Phys. Sin. 57, 7674-7679 (2008).

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Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Off-axis Gaussian Schell-model beam and partially coherent laser array beam in a turbulent atmosphere,” Opt. Commun. 278, 157-167 (2007).
[CrossRef]

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A: Pure Appl. Opt. 6, 390-395 (2004).
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L. Liu, M. A. Vorontsov, E. Polnau, T. Weyrauch, and L. A. Beresnev, “Adaptive phase-locked fiber array with wavefront phase tip-tilt compensation using piezoelectric fiber positioners,” Proc. SPIE 6708, 67080K (2007).
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P. Zhou, Z. Liu, X. Xu, and X. Chu, “Comparative study on the propagation performance of coherently combined and incoherently combined beams,” Opt. Commun. 282, 1640-1647(2009).
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P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248-256 (2009).
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X. Chu, Z. Liu, and Y. Wu, “Propagation of a general multi-Gaussian beam in turbulent atmosphere in a slant path,” J. Opt. Soc. Am. A 25, 74-49 (2008).
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P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
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G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460-472 (2007).
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T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

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C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17, 1753-1765 (2009).
[CrossRef] [PubMed]

Y. Cai, X. Lu, H. T. Eyyuboğlu, and Y. Baykal, “Paraxial propagation of a partially coherent flattened Gaussian beam through apertured ABCD optical systems,” Opt. Commun. 281, 3221-3229 (2008).
[CrossRef]

Lü, B.

E. Zhang, X. Ji, and B. Lü, “Propagation of the off-axis superposition of partially coherent beams through atmospheric turbulence,” Chin. Phys. B 18, 571-580 (2008).

X. Xiao, X. Ji, and B. Lü, “The influence of turbulence on propagation properties of partially coherent sinh-Gaussian beams and their beam quality in the far field,” Opt. Laser Technol. 40, 129-136 (2008).
[CrossRef]

X. Ji, E. Zhang, and B. Lü, “Superimposed partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. B 25, 825-833 (2008).
[CrossRef]

X. Ji and B. Lü, “Turbulence-induced quality degradation of partially coherent beams,” Opt. Commun. 251, 231-236(2005).
[CrossRef]

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T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
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P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248-256 (2009).
[CrossRef]

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P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248-256 (2009).
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J. Marmo, H. Injeyan, H. Komine, S. McNaught, J. Machan, and J. Sollee, “Joint high power solid state laser program advancements at Northrop Grumman,” Proc. SPIE 7195, 719507 (2009).
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G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460-472 (2007).
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[CrossRef]

Payne, D. N.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13, 546-551 (2007).
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Polnau, E.

L. Liu, M. A. Vorontsov, E. Polnau, T. Weyrauch, and L. A. Beresnev, “Adaptive phase-locked fiber array with wavefront phase tip-tilt compensation using piezoelectric fiber positioners,” Proc. SPIE 6708, 67080K (2007).
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Pourtal, E.

P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
[CrossRef]

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X. Ji and Z. Pu, “Angular spread of Gaussian Schell-model array beams propagating through atmospheric turbulence,” Appl. Phys. B 93, 915-923 (2008).
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T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
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Ranka, J.

Redmond, S.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460-472 (2007).
[CrossRef]

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in CLEO/QELS/PHAST (Optical Society of America, 2007).

Rice, R. R.

Robin, C. A.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

Rogers, J. L.

Rothenberg, J.

Rothenberg, J. E.

J. E. Rothenberg, “Passive coherent phasing of fiber laser arrays,” Proc. SPIE 6873, 687315 (2008).
[CrossRef]

Roy, P.

A. Desfarges-Berthelemot, V. Kermène, D. Sabourdy, J. Boullet, P. Roy, J. Lhermite, and A. Barthélémy, “Coherent combining of fibre lasers,” C. R. Physique 7, 244-253 (2006).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic , 1980).

Sabourdy, D.

A. Desfarges-Berthelemot, V. Kermène, D. Sabourdy, J. Boullet, P. Roy, J. Lhermite, and A. Barthélémy, “Coherent combining of fibre lasers,” C. R. Physique 7, 244-253 (2006).
[CrossRef]

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Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13, 546-551 (2007).
[CrossRef]

Saitou, T.

Sanchez, A.

Sanchez, A. D.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

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Sekiguchi, T.

Shay, T. M.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

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L. Michaille, C. R. Bennett, D. M. Taylor, and T. J. Shepherd, “Multi-core photonic crystal fibers for high power lasers and amplifiers,” Proc. SPIE 6102, 61020W (2006).
[CrossRef]

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Shirakawa, A.

Simpson, R.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460-472 (2007).
[CrossRef]

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Sollee, J.

J. Marmo, H. Injeyan, H. Komine, S. McNaught, J. Machan, and J. Sollee, “Joint high power solid state laser program advancements at Northrop Grumman,” Proc. SPIE 7195, 719507 (2009).
[CrossRef]

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460-472 (2007).
[CrossRef]

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S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in CLEO/QELS/PHAST (Optical Society of America, 2007).

Taylor, D. M.

L. Michaille, C. R. Bennett, D. M. Taylor, and T. J. Shepherd, “Multi-core photonic crystal fibers for high power lasers and amplifiers,” Proc. SPIE 6102, 61020W (2006).
[CrossRef]

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Turner, P. W.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13, 546-551 (2007).
[CrossRef]

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Vasseur, O.

P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
[CrossRef]

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T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

Vorontsov, M. A.

L. Liu, M. A. Vorontsov, E. Polnau, T. Weyrauch, and L. A. Beresnev, “Adaptive phase-locked fiber array with wavefront phase tip-tilt compensation using piezoelectric fiber positioners,” Proc. SPIE 6708, 67080K (2007).
[CrossRef]

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Wang, S. C. H.

Wang, X.

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248-256 (2009).
[CrossRef]

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E. C. Cheung, J. G. Ho, G. D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wickham, “Diffractive-optics-based beam combination of a phase-locked fiber laser array,” Opt. Lett. 33, 354-356 (2008).
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G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460-472 (2007).
[CrossRef]

J. Anderegg, S. Brosnan, E. C. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Weiss, S. B.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460-472 (2007).
[CrossRef]

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in CLEO/QELS/PHAST (Optical Society of America, 2007).

Weyrauch, T.

L. Liu, M. A. Vorontsov, E. Polnau, T. Weyrauch, and L. A. Beresnev, “Adaptive phase-locked fiber array with wavefront phase tip-tilt compensation using piezoelectric fiber positioners,” Proc. SPIE 6708, 67080K (2007).
[CrossRef]

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S. J. Brosnan, M. G. Wichham, and H. Komine, “Method and apparatus for optimizing the target intensity distribution transmitted from a fiber coupled array,” U.S. patent 7,283,702 (16 October 2007).

Wickham, M.

E. C. Cheung, J. G. Ho, G. D. Goodno, R. R. Rice, J. Rothenberg, P. Thielen, M. Weber, and M. Wickham, “Diffractive-optics-based beam combination of a phase-locked fiber laser array,” Opt. Lett. 33, 354-356 (2008).
[CrossRef] [PubMed]

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460-472 (2007).
[CrossRef]

J. Anderegg, S. Brosnan, E. C. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

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Wu, Y.

Xiao, X.

X. Xiao, X. Ji, and B. Lü, “The influence of turbulence on propagation properties of partially coherent sinh-Gaussian beams and their beam quality in the far field,” Opt. Laser Technol. 40, 129-136 (2008).
[CrossRef]

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P. Zhou, Z. Liu, X. Xu, and X. Chu, “Comparative study on the propagation performance of coherently combined and incoherently combined beams,” Opt. Commun. 282, 1640-1647(2009).
[CrossRef]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248-256 (2009).
[CrossRef]

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Zamel, J.

S. Redmond, S. McNaught, J. Zamel, L. Iwaki, S. Bammert, R. Simpson, S. B. Weiss, J. Szot, B. Flegal, T. Lee, H. Komine, and H. Injeyan, “15 kW near-diffraction-limited single-frequency Nd:YAG laser,” in CLEO/QELS/PHAST (Optical Society of America, 2007).

Zerinque, C.

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
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E. Zhang, X. Ji, and B. Lü, “Propagation of the off-axis superposition of partially coherent beams through atmospheric turbulence,” Chin. Phys. B 18, 571-580 (2008).

X. Ji, E. Zhang, and B. Lü, “Superimposed partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. B 25, 825-833 (2008).
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Zhao, D.

Zhou, P.

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248-256 (2009).
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P. Zhou, Z. Liu, X. Xu, and X. Chu, “Comparative study on the propagation performance of coherently combined and incoherently combined beams,” Opt. Commun. 282, 1640-1647(2009).
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Acta Phys. Sin.

X. Ji and X. Li, “Influence of turbulence on the coherent and incoherent combinations of off-axis Gaussian beams,” Acta Phys. Sin. 57, 7674-7679 (2008).

Appl. Phys. B

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A. Desfarges-Berthelemot, V. Kermène, D. Sabourdy, J. Boullet, P. Roy, J. Lhermite, and A. Barthélémy, “Coherent combining of fibre lasers,” C. R. Physique 7, 244-253 (2006).
[CrossRef]

Chin. Phys. B

E. Zhang, X. Ji, and B. Lü, “Propagation of the off-axis superposition of partially coherent beams through atmospheric turbulence,” Chin. Phys. B 18, 571-580 (2008).

IEEE J. Sel. Top. Quantum Electron.

Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-amplifier sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. 13, 546-551 (2007).
[CrossRef]

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, Jr., M. McClellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-scaling potential of actively phase-locked solid-state laser arrays,” IEEE J. Sel. Top. Quantum Electron. 13, 460-472 (2007).
[CrossRef]

P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. 15, 248-256 (2009).
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[CrossRef]

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L. Michaille, C. R. Bennett, D. M. Taylor, and T. J. Shepherd, “Multi-core photonic crystal fibers for high power lasers and amplifiers,” Proc. SPIE 6102, 61020W (2006).
[CrossRef]

Proc. SPIE

L. Liu, M. A. Vorontsov, E. Polnau, T. Weyrauch, and L. A. Beresnev, “Adaptive phase-locked fiber array with wavefront phase tip-tilt compensation using piezoelectric fiber positioners,” Proc. SPIE 6708, 67080K (2007).
[CrossRef]

Proc. SPIE

J. E. Rothenberg, “Passive coherent phasing of fiber laser arrays,” Proc. SPIE 6873, 687315 (2008).
[CrossRef]

T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “High power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

Proc. SPIE

J. Anderegg, S. Brosnan, E. C. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[CrossRef]

Proc. SPIE

J. Marmo, H. Injeyan, H. Komine, S. McNaught, J. Machan, and J. Sollee, “Joint high power solid state laser program advancements at Northrop Grumman,” Proc. SPIE 7195, 719507 (2009).
[CrossRef]

P. Bourdon, V. Jolivet, B. Bennai, L. Lombard, G. Canat, E. Pourtal, Y. Jaouen, and O. Vasseur, “Coherent beam combining of fiber amplifier arrays and application to laser beam propagation through turbulent atmosphere,” Proc. SPIE 6873, 687316 (2008).
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Figures (9)

Fig. 1
Fig. 1

Near-field intensity distributions of a partially coherent rectangular flat-topped laser array: (a) normalized 3-D intensity distribution of a single laser and (b) geometric arrangement of the laser array.

Fig. 2
Fig. 2

Average intensity distribution of a partially coherent rectangular flat-topped laser array at the receiving plane: (a)  σ 0 = 3 ω 0 y , C n 2 = 10 16 m 2 / 3 ; (b)  σ 0 = ω 0 y , C n 2 = 10 16 m 2 / 3 ; (c)  σ 0 = 0.5 ω 0 y , C n 2 = 10 16 m 2 / 3 ; (d)  σ 0 = 3 ω 0 y , (e)  σ 0 = ω 0 y free space (f)  σ 0 = 0.5 ω 0 y free space.

Fig. 3
Fig. 3

PIB curves for different values of σ 0 in different turbulent atmospheres: (a) free space, (b)  C n 2 = 10 16 m 2 / 3 , (c)  C n 2 = 10 15 m 2 / 3 .

Fig. 4
Fig. 4

Average intensity distribution of a partially coherent rectangular flat-topped laser array with different beam orders at the receiving plane: (a)  N = 3 , (b)  N = 5 , (c)  N = 7 .

Fig. 5
Fig. 5

PIB curves at the receiving plane for different beam orders.

Fig. 6
Fig. 6

Average intensity distribution of a partially coherent rectangular flat-topped laser array with different source sizes: (a)  ω 0 x = 0.75 cm , ω 0 y = 2 cm and (b)  ω 0 x = 3 cm , ω 0 y = 8 cm .

Fig. 7
Fig. 7

PIB curves at the receiving plane for a laser array with different source sizes.

Fig. 8
Fig. 8

Average intensity distribution of a partially coherent rectangular flat-topped laser array made up of six lasers at the receiving plane: (a) geometric arrangement of the laser array and (b) average intensity distribution.

Fig. 9
Fig. 9

PIB curves at the receiving plane for a laser array created by a different number of lasers.

Equations (11)

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E single ( x , y , z = 0 ) = Q x = 1 N x Q y = 1 N y ( 1 ) Q x + Q y N x N y ( N x Q x ) ( N y Q y ) exp { Q x ( x a ) 2 ω 0 x 2 Q y ( y b ) 2 ω 0 y 2 } ,
G ( x , ξ , y , η ) = exp { ( x ξ ) 2 / 2 σ x 2 ( y η ) 2 / 2 σ y 2 } ,
W ( x , y , ξ , η , z = 0 ) = α = 1 M β = 1 M E α ( x , y , z = 0 ) E β * ( ξ , η , z = 0 ) exp { [ ( x a α ) ( ξ a β ) ] 2 / 2 σ x 2 [ ( y b α ) ( η b β ) ] 2 / 2 σ y 2 } = α = 1 M β = 1 M Q x 1 = 1 N x Q y 1 = 1 N y Q x 2 = 1 N x Q y 2 = 1 N y ( 1 ) Q x 1 + Q y 1 + Q x 2 + Q y 2 N x N x N y N y ( N x Q x 1 ) ( N X Q x 2 ) ( N y Q y 1 ) ( N y Q y 2 ) exp { Q x 1 ( x a α ) 2 ω 0 x 2 Q y 1 ( y b α ) 2 ω 0 y 2 Q x 2 ( x a β ) 2 ω 0 x 2 Q y 2 ( y b β ) 2 ω 0 y 2 } exp { [ ( x a α ) ( ξ a β ) ] 2 / 2 σ x 2 [ ( y b α ) ( η b β ) ] 2 / 2 σ y 2 } ,
I ( p , q , z ) = k 2 ( 2 π z ) 2 W ( x , y , ξ , η , z = 0 ) exp { i k 2 z [ ( p x ) 2 + ( q y ) 2 ( p ξ ) 2 ( q η ) 2 ] } exp [ ψ ( x , y , p , q ) + ψ * ( ξ , η , p , q ) ] d x d y d ξ d η ,
exp [ ψ ( x , y , p , q ) + ψ * ( ξ , η , p , q ) ] = exp { 1 ρ 0 2 [ ( x ξ ) 2 + ( y η ) 2 ] } .
I N ( p , q , z ) = k 2 ( 2 π z ) 2 α = 1 M β = 1 M Q x 1 = 1 N x Q x 2 = 1 N x Q y 1 = 1 N y Q y 2 = 1 N y Γ α β Q x 1 Q x 2 Q y 1 Q y 2 ( p , q , z ) ,
Γ α β Q x 1 Q x 2 Q y 1 Q y 2 ( p , q , z ) = k 2 ( 2 π z ) 2 ( 1 ) Q x 1 + Q y 1 + Q x 2 + Q y 2 N x 2 N y 2 ( N x Q x 1 ) ( N x Q x 2 ) ( N y Q y 1 ) ( N y Q y 2 ) × exp { [ Q x 1 ( x a α ) 2 + Q x 2 ( ξ a β ) 2 ] ω 0 x 2 [ Q y 1 ( y b α ) 2 + Q y 2 ( η b β ) 2 ] ω 0 y 2 } × exp { i k 2 z [ ( p x ) 2 + ( q y ) 2 ( p ξ ) 2 ( q η ) 2 ] } exp { 1 ρ 0 2 [ ( x ξ ) 2 + ( y η ) 2 ] } × exp { [ ( x a α ) ( ξ a β ) ] 2 2 σ x 2 [ ( y b α ) ( η b β ) ] 2 2 σ y 2 } d x d y d ξ d η .
Γ α β Q x 1 Q x 2 Q y 1 Q y 2 ( p , q , z ) = k 2 ( 2 z ) 2 ( 1 ) Q x 1 + Q y 1 + Q x 2 + Q y 2 N x 2 N y 2 ( N x Q x 1 ) ( N x Q x 2 ) ( N y Q y 1 ) ( N y Q y 2 ) H ( p , z ) H ( q , z ) ,
H ( p , z ) = π β 1 x β 2 x + 4 ρ 1 x 4 exp ( Q x 1 a α 2 ω 0 x 2 Q x 2 a β 2 ω 0 x 2 ( a α a β ) 2 2 σ 0 x 2 + ( a β K 2 x + p K 3 β 1 x ) 2 ) × exp ( ( 2 a α β 2 x K 1 x ρ 1 x 2 + 2 p β 2 x K 3 ρ 1 x 2 2 a β K 2 x 2 p K 3 ) 2 4 ( β 1 x + 4 / β 2 x ρ 1 x 4 ) β 2 x 2 ρ 1 x 4 ) ,
H ( q , z ) = π β 1 y β 2 y + 4 ρ 1 y 4 exp ( Q y 1 b α 2 ω 0 y 2 Q y 2 b β 2 ω 0 y 2 ( b α b β ) 2 2 σ 0 y 2 + ( b β K 2 y + q K 3 β 1 y ) 2 ) × exp ( ( 2 b α β 2 y K 1 y ρ 1 y 2 + 2 q β 2 y K 3 ρ 1 y 2 2 b β K 2 y 2 q K 3 ) 2 4 ( β 1 y + 4 / β 2 y ρ 1 y 4 ) β 2 y 2 ρ 1 y 4 ) ,
PIB ( r , z ) = 0 2 π 0 r I ( ρ , z ) ρ d ρ d θ 0 2 π 0 I ( ρ , z ) ρ d ρ d θ ,

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