Abstract

A Lorentz beam array presents a good model to study a coherent diode laser array, which is an efficient radiation source for high-power beaming use. We present a detailed study of the propagation properties of a Lorentz beam array. Closed-form intensity distribution in the spatial frequency domain is deduced, and the effect of phase errors on the far-field intensity pattern is studied in detail. M2 factors for two limiting cases, i.e., coherent and incoherent combined laser arrays, are derived and compared. The pro pagation property of the Lorentz beam array is also compared with Gaussian and flat-topped beam arrays.

© 2010 Optical Society of America

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    [Crossref]

2009 (8)

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. 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]

P. Zhou, Y. Ma, X. Wang, H. Ma, X. Xu, and Z. Liu, “Average intensity of a partially coherent rectangular flat-topped laser array propagating in a turbulent atmosphere,” Appl. Opt. 48, 5251–5258 (2009).
[Crossref]

G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz beam,” Opt. Laser Technol. 41, 953–955 (2009).
[Crossref]

G. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B 26, 141–147 (2009).
[Crossref]

G. Zhou, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B 96, 149–153 (2009).
[Crossref]

S. P. Ng and P. B. Phua, “Coherent polarization locking of a diode emitter array,” Opt. Lett. 34, 2042–2044 (2009).
[Crossref]

P. Zhou, Z. Liu, X. Xu, and X. Chu, “Propagation of coherently combined flattened laser beam array in turbulent atmosphere,” Opt. Laser Technol. 41, 403–407 (2009).
[Crossref]

2008 (6)

G. Zhou, “Analytical vectorial structure of a Lorentz-Gauss beam in the far field,” Appl. Phys. B 93, 891–899(2008).
[Crossref]

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]

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

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]

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

2007 (7)

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]

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]

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]

O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangularly symmetric optical fields,” Opt. Commun. 269, 274–284 (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]

W. Liang, A. Yariv, A. Kewitsch, and G. Rakuljic, “Coherent combining of the output of two semiconductor lasers using optical phase-lock loops,” Opt. Lett. 32, 370–372(2007).
[Crossref]

2006 (2)

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A Pure Appl. Opt. 8, 409–414 (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]

2003 (1)

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]

2000 (1)

1994 (1)

1992 (1)

1990 (1)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[Crossref]

1989 (1)

J. P. Hobimer, D. R. Myer, T. M. Brennan, and B. E. Hammons, “Integrated injection-locked high-power cw diode laser arrays,” Appl. Phys. Lett. 55, 531–533 (1989).
[Crossref]

1988 (1)

J. R. Leger, M. L. Scott, and W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771–1773 (1988).
[Crossref]

1982 (1)

D. S. Elliot, R. Roy, and S. J. Smith, “Extracavity laser band-shape and bandwidth modification,” Phys. Rev. A 26, 12–18(1982).
[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]

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]

Baykal, Y.

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

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]

Brennan, T. M.

J. P. Hobimer, D. R. Myer, T. M. Brennan, and B. E. Hammons, “Integrated injection-locked high-power cw diode laser arrays,” Appl. Phys. Lett. 55, 531–533 (1989).
[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]

Cai, Y.

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

Chen, Y.

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]

Cheung, E. C.

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]

Chu, X.

P. Zhou, Z. Liu, X. Xu, and X. Chu, “Propagation of coherently combined flattened laser beam array in turbulent atmosphere,” Opt. Laser Technol. 41, 403–407 (2009).
[Crossref]

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]

Du, X.

Elliot, D. S.

D. S. Elliot, R. Roy, and S. J. Smith, “Extracavity laser band-shape and bandwidth modification,” Phys. Rev. A 26, 12–18(1982).
[Crossref]

Epp, P.

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]

Eyyuboglu, H. T.

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

Fan, T. Y.

Gallant, 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]

Gawhary, O. E.

O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangularly symmetric optical fields,” Opt. Commun. 269, 274–284 (2007).
[Crossref]

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A Pure Appl. Opt. 8, 409–414 (2006).
[Crossref]

Goodno, G. D.

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]

Gradshteyn, I. S.

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

Guo, S.

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]

Hammons, B. E.

J. P. Hobimer, D. R. Myer, T. M. Brennan, and B. E. Hammons, “Integrated injection-locked high-power cw diode laser arrays,” Appl. Phys. Lett. 55, 531–533 (1989).
[Crossref]

Hammons, D.

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]

Hobimer, J. P.

J. P. Hobimer, D. R. Myer, T. M. Brennan, and B. E. Hammons, “Integrated injection-locked high-power cw diode laser arrays,” Appl. Phys. Lett. 55, 531–533 (1989).
[Crossref]

Injeyan, H.

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]

Ji, X.

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

Kewitsch, A.

Komine, H.

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]

Leger, J. R.

J. R. Leger, M. L. Scott, and W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771–1773 (1988).
[Crossref]

Li, B.

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]

Liang, W.

Lin, Q.

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]

Liu, L.

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]

Liu, Z.

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]

P. Zhou, Z. Liu, X. Xu, and X. Chu, “Propagation of coherently combined flattened laser beam array in turbulent atmosphere,” Opt. Laser Technol. 41, 403–407 (2009).
[Crossref]

P. Zhou, Y. Ma, X. Wang, H. Ma, X. Xu, and Z. Liu, “Average intensity of a partially coherent rectangular flat-topped laser array propagating in a turbulent atmosphere,” Appl. Opt. 48, 5251–5258 (2009).
[Crossref]

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]

Long, W. H.

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]

Lu, 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]

Lü, B.

Lucero, 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]

Ma, H.

Ma, Y.

P. Zhou, Y. Ma, X. Wang, H. Ma, X. Xu, and Z. Liu, “Average intensity of a partially coherent rectangular flat-topped laser array propagating in a turbulent atmosphere,” Appl. Opt. 48, 5251–5258 (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]

McClellan, M.

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]

McNaught, S. 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]

Myer, D. R.

J. P. Hobimer, D. R. Myer, T. M. Brennan, and B. E. Hammons, “Integrated injection-locked high-power cw diode laser arrays,” Appl. Phys. Lett. 55, 531–533 (1989).
[Crossref]

Nabors, C. D.

Nakagawa, K.

Ng, S. P.

Phua, P. B.

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).
[Crossref]

Pu, Z.

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]

Pulford, B.

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]

Rakuljic, G.

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]

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]

Roy, R.

D. S. Elliot, R. Roy, and S. J. Smith, “Extracavity laser band-shape and bandwidth modification,” Phys. Rev. A 26, 12–18(1982).
[Crossref]

Ryzhik, I. M.

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

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]

Sayama, S.

Scott, M. L.

J. R. Leger, M. L. Scott, and W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771–1773 (1988).
[Crossref]

Severini, S.

O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangularly symmetric optical fields,” Opt. Commun. 269, 274–284 (2007).
[Crossref]

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A Pure Appl. Opt. 8, 409–414 (2006).
[Crossref]

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]

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[Crossref]

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]

Smith, S. J.

D. S. Elliot, R. Roy, and S. J. Smith, “Extracavity laser band-shape and bandwidth modification,” Phys. Rev. A 26, 12–18(1982).
[Crossref]

Sollee, 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]

Veldkamp, W. B.

J. R. Leger, M. L. Scott, and W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771–1773 (1988).
[Crossref]

Vergien, C. L.

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]

Wang, W.

Wang, X.

P. Zhou, Y. Ma, X. Wang, H. Ma, X. Xu, and Z. Liu, “Average intensity of a partially coherent rectangular flat-topped laser array propagating in a turbulent atmosphere,” Appl. Opt. 48, 5251–5258 (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]

Weber, M.

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]

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]

Wickham, M.

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]

Wu, Y.

Xu, X.

P. Zhou, Z. Liu, X. Xu, and X. Chu, “Propagation of coherently combined flattened laser beam array in turbulent atmosphere,” Opt. Laser Technol. 41, 403–407 (2009).
[Crossref]

P. Zhou, Y. Ma, X. Wang, H. Ma, X. Xu, and Z. Liu, “Average intensity of a partially coherent rectangular flat-topped laser array propagating in a turbulent atmosphere,” Appl. Opt. 48, 5251–5258 (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]

Yariv, A.

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).
[Crossref]

Zhang, E.

Zhao, D.

Zhou, G.

G. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B 26, 141–147 (2009).
[Crossref]

G. Zhou, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B 96, 149–153 (2009).
[Crossref]

G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz beam,” Opt. Laser Technol. 41, 953–955 (2009).
[Crossref]

G. Zhou, “Analytical vectorial structure of a Lorentz-Gauss beam in the far field,” Appl. Phys. B 93, 891–899(2008).
[Crossref]

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).
[Crossref]

P. Zhou, Z. Liu, X. Xu, and X. Chu, “Propagation of coherently combined flattened laser beam array in turbulent atmosphere,” Opt. Laser Technol. 41, 403–407 (2009).
[Crossref]

P. Zhou, Y. Ma, X. Wang, H. Ma, X. Xu, and Z. Liu, “Average intensity of a partially coherent rectangular flat-topped laser array propagating in a turbulent atmosphere,” Appl. Opt. 48, 5251–5258 (2009).
[Crossref]

Zhu, Y.

Appl. Opt. (3)

Appl. Phys. B (5)

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]

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]

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

G. Zhou, “Analytical vectorial structure of a Lorentz-Gauss beam in the far field,” Appl. Phys. B 93, 891–899(2008).
[Crossref]

G. Zhou, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B 96, 149–153 (2009).
[Crossref]

Appl. Phys. Lett. (2)

J. R. Leger, M. L. Scott, and W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771–1773 (1988).
[Crossref]

J. P. Hobimer, D. R. Myer, T. M. Brennan, and B. E. Hammons, “Integrated injection-locked high-power cw diode laser arrays,” Appl. Phys. Lett. 55, 531–533 (1989).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

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]

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. Opt. A Pure Appl. Opt. (2)

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]

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A Pure Appl. Opt. 8, 409–414 (2006).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (3)

Opt. Commun. (2)

O. E. Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangularly symmetric optical fields,” Opt. Commun. 269, 274–284 (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]

Opt. Express (1)

Opt. Laser Technol. (2)

P. Zhou, Z. Liu, X. Xu, and X. Chu, “Propagation of coherently combined flattened laser beam array in turbulent atmosphere,” Opt. Laser Technol. 41, 403–407 (2009).
[Crossref]

G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz beam,” Opt. Laser Technol. 41, 953–955 (2009).
[Crossref]

Opt. Lett. (3)

Phys. Rev. A (1)

D. S. Elliot, R. Roy, and S. J. Smith, “Extracavity laser band-shape and bandwidth modification,” Phys. Rev. A 26, 12–18(1982).
[Crossref]

Proc. SPIE (4)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[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]

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]

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]

Other (2)

Northrop Grumman Space Technology, El Segundo, Calif., http://www.irconnect.com/noc/press/pages/news_releases.html?d=161575.

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

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Figures (5)

Fig. 1
Fig. 1

Two-dimensional sketch map of the P × Q Lorentz beams array.

Fig. 2
Fig. 2

Intensity distribution of the Lorentz beam array in the spatial frequency domain with different phase errors, λ = 800 nm , ω x = ω y = 5 λ , d x = d y = 3 ω 0 , and P = Q = 5 : (a)  σ 2 = 0 , (b)  σ 2 = 0.5 , (c)  σ 2 = 1 , (d)  σ 2 = 1.5 , (e)  σ 2 = 2.5 , (f)  σ 2 = π , (g)  σ 2 = 2 π , and (h)  σ 2 = 4 π .

Fig. 3
Fig. 3

Intensity distribution of the Lorentz beam array in the spatial frequency domain with different separation distances between adjacent elements, λ = 800 nm , ω x = ω y = 5 λ , and P = Q = 5 : (a) coherent beam combining, d x = d y = 5 ω 0 ; (b) incoherent beam combining, d x = d y = 5 ω 0 ; (c) coherent beam combining, d x = d y = 7 ω 0 ; and (d) incoherent beam combining, d x = d y = 7 ω 0 .

Fig. 4
Fig. 4

M 2 factor as a function of array numbers in the cases of coherent and incoherent beam combining, λ = 800 nm , and ω x = ω y = 5 λ : (a)  d x = d y = 3 ω 0 , (b)  d x = d y = 4 ω 0 , (c)  d x = d y = 5 ω 0 , and (d)  d x = d y = 6 ω 0 .

Fig. 5
Fig. 5

M 2 factor as a function of array numbers in the cases of coherent and incoherent beam combining for the Lorentz, Gaussian, and flat-topped beam arrays, λ = 800 nm , ω x = ω y = 3 λ , and d x = d y = 5 ω 0 : (a) coherent array and (b) incoherent array.

Equations (16)

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E j ( x , y , 0 ) = 1 ω x ω y 1 { 1 + [ ( x a j ) / ω x ] 2 } 1 { 1 + [ ( y b j ) / ω y ] 2 } exp ( i ϕ j ) ,
I incoherent ( x , y , 0 ) = j = 1 N | E j ( x , y , 0 ) | 2 = j = 1 N | 1 ω x ω y 1 { 1 + [ ( x a j ) / ω x ] 2 } 1 { 1 + [ ( y b j ) / ω y ] 2 } | 2 .
σ 0 x _ incoherent 2 = x 2 I incoherent ( x , y , 0 ) d x d y I incoherent ( x , y , 0 ) d x d y = j = 1 N 1 ω x 2 ω y 2 π ω y 2 ( π ω x 3 2 + π a j 2 ω x 2 ) j = 1 N 1 ω x 2 ω y 2 π ω y 2 π ω x 2 = N ω x 2 + j = 1 N a j 2 N .
σ 0 y _ incoherent 2 = y 2 I incoherent ( x , y , 0 ) d x d y I incoherent ( x , y , 0 ) d x d y = j = 1 N 1 ω x 2 ω y 2 π ω x 2 ( π ω y 3 2 + π b j 2 ω y 2 ) j = 1 N 1 ω x 2 ω y 2 π ω y 2 π ω x 2 = N ω y 2 + j = 1 N b j 2 N .
I coherent ( x , y , 0 ) = | j = 1 N E j ( x , y , 0 ) | 2 = | j = 1 N 1 ω x ω y 1 { 1 + [ ( x a j ) / ω x ] 2 } 1 { 1 + [ ( y b j ) / ω y ] 2 } | 2 .
F j ( f x , f y ) = I [ E j ( f x , f y ) ] = π 2 exp ( 2 π ω x | f x | ) exp ( 2 π ω y | f y | ) exp [ i 2 π ( f x a j + f y b j ) ] exp ( i ϕ j ) ,
E ( f x , f y ) = j = 1 N F j ( f x , f y ) = π 2 j = 1 N exp ( 2 π ω x | f x | ) exp ( 2 π ω y | f y | ) exp [ i 2 π ( f x a j + f y b j ) ] exp ( i ϕ j ) .
I ( f x , f y ) = π 4 | j = 1 N exp ( 2 π ω x | f x | ) exp ( 2 π ω y | f y | ) exp [ i 2 π ( f x a j + f y b j ) ] exp ( i ϕ j ) | 2 = π 4 j = 1 N exp ( 4 π ω x | f x | ) exp ( 4 π ω y | f y | ) + 2 π 4 j = 1 N 1 l = j + 1 N exp ( 4 π ω x | f x | ) exp ( 4 π ω y | f y | ) exp { i 4 π [ f x ( a j a l ) + f y ( b j b l ) ] } exp [ i ( ϕ j ϕ l ) ] .
exp { j ( ϕ m ϕ n ) } = exp ( σ 2 / 2 ) .
I ( f x , f y ) = π 4 | j = 1 N exp ( 2 π ω x | f x | ) exp ( 2 π ω y | f y | ) exp [ i 2 π ( f x a j + f y b j ) ] exp ( i ϕ j ) | 2 = π 4 j = 1 N exp ( 4 π ω x | f x | ) exp ( 4 π ω y | f y | ) + 2 π 4 j = 1 N 1 l = j + 1 N exp ( 4 π ω x | f x | ) exp ( 4 π ω y | f y | ) exp { i 2 π [ f x ( a j a l ) + f y ( b j b l ) ] } exp ( σ 2 / 2 ) .
σ f x 2 = f x 2 I ( f x ) d f x I ( f x ) d f x = j = 1 N 1 32 π 3 ω x 3 + 2 j = 1 N 1 l = j + 1 N [ 2 ( 4 π ω x β ) 3 + 2 ( 4 π ω x + β ) 3 ] exp ( σ 2 / 2 ) j = 1 N 1 4 π ω x + 2 j = 1 N 1 l = j + 1 N ( 1 4 π ω x β + 1 4 π ω x + β ) exp ( σ 2 / 2 ) ,
β = 2 i π ( a j a l ) .
M x _ coherent 2 = 4 π σ 0 x σ f x ,
σ f x = { j = 1 N 1 32 π 3 ω x 3 + 2 j = 1 N 1 l = j + 1 N [ 2 ( 4 π ω x β ) 3 + 2 ( 4 π ω x + β ) 3 ] j = 1 N 1 4 π ω x + 2 j = 1 N 1 l = j + 1 N ( 1 4 π ω x β + 1 4 π ω x + β ) } 1 / 2 ,
M x _ incoherent 2 = 4 π σ 0 x σ f x = 4 π [ ( N ω x 2 + j = 1 N a j 2 ) N × 1 8 π 2 ω x 2 ] 1 / 2 .
M x 2 = 4 π σ 0 x σ f x = 2 ,

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