Abstract

We show that the refractive index modification photoinduced in a biased nonlinear photorefractive crystal can be accurately measured and controlled by means of a background incoherent illumination and an external electric field. The proposed easy-to-implement method is based on the far-field measurement of the diffraction patterns of a laser beam propagating through a self-defocusing medium undergoing spatial self-phase modulation. For various experimental conditions, both saturation intensity and maximum refractive index modification have been measured. We also clearly evidence and characterise the anisotropic nonlinear response of the crystal in the stationary regime.

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

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2019 (3)

D. Song, D. Leykam, J. Su, X. Liu, L. Tang, S. Liu, J. Zhao, N. K. Efremidis, J. Xu, and Z. Chen, “Valley vortex states and degeneracy lifting via photonic higher-band excitation,” Phys. Rev. Lett. 122(12), 123903 (2019).
[Crossref]

V. M. di Pietro, A. Jullien, U. Bortolozzo, N. Forget, and S. Residori, “Thermally-induced nonlinear spatial shaping of infrared femtosecond pulses in nematic liquid crystals,” Laser Phys. Lett. 16(1), 015301 (2019).
[Crossref]

Y. Shan, J. Tang, L. Wu, S. Lu, X. Dai, and Y. Xiang, “Spatial self-phase modulation and all-optical switching of graphene oxide dispersions,” J. Alloys Compd. 771, 900–904 (2019).
[Crossref]

2018 (4)

N. Santic, A. Fusaro, S. Salem, J. Garnier, A. Picozzi, and R. Kaiser, “Nonequilibrium precondensation of classical waves in two dimensions propagating through atomic vapors,” Phys. Rev. Lett. 120(5), 055301 (2018).
[Crossref]

Q. Fontaine, T. Bienaimé, S. Pigeon, E. Giacobino, A. Bramati, and Q. Glorieux, “Observation of the bogoliubov dispersion in a fluid of light,” Phys. Rev. Lett. 121(18), 183604 (2018).
[Crossref]

C. Michel, O. Boughdad, M. Albert, P.-É. Larre, and M. Belléc, “Superfluid motion and drag-force cancellation in a fluid of light,” Nat. Commun. 9(1), 2108 (2018).
[Crossref]

C. Finot, F. Chaussard, and S. Boscolo, “Simple guidelines to predict self-phase modulation patterns,” J. Opt. Soc. Am. B 35(12), 3143–3152 (2018).
[Crossref]

2017 (1)

M. Boguslawski, S. Brake, D. Leykam, A. S. Desyatnikov, and C. Denz, “Observation of transverse coherent backscattering in disordered photonic structures,” Sci. Rep. 7(1), 10439–55 (2017).
[Crossref]

2016 (2)

M. Boguslawski, N. M. Lučić, F. Diebel, D. V. Timotijević, C. Denz, and D. M. J. Savić, “Light localization in optically induced deterministic aperiodic fibonacci lattices,” Optica 3(7), 711–717 (2016).
[Crossref]

D. Pierangeli, F. Di Mei, G. Di Domenico, A. J. Agranat, C. Conti, and E. DelRe, “Turbulent transitions in optical wave propagation,” Phys. Rev. Lett. 117(18), 183902 (2016).
[Crossref]

2014 (2)

F. Diebel, D. Leykam, M. Boguslawski, P. Rose, C. Denz, and A. S. Desyatnikov, “All-optical switching in optically induced nonlinear waveguide couplers,” Appl. Phys. Lett. 104(26), 261111 (2014).
[Crossref]

J. Armijo, R. Allio, and C. Mejía-Cortés, “Absolute calibration of the refractive index in photo-induced photonic lattices,” Opt. Express 22(17), 20574 (2014).
[Crossref]

2012 (2)

V. Caullet, N. Marsal, D. Wolfersberger, and M. Sciamanna, “Vortex induced rotation dynamics of optical patterns,” Phys. Rev. Lett. 108(26), 263903 (2012).
[Crossref]

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. W. Fleischer, “Observation of the kinetic condensation of classical waves,” Nat. Phys. 8(6), 470–474 (2012).
[Crossref]

2011 (1)

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photonics 5(1), 39–42 (2011).
[Crossref]

2010 (2)

2009 (1)

2008 (1)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[Crossref]

2007 (3)

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[Crossref]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[Crossref]

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3(1), 46–51 (2007).
[Crossref]

2005 (2)

J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13(6), 1780–1796 (2005).
[Crossref]

L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A: Pure Appl. Opt. 7(8), 409–415 (2005).
[Crossref]

2003 (1)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[Crossref]

2000 (1)

1998 (2)

1997 (1)

S. Bian and J. Frejlich, “Z-scan measurements of photorefractive nonlinearities for a sbn: Ce crystal,” Appl. Phys. B: Lasers Opt. 64(5), 539–546 (1997).
[Crossref]

1996 (1)

1995 (2)

A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A 51(2), 1520–1531 (1995).
[Crossref]

D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. Am. B 12(9), 1628–1633 (1995).
[Crossref]

1991 (1)

H. Zhang, X. He, Y. Shih, and S. Tang, “A new method for measuring the electro-optic coefficients with higher sensitivity and higher accuracy,” Opt. Commun. 86(6), 509–512 (1991).
[Crossref]

1989 (1)

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25(3), 484–519 (1989).
[Crossref]

1981 (1)

1978 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. i. steady state,” Ferroelectrics 22(1), 949–960 (1978).
[Crossref]

Agranat, A. J.

D. Pierangeli, F. Di Mei, G. Di Domenico, A. J. Agranat, C. Conti, and E. DelRe, “Turbulent transitions in optical wave propagation,” Phys. Rev. Lett. 117(18), 183902 (2016).
[Crossref]

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photonics 5(1), 39–42 (2011).
[Crossref]

Agrawal, G.

G. Agrawal, Nonlinear fiber optics, 5th ed. (Academic Press, 2012).

Aguilar, P. A. M.

Albert, M.

C. Michel, O. Boughdad, M. Albert, P.-É. Larre, and M. Belléc, “Superfluid motion and drag-force cancellation in a fluid of light,” Nat. Commun. 9(1), 2108 (2018).
[Crossref]

Allio, R.

Anderson, D. Z.

A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A 51(2), 1520–1531 (1995).
[Crossref]

Arakelian, S. M.

Armijo, J.

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[Crossref]

Barsi, C.

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. W. Fleischer, “Observation of the kinetic condensation of classical waves,” Nat. Phys. 8(6), 470–474 (2012).
[Crossref]

Bartal, G.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[Crossref]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[Crossref]

J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13(6), 1780–1796 (2005).
[Crossref]

Belléc, M.

C. Michel, O. Boughdad, M. Albert, P.-É. Larre, and M. Belléc, “Superfluid motion and drag-force cancellation in a fluid of light,” Nat. Commun. 9(1), 2108 (2018).
[Crossref]

Bian, S.

S. Bian and J. Frejlich, “Z-scan measurements of photorefractive nonlinearities for a sbn: Ce crystal,” Appl. Phys. B: Lasers Opt. 64(5), 539–546 (1997).
[Crossref]

Bienaimé, T.

Q. Fontaine, T. Bienaimé, S. Pigeon, E. Giacobino, A. Bramati, and Q. Glorieux, “Observation of the bogoliubov dispersion in a fluid of light,” Phys. Rev. Lett. 121(18), 183604 (2018).
[Crossref]

Boguslawski, M.

M. Boguslawski, S. Brake, D. Leykam, A. S. Desyatnikov, and C. Denz, “Observation of transverse coherent backscattering in disordered photonic structures,” Sci. Rep. 7(1), 10439–55 (2017).
[Crossref]

M. Boguslawski, N. M. Lučić, F. Diebel, D. V. Timotijević, C. Denz, and D. M. J. Savić, “Light localization in optically induced deterministic aperiodic fibonacci lattices,” Optica 3(7), 711–717 (2016).
[Crossref]

F. Diebel, D. Leykam, M. Boguslawski, P. Rose, C. Denz, and A. S. Desyatnikov, “All-optical switching in optically induced nonlinear waveguide couplers,” Appl. Phys. Lett. 104(26), 261111 (2014).
[Crossref]

Bortolozzo, U.

V. M. di Pietro, A. Jullien, U. Bortolozzo, N. Forget, and S. Residori, “Thermally-induced nonlinear spatial shaping of infrared femtosecond pulses in nematic liquid crystals,” Laser Phys. Lett. 16(1), 015301 (2019).
[Crossref]

Boscolo, S.

Bosshard, C.

Boughdad, O.

C. Michel, O. Boughdad, M. Albert, P.-É. Larre, and M. Belléc, “Superfluid motion and drag-force cancellation in a fluid of light,” Nat. Commun. 9(1), 2108 (2018).
[Crossref]

Brake, S.

M. Boguslawski, S. Brake, D. Leykam, A. S. Desyatnikov, and C. Denz, “Observation of transverse coherent backscattering in disordered photonic structures,” Sci. Rep. 7(1), 10439–55 (2017).
[Crossref]

Bramati, A.

Q. Fontaine, T. Bienaimé, S. Pigeon, E. Giacobino, A. Bramati, and Q. Glorieux, “Observation of the bogoliubov dispersion in a fluid of light,” Phys. Rev. Lett. 121(18), 183604 (2018).
[Crossref]

Breitling, D.

Buljan, H.

Carrasco, M. L. A.

Carvalho, M. I.

Castillo, M. D. I.

Caullet, V.

V. Caullet, N. Marsal, D. Wolfersberger, and M. Sciamanna, “Vortex induced rotation dynamics of optical patterns,” Phys. Rev. Lett. 108(26), 263903 (2012).
[Crossref]

Cerda, S. C.

Chaussard, F.

Chen, Z.

D. Song, D. Leykam, J. Su, X. Liu, L. Tang, S. Liu, J. Zhao, N. K. Efremidis, J. Xu, and Z. Chen, “Valley vortex states and degeneracy lifting via photonic higher-band excitation,” Phys. Rev. Lett. 122(12), 123903 (2019).
[Crossref]

Chi, M.

Christodoulides, D. N.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[Crossref]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[Crossref]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
[Crossref]

D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. Am. B 12(9), 1628–1633 (1995).
[Crossref]

Cohen, O.

Conti, C.

D. Pierangeli, F. Di Mei, G. Di Domenico, A. J. Agranat, C. Conti, and E. DelRe, “Turbulent transitions in optical wave propagation,” Phys. Rev. Lett. 117(18), 183902 (2016).
[Crossref]

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photonics 5(1), 39–42 (2011).
[Crossref]

Dai, X.

Y. Shan, J. Tang, L. Wu, S. Lu, X. Dai, and Y. Xiang, “Spatial self-phase modulation and all-optical switching of graphene oxide dispersions,” J. Alloys Compd. 771, 900–904 (2019).
[Crossref]

DelRe, E.

D. Pierangeli, F. Di Mei, G. Di Domenico, A. J. Agranat, C. Conti, and E. DelRe, “Turbulent transitions in optical wave propagation,” Phys. Rev. Lett. 117(18), 183902 (2016).
[Crossref]

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photonics 5(1), 39–42 (2011).
[Crossref]

Deng, L.

L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A: Pure Appl. Opt. 7(8), 409–415 (2005).
[Crossref]

Denz, C.

M. Boguslawski, S. Brake, D. Leykam, A. S. Desyatnikov, and C. Denz, “Observation of transverse coherent backscattering in disordered photonic structures,” Sci. Rep. 7(1), 10439–55 (2017).
[Crossref]

M. Boguslawski, N. M. Lučić, F. Diebel, D. V. Timotijević, C. Denz, and D. M. J. Savić, “Light localization in optically induced deterministic aperiodic fibonacci lattices,” Optica 3(7), 711–717 (2016).
[Crossref]

F. Diebel, D. Leykam, M. Boguslawski, P. Rose, C. Denz, and A. S. Desyatnikov, “All-optical switching in optically induced nonlinear waveguide couplers,” Appl. Phys. Lett. 104(26), 261111 (2014).
[Crossref]

C. Denz, M. Schwab, M. Sedlatschek, T. Tschudi, and T. Honda, “Pattern dynamics and competition in a photorefractive feedback system,” J. Opt. Soc. Am. B 15(7), 2057–2064 (1998).
[Crossref]

C. Denz, M. Schwab, and C. Weilnau, Transverse-Pattern Formation in Photorefractive Optics (Springer, 2003).

Desyatnikov, A. S.

M. Boguslawski, S. Brake, D. Leykam, A. S. Desyatnikov, and C. Denz, “Observation of transverse coherent backscattering in disordered photonic structures,” Sci. Rep. 7(1), 10439–55 (2017).
[Crossref]

F. Diebel, D. Leykam, M. Boguslawski, P. Rose, C. Denz, and A. S. Desyatnikov, “All-optical switching in optically induced nonlinear waveguide couplers,” Appl. Phys. Lett. 104(26), 261111 (2014).
[Crossref]

Di Domenico, G.

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J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13(6), 1780–1796 (2005).
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C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. W. Fleischer, “Observation of the kinetic condensation of classical waves,” Nat. Phys. 8(6), 470–474 (2012).
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W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3(1), 46–51 (2007).
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J. W. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13(6), 1780–1796 (2005).
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J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003).
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Huignard, J.-P.

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W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3(1), 46–51 (2007).
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N. Santic, A. Fusaro, S. Salem, J. Garnier, A. Picozzi, and R. Kaiser, “Nonequilibrium precondensation of classical waves in two dimensions propagating through atomic vapors,” Phys. Rev. Lett. 120(5), 055301 (2018).
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D. Pierangeli, F. Di Mei, G. Di Domenico, A. J. Agranat, C. Conti, and E. DelRe, “Turbulent transitions in optical wave propagation,” Phys. Rev. Lett. 117(18), 183902 (2016).
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Q. Fontaine, T. Bienaimé, S. Pigeon, E. Giacobino, A. Bramati, and Q. Glorieux, “Observation of the bogoliubov dispersion in a fluid of light,” Phys. Rev. Lett. 121(18), 183604 (2018).
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Residori, S.

V. M. di Pietro, A. Jullien, U. Bortolozzo, N. Forget, and S. Residori, “Thermally-induced nonlinear spatial shaping of infrared femtosecond pulses in nematic liquid crystals,” Laser Phys. Lett. 16(1), 015301 (2019).
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C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. W. Fleischer, “Observation of the kinetic condensation of classical waves,” Nat. Phys. 8(6), 470–474 (2012).
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F. Diebel, D. Leykam, M. Boguslawski, P. Rose, C. Denz, and A. S. Desyatnikov, “All-optical switching in optically induced nonlinear waveguide couplers,” Appl. Phys. Lett. 104(26), 261111 (2014).
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N. Santic, A. Fusaro, S. Salem, J. Garnier, A. Picozzi, and R. Kaiser, “Nonequilibrium precondensation of classical waves in two dimensions propagating through atomic vapors,” Phys. Rev. Lett. 120(5), 055301 (2018).
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N. Santic, A. Fusaro, S. Salem, J. Garnier, A. Picozzi, and R. Kaiser, “Nonequilibrium precondensation of classical waves in two dimensions propagating through atomic vapors,” Phys. Rev. Lett. 120(5), 055301 (2018).
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V. Caullet, N. Marsal, D. Wolfersberger, and M. Sciamanna, “Vortex induced rotation dynamics of optical patterns,” Phys. Rev. Lett. 108(26), 263903 (2012).
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N. Marsal, D. Wolfersberger, M. Sciamanna, and G. Montemezzani, “Noise- and dynamics-sustained patterns in a nonlinear photorefractive system,” Phys. Rev. A 81(3), 031804 (2010).
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Y. Shan, J. Tang, L. Wu, S. Lu, X. Dai, and Y. Xiang, “Spatial self-phase modulation and all-optical switching of graphene oxide dispersions,” J. Alloys Compd. 771, 900–904 (2019).
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D. Song, D. Leykam, J. Su, X. Liu, L. Tang, S. Liu, J. Zhao, N. K. Efremidis, J. Xu, and Z. Chen, “Valley vortex states and degeneracy lifting via photonic higher-band excitation,” Phys. Rev. Lett. 122(12), 123903 (2019).
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N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. i. steady state,” Ferroelectrics 22(1), 949–960 (1978).
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F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
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D. Song, D. Leykam, J. Su, X. Liu, L. Tang, S. Liu, J. Zhao, N. K. Efremidis, J. Xu, and Z. Chen, “Valley vortex states and degeneracy lifting via photonic higher-band excitation,” Phys. Rev. Lett. 122(12), 123903 (2019).
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Y. Shan, J. Tang, L. Wu, S. Lu, X. Dai, and Y. Xiang, “Spatial self-phase modulation and all-optical switching of graphene oxide dispersions,” J. Alloys Compd. 771, 900–904 (2019).
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D. Song, D. Leykam, J. Su, X. Liu, L. Tang, S. Liu, J. Zhao, N. K. Efremidis, J. Xu, and Z. Chen, “Valley vortex states and degeneracy lifting via photonic higher-band excitation,” Phys. Rev. Lett. 122(12), 123903 (2019).
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D. Song, D. Leykam, J. Su, X. Liu, L. Tang, S. Liu, J. Zhao, N. K. Efremidis, J. Xu, and Z. Chen, “Valley vortex states and degeneracy lifting via photonic higher-band excitation,” Phys. Rev. Lett. 122(12), 123903 (2019).
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Appl. Opt. (1)

Appl. Phys. B: Lasers Opt. (1)

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Appl. Phys. Lett. (1)

F. Diebel, D. Leykam, M. Boguslawski, P. Rose, C. Denz, and A. S. Desyatnikov, “All-optical switching in optically induced nonlinear waveguide couplers,” Appl. Phys. Lett. 104(26), 261111 (2014).
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Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. i. steady state,” Ferroelectrics 22(1), 949–960 (1978).
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IEEE J. Quantum Electron. (1)

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J. Opt. A: Pure Appl. Opt. (1)

L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field diffraction patterns of divergent and convergent Gaussian beams passing through self-focusing and self-defocusing media,” J. Opt. A: Pure Appl. Opt. 7(8), 409–415 (2005).
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J. Opt. Soc. Am. B (5)

Laser Phys. Lett. (1)

V. M. di Pietro, A. Jullien, U. Bortolozzo, N. Forget, and S. Residori, “Thermally-induced nonlinear spatial shaping of infrared femtosecond pulses in nematic liquid crystals,” Laser Phys. Lett. 16(1), 015301 (2019).
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Nat. Commun. (1)

C. Michel, O. Boughdad, M. Albert, P.-É. Larre, and M. Belléc, “Superfluid motion and drag-force cancellation in a fluid of light,” Nat. Commun. 9(1), 2108 (2018).
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Nat. Photonics (1)

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photonics 5(1), 39–42 (2011).
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Nat. Phys. (2)

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3(1), 46–51 (2007).
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C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. W. Fleischer, “Observation of the kinetic condensation of classical waves,” Nat. Phys. 8(6), 470–474 (2012).
[Crossref]

Nature (2)

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Figures (5)

Fig. 1.
Fig. 1. Sketch of the experimental setup. A focused cw laser beam propagates in a biased SBN photorefractive crystal. The beam is linearly polarised along $x$, corresponding to the $c$-axis. The crystal is thinner than the Rayleigh length and placed out of focus which garanties a quasi-constant beam curvature radius. The spatial (2D) spectrum is measured in the far-field. The external voltage (resp. the white light illumination) controls the maximum value of the photoinduced refractive index modification $\Delta n_\textrm {max}$ (resp. the saturation intensity $I_\textrm {sat}$).
Fig. 2.
Fig. 2. Intensity distribution of the spatial spectrum measured in the far-field, i.e. ($k_x, k_y$) plane, for different initial laser intensities. (a) Experimental measurements. The white rectangle corresponds to the part of the image used to extract the profile of the spatial spectrum along the $k_x$-direction. (b) Numerical image obtained for $I = 100$ mW.cm$^{-2}$, by solving the coupled Eqs. (4) and (2). (c) Numerical image obtained for the same optical intensity by solving the Eq. (4) in the isotropic approximation. For each panel, $k_x$ (resp. $k_y$) spans from -52.5 mm$^{-1}$ to 52.5 mm$^{-1}$ (resp. -36 mm$^{-1}$ to 36 mm$^{-1}$).
Fig. 3.
Fig. 3. (a) Experimental spectrum profiles (cut along $k_x$) for various input intensities. Black (resp. white) corresponds to the maximum (resp. minimum) spectral power which is normalised to 1 for each measurement. The widths of the profiles have been artificially increased for clarity. (b-c) Typical profiles at $I=9$ mW.cm$^{-2}$ and $I=49$ mW.cm$^{-2}$ (red and blue curves respectively). The gray points count twice the number of rings $N_\textrm {ring}$. The concave (resp. convexe) behaviour at $k_x=0$ (dashed gray line) in the red (resp. blue) curve defines the nonlinear phase shift as $\Delta \Psi _0 = 2\pi {}N_{\mathrm {rings}}$ [resp. $\Delta \Psi _0 = (2N_{\mathrm {rings}} - 1)\pi$], see text for details.
Fig. 4.
Fig. 4. Nonlinear index of refraction along the $c$-axis for a given set of parameters of white light intensity and external electric field with respect to the incident optical intensity and corresponding fits using a isotopic saturable model (Eq. (6)). (a) Same applied external electric field of $1.5\times 10^{5}\,\mathrm {V.m}^{-1}$ and different white light power $[0.36,\,0.96,\,1.92]\,\mathrm {W}$. The black-dotted line corresponds to the mean value of the plateau extracted from the data when the saturation regime is reached, $\Delta n_\textrm {max, mean} = (2.59 \pm 0.07) \times 10^{-4}$. (b) Same applied white light power (1.92 W), giving $I_{\mathrm {sat}} \simeq 25\,\mathrm {mW.cm}^{2}$ and different electric field $[0.5,\,1,\,1.5] \times 10^{5}\,\mathrm {V.m}^{-1}$, from left to right. On both panels, vertical error bars are estimated looking at the cuts of the images and represent an uncertainty of one ring on the rings count at high intensity. Horizontal errors bars are smaller than the symbols.
Fig. 5.
Fig. 5. Measurement of the anisotropy of SBN. Blue circles: ratio of the fitted $|\Delta n_{\textrm {max}}|$ along the $k_x$ direction and the orthogonal direction, errorbars: uncertainty given by the fitting procedure. Blue solid line: mean value of the data, light-blue area: standard deviation of the data. Red diamonds: aspect ratios extracted directly from the images, averaged over the 10 images having the highest intensity. Error bars correspond to the standard deviation.

Tables (1)

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Table 1. Summary of the fitted parameters extracted using an isotropic saturable model on the data presented on Fig. 4(a) and Fig. 4(b).

Equations (9)

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Δ n ( I ) = 1 2 n 0 3 r 33 E sc [ I ( x , y ) ] ,
2 ϕ 0 + ln ( 1 + I ~ ) ϕ 0 = | E ext | x ln ( 1 + I ~ ) ,
Δ n ( I ) = 1 2 n 0 3 r 33 E ext 1 1 + I ~ .
i z E = 1 2 k 0 n 0 2 E i α E + k 0 Δ n ( I ) E ,
i z A = 1 2 k 0 n 0 2 A i α A + k 0 Δ n max ( I ~ 1 + I ~ ) A .
Δ n ( I ) = 1 2 n 0 3 r 33 E ext I ~ 1 + I ~ = Δ n max I ~ 1 + I ~ ,
Δ Ψ NL ( x , y ) = k 0 0 L z Δ n ( r , z ) d z ,
Δ Ψ NL ( x , y ) = Δ Ψ 0 exp ( r 2 / w p 2 ) ,
N r i n g s Δ Ψ 0 2 π .

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