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Gray spatial solitons in biased two-photon photovoltaic photorefractive crystals Guangyong Zhang a,*, Yongjin Cheng a, Zhongjie Luo a, Tao Lv a,b, Qiujiao Du a,b a b

Department of Physics, China University of Geosciences, Wuhan 430074, PR China Wuhan National Laboratory for Optoelectronics, School of Optoelectronic Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China

a r t i c l e

i n f o

Article history: Received 16 July 2009 Received in revised form 29 September 2009 Accepted 29 September 2009

PACS: 42.65.Jx 42.65.Hw 42.65.Tg

a b s t r a c t This paper predicts that gray spatial solitons can exist in biased two-photon photovoltaic photorefractive crystals. Under appropriate conditions and in the steady state, the gray spatial solitons solution of the optical evolution equation is obtained. The properties associated with these solitons, such as their intensity proﬁle, intensity full width at half-maximum, width, transverse velocity and phase distribution, are discussed as functions of their normalized intensity and degree of ‘‘grayness”. Relevant examples are provided. Ó 2009 Elsevier B.V. All rights reserved.

Keywords: Nonlinear optics Spatial optical solitons Photorefractive effects

1. Introduction Since their theoretical prediction and ﬁrst experimental observation [1,2], photorefractive (PR) spatial optical solitons have attracted much special interest, for these solitons can be formed at low light intensity and in two dimensions, and are potentially useful for all-optical switching, beam steering, and optical interconnects [3,4]. Most often, the photorefractive nonlinearity responsible for the self-trapping of solitary beams relies on the application of the external electric ﬁeld, the photovoltaic ﬁeld of the photorefractive materials, or both of them. Thus, screening solitons have been investigated in biased PR crystals due to the nonuniform screening of the bias ﬁeld [5–9], photovoltaic solitons have been investigated in PR crystals resulting from the photovoltaic effect of the crystals [10–12], and screening-photovoltaic solitons have also been proved in biased photovoltaic PR crystals owning their existence to both photovoltaic effect and spatially non-uniform screening of the applied ﬁeld [13,14]. Moreover, it has been proved that holographic solitons [15,16], counter-propagating solitons [17], elliptical solitons [18], discrete solitons in waveguide arrays [19], and solitons in anisotropic media [20] could also form in biased or unbiased PR crystals.

* Corresponding author. E-mail address: [email protected] (G. Zhang). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.09.078

In 2003, for the ﬁrst time, Ramadan et al. observed the self-conﬁnement of light beams at 633 nm via two-step absorption processes [21]. Castro-Camus and Magana provided a theoretical model to describe the two-photon PR effect [22]. 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 gives rise to nonlinear changes of refractive index in the medium. Based on this model, several groups have investigated spatial solitons due to two-photon PR effect in biased or unbiased crystal [23–26]. Their results not only show the existence of dark and bright spatial solitons, but gray solitons and soliton pair as well. Very recently, we had just predicted that spatial solitons are possible in biased two-photon photovoltaic PR crystals [27]. Inasmuch as these spatial optical solitons result from both the spatially non-uniform screening of the bias electric ﬁeld and the photovoltaic effect, we termed these solitons two-photon screening-photovoltaic (TPSP) solitons. In this paper, we demonstrate the existence of gray TPSP solitons in biased two-photon photovoltaic PR crystal. In the steady state and under appropriate conditions, the gray solitons solution of the optical wave evolution equation is obtained. The properties associated with these solitons, such as their intensity proﬁle, inten-

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G. Zhang et al. / Optics Communications 283 (2010) 335–339

sity full width at half-maximum (FWHM), transverse velocity and phase distribution are discussed in detail.

1 ðq þ 1ÞðjUj2 þ 1 þ rÞ ðg q jUj2 ÞðjUj2 þ 1 þ rÞ U as U ¼ 0: iU n þ U ss gb 2 2 ðjUj þ 1Þðq þ 1 þ rÞ jUj2 þ 1 ð3Þ 2

2. Dynamical evolution equation of TPSP solitons We start our analysis by considering an optical beam propagating in a biased two-photon photovoltaic PR crystal along the z axis. The crystal, an external electric ﬁeld with voltage bias Va, and a resistor are connected in a chain by electrode leads. The beam is permitted to diffract only along the x direction. The crystal is put with its optical axis along the x coordinate and is illuminated by the gating beam. Moreover, let us assume that the incident optical beam is linearly polarized along the x direction, and the external bias ﬁeld is applied in the same direction. As usual, we express the optical ﬁeld of the incident beam in terms of the slowly varying envelope /, i.e. E ¼ ^ x/ðx; zÞ expðikzÞ, where k = k0ne, k0 ¼ 2p=k0 , ne is the unperturbed extraordinary index of refraction and k0 is the free-space wavelength of the light wave employed. Under these conditions the optical beam satisﬁes the following envelope evolution equation [6,27].

i/z þ

1 k0 / ðn3 r 33 Esc Þ/ ¼ 0; 2k xx 2 e

ð1Þ

where /z = o//oz, /xx = o2//ox2, r33 is the electro-optic coefﬁcient of the TPPR crystal, and Esc is the induced space-charge ﬁeld. 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 [22], which describes the two-photon PR effect in a medium in which the photovoltaic current is nonzero. In steady state, by neglecting the diffusion and losses effects, the spacecharge ﬁeld Esc can be obtained as follows [27].

Esc ¼ gEa

ðI21 þ I2d ÞðI2 þ I2d þ c1 NA =s2 Þ þ EP ðI2 þ I2d ÞðI21 þ I2d þ c1 NA =s2 Þ

s2 ðgI21 I2 ÞðI2 þ I2d þ c1 NA =s2 Þ ; ðs1 I1 þ b1 ÞðI2 þ I2d Þ

ð2Þ

where g = 1/(1 + p), p ¼ el l and e are the electron mobility and the charge, n1 = n(x ? ±1) is the electron density in the regions x ? ±1; D is the resistance, W is the x width of the PR crystal and S is the surface area of the crystal’s electrodes. From the expression of g we can see that it is a positive parameter associated with the resistance and is bounded between 0 6 g 6 1. For the case of g = 1, which corresponding to the case of D = 0, which implies that Ea can be totally applied to the crystal. For the case of g = 0, which corresponding to the open-circuit condition with D ? 1, and no bias ﬁeld is applied to the crystal in this case; Ea = Va/W, Va is the external bias voltage, when the spatial extent of the soliton beam is much narrower than the width W of the PR crystal, this expression can hold [6]; EP = jcNA/el is the photovoltaic ﬁeld, NA is the acceptor or trap density, j is the photovoltaic constant; c and c1 are the recombination factors of the CB–VB and IL–VB transitions, respectively; s1 and s2 are photoexcitation crosses; I2d = b2/s2 is the dark irradiance intensity; b1 and b2 are the thermoionization probability constants for the transitions of VB–IL and IL–CB; I1 denotes the intensity of the gating beam, which can be considered as a constant; I21 = I2(x ? ±1, z), I2 denotes the intensity of the soliton beam. According to Poynting’s theorem, I2 can be expressed as I2 = (ne/2g0)|/|2, where g0 = (l0/e0)1/2. By substituting Eq. (2) into Eq. (1), we can establish the envelope evolution equation of TPSP solitons. For convenience, we adopt the following dimensionless coordinates and variables, i.e. 2 n ¼ z=ðkx0 Þ, s = x/x0 and / = (2g0I2d/ne)1/2U. Where x0 is an arbitrary spatial width. Under these conditions, the normalized envelope U satisﬁes the following dynamical evolution equation [27]: n1 SD , W

2

Where Un = oU/on, Uss = o U/os , q = I21/I2d is the intensity ratio of the soliton beam at g ? ±1 with respect to I2d, a ¼ ðk0 x0 Þ2 ðn4e r 33 =2ÞEp , b ¼ ðk0 x0 Þ2 ðn4e r 33 =2ÞEa , s = b2/(s1I1 + b1) and r = c1NA/ s2I2d = c1NA/b2. Eq. (3) represents the normalized dynamical evolution equation of TPSP solitons. As we can see, TPSP solitons result from both the spatially non-uniform screening of the applied ﬁeld (Ea or b) and the photovoltaic effect (Ep or a), they differ from both screening solitons in a biased non-photovoltaic two-photon PR crystal [23] and photovoltaic solitons in a two-photon PR photovoltaic crystal without an external bias ﬁeld [26]. In fact, Eq. (3) can be used to describe the evolution of two-photon screening solitons or two-photon photovoltaic solitons under appropriate conditions. 3. Solution of the gray TPSP solitons Now we look for the gray spatial solitons solution of Eq. (3), in doing so, we introduce the moving coordinates g = s tn, f = n, and substitute the transformation U(s, n) = A(g, f) exp(itg) exp(it2f/2) into Eq. (3), we can ﬁnd that the new envelope A(g, f) satisﬁes the same evolution equation of Eq. (3), i.e. 1 gbðq þ 1ÞðjAj2 þ 1 þ rÞ asðg q jAj2 ÞðjAj2 þ 1 þ rÞ A A ¼ 0: iAf þ Agg 2 2 2 ðjAj þ 1Þðq þ 1 þ rÞ jAj þ 1 ð4Þ 2

2

Where Af = oA/of, Agg = o A/og . In the new moving-coordinate system t represents the normalized transverse velocity of the gray TPSP solitons. Eq. (4) is just the Galilean invariance of Eq. (3), and the solution of Eq. (4) automatically satisﬁes Eq. (3) and vice versa. According to [8,9], the gray TPSP spatial solitons solution of Eq. (4) can be expressed as

Z A ¼ q1=2 yðgÞ exp ilf iJ 0

g

^ dg

g

y2 ð ^ Þ

þ iU0 ;

ð5Þ

where J is a real constant to be determined, U0 is an arbitrary phase, y(g) is a normalized real function bounded between jyðgÞj 6 1, y(g) satisﬁes the boundary conditions: y2(0) = m, y0 (0) = 0, y(g ? ±1) = 1 and all the derivatives of y(g) are also zero at inﬁnity. Note that the parameter m (0 < m < 1) describes the soliton grayness, i.e., the intensity at the beam centre is I2(0) = mI21, and also that m = 0 corresponds to a dark soliton. By employing the condition J = t, we obtain

Z g ~ t2 dg Uðs; nÞ ¼ q1=2 yðgÞ exp i l þ n þ it g þ iU0 : 2 ~Þ 2 0 y ðg ð6Þ We put the condition J = t so that the phase of the gray spatial solitons is constant when g or s ? ±1, which is consistent with the excitation conditions right at the origin [8]. By substituting Eq. (6) into Eq. (3) we ﬁnd that the normalized intensity proﬁle y(g) satisﬁes the following ordinary differential equation:

2 d y t2 gbð1 þ qÞ r y 2asqðg y2 Þ 2 l y 2 1 þ dg2 1þqþr y3 1 þ qy2 ð1 þ

r 1 þ qy2

Þy ¼ 0:

ð7Þ

According to the boundary conditions of y(g) at inﬁnity, we arrive at

t2 ¼ 2l 2gb 2as½g q q r þ rðg q þ 1Þ=ð1 þ qÞ:

ð8Þ

G. Zhang et al. / Optics Communications 283 (2010) 335–339

From Eqs. (7) and (8) we can see that the normalized intensity proﬁle and the transverse velocity of TPSP solitons have a relation with both parameters b and a. Integrating Eq. (7) once, and using the boundary conditions of y(g) at zero, we can get

l ¼ gbvð1 þ q þ rQ 1 Þ as½g q q r þ Q 2 þ rðg q þ 1Þ=ð1 þ qÞ; ð9Þ

2 dy ¼ 2½l þ gbvð1 þ qÞ þ asðg q rÞðy2 1Þ dg 1 2 asqðy4 1Þ t2 2 1 þ ½gbvrð1 þ qÞ y q þ asrð1 þ g qÞ ln

1 þ qy2 ; 1þq

ð10Þ

where v = 1/(1 + q + r), Q1 = 1/(1 m) + mx ln[(1 + mq)/(1 + q)]/ d ln½ð1þ (1 m)2, x = (1 + q)/q, Q 2 ¼ mq=2 þ md=ð1 mÞ þ m^ d ¼ rð1 þ g qÞ=q. mqÞ=ð1 þ qÞ=ð1 mÞ2 , d = r(1 + gq)/(1 + q) and ^ Again, Eq. (10) shows the relation between the normalized proﬁle of TPSP solitons and the parameters b and a. Substituting Eq. (9) into Eq. (8), we can obtain

t2 ¼ 2gbvrðQ 1 1Þ þ 2asQ 2 :

ð11Þ

By expanding ln[(1 + mq)/(1 + q)] = ln[1 q(1 m)/(1 + q)] into series, we can prove that Q1 and Q2 are negative. Therefore, by appropriately choosing the external bias ﬁeld and the photovoltaic ﬁeld so that t2 > 0. In this case, the gray TPSP solitons can exist in a biased twophoton photovoltaic PR crystal. Substituting Eqs. (9) and (11) into Eq. (10), integrating once again, we can get the normalized intensity proﬁle of the gray TPSP solitons satisﬁes the following equation

337

solutions and, therefore, the intensity FWHM of TPSP solitons must be obtained numerically by integrating this equation. In order to illustrate our prediction, we take the following parameters [26,27]: ne = 2.2, r33 = 30 1012 m V1, Ep = 4 106 V m1, Ea = 2.0 106 V m1, s1 = s2 = 3 104 m2 W1 s1, b1 = b2 = 0.05 s1, c1 = 3.3 1017 m3 s1 and NA = 1022 m3. Other parameters are k0 ¼ 0:5 lm, x0 = 10 lm, g = 1 and I1 = 1 106 Wm2. Using above parameters, we can calculate that a = 22.2, b = 11.1, s = 1.67 104 and r 106. Because the dark irradiance I2d can be modulate artiﬁcially by using incoherent uniform illumination [10], so the value of r can be adjusted. Here, we take r = 104. Fig. 1a shows the normalized intensity proﬁle of the gray TPSP solitons as a function of g (or s) for different values of the grayness parameter m when q = 10. It is obvious from Fig. 1a that for a given q the spatial soliton becomes narrower as m decreases. From Eq. (13) we can see that the solitons width is a function of the intensity ration q and the grayness parameter m. The intensity FWHM of the gray TPSP solitons is depicted in Fig. 1b as a function of q for different values of the grayness parameter m. Fig. 1b clearly indicates that the soliton width generally decreases with q. At very low values of q, the FWHM decreases very rapidly, whereas it saturates for q P 10. From Fig. 1b, we also can see that the lower degree of grayness of the soliton is, the narrower width of the soliton is. The dimensionless transverse velocity of the gray TPSP solitons is depicted in Fig. 2a as a function of q for various values of the grayness parameter m. As one can see, the transverse velocity gen-

~2 1 1 1 þ qy 2 ~ 2gb r v Q þ 2 þ 1 x ln y 1 pﬃﬃﬃ ~2 ~2 y y 1þq m q ~4 1 ~ 2 Þ þ ðy 2as ðq Q 2 dÞð1 y 1Þ þ Q 2 2 1 ~ y 2 ~2 1=2 1 þ qy ^: ^d ln dy ð12Þ 1þq

g¼

Z

y

In the right hand of Eq. (12), the ﬁrst tern associated with b results from the spatially non-uniform of the applied ﬁeld and, the second term associated with a result from the photovoltaic effect. Thus, in biased two-photon photovoltaic PR crystals, given a set of parameters a, b, g and, the grayness m of this gray TPSP soliton as well as its maximum intensity ratio q, one can the uniquely determine the parameter l from Eq. (9) and, in turn, its transverse velocity t from Eq. (11). From here one can obtain the normalized intensity proﬁle y(g) of TPSP solitons by numerically integrating Eq. (12). From Eq. (12) we can directly obtain the intensity FWHM of the gray TPSP solitons in based two-photon photovoltaic PR crystal as follow

Z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1þmÞ=2 1 1 ~2 þ 2 þ 1 Dg ¼ 2 pﬃﬃﬃ 2gbrv Q 1 y ~2 ~2 y y m h 2 ~ 1 þ qy q ~4 ~ 2 Þ þ ðy 1Þ x ln 2as ðq Q 2 dÞð1 y 1þq 2 ~2 1=2 1 1 þ qy ^; dy ð13Þ þQ 2 2 1 ^d ln ~ y 1þq where Dg is deﬁned as the spatial FWHM between the background and the lowest value of the intensity, i.e., the points where y2(g) = (1 + m)/2. Just as Eq. (12), Eq. (13) also does not closed-form

Fig. 1. (a) The normalized intensity proﬁle of the gray TPSP solitons for various values of the grayness parameter m when a = 22.2, b = 11.1 and q = 10. (b) The FWHM of the gray TPSP soliton as a function of q.

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G. Zhang et al. / Optics Communications 283 (2010) 335–339

Fig. 2. (a) The transverse velocity of the gray TPSP solitons as a function of the grayness parameter m and q. (b) The transverse phase proﬁle of the gray TPSP solitons for various values of the grayness parameter m.

erally increases with q. Moreover, the transverse velocity increases very rapidly when q < 10 and becomes almost constant at higher values of qðq P 40Þ. It is also interesting to note from Eqs. (9) and (11) that the soliton transverse velocity m is approaching zero as m ? 0. This should have been anticipated, since this case corresponds to the dark spatial soliton, the transverse velocity of which is zero. According to Eq. (8), the transverse phase distribution of the Rg ~ ½1=y2 ðg ~ Þg. The gray TPSP solitons is given by Du ¼ tfg 0 dg transverse phase proﬁle of the gray TPSP solitons is shown in Fig. 2b for different values of m. Unlike the bright/dark soliton, the gray TPSP soliton’s phase is no longer constant across g but stead varies in a very involved fashion. As we can see from Fig. 2b, the soliton total phase jump, i.e., the phase difference between the two inﬁnite tails of the localized structure, exceeds p for relatively low degrees of grayness (when the soliton is close to being black). These solitons can be called ‘‘darker than black,” and the similar results were pointed out in the case of gray screening solitons [8]. 4. Discussion The above results indicate that gray TPSP spatial optical solitons are possible in biased two-photon photovoltaic PR crystals. In our previous work [27], we have predicted that bright and dark spatial solitons can be formed in biased two-photon photovoltaic PR crystals. Although gray TPSP solitons satisfy the same dynamical evolution equation as bright and dark solitons, they have different

boundary conditions, and then they have different properties. One of interesting phenomena is that the transverse velocity of gray TPSP solitons depends on the normalized intensity ratio q as we have shown in this paper, however, the transverse velocity of dark TPSP solitons does not depend on the normalized intensity ratio and it is always zero. What is more, the normalized intensity proﬁle of the dark TPSP solitons can be achieved by taking m = 0 in Eq. (12), which has a same form as Eq. (25) in [27]. In this case, the gray TPSP optical solitons degenerate into the dark ones, i.e., the dark TPSP solitons is the special case of the gray TPSP solitons when the grayness is zero. Before closing, it would be appropriate at this stage to compare our results with those available in literature. Recently, several workers have investigated two-photon PR spatial solitons. Notable among those are works of Hou et al. [23,26] and Asif et al. [24]. In the steady state, Hou et al. have predicted that dark and bright spatial solitons are possible in biased two-photon PR crystals [23] and in two-photon photovoltaic PR crystals under open-circuit case [26], and the same group have investigated the properties of the gray photovoltaic spatial solitons under open-circuit case [25]. Three key features distinguish our work with that of Hou et al. The ﬁrst difference is that while we have treated these spatial optical solitons resulted from both the spatially non-uniform screening of the bias electric ﬁeld and the photovoltaic effect, they have studied these solitons resulted from either the spatially non-uniform screening of the bias ﬁeld or the photovoltaic effect. Second and perhaps the most important point is that, in the present paper we have provided a universal formula for gray spatial solitons in two-photon PR crystals, which allows one can obtain gray photovoltaic solitons and screening solitons under the appreciate conditions. By taking b = 0 and g = 0, we can see that Eq. (12) has a same form as Eq. (15) in [25], in which the gray photovoltaic solitons are discussed under open-circuit conditions. By taking a = 0 and g = 1 in Eq. (3), we can obtain Eq. (14) in [23], in which the two-photon screening solitons discussed in detail. Moreover, if we take b = 0 and g = 1 in Eq. (12), we can also obtain the expression of the gray photovoltaic solitons in unbiased two-photon PR crystal in closedcircuit realization, which to our knowledge has not ever been studied. Thirdly, for the existence of spatial solitons that reported by Hou et al. the dark and gray (bright) solitons are possible only when the photovoltaic ﬁeld is set up in the opposite (same) direction with respect to the optical c axis of the crystals, i.e., Ep < 0 (Ep > 0).The works of Asif et al. also suffer from this deﬁciency [24]. Whereas in our case there is no such restriction, for example, we take Ep = 2 106 V m1 (a = 11.1), Ea = 4.0 106 V m1 (b = 22.2), q = 10 and m = 0.1, other parameters taken as in Section 3, these gray TPSP solitons can be determined from Eq. (12), and the solitons intensity FWHM is about 60 lm.

5. Conclusion We have presented the gray TPSP solitons solution of the optical wave evolution equation in biased two-photon photovoltaic PR crystals. The properties of these gray solitons, such as their normalized intensity proﬁle, intensity FWHM, transverse velocity and transverse phase proﬁle have been discussed in detail. It is shown that the gray TPSP spatial solitons can be formed in a biased twophoton photovoltaic PR crystal. The gray soliton becomes narrower as the grayness parameter m decreases for a given normalized intensity ratio q, the soliton width generally decreases with intensity ration q, the soliton transverse velocity generally increases with intensity ration q, and the soliton phase varies in a very involved fashion across g, the soliton total phase jump exceeds p for the relatively low values of the grayness parameter m. The gray TPSP solitons can be thinking of as the unity of the gray screening

G. Zhang et al. / Optics Communications 283 (2010) 335–339

solitons and the gray photovoltaic solitons in two-photon PR crystals. The gray TPSP solitons degenerate into the dark ones when the degree of grayness is zero. Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant No. 10570251) and the Natural Science Foundation of Hubei Province of China (Grant No. 2008CDB005). References [1] M. Segev, B. Crosignani, A. Yariv, B. Fischer, Phys. Rev. Lett. 68 (1992) 923. [2] G.C. Duree, J.L. Shultz, G.J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E.J. Sharp, R.R. Neurgaonkar, Phys. Rev. Lett. 71 (1993) 533. [3] G.I. Stegeman1, M. Segev, Science 286 (1999) 1518. [4] W. Krolikowski, B. Luther-Davies, C. Denz, IEEE J. Quantum Electron. 39 (2003) 3. [5] M. Segev, G.C. Valley, B. Crosignani, P. Di Porto, A. Yariv, Phys. Rev. Lett. 73 (1994) 3211. [6] D.N. Christodoulides, M.I. Carvalho, J. Opt. Soc. Am. B 12 (1995) 1628. [7] Z.G. Chen, M. Mitchell, M.F. Shih, M. Segev, M.H. Garrett, G.C. Valley, Opt. Lett. 21 (1996) 629. [8] A.G. Grandpierre, D.N. Christodoulides, T.H. Coskun, M. Segev, Y.S. Kivshar, J. Opt. Soc. Am. B 18 (2001) 55.

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