Abstract

We propose to combine multiple laser beams into a single diffraction-limited beam by beam self-focusing (collapse) in a Kerr medium. Beams with total power above critical are first combined in the near field and then propagated in the optical fiber/waveguide with Kerr nonlinearity. Random fluctuations during propagation eventually trigger a strong self-focusing event and produce a diffraction-limited beam carrying the critical power.

© 2014 Optical Society of America

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  1. D. J. Richardson, J. Nilsson, and W. A. Clarkson, J. Opt. Soc. Am. B 27, B63 (2010).
    [CrossRef]
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    [CrossRef]
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  4. http://www.ipgphotonics.com .
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    [CrossRef]
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    [CrossRef]
  7. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007).
    [CrossRef]
  8. S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 14, 1353 (1971).
  9. V. E. Zakharov, Sov. Phys. JETP 35, 908 (1972).
  10. C. Sulem and P. L. Sulem, Nonlinear Schrödinger Equations: Self-Focusing and Wave Collapse (World Scientific, 1999).
  11. P. M. Lushnikov, S. A. Dyachenko, and N. Vladimirova, Phys. Rev. A 88, 013845 (2013).
    [CrossRef]
  12. P. M. Lushnikov and H. A. Rose, Phys. Rev. Lett. 92, 255003 (2004).
    [CrossRef]
  13. P. M. Lushnikov and H. A. Rose, Plasma Phys. Controlled Fusion 48, 1501 (2006).
    [CrossRef]
  14. P. M. Lushnikov and N. Vladimirova, Opt. Lett. 35, 1965 (2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  18. S. K. Turitsyn, Aston University, Birmingham, UK (private communication, 2012).
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    [CrossRef]
  20. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. Barty, Opt. Commun. 16, 13240 (2008).
    [CrossRef]
  21. A. V. Smith and B. T. Do, Appl. Opt. 47, 4812 (2008).
    [CrossRef]
  22. G. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2012).

2013

C. Jauregui, J. Limpert, and A. Tünnermann, Nat. Photonics 7, 861 (2013).
[CrossRef]

P. M. Lushnikov, S. A. Dyachenko, and N. Vladimirova, Phys. Rev. A 88, 013845 (2013).
[CrossRef]

2012

2011

2010

2008

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. Barty, Opt. Commun. 16, 13240 (2008).
[CrossRef]

A. V. Smith and B. T. Do, Appl. Opt. 47, 4812 (2008).
[CrossRef]

2007

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

2006

P. M. Lushnikov and H. A. Rose, Plasma Phys. Controlled Fusion 48, 1501 (2006).
[CrossRef]

2005

T. Y. Fan, IEEE J. Sel. Top. Quantum Electron. 11, 567 (2005).
[CrossRef]

2004

P. M. Lushnikov and H. A. Rose, Phys. Rev. Lett. 92, 255003 (2004).
[CrossRef]

2003

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, Opt. Commun. 219, 427 (2003).
[CrossRef]

2002

1972

V. E. Zakharov, Sov. Phys. JETP 35, 908 (1972).

1971

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 14, 1353 (1971).

Abramov, M.

V. Gapontsev, F. A. Fomin, and M. Abramov, in Proceedings of Advanced Solid-State Photonics (Optical Society of America, 2010), paper AWA1.

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2012).

Augst, S. J.

Barty, C.

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. Barty, Opt. Commun. 16, 13240 (2008).
[CrossRef]

Beach, R. J.

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. Barty, Opt. Commun. 16, 13240 (2008).
[CrossRef]

Bergé, L.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

Boudebs, G.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, Opt. Commun. 219, 427 (2003).
[CrossRef]

Cherukulappurath, S.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, Opt. Commun. 219, 427 (2003).
[CrossRef]

Chin, S. L.

Chung, Y.

Y. Chung and P. M. Lushnikov, Phys. Rev. E 84, 036602 (2011).
[CrossRef]

Clarkson, W. A.

Daigle, J.-F.

Dawson, J. W.

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. Barty, Opt. Commun. 16, 13240 (2008).
[CrossRef]

Do, B. T.

Dyachenko, S. A.

P. M. Lushnikov, S. A. Dyachenko, and N. Vladimirova, Phys. Rev. A 88, 013845 (2013).
[CrossRef]

Fan, T. Y.

Fomin, F. A.

V. Gapontsev, F. A. Fomin, and M. Abramov, in Proceedings of Advanced Solid-State Photonics (Optical Society of America, 2010), paper AWA1.

Gabitov, I. R.

Gapontsev, V.

V. Gapontsev, F. A. Fomin, and M. Abramov, in Proceedings of Advanced Solid-State Photonics (Optical Society of America, 2010), paper AWA1.

Goodno, G. D.

Heebner, J. E.

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. Barty, Opt. Commun. 16, 13240 (2008).
[CrossRef]

Hosseini, S.

Jauregui, C.

C. Jauregui, J. Limpert, and A. Tünnermann, Nat. Photonics 7, 861 (2013).
[CrossRef]

Kasparian, J.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

Kosareva, O.

Leblond, H.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, Opt. Commun. 219, 427 (2003).
[CrossRef]

Limpert, J.

C. Jauregui, J. Limpert, and A. Tünnermann, Nat. Photonics 7, 861 (2013).
[CrossRef]

Lushnikov, P. M.

P. M. Lushnikov, S. A. Dyachenko, and N. Vladimirova, Phys. Rev. A 88, 013845 (2013).
[CrossRef]

Y. Chung and P. M. Lushnikov, Phys. Rev. E 84, 036602 (2011).
[CrossRef]

P. M. Lushnikov and N. Vladimirova, Opt. Lett. 35, 1965 (2010).
[CrossRef]

P. M. Lushnikov and H. A. Rose, Plasma Phys. Controlled Fusion 48, 1501 (2006).
[CrossRef]

P. M. Lushnikov and H. A. Rose, Phys. Rev. Lett. 92, 255003 (2004).
[CrossRef]

I. R. Gabitov and P. M. Lushnikov, Opt. Lett. 27, 113 (2002).
[CrossRef]

Makarov, V.

Messerly, M. J.

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. Barty, Opt. Commun. 16, 13240 (2008).
[CrossRef]

Nilsson, J.

Nuter, R.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

Panov, N.

Pax, P. H.

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. Barty, Opt. Commun. 16, 13240 (2008).
[CrossRef]

Petrishchev, V. A.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 14, 1353 (1971).

Redmond, S. M.

Richardson, D. J.

Ripin, D. J.

Rose, H. A.

P. M. Lushnikov and H. A. Rose, Plasma Phys. Controlled Fusion 48, 1501 (2006).
[CrossRef]

P. M. Lushnikov and H. A. Rose, Phys. Rev. Lett. 92, 255003 (2004).
[CrossRef]

Rothenberg, J. E.

Roy, G.

Sanchez, F.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, Opt. Commun. 219, 427 (2003).
[CrossRef]

Shverdin, M. Y.

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. Barty, Opt. Commun. 16, 13240 (2008).
[CrossRef]

Siders, C. W.

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. Barty, Opt. Commun. 16, 13240 (2008).
[CrossRef]

Skupin, S.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

Smektala, F.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, Opt. Commun. 219, 427 (2003).
[CrossRef]

Smith, A. V.

Sridharan, A. K.

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. Barty, Opt. Commun. 16, 13240 (2008).
[CrossRef]

Stappaerts, E. A.

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. Barty, Opt. Commun. 16, 13240 (2008).
[CrossRef]

Sulem, C.

C. Sulem and P. L. Sulem, Nonlinear Schrödinger Equations: Self-Focusing and Wave Collapse (World Scientific, 1999).

Sulem, P. L.

C. Sulem and P. L. Sulem, Nonlinear Schrödinger Equations: Self-Focusing and Wave Collapse (World Scientific, 1999).

Talanov, V. I.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 14, 1353 (1971).

Thielen, P. A.

Troles, J.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, Opt. Commun. 219, 427 (2003).
[CrossRef]

Tünnermann, A.

C. Jauregui, J. Limpert, and A. Tünnermann, Nat. Photonics 7, 861 (2013).
[CrossRef]

Turitsyn, S. K.

S. K. Turitsyn, Aston University, Birmingham, UK (private communication, 2012).

Vladimirova, N.

P. M. Lushnikov, S. A. Dyachenko, and N. Vladimirova, Phys. Rev. A 88, 013845 (2013).
[CrossRef]

P. M. Lushnikov and N. Vladimirova, Opt. Lett. 35, 1965 (2010).
[CrossRef]

Vlasov, S. N.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 14, 1353 (1971).

Wang, T.

Wolf, J.-P.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

Yu, C. X.

Yuan, S.

Zakharov, V. E.

V. E. Zakharov, Sov. Phys. JETP 35, 908 (1972).

Appl. Opt.

IEEE J. Sel. Top. Quantum Electron.

T. Y. Fan, IEEE J. Sel. Top. Quantum Electron. 11, 567 (2005).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved., Radiofiz.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 14, 1353 (1971).

J. Opt. Soc. Am. B

Nat. Photonics

C. Jauregui, J. Limpert, and A. Tünnermann, Nat. Photonics 7, 861 (2013).
[CrossRef]

Opt. Commun.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, Opt. Commun. 219, 427 (2003).
[CrossRef]

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. Barty, Opt. Commun. 16, 13240 (2008).
[CrossRef]

Opt. Lett.

Phys. Rev. A

P. M. Lushnikov, S. A. Dyachenko, and N. Vladimirova, Phys. Rev. A 88, 013845 (2013).
[CrossRef]

Phys. Rev. E

Y. Chung and P. M. Lushnikov, Phys. Rev. E 84, 036602 (2011).
[CrossRef]

Phys. Rev. Lett.

P. M. Lushnikov and H. A. Rose, Phys. Rev. Lett. 92, 255003 (2004).
[CrossRef]

Plasma Phys. Controlled Fusion

P. M. Lushnikov and H. A. Rose, Plasma Phys. Controlled Fusion 48, 1501 (2006).
[CrossRef]

Rep. Prog. Phys.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007).
[CrossRef]

Sov. Phys. JETP

V. E. Zakharov, Sov. Phys. JETP 35, 908 (1972).

Other

C. Sulem and P. L. Sulem, Nonlinear Schrödinger Equations: Self-Focusing and Wave Collapse (World Scientific, 1999).

V. Gapontsev, F. A. Fomin, and M. Abramov, in Proceedings of Advanced Solid-State Photonics (Optical Society of America, 2010), paper AWA1.

http://www.ipgphotonics.com .

S. K. Turitsyn, Aston University, Birmingham, UK (private communication, 2012).

G. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2012).

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of nonlinear beam combining. An array of beams with noncorrelated phases enters a nonlinear optical fiber at z=0. Inside the fiber, the laser field is randomized due to nonlinear interactions (as shown for a typical cross section at z=z1). A large fluctuation of that random field triggers a strong self-focusing event producing a nearly diffraction-limited hot spot at z=z2 (shown by the long arrow), which carries the critical power Pc.

Fig. 2.
Fig. 2.

Simulation of nonlinear beam combining in NLSE in Eq. (2). The snapshots of the distributions (vertical axis) in (x,y) of the amplitude |ψ| (top row) and the phase arg(ψ) (bottom row) for different values of z. Left column: array of Gaussian beams with random phases are used as initial conditions (z=0). Middle column: Kerr nonlinearity results in randomization of phases and amplitudes after the propagation distance zznl, as shown for z=z1=10 (znl=5.6 in that case). Right column: random fluctuations of amplitudes trigger the strong self-focusing collapse event (z=z2=15).

Fig. 3.
Fig. 3.

max(x,y)|ψ| in the waveguide’s cross section versus z. Dashed line shows the result of the same simulation of the NLSE in Eq. (2) as in Fig. 2. The solid line shows the simulation of the regularized NLSE in Eq. (5) with a1=0.001 and the same initial condition as for the dashed curve. Thick dots correspond to z=z1 and z=z2 of Fig. 2.

Fig. 4.
Fig. 4.

Probability density functions (PDFs) of the catastrophic self-focusing distance zsf collected over 360 simulations with random initial phases and the total power 10Pc. (a) N=10×10 combined beams with r0=1.13. (b) N=8×8 combined beams with r0=1.41.

Fig. 5.
Fig. 5.

Simulation similar to Fig. 2 with 91 beams of the power 0.1Pc and r0=1.13, but with the added circular barrier at r=0.45L to represent the total internal reflection of the circular waveguide. z1=1 and z2=9.7.

Equations (5)

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

izψ+12k2ψ+kn2n0|ψ|2ψ=0,
izψ+2ψ+|ψ|2ψ=0,
Pc=Ncλ028π2n2n011.70λ028π2n2n0.
|ψ(x,y,z)|L(z)1R(ρ),ρr/L(z),|r|r,
izψ+2ψ+|ψ|2ψa1|ψ|4ψ=0,

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