## Localized surface waves at the interface between linear dielectric and biased centrosymmetric photorefractive crystals |

Optics Express, Vol. 21, Issue 13, pp. 15595-15602 (2013)

http://dx.doi.org/10.1364/OE.21.015595

Acrobat PDF (3343 KB)

### Abstract

We study localized surface waves at the interface between linear dielectric and biased centrosymmetric photorefractive (CP) crystals. If the propagation constant *b* is fixed, the energy of localized surface waves increases with the order of localized surface waves. For low *b* values, the considerable part of the energy of localized surface waves is concentrated in the linear dielectric and decreases with an increase in *b*. For high *b* values, the part of the energy of localized surface waves concentrated in the nonlinear CP crystals is always higher than that in the linear dielectric and increases with *b*. The stability properties of these localized surface waves are also discussed in detail.

© 2013 OSA

## 1. Introduction

1. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. **73**(24), 3211–3214 (1994) [CrossRef] [PubMed] .

2. 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] .

3. M. Segev, G. C. Valley, M. C. Bashaw, M. Taya, and M. M. Fejer, “Photovoltaic spatial solitons,” J. Opt. Soc. Am. B **14**(7), 1772–1781 (1997) [CrossRef] .

4. H. Wang and W. She, “Incoherently coupled grey photovoltaic spatial soliton families,” Chin. Phys. Lett. **22**(1), 128–131 (2005) [CrossRef] .

5. K. Lu, T. Tang, and Y. Zhang, “One-dimensional steady-state spatial solitons in photovoltaic photorefractive materials with an external applied field,” Phys. Rev. A **61**(5), 053822 (2000) [CrossRef] .

6. E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. I. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. **85**(12), 2193–2195 (2004) [CrossRef] .

7. M. Segev and A. J. Agranat, “Spatial solitons in centrosymmetric photorefractive media,” Opt. Lett. **22**(17), 1299–1301 (1997) [CrossRef] .

10. A. Ciattoni, A. Marini, C. Rizza, and E. DelRe, “Collision and fusion of counterpropagating micrometer-sized optical beams in periodically biased photorefractive crystals,” Opt. Lett. **34**(7), 911–913 (2009) [CrossRef] [PubMed] .

11. Y. V. Kartashov, L. Torner, V. A. Vysloukh, and D. Mihalache, “Multipole vector solitons in nonlocal nonlinear media,” Opt. Lett. **31**(10), 1483–1485 (2006) [CrossRef] [PubMed] .

22. V. A. Aleshkevich, V. A. Vysloukh, and Y. V. Kartashev, “Optical surface waves at the interface between a linear dielectric and a photorefractive crystal,” Quantum Electron. **30**(10), 905–910 (2000) [CrossRef] .

18. T. Zhang, X. Ren, B. Wang, C. Lou, Z. Hu, W. Shao, Y. Xu, H. Kang, J. Yang, D. Yang, L. Feng, and J. Xu, “Surface waves with photorefractive nonlinearity,” Phys. Rev. A **76**(1), 013827 (2007) [CrossRef] .

19. 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 **51**(2), 1571–1577 (1995) [CrossRef] [PubMed] .

20. M. Cronin-Golomb, “Photorefractive surface waves,” Opt. Lett. **20**(20), 2075–2077 (1995) [CrossRef] [PubMed] .

20. M. Cronin-Golomb, “Photorefractive surface waves,” Opt. Lett. **20**(20), 2075–2077 (1995) [CrossRef] [PubMed] .

21. V. Aleshkevich, Y. Kartashov, A. Egorov, and V. Vysloukh, “Stability and formation of localized surface waves at the dielectric photorefractive crystal boundary,” Phys. Rev. E **64**(5), 056610 (2001) [CrossRef] .

22. V. A. Aleshkevich, V. A. Vysloukh, and Y. V. Kartashev, “Optical surface waves at the interface between a linear dielectric and a photorefractive crystal,” Quantum Electron. **30**(10), 905–910 (2000) [CrossRef] .

*b*is equal to a certain threshold value. When

*b*is less than such a certain threshold value, the considerable part of the energy of localized surface waves is concentrated in the linear dielectric and decreases with an increase in

*b*. When

*b*is greater than such a certain threshold value, the part of the energy of localized surface waves concentrated in the nonlinear CP crystals is always higher than that in the linear dielectric and increases with

*b*. The stability properties of these localized surface waves have been investigated numerically and we have found that they are stable.

## 2. Theoretical model

*x*≥ 0 and centrosymmetric photorefractive crystals occupying the half-space

*x*< 0 along the

*z*-axis and is allowed to diffract only along the

*x*direction. Moreover, let us assume that the beam is linearly polarized along the

*x*-axis and that the external bias electric field is applied in the same direction. For illustration purposes, let the centrosymmetric photorefractive crystal be potassium lithium tantalate niobate (KLTN) with its optical

*c*-axis oriented along the

*x*-axis. In this case, the propagation of the optical beam is described by the standard shortened wave equations for the complex amplitude

*A*(

*x*,

*z*) of the light field: where

*n*is the unperturbed refractive index of the CP crystal,

*g*is the effective quadratic electro-optic coefficient,

_{eff}*ε*

_{0}and

*ε*, respectively, are the vacuum and relative dielectric constants,

_{r}*E*is the space-charge field inside the CP crystal,

_{sc}*k*= 2

*πn/λ*is the wave number in the area of the CP crystal,

*λ*is the free-space wavelength of the lightwave used,

*k*

_{0}= 2

*πn*

_{0}/

*λ*is the wave number in the area of the linear dielectric,

*n*

_{0}is the dielectric refractive index. For relatively broad beam configurations, the space-charge field can be obtained from the Kukhtarev-Vinetskii model and is approximately given by [2

2. 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] .

*I*= |

*A*|

^{2}is the intensity of the light beam,

*K*is the Boltzmann constant,

_{B}*T*is the absolute temperature,

*e*is the electron charge, and

*I*is the dark irradiance of the crystal,

_{d}*E*

_{0}is the value of the space-charge field at

*x*→ ±∞. If the spatial extent of the optical wave is much less than the

*x*-width

*l*of the CP crystal, then under a constant voltage bias

*V*,

*E*

_{0}is approximately given by ±

*V/l*. Moreover, for simplicity, let us adopt the following dimensionless coordinates, i.e., let

*s*=

*x/x*

_{0}, where

*x*

_{0}is an arbitrary transverse scale. By employing these latter transformations and by substituting Eq. (2) into Eq. (1b), the complex amplitude of the light field is found to satisfy:

*E*

_{0}= 2×10

^{5}V/m,

*λ*= 0.5μm, and

*x*

_{0}= 9μm. The KLTN parameters [9

9. E. DelRe, B. Crosignani, M. Tamburrini, M. Segev, M. Mitchell, E. Pefaeli, and A. J. Agranat, “One-dimensional steady-state photorefractive spatial solitons in centrosymmetric paraelectric potassium lithium tantalate niobate,” Opt. Lett. **23**(6), 421–423 (1998) [CrossRef] .

*n*= 2.2,

*T*= 21°C, and

*g*= 0.12m4C

_{eff}^{−2}. For this set of values,

*β*= 3.6,

*μ*

_{1}= 0.1, and

*μ*

_{2}= 0.00071.

## 3. Numerical results

*A*can be expressed as

*b*is the real propagation constant and the envelope

*u*(

*s*) is the real function. Substitution of this form of

*A*into Eqs. (3a) and (3b) yields where both

*u*and

*du/ds*should match the continuity conditions at the boundary point

*s*= 0. In the area of nonlinear CP medium, Eq. (4b) cannot be solved analytically and should be integrated numerically, for example, by the shooting method that reduces a two-point boundary problem to the Cauchy problem. The initial conditions are chosen by using the fact that in the area of linear dielectric the solutions of Eq. (4a) can be readily obtained and are given by

*u*=

*m*exp[−(2

*b*)

^{1/2}

*s*], where

*m*is the free parameter describing the strength of the nonlinear effects. By varying parameters

*b*and

*m*, and integrating Eq. (4b), we obtained various profiles of surface waves at the interface between linear dielectric and biased CP crystals.

*u*is equivalent to the particle position and the transverse coordinate

*s*is equivalent to time. In the area of the nonlinear CP crystal (

*s*< 0) Eq. (4b) can be readily written in the following form: where

*U*= (

*p*−

*b*)

*u*

^{2}+

*β*(1 +

*u*

^{2})

^{−1}−

*β*and

*T*= (1/2)(

*du/ds*)

^{2}are the potential and kinetic energies of the particle with unit mass, respectively, and the right-hand side of Eq. (5) shows the nonlinear friction force. Figure 1 depicts typical profiles of the potential well

*U*for different values of

*p*−

*b*when

*β*= 3.6. The potential well is symmetric with respect to the point

*u*= 0. This figure shows only the right part of the potential well corresponding to positive values of

*u*.

*p*−

*b*>

*β*, the potential well has a single stable stationary point

*u*= 0 (see dashed curve in Fig. 1). In this case, a mechanical particle with a nonzero initial energy

*U*+

*T*performs damped oscillations (as

*s*varies from 0 to −∞), passing from the right wing of the potential well to the left wing and losing its energy because of the influence of nonlinear friction. When

*s*→ −∞, this particle asymptotically approaches the stable equilibrium position

*u*= 0. This type of particle motion corresponds to the well-known delocalized surface waves [21

21. V. Aleshkevich, Y. Kartashov, A. Egorov, and V. Vysloukh, “Stability and formation of localized surface waves at the dielectric photorefractive crystal boundary,” Phys. Rev. E **64**(5), 056610 (2001) [CrossRef] .

*m*when

*β*= 3.6,

*p*= 6,

*b*= 1,

*μ*

_{1}= 0.1, and

*μ*

_{2}= 0.00071.

*p*−

*b*<

*β*, the potential well has two stable (

*u*= 0) stationary points, as shown in Fig. 1. A particle with a nonzero initial energy

*U*+

*T*will be periodically transferred from the right wing of the potential well (corresponding to positive

*u*) into the left wing of the well (corresponding to negative

*u*) until it stops at one of the two stable stationary points

*u*= 0 because of the energy loss. The former case corresponds to the shock surface waves with an infinite energy [21

21. V. Aleshkevich, Y. Kartashov, A. Egorov, and V. Vysloukh, “Stability and formation of localized surface waves at the dielectric photorefractive crystal boundary,” Phys. Rev. E **64**(5), 056610 (2001) [CrossRef] .

*s*→ −∞. Figure 3 depicts typical profiles of the shock surface waves of the first three orders. Note that the order of a wave is defined by the number of intersections of its envelope with the

*s*-axis, including the point

*s*= 0. When

*s*→ −∞, shock surface waves represent damped oscillations superimposed on the constant background where height is given by

*u*= 0. This situation corresponds to the formation of localized surface waves without oscillating tails in the volume of the CP crystal. For low

*b*values, Fig. 4(a) depicts the profiles of localized surface waves of the first three orders, whereas Fig. 4(b) depicts the profiles of the second order localized surface waves for three different values of

*b*. The energy

*b*is equal to a certain threshold value and the energy of localized surface waves increases with the order of localized surface waves when

*b*is fixed. Moreover, Figs. 4(a) and 4(b) indicate that the considerable part of the energy of localized surface waves can be concentrated in the linear dielectric and decreases with an increase in

*b*. For high

*b*values, the profiles of localized surface waves of the first three orders are shown in Fig. 6(a), whereas the profiles of the third order localized surface waves for three different values of

*b*are shown in Fig. 6(b). It is clearly seen from Figs. 6(a) and 6(b) that the part of the energy of localized surface waves concentrated in the nonlinear CP crystal is always higher than that in the linear dielectric and increases with

*b*.

*b*= 0.01 and 0.03 in Fig. 4(b). Figures 7(a) and 7(b) depict the evolution of the second order localized surface modes for

*b*= 0.01 and 0.03 in Fig. 4(b) when their input amplitude have been perturbed by 20%. Moreover, consider the third order localized surface modes for

*b*= 0.06 and 1 in Fig. 6(b). Figures 7(c) and 7(d) show the evolution of the third order localized surface modes for

*b*= 0.06 and 1 in Fig. 6(b) when their input amplitude have been perturbed by 20%. The solitary behavior of these localized surface modes is evident in these figures since they propagate unchanged. Similarly, a stability study of other localized surface modes in Figs. 4(b) and 6(b) shows that they are also stable.

*p*−

*b*< 0, the potential well has a single unstable stationary point

*u*= 0 (see dash-dot curve in Fig. 1). In this case, the finite motion of a particle with a nonzero energy

*U*+

*T*is not possible and it is useless to speak about surface waves.

## 4. Conclusion

*b*is fixed, the energy of localized surface waves increases with the order of localized surface waves. We have found that for low

*b*values the considerable part of the energy of localized surface waves is concentrated in the dielectric and decreases with an increase in

*b*and that for high

*b*values the part of the energy of localized surface waves concentrated in the nonlinear CP crystals is always higher than that in the linear dielectric and increases with

*b*. The stability of these localized surface waves have been investigated numerically and it has been found that they are stable.

## Acknowledgment

## References and links

1. | M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. |

2. | D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. Am. B |

3. | M. Segev, G. C. Valley, M. C. Bashaw, M. Taya, and M. M. Fejer, “Photovoltaic spatial solitons,” J. Opt. Soc. Am. B |

4. | H. Wang and W. She, “Incoherently coupled grey photovoltaic spatial soliton families,” Chin. Phys. Lett. |

5. | K. Lu, T. Tang, and Y. Zhang, “One-dimensional steady-state spatial solitons in photovoltaic photorefractive materials with an external applied field,” Phys. Rev. A |

6. | E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. I. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. |

7. | M. Segev and A. J. Agranat, “Spatial solitons in centrosymmetric photorefractive media,” Opt. Lett. |

8. | K. Zhan, C. Hou, and S. Pu, “Temporal behavior of spatial solitons in centrosymmetric photorefractive crystals,” Optics & Laser Technology |

9. | E. DelRe, B. Crosignani, M. Tamburrini, M. Segev, M. Mitchell, E. Pefaeli, and A. J. Agranat, “One-dimensional steady-state photorefractive spatial solitons in centrosymmetric paraelectric potassium lithium tantalate niobate,” Opt. Lett. |

10. | A. Ciattoni, A. Marini, C. Rizza, and E. DelRe, “Collision and fusion of counterpropagating micrometer-sized optical beams in periodically biased photorefractive crystals,” Opt. Lett. |

11. | Y. V. Kartashov, L. Torner, V. A. Vysloukh, and D. Mihalache, “Multipole vector solitons in nonlocal nonlinear media,” Opt. Lett. |

12. | W. Chen, Y. He, and H. Wang, “Surface defect superlattice solitons,” J. Opt. Soc. Am. B |

13. | K. G. Makris, J. Hudock, D. N. Christodoulides, G. I. Stegeman, O. Manela, and M. Segev, “Surface lattice solitons,” Opt. Lett. |

14. | W. Q. Chen, X. Yang, S. Y. Zhong, Z. Yan, T. H. Zhang, J. G. Tian, and J. J. Xu, “Surface dark screening solitons,” Opt. Lett. |

15. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface lattice solitons in diffusive nonlinear media,” Opt. Lett. |

16. | J. Safioui, E. Fazio, F. Devaux, and M. Chauvet, “Surface-wave pyroelectric photorefractive soliton,” Opt. Lett. |

17. | G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Observation of discrete quadratic surface solitons,” Opt. Express. |

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

19. | 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 |

20. | M. Cronin-Golomb, “Photorefractive surface waves,” Opt. Lett. |

21. | V. Aleshkevich, Y. Kartashov, A. Egorov, and V. Vysloukh, “Stability and formation of localized surface waves at the dielectric photorefractive crystal boundary,” Phys. Rev. E |

22. | V. A. Aleshkevich, V. A. Vysloukh, and Y. V. Kartashev, “Optical surface waves at the interface between a linear dielectric and a photorefractive crystal,” Quantum Electron. |

**OCIS Codes**

(190.4350) Nonlinear optics : Nonlinear optics at surfaces

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: April 25, 2013

Revised Manuscript: May 21, 2013

Manuscript Accepted: May 27, 2013

Published: June 21, 2013

**Citation**

Weijun Chen, Keqing Lu, Juanli Hui, Tianrun Feng, Shuqin Liu, Pingjuan Niu, and Liyuan Yu, "Localized surface waves at the interface between linear dielectric and biased centrosymmetric photorefractive crystals," Opt. Express **21**, 15595-15602 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15595

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### References

- M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett.73(24), 3211–3214 (1994). [CrossRef] [PubMed]
- D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. Am. B12(9), 1628–1633 (1995). [CrossRef]
- M. Segev, G. C. Valley, M. C. Bashaw, M. Taya, and M. M. Fejer, “Photovoltaic spatial solitons,” J. Opt. Soc. Am. B14(7), 1772–1781 (1997). [CrossRef]
- H. Wang and W. She, “Incoherently coupled grey photovoltaic spatial soliton families,” Chin. Phys. Lett.22(1), 128–131 (2005). [CrossRef]
- K. Lu, T. Tang, and Y. Zhang, “One-dimensional steady-state spatial solitons in photovoltaic photorefractive materials with an external applied field,” Phys. Rev. A61(5), 053822 (2000). [CrossRef]
- E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. I. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett.85(12), 2193–2195 (2004). [CrossRef]
- M. Segev and A. J. Agranat, “Spatial solitons in centrosymmetric photorefractive media,” Opt. Lett.22(17), 1299–1301 (1997). [CrossRef]
- K. Zhan, C. Hou, and S. Pu, “Temporal behavior of spatial solitons in centrosymmetric photorefractive crystals,” Optics & Laser Technology43(7), 1274–1278 (2011). [CrossRef]
- E. DelRe, B. Crosignani, M. Tamburrini, M. Segev, M. Mitchell, E. Pefaeli, and A. J. Agranat, “One-dimensional steady-state photorefractive spatial solitons in centrosymmetric paraelectric potassium lithium tantalate niobate,” Opt. Lett.23(6), 421–423 (1998). [CrossRef]
- A. Ciattoni, A. Marini, C. Rizza, and E. DelRe, “Collision and fusion of counterpropagating micrometer-sized optical beams in periodically biased photorefractive crystals,” Opt. Lett.34(7), 911–913 (2009). [CrossRef] [PubMed]
- Y. V. Kartashov, L. Torner, V. A. Vysloukh, and D. Mihalache, “Multipole vector solitons in nonlocal nonlinear media,” Opt. Lett.31(10), 1483–1485 (2006). [CrossRef] [PubMed]
- W. Chen, Y. He, and H. Wang, “Surface defect superlattice solitons,” J. Opt. Soc. Am. B24(10), 2584–2588 (2007). [CrossRef]
- K. G. Makris, J. Hudock, D. N. Christodoulides, G. I. Stegeman, O. Manela, and M. Segev, “Surface lattice solitons,” Opt. Lett.31(18), 2774–2776 (2006). [CrossRef] [PubMed]
- W. Q. Chen, X. Yang, S. Y. Zhong, Z. Yan, T. H. Zhang, J. G. Tian, and J. J. Xu, “Surface dark screening solitons,” Opt. Lett.36(19), 3801–3803 (2011). [CrossRef] [PubMed]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface lattice solitons in diffusive nonlinear media,” Opt. Lett.33(8), 773–775 (2008). [CrossRef] [PubMed]
- J. Safioui, E. Fazio, F. Devaux, and M. Chauvet, “Surface-wave pyroelectric photorefractive soliton,” Opt. Lett.35(8), 1254–1256 (2010). [CrossRef] [PubMed]
- G. A. Siviloglou, K. G. Makris, R. Iwanow, R. Schiek, D. N. Christodoulides, G. I. Stegeman, Y. Min, and W. Sohler, “Observation of discrete quadratic surface solitons,” Opt. Express.14(12), 5508–5516 (2006). [CrossRef] [PubMed]
- T. Zhang, X. Ren, B. Wang, C. Lou, Z. Hu, W. Shao, Y. Xu, H. Kang, J. Yang, D. Yang, L. Feng, and 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]
- M. Cronin-Golomb, “Photorefractive surface waves,” Opt. Lett.20(20), 2075–2077 (1995). [CrossRef] [PubMed]
- V. Aleshkevich, Y. Kartashov, A. Egorov, and V. Vysloukh, “Stability and formation of localized surface waves at the dielectric photorefractive crystal boundary,” Phys. Rev. E64(5), 056610 (2001). [CrossRef]
- V. A. Aleshkevich, V. A. Vysloukh, and Y. V. Kartashev, “Optical surface waves at the interface between a linear dielectric and a photorefractive crystal,” Quantum Electron.30(10), 905–910 (2000). [CrossRef]

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