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Bright–dark self-coupled vector spatial solitons in biased two-photon photovoltaic photorefractive crystals
Optics Communications 283 (2010) 3512–3515
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Bright–dark self-coupled vector spatial solitons in biased two-photon photovoltaic photorefractive crystals Qichang Jiang ⁎, Yanli Su, Xuanmang Ji Department of Physics and Electronic Engineering, Yuncheng University, Yuncheng, 044000, China
a r t i c l e
i n f o
Article history: Received 2 December 2009 Received in revised form 23 April 2010 Accepted 23 April 2010 Keywords: Nonlinear optics Bright–dark vector spatial solitons Two-photon photorefractive effect
a b s t r a c t This paper predicts that bright–dark self-coupled vector solitons are possible in biased two-photon photovoltaic photorefractive crystals under steady-state conditions. The solutions of these vector solitons can be determined by use of simple numerical integration procedures. When the photovoltaic effect is neglectable, these vector solitons are bright–dark vector screening solitons. When the external bias field is absent, these vector solitons degenerate the bright–dark vector photovoltaic solitons. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved.
1. Introduction Photorefractive spatial solitons (PRSS) were first observed experimentally by Duree et al. [1] just a year after their theoretical prediction by Segev et al. [2] in 1992 and have been the subject of active research both theoretically and experimentally since then [3–24]. To date, three different types of steady-state PRSS (screening solitons [3,4], photovoltaic (PV) solitons [5,6] and screening photovoltaic (SP) solitons [7,8]) have been predicted and have been observed experimentally. Moreover, PRSS can be denoted as scalar solitons (one component) and vector solitons (multi-components) according to the number of components of solitons [9]. The most important prerequisite for the generation of the vector solitons is the absence of any interference between the single components. In general, there exist three ways to achieve the requirements. The original suggestion of Manakov [10] is based on two beams with orthogonal states of polarization. A second approach can be realized by applying two beams of different wavelength as for the case of all quadratic solitons [11]. Finally, the vector solitons can be formed using mutually incoherent beams which have the same polarization, wavelength [12]. Vector screening solitons [13,14] and vector SP solitons [15] in a biased PR crystal have been predicted, which involve the two polarization components of an optical beam that are orthogonal to one another. Of particular interest are bright–dark self-coupled vector solitons. Bright–dark self-coupled vector screening solitons which occur in steady-state when the intensities of the two optical beams are approximately equal [14]. Bright–dark self-coupled vector SP solitons are possible in biased PR-PV crystals under steady-state conditions, its
⁎ Corresponding author. E-mail address: [email protected] (Q. Jiang).
analytical solutions can be obtained when the intensities of the two optical beams are approximately equal and these vector solitons can also be determined by use of simple numerical integration procedures when the intensities of the two optical beams make a great difference [15]. All of the above-mentioned solitons result from the single-photon process. In 2003, a new model was introduced by Castro-Camus and Magana [16], which involves two-photon PR effect. This model includes a valance band (VB), a conduction band (CB) and an intermediate allowed level (IL). A gating beam is used to maintain a fixed quantity of excited electrons from the VB, which are then excited to the CB by signal beam. The single beam induces a charge distribution identical to its intensity distribution, which in turn gives rise to a nonlinear change of refractive index through space charge field. At one time, the two-photon process was observed experimentally by W Ramadan et al. [17]. Based on this model, screening solitons [18], PV solitons [19] and SP solitons [20] in two-photon PR crystals have been predicted. On the other hand, incoherently coupled bright–bright, dark–dark, bright–dark, and grey–grey soliton pairs whose carrier beams share the same polarization, wavelength, and are mutually incoherent have been predicted for screening solitons or PV solitons [21–25] that result from the two-photon PR effect. In this paper, we show that bright–dark self-coupled vector SP solitons are possible in biased PV-PR crystals with two-photon PR effect. Moreover, the stability of the bright–dark vector solitons is investigated numerically.
2. Theoretical model As shown in Fig. 1, an envisaged experiment is arranged as follows. Two collimated CW laser beams produced by two separate lasers L1 and L2. The gating beam L1 is expanded and then sent to the crystal along the y direction. The signal beam L2 is focused on the crystal input face
0030-4018/$ – see front matter. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.04.094
Q. Jiang et al. / Optics Communications 283 (2010) 3512–3515
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where I1is the intensity of the gating beam; I2 = I2(x, z) is total power density of the extraordinary and ordinary components, can be obtained by summing the two Poynting fluxes, i.e., I2 = (n̂e / 2η0)|ϕe|2 + (no / 2η0)|ϕo|2; g =
Fig. 1. An envisaged experiment arrangement, L1 and L2 are the lasers, Pc is the Pockels cell.
along the z axis. Two Pockels cells, one for the gating beam and one for the signal beam respectively, allow one to regulate the intensity and polarization of the two beams. The beam profiles can be detected by the charge coupled device (CCD). The signal beam propagates in a PV-PR crystal with two-photon PR effect and is allowed to diffract only along the x direction. Moreover, let us assume that the external bias electric field is also applied along x. For demonstration purposes, let the PV-PR crystal be LiNbO3, which is illuminated by the gating beam. As previously pointed out, this crystal is a good candidate for the observation of the self-coupled or cross-coupled vector solitons [13–15]. More specifically, for the self-coupled case, the permittivity changes in LiNbO3 along the extraordinary and ordinary components of the optical beam are equal, i.e., Δɛee = Δɛoo, provided that the optical c axis of the crystal makes an angle θ ≈ 11.9° with respect to the z axis in the xoz coordinate plane. Δɛee and Δɛoo represent the diagonal perturbations on the relative permittivity tensor. Moreover, in this case the off-diagonal elements, i.e., Δɛeo and Δɛoe, are zero. By associating slowly varying envelopes with the extraordinary and ordinary polarizations, φe(x, z) and φo (x, z), then one quickly finds the following set of self-coupled nonlinear evolution equations [14,15]: 2
2ike
∂ϕe ∂ ϕe 2 + k Δεϕe = 0 + ∂z ∂x2
ð1aÞ
2iko
∂ϕo ∂2 ϕo 2 + k Δεϕo = 0 + ∂z ∂x2
ð1bÞ
ESC
+ Ep ðgI2∞ −I2 Þ
s2 ðI2 + I2d + γ1 NA = s2 Þ ðs1 I1 + β1 ÞðI2 + I2d Þ
p=
eωðs1 I1 + β1 ÞðN−NA Þ WγNA ðI2∞ + I2d + γ1 NA = s2 Þ,
where
S is the surface area of the crystal's electrodes, R is resistance, W is the distance between the crystal's electrodes; In general,0 b g b 1, which implies that only part of the bias field EA can be applied to the crystal. For example, the short-circuit condition, R = 0 and g = 1, which implies that EA can be totally applied to the crystal. For the open-circuit condition, R → ∞ then g = 0, this implies that no bias field is applied to the crystal. I2d = β2 / s2 is the so-called dark irradiance, NA is the acceptor or trap density; γ,γ1 are the recombination factor of the CBVB, IL-VB transition, respectively; β1 and β2 are the thermoionization probability constant for transitions of VB-IL and IL-CB; s1 and s2 are photoexcitation crosses; Ep = κγNA / eω is the PV field, κ, ω, and e are, respectively, the PV constant, the electron mobility, and the charge. For the sake of convenience, let us adopt the following dimensionless coordinates and variables: s = x / x0, ξ = z / (k0x20), U = (2η0I2d / n̂e)− 1/2ϕe and V = (2η0I2d / no)− 1/2φo. x0 is an arbitrary spatial width, and the power densities of the optical beams have been scaled with respect to the dark irradianceI2d. By employing these latter transformations and by substituting expressing Eq. (2) into Eqs. (1a) and (1b), and after appropriate normalization, we find that the normalized planar envelopes U and V satisfy 2 1∂ U gβð1 + ρÞ σ nˆ ∂U U − + 1+ i e 2 2 2 no ∂ξ 2 ∂s ð1 + ρ + σ Þ 1 + j U j + jVj − αη
i
2 2 2 2 gρ−jUj −jVj 1 + jUj + jVj + σ 1 + jUj2 + jVj2
U=0
∂V 1 ∂2 V gβð1 + ρÞ σ V − + 1 + 2 ∂s2 ð1 + ρ + σ Þ ∂ξ 1 + j U j 2 + jVj2 − αη
gρ−jUj2 −jVj2 1 + jUj2 + jVj2 + σ 1 + jUj2 + jVj2
ð3aÞ
ð3bÞ
V=0
where α = −(235.85 × 10− 12/2)(kx0)2Ep, β = −(235.85 × 10− 12/2) (kx0)2EA, η = s I β+2 β , σ = γ1NA / β2, ρ = I2∞ / I2d. In what follows, we 1 1 1 will discuss the possible bright–dark vector SP soliton solutions of Eq. (3a). 3. Bright–dark self-coupled vector solitons
where k = 2π / λ and λ is the free-space wavelength of the light wave used, and Δɛ = Δɛee = Δɛoo. The wave numbers ke and ko are defined as ke = kn̂e and ko = kno, where n̂e and no are the refractive indices seen by the extraordinary and ordinary components. The relative permittivity changes Δɛee and Δɛoo can be expressed as Δεee = −reff,en̂ 4eESC and Δεoo = −reff,on̂ 4oESC, where reff,e and reff,o are the effective electro-optic coefficients for the extraordinary and ordinary polarizations, respectively. When the optical beam propagates in LiNbO3 along the z axis at an angle θ=11.9° with respect to the c axis, Δɛee =Δɛoo =235.85×10− 12ESC and ESC represents the space-charge field [14]. Under strong electro-field, the drift component will be dominant. In this case, we can neglect the diffusion effect, thus ESC can be approximately given by [20] ðI + I2d ÞðI2 + I2d + γ1 NA = s2 Þ = gEA 2∞ ðI2∞ + I2d + γ1 NA = s2 ÞðI2 + I2d Þ
1 1 + pSRðI2∞ + I2d Þ,
ð2Þ
To find the bright–dark self-coupled vector solitons solutions of Eq. (3a) let us express the normalized envelopes U and V in the following way: U = r− 1/2f(s)exp[i(n0/n̂e)μξ], V = ρ1/2q(s) exp (iνξ), where f(s) corresponds to a bright beam envelope and q(s)to a dark one. Here we have assumed, without and loss effects, that the extraordinary envelope U is bright, whereas the ordinary envelope V is dark. Hence, one requires that f(0) = 1,f′(0) = 0,f(s → ± ∞) = 0,q(0) = 0, and q(s → ± ∞) = ±1, and that all the derivatives of f(s) and q(s) vanish at infinity (s → ± ∞). The positive variables r and ρ represent the ratios of their maximum power density with respect to the dark irradiance I2d. Substitution of these forms of U and V into Eq. (3a), we get 9 8 gβð1 + ρÞ gβσ 1+ρ > > > > > > = < μ + 1 + σ + ρ + 1 + σ + ρ 1 + rf 2 + ρq2 f″ = 2 f > > ð1 + gρÞσ > > 2 2 > − 1 + rf + ρq −σ > ; : + αη ð1 + gρÞ + 1 + rf 2 + ρq2 ð4aÞ
3514
Q. Jiang et al. / Optics Communications 283 (2010) 3512–3515
9 8 gβð1 + ρÞ gβσ 1+ρ > > > > + > >ν + = < 2 2 1+σ+ρ 1 + σ + ρ 1 + rf + ρq q q″ = 2 > > ð1 + gρÞσ > > 2 2 > > −σ − 1 + rf + ρq ; : + αη ð1 + gρÞ + 1 + rf 2 + ρq2
ð4bÞ Where f″ = d2f / ds2, q″ = d2q / ds2. We now look for particular solutions which also satisfy the condition f2 + q2 = 1. In this case, Eq. (4a) take the forms of 8 9 gβð1 + ρÞ gβσ 1 > > > > μ + + > > < = 1+σ+ρ 1 + σ + ρ 1 + δf 2 f″ = 2 f > > 1 + gρ 1 > > > −1 + ρðg−1Þ−ð1 + ρÞδf 2 > : + αη σ ; 1 + ρ 1 + δf 2 ð5aÞ 8 9 gβð1 + ρÞ gβσ 1 > > > > > > < ν + 1 + σ + ρ + 1 + σ + ρ δ−δq2 + 1 = q″ = 2 >q > 1 + gρ 1 > > > −1 + ρðg−1Þ−ð1 + ρÞδ 1−q2 > : + αη σ ; 1 + ρ δ−δq2 + 1
ð5bÞ where δ = (r − ρ) / (1 + ρ). Form Eq. (5a) and by the use of the f and q boundary conditions, the values of μ and ν can be readily obtained and given by gβð1 + ρÞ gβσ lnð1 + δÞ − − αηρðg−1Þ 1+σ+ρ 1+σ+ρ δ
μ=−
+ αησ −
ð6Þ
αησ ð1 + gρÞ lnð1 + δÞ αηδð1 + ρÞ + 1+ρ δ 2
ν = − βg − αη
ðg−1Þρð1 + ρ + σ Þ 1+ρ
ð7Þ
Substituting the values of μ, ν in the Eq. (5b), and integrating once we get
2gβσ 2αησ 1 + gρ + ð1 + σ + ρÞδ δ 1+ρ
2
ðf ′Þ =
ð8Þ
h i 2 2 2 2 × ln 1 + δf −f ln ð1 + δÞ −αηδð1 + ρÞf f −1 n h io 2βσ 2 2 δ 1−q − ln 1 + δ 1−q ð1 + σ + ρÞδ 1 + gρ h i 2αησ 2 2 + δ 1−q − ln 1 + δ 1−q δ 1+ρ
2
ðq′Þ =
ð9Þ
Fig. 2. Intensity profiles of bright–dark vector solitons under different conditions (a) α = − 52.1, β = − 27.9 and g = 1 (dotted curves) (b) α = 0, β = − 27.9 and g = 1 (solid curves) (c) α = −52.1, β = 0 and g = 0 (dashed curves).
Eq. (10) can be solved numerically to reveal existence of bright– dark self-coupled vector solitons. To illustrate our results, the parameters are taken here to be [16,17,20] Ep = 2.8 × 106 Vm− 1, EA = 1.5 × 106 V/m, λ0 = 0.633 μm, x0 = 40 μm. We can find that α = − 52.1,β = − 27.9. Moreover, we take η = 1.67 × 10− 4, σ = 104, g = 1, δ = − 0.05, r = 9.45, ρ = 10.Fig. 2 depicts the normalized intensity profiles of the bright–dark self-coupled vector solitons for different conditions. It shows that from Eq. (10), we can obtain the bright–dark self-coupled SP, screening and PV vector solitons by adjusting the values of α, β and g. As shown in Refs [16,17], the electron density at the IL grows as a result of photoionization and thermoionization from VB and by recombination from CB and decreases by recombination to the VB and by photoionization to the CB. In other words, we can control the energy level and pump efficiency of the IL by adjusting the wavelength and the intensity values of the gating beam, respectively. Thus, the property of vector solitons is influenced by the intensity of the gating beam. In order to examine the influence of the gating beam on the bright–dark vector solitons, we computed profiles of the SP vector solitons under different values of the gating beam intensity. They have been shown in Fig. 3. With increasing the value of I1, the width of each component increase, which is agree with the experiment in Ref [17]. Finally, to investigate the stability of these solitons, Eqs. (3a) and (3b) have been solved numerically using a finite-difference method. The normalized envelopes given by dotted curves of Fig. 2 have been used as the input beam profiles of bright and dark components. Fig. 4 depicts the evolution of the bright and dark solitons, respectively.
2 2αησρðg−1Þ 2 2 q −1 + αηð1 + ρÞδ q −1 1+ρ
−
We can choose appropriate the values and signs of α and β and δ to make the right hand of Eqs. (8) and (9) to be positive so that they can be meet the mathematics' requirement. In this case, Eqs. (8) and (9) can be solved numerically to find out bright–dark self-coupled solitons profile. Further integrating Eq. (8) once we get 1
s = F∫ f
f
2gβσ 2αησ 1 + gρ + ð1 + σ + ρÞδ δ 1+ρ
g
h i 2 2 2 2 × ln 1 + δf −f ln ð1 + δÞ −αηδð1 + ρÞf f −1
−1 = 2
df ð10Þ
Fig. 3. Intensity profiles of bright–dark vector solitons under different values of the gating beam intensity I1 (a) α = − 52.1, β = − 27.9 and I = 106 W/m2 (dashed curves) (b) α = –52.1, β = − 27.9 and I = 105 W/m2 (solid curves).
Q. Jiang et al. / Optics Communications 283 (2010) 3512–3515
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bright–dark self-coupled vector SP solitons as a unity form of the vector screening solitons and the PV solitons. Based on our numerically results, we can predict that the experiment observation is possible when the values of α and β are negative in a certain regime and the gating beam takes appropriate wavelength and intensity. Acknowledgements This work was supported by the Science and Technology Development Foundation of Higher Education of Shanxi Province, China (Grant No. 200611042) and the Basic Research Foundation of Yuncheng University, China (Grant No. JC-2009003). References
Fig. 4. Stable propagation of the (a) bright and (b) dark components of the vector solitons when α = −52.1, β = −27.9, δ = − 0.05and g = 1.
Note that modulational instability will emerge when α and β are positive. We also investigated the stability of a single component of such a pair in the absence of the other, that is, when either U or V is zero. We have found that the bright component or the dark component cannot propagate stably in the absence of the other. 4. Conclusion In conclusion, we have shown that the self-coupled bright–dark vector solitons is possible in biased two-photon PV-PR crystals under steady-state conditions. These vector solitons are stable only when α and β are negative. The bright–dark self-coupled vector screening solitons and PV solitons can be obtained from the vector SP solitons by adjusting the values of α, β and g, that is to say, we can think of the
[1] G.C. Duree, J.L. Schultz, G.J. Salamo, M. Segev, A. Yariv, B. Crosignani, Phys. Rev. Lett. 71 (1993) 533. [2] M. Segev, B. Crosignani, A. Yariv, B. Fischer, Phys. Rev. Lett. 68 (1992) 923. [3] M. Segev, G.C. Valley, B. Crosignani, P. Diporto, A. Yariv, Phys. Rev. Lett. 73 (1994) 3211. [4] M.F. Shih, M. Segev, G.C. Valley, G. Salamo, B. Crosignani, P. Diporto, Electron. Lett. 31 (1995) 826. [5] G.C. Valley, M. Segev, B. Crosighani, A. Yariv, M.M. Fejer, M.C. Bashaw, Phys. Rev. A 50 (1994) R4457. [6] M. Taya, M.C. Bashaw, M.M. Fejer, M. Segev, G.C. Valley, Phys. Rev. A 52 (1995) 3095. [7] J.S. Liu, K.Q. Lu, J. Opt. Soc. Am. B 16 (1999) 550. [8] E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, V.I. Vlad, Appl. Phys. Lett. 85 (2004) 2193. [9] W. Carsten, A. Marcus, P. Jurgen, T. Denis, S. Jochen, D. Cornelia, Ann. Phys. 11 (2002) 573. [10] S.V. Manakov, Sov. Phys. JETP 38 (1974) 248. [11] W. Torruellas, Y.S. Kivshar, G.I. Stegeman, Quadratic Solitons in Spatial Solitons. Berlin, 2001, pp. 127–168. [12] D.N. Christodoulides, S.R. Singh, M.I. Carvalho, M. Segev, Appl. Phys. Lett 68 (1996) 1763. [13] M. Segev, G.C. Valley, S.R. Singh, M.I. Carvalho, D.N. Christodoulides, Opt. Lett. 20 (1995) 1764. [14] M.I. Carvalho, S.R. Singh, D.N. Christodoulides, R.I. Joseph, Phys. Rev. E 53 (1996) R53. [15] K.Q. Lu, S.X. Qian, W. Zhao, Y.P. Zhang, Z.S. Wu, Opt. Commun. 209 (2002) 437. [16] E. Castro-Camus, L.F. Magana, Opt. Lett. 28 (2003) 1129. [17] W. Ramadan, E. Fazio, A. Mascioletti, F. Inam, R. Rinaldi, et al., J. Opt. A: Pure Appl. Opt. 5 (2003) S432. [18] C.F. Hou, Y.B. Pei, Z.X. Zhou, Phys. Rev. A 71 (2005) 053817. [19] C.F. Hou, Y. Zhang, Y.Y. Jiang, Opt. Commun. 273 (2007) 544. [20] G.Y. Zhang, J.S. Liu, J. Opt. Soc. Am. B 26 (2009) 113. [21] Y. Zhang, C.F. Hou, X.D. Sun, Acta Phys. Sin. 56 (2007) 3261. [22] K.Q. Lu, W. Zhao, Y.L. Yang, Appl. Phys. B 87 (2007) 469. [23] Y. Zhang, C.F. Hou, F. Wang, X.D. Sun, Optik 119 (2008) 700. [24] N. Asif, S. Shwetanshumala, S. Konar, Phys. Lett. A 372 (2008) 735. [25] S. Srivasrava, S. Konar, Opt. Laser Tech. 41 (2009) 419.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Bright–dark self-coupled vector spatial solitons in biased two-photon photovoltaic photorefractive crystals Qichang Jiang ⁎, Yanli Su, Xuanmang Ji Department of Physics and Electronic Engineering, Yuncheng University, Yuncheng, 044000, China
a r t i c l e
i n f o
Article history: Received 2 December 2009 Received in revised form 23 April 2010 Accepted 23 April 2010 Keywords: Nonlinear optics Bright–dark vector spatial solitons Two-photon photorefractive effect
a b s t r a c t This paper predicts that bright–dark self-coupled vector solitons are possible in biased two-photon photovoltaic photorefractive crystals under steady-state conditions. The solutions of these vector solitons can be determined by use of simple numerical integration procedures. When the photovoltaic effect is neglectable, these vector solitons are bright–dark vector screening solitons. When the external bias field is absent, these vector solitons degenerate the bright–dark vector photovoltaic solitons. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved.
1. Introduction Photorefractive spatial solitons (PRSS) were first observed experimentally by Duree et al. [1] just a year after their theoretical prediction by Segev et al. [2] in 1992 and have been the subject of active research both theoretically and experimentally since then [3–24]. To date, three different types of steady-state PRSS (screening solitons [3,4], photovoltaic (PV) solitons [5,6] and screening photovoltaic (SP) solitons [7,8]) have been predicted and have been observed experimentally. Moreover, PRSS can be denoted as scalar solitons (one component) and vector solitons (multi-components) according to the number of components of solitons [9]. The most important prerequisite for the generation of the vector solitons is the absence of any interference between the single components. In general, there exist three ways to achieve the requirements. The original suggestion of Manakov [10] is based on two beams with orthogonal states of polarization. A second approach can be realized by applying two beams of different wavelength as for the case of all quadratic solitons [11]. Finally, the vector solitons can be formed using mutually incoherent beams which have the same polarization, wavelength [12]. Vector screening solitons [13,14] and vector SP solitons [15] in a biased PR crystal have been predicted, which involve the two polarization components of an optical beam that are orthogonal to one another. Of particular interest are bright–dark self-coupled vector solitons. Bright–dark self-coupled vector screening solitons which occur in steady-state when the intensities of the two optical beams are approximately equal [14]. Bright–dark self-coupled vector SP solitons are possible in biased PR-PV crystals under steady-state conditions, its
⁎ Corresponding author. E-mail address: [email protected] (Q. Jiang).
analytical solutions can be obtained when the intensities of the two optical beams are approximately equal and these vector solitons can also be determined by use of simple numerical integration procedures when the intensities of the two optical beams make a great difference [15]. All of the above-mentioned solitons result from the single-photon process. In 2003, a new model was introduced by Castro-Camus and Magana [16], which involves two-photon PR effect. This model includes a valance band (VB), a conduction band (CB) and an intermediate allowed level (IL). A gating beam is used to maintain a fixed quantity of excited electrons from the VB, which are then excited to the CB by signal beam. The single beam induces a charge distribution identical to its intensity distribution, which in turn gives rise to a nonlinear change of refractive index through space charge field. At one time, the two-photon process was observed experimentally by W Ramadan et al. [17]. Based on this model, screening solitons [18], PV solitons [19] and SP solitons [20] in two-photon PR crystals have been predicted. On the other hand, incoherently coupled bright–bright, dark–dark, bright–dark, and grey–grey soliton pairs whose carrier beams share the same polarization, wavelength, and are mutually incoherent have been predicted for screening solitons or PV solitons [21–25] that result from the two-photon PR effect. In this paper, we show that bright–dark self-coupled vector SP solitons are possible in biased PV-PR crystals with two-photon PR effect. Moreover, the stability of the bright–dark vector solitons is investigated numerically.
2. Theoretical model As shown in Fig. 1, an envisaged experiment is arranged as follows. Two collimated CW laser beams produced by two separate lasers L1 and L2. The gating beam L1 is expanded and then sent to the crystal along the y direction. The signal beam L2 is focused on the crystal input face
0030-4018/$ – see front matter. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.04.094
Q. Jiang et al. / Optics Communications 283 (2010) 3512–3515
3513
where I1is the intensity of the gating beam; I2 = I2(x, z) is total power density of the extraordinary and ordinary components, can be obtained by summing the two Poynting fluxes, i.e., I2 = (n̂e / 2η0)|ϕe|2 + (no / 2η0)|ϕo|2; g =
Fig. 1. An envisaged experiment arrangement, L1 and L2 are the lasers, Pc is the Pockels cell.
along the z axis. Two Pockels cells, one for the gating beam and one for the signal beam respectively, allow one to regulate the intensity and polarization of the two beams. The beam profiles can be detected by the charge coupled device (CCD). The signal beam propagates in a PV-PR crystal with two-photon PR effect and is allowed to diffract only along the x direction. Moreover, let us assume that the external bias electric field is also applied along x. For demonstration purposes, let the PV-PR crystal be LiNbO3, which is illuminated by the gating beam. As previously pointed out, this crystal is a good candidate for the observation of the self-coupled or cross-coupled vector solitons [13–15]. More specifically, for the self-coupled case, the permittivity changes in LiNbO3 along the extraordinary and ordinary components of the optical beam are equal, i.e., Δɛee = Δɛoo, provided that the optical c axis of the crystal makes an angle θ ≈ 11.9° with respect to the z axis in the xoz coordinate plane. Δɛee and Δɛoo represent the diagonal perturbations on the relative permittivity tensor. Moreover, in this case the off-diagonal elements, i.e., Δɛeo and Δɛoe, are zero. By associating slowly varying envelopes with the extraordinary and ordinary polarizations, φe(x, z) and φo (x, z), then one quickly finds the following set of self-coupled nonlinear evolution equations [14,15]: 2
2ike
∂ϕe ∂ ϕe 2 + k Δεϕe = 0 + ∂z ∂x2
ð1aÞ
2iko
∂ϕo ∂2 ϕo 2 + k Δεϕo = 0 + ∂z ∂x2
ð1bÞ
ESC
+ Ep ðgI2∞ −I2 Þ
s2 ðI2 + I2d + γ1 NA = s2 Þ ðs1 I1 + β1 ÞðI2 + I2d Þ
p=
eωðs1 I1 + β1 ÞðN−NA Þ WγNA ðI2∞ + I2d + γ1 NA = s2 Þ,
where
S is the surface area of the crystal's electrodes, R is resistance, W is the distance between the crystal's electrodes; In general,0 b g b 1, which implies that only part of the bias field EA can be applied to the crystal. For example, the short-circuit condition, R = 0 and g = 1, which implies that EA can be totally applied to the crystal. For the open-circuit condition, R → ∞ then g = 0, this implies that no bias field is applied to the crystal. I2d = β2 / s2 is the so-called dark irradiance, NA is the acceptor or trap density; γ,γ1 are the recombination factor of the CBVB, IL-VB transition, respectively; β1 and β2 are the thermoionization probability constant for transitions of VB-IL and IL-CB; s1 and s2 are photoexcitation crosses; Ep = κγNA / eω is the PV field, κ, ω, and e are, respectively, the PV constant, the electron mobility, and the charge. For the sake of convenience, let us adopt the following dimensionless coordinates and variables: s = x / x0, ξ = z / (k0x20), U = (2η0I2d / n̂e)− 1/2ϕe and V = (2η0I2d / no)− 1/2φo. x0 is an arbitrary spatial width, and the power densities of the optical beams have been scaled with respect to the dark irradianceI2d. By employing these latter transformations and by substituting expressing Eq. (2) into Eqs. (1a) and (1b), and after appropriate normalization, we find that the normalized planar envelopes U and V satisfy 2 1∂ U gβð1 + ρÞ σ nˆ ∂U U − + 1+ i e 2 2 2 no ∂ξ 2 ∂s ð1 + ρ + σ Þ 1 + j U j + jVj − αη
i
2 2 2 2 gρ−jUj −jVj 1 + jUj + jVj + σ 1 + jUj2 + jVj2
U=0
∂V 1 ∂2 V gβð1 + ρÞ σ V − + 1 + 2 ∂s2 ð1 + ρ + σ Þ ∂ξ 1 + j U j 2 + jVj2 − αη
gρ−jUj2 −jVj2 1 + jUj2 + jVj2 + σ 1 + jUj2 + jVj2
ð3aÞ
ð3bÞ
V=0
where α = −(235.85 × 10− 12/2)(kx0)2Ep, β = −(235.85 × 10− 12/2) (kx0)2EA, η = s I β+2 β , σ = γ1NA / β2, ρ = I2∞ / I2d. In what follows, we 1 1 1 will discuss the possible bright–dark vector SP soliton solutions of Eq. (3a). 3. Bright–dark self-coupled vector solitons
where k = 2π / λ and λ is the free-space wavelength of the light wave used, and Δɛ = Δɛee = Δɛoo. The wave numbers ke and ko are defined as ke = kn̂e and ko = kno, where n̂e and no are the refractive indices seen by the extraordinary and ordinary components. The relative permittivity changes Δɛee and Δɛoo can be expressed as Δεee = −reff,en̂ 4eESC and Δεoo = −reff,on̂ 4oESC, where reff,e and reff,o are the effective electro-optic coefficients for the extraordinary and ordinary polarizations, respectively. When the optical beam propagates in LiNbO3 along the z axis at an angle θ=11.9° with respect to the c axis, Δɛee =Δɛoo =235.85×10− 12ESC and ESC represents the space-charge field [14]. Under strong electro-field, the drift component will be dominant. In this case, we can neglect the diffusion effect, thus ESC can be approximately given by [20] ðI + I2d ÞðI2 + I2d + γ1 NA = s2 Þ = gEA 2∞ ðI2∞ + I2d + γ1 NA = s2 ÞðI2 + I2d Þ
1 1 + pSRðI2∞ + I2d Þ,
ð2Þ
To find the bright–dark self-coupled vector solitons solutions of Eq. (3a) let us express the normalized envelopes U and V in the following way: U = r− 1/2f(s)exp[i(n0/n̂e)μξ], V = ρ1/2q(s) exp (iνξ), where f(s) corresponds to a bright beam envelope and q(s)to a dark one. Here we have assumed, without and loss effects, that the extraordinary envelope U is bright, whereas the ordinary envelope V is dark. Hence, one requires that f(0) = 1,f′(0) = 0,f(s → ± ∞) = 0,q(0) = 0, and q(s → ± ∞) = ±1, and that all the derivatives of f(s) and q(s) vanish at infinity (s → ± ∞). The positive variables r and ρ represent the ratios of their maximum power density with respect to the dark irradiance I2d. Substitution of these forms of U and V into Eq. (3a), we get 9 8 gβð1 + ρÞ gβσ 1+ρ > > > > > > = < μ + 1 + σ + ρ + 1 + σ + ρ 1 + rf 2 + ρq2 f″ = 2 f > > ð1 + gρÞσ > > 2 2 > − 1 + rf + ρq −σ > ; : + αη ð1 + gρÞ + 1 + rf 2 + ρq2 ð4aÞ
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9 8 gβð1 + ρÞ gβσ 1+ρ > > > > + > >ν + = < 2 2 1+σ+ρ 1 + σ + ρ 1 + rf + ρq q q″ = 2 > > ð1 + gρÞσ > > 2 2 > > −σ − 1 + rf + ρq ; : + αη ð1 + gρÞ + 1 + rf 2 + ρq2
ð4bÞ Where f″ = d2f / ds2, q″ = d2q / ds2. We now look for particular solutions which also satisfy the condition f2 + q2 = 1. In this case, Eq. (4a) take the forms of 8 9 gβð1 + ρÞ gβσ 1 > > > > μ + + > > < = 1+σ+ρ 1 + σ + ρ 1 + δf 2 f″ = 2 f > > 1 + gρ 1 > > > −1 + ρðg−1Þ−ð1 + ρÞδf 2 > : + αη σ ; 1 + ρ 1 + δf 2 ð5aÞ 8 9 gβð1 + ρÞ gβσ 1 > > > > > > < ν + 1 + σ + ρ + 1 + σ + ρ δ−δq2 + 1 = q″ = 2 >q > 1 + gρ 1 > > > −1 + ρðg−1Þ−ð1 + ρÞδ 1−q2 > : + αη σ ; 1 + ρ δ−δq2 + 1
ð5bÞ where δ = (r − ρ) / (1 + ρ). Form Eq. (5a) and by the use of the f and q boundary conditions, the values of μ and ν can be readily obtained and given by gβð1 + ρÞ gβσ lnð1 + δÞ − − αηρðg−1Þ 1+σ+ρ 1+σ+ρ δ
μ=−
+ αησ −
ð6Þ
αησ ð1 + gρÞ lnð1 + δÞ αηδð1 + ρÞ + 1+ρ δ 2
ν = − βg − αη
ðg−1Þρð1 + ρ + σ Þ 1+ρ
ð7Þ
Substituting the values of μ, ν in the Eq. (5b), and integrating once we get
2gβσ 2αησ 1 + gρ + ð1 + σ + ρÞδ δ 1+ρ
2
ðf ′Þ =
ð8Þ
h i 2 2 2 2 × ln 1 + δf −f ln ð1 + δÞ −αηδð1 + ρÞf f −1 n h io 2βσ 2 2 δ 1−q − ln 1 + δ 1−q ð1 + σ + ρÞδ 1 + gρ h i 2αησ 2 2 + δ 1−q − ln 1 + δ 1−q δ 1+ρ
2
ðq′Þ =
ð9Þ
Fig. 2. Intensity profiles of bright–dark vector solitons under different conditions (a) α = − 52.1, β = − 27.9 and g = 1 (dotted curves) (b) α = 0, β = − 27.9 and g = 1 (solid curves) (c) α = −52.1, β = 0 and g = 0 (dashed curves).
Eq. (10) can be solved numerically to reveal existence of bright– dark self-coupled vector solitons. To illustrate our results, the parameters are taken here to be [16,17,20] Ep = 2.8 × 106 Vm− 1, EA = 1.5 × 106 V/m, λ0 = 0.633 μm, x0 = 40 μm. We can find that α = − 52.1,β = − 27.9. Moreover, we take η = 1.67 × 10− 4, σ = 104, g = 1, δ = − 0.05, r = 9.45, ρ = 10.Fig. 2 depicts the normalized intensity profiles of the bright–dark self-coupled vector solitons for different conditions. It shows that from Eq. (10), we can obtain the bright–dark self-coupled SP, screening and PV vector solitons by adjusting the values of α, β and g. As shown in Refs [16,17], the electron density at the IL grows as a result of photoionization and thermoionization from VB and by recombination from CB and decreases by recombination to the VB and by photoionization to the CB. In other words, we can control the energy level and pump efficiency of the IL by adjusting the wavelength and the intensity values of the gating beam, respectively. Thus, the property of vector solitons is influenced by the intensity of the gating beam. In order to examine the influence of the gating beam on the bright–dark vector solitons, we computed profiles of the SP vector solitons under different values of the gating beam intensity. They have been shown in Fig. 3. With increasing the value of I1, the width of each component increase, which is agree with the experiment in Ref [17]. Finally, to investigate the stability of these solitons, Eqs. (3a) and (3b) have been solved numerically using a finite-difference method. The normalized envelopes given by dotted curves of Fig. 2 have been used as the input beam profiles of bright and dark components. Fig. 4 depicts the evolution of the bright and dark solitons, respectively.
2 2αησρðg−1Þ 2 2 q −1 + αηð1 + ρÞδ q −1 1+ρ
−
We can choose appropriate the values and signs of α and β and δ to make the right hand of Eqs. (8) and (9) to be positive so that they can be meet the mathematics' requirement. In this case, Eqs. (8) and (9) can be solved numerically to find out bright–dark self-coupled solitons profile. Further integrating Eq. (8) once we get 1
s = F∫ f
f
2gβσ 2αησ 1 + gρ + ð1 + σ + ρÞδ δ 1+ρ
g
h i 2 2 2 2 × ln 1 + δf −f ln ð1 + δÞ −αηδð1 + ρÞf f −1
−1 = 2
df ð10Þ
Fig. 3. Intensity profiles of bright–dark vector solitons under different values of the gating beam intensity I1 (a) α = − 52.1, β = − 27.9 and I = 106 W/m2 (dashed curves) (b) α = –52.1, β = − 27.9 and I = 105 W/m2 (solid curves).
Q. Jiang et al. / Optics Communications 283 (2010) 3512–3515
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bright–dark self-coupled vector SP solitons as a unity form of the vector screening solitons and the PV solitons. Based on our numerically results, we can predict that the experiment observation is possible when the values of α and β are negative in a certain regime and the gating beam takes appropriate wavelength and intensity. Acknowledgements This work was supported by the Science and Technology Development Foundation of Higher Education of Shanxi Province, China (Grant No. 200611042) and the Basic Research Foundation of Yuncheng University, China (Grant No. JC-2009003). References
Fig. 4. Stable propagation of the (a) bright and (b) dark components of the vector solitons when α = −52.1, β = −27.9, δ = − 0.05and g = 1.
Note that modulational instability will emerge when α and β are positive. We also investigated the stability of a single component of such a pair in the absence of the other, that is, when either U or V is zero. We have found that the bright component or the dark component cannot propagate stably in the absence of the other. 4. Conclusion In conclusion, we have shown that the self-coupled bright–dark vector solitons is possible in biased two-photon PV-PR crystals under steady-state conditions. These vector solitons are stable only when α and β are negative. The bright–dark self-coupled vector screening solitons and PV solitons can be obtained from the vector SP solitons by adjusting the values of α, β and g, that is to say, we can think of the
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