Abstract

Numerical analysis of the effects of aberrations on coherently combined fiber laser beams is presented. We prove that traditional beam quality criteria, such as the M2 factor and the Strehl ratio, do not consider necessary information to evaluate the quality of a coherently combined laser beam. The beam propagation factor (BPF) is introduced and studied as a proper beam quality factor for the coherently combined beam. Two main categories of aberrations, geometry and nongeometry factors, are numerically studied to investigate their effect on beam quality by using the BPF criterion. For a ring-distributed fiber laser array with certain vacancy factor and a RMS value of tilt error, we obtain a semianalytical equation to evaluate their effect on beam quality. We present a brief discussion of those aberrations at the end of this paper. Our generalized methodology offers a good reference for investigating coherent combining of fiber laser beams in a comprehensive way.

© 2008 Optical Society of America

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  1. Y. Jeong, J. K. Sahu, D. N. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12, 6088-6092 (2004).
    [CrossRef] [PubMed]
  2. T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567-577 (2005).
    [CrossRef]
  3. 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, 61020U-1-61020U-5 (2006).
  4. S. J. Augst, J. K. Ranka, T. Y. Fan, and A. Sanchez, “Beam combining of ytterbium fiber amplifiers,” J. Opt. Soc. Am. B 24, 1707-1715 (2007).
    [CrossRef]
  5. J. B. Shellan, “Phased-array performance degradation due to mirror misfigures, piston errors, jitter, and polarization errors,” J. Opt. Soc. Am. A 2, 555-567 (1985).
    [CrossRef]
  6. C. B. Hogge, H. F. Schultz, D. B. Mason, and W. E. Thompson, “Physical optics of multiaperture systems,” Appl. Opt. 27, 5127-5134 (1988).
    [CrossRef] [PubMed]
  7. C. D. Nabors, “Effects of phased errors on coherent emitter arrays,” App. Opt 33, 2284-2289 (1994).
    [CrossRef]
  8. B. Lü and H. Ma, “Coherent and incoherent off-axis Hermite-Gaussian beam combinations,” App. Opt 39, 1279-1289(2000).
    [CrossRef]
  9. B. Lü and H. Ma, “Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185-194 (1999).
    [CrossRef]
  10. Y. Li, L. Qian, D. Lu, D. Fan, and S. Wen, “Coherent and incoherent combining of fiber array with hexagonal ring distribution,” Opt. Laser Technol. 39, 957-963 (2007).
    [CrossRef]
  11. B. Lü and H. Ma, “Beam propagation properties of radial laser arrays,” J. Opt. Soc. Am. A 17, 2005-2009 (2000).
    [CrossRef]
  12. A. E. Siegman, “How to (maybe) measure laser beam quality,” presented at the OSA Annual Meeting, October 1997.
  13. A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 1-4 (1990).
  14. http://www.darpa.mil/mto/programs/adhels/index.htm, “Architecture for diode high energy laser systems”
  15. K. Koyabu, F. Ohira, and T. Yamamoto, “Fabrication of two-dimensional fiber arrays using microferrules,” IEEE Trans. Compon. Packag. Manuf. Technol. C 21, 11-19 (1998).
    [CrossRef]
  16. J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. J. Shaw, and R. C. Lawrence, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count ,” Proc. SPIE 6306, 63060G-1-63060G-11 (2006).
  17. P. Zhou, Z. Chen, X. Wang, Z. Liu, and X. Xu, “Tolerance on tilt error for coherent combining of fiber lasers,” submitted to Chin. Opt. Lett.
  18. B. Li, B. Lü, “The polarization property and irradiance distribution of incoherent and coherent Gaussian beam combinations,” Optik (Jena) 113, 535-540 (2003).
    [CrossRef]
  19. S. Norman, M. Zervas, A. Appleyard, M. Durkin, R. Horley, M. Varnham, J. Nilsson, and Y. Jeong, “Latest development of high power fiber lasers in SPI,” Proc. SPIE 5335, 229-237 (2004).
    [CrossRef]
  20. F. Gori, M. Santarsiero, and R. Borghi, “Intensity based modal analysis of partially coherent beams with Hermite-Gaussian modes,” Opt. Lett 23, 989-991 (1998).
    [CrossRef]
  21. S. J. Augst, T. Y. Fan, and A. Sanchez, “Coherent beam combining and phase noise measurements of ytterbium fiber amplifiers,” Opt. Lett. 29, 474-476 (2004).
    [CrossRef] [PubMed]

2007 (2)

Y. Li, L. Qian, D. Lu, D. Fan, and S. Wen, “Coherent and incoherent combining of fiber array with hexagonal ring distribution,” Opt. Laser Technol. 39, 957-963 (2007).
[CrossRef]

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

2006 (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, 61020U-1-61020U-5 (2006).

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. J. Shaw, and R. C. Lawrence, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count ,” Proc. SPIE 6306, 63060G-1-63060G-11 (2006).

2005 (1)

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

2004 (3)

2003 (1)

B. Li, B. Lü, “The polarization property and irradiance distribution of incoherent and coherent Gaussian beam combinations,” Optik (Jena) 113, 535-540 (2003).
[CrossRef]

2000 (2)

B. Lü and H. Ma, “Beam propagation properties of radial laser arrays,” J. Opt. Soc. Am. A 17, 2005-2009 (2000).
[CrossRef]

B. Lü and H. Ma, “Coherent and incoherent off-axis Hermite-Gaussian beam combinations,” App. Opt 39, 1279-1289(2000).
[CrossRef]

1999 (1)

B. Lü and H. Ma, “Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185-194 (1999).
[CrossRef]

1998 (2)

K. Koyabu, F. Ohira, and T. Yamamoto, “Fabrication of two-dimensional fiber arrays using microferrules,” IEEE Trans. Compon. Packag. Manuf. Technol. C 21, 11-19 (1998).
[CrossRef]

F. Gori, M. Santarsiero, and R. Borghi, “Intensity based modal analysis of partially coherent beams with Hermite-Gaussian modes,” Opt. Lett 23, 989-991 (1998).
[CrossRef]

1994 (1)

C. D. Nabors, “Effects of phased errors on coherent emitter arrays,” App. Opt 33, 2284-2289 (1994).
[CrossRef]

1990 (1)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 1-4 (1990).

1988 (1)

1985 (1)

Anderegg, J.

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, 61020U-1-61020U-5 (2006).

Appleyard, A.

S. Norman, M. Zervas, A. Appleyard, M. Durkin, R. Horley, M. Varnham, J. Nilsson, and Y. Jeong, “Latest development of high power fiber lasers in SPI,” Proc. SPIE 5335, 229-237 (2004).
[CrossRef]

Augst, S. J.

B.,

B. Li, B. Lü, “The polarization property and irradiance distribution of incoherent and coherent Gaussian beam combinations,” Optik (Jena) 113, 535-540 (2003).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, and R. Borghi, “Intensity based modal analysis of partially coherent beams with Hermite-Gaussian modes,” Opt. Lett 23, 989-991 (1998).
[CrossRef]

Brosnan, S.

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, 61020U-1-61020U-5 (2006).

Chen, Z.

P. Zhou, Z. Chen, X. Wang, Z. Liu, and X. Xu, “Tolerance on tilt error for coherent combining of fiber lasers,” submitted to Chin. Opt. Lett.

Cheung, E.

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, 61020U-1-61020U-5 (2006).

D.,

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, 61020U-1-61020U-5 (2006).

Durkin, M.

S. Norman, M. Zervas, A. Appleyard, M. Durkin, R. Horley, M. Varnham, J. Nilsson, and Y. Jeong, “Latest development of high power fiber lasers in SPI,” Proc. SPIE 5335, 229-237 (2004).
[CrossRef]

Epp, P.

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, 61020U-1-61020U-5 (2006).

Fan, D.

Y. Li, L. Qian, D. Lu, D. Fan, and S. Wen, “Coherent and incoherent combining of fiber array with hexagonal ring distribution,” Opt. Laser Technol. 39, 957-963 (2007).
[CrossRef]

Fan, T. Y.

Gori, F.

F. Gori, M. Santarsiero, and R. Borghi, “Intensity based modal analysis of partially coherent beams with Hermite-Gaussian modes,” Opt. Lett 23, 989-991 (1998).
[CrossRef]

Hammons,

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, 61020U-1-61020U-5 (2006).

Hogge, C. B.

Horley, R.

S. Norman, M. Zervas, A. Appleyard, M. Durkin, R. Horley, M. Varnham, J. Nilsson, and Y. Jeong, “Latest development of high power fiber lasers in SPI,” Proc. SPIE 5335, 229-237 (2004).
[CrossRef]

Jeong, Y.

S. Norman, M. Zervas, A. Appleyard, M. Durkin, R. Horley, M. Varnham, J. Nilsson, and Y. Jeong, “Latest development of high power fiber lasers in SPI,” Proc. SPIE 5335, 229-237 (2004).
[CrossRef]

Y. Jeong, J. K. Sahu, D. N. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12, 6088-6092 (2004).
[CrossRef] [PubMed]

Kansky, J. E.

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. J. Shaw, and R. C. Lawrence, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count ,” Proc. SPIE 6306, 63060G-1-63060G-11 (2006).

Komine, H.

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, 61020U-1-61020U-5 (2006).

Koyabu, K.

K. Koyabu, F. Ohira, and T. Yamamoto, “Fabrication of two-dimensional fiber arrays using microferrules,” IEEE Trans. Compon. Packag. Manuf. Technol. C 21, 11-19 (1998).
[CrossRef]

Lawrence, R. C.

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. J. Shaw, and R. C. Lawrence, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count ,” Proc. SPIE 6306, 63060G-1-63060G-11 (2006).

Li, B.

B. Li, B. Lü, “The polarization property and irradiance distribution of incoherent and coherent Gaussian beam combinations,” Optik (Jena) 113, 535-540 (2003).
[CrossRef]

Li, Y.

Y. Li, L. Qian, D. Lu, D. Fan, and S. Wen, “Coherent and incoherent combining of fiber array with hexagonal ring distribution,” Opt. Laser Technol. 39, 957-963 (2007).
[CrossRef]

Liu, Z.

P. Zhou, Z. Chen, X. Wang, Z. Liu, and X. Xu, “Tolerance on tilt error for coherent combining of fiber lasers,” submitted to Chin. Opt. Lett.

Lu, D.

Y. Li, L. Qian, D. Lu, D. Fan, and S. Wen, “Coherent and incoherent combining of fiber array with hexagonal ring distribution,” Opt. Laser Technol. 39, 957-963 (2007).
[CrossRef]

Lü, B.

B. Lü and H. Ma, “Beam propagation properties of radial laser arrays,” J. Opt. Soc. Am. A 17, 2005-2009 (2000).
[CrossRef]

B. Lü and H. Ma, “Coherent and incoherent off-axis Hermite-Gaussian beam combinations,” App. Opt 39, 1279-1289(2000).
[CrossRef]

B. Lü and H. Ma, “Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185-194 (1999).
[CrossRef]

Ma, H.

B. Lü and H. Ma, “Beam propagation properties of radial laser arrays,” J. Opt. Soc. Am. A 17, 2005-2009 (2000).
[CrossRef]

B. Lü and H. Ma, “Coherent and incoherent off-axis Hermite-Gaussian beam combinations,” App. Opt 39, 1279-1289(2000).
[CrossRef]

B. Lü and H. Ma, “Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185-194 (1999).
[CrossRef]

Mason, D. B.

Murphy, D. V.

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. J. Shaw, and R. C. Lawrence, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count ,” Proc. SPIE 6306, 63060G-1-63060G-11 (2006).

Nabors, C. D.

C. D. Nabors, “Effects of phased errors on coherent emitter arrays,” App. Opt 33, 2284-2289 (1994).
[CrossRef]

Nilsson, J.

Y. Jeong, J. K. Sahu, D. N. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12, 6088-6092 (2004).
[CrossRef] [PubMed]

S. Norman, M. Zervas, A. Appleyard, M. Durkin, R. Horley, M. Varnham, J. Nilsson, and Y. Jeong, “Latest development of high power fiber lasers in SPI,” Proc. SPIE 5335, 229-237 (2004).
[CrossRef]

Norman, S.

S. Norman, M. Zervas, A. Appleyard, M. Durkin, R. Horley, M. Varnham, J. Nilsson, and Y. Jeong, “Latest development of high power fiber lasers in SPI,” Proc. SPIE 5335, 229-237 (2004).
[CrossRef]

Ohira, F.

K. Koyabu, F. Ohira, and T. Yamamoto, “Fabrication of two-dimensional fiber arrays using microferrules,” IEEE Trans. Compon. Packag. Manuf. Technol. C 21, 11-19 (1998).
[CrossRef]

Payne, D. N.

Qian, L.

Y. Li, L. Qian, D. Lu, D. Fan, and S. Wen, “Coherent and incoherent combining of fiber array with hexagonal ring distribution,” Opt. Laser Technol. 39, 957-963 (2007).
[CrossRef]

Ranka, J. K.

Sahu, J. K.

Sanchez, A.

Santarsiero, M.

F. Gori, M. Santarsiero, and R. Borghi, “Intensity based modal analysis of partially coherent beams with Hermite-Gaussian modes,” Opt. Lett 23, 989-991 (1998).
[CrossRef]

Schultz, H. F.

Shaw, S. E. J.

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. J. Shaw, and R. C. Lawrence, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count ,” Proc. SPIE 6306, 63060G-1-63060G-11 (2006).

Shellan, J. B.

Siegman, A. E.

A. E. Siegman, “How to (maybe) measure laser beam quality,” presented at the OSA Annual Meeting, October 1997.

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 1-4 (1990).

Thompson, W. E.

Varnham, M.

S. Norman, M. Zervas, A. Appleyard, M. Durkin, R. Horley, M. Varnham, J. Nilsson, and Y. Jeong, “Latest development of high power fiber lasers in SPI,” Proc. SPIE 5335, 229-237 (2004).
[CrossRef]

Wang, X.

P. Zhou, Z. Chen, X. Wang, Z. Liu, and X. Xu, “Tolerance on tilt error for coherent combining of fiber lasers,” submitted to Chin. Opt. Lett.

Weber, M.

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, 61020U-1-61020U-5 (2006).

Wen, S.

Y. Li, L. Qian, D. Lu, D. Fan, and S. Wen, “Coherent and incoherent combining of fiber array with hexagonal ring distribution,” Opt. Laser Technol. 39, 957-963 (2007).
[CrossRef]

Wickham, M.

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, 61020U-1-61020U-5 (2006).

Xu, X.

P. Zhou, Z. Chen, X. Wang, Z. Liu, and X. Xu, “Tolerance on tilt error for coherent combining of fiber lasers,” submitted to Chin. Opt. Lett.

Yamamoto, T.

K. Koyabu, F. Ohira, and T. Yamamoto, “Fabrication of two-dimensional fiber arrays using microferrules,” IEEE Trans. Compon. Packag. Manuf. Technol. C 21, 11-19 (1998).
[CrossRef]

Yu, C. X.

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. J. Shaw, and R. C. Lawrence, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count ,” Proc. SPIE 6306, 63060G-1-63060G-11 (2006).

Zervas, M.

S. Norman, M. Zervas, A. Appleyard, M. Durkin, R. Horley, M. Varnham, J. Nilsson, and Y. Jeong, “Latest development of high power fiber lasers in SPI,” Proc. SPIE 5335, 229-237 (2004).
[CrossRef]

Zhou, P.

P. Zhou, Z. Chen, X. Wang, Z. Liu, and X. Xu, “Tolerance on tilt error for coherent combining of fiber lasers,” submitted to Chin. Opt. Lett.

App. Opt (2)

C. D. Nabors, “Effects of phased errors on coherent emitter arrays,” App. Opt 33, 2284-2289 (1994).
[CrossRef]

B. Lü and H. Ma, “Coherent and incoherent off-axis Hermite-Gaussian beam combinations,” App. Opt 39, 1279-1289(2000).
[CrossRef]

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

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

IEEE Trans. Compon. Packag. Manuf. Technol. C (1)

K. Koyabu, F. Ohira, and T. Yamamoto, “Fabrication of two-dimensional fiber arrays using microferrules,” IEEE Trans. Compon. Packag. Manuf. Technol. C 21, 11-19 (1998).
[CrossRef]

J. Opt. Soc. Am. A (2)

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

Opt. Commun. (1)

B. Lü and H. Ma, “Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185-194 (1999).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (1)

Y. Li, L. Qian, D. Lu, D. Fan, and S. Wen, “Coherent and incoherent combining of fiber array with hexagonal ring distribution,” Opt. Laser Technol. 39, 957-963 (2007).
[CrossRef]

Opt. Lett (1)

F. Gori, M. Santarsiero, and R. Borghi, “Intensity based modal analysis of partially coherent beams with Hermite-Gaussian modes,” Opt. Lett 23, 989-991 (1998).
[CrossRef]

Opt. Lett. (1)

Optik (Jena) (1)

B. Li, B. Lü, “The polarization property and irradiance distribution of incoherent and coherent Gaussian beam combinations,” Optik (Jena) 113, 535-540 (2003).
[CrossRef]

Proc. SPIE (4)

S. Norman, M. Zervas, A. Appleyard, M. Durkin, R. Horley, M. Varnham, J. Nilsson, and Y. Jeong, “Latest development of high power fiber lasers in SPI,” Proc. SPIE 5335, 229-237 (2004).
[CrossRef]

J. E. Kansky, C. X. Yu, D. V. Murphy, S. E. J. Shaw, and R. C. Lawrence, “Beam control of a 2D polarization maintaining fiber optic phased array with high-fiber count ,” Proc. SPIE 6306, 63060G-1-63060G-11 (2006).

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, 61020U-1-61020U-5 (2006).

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 1-4 (1990).

Other (3)

http://www.darpa.mil/mto/programs/adhels/index.htm, “Architecture for diode high energy laser systems”

A. E. Siegman, “How to (maybe) measure laser beam quality,” presented at the OSA Annual Meeting, October 1997.

P. Zhou, Z. Chen, X. Wang, Z. Liu, and X. Xu, “Tolerance on tilt error for coherent combining of fiber lasers,” submitted to Chin. Opt. Lett.

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Figures (19)

Fig. 1
Fig. 1

Schematic diagram of the fiber laser array with ring distribution.

Fig. 2
Fig. 2

Contour plot of far-field intensity profile for a coherently combined ring-distributed fiber laser array (91 lasers).

Fig. 3
Fig. 3

Contour plot of far-field intensity profile for a coherently combined ring-distributed fiber laser array: (a) 61 lasers, (b) 91 lasers, and (c) 127 lasers.

Fig. 4
Fig. 4

Contour plot of far-field intensity profile for a coherently combined ring-distributed fiber laser with different vacancy factors (a)  t = 0.5 and (b)  t = 3 .

Fig. 5
Fig. 5

Far-field intensity profile along the x axis for coherently combined ring-distributed fiber laser with different vacancy factors.

Fig. 6
Fig. 6

BPF dependence on the vacancy factor and the number of lasers in a coherent fiber laser array.

Fig. 7
Fig. 7

Contour plot of far-field intensity profile for 36 combined fiber lasers with different array shape: (a) ring distribution and (b) square distribution.

Fig. 8
Fig. 8

Normalized far-field intensity profile along the x axis for 36 combined fiber lasers with different array shape.

Fig. 9
Fig. 9

Contour plot of far-field intensity profile for 169 combined fiber lasers with different array shape: (a) ring distribution and (b) square distribution.

Fig. 10
Fig. 10

Normalized far-field intensity profile along the x axis for 169 combined fiber lasers with different array shape.

Fig. 11
Fig. 11

Dependence of BPF on the vacancy factor.

Fig. 12
Fig. 12

Dependence of BPF on the normalized tilt error.

Fig. 13
Fig. 13

Dependence of normalized BPF on the normalized tilt error.

Fig. 14
Fig. 14

BPF dependence on phase error.

Fig. 15
Fig. 15

BPF dependence on polarization angle error.

Fig. 16
Fig. 16

BPF dependence on jitter.

Fig. 17
Fig. 17

BPF dependence on beam quality of each beamlet.

Fig. 18
Fig. 18

Contour plot of far-field irradiance profile for a coherently combined fiber laser array (91 lasers) with some broken elements: (a) 11 lasers not working, (b) 21 lasers not working, and (c) 31 lasers not working.

Fig. 19
Fig. 19

Far-field irradiance profile for coherently combined fiber laser array with some broken elements along the x axis.

Equations (11)

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E n ( x , y , z ) = exp ( ( ( x x n ) 2 + ( y y n ) 2 ) w 2 ( z ) ) · exp ( i { k [ ( ( x x n ) 2 + ( y y n ) 2 ) 2 R ( z ) + z ] ψ + ψ n 0 } ) ,
k = 2 π λ Z 0 = π w 0 2 λ , w ( z ) = w 0 1 + ( z Z 0 ) 2 , R ( z ) = Z 0 ( z Z 0 + Z 0 z ) , ψ = arctan ( z Z 0 ) .
I ( x , y , z ) = ( n E n ( x , y , z ) ) ( n E n ( x , y , z ) ) * .
BPF fill _ factor ( t ) = 0.000639 t 6 + 0.01124 t 5 0.07675 t 4 + 0.2485 t 3 0.3302 t 2 0.1388 t + 0.838.
E n _ tilt ( x , y , z ) = n exp [ ( η n 2 + ξ n 2 ) w 2 ( χ n ) ] exp { i × k [ ( η n 2 + ξ n 2 ) 2 Δ R n + χ n ] Δ ψ n } ,
{ η n = cos θ x , n x + sin θ y , n sin θ x , n y sin θ x , n cos θ y , n z ξ n = cos θ y , n y + sin θ y , n z χ n = sin θ x , n x sin θ y , n cos θ x , n y + cos θ x , n cos θ y , n z Δ R n = Z 0 ( χ n / Z 0 + Z 0 / χ n ) Δ ψ n = arctan ( χ n / Z 0 ) w ( χ n ) = w 0 1 + ( χ n / Z 0 ) 2 .
BPF tilt_ error ( r ) = 0.25563 r 6 1.4233 r 5 + 2.5507 r 4 1.0614 r 3 0.99629 r 2 0.04239 r + 0.9998.
BPF = BPF fill _ factor ( t ) × BPF fill _ factor ( r ) .
I ( x , y , z ) = ( n E n ( x , y , z ) × exp ( i × δ φ n ) ) ( n E n ( x , y , z ) × exp ( i × δ φ n ) ) * .
I ( x , y , z ) = ( n E n x ( x , y , z ) ) ( n E n x ( x , y , z ) ) * + ( n E n y ( x , y , z ) ) ( n E n y ( x , y , z ) ) * ,
E n x ( x , y , z ) = E n ( x , y , z ) cos θ n E n y ( x , y , z ) = E n ( x , y , z ) sin θ n ,

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