Abstract

Traditionally, interactions between laser beams or filaments were considered to be deterministic. We show, however, that in most physical settings, these interactions ultimately become stochastic. Specifically, we show that in the nonlinear propagation of laser beams, the shot-to-shot variation of the nonlinear phase shift increases with distance, and ultimately becomes uniformly distributed in [0, 2π]. Therefore, if two beams travel a sufficiently long distance before interacting, it is not possible to predict whether they would intersect in- or out-of-phase. Hence, if the underlying propagation model is non-integrable, deterministic predictions and control of the outcome of the interaction become impossible. Because the relative phase between the two beams becomes uniformly distributed in [0, 2π], however, the statistics of these stochastic interactions are universal and fully predictable. These statistics can be efficiently computed using a novel universal model for stochastic interactions, even when the noise distribution is unknown.

© 2017 Optical Society of America

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References

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2016 (1)

2015 (2)

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, É. Lallier, C. Larat, M. Antier-Murgey, J. Chanteloup, A. Brignon, and C. Simon-Boisson, “Xcan—a coherent amplification network of femtosecond fiber chirped-pulse amplifiers,” Europ. Phys. J. Special Topics 224, 2609–2613 (2015).
[Crossref]

M. Wimmer, A. Regensburger, M.A. Miri, C. Bersch, D.N. Christdoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Comm. 6, 7782 (2015)
[Crossref]

2014 (2)

J. H. Nguyen, P. Dyke, D. Luo, B. Malomed, and R. Hulet, “Collisions of matter-wave solitons,” Nat. Phys. 10, 918–922 (2014).
[Crossref]

G. Mourou, T. Tajima, M. Quinn, B. Brocklesby, and J. Limpert, “Are fiber-based lasers the future of accelerators?” Nucl. Instrum. Methods Phys. Res. Sect. A 740, 17–20 (2014).
[Crossref]

2013 (3)

G. Mourou, B. Brocklesby, T. Tajima, and J. Limpert, “The future is fibre accelerators,” Nat. Photon. 7, 258–261 (2013).
[Crossref]

A. Goy and D. Psaltis, “Imaging in focusing Kerr media using reverse propagation [invited],” Phot. Res. 1, 96–101 (2013).
[Crossref]

A. O’Hagan, “Polynomial chaos: A tutorial and critique from a statistician’s perspective,” SIAM/ASA J. Uncertainty Quantification 20, 1–20 (2013).

2012 (1)

B. Shim, S. Schrauth, A. Gaeta, M. Klein, and G. Fibich, “Loss of phase of collapsing beams,” Phys. Rev. Lett. 108, 043902 (2012).
[Crossref] [PubMed]

2011 (3)

K.G. Makris, R. El-Ganainy, D.N. Christodoulides, and Z.H. Musslimani, “PT-Symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011)
[Crossref]

F.K. Abdullaev, Y.V. Kartashov, V.V. Konotop, and D.A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011)
[Crossref]

G. Fibich and M. Klein, “Continuations of the nonlinear Schrödinger equation beyond the singularity,” Nonlinearity 24, 519–552 (2011).
[Crossref]

2010 (1)

2009 (1)

C. Barsi, W. Wan, and J. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photon. 3, 211–215 (2009).
[Crossref]

2007 (2)

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Physics reports 441, 47–189 (2007).
[Crossref]

A. Ishaaya, T. Grow, S. Ghosh, L. Vuong, and A. Gaeta, “Self-focusing dynamics of coupled optical beams,” Phys. Rev. A 75, 023813 (2007).
[Crossref]

2005 (3)

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

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite Range of nonlocality: First observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95213904 (2005)
[Crossref] [PubMed]

D. Day and L. Romero, “Roots of polynomials expressed in terms of orthogonal polynomials,” SIAM J. Numer. Anal. 43, 1969–1987 (2005).
[Crossref]

2004 (2)

Q. Quo, B. Luo, F. Yi, S. Chi, and Y Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69, 016602 (2004)
[Crossref]

J. Meier, G. Stegeman, Y. Silberberg, R. Morandotti, and J. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. 93, 093903 (2004).
[Crossref] [PubMed]

2001 (2)

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Breakup and fusion of self-guided femtosecond light pulses in air,” Phys. Rev. Lett. 86, 5470–5473 (2001).
[Crossref] [PubMed]

Z. Musslimani, M. Soljačić, M. Segev, and D. Christodoulides, “Delayed-action interaction and spin-orbit coupling between solitons,” Phys. Rev. Lett. 86, 799 (2001).
[Crossref] [PubMed]

1999 (1)

G. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999).
[Crossref] [PubMed]

1997 (2)

1996 (2)

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76, 2698 (1996).
[Crossref] [PubMed]

W. Kruer, S. Wilks, B. Afeyan, and R. Kirkwood, “Energy transfer between crossing laser beams,” Phys. Plasmas 3, 382–385 (1996).
[Crossref]

1993 (1)

1992 (1)

F. Merle, “On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass,” Comm. Pure Appl. Math. 45, 203–254 (1992).
[Crossref]

1989 (1)

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
[Crossref]

1987 (1)

1985 (1)

M.I. Weinstein, “Modulational stability of ground states of nonlinear Schrödinger equations,” SIAM J. Math. Anal. 16, 472–491 (1985)
[Crossref]

1983 (1)

1982 (2)

C. Canuto and A. Quarteroni, “Approximation results for orthogonal polynomials in Sobolev spaces,” Math. Comp. 38, 67–86 (1982).
[Crossref]

T. Cazenave and P.L. Lions, “Orbital stability of standing waves for some nonlinear Schrödinger equations,” Comm. Math. Phys. 85, 549–561 (1982)
[Crossref]

1980 (1)

C. Su and R. M. Mirie, “On head-on collisions between two solitary waves,” J. Fluid Mech. 98, 509–525 (1980).
[Crossref]

1973 (2)

A. C. Scott, F. Chu, and D. W. McLaughlin, “The soliton: A new concept in applied science,” Proc. IEEE 61, 1443–1483 (1973).
[Crossref]

V. Zakharov and A. Shabat, “Interaction between solitons in a stable medium,” Sov. Phys. JETP 37, 823–828 (1973).

1965 (1)

N. Zabusky and M. Kruskal, “Interaction of solitons in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15, 240 (1965).
[Crossref]

Abdullaev, F.K.

F.K. Abdullaev, Y.V. Kartashov, V.V. Konotop, and D.A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011)
[Crossref]

Ablowitz, M.

M. Ablowitz, B. Prinari, and A. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems (Cambridge University, 2004).

Afeyan, B.

W. Kruer, S. Wilks, B. Afeyan, and R. Kirkwood, “Energy transfer between crossing laser beams,” Phys. Plasmas 3, 382–385 (1996).
[Crossref]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

Aitchison, J.

J. Meier, G. Stegeman, Y. Silberberg, R. Morandotti, and J. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. 93, 093903 (2004).
[Crossref] [PubMed]

Antier-Murgey, M.

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, É. Lallier, C. Larat, M. Antier-Murgey, J. Chanteloup, A. Brignon, and C. Simon-Boisson, “Xcan—a coherent amplification network of femtosecond fiber chirped-pulse amplifiers,” Europ. Phys. J. Special Topics 224, 2609–2613 (2015).
[Crossref]

Barsi, C.

C. Barsi, W. Wan, and J. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photon. 3, 211–215 (2009).
[Crossref]

Bellanger, C.

Bellanger, S.

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, É. Lallier, C. Larat, M. Antier-Murgey, J. Chanteloup, A. Brignon, and C. Simon-Boisson, “Xcan—a coherent amplification network of femtosecond fiber chirped-pulse amplifiers,” Europ. Phys. J. Special Topics 224, 2609–2613 (2015).
[Crossref]

Bergé, L.

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Breakup and fusion of self-guided femtosecond light pulses in air,” Phys. Rev. Lett. 86, 5470–5473 (2001).
[Crossref] [PubMed]

Bersch, C.

M. Wimmer, A. Regensburger, M.A. Miri, C. Bersch, D.N. Christdoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Comm. 6, 7782 (2015)
[Crossref]

Bourderionnet, J.

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, É. Lallier, C. Larat, M. Antier-Murgey, J. Chanteloup, A. Brignon, and C. Simon-Boisson, “Xcan—a coherent amplification network of femtosecond fiber chirped-pulse amplifiers,” Europ. Phys. J. Special Topics 224, 2609–2613 (2015).
[Crossref]

C. Bellanger, B. Toulon, J. Primot, L. Lombard, J. Bourderionnet, and A. Brignon, “Collective phase measurement of an array of fiber lasers by quadriwave lateral shearing interferometry for coherent beam combining,” Opt. Lett. 35, 3931–3933 (2010).
[Crossref] [PubMed]

Brignon, A.

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, É. Lallier, C. Larat, M. Antier-Murgey, J. Chanteloup, A. Brignon, and C. Simon-Boisson, “Xcan—a coherent amplification network of femtosecond fiber chirped-pulse amplifiers,” Europ. Phys. J. Special Topics 224, 2609–2613 (2015).
[Crossref]

C. Bellanger, B. Toulon, J. Primot, L. Lombard, J. Bourderionnet, and A. Brignon, “Collective phase measurement of an array of fiber lasers by quadriwave lateral shearing interferometry for coherent beam combining,” Opt. Lett. 35, 3931–3933 (2010).
[Crossref] [PubMed]

Brocklesby, B.

G. Mourou, T. Tajima, M. Quinn, B. Brocklesby, and J. Limpert, “Are fiber-based lasers the future of accelerators?” Nucl. Instrum. Methods Phys. Res. Sect. A 740, 17–20 (2014).
[Crossref]

G. Mourou, B. Brocklesby, T. Tajima, and J. Limpert, “The future is fibre accelerators,” Nat. Photon. 7, 258–261 (2013).
[Crossref]

Canuto, C.

C. Canuto and A. Quarteroni, “Approximation results for orthogonal polynomials in Sobolev spaces,” Math. Comp. 38, 67–86 (1982).
[Crossref]

Carmon, T.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite Range of nonlocality: First observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95213904 (2005)
[Crossref] [PubMed]

Cazenave, T.

T. Cazenave and P.L. Lions, “Orbital stability of standing waves for some nonlinear Schrödinger equations,” Comm. Math. Phys. 85, 549–561 (1982)
[Crossref]

Chanteloup, J.

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, É. Lallier, C. Larat, M. Antier-Murgey, J. Chanteloup, A. Brignon, and C. Simon-Boisson, “Xcan—a coherent amplification network of femtosecond fiber chirped-pulse amplifiers,” Europ. Phys. J. Special Topics 224, 2609–2613 (2015).
[Crossref]

Chi, S.

Q. Quo, B. Luo, F. Yi, S. Chi, and Y Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69, 016602 (2004)
[Crossref]

Christdoulides, D.N.

M. Wimmer, A. Regensburger, M.A. Miri, C. Bersch, D.N. Christdoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Comm. 6, 7782 (2015)
[Crossref]

Christodoulides, D.

Z. Musslimani, M. Soljačić, M. Segev, and D. Christodoulides, “Delayed-action interaction and spin-orbit coupling between solitons,” Phys. Rev. Lett. 86, 799 (2001).
[Crossref] [PubMed]

Christodoulides, D.N.

K.G. Makris, R. El-Ganainy, D.N. Christodoulides, and Z.H. Musslimani, “PT-Symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011)
[Crossref]

Christou, J.

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76, 2698 (1996).
[Crossref] [PubMed]

Chu, F.

A. C. Scott, F. Chu, and D. W. McLaughlin, “The soliton: A new concept in applied science,” Proc. IEEE 61, 1443–1483 (1973).
[Crossref]

Cohen, O.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite Range of nonlocality: First observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95213904 (2005)
[Crossref] [PubMed]

Couairon, A.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Physics reports 441, 47–189 (2007).
[Crossref]

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Breakup and fusion of self-guided femtosecond light pulses in air,” Phys. Rev. Lett. 86, 5470–5473 (2001).
[Crossref] [PubMed]

Craik, A.

A. Craik, Wave Interactions and Fluid Flows (Cambridge University, 1988).

Daniault, L.

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, É. Lallier, C. Larat, M. Antier-Murgey, J. Chanteloup, A. Brignon, and C. Simon-Boisson, “Xcan—a coherent amplification network of femtosecond fiber chirped-pulse amplifiers,” Europ. Phys. J. Special Topics 224, 2609–2613 (2015).
[Crossref]

Day, D.

D. Day and L. Romero, “Roots of polynomials expressed in terms of orthogonal polynomials,” SIAM J. Numer. Anal. 43, 1969–1987 (2005).
[Crossref]

Derevyanko, S.

H. Frostig, E. Small, S. Derevyanko, and Y. Silberberg, “Focusing coherent light through a nonlinear scattering medium,” arXiv preprint arXiv:1607.08105 (2016).

Dutt, A.

G. Patwardhan, X. Gao, A. Dutt, J. Ginsberg, and A. Gaeta, “Loss of polarization in collapsing beams of elliptical polarization,” in CLEO: QELS_Fundamental Science, (Optical Society of America, 2017), pp. FM3F–7.

Dyke, P.

J. H. Nguyen, P. Dyke, D. Luo, B. Malomed, and R. Hulet, “Collisions of matter-wave solitons,” Nat. Phys. 10, 918–922 (2014).
[Crossref]

El-Ganainy, R.

K.G. Makris, R. El-Ganainy, D.N. Christodoulides, and Z.H. Musslimani, “PT-Symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011)
[Crossref]

Fan, T. Y.

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

Fedoruk, M.

Fibich, G.

B. Shim, S. Schrauth, A. Gaeta, M. Klein, and G. Fibich, “Loss of phase of collapsing beams,” Phys. Rev. Lett. 108, 043902 (2012).
[Crossref] [PubMed]

G. Fibich and M. Klein, “Continuations of the nonlinear Schrödinger equation beyond the singularity,” Nonlinearity 24, 519–552 (2011).
[Crossref]

G. Fibich, The Nonlinear Schrödinger Equation (Springer, 2015).
[Crossref]

Fleischer, J.

C. Barsi, W. Wan, and J. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photon. 3, 211–215 (2009).
[Crossref]

Franco, M.

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Breakup and fusion of self-guided femtosecond light pulses in air,” Phys. Rev. Lett. 86, 5470–5473 (2001).
[Crossref] [PubMed]

Frostig, H.

H. Frostig, E. Small, S. Derevyanko, and Y. Silberberg, “Focusing coherent light through a nonlinear scattering medium,” arXiv preprint arXiv:1607.08105 (2016).

Gaeta, A.

B. Shim, S. Schrauth, A. Gaeta, M. Klein, and G. Fibich, “Loss of phase of collapsing beams,” Phys. Rev. Lett. 108, 043902 (2012).
[Crossref] [PubMed]

A. Ishaaya, T. Grow, S. Ghosh, L. Vuong, and A. Gaeta, “Self-focusing dynamics of coupled optical beams,” Phys. Rev. A 75, 023813 (2007).
[Crossref]

G. Patwardhan, X. Gao, A. Dutt, J. Ginsberg, and A. Gaeta, “Loss of polarization in collapsing beams of elliptical polarization,” in CLEO: QELS_Fundamental Science, (Optical Society of America, 2017), pp. FM3F–7.

Gao, X.

G. Patwardhan, X. Gao, A. Dutt, J. Ginsberg, and A. Gaeta, “Loss of polarization in collapsing beams of elliptical polarization,” in CLEO: QELS_Fundamental Science, (Optical Society of America, 2017), pp. FM3F–7.

Garcia-Quirino, G.

Ghosh, S.

A. Ishaaya, T. Grow, S. Ghosh, L. Vuong, and A. Gaeta, “Self-focusing dynamics of coupled optical beams,” Phys. Rev. A 75, 023813 (2007).
[Crossref]

Ginsberg, J.

G. Patwardhan, X. Gao, A. Dutt, J. Ginsberg, and A. Gaeta, “Loss of polarization in collapsing beams of elliptical polarization,” in CLEO: QELS_Fundamental Science, (Optical Society of America, 2017), pp. FM3F–7.

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Goy, A.

A. Goy and D. Psaltis, “Imaging in focusing Kerr media using reverse propagation [invited],” Phot. Res. 1, 96–101 (2013).
[Crossref]

Grow, T.

A. Ishaaya, T. Grow, S. Ghosh, L. Vuong, and A. Gaeta, “Self-focusing dynamics of coupled optical beams,” Phys. Rev. A 75, 023813 (2007).
[Crossref]

Holmstrom, S. A.

Hulet, R.

J. H. Nguyen, P. Dyke, D. Luo, B. Malomed, and R. Hulet, “Collisions of matter-wave solitons,” Nat. Phys. 10, 918–922 (2014).
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Ishaaya, A.

A. Ishaaya, T. Grow, S. Ghosh, L. Vuong, and A. Gaeta, “Self-focusing dynamics of coupled optical beams,” Phys. Rev. A 75, 023813 (2007).
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Kartashov, Y.V.

F.K. Abdullaev, Y.V. Kartashov, V.V. Konotop, and D.A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011)
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Kirkwood, R.

W. Kruer, S. Wilks, B. Afeyan, and R. Kirkwood, “Energy transfer between crossing laser beams,” Phys. Plasmas 3, 382–385 (1996).
[Crossref]

Kivshar, Y. S.

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
[Crossref]

Klein, M.

B. Shim, S. Schrauth, A. Gaeta, M. Klein, and G. Fibich, “Loss of phase of collapsing beams,” Phys. Rev. Lett. 108, 043902 (2012).
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G. Fibich and M. Klein, “Continuations of the nonlinear Schrödinger equation beyond the singularity,” Nonlinearity 24, 519–552 (2011).
[Crossref]

Konotop, V.V.

F.K. Abdullaev, Y.V. Kartashov, V.V. Konotop, and D.A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011)
[Crossref]

Królikowski, W.

Kruer, W.

W. Kruer, S. Wilks, B. Afeyan, and R. Kirkwood, “Energy transfer between crossing laser beams,” Phys. Plasmas 3, 382–385 (1996).
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Kruskal, M.

N. Zabusky and M. Kruskal, “Interaction of solitons in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15, 240 (1965).
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Lallier, É.

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, É. Lallier, C. Larat, M. Antier-Murgey, J. Chanteloup, A. Brignon, and C. Simon-Boisson, “Xcan—a coherent amplification network of femtosecond fiber chirped-pulse amplifiers,” Europ. Phys. J. Special Topics 224, 2609–2613 (2015).
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Larat, C.

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, É. Lallier, C. Larat, M. Antier-Murgey, J. Chanteloup, A. Brignon, and C. Simon-Boisson, “Xcan—a coherent amplification network of femtosecond fiber chirped-pulse amplifiers,” Europ. Phys. J. Special Topics 224, 2609–2613 (2015).
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L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, É. Lallier, C. Larat, M. Antier-Murgey, J. Chanteloup, A. Brignon, and C. Simon-Boisson, “Xcan—a coherent amplification network of femtosecond fiber chirped-pulse amplifiers,” Europ. Phys. J. Special Topics 224, 2609–2613 (2015).
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G. Mourou, T. Tajima, M. Quinn, B. Brocklesby, and J. Limpert, “Are fiber-based lasers the future of accelerators?” Nucl. Instrum. Methods Phys. Res. Sect. A 740, 17–20 (2014).
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G. Mourou, B. Brocklesby, T. Tajima, and J. Limpert, “The future is fibre accelerators,” Nat. Photon. 7, 258–261 (2013).
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T. Cazenave and P.L. Lions, “Orbital stability of standing waves for some nonlinear Schrödinger equations,” Comm. Math. Phys. 85, 549–561 (1982)
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Lugo-Martinez, G.

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Q. Quo, B. Luo, F. Yi, S. Chi, and Y Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69, 016602 (2004)
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Luo, D.

J. H. Nguyen, P. Dyke, D. Luo, B. Malomed, and R. Hulet, “Collisions of matter-wave solitons,” Nat. Phys. 10, 918–922 (2014).
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V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76, 2698 (1996).
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K.G. Makris, R. El-Ganainy, D.N. Christodoulides, and Z.H. Musslimani, “PT-Symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011)
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J. H. Nguyen, P. Dyke, D. Luo, B. Malomed, and R. Hulet, “Collisions of matter-wave solitons,” Nat. Phys. 10, 918–922 (2014).
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Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
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C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite Range of nonlocality: First observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95213904 (2005)
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A. C. Scott, F. Chu, and D. W. McLaughlin, “The soliton: A new concept in applied science,” Proc. IEEE 61, 1443–1483 (1973).
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J. Meier, G. Stegeman, Y. Silberberg, R. Morandotti, and J. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. 93, 093903 (2004).
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F. Merle, “On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass,” Comm. Pure Appl. Math. 45, 203–254 (1992).
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M. Wimmer, A. Regensburger, M.A. Miri, C. Bersch, D.N. Christdoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Comm. 6, 7782 (2015)
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Mollenauer, L. F.

Morandotti, R.

J. Meier, G. Stegeman, Y. Silberberg, R. Morandotti, and J. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. 93, 093903 (2004).
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G. Mourou, T. Tajima, M. Quinn, B. Brocklesby, and J. Limpert, “Are fiber-based lasers the future of accelerators?” Nucl. Instrum. Methods Phys. Res. Sect. A 740, 17–20 (2014).
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G. Mourou, B. Brocklesby, T. Tajima, and J. Limpert, “The future is fibre accelerators,” Nat. Photon. 7, 258–261 (2013).
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Z. Musslimani, M. Soljačić, M. Segev, and D. Christodoulides, “Delayed-action interaction and spin-orbit coupling between solitons,” Phys. Rev. Lett. 86, 799 (2001).
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Musslimani, Z.H.

K.G. Makris, R. El-Ganainy, D.N. Christodoulides, and Z.H. Musslimani, “PT-Symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011)
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A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Physics reports 441, 47–189 (2007).
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S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Breakup and fusion of self-guided femtosecond light pulses in air,” Phys. Rev. Lett. 86, 5470–5473 (2001).
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Nguyen, J. H.

J. H. Nguyen, P. Dyke, D. Luo, B. Malomed, and R. Hulet, “Collisions of matter-wave solitons,” Nat. Phys. 10, 918–922 (2014).
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A. O’Hagan, “Polynomial chaos: A tutorial and critique from a statistician’s perspective,” SIAM/ASA J. Uncertainty Quantification 20, 1–20 (2013).

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G. Patwardhan, X. Gao, A. Dutt, J. Ginsberg, and A. Gaeta, “Loss of polarization in collapsing beams of elliptical polarization,” in CLEO: QELS_Fundamental Science, (Optical Society of America, 2017), pp. FM3F–7.

Peschel, U.

M. Wimmer, A. Regensburger, M.A. Miri, C. Bersch, D.N. Christdoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Comm. 6, 7782 (2015)
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Prade, B.

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Breakup and fusion of self-guided femtosecond light pulses in air,” Phys. Rev. Lett. 86, 5470–5473 (2001).
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Prinari, B.

M. Ablowitz, B. Prinari, and A. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems (Cambridge University, 2004).

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A. Goy and D. Psaltis, “Imaging in focusing Kerr media using reverse propagation [invited],” Phot. Res. 1, 96–101 (2013).
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C. Canuto and A. Quarteroni, “Approximation results for orthogonal polynomials in Sobolev spaces,” Math. Comp. 38, 67–86 (1982).
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G. Mourou, T. Tajima, M. Quinn, B. Brocklesby, and J. Limpert, “Are fiber-based lasers the future of accelerators?” Nucl. Instrum. Methods Phys. Res. Sect. A 740, 17–20 (2014).
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Q. Quo, B. Luo, F. Yi, S. Chi, and Y Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69, 016602 (2004)
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M. Wimmer, A. Regensburger, M.A. Miri, C. Bersch, D.N. Christdoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Comm. 6, 7782 (2015)
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C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite Range of nonlocality: First observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95213904 (2005)
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Sanchez-Mondragon, J.

Schrauth, S.

B. Shim, S. Schrauth, A. Gaeta, M. Klein, and G. Fibich, “Loss of phase of collapsing beams,” Phys. Rev. Lett. 108, 043902 (2012).
[Crossref] [PubMed]

Scott, A. C.

A. C. Scott, F. Chu, and D. W. McLaughlin, “The soliton: A new concept in applied science,” Proc. IEEE 61, 1443–1483 (1973).
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Segev, M.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite Range of nonlocality: First observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95213904 (2005)
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Z. Musslimani, M. Soljačić, M. Segev, and D. Christodoulides, “Delayed-action interaction and spin-orbit coupling between solitons,” Phys. Rev. Lett. 86, 799 (2001).
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G. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999).
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V. Zakharov and A. Shabat, “Interaction between solitons in a stable medium,” Sov. Phys. JETP 37, 823–828 (1973).

Sheppard, A.

Shim, B.

B. Shim, S. Schrauth, A. Gaeta, M. Klein, and G. Fibich, “Loss of phase of collapsing beams,” Phys. Rev. Lett. 108, 043902 (2012).
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Sidelnikov, O.

Silberberg, Y.

J. Meier, G. Stegeman, Y. Silberberg, R. Morandotti, and J. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. 93, 093903 (2004).
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H. Frostig, E. Small, S. Derevyanko, and Y. Silberberg, “Focusing coherent light through a nonlinear scattering medium,” arXiv preprint arXiv:1607.08105 (2016).

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L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, É. Lallier, C. Larat, M. Antier-Murgey, J. Chanteloup, A. Brignon, and C. Simon-Boisson, “Xcan—a coherent amplification network of femtosecond fiber chirped-pulse amplifiers,” Europ. Phys. J. Special Topics 224, 2609–2613 (2015).
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Small, E.

H. Frostig, E. Small, S. Derevyanko, and Y. Silberberg, “Focusing coherent light through a nonlinear scattering medium,” arXiv preprint arXiv:1607.08105 (2016).

Snyder, A.

Soljacic, M.

Z. Musslimani, M. Soljačić, M. Segev, and D. Christodoulides, “Delayed-action interaction and spin-orbit coupling between solitons,” Phys. Rev. Lett. 86, 799 (2001).
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Stegeman, G.

J. Meier, G. Stegeman, Y. Silberberg, R. Morandotti, and J. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. 93, 093903 (2004).
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G. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999).
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Su, C.

C. Su and R. M. Mirie, “On head-on collisions between two solitary waves,” J. Fluid Mech. 98, 509–525 (1980).
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G. Mourou, T. Tajima, M. Quinn, B. Brocklesby, and J. Limpert, “Are fiber-based lasers the future of accelerators?” Nucl. Instrum. Methods Phys. Res. Sect. A 740, 17–20 (2014).
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G. Mourou, B. Brocklesby, T. Tajima, and J. Limpert, “The future is fibre accelerators,” Nat. Photon. 7, 258–261 (2013).
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V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. 76, 2698 (1996).
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Toulon, B.

Trubatch, A.

M. Ablowitz, B. Prinari, and A. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems (Cambridge University, 2004).

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Tzortzakis, S.

S. Tzortzakis, L. Bergé, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Breakup and fusion of self-guided femtosecond light pulses in air,” Phys. Rev. Lett. 86, 5470–5473 (2001).
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A. Ishaaya, T. Grow, S. Ghosh, L. Vuong, and A. Gaeta, “Self-focusing dynamics of coupled optical beams,” Phys. Rev. A 75, 023813 (2007).
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W. Kruer, S. Wilks, B. Afeyan, and R. Kirkwood, “Energy transfer between crossing laser beams,” Phys. Plasmas 3, 382–385 (1996).
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M. Wimmer, A. Regensburger, M.A. Miri, C. Bersch, D.N. Christdoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Comm. 6, 7782 (2015)
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Q. Quo, B. Luo, F. Yi, S. Chi, and Y Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69, 016602 (2004)
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Q. Quo, B. Luo, F. Yi, S. Chi, and Y Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69, 016602 (2004)
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N. Zabusky and M. Kruskal, “Interaction of solitons in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15, 240 (1965).
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V. Zakharov and A. Shabat, “Interaction between solitons in a stable medium,” Sov. Phys. JETP 37, 823–828 (1973).

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F.K. Abdullaev, Y.V. Kartashov, V.V. Konotop, and D.A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011)
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Comm. Math. Phys. (1)

T. Cazenave and P.L. Lions, “Orbital stability of standing waves for some nonlinear Schrödinger equations,” Comm. Math. Phys. 85, 549–561 (1982)
[Crossref]

Comm. Pure Appl. Math. (1)

F. Merle, “On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass,” Comm. Pure Appl. Math. 45, 203–254 (1992).
[Crossref]

Europ. Phys. J. Special Topics (1)

L. Daniault, S. Bellanger, J. Le Dortz, J. Bourderionnet, É. Lallier, C. Larat, M. Antier-Murgey, J. Chanteloup, A. Brignon, and C. Simon-Boisson, “Xcan—a coherent amplification network of femtosecond fiber chirped-pulse amplifiers,” Europ. Phys. J. Special Topics 224, 2609–2613 (2015).
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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).
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Int. J. Theor. Phys. (1)

K.G. Makris, R. El-Ganainy, D.N. Christodoulides, and Z.H. Musslimani, “PT-Symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011)
[Crossref]

J. Fluid Mech. (1)

C. Su and R. M. Mirie, “On head-on collisions between two solitary waves,” J. Fluid Mech. 98, 509–525 (1980).
[Crossref]

Math. Comp. (1)

C. Canuto and A. Quarteroni, “Approximation results for orthogonal polynomials in Sobolev spaces,” Math. Comp. 38, 67–86 (1982).
[Crossref]

Nat. Comm. (1)

M. Wimmer, A. Regensburger, M.A. Miri, C. Bersch, D.N. Christdoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Comm. 6, 7782 (2015)
[Crossref]

Nat. Photon. (2)

C. Barsi, W. Wan, and J. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photon. 3, 211–215 (2009).
[Crossref]

G. Mourou, B. Brocklesby, T. Tajima, and J. Limpert, “The future is fibre accelerators,” Nat. Photon. 7, 258–261 (2013).
[Crossref]

Nat. Phys. (1)

J. H. Nguyen, P. Dyke, D. Luo, B. Malomed, and R. Hulet, “Collisions of matter-wave solitons,” Nat. Phys. 10, 918–922 (2014).
[Crossref]

Nonlinearity (1)

G. Fibich and M. Klein, “Continuations of the nonlinear Schrödinger equation beyond the singularity,” Nonlinearity 24, 519–552 (2011).
[Crossref]

Nucl. Instrum. Methods Phys. Res. Sect. A (1)

G. Mourou, T. Tajima, M. Quinn, B. Brocklesby, and J. Limpert, “Are fiber-based lasers the future of accelerators?” Nucl. Instrum. Methods Phys. Res. Sect. A 740, 17–20 (2014).
[Crossref]

Opt. Express (1)

Opt. Lett. (6)

Phot. Res. (1)

A. Goy and D. Psaltis, “Imaging in focusing Kerr media using reverse propagation [invited],” Phot. Res. 1, 96–101 (2013).
[Crossref]

Phys. Plasmas (1)

W. Kruer, S. Wilks, B. Afeyan, and R. Kirkwood, “Energy transfer between crossing laser beams,” Phys. Plasmas 3, 382–385 (1996).
[Crossref]

Phys. Rev. A (2)

A. Ishaaya, T. Grow, S. Ghosh, L. Vuong, and A. Gaeta, “Self-focusing dynamics of coupled optical beams,” Phys. Rev. A 75, 023813 (2007).
[Crossref]

F.K. Abdullaev, Y.V. Kartashov, V.V. Konotop, and D.A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011)
[Crossref]

Phys. Rev. E (1)

Q. Quo, B. Luo, F. Yi, S. Chi, and Y Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69, 016602 (2004)
[Crossref]

Phys. Rev. Lett. (7)

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite Range of nonlocality: First observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95213904 (2005)
[Crossref] [PubMed]

B. Shim, S. Schrauth, A. Gaeta, M. Klein, and G. Fibich, “Loss of phase of collapsing beams,” Phys. Rev. Lett. 108, 043902 (2012).
[Crossref] [PubMed]

J. Meier, G. Stegeman, Y. Silberberg, R. Morandotti, and J. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. 93, 093903 (2004).
[Crossref] [PubMed]

N. Zabusky and M. Kruskal, “Interaction of solitons in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15, 240 (1965).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 The cubic-quintic NLS (2) with d = 1, = 10−3, and the initial condition (5) at (a1)–(a4) z = 0.15, (b1)–(b4) z = 3, and (c1c4) z = 11. (a1)–(c1) Cumulative on-axis phase as a function of α. (a2)–(c2) Non-cumulative on-axis phase. (a3)–(c3) The PDF of φ̃. (a4)–(c4) Transverse profile for α = 1 (solid) and α = −1 (dot-dash).
Fig. 2
Fig. 2 Same as Fig. 1. (a) The intensity |ψ(z, x)|2 for α = 0. (b) The on-axis phase for α = 1 (solid) and α = −1 (dots).
Fig. 3
Fig. 3 The 1d cubic-quintic NLS (2) with = 2· 10−2 and the initial condition (9) with κ0 = 8. (a) κ1 = 8, η0 = 0. (b) κ1 = 8.1, η0 = 0. (c) κ1 = 8.1, η0 ≈ −0.48π.
Fig. 4
Fig. 4 Solutions of the 1d cubic-quintic NLS (2) with = 2 · 10−2. (a1)–(d1): the exit intensity |ψ(zf = 17, x; α)|2, (a2)–(d2): the probability of the number of output beams, and (e) the mean (▵, ★, ○, □) and standard deviation of the lateral location of the output beams, for the noisy initial conditions (11a)(11d), respectively. Here a = 12, θ = π 8, and κ0 = 8.
Fig. 5
Fig. 5 Same as figure 1 for d = 2.
Fig. 6
Fig. 6 The cubic-quintic NLS (2) with = 10−3, and the initial condition (5) in one (dot-dash) and two (solid) dimensions. (a) Distance between the PDF of φ̃ and the uniform distribution on [0, 2π]. (b) The propagation constant of the beam core, see (6), as a function of α.

Equations (36)

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i z ψ ( z , x ) + 2 ψ + N ( | ψ | ) ψ = 0 ,
i z ψ ( z , x ) + 2 ψ + | ψ | 2 ψ | ψ | 4 ψ = 0 ,
i z ψ ( z , x ) + 2 ψ + | ψ | 2 1 + | ψ | 2 ψ = 0 ,
ψ ( z = 0 , x ; α ) = ψ 0 ( x ; α ) ,
φ ˜ ( z ; α ) = φ ( z ; α ) mod ( 2 π ) .
ψ 0 ( x ; α ) = 3.4 ( 1 + 0.1 ) e x 2 , α ~ U ( 1 , 1 ) ,
ψ ( z , x ; α ) e i η 0 ( α ) e i κ z R κ ( x ) + radiation ,
φ ( z ; α ) η 0 ( α ) + z κ ( α ) .
lim z φ ( z ; α ) mod ( 2 π ) ~ U ( [ 0 , 2 π ] ) .
Δ φ ( z ) : = φ ( z ; α max ) φ ( z ; α min ) Δ η 0 + z Δ κ ,
Z lop : = 2 π Δ κ .
ψ 0 ( x ) = e i θ x R κ 0 ( x a ) + e i η 0 e i θ x R κ 1 ( x + a ) ,
ψ ( z , x ) e i κ 0 z e i θ x i θ 2 z R κ 0 ( x a 2 θ z ) + e i η 0 e i κ 1 z e i θ x i θ 2 R κ 1 ( x + a + 2 θ z ) .
Δ φ ( κ 1 κ 0 ) z cross + η 0 .
ψ 0 ( x ) = e i θ x R κ 0 ( x a ) + c e i η 0 e i θ x R κ 1 ( x + a )
random κ : η 0 = 0 , c = 1 , κ 1 = κ 0 ( 1 + α 8 ) ,
random κ , out of phase : η 0 = π , c = 1 , κ 1 = κ 0 ( 1 + α 8 ) ,
random power : η 0 = 0 , c = 1 + 0.1 α , κ 1 = κ 0 ,
random phase : η 0 = π α , c = 1 , κ 1 = κ 0 ,
ψ 0 universal = ψ 0 ( 1 ) + e i π α ψ 0 ( 2 ) , α ~ U ( 1 , 1 )
g ( α ) g N ( α ) : = n = 0 N 1 g ^ N ( n ) p n ( α ) ,
g ^ N ( n ) = j = 1 N p n ( α j N ) g ( α j N ) w j N , n = 0 , , N 1 .
𝔼 α [ g ( α ) ] 1 p 0 g ^ N ( 0 ) , σ [ g ( α ) ] n = 0 N 1 | g ^ N ( n ) | 2 | g ^ N ( 0 ) | 2 p 0 2 .
lim z μ ( φ ˜ 1 ( [ a , b ] ) ) = b a 2 π ,
φ z ( x k z ) = 2 k π + a z , φ z ( y k z ) = 2 k π + b z , k .
μ ( φ ˜ 1 ( [ a , b ] ) ) = k = k min k max μ ( x k z , y k z ) + E ( z ) = = k = k min k max μ ( φ z 1 ( 2 π k + a z ) , φ z 1 ( 2 π k + b z ) ) + E ( z ) ,
E ( z ) = μ ( [ α min , y k min 1 z ] ) + μ ( [ x k max + 1 z , α max ] ) .
z φ z ( α min + δ ) z φ z ( α min ) = ( η 0 ( α min + δ ) η 0 ( α min ) ) + z ( κ ( α min + δ ) κ min )
α min lim z x k min z < α min + δ .
μ ( φ z 1 ( 2 π k + a z ) , φ z 1 ( 2 π k + b z ) ) = φ z 1 ( 2 π k + a z ) φ z 1 ( 2 π k + b z ) c ( α ) d α = 2 π k + a z 2 π k + b z g z ( y ) d y ,
μ ( φ z 1 ( 2 π k + a z ) , φ z 1 ( 2 π k + b z ) ) = g z ( 2 π k + ξ k z k ) b a z .
μ ( φ z 1 ( [ a , b ] ) ) = b a z k = k min k max g z ( 2 π k + ξ k z z ) + E ( z ) .
I : = α min α max c ( α ) d y = μ ( α min , α max ) = 1 , I 2 : = x k min z y k max z c ( α ) d α = 2 π k min + a z 2 π k max + b z g z ( y ) d y .
I 2 = 2 π z k = k min k max g z ( 2 π k + ξ k z z ) + O ( z 1 ) .
φ z , min φ z , max = φ z , min φ z ( x k min z ) + φ z ( x k min z ) φ z ( y k max z ) + φ z ( y k max z ) φ z , max ,
μ ( φ z 1 ( [ a , b ] ) ) = b a 2 π + o ( 1 ) ,

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