Photovoltaic solitons in two-photon photorefractive materials under open-circuit conditions

Optics Communications 273 (2007) 544–548 www.elsevier.com/locate/optcom

Photovoltaic solitons in two-photon photorefractive materials under open-circuit conditions Chunfeng Hou *, Yu Zhang, Yongyuan Jiang, Yanbo Pei Department of Physics, Harbin Institute of Technology, Harbin 150001, PR China Received 8 November 2006; received in revised form 22 December 2006; accepted 17 January 2007

Abstract We present the evolution equation of one-dimensional spatial soliton in two-photon photorefractive media under open-circuit conditions. In the steady state regime, our solutions show that the dark and bright photovoltaic spatial solitons can be supported in twophoton photorefractive media under open-circuit conditions. Ó 2007 Elsevier B.V. All rights reserved. PACS: 42.65.Tg; 42.65.Hw; 42.70.a Keywords: Spatial soliton; Photorefractive effect; Photorefractive material

Illuminated by light, photorefractive materials can set up space charge field in their interior, which will induce non-linear changes in the refractive index of the materials by means of the electro-optic (Pockels) effect. The latter process is then capable of counteracting the effects of diffraction, and thus the non-diffracting optical beam or photorefractive soliton forms [1,2]. So far, several different types of photorefractive spatial solitons have been investigated, namely, quasisteady-state solitons [3–5], screening solitons [6–9], photovoltaic solitons [10–13], and screening-photovoltaic solitons [14–17]. The quasi-steady-state solitons can occur in biased photorefractive crystal during the finite time in which the externally applied field is slowly being screened by the space charge field. The screening solitons can exist in steady-state in biased non-photovoltaic photorefractive crystal. They result from non-uniform screening of the bias field. The photovoltaic solitons can be formed in steady-state in photovoltaic photorefractive materials. These photovoltaic solitons rely on the photovoltaic effect to create the space charge field. The screening-photovoltaic solitons are possible in steadystate in biased photovoltaic photorefractive materials, they *

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0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.01.024

stem from both photovoltaic effect and non-uniform screening of the bias field. Moreover, several years ago Vlad et al. [18–22] studied the self-confinement and breathing solitonlike propagation in BSO crystal with strong optical activity. All of the above mentioned photorefractive solitons are result from single-photon photorefractive effect. In 2003, Ramadan et al. [23] demonstrated that it was possible to realize bright spatial solitons at not directly absorbed wavelengths by using two-step excitation in BSO crystal applied with bias field. Later on, in 2006 Vlad et al. [24] created soliton waveguides in lithium niobate crystals with low-power cw green laser and with high-repetition-rate femtosecond laser pulses, at 800 nm, assisted by a green background and an external bias electrical field. Recently, Castro-Camus and Magana [25] provided a new model of the two-photon photorefractive effect. Castro-Camus model includes a valence band (VB), a conduction band (CB), and an intermediate allowed level (IL). The intermediate allowed level is used to maintain a quantity of excited electrons from the valence band by the gating beam. These electrons are then excited again to the conduction band by the signal beam. The pattern of the signal beam can induce a spatial dependent charge distribution that give rise to non-linear changes of refractive index in the medium.

C. Hou et al. / Optics Communications 273 (2007) 544–548

Based on Castro-Camus model, we have just shown that screening solitons can be formed in two-photon photorefractive media [26]. In this letter, the evolution equation of one-dimensional spatial solitons in the photovoltaic photorefractive media with two-photon photorefractive effect is presented by using Castro-Camus model, the solutions predict that dark and bright photovoltaic spatial solitons can also be supported in two-photon photorefractive media in the steady state and under open-circuit conditions. Compared to screening solitons, the external bias field is not required to realize photovoltaic solitons in two-photon photorefractive media, which is convenient for practical applications. The establishment of the photovoltaic solitons in two-photon photorefractive media requires two laser beams, without the gating beam, the signal beam can not evolve into spatial soliton. This aspect of the photovoltaic solitons in two-photon photorefractive media may be useful for alloptical switching and beam steering. Moreover, these two-photon photovoltaic solitons have the advantage of creating permanent waveguides with no erasure problem as a result of re-excitation of the electrons by the signal beam, though the producing process of these two-photon photovoltaic solitons is a bit more complicated than the single-photon photovoltaic solitons. To study the photovoltaic spatial solitons in two-photon photorefractive materials, we consider an optical beam that propagates in a photorefractive medium with two-photon photorefractive effect along the z axis and is permitted to diffract only along the x direction. The photorefractive medium is put with its optical c axis oriented along x coordinate and is illuminated by the gating beam. Moreover, let us assume that the polarization of the incident optical beam is parallel to the c axis. As usual, we express the optical field of the incident beam in terms of slowly varying envelope /, i.e., E ¼ ^x/ðx; zÞ expðikzÞ, where k ¼ k 0 ne ¼ ð2p=k0 Þne , ne is the unperturbed extraordinary index of refraction, and k0 is the free-space wavelength. Under these conditions the optical beam satisfies the following envelope evolution equation [8,26]: i/z þ

1 /  2k xx

k 0 n3e r33 Esc 2

/ ¼ 0;

ð1Þ 2

where /z ¼ o/=oz, /xx ¼ o /=ox , r33 is the electro-optic coefficient, and ~ Esc ¼ Esc^x is the space charge field in the medium. The space charge field in Eq. (1) can be obtained from the set of rate, current, and Poisson’s equations proposed by Castro-Camus and Magana to describe the two-photon photorefractive effect. In the steady-state and under opencircuit conditions these equations are [25] þ

þ

ðs1 I 1 þ b1 ÞðN  N Þ  c1 n1 N  cnN ¼ 0; þ

1 oJ  cnN þ  c2 nðn01  n1 Þ ¼ 0; e ox

oEsc ¼ eðN þ  n  n1  N A Þ; ox on J ¼ elnEsc þ eD þ js2 ðN  N þ ÞI 2 ¼ 0; ox

e0 e

ð4Þ ð5Þ ð6Þ

where N is the donor density, N þ is the ionized donor density, N A is the acceptor or trap density, and n is the density of the electrons in the CB; n1 is the density of the electron in the intermediate; n01 is the density of traps in the intermediate state; s1 and s2 are photoexcitation crosses; b1 and b2 are the thermoionization probability constants for the transitions of VB–IL and IL–CB; c, c1 , and c2 are the recombination factors of the CB–VB, IL–VB, and CB–IL transitions, respectively; D is the diffusion coefficient; j is the photovoltaic constant; l and e are, respectively, the electron mobility and the charge; e0 and e are the vacuum and relative dielectric constants, respectively; J is the current density; I 1 is the intensity of the gating beam, which can be considered as a constant; I 2 is the intensity of the soliton beam; and Esc is the space charge field in the medium. According to Poynting’s theorem, I 2 can be expressed 2 in terms of the envelope /, that is, I 2 ¼ ðne =2g0 Þj/j , where 1=2 g0 ¼ ðl0 =e0 Þ . One can adopt the approximation N þ  N A and neglect the term ðn01  nn Þ  N A with respect to the other terms. In this case, from Eqs. (2) and (3) we obtain n1 ¼

cN A n : s 2 I 2 þ b2

ð7Þ

Substituting Eq. (7) into Eq. (2), we get n¼

ðs1 I 1 þ b1 Þðs2 I 2 þ b2 ÞðN  N A Þ : cN A ðs2 I 2 þ b2 þ c1 N A Þ

ð8Þ

From Eq. (6) we have Esc ¼ 

js2 ðN  N A ÞI 2 D on :  ln ox eln

ð9Þ

The insertion of Eq. (8) into Eq. (9) yields s2 I 2 ðI 2 þ I 2d þ c1 N A =s2 Þ ðs1 I 1 þ b1 ÞðI 2 þ I 2d Þ Dc1 N A oI 2 ;  ls2 ðI 2 þ I 2d þ c1 N A =s2 ÞðI 2 þ I 2d Þ ox

Esc ¼ Ep

2

þ

ðs2 I 2 þ b2 Þn1 þ

545

ðs1 I 1 þ b1 ÞðN  N Þ þ c2 nðn01  n1 Þ  c1 n1 N  ðs2 I 2 þ b2 Þn1 ¼ 0;

ð2Þ þ

ð3Þ

ð10Þ

A where Ep ¼ jcN is the photovoltaic field, and I 2d ¼ b2 =s2 is el the so-called dark irradiance. In photovoltaic materials, the photovoltaic field will be dominant. In this case, the diffusion effect (D term) can be considered small and can be dropped out. Thus, the expression for the space charge field can be simplified as

Esc ¼ Ep

s2 I 2 ðI 2 þ I 2d þ c1 N A =s2 Þ : ðs1 I 1 þ b1 ÞðI 2 þ I 2d Þ

ð11Þ

By substituting expression (11) into Eq. (1), we can establish the envelope evolution equation of the soliton. For convenience, the following dimensionless coordinates and

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C. Hou et al. / Optics Communications 273 (2007) 544–548

variables are adopted, i.e., s ¼ x=x0 , n ¼ z=ðkx20 Þ, 1=2 / ¼ ð2g0 I 2d =ne Þ U , where x0 is an arbitrary spatial width for scaling. Under these conditions, the following dynamical evolution equation can be obtained: 1 jU j2 ðjU j2 þ 1 þ rÞ iU n þ U ss þ ag U ¼ 0; 2 2 jU j þ 1

ð12Þ

that is 1 agrU 2 iU n þ U ss þ agrU þ agjU j U  ¼ 0; 2 2 jU j þ 1

ð13Þ

2

2 where a ¼ ðk 0 x0 Þ ðn4e r33 =2ÞEp , g ¼ s1 I b1 þb , r ¼ c1 N A =s2 I 2d 1 ¼ c1 N A =b2 . Eqs. (12) and (13) are general and they will give rise to dark or bright spatial solitons for different photorefractive materials, which will be discussed infra. The dark solitons exhibit an anti-symmetric field profile (with respect to x), and, moreover, they correspond to dark notches that are embedded in a constant intensity background. It can be assumed that the intensity of the dark soliton beam attains asymptotically a constant value at infinity, that is I 2 ðx ! 1; zÞ ¼ I 21 . To get dark soliton solution, we put U ¼ q1=2 yðsÞ expðiunÞ, where q ¼ I 21 =I 2d , u represents a non-linear shift of the propagation constant, yðsÞ is a normalized odd function of s and satisfies the following boundary conditions: yð0Þ ¼ 0, yðs ! 1Þ ¼ 1, and all the derivatives of yðsÞ vanish at infinity. Substitution of this form of U into Eq. (13) leads to the following equation:

d2 y 2agry : ¼ 2ðu  agrÞy  2agqy 3 þ 2 ds2 qy þ 1

ð14Þ

Using the boundary conditions at infinity, we can deduce from Eq. (14) that   r u ¼ agq 1 þ : ð15Þ qþ1

To illustrate our results, we consider LiNbO3 crystal, the LiNbO3 parameters are taken here to be [14,27] ne ¼ 2:2, r33 ¼ 30  1012 m V1 , Ep ¼ 4  106 V m1 , s1 ¼ 3 104 m2 W1 s1 , s2 ¼ 3  104 m2 W1 s1 , b1 ¼ 0:05 s1 , b2 ¼ 0:05 s1 , c1 ¼ 3:3  1017 m3 s1 , and N A ¼ 1022 m3 . Other parameters are k0 ¼ 0:5 lm, x0 ¼ 10 lm, and I 1 ¼ 1  106 W m2 . Based on the above parameters, we can calculate that a ¼ 22:2, g ¼ 1:67  104 , and r  106 . The dark irradiance I 2d can be modulated artificially by using incoherent uniform illumination [10], so that r can be adjusted. Here, we take r ¼ 104 . Fig. 1 shows the normalized intensity profiles of the dark photovoltaic solitons in two-photon photorefractive medium for q = 1, 10, and 100. The bright solitons can be analyzed in a similar way. We can obtain the bright soliton solutions of Eq. (13) by expressing the normalized envelope U in the form of U ¼ r1=2 yðsÞ expðivnÞ, where v represents a non-linear shift of the propagation constant, r stands for the ratio of the peak value of the intensity of the soliton to the dark irradiance I 2d , yðsÞ is a normalized real function bounded between 0 6 yðsÞ 6 1. Furthermore, since the optical intensity of the bright soliton attains to maximum at the beam center ðs ¼ 0Þ and vanishes at infinity ðs ! 1Þ, yðsÞ is required to satisfy the boundary conditions of yð0Þ ¼ 1, y_ ð0Þ ¼ 0, and yðs ! 1Þ ¼ 0. Substitution of this form of U into Eq. (13) leads to the following equation: d2 y 2agry : ¼ 2ðv  agrÞy  2agry 3 þ 2 ds2 ry þ 1

ð19Þ

Eq. (19) can be integrated and yields  2 dy ¼ 2ðv  agrÞðy 2  1Þ  agrðy 4  1Þ ds   2agr 1 þ ry 2 ln þ : r 1þr

ð20Þ

By integrating Eq. (14) once, we get  2     dy 2r 2 2r 1 þ qy 2 2 2 ðy  1Þ  ln þ qðy  1Þ : ¼ ag ds qþ1 q 1þq 1.0

It can be proved that    2r 2r  2 1 þ qy 2 2 y 1  ln þ qðy 2  1Þ > 0 qþ1 q 1þq

ð17Þ

for 0 < y 2 < 1. So we can know that the dark soliton requires a < 0, i.e., Ep < 0. Further integration of Eq. (16) leads to    Z 0 2r 2r 1 þ q~y 2 1=2 2 ð~y  1Þ  ln ðagÞ s¼ qþ1 q 1þq y 1=2 2 d~y ; ð18Þ þqð~y 2  1Þ from which the normalized envelope y(s) of the dark soliton can be obtained by use of numerical integration procedures.

Normalized Intensity

ð16Þ

ρ =1 ρ =10 ρ =100

0.5

0.0

-1.0

-0.5

0.0

0.5

1.0

s Fig. 1. The normalized intensity profiles of the dark photovoltaic solitons in two-photon photorefractive medium for q = 1, 10, and 100.

C. Hou et al. / Optics Communications 273 (2007) 544–548

According to the boundary conditions at infinity, we can know from Eq. (20) that   lnð1 þ rÞ agr þ : ð21Þ v ¼ agr 1  r 2 The insertion of Eq. (21) into Eq. (20) leads to  2 dy 2agr  lnð1 þ ry 2 Þ  y 2 lnð1 þ rÞ þ agry 2 ð1  y 2 Þ: ¼ ds r ð22Þ 2

2

We can show that lnð1 þ ry Þ  y lnð1 þ rÞ > 0 for 0 < y 2 < 1, and thus we can know that the bright soliton requires a > 0, i.e., Ep > 0. By further integrating Eq. (22) we get

Z 1 1=2 2r  lnð1 þ r~y 2 Þ  ~y 2 lnð1 þ rÞ s¼ ðagÞ r y 1=2 d~y : ð23Þ þr~y 2 ð1  ~y 2 Þ From Eq. (23) we can obtain the normalized bright soliton field profile yðsÞ by numerical integration. Bright spatial solitons are possible if the perturbation of the crystal refractive index is positive. She et al. [13] found that the perturbation of the refractive index in Cu: KNSBN was positive. Moreover, according to Ref. [28], intermediate energy level is included in Cu: KNSBN crystal. So we take Cu: KNSBN crystal as example to illustrate our results, the crystal parameters are taken here to be [13]: ne ¼ 2:27, r33 ¼ 200  1012 m V1 , Ep ¼ 2:8  106 V m1 . Other parameters are k0 ¼ 0:5 lm, and x0 ¼ 10 lm. Using above parameters, we can calculate that a ¼ 117:3. Moreover, we take g ¼ 1:5  104 and r ¼ 104 . Fig. 2 depicts the normalized intensity profiles of the bright photovoltaic solitons in two-photon photorefractive medium for r = 50, 100, and 200. In the low amplitude regime, that is, when jU j2  1, Eqs. (12) or (13) can be simplified as

Normalized Intensity

1.0 r=50 r=100 r=200

ð24Þ

Eq. (24) is the non-linear Schro¨dinger equation, and can be solved analytically. When a < 0 Eq. (24) has dark soliton solution and can be given by n o 1=2 U ðs; nÞ ¼ q1=2 tanh ½agqð1 þ rÞ s exp½iagqð1 þ rÞn; ð25Þ from which the dimensionless intensity full width at half maximum (FWHM) of the dark soliton can be obtained directly: pffiffiffi 2 lnð1 þ 2Þ Dsd ¼ : ð26Þ 1=2 ½agqð1 þ rÞ On the other hand, Eq. (24) also has bright soliton solution which reads n o U ðs; nÞ ¼ r1=2 sech ½agrð1 þ rÞ1=2 s exp½iagrð1 þ rÞn=2; ð27Þ according to which the FWHM can be expressed as pffiffiffi 2 lnð1 þ 2Þ : ð28Þ Dsb ¼ ½agrð1 þ rÞ1=2 In conclusion, we present a theory based on the CastroCamus model that give rise to the evolution equation of one-dimensional photovoltaic spatial solitons in two-photon photorefractive media. In the steady state and under open-circuit conditions, our solutions predict that dark and bright photovoltaic spatial solitons can be supported in two-photon photorefractive media. It is shown that the dark photovoltaic spatial solitons are possible in two-photon photorefractive medium only when the photovoltaic field is set up in the opposite direction with respect to the optical c axis of the medium (i.e., Ep < 0Þ, whereas the bright photovoltaic spatial solitons require the photovoltaic field orients in the same direction with respect to the optical c axis of the medium (that is, Ep > 0Þ. It is found that the FWHM of these spatial solitons is inversely proportional to the square root of the absolute value of the photovoltaic field. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 60508005), and the Scientific Research Foundation of Harbin Institute of Technology (Grant No. HIT. 2003. 31)

0.5

0.0

1 iU n þ U ss þ agð1 þ rÞjU j2 U ¼ 0: 2

547

References -1.0

-0.5

0.0

0.5

1.0

s Fig. 2. The normalized intensity profiles of the bright photovoltaic solitons in two-photon photorefractive medium for r = 50, 100, and 200.

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