Abstract

Self-phasing due to spatial mode selection in a two-element passively coupled fiber laser is studied. We find that the addition of a second supermode in a coupled resonator results in a 90% increase in the average output power and nearly π/2 radians of passive phase adjustment versus applied phase errors between the gain elements. These results require a phase of zero (modulo 2π) between the beams in the external cavity. These findings are supported by an eigenmode analysis of the resonator and show that beam recycling is a useful resonator design feature but must be appropriately implemented to obtain beneficial results.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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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).
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  2. D. Sabourdy, V. Kermene, A. Desfarges-Berthelemot, M. Vampouille, and A. Barthelemy, “Coherent combining of two Nd:YAG lasers in a Vernier-Michelson-type cavity,” Appl. Phys. B: Lasers Opt. 75(4-5), 503–507 (2002).
    [Crossref]
  3. J. R. Leger, G. J. Swanson, and W. B. Veldkamp, “Coherent laser addition using binary phase gratings,” Appl. Opt. 26(20), 4391–4399 (1987).
    [Crossref]
  4. J. R. Leger, “Lateral mode control of an AlGaAs laser array in a talbot cavity,” Appl. Phys. Lett. 55(4), 334–336 (1989).
    [Crossref]
  5. R. H. Rediker, R. P. Schloss, and L. J. Van Ruyven, “Operation of individual diode lasers as a coherent ensemble controlled by a spatial filter within an external cavity,” Appl. Phys. Lett. 46(2), 133–135 (1985).
    [Crossref]
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    [Crossref]
  9. 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(2), 248–256 (2009).
    [Crossref]
  10. 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]
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  12. D. Kouznetsov, J.-F. Bisson, A. Shirakawa, and K.-I. Ueda, “Limits of coherent addition of lasers: Simple estimate,” Opt. Rev. 12(6), 445–447 (2005).
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    [Crossref]
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    [Crossref]
  17. E. J. Bochove, A. B. Aceves, Y. Braiman, P. Colet, R. Deiterding, A. Jacobo, C. A. Miller, C. Rhodes, and S. A. Shakir, “Model of the self-q-switching instability of passively phased fiber laser arrays,” IEEE J. Quantum Electron. 47(6), 777–785 (2011).
    [Crossref]
  18. W. M. Kunkel and J. R. Leger, “Gain dependent self-phasing in a two-core coherently combined fiber laser,” Opt. Express 26(8), 9373–9388 (2018).
    [Crossref]
  19. W. M. Kunkel and J. R. Leger, “Kramers-kronig self-phasing effect in passive beam combining resonators,” Proc. SPIE 10904, 109041N (2019).
    [Crossref]
  20. C. J. Corcoran and F. Durville, “Passive phasing in a coherent laser array,” IEEE J. Sel. Top. Quantum Electron. 15(2), 294–300 (2009).
    [Crossref]
  21. 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]
  22. M. Khajavikhan and J. 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]
  23. M. Khajavikhan, K. John, and J. R. Leger, “Experimental measurements of supermodes in superposition architectures for coherent laser beam combining,” IEEE J. Quantum Electron. 46(8), 1221–1231 (2010).
    [Crossref]
  24. W. M. Kunkel and J. R. Leger, “Passive coherent laser beam combining with spatial mode selecting feedback,” IEEE J. Quantum Electron. 55(4), 1–8 (2019).
    [Crossref]
  25. H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24(4), 505–515 (1977).
    [Crossref]
  26. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005), 3rd ed.
  27. A. E. Siegman, Lasers (University Science Books, 1986). pp. 567.
  28. W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36(8), 2487–2490 (1965).
    [Crossref]
  29. C. Wan, B. Tiffany, and J. R. Leger, “Analysis of path length sensitivity in coherent beam combining by spatial filtering,” IEEE J. Quantum Electron. 47(6), 770–776 (2011).
    [Crossref]
  30. C. Wan and J. R. Leger, “Experimental measurements of path length sensitivity in coherent beam combining by spatial filtering,” IEEE J. Quantum Electron. 48(8), 1045–1051 (2012).
    [Crossref]
  31. 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]
  32. W. M. Kunkel, “Passive coherent beam combining by gain effects in fiber lasers and spatial beam control in gradient refractive index media,” Ph.D. thesis, University of Minnesota (2018).

2019 (2)

W. M. Kunkel and J. R. Leger, “Kramers-kronig self-phasing effect in passive beam combining resonators,” Proc. SPIE 10904, 109041N (2019).
[Crossref]

W. M. Kunkel and J. R. Leger, “Passive coherent laser beam combining with spatial mode selecting feedback,” IEEE J. Quantum Electron. 55(4), 1–8 (2019).
[Crossref]

2018 (1)

2014 (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]

2013 (2)

2012 (1)

C. Wan and J. R. Leger, “Experimental measurements of path length sensitivity in coherent beam combining by spatial filtering,” IEEE J. Quantum Electron. 48(8), 1045–1051 (2012).
[Crossref]

2011 (3)

C. Wan, B. Tiffany, and J. R. Leger, “Analysis of path length sensitivity in coherent beam combining by spatial filtering,” IEEE J. Quantum Electron. 47(6), 770–776 (2011).
[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]

E. J. Bochove, A. B. Aceves, Y. Braiman, P. Colet, R. Deiterding, A. Jacobo, C. A. Miller, C. Rhodes, and S. A. Shakir, “Model of the self-q-switching instability of passively phased fiber laser arrays,” IEEE J. Quantum Electron. 47(6), 777–785 (2011).
[Crossref]

2010 (2)

D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives (invited),” J. Opt. Soc. Am. B 27(11), B63–B92 (2010).
[Crossref]

M. Khajavikhan, K. John, and J. R. Leger, “Experimental measurements of supermodes in superposition architectures for coherent laser beam combining,” IEEE J. Quantum Electron. 46(8), 1221–1231 (2010).
[Crossref]

2009 (3)

M. Khajavikhan and J. 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]

C. J. Corcoran and F. Durville, “Passive phasing in a coherent laser array,” IEEE J. Sel. Top. Quantum Electron. 15(2), 294–300 (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(2), 248–256 (2009).
[Crossref]

2008 (2)

2007 (1)

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-synchronous and self-referenced coherent beam combination for large optical arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 480–486 (2007).
[Crossref]

2005 (2)

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11(3), 567–577 (2005).
[Crossref]

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

2004 (1)

2003 (1)

2002 (1)

D. Sabourdy, V. Kermene, A. Desfarges-Berthelemot, M. Vampouille, and A. Barthelemy, “Coherent combining of two Nd:YAG lasers in a Vernier-Michelson-type cavity,” Appl. Phys. B: Lasers Opt. 75(4-5), 503–507 (2002).
[Crossref]

1989 (1)

J. R. Leger, “Lateral mode control of an AlGaAs laser array in a talbot cavity,” Appl. Phys. Lett. 55(4), 334–336 (1989).
[Crossref]

1987 (1)

1985 (1)

R. H. Rediker, R. P. Schloss, and L. J. Van Ruyven, “Operation of individual diode lasers as a coherent ensemble controlled by a spatial filter within an external cavity,” Appl. Phys. Lett. 46(2), 133–135 (1985).
[Crossref]

1977 (1)

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 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]

Aceves, A. B.

E. J. Bochove, A. B. Aceves, Y. Braiman, P. Colet, R. Deiterding, A. Jacobo, C. A. Miller, C. Rhodes, and S. A. Shakir, “Model of the self-q-switching instability of passively phased fiber laser arrays,” IEEE J. Quantum Electron. 47(6), 777–785 (2011).
[Crossref]

Augst, S. J.

Baker, J. T.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-synchronous and self-referenced coherent beam combination for large optical arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 480–486 (2007).
[Crossref]

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]

D. Sabourdy, V. Kermene, A. Desfarges-Berthelemot, M. Vampouille, and A. Barthelemy, “Coherent combining of two Nd:YAG lasers in a Vernier-Michelson-type cavity,” Appl. Phys. B: Lasers Opt. 75(4-5), 503–507 (2002).
[Crossref]

Benham, V.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-synchronous and self-referenced coherent beam combination for large optical arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 480–486 (2007).
[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. 12(6), 445–447 (2005).
[Crossref]

Bochove, E. J.

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]

E. J. Bochove, A. B. Aceves, Y. Braiman, P. Colet, R. Deiterding, A. Jacobo, C. A. Miller, C. Rhodes, and S. A. Shakir, “Model of the self-q-switching instability of passively phased fiber laser arrays,” IEEE J. Quantum Electron. 47(6), 777–785 (2011).
[Crossref]

E. J. Bochove, P. K. Cheo, and G. G. King, “Self-organization in a multicore fiber laser array,” Opt. Lett. 28(14), 1200–1202 (2003).
[Crossref]

Braiman, Y.

E. J. Bochove, A. B. Aceves, Y. Braiman, P. Colet, R. Deiterding, A. Jacobo, C. A. Miller, C. Rhodes, and S. A. Shakir, “Model of the self-q-switching instability of passively phased fiber laser arrays,” IEEE J. Quantum Electron. 47(6), 777–785 (2011).
[Crossref]

Cheo, P. K.

Cheung, E. C.

Chiang, H.-S.

Clarkson, W. A.

Colet, P.

E. J. Bochove, A. B. Aceves, Y. Braiman, P. Colet, R. Deiterding, A. Jacobo, C. A. Miller, C. Rhodes, and S. A. Shakir, “Model of the self-q-switching instability of passively phased fiber laser arrays,” IEEE J. Quantum Electron. 47(6), 777–785 (2011).
[Crossref]

Corcoran, C. J.

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]

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 and F. Durville, “Passive phasing in a coherent laser array,” IEEE J. Sel. Top. Quantum Electron. 15(2), 294–300 (2009).
[Crossref]

Dammann, H.

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

Deiterding, R.

E. J. Bochove, A. B. Aceves, Y. Braiman, P. Colet, R. Deiterding, A. Jacobo, C. A. Miller, C. Rhodes, and S. A. Shakir, “Model of the self-q-switching instability of passively phased fiber laser arrays,” IEEE J. Quantum Electron. 47(6), 777–785 (2011).
[Crossref]

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]

D. Sabourdy, V. Kermene, A. Desfarges-Berthelemot, M. Vampouille, and A. Barthelemy, “Coherent combining of two Nd:YAG lasers in a Vernier-Michelson-type cavity,” Appl. Phys. B: Lasers Opt. 75(4-5), 503–507 (2002).
[Crossref]

Durville, F.

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 and F. Durville, “Passive phasing in a coherent laser array,” IEEE J. Sel. Top. Quantum Electron. 15(2), 294–300 (2009).
[Crossref]

Fan, T. Y.

T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11(3), 567–577 (2005).
[Crossref]

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]

Goodman, J. W.

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

Goodno, G. D.

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(2), 248–256 (2009).
[Crossref]

Ho, J. G.

Jacobo, A.

E. J. Bochove, A. B. Aceves, Y. Braiman, P. Colet, R. Deiterding, A. Jacobo, C. A. Miller, C. Rhodes, and S. A. Shakir, “Model of the self-q-switching instability of passively phased fiber laser arrays,” IEEE J. Quantum Electron. 47(6), 777–785 (2011).
[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]

John, K.

M. Khajavikhan, K. John, and J. R. Leger, “Experimental measurements of supermodes in superposition architectures for coherent laser beam combining,” IEEE J. Quantum Electron. 46(8), 1221–1231 (2010).
[Crossref]

Kermene, V.

D. Sabourdy, V. Kermene, A. Desfarges-Berthelemot, M. Vampouille, and A. Barthelemy, “Coherent combining of two Nd:YAG lasers in a Vernier-Michelson-type cavity,” Appl. Phys. B: Lasers Opt. 75(4-5), 503–507 (2002).
[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, K. John, and J. R. Leger, “Experimental measurements of supermodes in superposition architectures for coherent laser beam combining,” IEEE J. Quantum Electron. 46(8), 1221–1231 (2010).
[Crossref]

M. Khajavikhan and J. 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]

King, G. G.

Klotz, E.

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 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. 12(6), 445–447 (2005).
[Crossref]

Kunkel, W. M.

W. M. Kunkel and J. R. Leger, “Passive coherent laser beam combining with spatial mode selecting feedback,” IEEE J. Quantum Electron. 55(4), 1–8 (2019).
[Crossref]

W. M. Kunkel and J. R. Leger, “Kramers-kronig self-phasing effect in passive beam combining resonators,” Proc. SPIE 10904, 109041N (2019).
[Crossref]

W. M. Kunkel and J. R. Leger, “Gain dependent self-phasing in a two-core coherently combined fiber laser,” Opt. Express 26(8), 9373–9388 (2018).
[Crossref]

W. M. Kunkel, “Passive coherent beam combining by gain effects in fiber lasers and spatial beam control in gradient refractive index media,” Ph.D. thesis, University of Minnesota (2018).

Leger, J.

M. Khajavikhan and J. 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]

Leger, J. R.

W. M. Kunkel and J. R. Leger, “Kramers-kronig self-phasing effect in passive beam combining resonators,” Proc. SPIE 10904, 109041N (2019).
[Crossref]

W. M. Kunkel and J. R. Leger, “Passive coherent laser beam combining with spatial mode selecting feedback,” IEEE J. Quantum Electron. 55(4), 1–8 (2019).
[Crossref]

W. M. Kunkel and J. R. Leger, “Gain dependent self-phasing in a two-core coherently combined fiber laser,” Opt. Express 26(8), 9373–9388 (2018).
[Crossref]

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]

C. Wan and J. R. Leger, “Experimental measurements of path length sensitivity in coherent beam combining by spatial filtering,” IEEE J. Quantum Electron. 48(8), 1045–1051 (2012).
[Crossref]

C. Wan, B. Tiffany, and J. R. Leger, “Analysis of path length sensitivity in coherent beam combining by spatial filtering,” IEEE J. Quantum Electron. 47(6), 770–776 (2011).
[Crossref]

M. Khajavikhan, K. John, and J. R. Leger, “Experimental measurements of supermodes in superposition architectures for coherent laser beam combining,” IEEE J. Quantum Electron. 46(8), 1221–1231 (2010).
[Crossref]

J. R. Leger, “Lateral mode control of an AlGaAs laser array in a talbot cavity,” Appl. Phys. Lett. 55(4), 334–336 (1989).
[Crossref]

J. R. Leger, G. J. Swanson, and W. B. Veldkamp, “Coherent laser addition using binary phase gratings,” Appl. Opt. 26(20), 4391–4399 (1987).
[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(2), 248–256 (2009).
[Crossref]

Lu, C. A.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-synchronous and self-referenced coherent beam combination for large optical arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 480–486 (2007).
[Crossref]

Ma, H.

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(2), 248–256 (2009).
[Crossref]

Ma, Y.

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(2), 248–256 (2009).
[Crossref]

Miller, C. A.

E. J. Bochove, A. B. Aceves, Y. Braiman, P. Colet, R. Deiterding, A. Jacobo, C. A. Miller, C. Rhodes, and S. A. Shakir, “Model of the self-q-switching instability of passively phased fiber laser arrays,” IEEE J. Quantum Electron. 47(6), 777–785 (2011).
[Crossref]

Nilsson, J.

Pilkington, D.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-synchronous and self-referenced coherent beam combination for large optical arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 480–486 (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]

Rediker, R. H.

R. H. Rediker, R. P. Schloss, and L. J. Van Ruyven, “Operation of individual diode lasers as a coherent ensemble controlled by a spatial filter within an external cavity,” Appl. Phys. Lett. 46(2), 133–135 (1985).
[Crossref]

Rhodes, C.

E. J. Bochove, A. B. Aceves, Y. Braiman, P. Colet, R. Deiterding, A. Jacobo, C. A. Miller, C. Rhodes, and S. A. Shakir, “Model of the self-q-switching instability of passively phased fiber laser arrays,” IEEE J. Quantum Electron. 47(6), 777–785 (2011).
[Crossref]

Rice, R. R.

Richardson, D. J.

Rigrod, W. W.

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

Rothenberg, J.

Rothenberg, J. E.

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

Sabourdy, D.

D. Sabourdy, V. Kermene, A. Desfarges-Berthelemot, M. Vampouille, and A. Barthelemy, “Coherent combining of two Nd:YAG lasers in a Vernier-Michelson-type cavity,” Appl. Phys. B: Lasers Opt. 75(4-5), 503–507 (2002).
[Crossref]

Sahu, J.

Sanchez, A.

Sanchez, A. D.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-synchronous and self-referenced coherent beam combination for large optical arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 480–486 (2007).
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[Crossref]

Shay, T. M.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-synchronous and self-referenced coherent beam combination for large optical arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 480–486 (2007).
[Crossref]

Shirakawa, A.

D. Kouznetsov, J.-F. Bisson, A. Shirakawa, and K.-I. Ueda, “Limits of coherent addition of lasers: Simple estimate,” Opt. Rev. 12(6), 445–447 (2005).
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C. Wan, B. Tiffany, and J. R. Leger, “Analysis of path length sensitivity in coherent beam combining by spatial filtering,” IEEE J. Quantum Electron. 47(6), 770–776 (2011).
[Crossref]

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

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D. Sabourdy, V. Kermene, A. Desfarges-Berthelemot, M. Vampouille, and A. Barthelemy, “Coherent combining of two Nd:YAG lasers in a Vernier-Michelson-type cavity,” Appl. Phys. B: Lasers Opt. 75(4-5), 503–507 (2002).
[Crossref]

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[Crossref]

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C. Wan and J. R. Leger, “Experimental measurements of path length sensitivity in coherent beam combining by spatial filtering,” IEEE J. Quantum Electron. 48(8), 1045–1051 (2012).
[Crossref]

C. Wan, B. Tiffany, and J. R. Leger, “Analysis of path length sensitivity in coherent beam combining by spatial filtering,” IEEE J. Quantum Electron. 47(6), 770–776 (2011).
[Crossref]

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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(2), 248–256 (2009).
[Crossref]

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Wickham, M.

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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(2), 248–256 (2009).
[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(2), 248–256 (2009).
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Appl. Phys. B: Lasers Opt. (1)

D. Sabourdy, V. Kermene, A. Desfarges-Berthelemot, M. Vampouille, and A. Barthelemy, “Coherent combining of two Nd:YAG lasers in a Vernier-Michelson-type cavity,” Appl. Phys. B: Lasers Opt. 75(4-5), 503–507 (2002).
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[Crossref]

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[Crossref]

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E. J. Bochove, A. B. Aceves, Y. Braiman, P. Colet, R. Deiterding, A. Jacobo, C. A. Miller, C. Rhodes, and S. A. Shakir, “Model of the self-q-switching instability of passively phased fiber laser arrays,” IEEE J. Quantum Electron. 47(6), 777–785 (2011).
[Crossref]

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[Crossref]

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W. M. Kunkel and J. R. Leger, “Passive coherent laser beam combining with spatial mode selecting feedback,” IEEE J. Quantum Electron. 55(4), 1–8 (2019).
[Crossref]

C. Wan, B. Tiffany, and J. R. Leger, “Analysis of path length sensitivity in coherent beam combining by spatial filtering,” IEEE J. Quantum Electron. 47(6), 770–776 (2011).
[Crossref]

C. Wan and J. R. Leger, “Experimental measurements of path length sensitivity in coherent beam combining by spatial filtering,” IEEE J. Quantum Electron. 48(8), 1045–1051 (2012).
[Crossref]

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C. J. Corcoran and F. Durville, “Passive phasing in a coherent laser array,” IEEE J. Sel. Top. Quantum Electron. 15(2), 294–300 (2009).
[Crossref]

M. Khajavikhan and J. 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]

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[Crossref]

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-synchronous and self-referenced coherent beam combination for large optical arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 480–486 (2007).
[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(2), 248–256 (2009).
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D. Kouznetsov, J.-F. Bisson, A. Shirakawa, and K.-I. Ueda, “Limits of coherent addition of lasers: Simple estimate,” Opt. Rev. 12(6), 445–447 (2005).
[Crossref]

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W. M. Kunkel and J. R. Leger, “Kramers-kronig self-phasing effect in passive beam combining resonators,” Proc. SPIE 10904, 109041N (2019).
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A. E. Siegman, “Resonant modes of linearly coupled multiple fiber laser structures,” Unpublished pp. 1–25 (2004).

W. M. Kunkel, “Passive coherent beam combining by gain effects in fiber lasers and spatial beam control in gradient refractive index media,” Ph.D. thesis, University of Minnesota (2018).

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Figures (7)

Fig. 1.
Fig. 1. Model of the Dammann grating resonator. $E_{A}$ and $E_{B}$ denote the field amplitudes at the gain media end mirrors. $\Delta \phi$ is the applied phase error between the gain elements obtained by shifting the Dammann grating. The combined beam is shown as the thick blue solid line, and the uncombined beams as red dashed lines. Recall $\theta _{1}=\lambda f_{0}$ and $\theta _{2}=2 \lambda f_{0}$.
Fig. 2.
Fig. 2. Illustration of beam recycling. The propagation angle of the uncombined beam from the upper gain medium (dotted line) is inverted after reflection (dashed line) from the end mirror. Similarly, the uncombined beam from the lower gain medium (not shown) couples back into the upper gain medium. The combined beam (not shown) exits the cavity along the optical axis.
Fig. 3.
Fig. 3. Supermode round trip power loss $L$ versus applied phase error $\Delta \phi$ for the recycling resonator with (a) $\phi _{r}=\pi$ and (b) $\phi _{r}=0$. The end mirror reflectivity is $r=0.048$.
Fig. 4.
Fig. 4. Diagram of experimental setup. Components inside dotted boxes are for measurement purposes. The spatial filtering method used to adjust the resonator feedback is also shown.
Fig. 5.
Fig. 5. Measured and theoretical (a) supermode relative phase $\phi _{Rel}$ and (b) combined beam power for the standard resonator.
Fig. 6.
Fig. 6. Measured and theoretical (a) supermode relative phase $\phi _{Rel}$ and (b) combined beam power for the recycling resonator.
Fig. 7.
Fig. 7. Measured (a) combined beam power and (b) supermode fringe visibility for recycle phase shifts of $\phi _{r} = 1.93$ and $\phi _{r} = \pi$. Data for $\phi _{r} = 0$ is also shown for reference.

Equations (22)

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T A ( f x ) = F { t A ( x ) } = δ ( f x ) n = sinc ( n 2 ) δ ( f x n f 0 )
T ~ A ( f x ) = F { t A ( x x 0 ) } = exp ( i 2 π f x x 0 ) T A ( f x )
= δ ( f x ) n = sinc ( n 2 ) δ ( f x n f 0 ) exp ( i ϕ n )
T ~ A ( f x ) n = 4 4 c n δ ( f x n f 0 )
c n = { 0 , if n even 2 π exp ( i n Δ ϕ 2 ) , n = 1 , 1 2 3 π exp ( i n Δ ϕ 2 ) , n = 3 , 3
Δ ϕ = ϕ 1 ϕ 1 = 4 π f 0 x 0 = 4 π x 0 d
D = [ c 0 c 1 c 2 c 3 c 4 c 1 c 0 c 1 c 2 c 3 c 2 c 1 c 0 c 1 c 2 c 3 c 2 c 1 c 0 c 1 c 4 c 3 c 2 c 1 c 0 ]
G = [ 0 0 0 0 0 0 g 0 0 0 0 0 0 0 0 0 0 0 g 0 0 0 0 0 0 ]
R = r [ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ]
M R T = G D T R D G = 4 g 2 r π 2 [ 0 0 0 0 0 0 m 1 0 1 0 0 0 0 0 0 0 1 0 m 2 0 0 0 0 0 0 ]
λ = 8 g 2 r π 2 cos ( Δ ϕ )
E = [ 0 E A 0 E B 0 ] = [ 0 1 0 exp ( i Δ ϕ ) 0 ]
L = 1 | λ | 2 = 1 64 r 2 π 4 cos 2 ( Δ ϕ )
ϕ R e l = E B E A
G s a t = g 2 = π 2 8 r | cos Δ ϕ |
P 2 = R 1 ( ln G 0 + ln R 1 R 2 ) ( R 1 + R 2 ) ( 1 R 1 R 2 )
P 1 = P 2 R 1 G s a t = G s a t ln ( G 0 G s a t ) ( 1 + R 1 G s a t ) ( G s a t 1 )
E c a v = [ E U 1 0 E C 0 E U 2 ] = D G P 1 E
R = r [ 0 0 0 0 1 0 0 0 0 0 0 0 exp ( i ϕ r ) 0 0 0 0 0 0 0 1 0 0 0 0 ] .
M R T = G D T R D G = 4 g 2 r 3 π 2 [ 0 0 0 0 0 0 m 1 0 19 3 0 0 0 0 0 0 0 19 3 0 m 2 0 0 0 0 0 0 ]
λ 1 , 2 = 4 g 2 r π 2 ( cos ( Δ ϕ ) 3 ± cos 2 ( Δ ϕ ) 9 + 352 81 )
E 1 , 2 = [ 0 19 9 0 i sin ( Δ ϕ ) 3 ± cos 2 ( Δ ϕ ) 9 + 352 81 0 ]