Abstract

Coherent combining efficiency is examined analytically for large arrays of non-ideal lasers combined using filled aperture elements with nonuniform splitting ratios. Perturbative expressions are developed for efficiency loss from combiner splitting ratios, power imbalance, spatial misalignments, beam profile nonuniformities, pointing and wavefront errors, depolarization, and temporal dephasing of array elements. It is shown that coupling efficiency of arrays is driven by non-common spatial aberrations, and that common-path aberrations have no impact on coherent combining efficiency. We derive expressions for misalignment losses of Gaussian beams, providing tolerancing metrics for co-alignment and uniformity of arrays of single-mode fiber lasers.

© 2010 OSA

Full Article  |  PDF Article

Errata

Gregory D. Goodno, Chun-Ching Shih, and Joshua E. Rothenberg, "Perturbative analysis of coherent combining efficiency with mismatched lasers: errata," Opt. Express 20, 23587-23588 (2012)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-20-21-23587

References

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  1. T. Y. Fan, “Laser Beam Combining for High-Power, High-Radiance Sources,” IEEE J. Sel. Top. Quantum Electron. 11(3), 567–577 (2005).
    [CrossRef]
  2. J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high-power fiber arrays,” Proc. SPIE 6102, 61020U1 (2006).
  3. G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. 31(9), 1247–1249 (2006), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-31-9-1247 .
    [CrossRef] [PubMed]
  4. S. J. McNaught, C. P. Asman, H. Injeyan, A. Jankevics, A. M. Johnson, G. C. Jones, H. Komine, J. Machan, J. Marmo, M. McClellan, R. Simpson, J. Sollee, M. M. Valley, M. Weber, and S. B. Weiss, “100-kW Coherently Combined Nd:YAG MOPA Laser Array,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThD2, http://www.opticsinfobase.org/abstract.cfm?URI=FiO-2009-FThD2
  5. T. H. Loftus, A. M. Thomas, M. Norsen, J. Minelly, P. Jones, E. Honea, S. A. Shakir, S. Hendow, W. Culver, B. Nelson, and M. Fitelson, “Four-Channel, High Power, Passively Phase Locked Fiber Array,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2008), paper WA4, http://www.opticsinfobase.org/abstract.cfm?URI=ASSP-2008-WA4
  6. T. M. Shay, J. T. Baker, A. D. Sancheza, C. A. Robin, C. L. Vergien, A. Flores, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “Phasing of High Power Fiber Amplifier Arrays,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2010), paper AMA1, http://www.opticsinfobase.org/abstract.cfm?URI=ASSP-2010-AMA1
  7. D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE 7580, 75801V (2010).
    [CrossRef]
  8. P. A. Thielen, J. G. Ho, M. Hemmat, G. D. Goodno, R. R. Rice, J. Rothenberg, M. Wickham, J. T. Baker, D. Gallant, C. Robin, C. Vergien, C. Zeringue, T. J. Bronder, T. M. Shay, and A. D. Sanchez, 400-W, High-Efficiency Coherent Combination of Fiber Lasers,” presented at 22nd Annual Solid State and Diode Laser Technology Review (Newton, MA 2009).
  9. C. X. Yu, S. J. Augst, S. Redmond, D. V. Murphy, A. Sanchez, and T. Y. Fan, “Phase Coherence, Phase Noise and Phase Control in High-Power Yb Fiber Amplifiers”, presented at 23rd Annual Solid State and Diode Laser Technology Review, Broomfield, CO (2010).
  10. B. Wang, E. Mies, M. Minden, and A. Sanchez, “All-fiber 50 W coherently combined passive laser array,” Opt. Lett. 34(7), 863–865 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-7-863 .
    [CrossRef] [PubMed]
  11. H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. 30(11), 1339–1341 (2005), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-30-11-1339 .
    [CrossRef] [PubMed]
  12. 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(10), 1542–1544 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-10-1542 .
    [CrossRef] [PubMed]
  13. 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(4), 354–356 (2008), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-33-4-354 .
    [CrossRef] [PubMed]
  14. J. R. Leger, G. J. Swanson, and W. B. Veldkamp, “Coherent laser addition using binary phase gratings,” Appl. Opt. 26(20), 4391–4399 (1987), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-26-20-4391 .
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  15. C. D. Nabors, “Effects of phase errors on coherent emitter arrays,” Appl. Opt. 33(12), 2284–2289 (1994), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-33-12-2284 .
    [CrossRef] [PubMed]
  16. 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(12), 2930–2939 (2007), http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-24-12-2930 .
    [CrossRef]
  17. T. Y. Fan, “The effect of amplitude (power) variations on beam combining efficiency for phased arrays,” IEEE J. Sel. Top. Quantum Electron. 15(2), 291–293 (2009).
    [CrossRef]
  18. S. E. Christensen, and O. Koski, “2-Dimensional Waveguide Coherent Beam Combiner,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper WC1, http://www.opticsinfobase.org/abstract.cfm?URI=ASSP-2007-WC1
  19. J. R. Andrews, “Interferometric power amplifiers,” Opt. Lett. 14(1), 33–35 (1989), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-14-1-33 .
    [CrossRef] [PubMed]
  20. R. Uberna, A. Bratcher, and B. G. Tiemann, “Coherent Polarization Beam Combination,” IEEE J. Quantum Electron. 46(8), 1191–1196 (2010).
    [CrossRef]
  21. D. D. Lowenthal, “Maréchal intensity criteria modified for gaussian beams,” Appl. Opt. 13(9), 2126–2133 (1974), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-13-9-2126 .
    [CrossRef] [PubMed]
  22. J. W. Goodman, Statistical Optics (Wiley, 2000), pp. 163–165.

2010 (3)

D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE 7580, 75801V (2010).
[CrossRef]

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(10), 1542–1544 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-10-1542 .
[CrossRef] [PubMed]

R. Uberna, A. Bratcher, and B. G. Tiemann, “Coherent Polarization Beam Combination,” IEEE J. Quantum Electron. 46(8), 1191–1196 (2010).
[CrossRef]

2009 (2)

T. Y. Fan, “The effect of amplitude (power) variations on beam combining efficiency for phased arrays,” IEEE J. Sel. Top. Quantum Electron. 15(2), 291–293 (2009).
[CrossRef]

B. Wang, E. Mies, M. Minden, and A. Sanchez, “All-fiber 50 W coherently combined passive laser array,” Opt. Lett. 34(7), 863–865 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-7-863 .
[CrossRef] [PubMed]

2008 (1)

2007 (1)

2006 (1)

2005 (2)

1994 (1)

1989 (1)

1987 (1)

1974 (1)

Aflatouni, F.

Andrews, J. R.

Bratcher, A.

R. Uberna, A. Bratcher, and B. G. Tiemann, “Coherent Polarization Beam Combination,” IEEE J. Quantum Electron. 46(8), 1191–1196 (2010).
[CrossRef]

Bruesselbach, H.

Cheung, E. C.

Clark, R. G.

D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE 7580, 75801V (2010).
[CrossRef]

Epp, P.

Fan, T. Y.

T. Y. Fan, “The effect of amplitude (power) variations on beam combining efficiency for phased arrays,” IEEE J. Sel. Top. Quantum Electron. 15(2), 291–293 (2009).
[CrossRef]

T. Y. Fan, “Laser Beam Combining for High-Power, High-Radiance Sources,” IEEE J. Sel. Top. Quantum Electron. 11(3), 567–577 (2005).
[CrossRef]

Goodno, G. D.

Hashemi, H.

Ho, J. G.

Howland, D.

Injeyan, H.

Jones, D. C.

D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE 7580, 75801V (2010).
[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(11), 1339–1341 (2005), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-30-11-1339 .
[CrossRef] [PubMed]

Kewitsch, A.

Komine, H.

Leger, J. R.

Liang, W.

Long, W.

Lowenthal, D. D.

Mangir, M. S.

McClellan, M.

McComb, T. S.

McNaught, S. J.

Mies, E.

Minden, M.

Nabors, C. D.

Rakuljic, G.

Redmond, S.

Rice, R. R.

Rogers, J. L.

Rothenberg, J.

Rothenberg, J. E.

Sanchez, A.

Satyan, N.

Scott, A. M.

D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE 7580, 75801V (2010).
[CrossRef]

Simpson, R.

Sollee, J.

Stace, C.

D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE 7580, 75801V (2010).
[CrossRef]

Stacey, C. D.

D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE 7580, 75801V (2010).
[CrossRef]

Stone, S. M.

D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE 7580, 75801V (2010).
[CrossRef]

Swanson, G. J.

Thielen, P.

Thielen, P. A.

Tiemann, B. G.

R. Uberna, A. Bratcher, and B. G. Tiemann, “Coherent Polarization Beam Combination,” IEEE J. Quantum Electron. 46(8), 1191–1196 (2010).
[CrossRef]

Turner, A. J.

D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE 7580, 75801V (2010).
[CrossRef]

Uberna, R.

R. Uberna, A. Bratcher, and B. G. Tiemann, “Coherent Polarization Beam Combination,” IEEE J. Quantum Electron. 46(8), 1191–1196 (2010).
[CrossRef]

Veldkamp, W. B.

Wang, B.

Weber, M.

Weber, M. E.

Weiss, S. B.

Wickham, M.

Wickham, M. G.

Yariv, A.

Appl. Opt. (3)

IEEE J. Quantum Electron. (1)

R. Uberna, A. Bratcher, and B. G. Tiemann, “Coherent Polarization Beam Combination,” IEEE J. Quantum Electron. 46(8), 1191–1196 (2010).
[CrossRef]

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

T. Y. Fan, “The effect of amplitude (power) variations on beam combining efficiency for phased arrays,” IEEE J. Sel. Top. Quantum Electron. 15(2), 291–293 (2009).
[CrossRef]

T. Y. Fan, “Laser Beam Combining for High-Power, High-Radiance Sources,” IEEE J. Sel. Top. Quantum Electron. 11(3), 567–577 (2005).
[CrossRef]

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

Opt. Lett. (6)

B. Wang, E. Mies, M. Minden, and A. Sanchez, “All-fiber 50 W coherently combined passive laser array,” Opt. Lett. 34(7), 863–865 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-7-863 .
[CrossRef] [PubMed]

H. Bruesselbach, D. C. Jones, M. S. Mangir, M. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. 30(11), 1339–1341 (2005), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-30-11-1339 .
[CrossRef] [PubMed]

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(10), 1542–1544 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-10-1542 .
[CrossRef] [PubMed]

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(4), 354–356 (2008), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-33-4-354 .
[CrossRef] [PubMed]

G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. 31(9), 1247–1249 (2006), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-31-9-1247 .
[CrossRef] [PubMed]

J. R. Andrews, “Interferometric power amplifiers,” Opt. Lett. 14(1), 33–35 (1989), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-14-1-33 .
[CrossRef] [PubMed]

Proc. SPIE (1)

D. C. Jones, A. J. Turner, A. M. Scott, S. M. Stone, R. G. Clark, C. Stace, and C. D. Stacey, “A multi-channel phase locked fibre bundle laser,” Proc. SPIE 7580, 75801V (2010).
[CrossRef]

Other (8)

P. A. Thielen, J. G. Ho, M. Hemmat, G. D. Goodno, R. R. Rice, J. Rothenberg, M. Wickham, J. T. Baker, D. Gallant, C. Robin, C. Vergien, C. Zeringue, T. J. Bronder, T. M. Shay, and A. D. Sanchez, 400-W, High-Efficiency Coherent Combination of Fiber Lasers,” presented at 22nd Annual Solid State and Diode Laser Technology Review (Newton, MA 2009).

C. X. Yu, S. J. Augst, S. Redmond, D. V. Murphy, A. Sanchez, and T. Y. Fan, “Phase Coherence, Phase Noise and Phase Control in High-Power Yb Fiber Amplifiers”, presented at 23rd Annual Solid State and Diode Laser Technology Review, Broomfield, CO (2010).

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high-power fiber arrays,” Proc. SPIE 6102, 61020U1 (2006).

S. J. McNaught, C. P. Asman, H. Injeyan, A. Jankevics, A. M. Johnson, G. C. Jones, H. Komine, J. Machan, J. Marmo, M. McClellan, R. Simpson, J. Sollee, M. M. Valley, M. Weber, and S. B. Weiss, “100-kW Coherently Combined Nd:YAG MOPA Laser Array,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThD2, http://www.opticsinfobase.org/abstract.cfm?URI=FiO-2009-FThD2

T. H. Loftus, A. M. Thomas, M. Norsen, J. Minelly, P. Jones, E. Honea, S. A. Shakir, S. Hendow, W. Culver, B. Nelson, and M. Fitelson, “Four-Channel, High Power, Passively Phase Locked Fiber Array,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2008), paper WA4, http://www.opticsinfobase.org/abstract.cfm?URI=ASSP-2008-WA4

T. M. Shay, J. T. Baker, A. D. Sancheza, C. A. Robin, C. L. Vergien, A. Flores, C. Zerinque, D. Gallant, C. A. Lu, B. Pulford, T. J. Bronder, and A. Lucero, “Phasing of High Power Fiber Amplifier Arrays,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2010), paper AMA1, http://www.opticsinfobase.org/abstract.cfm?URI=ASSP-2010-AMA1

S. E. Christensen, and O. Koski, “2-Dimensional Waveguide Coherent Beam Combiner,” in Advanced Solid-State Photonics, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper WC1, http://www.opticsinfobase.org/abstract.cfm?URI=ASSP-2007-WC1

J. W. Goodman, Statistical Optics (Wiley, 2000), pp. 163–165.

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Figures (4)

Fig. 1
Fig. 1

Power splitting ratios for a 1 x N beamsplitter / combiner.

Fig. 2
Fig. 2

Effect of channel-by-channel correlations between input powers and BC splitting fractions on the combining efficiency for a lossless BC and perfectly aligned plane waves. The parameter σ refers to the fractional (normalized) standard deviations of A and D, which are assumed to be equal.

Fig. 3
Fig. 3

Comparison of the exact (Monte Carlo) and approximate solutions for combining efficiency of large arrays (N = 103) with nonuniform and uncorrelated input powers and BC splitting ratios, assuming a lossless splitter and perfectly aligned plane waves.

Fig. 4
Fig. 4

Co-alignment and uniformity tolerances for spatially and spectrally Gaussian beams with 1% allowance for combining loss for each effect. The BC is assumed to be lossless and uniform (ηBC = 1 and Dn = N -1/2). RMS parameter variation refers to beam-to-beam mismatch. Fractional parameters and values are relative to the array average FWHM parameter, colored black in each sketch.

Equations (34)

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η B C = D n 2 ,
η = | D n E n | 2 / | E n | 2 .
E n ( r , t ) = A n ( r ) [ cos ( χ n ) u x + sin ( χ n ) exp ( i Γ n ) u y ] × exp [ i ω 0 ( t + δ τ n ) + i ψ ( t + δ τ n ) + i ϕ n ( r ) ] ,
η ( r ) = | D n E n ( r , t ) | 2 / A n ( r ) 2 ,
η = η ( r ) A n ( r ) 2 d r / A n ( r ) 2 d r ,
η = 1 P t o t | D n E n ( r , t ) | 2 d r ,
η = 1 P t o t | D n A n ( r ) cos ( χ n ) exp [ i ω 0 ( t + δ τ n ) + i ψ ( t + δ τ n ) + i ϕ n ( r ) ] | 2 d r .
E n ( r , t ) = [ A ( r ) + δ A n ( r ) ] ( 1 δ χ n 2 / 2 ) × exp { i ω 0 ( t + δ τ n ) + i ψ ( t ) + i Δ ω ( t ) ( t + δ τ n ) + i [ ϕ ( r ) + δ ϕ n ( r ) ] } .
D n = η B C / N + δ D n ,
η = 1 P t o t | ( η B C / N + δ D n ) [ A ( r ) + δ A n ( r ) ] ( 1 δ χ n 2 / 2 ) exp [ i Δ ω ( t ) δ τ n + i δ ϕ n ( r ) ] | 2 d r .
η = η B C η B C P [ 1 N δ A n ( r ) 2 ( 1 N δ A n ( r ) ) 2 ] d r + 2 P η B C N A ( r ) δ D n δ A n ( r ) d r η B C P A ( r ) 2 [ 1 N δ ϕ n ( r ) 2 ( 1 N δ ϕ n ( r ) ) 2 ] d r N [ 1 N δ D n 2 ( 1 N δ D n ) 2 ] η B C Δ ω ( t ) 2 [ 1 N δ τ n 2 ( 1 N δ τ n ) 2 ] η B C [ 1 N δ χ n 2 ] ,
P t o t A n ( r ) 2 d r = ( N A ( r ) 2 + 2 A ( r ) δ A n ( r ) + δ A n ( r ) 2 ) d r ,
2 δ D n = N / η B C δ D n 2 .
σ u 2 = 1 N δ u n 2 ( 1 N δ u n ) 2 ,
σ A ( r ) , D = 1 N ( D n η B C / N ) [ A n ( r ) A ( r ) ] = 1 N δ D n δ A n ( r ) .
η = η B C [ 1 1 P ( σ A ( r ) 2 + A ( r ) 2 σ ϕ ( r ) 2 2 N / η B C A ( r ) σ A ( r ) , D ) d r Δ ω ( t ) 2 σ τ 2 σ χ 2 N σ D 2 / η B C ] .
1 P A ( r ) 2 d r = N A ( r ) 2 d r A n 2 ( r ) d r 1.
η = η B C ( 1 1 P A ( r ) 2 σ ϕ ( r ) 2 d r σ ε 2 σ χ 2 Δ ω ( t ) 2 σ τ 2 ) + 2 N η B C σ ε , D N σ D 2 .
η = η B C ( 1 σ A 2 / A 2 ) .
η = η B C ( 1 σ P 2 / 4 P 2 ) .
A n 2 = C 2 D n 2 = C 2 η B C ; C = A n 2 / η B C N / η B C A .
cor ( A , D ) = σ A , D σ A σ D = σ ε , D σ ε σ D .
η = η B C ( 1 σ ε 2 ) σ D 2 + 2 η B C σ ε , D .
η = 1 1 P A ( r ) 2 [ 1 N δ ϕ n ( r ) 2 ( 1 N δ ϕ n ( r ) ) 2 ] d r .
η = 1 1 N 1 P A ( r ) 2 δ φ n ( r ) 2 d r .
σ W F E , n 2 = 1 P A ( r ) 2 δ ϕ n ( r ) 2 d r .
A ( x ) = ( 2 / π ) 1 / 4 P / w exp [ x 2 / w 2 ] ,
A n ( x ) = ( 2 / π ) 1 / 4 P / w exp [ ( x δ x n ) 2 / w 2 ] .
δ A n ( x ) = d A n ( x ) d x δ x n = 2 ( 2 / π ) 1 / 4 P / w ( x δ x n ) w 2 exp [ ( x δ x n ) 2 / w 2 ] δ x n .
σ A 2 = 2 / π ( 4 P / w 3 ) x 2 exp [ 2 x 2 / w 2 ] σ x 2 / w 2 .
η = η B C ( 1 1 P σ A 2 d x ) = η B C ( 1 2 π 4 σ x 2 w 5 x 2 exp [ 2 x 2 / w 2 ] d x ) = η B C ( 1 σ x 2 / w 2 ) .
A n ( x ) = ( 2 π ) 1 / 4 P w + δ w n exp [ x 2 / ( w + δ w n ) 2 ] .
A ( θ ) = 1 2 π A ( x ) e i 2 π x θ / λ d x = ( 2 π ) 1 / 4 P w 2 exp ( θ 2 / θ 0 2 ) ,
η = η B C [ 1 Δ ω ( t ) 2 σ τ 2 ] = η B C [ 1 π 2 2 ln ( 2 ) Δ f F W H M 2 σ τ 2 ] .

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