## Self-bending of light in photorefractive semiconductors with hot-electron effect |

Optics Express, Vol. 22, Issue 4, pp. 4599-4605 (2014)

http://dx.doi.org/10.1364/OE.22.004599

Acrobat PDF (1153 KB)

### Abstract

This article analyzes nonlinear light propagation in semiconductors with bipolar conductivity and nonlinear transport of electrons. We show how the competition between electron and hole conductivity can influence light propagation, leading to the self-bending effect of optical beam trajectory, which depending on the value of trap compensation coefficient may be stationary or transient.

© 2014 Optical Society of America

## 1. Introduction

1. Z. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. **75**(8), 086401 (2012). [CrossRef] [PubMed]

2. Y. S. Kivshar and G. I. Stegeman, “Spatial optical solitons: Guiding light for future technologies,” Opt. Photonics News **13**(2), 59–63 (2002). [CrossRef]

3. E. DelRe, P. Di Porto, and B. Crosignani, “Photorefractive solitons and their underlying nonlocal physics,” Prog. Opt. **53**, 153–200 (2009). [CrossRef]

4. S. Lan, M. F. Shih, and M. Segev, “Self-trapping of one-dimensional and two-dimensional optical beams and induced waveguides in photorefractive KNbO_{3.},” Opt. Lett. **22**(19), 1467–1469 (1997). [CrossRef] [PubMed]

5. S. Lan, E. Delre, Z. G. Chen, M. F. Shih, and M. Segev, “Directional coupler with soliton-induced waveguides,” Opt. Lett. **24**(7), 475–477 (1999). [CrossRef] [PubMed]

6. K. Pismennaya, O. Kashin, V. Matusevich, A. Kiessling, and R. Kowarschik, “Beam self-trapping and self-bending dynamics in a strontium barium niobate crystal,” J. Opt. Soc. Am. B **25**(2), 136–139 (2008). [CrossRef]

7. J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. **170**(4-6), 291–297 (1999). [CrossRef]

9. D. Wolfersberger, N. Khelfaoui, C. Dan, N. Fressengeas, and H. Leblond, “Fast photorefractive self-focusing in InP:Fe semiconductor at infrared wavelengths,” Appl. Phys. Lett. **92**(2), 021106 (2008). [CrossRef]

10. M. Chauvet, S. A. Hawkins, G. J. Salamo, M. Segev, D. F. Bliss, and G. Bryant, “Self-trapping of planar optical beams by use of the photorefractive effect in InP:Fe,” Opt. Lett. **21**(17), 1333–1335 (1996). [CrossRef] [PubMed]

11. T. Schwartz, Y. Ganor, T. Carmon, R. Uzdin, S. Shwartz, M. Segev, and U. El-Hanany, “Photorefractive Solitons and Light-induced resonance control in semiconductor CdZnTe,” Opt. Lett. **27**(14), 1229–1231 (2002). [CrossRef] [PubMed]

## 2. Photorefractive hot-electron effect

*T*[12,13]:where

_{L}*q*is an electron charge,

*k*is Boltzmann's constant,

_{B}*v*(

_{n}*E*) is a drift velocity of electrons, that depends on the electric field

*E*,

*τ*is mean time of energy relaxation ranging from 0.1 ps to 1 ps [12,14

_{r}14. D. D. Nolte, S. Balasubramanian, and M. R. Melloch, “Nonlinear charge transport in photorefractive semiconductor quantum wells,” Opt. Mater. **18**(1), 199–203 (2001). [CrossRef]

*μ*and

_{nl}*μ*are, respectively, carrier mobilities in the central and side valleys,

_{nu}*f*(

*E*) is a distribution function describing the population of central valley, expressed by this relation:where Δ

*U*is the energy difference between the central and side minimum,

*R*is a ratio of state densities in the central and side valleys. For gallium arsenide

*R*= 94, Δ

*U*= 0.31 eV. This mechanism can be embedded in a traditional model of the photorefraction of semiconductors, describing the photogeneration, transport and trapping of the carriers. When traps are deep acceptors with a concentration

*N*compensating shallow donors with a concentration

_{A}*N*, we get this system of equations [15

_{D}15. Q. N. Wang, R. M. Brubaker, and D. D. Nolte, “Photorefractive phase shift induced by hot-electron transport: multiple-quantum-well structures,” J. Opt. Soc. Am. B **11**(9), 1773–1779 (1994). [CrossRef]

*n*is a concentration of electrons,

*p*is a concentration of holes,

*β*is the coefficient of thermal generation of carriers from deep traps,

*γ*is the coefficient of the recombination of carriers with traps,

*S*is the photoionization cross section divided by photon energy,

*J*is current density,

*I*is light intensity distribution. The indices

*n*and

*p*refer to quantities characterizing, respectively, electrons and holes.

**A**is an amplitude of optical wave propagating along axis

*z*,

14. D. D. Nolte, S. Balasubramanian, and M. R. Melloch, “Nonlinear charge transport in photorefractive semiconductor quantum wells,” Opt. Mater. **18**(1), 199–203 (2001). [CrossRef]

20. S. Balasubramanian, I. Lahiri, Y. Ding, M. Melloch, and D. Nolte, “Two-wave-mixing dynamics and nonlinear hot-electron transport in transverse-geometry photorefractive quantum wells studied by moving gratings,” Appl. Phys. B **68**(5), 863–869 (1999). [CrossRef]

*x*direction and propagating in the

*z*direction). Furthermore, I have selected an acceptor model, but note that the ratio of electron concentration to hole concentration is essential, causing the qualitative change of the obtained results. The ratio can be controlled by changing the trap compensation coefficient in acceptor and donor models alike.

## 3. Self-bending of light in hot-electron photorefractives

*r*

_{41}= 1.43x10

^{−12}m/V for

*λ*= 1.06μm). In this case changes in refractive index required for the compensation of diffraction are available in two ranges: high values of the external electric field and relatively wide optical beams. Another possibility arises in an intermediate case, at a boundary region of the two experimental cases. If a beam with Gaussian distribution of intensity and a radius

*w*= 14.5 μm (FWHM ≈24 μm) were to propagate in conditions in which a soliton with similar dimensions propagates, then the applied electric field should have an approximate value 35 kV/cm. Let us see which phenomena will accompany the propagation of such beam in semi-insulating gallium arsenide described by Eqs. (1)–(4) in conditions allowing for its trapping.

*r*=

*N*/

_{D}*N*= 0.88 (

_{A}*N*= 4.4 x 10

_{D}^{16}cm

^{−3},

*N*= 5 x 10

_{A}^{16}cm

^{−3}). For homogeneous illumination

*I*= 12 W/cm

_{B}^{2}, with the assumed parameters, the electron concentration to hole concentration ratio is

*n*

_{0}/

*p*

_{0}≈2.06.

*I*=

_{M}*I*. Although the numerical approach used allows to observe all variables in two dimensions, for clarity, in the case of space-charge field the response of the medium in one dimension is shown (Media 2 in Fig. 1(b)). In this case, the induced space-charge field and consequent changes in refractive index have an asymmetric distribution. Observed from the start of temporal evolution, the asymmetry remains in a stationary state, inducing self-bending of optical beam trajectory. The standard, approximate approach used in an analysis of transient states occurring during nonlinear propagation in photorefractive materials assumes two things: that the distribution of carrier concentration settles much faster than the distribution of space-charge field, and that this distribution is a linear function of light intensity [23

_{B}23. N. Fressengeas, J. Maufoy, and G. Kugel, “Temporal behavior of bidimensional photorefractive bright spatial solitons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **54**(6), 6866–6875 (1996). [CrossRef] [PubMed]

*t*= 0.1 μs). In case of electrons, a strong nonlocality relative to light intensity distribution is visible, caused by high drift length. As neither electron concentration nor hole concentration is dominant in the examined case, both types of carrier affect the propagation.

*n*

_{0}/

*p*

_{0}is sufficiently low, the effect caused by electrons does not influence the electric field distribution, and propagation is rectilinear, as illustrated by the trajectory V in Fig. 3(a) for

*r*= 0.4 (

*n*

_{0}/

*p*

_{0}≈0.017).

*r*= 0.88. In this case, however, the simulation has been made with an assumption that the electron transport is linear. In this situation one can observe self-bending of beam trajectory which occurs in transient states, but not in a stationary state. In this case, concentrations of carriers, despite a drift at the onset of time evolution, at a stationary state take the distributions that are approximately a linear function of light intensity distribution [17

17. A. Ziółkowski, “Temporal analysis of solitons in photorefractive semiconductors,” J. Opt. **14**(3), 035202 (2012). [CrossRef]

*r*= 0.4.

## 4. Conclusions

20. S. Balasubramanian, I. Lahiri, Y. Ding, M. Melloch, and D. Nolte, “Two-wave-mixing dynamics and nonlinear hot-electron transport in transverse-geometry photorefractive quantum wells studied by moving gratings,” Appl. Phys. B **68**(5), 863–869 (1999). [CrossRef]

24. E. DelRe, A. D’Ercole, and E. Palange, “Mechanisms supporting long propagation regimes of photorefractive solitons,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. **71**(3), 036610 (2005). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | Z. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. |

2. | Y. S. Kivshar and G. I. Stegeman, “Spatial optical solitons: Guiding light for future technologies,” Opt. Photonics News |

3. | E. DelRe, P. Di Porto, and B. Crosignani, “Photorefractive solitons and their underlying nonlocal physics,” Prog. Opt. |

4. | S. Lan, M. F. Shih, and M. Segev, “Self-trapping of one-dimensional and two-dimensional optical beams and induced waveguides in photorefractive KNbO |

5. | S. Lan, E. Delre, Z. G. Chen, M. F. Shih, and M. Segev, “Directional coupler with soliton-induced waveguides,” Opt. Lett. |

6. | K. Pismennaya, O. Kashin, V. Matusevich, A. Kiessling, and R. Kowarschik, “Beam self-trapping and self-bending dynamics in a strontium barium niobate crystal,” J. Opt. Soc. Am. B |

7. | J. Petter, C. Weilnau, C. Denz, A. Stepken, and F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. |

8. | P. Günter and J. P. Huignard, |

9. | D. Wolfersberger, N. Khelfaoui, C. Dan, N. Fressengeas, and H. Leblond, “Fast photorefractive self-focusing in InP:Fe semiconductor at infrared wavelengths,” Appl. Phys. Lett. |

10. | M. Chauvet, S. A. Hawkins, G. J. Salamo, M. Segev, D. F. Bliss, and G. Bryant, “Self-trapping of planar optical beams by use of the photorefractive effect in InP:Fe,” Opt. Lett. |

11. | T. Schwartz, Y. Ganor, T. Carmon, R. Uzdin, S. Shwartz, M. Segev, and U. El-Hanany, “Photorefractive Solitons and Light-induced resonance control in semiconductor CdZnTe,” Opt. Lett. |

12. | K. Seeger, |

13. | S. M. Sze, |

14. | D. D. Nolte, S. Balasubramanian, and M. R. Melloch, “Nonlinear charge transport in photorefractive semiconductor quantum wells,” Opt. Mater. |

15. | Q. N. Wang, R. M. Brubaker, and D. D. Nolte, “Photorefractive phase shift induced by hot-electron transport: multiple-quantum-well structures,” J. Opt. Soc. Am. B |

16. | M. Wichtowski and E. Weinert-Rączka, “Temporal response of photorefractive multiple quantum wells in Franz–Keldysh geometry,” Opt. Commun. |

17. | A. Ziółkowski, “Temporal analysis of solitons in photorefractive semiconductors,” J. Opt. |

18. | A. Ziółkowski, “A numerical approach to nonlinear propagation of light in photorefractive media,” Comput. Phys. Commun. |

19. | D. D. Nolte, |

20. | S. Balasubramanian, I. Lahiri, Y. Ding, M. Melloch, and D. Nolte, “Two-wave-mixing dynamics and nonlinear hot-electron transport in transverse-geometry photorefractive quantum wells studied by moving gratings,” Appl. Phys. B |

21. | C. Dari-Salisburgo, E. DelRe, and E. Palange, “Molding and stretched evolution of optical solitons in cumulative nonlinearities,” Phys. Rev. Lett. |

22. | E. DelRe and E. Palange, “Optical nonlinearity and existence conditions for quasi-steady-state photorefractive solitons,” J. Opt. Soc. Am. B |

23. | N. Fressengeas, J. Maufoy, and G. Kugel, “Temporal behavior of bidimensional photorefractive bright spatial solitons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

24. | E. DelRe, A. D’Ercole, and E. Palange, “Mechanisms supporting long propagation regimes of photorefractive solitons,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. |

**OCIS Codes**

(190.5330) Nonlinear optics : Photorefractive optics

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: November 25, 2013

Revised Manuscript: January 17, 2014

Manuscript Accepted: January 17, 2014

Published: February 20, 2014

**Citation**

Andrzej Ziółkowski, "Self-bending of light in photorefractive semiconductors with hot-electron effect," Opt. Express **22**, 4599-4605 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-4599

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### References

- Z. Chen, M. Segev, D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75(8), 086401 (2012). [CrossRef] [PubMed]
- Y. S. Kivshar, G. I. Stegeman, “Spatial optical solitons: Guiding light for future technologies,” Opt. Photonics News 13(2), 59–63 (2002). [CrossRef]
- E. DelRe, P. Di Porto, B. Crosignani, “Photorefractive solitons and their underlying nonlocal physics,” Prog. Opt. 53, 153–200 (2009). [CrossRef]
- S. Lan, M. F. Shih, M. Segev, “Self-trapping of one-dimensional and two-dimensional optical beams and induced waveguides in photorefractive KNbO3.,” Opt. Lett. 22(19), 1467–1469 (1997). [CrossRef] [PubMed]
- S. Lan, E. Delre, Z. G. Chen, M. F. Shih, M. Segev, “Directional coupler with soliton-induced waveguides,” Opt. Lett. 24(7), 475–477 (1999). [CrossRef] [PubMed]
- K. Pismennaya, O. Kashin, V. Matusevich, A. Kiessling, R. Kowarschik, “Beam self-trapping and self-bending dynamics in a strontium barium niobate crystal,” J. Opt. Soc. Am. B 25(2), 136–139 (2008). [CrossRef]
- J. Petter, C. Weilnau, C. Denz, A. Stepken, F. Kaiser, “Self-bending of photorefractive solitons,” Opt. Commun. 170(4-6), 291–297 (1999). [CrossRef]
- P. Günter and J. P. Huignard, Photorefractive Materials and Their Applications (Springer, 2007), Vol. III, Chap. 11.
- D. Wolfersberger, N. Khelfaoui, C. Dan, N. Fressengeas, H. Leblond, “Fast photorefractive self-focusing in InP:Fe semiconductor at infrared wavelengths,” Appl. Phys. Lett. 92(2), 021106 (2008). [CrossRef]
- M. Chauvet, S. A. Hawkins, G. J. Salamo, M. Segev, D. F. Bliss, G. Bryant, “Self-trapping of planar optical beams by use of the photorefractive effect in InP:Fe,” Opt. Lett. 21(17), 1333–1335 (1996). [CrossRef] [PubMed]
- T. Schwartz, Y. Ganor, T. Carmon, R. Uzdin, S. Shwartz, M. Segev, U. El-Hanany, “Photorefractive Solitons and Light-induced resonance control in semiconductor CdZnTe,” Opt. Lett. 27(14), 1229–1231 (2002). [CrossRef] [PubMed]
- K. Seeger, Semiconductor Physics (Springer, 2004), Chap. 4.
- S. M. Sze, Physics of Semiconductor Devices (Wiley-Interscience, 2006), Chap. 11.
- D. D. Nolte, S. Balasubramanian, M. R. Melloch, “Nonlinear charge transport in photorefractive semiconductor quantum wells,” Opt. Mater. 18(1), 199–203 (2001). [CrossRef]
- Q. N. Wang, R. M. Brubaker, D. D. Nolte, “Photorefractive phase shift induced by hot-electron transport: multiple-quantum-well structures,” J. Opt. Soc. Am. B 11(9), 1773–1779 (1994). [CrossRef]
- M. Wichtowski, E. Weinert-Rączka, “Temporal response of photorefractive multiple quantum wells in Franz–Keldysh geometry,” Opt. Commun. 281(5), 1233–1243 (2008). [CrossRef]
- A. Ziółkowski, “Temporal analysis of solitons in photorefractive semiconductors,” J. Opt. 14(3), 035202 (2012). [CrossRef]
- A. Ziółkowski, “A numerical approach to nonlinear propagation of light in photorefractive media,” Comput. Phys. Commun. 185(2), 504–511 (2014). [CrossRef]
- D. D. Nolte, Photorefractive Effects and Materials (Kluwer, 1995).
- S. Balasubramanian, I. Lahiri, Y. Ding, M. Melloch, D. Nolte, “Two-wave-mixing dynamics and nonlinear hot-electron transport in transverse-geometry photorefractive quantum wells studied by moving gratings,” Appl. Phys. B 68(5), 863–869 (1999). [CrossRef]
- C. Dari-Salisburgo, E. DelRe, E. Palange, “Molding and stretched evolution of optical solitons in cumulative nonlinearities,” Phys. Rev. Lett. 91(26), 263903 (2003). [CrossRef] [PubMed]
- E. DelRe, E. Palange, “Optical nonlinearity and existence conditions for quasi-steady-state photorefractive solitons,” J. Opt. Soc. Am. B 23(11), 2323–2327 (2006). [CrossRef]
- N. Fressengeas, J. Maufoy, G. Kugel, “Temporal behavior of bidimensional photorefractive bright spatial solitons,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(6), 6866–6875 (1996). [CrossRef] [PubMed]
- E. DelRe, A. D’Ercole, E. Palange, “Mechanisms supporting long propagation regimes of photorefractive solitons,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 71(3), 036610 (2005). [CrossRef] [PubMed]

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