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Стационарные множественные темные пространственные солитоны в фотогальванических средах с замкнутым электрическим контуром


© 2013 г. Y. H. Zhang*; X. H. Hu**; K. Q. Lu**; B. Y. Liu*; W. Y. Liu*; R. L. Guo*
** Xi’an Technological University, Xi’an, China 710032
** State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision ** Mechanics, Chinese Academic of Sciences, Xi’an, China 710119
** Е-mail: zhangyh1979@163.com
We theoretically study the formation of the steady state multiple dark photovoltaic solitons in the closed-circuit photovoltaic photorefractive crystal. The results indicate that the formation of the multiple dark photovoltaic solitons in the closed-circuit photovoltaic crystal is dependent on the initial width of the dark notch at the entrance face of the crystal. The number of the solitons generated increases with the initial width of the dark notch. If the initial width of the dark notch is small, only a fundamental soliton or Y-junction soliton pair is generated. As the initial width of the dark notch is increased, the dark notch tends to split into an odd (or even) number of multiple dark photovoltaic solitons sequence, which realizes a progressive transition from a lower-order soliton to a higher-order solitons sequence. When the multiple solitons are generated, the separations between adjacent dark solitons become slightly smaller. The soliton pairs far away from the center have bigger width and less visibility and they move away from each other as they propagate in the photorefractive nonlinear crystal.

Keywords: photorefractive spatial solitons, photovoltaic effect, multiple solitons splitting, beam propagation method, close-circuit condition.

Codes OCIS: 190.0190.

Submitted 20.12.2012.

Теоретически изучено формирование стационарных множественных темных фотогальванических солитонов в фотогальваническом кристалле с замкнутым электрическим контуром. Показано, что процесс формирования таких солитонов зависит от начальной ширины провала интенсивности на входной грани кристалла. При увеличении ширины провала число солитонов возрастает. Если начальная ширина провала мала, генерируется только фундаментальный солитон или солитонная пара Y-соединения. При возрастании начальной ширины провала проявляется тенденция к его расщеплению на серию нечетного (или четного) числа множественных темных фотогальванических солитонов, в которой реализуется последовательный переход от солитона низкого порядка к набору солитонов более высокого порядка. Когда генерируются множественные солитоны, расстояния между соседними темными солитонами слегка уменьшается. Солитонные пары вдали от центра обладают большей шириной и меньшим контрастом, и они удаляются друг от друга при распространении в фотогальваническом нелинейном кристалле.
Ключевые слова: фотогальванические пространственные солитоны, фотогальванический эффект, множественное расщепление солитонов, метод распространения излучения, условие замкнутого электрического контура.

1. Introduction
Photorefractive spatial solitons [1–4] are self-trapped non-diffracting light beams or dark notches. They are generated when diffraction

is exactly compensated by the nonlinear selffocusing (or self-defocusing) effect in nonlinear photorefractive media. In the past two decades, photorefractive spatial solitons have become a major field of research in nonlinear optics due

“Оптический журнал”, 80, 3, 2013


to their potential applications, such as all-optical beam switching and routing, optical interconnects, and reconfigurable soliton-induced waveguides [5–8]. At present, photorefractive spatial solitons are generally classified into four generic types: quasi-steady-state solitons, screening solitons, photovoltaic solitons and screening-photovoltaic solitons. Photovoltaic solitons are distinct from other photorefractive spatial solitons because they do not necessitate application of an external field but only rely on the photovoltaic effect in the photorefractive crystal. Since Valley [9] et al predicted the existence of the photovoltaic spatial solitons in 1994, the investigation on photovoltaic spatial solitons demonstrated much interest. Dark photovoltaic solitons can be formed in self-defocusing photovoltaic-photorefractive media, in which the refractive index of the medium decreases in the illuminated regions while the refractive index in the region of the dark notch remains unchanged. For the dark photovoltaic soliton, a Y-junction soliton and a fundamental soliton were firstly observed experimentally in steady state regime [10, 11]. Theoretically, the characteristics of the fundamental dark photovoltaic soliton in open-circuit and closed-circuit cases were analyzed [12]. However, there is no available theory to explain the formation of a Y-junction soliton. In addition, the quasi-steady state multi-dark-solitons [13, 14] were observed experimentally in LiNbO3:Fe crystal by increasing the illumination time with a fixed incident power. The phenomena occurred in both the amplitude mask and the phase mask. It is a pity that there was not a related theory to plain the phenomena. Up to 2003, M. Chauvet [15] built up a theoretical model to characterize the time-dependent formation of one-dimensional dark photovoltaic solitons under open-circuit conditions. A year later, based on the theoretical model for time-dependent photovoltaic solitons, G. Couton et al used beam propagation method (BPM) to simulate numerically the evolution of the notch-bearing beam inside photovoltaic crystal [16]. The numerical results demonstrated that the multiple solitons occurred in transient (i.e., quasi-steady state) regime. Here, the profile of the dark notch at the entrance of the crystal was described by hyperbolic tangent function. Meanwhile, they observed experimentally the self-formation of one-dimensional multiple dark-grey photovoltaic solitons in the quasi-steady state regime. In addition, Bodnar [17] reported that he also utilized BPM to simu-

late the propagation of the steady state multiple dark photovoltaic solitons by using three given especial functions describing the dark notch in 2007. The simulation results indicated that an odd (or even) number of dark solitons was formed in odd (or even) initial conditions. However, he hasn’t observed experimentally the multiple dark solitons sequence. Our research team also simulated numerically the evolution of the multiple dark photovoltaic spatial solitons in steady state and quasi-steady state under open-circuit conditions [18, 19]. We differ from them in that the function describing the dark notch is obtained by solving the solitons equation, rather than special function.
In fact, besides the multiple dark solitons observed on the photovoltaic media, the multiple dark solitons have been theoretically predicted and experimentally observed in Kerr nonlinear media [20–23] and biased photorefractive crystal [24–27] by using coherent light source.
All these investigations on multiple spatial solitons in photovoltaic media were performed in open-circuit conditions. In this paper, we aim to investigate theoretically the formation of steady state multiple dark photovoltaic solitons under closed-circuit conditions by using BPM. The simulation results indicate that a single dark notch with a properly input width can split into multiple dark notches in a closed-circuit self-defocusing nonlinear photovoltaic medium. The initial profile of the dark notch at the entrance of the photovoltaic crystal is obtained by solving the nonlinear Schro..dinger equation which describes the propagation of the photorefractive spatial soliton in photorefractive media. Firstly, we solve the nonlinear NSL to obtain a fundamental dark soliton profile by utilizing the fourth order Runge-Kutta algorithm. Then, we utilized the BPM to numerically simulate the evolution of closed-circuit dark photovoltaic solitons. If we expand the width of the dark notch, we find that the dark notch on the otherwise uniform optical beam can split into multiple dark notches, realizing the transformation from a fundamental soliton to an odd-number sequence of multiple dark solitons or from a Y-junction soliton to an even-number sequence of multiple dark solitons. The number of solitons increases with the width of the dark notch. The characteristics of the multiple dark photovoltaic solitons in closedcircuit photovoltaic nonlinear media are similar to that of the multiple dark photovoltaic solitons in open-circuit conditions.

14 “Оптический журнал”, 80, 3, 2013

2. Theoretical treatment

To start, let us assume that a coherent beam
uniformed along the y direction propagates in
a photovoltaic crystal along the z axis and is al-
lowed to diffract only along the x axis. In addi-
tion, we assume that the coherent beam is lin-
early polarized along the x direction. For dem-
onstration purposes, let the photorefractive
crystal be LiNbO3 with its optical c-axis oriented along the x coordinate. Under these conditions,
the perturbed refractive index along the x-axis is given by (ne)2 = ne2 – ne4r33Esc, where ne is the unperturbed extraordinary index of refraction,
r33 is the electro-optic coefficient of the LiNbO3 crystal, Esc is the induced space-charge electric field inside the crystal. Therefore, the refractive
index variation arising from the space charge field is ne = –1/2n3er33Esc. Under close-circuit condition the steady state equation for the 1D
space charge field [12] is given by


Esc Ep


J-I 1+ I

Id Id



where E is normalized space charge field,

I = I(x, z) is the intensity of the optical beam,

Ep is the photovoltaic field сonstant, Id is dark irradiance, J is the normalized current den-

sity inside the crystal. This expression for

the space-charge field is valid for any input

beam intensity distribution when the photore-

fractive effect is governed by the photovoltaic

effect in a crystal in close-circuit conditions.

Theoretically, the solution for a one-dimen-

sional dark photovoltaic spatial soliton propagat-

ing along the z axis is described by the nonlinear Schro..dinger equation

ççççæè ¶¶z


i 2k

¶2 ¶x2






k ne





where k = (2/0)ne is the wave number in the photorefractive crystal, A(x, z) is the slowly

varying amplitude of the input optical field.

The dimensionless parameters  = x/d and  = = z/(kd2) are the scaled transverse and longitudinal coordinates. Where d = ((kne)2r33Ep)–1/2 is the characteristic length scale. Therefore,

propagation Eq. (2) becomes

çèçççæ ¶¶


i 2

¶2 ¶2




i 2

J-r 1+r

2 2




where (, ) is the slowly varying amplitude of the optical field in units of (rId)1/2, r is the ratio of soliton peak intensity Imax to dark irradi-

ance Id. For a fundamental dark soliton, the solution can be written as (, ) = u()exp(ikd2),

where u() is the normalized soliton profile,

which depends only on , and  is the soliton

propagation constant. In this particular case,

Eq. (3) becomes




J-r 1+r

u() u()

2 2



The sign that is to be retained for the second part of Eq. (4) is identical to the sign of the product – r33ph, where ph is photovoltaic constant. A plus sign is selected for the purpose of generating dark soliton. In our calculation, we numerically integrate Eq. (4) and apply boundary conditions of dark soliton u() = 1 and u() = 0 to obtain









+ r







In addition, by applying boundary conditions of dark solitons u() = 1 and u() = 0 in Eq. (4), the soliton propagation constant can be obtained


J-r 2kd2 (1+





In order to yield the spatial profile of the
fundamental dark soliton, Eq. (4) is integrated
numerically by the fourth order Runge–Kutta
algorithm combined with initial conditions of
dark soliton u(0) and u(0) = 0. Here, for shortcircuit case J  lru2/[x(1 + ru2) + l] [12, 28] and for open circuit case J = 0. When l >> x case and r