Abstract

The influence of the Kramers-Kronig phase is demonstrated in a coherently combined fiber laser where other passive phasing mechanisms such as wavelength tuning have been suppressed. A mathematical model is developed to predict the lasing supermode and is supported by experimental measurements of the gain, phase, and power. The results show that the difference in Kramers-Kronig phase arising from a difference in gain between the two arms partially compensates for an externally applied phase error.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. Coherent Laser Beam Combining, A. Brignon, ed. (Wiley, 2013).
  2. A. E. Siegman, “Resonant modes of linearly coupled multiple fiber laser structures,” unpublished (2004).
  3. D. Kouznetsov, J. F. Bisson, A. Shirakawa, and K. I. Ueda, “Limits of coherent addition of lasers: Simple estimate,” Opt. Rev. Lett. 12(6), 445–447 (2005).
    [Crossref]
  4. J. E. Rothenberg, “Passive coherent phasing of fiber laser arrays,” Proc. SPIE 6873, 687315 (2008).
    [Crossref]
  5. F. Jeux, A. Desfarges-Berthelemot, V. Kermène, and A. Barthelemy, “Efficient passive phasing of an array of 20 ring fiber lasers,” Laser Phys. Lett. 11(9), 095003 (2014).
    [Crossref]
  6. C. J. Corcoran and F. Durville, “Passive coherent combination of a diode laser array with 35 elements,” Opt. Express 22(7), 8420–8425 (2014).
    [Crossref] [PubMed]
  7. W. Z. Chang, T. W. Wu, H. G. H. Winful, and A. Galvanauskas, “Array size scalability of passively coherently phased fiber laser arrays,” Opt. Express 18(9), 9634–9642 (2010).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  9. S. J. Augst, T. Y. Fan, and A. Sanchez, “Coherent beam combining and phase noise measurements of ytterbium fiber amplifiers,” Opt. Lett. 29(5), 474–476 (2004).
    [Crossref] [PubMed]
  10. C. J. Corcoran and K. A. Pasch, “Output phase characteristics of a nonlinear regenerative fiber amplifier,” IEEE J. Quantum Electron. 43(6), 437–439 (2007).
    [Crossref]
  11. W. Ray, C. J. Corcoran, and F. Durville, “Regenerative feedback opens the scaling bottleneck of passive fiber laser arrays,” presented at Solid State Diode Laser Technology Review, Santa Fe, New Mexico, USA 6–9 June 2011.
  12. J. W. Arkwright, P. Elango, G. R. Atkins, T. Whitbread, and M. J. Digonnet, “Experimental and theoretical analysis of the resonant nonlinearity in ytterbium-doped fiber,” J. Lightwave Technol. 16(5), 798–806 (1998).
    [Crossref]
  13. E. J. Bochove, M. R. Zunoubi, and C. J. Corcoran, “Effect of Kerr and resonant nonlinearities on phase locking of a multistable fiber amplifier array,” Opt. Lett. 38(23), 5016–5019 (2013).
    [Crossref] [PubMed]
  14. A. P. Napartovich, N. N. Elkin, and D. V. Vysotsky, “Asymptotic theory of a large fiber-laser array passive phase locking,” Appl. Opt. 53(31), I23–I30 (2014).
    [Crossref] [PubMed]
  15. H.-S. Chiang, J. R. Leger, J. Nilsson, and J. Sahu, “Direct observation of Kramers-Kronig self-phasing in coherently combined fiber lasers,” Opt. Lett. 38(20), 4104–4107 (2013).
    [Crossref] [PubMed]
  16. H.-S. Chiang, J. Nilsson, J. Sahu, and J. R. Leger, “Experimental measurements of the origin of self-phasing in passively coupled fiber lasers,” Opt. Lett. 40(6), 962–965 (2015).
    [Crossref] [PubMed]
  17. H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).
    [Crossref]
  18. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co. 2005), Problem 4.14.
  19. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
    [Crossref]
  20. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford, 2007).
  21. W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36(8), 2487–2490 (1965).
    [Crossref]
  22. A. E. Siegman, Lasers (University Science Books, 1986), Sect. 13.7.
  23. M. J. Adams, “Physics and applications of optical bistability in semiconductor laser amplifiers,” Solid-State Electron. 30(1), 43–51 (1987).
    [Crossref]
  24. C. J. Corcoran, F. Durville, and K. A. Pasch, “Coherent array of nonlinear regenerative fiber amplifiers,” IEEE J. Quantum Electron. 44(3), 275–282 (2008).
    [Crossref]
  25. C. J. Corcoran, F. Durville, and W. Ray, “Regenerative phase shift and its effect on coherent laser arrays,” IEEE J. Quantum Electron. 47(7), 1043–1048 (2011).
    [Crossref]
  26. M. Khajavikhan and J. R. Leger, “Modal analysis of path length sensitivity in superposition architectures for coherent laser beam combining,” IEEE J. Sel. Top. Quantum Electron. 15(2), 281–290 (2009).
    [Crossref]
  27. M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. 23(1), 9–29 (1987).
    [Crossref]

2015 (1)

2014 (3)

2013 (2)

2011 (1)

C. J. Corcoran, F. Durville, and W. Ray, “Regenerative phase shift and its effect on coherent laser arrays,” IEEE J. Quantum Electron. 47(7), 1043–1048 (2011).
[Crossref]

2010 (2)

2009 (1)

M. Khajavikhan and J. R. Leger, “Modal analysis of path length sensitivity in superposition architectures for coherent laser beam combining,” IEEE J. Sel. Top. Quantum Electron. 15(2), 281–290 (2009).
[Crossref]

2008 (2)

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

C. J. Corcoran, F. Durville, and K. A. Pasch, “Coherent array of nonlinear regenerative fiber amplifiers,” IEEE J. Quantum Electron. 44(3), 275–282 (2008).
[Crossref]

2007 (1)

C. J. Corcoran and K. A. Pasch, “Output phase characteristics of a nonlinear regenerative fiber amplifier,” IEEE J. Quantum Electron. 43(6), 437–439 (2007).
[Crossref]

2005 (1)

D. Kouznetsov, J. F. Bisson, A. Shirakawa, and K. I. Ueda, “Limits of coherent addition of lasers: Simple estimate,” Opt. Rev. Lett. 12(6), 445–447 (2005).
[Crossref]

2004 (1)

1998 (1)

1987 (2)

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. 23(1), 9–29 (1987).
[Crossref]

M. J. Adams, “Physics and applications of optical bistability in semiconductor laser amplifiers,” Solid-State Electron. 30(1), 43–51 (1987).
[Crossref]

1982 (1)

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
[Crossref]

1977 (1)

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).
[Crossref]

1965 (1)

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36(8), 2487–2490 (1965).
[Crossref]

Adams, M. J.

M. J. Adams, “Physics and applications of optical bistability in semiconductor laser amplifiers,” Solid-State Electron. 30(1), 43–51 (1987).
[Crossref]

Arkwright, J. W.

Atkins, G. R.

Augst, S. J.

Barthelemy, A.

F. Jeux, A. Desfarges-Berthelemot, V. Kermène, and A. Barthelemy, “Efficient passive phasing of an array of 20 ring fiber lasers,” Laser Phys. Lett. 11(9), 095003 (2014).
[Crossref]

Bisson, J. F.

D. Kouznetsov, J. F. Bisson, A. Shirakawa, and K. I. Ueda, “Limits of coherent addition of lasers: Simple estimate,” Opt. Rev. Lett. 12(6), 445–447 (2005).
[Crossref]

Bochove, E. J.

Buus, J.

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. 23(1), 9–29 (1987).
[Crossref]

Chang, W. Z.

Chiang, H.-S.

Corcoran, C. J.

C. J. Corcoran and F. Durville, “Passive coherent combination of a diode laser array with 35 elements,” Opt. Express 22(7), 8420–8425 (2014).
[Crossref] [PubMed]

E. J. Bochove, M. R. Zunoubi, and C. J. Corcoran, “Effect of Kerr and resonant nonlinearities on phase locking of a multistable fiber amplifier array,” Opt. Lett. 38(23), 5016–5019 (2013).
[Crossref] [PubMed]

C. J. Corcoran, F. Durville, and W. Ray, “Regenerative phase shift and its effect on coherent laser arrays,” IEEE J. Quantum Electron. 47(7), 1043–1048 (2011).
[Crossref]

C. J. Corcoran, F. Durville, and K. A. Pasch, “Coherent array of nonlinear regenerative fiber amplifiers,” IEEE J. Quantum Electron. 44(3), 275–282 (2008).
[Crossref]

C. J. Corcoran and K. A. Pasch, “Output phase characteristics of a nonlinear regenerative fiber amplifier,” IEEE J. Quantum Electron. 43(6), 437–439 (2007).
[Crossref]

Dammann, H.

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).
[Crossref]

Davidson, N.

Desfarges-Berthelemot, A.

F. Jeux, A. Desfarges-Berthelemot, V. Kermène, and A. Barthelemy, “Efficient passive phasing of an array of 20 ring fiber lasers,” Laser Phys. Lett. 11(9), 095003 (2014).
[Crossref]

Digonnet, M. J.

Durville, F.

C. J. Corcoran and F. Durville, “Passive coherent combination of a diode laser array with 35 elements,” Opt. Express 22(7), 8420–8425 (2014).
[Crossref] [PubMed]

C. J. Corcoran, F. Durville, and W. Ray, “Regenerative phase shift and its effect on coherent laser arrays,” IEEE J. Quantum Electron. 47(7), 1043–1048 (2011).
[Crossref]

C. J. Corcoran, F. Durville, and K. A. Pasch, “Coherent array of nonlinear regenerative fiber amplifiers,” IEEE J. Quantum Electron. 44(3), 275–282 (2008).
[Crossref]

Elango, P.

Elkin, N. N.

Fan, T. Y.

Fridman, M.

Friesem, A. A.

Galvanauskas, A.

Henry, C. H.

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
[Crossref]

Jeux, F.

F. Jeux, A. Desfarges-Berthelemot, V. Kermène, and A. Barthelemy, “Efficient passive phasing of an array of 20 ring fiber lasers,” Laser Phys. Lett. 11(9), 095003 (2014).
[Crossref]

Kermène, V.

F. Jeux, A. Desfarges-Berthelemot, V. Kermène, and A. Barthelemy, “Efficient passive phasing of an array of 20 ring fiber lasers,” Laser Phys. Lett. 11(9), 095003 (2014).
[Crossref]

Khajavikhan, M.

M. Khajavikhan and J. R. Leger, “Modal analysis of path length sensitivity in superposition architectures for coherent laser beam combining,” IEEE J. Sel. Top. Quantum Electron. 15(2), 281–290 (2009).
[Crossref]

Klotz, E.

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).
[Crossref]

Kouznetsov, D.

D. Kouznetsov, J. F. Bisson, A. Shirakawa, and K. I. Ueda, “Limits of coherent addition of lasers: Simple estimate,” Opt. Rev. Lett. 12(6), 445–447 (2005).
[Crossref]

Leger, J. R.

Napartovich, A. P.

Nilsson, J.

Nixon, M.

Osinski, M.

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. 23(1), 9–29 (1987).
[Crossref]

Pasch, K. A.

C. J. Corcoran, F. Durville, and K. A. Pasch, “Coherent array of nonlinear regenerative fiber amplifiers,” IEEE J. Quantum Electron. 44(3), 275–282 (2008).
[Crossref]

C. J. Corcoran and K. A. Pasch, “Output phase characteristics of a nonlinear regenerative fiber amplifier,” IEEE J. Quantum Electron. 43(6), 437–439 (2007).
[Crossref]

Ray, W.

C. J. Corcoran, F. Durville, and W. Ray, “Regenerative phase shift and its effect on coherent laser arrays,” IEEE J. Quantum Electron. 47(7), 1043–1048 (2011).
[Crossref]

Rigrod, W. W.

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36(8), 2487–2490 (1965).
[Crossref]

Rothenberg, J. E.

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

Sahu, J.

Sanchez, A.

Shirakawa, A.

D. Kouznetsov, J. F. Bisson, A. Shirakawa, and K. I. Ueda, “Limits of coherent addition of lasers: Simple estimate,” Opt. Rev. Lett. 12(6), 445–447 (2005).
[Crossref]

Siegman, A. E.

A. E. Siegman, “Resonant modes of linearly coupled multiple fiber laser structures,” unpublished (2004).

Ueda, K. I.

D. Kouznetsov, J. F. Bisson, A. Shirakawa, and K. I. Ueda, “Limits of coherent addition of lasers: Simple estimate,” Opt. Rev. Lett. 12(6), 445–447 (2005).
[Crossref]

Vysotsky, D. V.

Whitbread, T.

Winful, H. G. H.

Wu, T. W.

Zunoubi, M. R.

Appl. Opt. (1)

IEEE J. Quantum Electron. (5)

M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers—an overview,” IEEE J. Quantum Electron. 23(1), 9–29 (1987).
[Crossref]

C. J. Corcoran and K. A. Pasch, “Output phase characteristics of a nonlinear regenerative fiber amplifier,” IEEE J. Quantum Electron. 43(6), 437–439 (2007).
[Crossref]

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18(2), 259–264 (1982).
[Crossref]

C. J. Corcoran, F. Durville, and K. A. Pasch, “Coherent array of nonlinear regenerative fiber amplifiers,” IEEE J. Quantum Electron. 44(3), 275–282 (2008).
[Crossref]

C. J. Corcoran, F. Durville, and W. Ray, “Regenerative phase shift and its effect on coherent laser arrays,” IEEE J. Quantum Electron. 47(7), 1043–1048 (2011).
[Crossref]

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

M. Khajavikhan and J. R. Leger, “Modal analysis of path length sensitivity in superposition architectures for coherent laser beam combining,” IEEE J. Sel. Top. Quantum Electron. 15(2), 281–290 (2009).
[Crossref]

J. Appl. Phys. (1)

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36(8), 2487–2490 (1965).
[Crossref]

J. Lightwave Technol. (1)

Laser Phys. Lett. (1)

F. Jeux, A. Desfarges-Berthelemot, V. Kermène, and A. Barthelemy, “Efficient passive phasing of an array of 20 ring fiber lasers,” Laser Phys. Lett. 11(9), 095003 (2014).
[Crossref]

Opt. Acta (Lond.) (1)

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta (Lond.) 24(4), 505–515 (1977).
[Crossref]

Opt. Express (2)

Opt. Lett. (5)

Opt. Rev. Lett. (1)

D. Kouznetsov, J. F. Bisson, A. Shirakawa, and K. I. Ueda, “Limits of coherent addition of lasers: Simple estimate,” Opt. Rev. Lett. 12(6), 445–447 (2005).
[Crossref]

Proc. SPIE (1)

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

Solid-State Electron. (1)

M. J. Adams, “Physics and applications of optical bistability in semiconductor laser amplifiers,” Solid-State Electron. 30(1), 43–51 (1987).
[Crossref]

Other (6)

A. E. Siegman, Lasers (University Science Books, 1986), Sect. 13.7.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford, 2007).

Coherent Laser Beam Combining, A. Brignon, ed. (Wiley, 2013).

A. E. Siegman, “Resonant modes of linearly coupled multiple fiber laser structures,” unpublished (2004).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co. 2005), Problem 4.14.

W. Ray, C. J. Corcoran, and F. Durville, “Regenerative feedback opens the scaling bottleneck of passive fiber laser arrays,” presented at Solid State Diode Laser Technology Review, Santa Fe, New Mexico, USA 6–9 June 2011.

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Figures (6)

Fig. 1
Fig. 1 Experimental setup to measure gain-dependent self-phasing in a coherently combined fiber laser. The two core modes are combined by a Dammann grating. Measurements of the active resonator are performed using an external probe laser.
Fig. 2
Fig. 2 Beam splitting and combining with the Dammann grating. (a) A beam incident from the right is split into multiple orders; only the m=±1 orders enter the fiber cores. (b) The on-axis orders from the fiber cores are combined.
Fig. 3
Fig. 3 Total phase difference φT measured at the left-hand fiber facet, versus applied phase difference 2Δφ. Mode-matching efficiencies (a) highly imbalanced (a1/a2 = 0.6), and (b) nearly equal (a1/a2 = 0.9).
Fig. 4
Fig. 4 Power in the laser mode versus applied phase, observed at (a) the right-hand side of the resonator in the on-axis beam, and (b) each core at the left-hand side of the fiber.
Fig. 5
Fig. 5 Ratio of the gain saturated by the laser mode to the small-signal gain, plotted as a function of applied phase.
Fig. 6
Fig. 6 Adjustments to the alignment of Dammann and Littrow gratings. Views are shown looking downward at the external cavity optics (spatial filter and other components not shown), and looking at the fiber facet, where the fiber cores (dashed circles) and incident Dammann orders (solid circles) are depicted. Different alignment conditions: (a) “Perfect” alignment, (b) Dammann grating tilted, (c) Dammann and Littrow gratings tilted.

Equations (25)

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

b m =| b m |exp( jmΔφ ), | b m |={ 2 | m |π (m=odd) 0 (m=even) ,
M 1 =( g 1 exp( jΔ φ KK /2 ) 0 0 g 2 exp( +jΔ φ KK /2 ) ),
M 2 = 4 π 2 ( exp( j 2Δφ ) 1 1 exp( +j 2Δφ ) ),
M c =( a 1 0 0 a 2 )
M RT =r M 1 M c M 2 M 1 =r 4 π 2 ( g 1 2 a 1 exp( j φ T ) g 1 g 2 a 1 g 1 g 2 a 2 g 2 2 a 2 exp( +j φ T ) ),
M RT ψ n = λ n ψ n (n=1,2),
λ= 4 π 2 r[ g 1 2 a 1 exp( j φ T )+ g 2 2 a 2 exp( +j φ T ) ],
ψ=( E 1 E 2 )= 1 g 1 2 a 1 2 + g 2 2 a 2 2 ( g 1 a 1 exp( j φ T /2 ) g 2 a 2 exp( +j φ T /2 ) ).
1= | λ | 2 = 16 π 4 r 2 [ g 1 4 a 1 2 + g 2 4 a 2 2 +2 g 1 2 g 2 2 a 1 a 2 cos( 2 φ T ) ],
φ KK,i =k 0 L Δ n i ( z ) dz
G i =exp[ 2k 0 L Δ n i (z)dz ],
φ KK,i = α 2 ln G i ,
φ T = α 2 ln( G 2 G 1 )+2Δφ,
lnG= ln G 0 1+P/ P sat ,
P R P sat = R L ( ln G 0 + 1 2 ln R L R R ) ( R L + R R )( 1 R L R R ) ,
P R,i P sat = R L G i 2 ln( G 0 / G i ) ( R L G i +1 )( G i 1 ) ,
P L,i P sat = P net | E i | 2 P sat = G i ln( G 0 / G i ) ( R L G i +1 )( G i 1 ) .
| E 2 | 2 | E 1 | 2 = P L,2 P L,1 = G 2 a 2 2 G 1 a 1 2 .
ln( G 0 / G 1 ) a 1 2 ( R L G 1 +1 )( G 1 1 ) = ln( G 0 / G 2 ) a 2 2 ( R L G 2 +1 )( G 2 1 ) ,
M 2 =DBD ,
D=( b 0 b 1 b 2 b 1 b 0 b 1 b 2 b 1 b 0 )= 2 π ( 0 e jΔφ 0 e jΔφ 0 e jΔφ 0 e jΔφ 0 ) .
B=( 0 0 0 0 1 0 0 0 0 ) .
M 2 = 4 π 2 ( e j2Δφ 0 1 0 0 0 1 0 e j2Δφ ) .
u R = 2 π ( 0 e jΔφ 0 e jΔφ 0 e jΔφ 0 e jΔφ 0 )( G 1 e jΔ φ KK /2 0 0 0 0 0 0 0 G 2 e jΔ φ KK /2 )( P L,1 e j φ T /2 0 P L,2 e jΔ φ T /2 ) ,
P on-axis = ( 2/π ) 2 [ G 1 P L,1 + G 2 P L,2 +2 G 1 G 2 P L,1 P L,2 cos( 2 φ T ) ] ,

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