Abstract

The combination effect of multiwavelength active coherent beam combination (CBC) is investigated theoretically. The dependence of the combination effect on the optical path control precision, spectral width, wavelength number, and channel number is revealed. In the case of small optical path variance, the combination effect approximately decreases in quadratic form with wavelength number N, spectral width Δν, and optical path variance σ increasing. In the case of large optical path variance, the combination effect is independent of the optical path variance and the spectral width. The larger the wavelength number is, the smaller the Strehl ratio expectation is, and it finally degenerates to the incoherent combination. The necessity of optical path control is discussed. This study is helpful for multiwavelength CBC system design and the combination effect estimation.

© 2012 Optical Society of America

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References

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    [CrossRef]
  2. T. Y. Fan, “Laser beam combining for high-power, high radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
    [CrossRef]
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    [CrossRef]
  5. T. M. Shay, J. T. Baker, A. D. Sanchez, C. A. Robin, C. L. Vergien, C. Zerinque, A. D. Sanchez, D. Gallant, C. A. Lu, B. Pulford, C. T. J. Bronder, and A. Lucero, “High power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
    [CrossRef]
  6. S. J. McNaught, J. E. Rothenberg, P. A. Thielen, M. G. Wickham, M. E. Weber, and G. D. Goodno, “Coherent combining of a 1.26 kW fiber amplifier,” presented at Advanced Solid-State Photonics, San Diego, CA, February 3, 2010paper AMA2.
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    [CrossRef]
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    [CrossRef]
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2011

2010

2009

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

2007

2006

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

2005

T. Y. Fan, “Laser beam combining for high-power, high radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[CrossRef]

2004

1992

Anderegg, J.

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

Augst, S. J.

Baker, J. T.

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

Bronder, C. T. J.

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

Brosnan, S.

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

Burnett, J. G.

Cheung, E.

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

Dong, X.

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36, 951–953 (2011).
[CrossRef]

X. Wang, P. Zhou, J. Leng, W. Du, Y. Ma, H. Xiao, J. Zhu, X. Dong, X. Xu, Z. Liu, and Y. Zhao, “A 275 W multitone driven all-fiber amplifier seeded by a phase-modulated single-frequency laser for coherent beam combining,” IEEE Photon. Technol. Lett. 23, 980–982 (2011).
[CrossRef]

Du, W.

X. Wang, P. Zhou, J. Leng, W. Du, Y. Ma, H. Xiao, J. Zhu, X. Dong, X. Xu, Z. Liu, and Y. Zhao, “A 275 W multitone driven all-fiber amplifier seeded by a phase-modulated single-frequency laser for coherent beam combining,” IEEE Photon. Technol. Lett. 23, 980–982 (2011).
[CrossRef]

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36, 951–953 (2011).
[CrossRef]

Epp, P.

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

Fan, T. Y.

Flores, A.

A. Flores, T. M. Shay, C. A. Lu, C. Robin, B. Pulford, A. D. Sanchez, D. W. Hult, and K. B. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” paper presented at Conference on Lasers and Electro-Optics, Kirtland Air Force Base, NM, May 1–6, 2011.

Gallant, D.

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

Goldizen, K. C.

Goodno, G. D.

G. D. Goodno, S. J. McNaught, J. E. Rothenberg, T. S. McComb, P. A. Thielen, M. G. Wickham, and M. E. Weber, “Active phase and polarization locking of a 1.4 kW fiber amplifier,” Opt. Lett. 35, 1542–1544 (2010).
[CrossRef]

S. J. McNaught, J. E. Rothenberg, P. A. Thielen, M. G. Wickham, M. E. Weber, and G. D. Goodno, “Coherent combining of a 1.26 kW fiber amplifier,” presented at Advanced Solid-State Photonics, San Diego, CA, February 3, 2010paper AMA2.

Hammons, D.

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

Hult, D. W.

A. Flores, T. M. Shay, C. A. Lu, C. Robin, B. Pulford, A. D. Sanchez, D. W. Hult, and K. B. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” paper presented at Conference on Lasers and Electro-Optics, Kirtland Air Force Base, NM, May 1–6, 2011.

Jones, J. D. C.

Komine, H.

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

Korotky, S. K.

S. K. Korotky and T. River, “Multifrequency lightwave source using phase modulation for suppressing stimulated Brillouin scattering in optical fibers,” U. S. Patent 5,566,381 (15October1996).

Leng, J.

Liu, Z.

Lu, C. A.

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

A. Flores, T. M. Shay, C. A. Lu, C. Robin, B. Pulford, A. D. Sanchez, D. W. Hult, and K. B. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” paper presented at Conference on Lasers and Electro-Optics, Kirtland Air Force Base, NM, May 1–6, 2011.

Lucero, A.

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

Ma, Y.

McComb, T. S.

McNaught, S. J.

G. D. Goodno, S. J. McNaught, J. E. Rothenberg, T. S. McComb, P. A. Thielen, M. G. Wickham, and M. E. Weber, “Active phase and polarization locking of a 1.4 kW fiber amplifier,” Opt. Lett. 35, 1542–1544 (2010).
[CrossRef]

S. J. McNaught, J. E. Rothenberg, P. A. Thielen, M. G. Wickham, M. E. Weber, and G. D. Goodno, “Coherent combining of a 1.26 kW fiber amplifier,” presented at Advanced Solid-State Photonics, San Diego, CA, February 3, 2010paper AMA2.

Murphy, D. V.

Pulford, B.

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

A. Flores, T. M. Shay, C. A. Lu, C. Robin, B. Pulford, A. D. Sanchez, D. W. Hult, and K. B. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” paper presented at Conference on Lasers and Electro-Optics, Kirtland Air Force Base, NM, May 1–6, 2011.

Ranka, J. K.

Redmond, S. M.

River, T.

S. K. Korotky and T. River, “Multifrequency lightwave source using phase modulation for suppressing stimulated Brillouin scattering in optical fibers,” U. S. Patent 5,566,381 (15October1996).

Robin, C.

A. Flores, T. M. Shay, C. A. Lu, C. Robin, B. Pulford, A. D. Sanchez, D. W. Hult, and K. B. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” paper presented at Conference on Lasers and Electro-Optics, Kirtland Air Force Base, NM, May 1–6, 2011.

Robin, C. A.

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

Rothenberg, J. E.

G. D. Goodno, S. J. McNaught, J. E. Rothenberg, T. S. McComb, P. A. Thielen, M. G. Wickham, and M. E. Weber, “Active phase and polarization locking of a 1.4 kW fiber amplifier,” Opt. Lett. 35, 1542–1544 (2010).
[CrossRef]

S. J. McNaught, J. E. Rothenberg, P. A. Thielen, M. G. Wickham, M. E. Weber, and G. D. Goodno, “Coherent combining of a 1.26 kW fiber amplifier,” presented at Advanced Solid-State Photonics, San Diego, CA, February 3, 2010paper AMA2.

Rowland, K. B.

A. Flores, T. M. Shay, C. A. Lu, C. Robin, B. Pulford, A. D. Sanchez, D. W. Hult, and K. B. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” paper presented at Conference on Lasers and Electro-Optics, Kirtland Air Force Base, NM, May 1–6, 2011.

Sanchez, A.

Sanchez, A. D.

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

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

A. Flores, T. M. Shay, C. A. Lu, C. Robin, B. Pulford, A. D. Sanchez, D. W. Hult, and K. B. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” paper presented at Conference on Lasers and Electro-Optics, Kirtland Air Force Base, NM, May 1–6, 2011.

Schiller, J.

M. R. Spiegel, J. Schiller, and R. A. Srinivasan, Outline of Theory and Problems of Probability and Statistics (McGraw-Hill, 2000).

Shay, T. M.

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

A. Flores, T. M. Shay, C. A. Lu, C. Robin, B. Pulford, A. D. Sanchez, D. W. Hult, and K. B. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” paper presented at Conference on Lasers and Electro-Optics, Kirtland Air Force Base, NM, May 1–6, 2011.

Si, L.

Spiegel, M. R.

M. R. Spiegel, J. Schiller, and R. A. Srinivasan, Outline of Theory and Problems of Probability and Statistics (McGraw-Hill, 2000).

Srinivasan, R. A.

M. R. Spiegel, J. Schiller, and R. A. Srinivasan, Outline of Theory and Problems of Probability and Statistics (McGraw-Hill, 2000).

Thielen, P. A.

G. D. Goodno, S. J. McNaught, J. E. Rothenberg, T. S. McComb, P. A. Thielen, M. G. Wickham, and M. E. Weber, “Active phase and polarization locking of a 1.4 kW fiber amplifier,” Opt. Lett. 35, 1542–1544 (2010).
[CrossRef]

S. J. McNaught, J. E. Rothenberg, P. A. Thielen, M. G. Wickham, M. E. Weber, and G. D. Goodno, “Coherent combining of a 1.26 kW fiber amplifier,” presented at Advanced Solid-State Photonics, San Diego, CA, February 3, 2010paper AMA2.

Vergien, C. L.

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

Wang, X.

Weber, M.

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

Weber, M. E.

G. D. Goodno, S. J. McNaught, J. E. Rothenberg, T. S. McComb, P. A. Thielen, M. G. Wickham, and M. E. Weber, “Active phase and polarization locking of a 1.4 kW fiber amplifier,” Opt. Lett. 35, 1542–1544 (2010).
[CrossRef]

S. J. McNaught, J. E. Rothenberg, P. A. Thielen, M. G. Wickham, M. E. Weber, and G. D. Goodno, “Coherent combining of a 1.26 kW fiber amplifier,” presented at Advanced Solid-State Photonics, San Diego, CA, February 3, 2010paper AMA2.

Wickham, M.

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

Wickham, M. G.

G. D. Goodno, S. J. McNaught, J. E. Rothenberg, T. S. McComb, P. A. Thielen, M. G. Wickham, and M. E. Weber, “Active phase and polarization locking of a 1.4 kW fiber amplifier,” Opt. Lett. 35, 1542–1544 (2010).
[CrossRef]

S. J. McNaught, J. E. Rothenberg, P. A. Thielen, M. G. Wickham, M. E. Weber, and G. D. Goodno, “Coherent combining of a 1.26 kW fiber amplifier,” presented at Advanced Solid-State Photonics, San Diego, CA, February 3, 2010paper AMA2.

Xiao, H.

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36, 951–953 (2011).
[CrossRef]

X. Wang, P. Zhou, J. Leng, W. Du, Y. Ma, H. Xiao, J. Zhu, X. Dong, X. Xu, Z. Liu, and Y. Zhao, “A 275 W multitone driven all-fiber amplifier seeded by a phase-modulated single-frequency laser for coherent beam combining,” IEEE Photon. Technol. Lett. 23, 980–982 (2011).
[CrossRef]

Xu, X.

Yu, C. X.

Zerinque, C.

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

Zhao, Y.

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36, 951–953 (2011).
[CrossRef]

X. Wang, P. Zhou, J. Leng, W. Du, Y. Ma, H. Xiao, J. Zhu, X. Dong, X. Xu, Z. Liu, and Y. Zhao, “A 275 W multitone driven all-fiber amplifier seeded by a phase-modulated single-frequency laser for coherent beam combining,” IEEE Photon. Technol. Lett. 23, 980–982 (2011).
[CrossRef]

Zhou, P.

Zhu, J.

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36, 951–953 (2011).
[CrossRef]

X. Wang, P. Zhou, J. Leng, W. Du, Y. Ma, H. Xiao, J. Zhu, X. Dong, X. Xu, Z. Liu, and Y. Zhao, “A 275 W multitone driven all-fiber amplifier seeded by a phase-modulated single-frequency laser for coherent beam combining,” IEEE Photon. Technol. Lett. 23, 980–982 (2011).
[CrossRef]

Appl. Opt.

IEEE J. Sel. Top. Quantum Electron.

T. Y. Fan, “Laser beam combining for high-power, high radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005).
[CrossRef]

IEEE Photon. Technol. Lett.

X. Wang, P. Zhou, J. Leng, W. Du, Y. Ma, H. Xiao, J. Zhu, X. Dong, X. Xu, Z. Liu, and Y. Zhao, “A 275 W multitone driven all-fiber amplifier seeded by a phase-modulated single-frequency laser for coherent beam combining,” IEEE Photon. Technol. Lett. 23, 980–982 (2011).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Proc. SPIE

M. Wickham, J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, and M. Weber, “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, A. D. Sanchez, D. Gallant, C. A. Lu, B. Pulford, C. T. J. Bronder, and A. Lucero, “High power phase locking of a fiber amplifier array,” Proc. SPIE 7195, 71951M (2009).
[CrossRef]

Other

S. J. McNaught, J. E. Rothenberg, P. A. Thielen, M. G. Wickham, M. E. Weber, and G. D. Goodno, “Coherent combining of a 1.26 kW fiber amplifier,” presented at Advanced Solid-State Photonics, San Diego, CA, February 3, 2010paper AMA2.

A. Flores, T. M. Shay, C. A. Lu, C. Robin, B. Pulford, A. D. Sanchez, D. W. Hult, and K. B. Rowland, “Coherent beam combining of fiber amplifiers in a kW regime,” paper presented at Conference on Lasers and Electro-Optics, Kirtland Air Force Base, NM, May 1–6, 2011.

S. K. Korotky and T. River, “Multifrequency lightwave source using phase modulation for suppressing stimulated Brillouin scattering in optical fibers,” U. S. Patent 5,566,381 (15October1996).

M. R. Spiegel, J. Schiller, and R. A. Srinivasan, Outline of Theory and Problems of Probability and Statistics (McGraw-Hill, 2000).

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Figures (4)

Fig. 1.
Fig. 1.

Dependence of Strehl ratio on the optical path deviation in multiwavelength CBC.

Fig. 2.
Fig. 2.

Three-wavelength, three-channel MOPA-configuration CBC system. VDL, variable delay line; PM, phase modulator; FG, function generator; F-P I, Fabry–Perot interferometer; FA; fiber amplifier; CO, collimator.

Fig. 3.
Fig. 3.

Far-field intensity patterns of the three-wavelength, three-channel CBC system with different spectra at various optical path deviations.

Fig. 4.
Fig. 4.

Experimental Strehl ratio of the three-wavelength, three-channel CBC system with different spectra at various optical path deviations.

Equations (25)

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

Δν=(N1)×dν.
νn=ν0+(nN+12)×dν,
Um,νn=Im,νn·exp[j·(2πc·νn·lm+φm)].
S=1N·M2·n=1N|m=1Mexp(j·2πc·νn·lm)|2.
S=1N·M2·n=1Np,q=1Mexp(j·2πc·νn·lpq)=1N·M2·p,q=1Mn=1Nexp(j·2πc·νn·lpq)=1N·M2·p,q=1Mn=1Nexp[j·lpq·2πc·(ν0+2nN12×dν)]=1N·M2·p,q=1Msin(N·π/c·Δν·lpq)sin(π/c·Δν·lpq)exp(j·2πc·ν0·lpq)=1M+1M2·p>q1N·sin(N·π/c·Δν·lpq)sin(π/c·Δν·lpq)·2cos(2πc·ν0·lpq).
S=1M+2M2·p>q|1N·sin(N·π/c·Δν·lpq)sin(π/c·Δν·lpq)|.
1N·sin(N·π/c·Δν·lpq)sin(π/c·Δν·lpq)=116(N21)·(πc·Δν)2·lpq2+.
S1M+2M2·p>q[116(N21)·(πc·Δν)2·lpq2]=1M+2M2·M(M1)22M2·16(N21)·(πc·Δν)2·p>qlpq2=1(N21)3M2·(πc·Δν)2·p>qlpq2.
f(x)={1(4σ2)12·Γ(12)·x121·exp(x4σ2)x>00x0.
f(S)={1(4σ2)M12·Γ(M12)·[6M·(1S)(N21)·(π/c·Δν)2]M121·exp[6M·(1S)(N21)·(π/c·Δν)2·14σ2]1S>001S0,
f(S)={1(4σ2)M12·Γ(M12)·[6M·(N21)·(π/c·Δν)2]M121·(1S)M121·exp[3M·(1S)2(N21)·(π/c·Δν)2·σ2]S<10S1.
E(S)=S·f(S)dS=1(N21)·(π/c·Δν)2·σ23·M1M,
V(S)=[SE(S)]2·f(S)dS=2(N21)2·(π/c·Δν)4·σ49·M1M2.
D(S)=V(S)=2(N21)·(π/c·Δν)2·σ23·M1M.
E(S)1π23·c2(N·Δν·σ)2,
D(S)2·π23·c2·(N·Δν·σ)2M.
E(S)=1M+M1M·E(Apq),
V(S)=M·(M1)2·4M4·V(Apq)=2·(M1)M3·V(Apq).
E(Apq)=|1N·sin(N·π/c·Δν·x)sin(π/c·Δν·x)|·12π·σ·exp(x24σ2)dx=R=0,±1cΔν·(R12)·cΔν(R+12)·cΔν|Δνc·1N·sin(N·π/c·Δν·x)sin(π/c·Δν·x)|·12π·σ·exp(x24σ2)dx.
R=0,±1cΔν·12π·σ·exp[(R·cΔν)24σ2]·(R12)·cΔν(R+12)·cΔν|Δνc·1N·sin(N·π/c·Δν·x)sin(π/c·Δν·x)|dx=12·cΔν12·cΔν|Δνc·1N·sin(N·π/c·Δν·x)sin(π/c·Δν·x)|dx·R=0,±1cΔν·12π·σ·exp[(R·cΔν)24σ2]=12·cΔν12·cΔν|Δνc·1N·sin(N·π/c·Δν·x)sin(π/c·Δν·x)|dx.
E(Apq2)=|1N·sin(N·π/c·Δν·x)sin(π/c·Δν·x)|2·12π·σ·exp(x24σ2)dx12·cΔν12·cΔνΔνc·|1N·sin(N·π/c·Δν·x)sin(π/c·Δν·x)|2dx=1N,
V(Apq)=E(Apq2)E2(Apq)1N[52·(N+2)]2=1N254·(N+2)2.
E(S)=1M+M1M·E(Apq)1M+52·M1M·1N+2,
V(S)=2·(M1)M3·V(Apq)2·(M1)M3·[1N254·(N+2)2],
D(S)=V(S)=2·(M1)12M32·[1N254·(N+2)2]12.

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