## Dynamic of total internal reflection (2+1)D bright beams on photorefractive SBN61:Ce crystal |

Optics Express, Vol. 20, Issue 2, pp. 754-763 (2012)

http://dx.doi.org/10.1364/OE.20.000754

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### Abstract

We study experimentally and numerically the transient behavior of a (2+1)D beam when it is totally reflected by nonlinear interface formed by SBN61:Ce photorefractive crystal. The dynamics give rise to observation of new beams. Due to modulation instability of the beam, the nonlinear interface stimulates the break of the beam into new beams that are reflected to different angles.

© 2012 OSA

## 1. Introduction

7. W. Nasalski, “Modelling of beam reflection at a nonlinear-linear interface,” J. Opt. A, Pure Appl. Opt. **2**(5), 433–441 (2000). [CrossRef]

8. E. Alvarado-Méndez, R. Rojas-Laguna, J. G. Aviña-Cervantes, M. Torres-Cisneros, J. A. Andrade-Lucio, J. C. Pedraza-Ortega, E. A. Kuzin, J. J. Sánchez-Mondragón, and V. Vysloukh, “Total internal reflection of spatial solitons at interface formed by nonlinear saturable and linear medium,” Opt. Commun. **193**(1-6), 267–276 (2001). [CrossRef]

14. H. Gilles, S. S. Girard, and J. Hamel, “Simple technique for measuring the Goos-Hänchen effect with polarization modulation and a position-sensitive detector,” Opt. Lett. **27**(16), 1421–1423 (2002). [CrossRef] [PubMed]

*et al.*[15

15. J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, “Black and gray Helmholtz-Kerr soliton refraction,” Phys. Rev. A **83**(1), 013828 (2011). [CrossRef]

16. L. Jankovic, H. Kim, G. Stegeman, S. Carrasco, Ll. Torner, and M. Katz, “Quadratic soliton self-reflection at a quadratically nonlinear interface,” Opt. Lett. **28**(21), 2103–2105 (2003). [CrossRef] [PubMed]

17. F. Baronio, C. De Angelis, P. H. Pioger, V. Couderc, and A. Barthélémy, “Reflection of quadratic solitons at the boundary of nonlinear media,” Opt. Lett. **29**(9), 986–988 (2004). [CrossRef] [PubMed]

_{3}, and linear beam transmission or soliton reflection was observed. Effects of remote boundaries on soliton dynamics in nonlinear media with a large range of nonlocality can lead to soliton steering and oscillation in predetermined trajectories [18

18. B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Boundary force effects exerted on solitons in highly nonlinear media,” Opt. Lett. **32**(2), 154–156 (2007). [CrossRef] [PubMed]

*et al.*[19

19. J. Scheuer and M. Orenstein, “Oscillation modes of spatial soliton arrays in waveguides with nonlinear boundaries,” J. Opt. Soc. Am. B **19**(4), 732–739 (2002). [CrossRef]

*et al.*[20

20. M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. **2**(11), 737–742 (2006). [CrossRef]

*et al.*[21

21. M. Saffman and A. A. Zozulya, “Circular solitons do not exist in photorefractive media,” Opt. Lett. **23**(20), 1579–1581 (1998). [CrossRef] [PubMed]

22. M. F. Shih, M. Segev, and G. Salamo, “Circular waveguides induced by two-dimensional bright steady-state photorefractive spatial screening solitons,” Opt. Lett. **21**(13), 931–934 (1996). [CrossRef] [PubMed]

*et al.*[23

23. A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, “Solitary attractors and low-order filamentation in anisotropic self-defocusing media,” Phys. Rev. A **57**(1), 522–534 (1998). [CrossRef]

*et al.*[24

24. M. Saffman, G. McCarthy, and W. Krolikowski, “Two-dimensional modulational instability in photorefractive media,” J. Opt. B Quantum Semiclassical Opt. **6**(5), S397–S403 (2004). [CrossRef]

*et al.*[25

25. S. Carrasco, S. Polyakov, H. Kim, L. Jankovic, G. I. Stegeman, J. P. Torres, L. Torner, M. Katz, and D. Eger, “Observation of multiple soliton generation mediated by amplification of asymmetries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **67**(4), 046616 (2003). [CrossRef] [PubMed]

## 2. Mathematical model for the nonlinear interface

8. E. Alvarado-Méndez, R. Rojas-Laguna, J. G. Aviña-Cervantes, M. Torres-Cisneros, J. A. Andrade-Lucio, J. C. Pedraza-Ortega, E. A. Kuzin, J. J. Sánchez-Mondragón, and V. Vysloukh, “Total internal reflection of spatial solitons at interface formed by nonlinear saturable and linear medium,” Opt. Commun. **193**(1-6), 267–276 (2001). [CrossRef]

*V*characterizes the incidence angle in normalized coordinate. Numerically we found that angles of incidence

*V*> 0.5 the beam is split into reflected and transmitted and its energy is not conserved [26

26. E. Alvarado-Méndez, G. E. Torres-Cisneros, M. Torres-Cisneros, J. J. Sánchez-Mondragón, and V. Vysloukh, “Internal reflection of one-dimensional bright spatial solitons,” Opt. Quantum Electron. **30**(7/10), 687–696 (1998). [CrossRef]

*V*parameter is small enough. In consequence, the energy is approximately conserved during the reflection at the nonlinear interface [8

8. E. Alvarado-Méndez, R. Rojas-Laguna, J. G. Aviña-Cervantes, M. Torres-Cisneros, J. A. Andrade-Lucio, J. C. Pedraza-Ortega, E. A. Kuzin, J. J. Sánchez-Mondragón, and V. Vysloukh, “Total internal reflection of spatial solitons at interface formed by nonlinear saturable and linear medium,” Opt. Commun. **193**(1-6), 267–276 (2001). [CrossRef]

27. A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, “Self-focusing and soliton formation in media with anisotropic nonlocal material response,” Europhys. Lett. **36**(6), 419–424 (1996). [CrossRef]

*et al.*in photorefractive media [24

24. M. Saffman, G. McCarthy, and W. Krolikowski, “Two-dimensional modulational instability in photorefractive media,” J. Opt. B Quantum Semiclassical Opt. **6**(5), S397–S403 (2004). [CrossRef]

*x-y*plane to different propagation distance (Z), only for one particular case (

**193**(1-6), 267–276 (2001). [CrossRef]

28. V. A. Aleshkevich, Y. V. Kartashov, A. S. Zelenina, V. A. Vysloukh, J. P. Torres, and Ll. Torner, “Eigenvalue control and switching by fission of multisoliton bound states in planar waveguides,” Opt. Lett. **29**(5), 483–485 (2004). [CrossRef] [PubMed]

*et al.*[29

29. F. Ye, Y. V. Kartashov, and Ll. Torner, “Vector soliton fission by reflection at nonlinear interfaces,” Opt. Lett. **32**(4), 394–396 (2007). [CrossRef] [PubMed]

## 3. Experimental setup

**193**(1-6), 267–276 (2001). [CrossRef]

*a*(632.8 nm, 10 mW) was used to illuminate uniformly the photorefractive crystal (PRC) and another laser

*b*(632.8 nm 20 mW) was used to generate a circular Gaussian beam of 200 μm of diameter. We use SBN61:Ce PRC with electrooptic coefficient (

^{3}. The focusing beam was propagated along 10 mm of crystal, perpendicular to the

*c*-axis with a parallel polarization to it. A high voltage power supply was used to apply an external electric field. The input and output intensity distributions were captured with a CCD camera and a frame grabber system.

*et al.*[23

23. A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, “Solitary attractors and low-order filamentation in anisotropic self-defocusing media,” Phys. Rev. A **57**(1), 522–534 (1998). [CrossRef]

24. M. Saffman, G. McCarthy, and W. Krolikowski, “Two-dimensional modulational instability in photorefractive media,” J. Opt. B Quantum Semiclassical Opt. **6**(5), S397–S403 (2004). [CrossRef]

## 3. Conclusion

## Acknowledgments

## References and links

1. | A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Phys. JETP |

2. | P. W. Smith, J.-P. Hermann, W. J. Tomlinson, and P. J. Maloney, “Optical bistability at a nonlinear interface,” Appl. Phys. Lett. |

3. | N. N. Akhmediev, V. I. Korneev, and Yu. V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams,” Sov. Phys. JETP |

4. | A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A |

5. | A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. II. Multiple-particle and multiple-interface extensions,” Phys. Rev. A |

6. | Y. S. Kivshar, A. M. Kosevich, and O. A. Chubykalo, “Radiative effects in the theory of beam propagation at nonlinear interfaces,” Phys. Rev. A |

7. | W. Nasalski, “Modelling of beam reflection at a nonlinear-linear interface,” J. Opt. A, Pure Appl. Opt. |

8. | E. Alvarado-Méndez, R. Rojas-Laguna, J. G. Aviña-Cervantes, M. Torres-Cisneros, J. A. Andrade-Lucio, J. C. Pedraza-Ortega, E. A. Kuzin, J. J. Sánchez-Mondragón, and V. Vysloukh, “Total internal reflection of spatial solitons at interface formed by nonlinear saturable and linear medium,” Opt. Commun. |

9. | W. J. Tomlinson, J. P. Gordon, P. W. Smith, and A. E. Kaplan, “Reflection of a Gaussian beam at a nonlinear interface,” Appl. Opt. |

10. | P. Smith, W. Tomlinson, P. Maloney, and J.-P. Hermann, “Experimental studies of a nonlinear interface,” IEEE J. Quantum Electron. |

11. | H. T. Tran, “Quadratic nonlinear surface waves,” J. Nonlinear Opt. Phys. Mater. |

12. | T. H. Zhang, X. K. Ren, B. H. Wang, C. B. Lou, Z. J. Hu, W. W. Shao, Y. H. Xu, H. Z. Kang, J. Yang, L. Feng, and J. J. Xu, “Surface waves with photorefractive nonlinearity,” Phys. Rev. A |

13. | G. S. Garcia Quirino, J. J. Sanchez-Mondragon, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A |

14. | H. Gilles, S. S. Girard, and J. Hamel, “Simple technique for measuring the Goos-Hänchen effect with polarization modulation and a position-sensitive detector,” Opt. Lett. |

15. | J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, “Black and gray Helmholtz-Kerr soliton refraction,” Phys. Rev. A |

16. | L. Jankovic, H. Kim, G. Stegeman, S. Carrasco, Ll. Torner, and M. Katz, “Quadratic soliton self-reflection at a quadratically nonlinear interface,” Opt. Lett. |

17. | F. Baronio, C. De Angelis, P. H. Pioger, V. Couderc, and A. Barthélémy, “Reflection of quadratic solitons at the boundary of nonlinear media,” Opt. Lett. |

18. | B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Boundary force effects exerted on solitons in highly nonlinear media,” Opt. Lett. |

19. | J. Scheuer and M. Orenstein, “Oscillation modes of spatial soliton arrays in waveguides with nonlinear boundaries,” J. Opt. Soc. Am. B |

20. | M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys. |

21. | M. Saffman and A. A. Zozulya, “Circular solitons do not exist in photorefractive media,” Opt. Lett. |

22. | M. F. Shih, M. Segev, and G. Salamo, “Circular waveguides induced by two-dimensional bright steady-state photorefractive spatial screening solitons,” Opt. Lett. |

23. | A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, “Solitary attractors and low-order filamentation in anisotropic self-defocusing media,” Phys. Rev. A |

24. | M. Saffman, G. McCarthy, and W. Krolikowski, “Two-dimensional modulational instability in photorefractive media,” J. Opt. B Quantum Semiclassical Opt. |

25. | S. Carrasco, S. Polyakov, H. Kim, L. Jankovic, G. I. Stegeman, J. P. Torres, L. Torner, M. Katz, and D. Eger, “Observation of multiple soliton generation mediated by amplification of asymmetries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

26. | E. Alvarado-Méndez, G. E. Torres-Cisneros, M. Torres-Cisneros, J. J. Sánchez-Mondragón, and V. Vysloukh, “Internal reflection of one-dimensional bright spatial solitons,” Opt. Quantum Electron. |

27. | A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, “Self-focusing and soliton formation in media with anisotropic nonlocal material response,” Europhys. Lett. |

28. | V. A. Aleshkevich, Y. V. Kartashov, A. S. Zelenina, V. A. Vysloukh, J. P. Torres, and Ll. Torner, “Eigenvalue control and switching by fission of multisoliton bound states in planar waveguides,” Opt. Lett. |

29. | F. Ye, Y. V. Kartashov, and Ll. Torner, “Vector soliton fission by reflection at nonlinear interfaces,” Opt. Lett. |

**OCIS Codes**

(060.1810) Fiber optics and optical communications : Buffers, couplers, routers, switches, and multiplexers

(190.4350) Nonlinear optics : Nonlinear optics at surfaces

(190.5330) Nonlinear optics : Photorefractive optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: October 7, 2011

Revised Manuscript: November 12, 2011

Manuscript Accepted: December 10, 2011

Published: January 3, 2012

**Citation**

E. Alvarado-Méndez, M. Trejo-Durán, J. M. Estudillo-Ayala, J. A. Andrade-Lucio, E. Vargas-Rodríguez, and M. de G. García-Hernández, "Dynamic of total internal reflection (2+1)D bright beams on photorefractive SBN61:Ce crystal," Opt. Express **20**, 754-763 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-754

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### References

- A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Phys. JETP45, 896–905 (1977).
- P. W. Smith, J.-P. Hermann, W. J. Tomlinson, and P. J. Maloney, “Optical bistability at a nonlinear interface,” Appl. Phys. Lett.35(11), 846–848 (1979). [CrossRef]
- N. N. Akhmediev, V. I. Korneev, and Yu. V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams,” Sov. Phys. JETP61, 62–66 (1985).
- A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. I. Equivalent-particle theory for a single interface,” Phys. Rev. A39(4), 1809–1827 (1989). [CrossRef] [PubMed]
- A. B. Aceves, J. V. Moloney, and A. C. Newell, “Theory of light-beam propagation at nonlinear interfaces. II. Multiple-particle and multiple-interface extensions,” Phys. Rev. A39(4), 1828–1840 (1989). [CrossRef] [PubMed]
- Y. S. Kivshar, A. M. Kosevich, and O. A. Chubykalo, “Radiative effects in the theory of beam propagation at nonlinear interfaces,” Phys. Rev. A41(3), 1677–1688 (1990). [CrossRef] [PubMed]
- W. Nasalski, “Modelling of beam reflection at a nonlinear-linear interface,” J. Opt. A, Pure Appl. Opt.2(5), 433–441 (2000). [CrossRef]
- E. Alvarado-Méndez, R. Rojas-Laguna, J. G. Aviña-Cervantes, M. Torres-Cisneros, J. A. Andrade-Lucio, J. C. Pedraza-Ortega, E. A. Kuzin, J. J. Sánchez-Mondragón, and V. Vysloukh, “Total internal reflection of spatial solitons at interface formed by nonlinear saturable and linear medium,” Opt. Commun.193(1-6), 267–276 (2001). [CrossRef]
- W. J. Tomlinson, J. P. Gordon, P. W. Smith, and A. E. Kaplan, “Reflection of a Gaussian beam at a nonlinear interface,” Appl. Opt.21(11), 2041–2051 (1982). [CrossRef] [PubMed]
- P. Smith, W. Tomlinson, P. Maloney, and J.-P. Hermann, “Experimental studies of a nonlinear interface,” IEEE J. Quantum Electron.17(3), 340–348 (1981). [CrossRef]
- H. T. Tran, “Quadratic nonlinear surface waves,” J. Nonlinear Opt. Phys. Mater.5(1), 133–138 (1996). [CrossRef]
- T. H. Zhang, X. K. Ren, B. H. Wang, C. B. Lou, Z. J. Hu, W. W. Shao, Y. H. Xu, H. Z. Kang, J. Yang, L. Feng, and J. J. Xu, “Surface waves with photorefractive nonlinearity,” Phys. Rev. A76(1), 013827 (2007). [CrossRef]
- G. S. Garcia Quirino, J. J. Sanchez-Mondragon, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A51(2), 1571–1577 (1995). [CrossRef] [PubMed]
- H. Gilles, S. S. Girard, and J. Hamel, “Simple technique for measuring the Goos-Hänchen effect with polarization modulation and a position-sensitive detector,” Opt. Lett.27(16), 1421–1423 (2002). [CrossRef] [PubMed]
- J. Sánchez-Curto, P. Chamorro-Posada, and G. S. McDonald, “Black and gray Helmholtz-Kerr soliton refraction,” Phys. Rev. A83(1), 013828 (2011). [CrossRef]
- L. Jankovic, H. Kim, G. Stegeman, S. Carrasco, Ll. Torner, and M. Katz, “Quadratic soliton self-reflection at a quadratically nonlinear interface,” Opt. Lett.28(21), 2103–2105 (2003). [CrossRef] [PubMed]
- F. Baronio, C. De Angelis, P. H. Pioger, V. Couderc, and A. Barthélémy, “Reflection of quadratic solitons at the boundary of nonlinear media,” Opt. Lett.29(9), 986–988 (2004). [CrossRef] [PubMed]
- B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Boundary force effects exerted on solitons in highly nonlinear media,” Opt. Lett.32(2), 154–156 (2007). [CrossRef] [PubMed]
- J. Scheuer and M. Orenstein, “Oscillation modes of spatial soliton arrays in waveguides with nonlinear boundaries,” J. Opt. Soc. Am. B19(4), 732–739 (2002). [CrossRef]
- M. Peccianti, A. Dyadyusha, M. Kaczmarek, and G. Assanto, “Tunable refraction and reflection of self-confined light beams,” Nat. Phys.2(11), 737–742 (2006). [CrossRef]
- M. Saffman and A. A. Zozulya, “Circular solitons do not exist in photorefractive media,” Opt. Lett.23(20), 1579–1581 (1998). [CrossRef] [PubMed]
- M. F. Shih, M. Segev, and G. Salamo, “Circular waveguides induced by two-dimensional bright steady-state photorefractive spatial screening solitons,” Opt. Lett.21(13), 931–934 (1996). [CrossRef] [PubMed]
- A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, “Solitary attractors and low-order filamentation in anisotropic self-defocusing media,” Phys. Rev. A57(1), 522–534 (1998). [CrossRef]
- M. Saffman, G. McCarthy, and W. Krolikowski, “Two-dimensional modulational instability in photorefractive media,” J. Opt. B Quantum Semiclassical Opt.6(5), S397–S403 (2004). [CrossRef]
- S. Carrasco, S. Polyakov, H. Kim, L. Jankovic, G. I. Stegeman, J. P. Torres, L. Torner, M. Katz, and D. Eger, “Observation of multiple soliton generation mediated by amplification of asymmetries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.67(4), 046616 (2003). [CrossRef] [PubMed]
- E. Alvarado-Méndez, G. E. Torres-Cisneros, M. Torres-Cisneros, J. J. Sánchez-Mondragón, and V. Vysloukh, “Internal reflection of one-dimensional bright spatial solitons,” Opt. Quantum Electron.30(7/10), 687–696 (1998). [CrossRef]
- A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, “Self-focusing and soliton formation in media with anisotropic nonlocal material response,” Europhys. Lett.36(6), 419–424 (1996). [CrossRef]
- V. A. Aleshkevich, Y. V. Kartashov, A. S. Zelenina, V. A. Vysloukh, J. P. Torres, and Ll. Torner, “Eigenvalue control and switching by fission of multisoliton bound states in planar waveguides,” Opt. Lett.29(5), 483–485 (2004). [CrossRef] [PubMed]
- F. Ye, Y. V. Kartashov, and Ll. Torner, “Vector soliton fission by reflection at nonlinear interfaces,” Opt. Lett.32(4), 394–396 (2007). [CrossRef] [PubMed]

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