Abstract

A relatively simple technique for coupling lasers in an array is presented. It is based on the insertion of an intracavity optical element in the far-field plane of a degenerate cavity laser that is used to form an array of lasers. We show that it is possible to control the selection of the lasers to couple regardless of the array geometry. An intracavity spherical lens in the far-field plane is numerically and experimentally investigated and the results compared with those from the more complicated Talbot diffraction for coupling lasers. With an intracavity cylindrical lens in a two dimensional square array geometry, it is possible to obtain controlled one-dimensional coupling, and with an intracavity binary phase element it is possible to obtain versatile couplings.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent ising machine,” Nat. Photonics 8(12), 937–942 (2014).
    [Crossref]
  2. M. Nixon, E. Ronen, A. A. Friesem, and N. Davidson, “Observing geometric frustration with thousands of coupled lasers,” Phys. Rev. Lett. 110(18), 184102 (2013).
    [Crossref]
  3. V. Pal, C. Tradonsky, R. Chriki, A. A. Friesem, and N. Davidson, “Observing dissipative topological defects with coupled lasers,” Phys. Rev. Lett. 119(1), 013902 (2017).
    [Crossref]
  4. S. Tamate, Y. Yamamoto, A. Marandi, P. McMahon, and S. Utsunomiya, “Simulating the classical XY model with a laser network,” arXiv e-prints Physics Optics, 1608.00358 (2016).
  5. C. Tradonsky, O. Raz, V. Pal, R. Chriki, A. A. Friesem, and N. Davidson, “Rapid Phase Retrieval by Lasing,”, arXiv e-prints arXiv:1805.10967 (2018).
  6. A. Shirakawa, T. Saitou, T. Sekiguchi, and K. ichi Ueda, “Coherent addition of fiber lasers by use of a fiber coupler,” Opt. Express 10(21), 1167–1172 (2002).
    [Crossref]
  7. D. Botez, M. Jansen, L. J. Mawst, G. Peterson, and T. J. Roth, “Watt-range, coherent, uniphase powers from phase-locked arrays of antiguided diode lasers,” Appl. Phys. Lett. 58(19), 2070–2072 (1991).
    [Crossref]
  8. M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, “Passive phase locking of 25 fiber lasers,” Opt. Lett. 35(9), 1434–1436 (2010).
    [Crossref]
  9. F. X. D’Amato, E. T. Siebert, and C. Roychoudhuri, “Coherent operation of an array of diode lasers using a spatial filter in a talbot cavity,” Appl. Phys. Lett. 55(9), 816–818 (1989).
    [Crossref]
  10. C. Tradonsky, V. Pal, R. Chriki, N. Davidson, and A. A. Friesem, “Talbot diffraction and fourier filtering for phase locking an array of lasers,” Appl. Opt. 56(1), A126–A132 (2017).
    [Crossref]
  11. Y. Zhou, L. Liu, C. Etson, Y. Abranyos, A. Padilla, and Y. C. Chen, “Phase locking of a two-dimensional laser array by controlling the far-field pattern,” Appl. Phys. Lett. 84(16), 3025–3027 (2004).
    [Crossref]
  12. A. F. Glova, “Phase locking of optically coupled lasers,” Quantum Electron. 33(4), 283–306 (2003).
    [Crossref]
  13. 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]
  14. J. A. Arnaud, “Degenerate optical cavities,” Appl. Opt. 8(1), 189–196 (1969).
    [Crossref]
  15. M. Nixon, B. Redding, A. A. Friesem, H. Cao, and N. Davidson, “Efficient method for controlling the spatial coherence of a laser,” Opt. Lett. 38(19), 3858–3861 (2013).
    [Crossref]
  16. J. W. Goodman, Introduction to Fourier Optics, 3rd edition (Roberts and Company, 2005).
  17. H. Talbot, “Lxxvi. facts relating to optical science. no. iv,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 9(56), 401–407 (1836).
    [Crossref]
  18. L. Rayleigh, “Xxv. on copying diffraction-gratings, and on some phenomena connected therewith,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 11(67), 196–205 (1881).
    [Crossref]
  19. D. Mehuys, W. Streifer, R. G. Waarts, and D. F. Welch, “Modal analysis of linear talbot-cavity semiconductor lasers,” Opt. Lett. 16(11), 823–825 (1991).
    [Crossref]
  20. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
    [Crossref]
  21. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  22. 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).
    [Crossref]

2017 (2)

V. Pal, C. Tradonsky, R. Chriki, A. A. Friesem, and N. Davidson, “Observing dissipative topological defects with coupled lasers,” Phys. Rev. Lett. 119(1), 013902 (2017).
[Crossref]

C. Tradonsky, V. Pal, R. Chriki, N. Davidson, and A. A. Friesem, “Talbot diffraction and fourier filtering for phase locking an array of lasers,” Appl. Opt. 56(1), A126–A132 (2017).
[Crossref]

2014 (1)

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

2013 (2)

M. Nixon, E. Ronen, A. A. Friesem, and N. Davidson, “Observing geometric frustration with thousands of coupled lasers,” Phys. Rev. Lett. 110(18), 184102 (2013).
[Crossref]

M. Nixon, B. Redding, A. A. Friesem, H. Cao, and N. Davidson, “Efficient method for controlling the spatial coherence of a laser,” Opt. Lett. 38(19), 3858–3861 (2013).
[Crossref]

2010 (1)

2008 (1)

2004 (1)

Y. Zhou, L. Liu, C. Etson, Y. Abranyos, A. Padilla, and Y. C. Chen, “Phase locking of a two-dimensional laser array by controlling the far-field pattern,” Appl. Phys. Lett. 84(16), 3025–3027 (2004).
[Crossref]

2003 (1)

A. F. Glova, “Phase locking of optically coupled lasers,” Quantum Electron. 33(4), 283–306 (2003).
[Crossref]

2002 (1)

1991 (2)

D. Botez, M. Jansen, L. J. Mawst, G. Peterson, and T. J. Roth, “Watt-range, coherent, uniphase powers from phase-locked arrays of antiguided diode lasers,” Appl. Phys. Lett. 58(19), 2070–2072 (1991).
[Crossref]

D. Mehuys, W. Streifer, R. G. Waarts, and D. F. Welch, “Modal analysis of linear talbot-cavity semiconductor lasers,” Opt. Lett. 16(11), 823–825 (1991).
[Crossref]

1989 (1)

F. X. D’Amato, E. T. Siebert, and C. Roychoudhuri, “Coherent operation of an array of diode lasers using a spatial filter in a talbot cavity,” Appl. Phys. Lett. 55(9), 816–818 (1989).
[Crossref]

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]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1969 (1)

1961 (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

1881 (1)

L. Rayleigh, “Xxv. on copying diffraction-gratings, and on some phenomena connected therewith,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 11(67), 196–205 (1881).
[Crossref]

1836 (1)

H. Talbot, “Lxxvi. facts relating to optical science. no. iv,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 9(56), 401–407 (1836).
[Crossref]

Abranyos, Y.

Y. Zhou, L. Liu, C. Etson, Y. Abranyos, A. Padilla, and Y. C. Chen, “Phase locking of a two-dimensional laser array by controlling the far-field pattern,” Appl. Phys. Lett. 84(16), 3025–3027 (2004).
[Crossref]

Arnaud, J. A.

Botez, D.

D. Botez, M. Jansen, L. J. Mawst, G. Peterson, and T. J. Roth, “Watt-range, coherent, uniphase powers from phase-locked arrays of antiguided diode lasers,” Appl. Phys. Lett. 58(19), 2070–2072 (1991).
[Crossref]

Byer, R. L.

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Cao, H.

Chen, Y. C.

Y. Zhou, L. Liu, C. Etson, Y. Abranyos, A. Padilla, and Y. C. Chen, “Phase locking of a two-dimensional laser array by controlling the far-field pattern,” Appl. Phys. Lett. 84(16), 3025–3027 (2004).
[Crossref]

Cheung, E. C.

Chriki, R.

C. Tradonsky, V. Pal, R. Chriki, N. Davidson, and A. A. Friesem, “Talbot diffraction and fourier filtering for phase locking an array of lasers,” Appl. Opt. 56(1), A126–A132 (2017).
[Crossref]

V. Pal, C. Tradonsky, R. Chriki, A. A. Friesem, and N. Davidson, “Observing dissipative topological defects with coupled lasers,” Phys. Rev. Lett. 119(1), 013902 (2017).
[Crossref]

C. Tradonsky, O. Raz, V. Pal, R. Chriki, A. A. Friesem, and N. Davidson, “Rapid Phase Retrieval by Lasing,”, arXiv e-prints arXiv:1805.10967 (2018).

D’Amato, F. X.

F. X. D’Amato, E. T. Siebert, and C. Roychoudhuri, “Coherent operation of an array of diode lasers using a spatial filter in a talbot cavity,” Appl. Phys. Lett. 55(9), 816–818 (1989).
[Crossref]

Davidson, N.

C. Tradonsky, V. Pal, R. Chriki, N. Davidson, and A. A. Friesem, “Talbot diffraction and fourier filtering for phase locking an array of lasers,” Appl. Opt. 56(1), A126–A132 (2017).
[Crossref]

V. Pal, C. Tradonsky, R. Chriki, A. A. Friesem, and N. Davidson, “Observing dissipative topological defects with coupled lasers,” Phys. Rev. Lett. 119(1), 013902 (2017).
[Crossref]

M. Nixon, E. Ronen, A. A. Friesem, and N. Davidson, “Observing geometric frustration with thousands of coupled lasers,” Phys. Rev. Lett. 110(18), 184102 (2013).
[Crossref]

M. Nixon, B. Redding, A. A. Friesem, H. Cao, and N. Davidson, “Efficient method for controlling the spatial coherence of a laser,” Opt. Lett. 38(19), 3858–3861 (2013).
[Crossref]

M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, “Passive phase locking of 25 fiber lasers,” Opt. Lett. 35(9), 1434–1436 (2010).
[Crossref]

C. Tradonsky, O. Raz, V. Pal, R. Chriki, A. A. Friesem, and N. Davidson, “Rapid Phase Retrieval by Lasing,”, arXiv e-prints arXiv:1805.10967 (2018).

Etson, C.

Y. Zhou, L. Liu, C. Etson, Y. Abranyos, A. Padilla, and Y. C. Chen, “Phase locking of a two-dimensional laser array by controlling the far-field pattern,” Appl. Phys. Lett. 84(16), 3025–3027 (2004).
[Crossref]

Fox, A. G.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Fridman, M.

Friesem, A. A.

V. Pal, C. Tradonsky, R. Chriki, A. A. Friesem, and N. Davidson, “Observing dissipative topological defects with coupled lasers,” Phys. Rev. Lett. 119(1), 013902 (2017).
[Crossref]

C. Tradonsky, V. Pal, R. Chriki, N. Davidson, and A. A. Friesem, “Talbot diffraction and fourier filtering for phase locking an array of lasers,” Appl. Opt. 56(1), A126–A132 (2017).
[Crossref]

M. Nixon, B. Redding, A. A. Friesem, H. Cao, and N. Davidson, “Efficient method for controlling the spatial coherence of a laser,” Opt. Lett. 38(19), 3858–3861 (2013).
[Crossref]

M. Nixon, E. Ronen, A. A. Friesem, and N. Davidson, “Observing geometric frustration with thousands of coupled lasers,” Phys. Rev. Lett. 110(18), 184102 (2013).
[Crossref]

M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, “Passive phase locking of 25 fiber lasers,” Opt. Lett. 35(9), 1434–1436 (2010).
[Crossref]

C. Tradonsky, O. Raz, V. Pal, R. Chriki, A. A. Friesem, and N. Davidson, “Rapid Phase Retrieval by Lasing,”, arXiv e-prints arXiv:1805.10967 (2018).

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Glova, A. F.

A. F. Glova, “Phase locking of optically coupled lasers,” Quantum Electron. 33(4), 283–306 (2003).
[Crossref]

Goodman, J. W.

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

Goodno, G. D.

Ho, J. G.

ichi Ueda, K.

Jansen, M.

D. Botez, M. Jansen, L. J. Mawst, G. Peterson, and T. J. Roth, “Watt-range, coherent, uniphase powers from phase-locked arrays of antiguided diode lasers,” Appl. Phys. Lett. 58(19), 2070–2072 (1991).
[Crossref]

Li, T.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Liu, L.

Y. Zhou, L. Liu, C. Etson, Y. Abranyos, A. Padilla, and Y. C. Chen, “Phase locking of a two-dimensional laser array by controlling the far-field pattern,” Appl. Phys. Lett. 84(16), 3025–3027 (2004).
[Crossref]

Marandi, A.

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

S. Tamate, Y. Yamamoto, A. Marandi, P. McMahon, and S. Utsunomiya, “Simulating the classical XY model with a laser network,” arXiv e-prints Physics Optics, 1608.00358 (2016).

Mawst, L. J.

D. Botez, M. Jansen, L. J. Mawst, G. Peterson, and T. J. Roth, “Watt-range, coherent, uniphase powers from phase-locked arrays of antiguided diode lasers,” Appl. Phys. Lett. 58(19), 2070–2072 (1991).
[Crossref]

McMahon, P.

S. Tamate, Y. Yamamoto, A. Marandi, P. McMahon, and S. Utsunomiya, “Simulating the classical XY model with a laser network,” arXiv e-prints Physics Optics, 1608.00358 (2016).

Mehuys, D.

Nixon, M.

Padilla, A.

Y. Zhou, L. Liu, C. Etson, Y. Abranyos, A. Padilla, and Y. C. Chen, “Phase locking of a two-dimensional laser array by controlling the far-field pattern,” Appl. Phys. Lett. 84(16), 3025–3027 (2004).
[Crossref]

Pal, V.

C. Tradonsky, V. Pal, R. Chriki, N. Davidson, and A. A. Friesem, “Talbot diffraction and fourier filtering for phase locking an array of lasers,” Appl. Opt. 56(1), A126–A132 (2017).
[Crossref]

V. Pal, C. Tradonsky, R. Chriki, A. A. Friesem, and N. Davidson, “Observing dissipative topological defects with coupled lasers,” Phys. Rev. Lett. 119(1), 013902 (2017).
[Crossref]

C. Tradonsky, O. Raz, V. Pal, R. Chriki, A. A. Friesem, and N. Davidson, “Rapid Phase Retrieval by Lasing,”, arXiv e-prints arXiv:1805.10967 (2018).

Peterson, G.

D. Botez, M. Jansen, L. J. Mawst, G. Peterson, and T. J. Roth, “Watt-range, coherent, uniphase powers from phase-locked arrays of antiguided diode lasers,” Appl. Phys. Lett. 58(19), 2070–2072 (1991).
[Crossref]

Rayleigh, L.

L. Rayleigh, “Xxv. on copying diffraction-gratings, and on some phenomena connected therewith,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 11(67), 196–205 (1881).
[Crossref]

Raz, O.

C. Tradonsky, O. Raz, V. Pal, R. Chriki, A. A. Friesem, and N. Davidson, “Rapid Phase Retrieval by Lasing,”, arXiv e-prints arXiv:1805.10967 (2018).

Redding, B.

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]

Rice, R. R.

Ronen, E.

M. Nixon, E. Ronen, A. A. Friesem, and N. Davidson, “Observing geometric frustration with thousands of coupled lasers,” Phys. Rev. Lett. 110(18), 184102 (2013).
[Crossref]

Roth, T. J.

D. Botez, M. Jansen, L. J. Mawst, G. Peterson, and T. J. Roth, “Watt-range, coherent, uniphase powers from phase-locked arrays of antiguided diode lasers,” Appl. Phys. Lett. 58(19), 2070–2072 (1991).
[Crossref]

Rothenberg, J.

Roychoudhuri, C.

F. X. D’Amato, E. T. Siebert, and C. Roychoudhuri, “Coherent operation of an array of diode lasers using a spatial filter in a talbot cavity,” Appl. Phys. Lett. 55(9), 816–818 (1989).
[Crossref]

Saitou, T.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schloss, R. P.

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]

Sekiguchi, T.

Shirakawa, A.

Siebert, E. T.

F. X. D’Amato, E. T. Siebert, and C. Roychoudhuri, “Coherent operation of an array of diode lasers using a spatial filter in a talbot cavity,” Appl. Phys. Lett. 55(9), 816–818 (1989).
[Crossref]

Streifer, W.

Takata, K.

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Talbot, H.

H. Talbot, “Lxxvi. facts relating to optical science. no. iv,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 9(56), 401–407 (1836).
[Crossref]

Tamate, S.

S. Tamate, Y. Yamamoto, A. Marandi, P. McMahon, and S. Utsunomiya, “Simulating the classical XY model with a laser network,” arXiv e-prints Physics Optics, 1608.00358 (2016).

Thielen, P.

Tradonsky, C.

V. Pal, C. Tradonsky, R. Chriki, A. A. Friesem, and N. Davidson, “Observing dissipative topological defects with coupled lasers,” Phys. Rev. Lett. 119(1), 013902 (2017).
[Crossref]

C. Tradonsky, V. Pal, R. Chriki, N. Davidson, and A. A. Friesem, “Talbot diffraction and fourier filtering for phase locking an array of lasers,” Appl. Opt. 56(1), A126–A132 (2017).
[Crossref]

C. Tradonsky, O. Raz, V. Pal, R. Chriki, A. A. Friesem, and N. Davidson, “Rapid Phase Retrieval by Lasing,”, arXiv e-prints arXiv:1805.10967 (2018).

Utsunomiya, S.

S. Tamate, Y. Yamamoto, A. Marandi, P. McMahon, and S. Utsunomiya, “Simulating the classical XY model with a laser network,” arXiv e-prints Physics Optics, 1608.00358 (2016).

Van Ruyven, L. J.

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]

Waarts, R. G.

Wang, Z.

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Weber, M.

Welch, D. F.

Wickham, M.

Yamamoto, Y.

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

S. Tamate, Y. Yamamoto, A. Marandi, P. McMahon, and S. Utsunomiya, “Simulating the classical XY model with a laser network,” arXiv e-prints Physics Optics, 1608.00358 (2016).

Zhou, Y.

Y. Zhou, L. Liu, C. Etson, Y. Abranyos, A. Padilla, and Y. C. Chen, “Phase locking of a two-dimensional laser array by controlling the far-field pattern,” Appl. Phys. Lett. 84(16), 3025–3027 (2004).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (4)

Y. Zhou, L. Liu, C. Etson, Y. Abranyos, A. Padilla, and Y. C. Chen, “Phase locking of a two-dimensional laser array by controlling the far-field pattern,” Appl. Phys. Lett. 84(16), 3025–3027 (2004).
[Crossref]

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]

D. Botez, M. Jansen, L. J. Mawst, G. Peterson, and T. J. Roth, “Watt-range, coherent, uniphase powers from phase-locked arrays of antiguided diode lasers,” Appl. Phys. Lett. 58(19), 2070–2072 (1991).
[Crossref]

F. X. D’Amato, E. T. Siebert, and C. Roychoudhuri, “Coherent operation of an array of diode lasers using a spatial filter in a talbot cavity,” Appl. Phys. Lett. 55(9), 816–818 (1989).
[Crossref]

Bell Syst. Tech. J. (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Nat. Photonics (1)

A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent ising machine,” Nat. Photonics 8(12), 937–942 (2014).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Phys. Rev. Lett. (2)

M. Nixon, E. Ronen, A. A. Friesem, and N. Davidson, “Observing geometric frustration with thousands of coupled lasers,” Phys. Rev. Lett. 110(18), 184102 (2013).
[Crossref]

V. Pal, C. Tradonsky, R. Chriki, A. A. Friesem, and N. Davidson, “Observing dissipative topological defects with coupled lasers,” Phys. Rev. Lett. 119(1), 013902 (2017).
[Crossref]

Quantum Electron. (1)

A. F. Glova, “Phase locking of optically coupled lasers,” Quantum Electron. 33(4), 283–306 (2003).
[Crossref]

The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science (2)

H. Talbot, “Lxxvi. facts relating to optical science. no. iv,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 9(56), 401–407 (1836).
[Crossref]

L. Rayleigh, “Xxv. on copying diffraction-gratings, and on some phenomena connected therewith,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 11(67), 196–205 (1881).
[Crossref]

Other (3)

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

S. Tamate, Y. Yamamoto, A. Marandi, P. McMahon, and S. Utsunomiya, “Simulating the classical XY model with a laser network,” arXiv e-prints Physics Optics, 1608.00358 (2016).

C. Tradonsky, O. Raz, V. Pal, R. Chriki, A. A. Friesem, and N. Davidson, “Rapid Phase Retrieval by Lasing,”, arXiv e-prints arXiv:1805.10967 (2018).

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Figures (5)

Fig. 1.
Fig. 1. Degenerate cavity laser (DCL) arrangements. (a) Basic arrangement for forming an array of lasers. (b) Coupling of the lasers by means of fractional Talbot diffraction. (c) Coupling of the lasers by means of a lens with focal distance $f'$ in the far-field plane (Fourier plane at focal distance $f$ from Lens $1$ and Lens $2$). The lasers in (a) are uncoupled, while the lasers in (b) and (c) are coupled. O.C: Output coupler.
Fig. 2.
Fig. 2. Experimental and simulated near-field and far-field intensity distributions in the three different DCL arrangements shown in Fig. 1. No significant differences are observed in the near-field intensity distributions of the three arrangements, whereas differences are observed in the far-field intensity distributions. The far-field intensity distributions in the top row (a) indicate that there is no coupling between individual lasers, whereas in the lower rows (b) and (c) there are couplings, which are essentially identical.
Fig. 3.
Fig. 3. Experimental and simulated near-field and far-field intensity distributions for the DCL arrangement shown in Fig. 1 (c), with an intracavity cylindrical lens in the far-field plane, at different angular orientation. Two different lenses were used, one with focal length of $f'=1m$ to couple lasers vertically and horizontally and the other with $f'=0.5m$ to couple lasers diagonally. As evident, the coupling is along one coordinate. By rotating the cylindrical lens, it is possible to select which lasers are coupled.
Fig. 4.
Fig. 4. Representative experimental far-field intensity distributions for the DCL arrangement shown in Fig. 1 (c), with different intracavity binary phase elements (BPEs) centered along the axis of the DCL.
Fig. 5.
Fig. 5. Representative experimental far-field intensity distributions for the DCL arrangement shown in Fig. 1 (c), obtained with an eight sectors BPE at different angular orientations and/or different displacements from axis of DCL.

Equations (4)

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f = f 2 Z T / 4 ,
E b ( x , y , z F ) e i k 2 Z T / 4 ( ( x x ) 2 + ( y y ) 2 ) E ( x , y , 0 ) d x d y ,
E c ( x , y , z F ) F ( ν [ λ f ] ( e i k 2 f ( x 2 + y 2 ) F ( ν [ λ f ] E ( x , y , 0 ) ) ) ) .
E c ( x , y , z F ) e i k f 2 f 2 ( ( x x ) 2 + ( y y ) 2 ) E ( x , y , 0 ) d x d y ,

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