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Internal oscillation of vector solitons and necklace solitons
Optics Communications 283 (2010) 3342–3347
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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
Internal oscillation of vector solitons and necklace solitons Zebin Cai a,b,⁎, Jianchu Liang a, Lin Yi a, Lei Deng b a b
Department of Physics, Huazhong University of Science and Technology, Wuhan (430074), China Department of Basis, Air Force Radar Academy, Wuhan (430019), China
a r t i c l e
i n f o
Article history: Received 9 December 2009 Received in revised form 9 March 2010 Accepted 9 March 2010 Keywords: Internal mode Photoisomerization Internal oscillation Vector soliton Necklace solitons
a b s t r a c t Internal modes and internal oscillation of vector solitons associated with photoisomerization and necklace solitons in Bessel lattices are researched. While white noise gives rise to the unsmoothness of the vector solitons, the perturbation of internal modes results in the long-distance quasi-periodic oscillation of soliton shape. Internal modes of two-dimensional necklace solitons in Bessel lattices have both real and imaginary parts, which is different with the internal modes of one-dimensional solitons which have only real part. © 2010 Elsevier B.V. All rights reserved.
1. Introduction One of the methods for investigating solitons is to research their internal modes [1,2]. While integrable solitons, such as onedimensional (1-D) Kerr solitons do not have internal mode, stable non-integrable solitons, such as saturable photorefractive solitons have internal modes. Internal modes are responsible for long-lived quasi-periodic oscillation of soliton shape. A small perturbation to such solitons may turn into its internal mode and create internal oscillation [3]. Internal oscillation of stable solitons is helpful for understanding of soliton dynamics under perturbation and collision dynamics of such solitons. If the internal modes exist, OSSs can temporarily store some of the translational energy in the internal oscillation during collision and retrieve this energy after the collision. This energy exchange mechanism can induce very complex collision structures such as window sequences and even fractal structures [4,5]. A better understanding of such collision process requires a good knowledge of internal oscillation in these solitons. Internal modes and internal oscillation have been analyzed for Klein-Gordon solitons [1], Sine-Gordon solitons [6], fundamental solitons in saturable nonlinear media [7], and bright vortex solitons in cubic–quintic nonlinear media [8]. The internal modes mentioned above are internal modes of scalar solitons. The existence, stability, collision of vector solitons [9–16] have been investigated in detail. As for the internal modes of vector solitons, Kaup et al. investigated the internal modes of vector tem-
⁎ Corresponding author. Department of Physics, Huazhong University of Science and Technology, Wuhan (430074), China. E-mail address: [email protected] (Z. Cai). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.03.019
poral solitons in optical fiber with an arbitrary polarization. They found that internal oscillation eigenmode gave rise to antisymmetric oscillations of the symmetric solitons. Malomed and Tasgal [17] studied internal vibrations of a vector soliton in optical fiber via the variational approximation. On the other hand, all these investigations on internal modes are restricted to 1-D solitons in homogeneous material. Refs. [7, 8] can be regarded as the case of 1-D solitons, as the solitons are azimuth-independent. The investigation of internal modes of multi-pole solitons in photonic lattices remains a challenge. The aim of this paper is to investigate internal modes of 1-D vector solitons and 2-D multi-pole solitons in optical lattices. While vector solitons are multi-component solitons in which all the components interact, 2-D multi-pole solitons in optical lattices are higherdimensional solitons in which the solitons are affected by the linear optical lattices. Both the cases attracted much interest recently and the investigation of internal modes of these solitons will help comprehending the essentiality of such solitons. For the first case, we take the solitons in organic material associated with photoisomerization for example. The photorefractive effects in inorganic crystal and photoisomerization nonlinear effects in organic polymer are ideal nonlinear effects for soliton formation, as they can support optical solitons in power of mW/cm2. Photoisomerization nonlinearity was researched recently [18,19], and then photoisomerization spatial solitons were predicted [20–24] and realized [25,26]. Photoisomerization results in negative nonlinearity spontaneously, and thus only dark or gray OSSs can be formed by photoisomerization when no other light is present. Later it was demonstrated that bright isomerization optical spatial solitons (IOSSs) could be obtained by adding a background light [27–29]. The IOSSs mentioned above are scalar solitons, and no vector isomerization optical spatial solitons (VIOSSs) was researched except that the solution of spontaneous bright-
Z. Cai et al. / Optics Communications 283 (2010) 3342–3347
dark soliton pair was given [28]. We will obtain the solution of brightbright vector soliton, and investigate the initial modes and internal oscillation of VIOSSs. We will see that similar to the case of scalar solitons, the perturbation of internal modes of VIOSS results in the long-distance quasi-periodic oscillation. For the second case, we will investigate internal modes of 2-D multi-pole solitons in Bessel photonic crystal. Such solitons are usually called necklace solitons which are constituted of many pearls. As multi-humped solitons in homogeneous materials are unstable, an optical lattice, such as a Bessel photonic lattice should be presented to stabilize the propagation of the solitons. The study of wave propagation in optical periodic structures such as photonic crystals has attracted great interest in recent years. Periodic systems exhibit unique behavior that differs fundamentally from that of their homogeneous counterparts. In these periodic nonlinear lattices, the underlying dynamics are dominated by the interplay between linear coupling effects among adjacent optical potential and nonlinearity; an exact balance between these two processes results in a self-localized state, namely lattice soliton (LS) or discrete soliton. As eigenmodes of linear Schrödinger equation, Bessel beams can propagate in the free space without diffraction; in addition, Bessel lattices can be induced by nondiffracting Bessel beams which are created in experiments by a conical prism (axicon) [30] or computer-generated hologram [31]. So Bessel lattices become an important concentric optical photonic lattice supporting lattice solitons and attracted much interest [32,33]. We will see, internal modes of necklace solitons in Bessel optical lattices have both real part and imaginary part, which is different with the case of one-dimensional solitons. 2. Bright–bright VIOSSs When the bulk material is homogeneously illuminated by an arbitrarily polarized background light, and two linearly polarized light beams with the same polarization direction propagate collinearly through the material, the mechanic equations governing photoisomerization can be written as [30] h i dNT 2 = −NT q ðσ1 I1 + σ2 I2 Þ cos θ + σb Ib dt + NC q′ σ ′1 I1 + σ ′2 I2 + σ ′b Ib + γNC ; N = NT ðθÞ + NC ðθÞ:
ð1Þ
ð2Þ
where θ is the angle between the orientation of the trans molecule and the polarization direction of the signal laser, I is intensity, subscripts 1, 2 and b denote beam 1, beam 2 and background light; N is the initial distribution of trans molecules, NT(θ) is the density of the trans molecules and NC(θ) is the density of the cis molecules at the angle θ, σ1(σ2) is the absorption cross section of the signal light, σb is the absorption cross section of the background light by trans molecules parallel to the electric field. σ′1(σ′2) is the absorption cross section of the signal light by cis molecules, σ′b is the absorption cross section of the background light by cis molecules. q(q′) is called the quantum yields for the trans-cis (cis-trans) transition. γ is the thermal relaxation rate from cis to trans. To achieve low-loss solitons we choose the wavelength of the signal light in the long wavelength region that is far away from the absorption band. Thus the molecules isomerized to be cis ones are much less than the original trans molecules, and the stationary-state distribution of the concentration of trans molecules NT, given by Eqs. (1) and (2) can be expressed as " NT ðθÞ≈N 1−
# qðσ1 I1 + σ2 I2 Þ cos2 θ + qσb Ib : q′ σ ′1 I1 + σ ′2 I2 + σ ′b Ib + γ
ð3Þ
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The change of refractive index can be further derived [30]: 2
Δ n˜ j = αj ∫½NT −NT ðI1 = I2 = 0Þ cos θdΩ = 2παj N
b a1 I1 + a2 I2 + b ; − d c1 I1 + c2 I2 + d
ð4Þ where a1 = 25 qσ1 , a2 = 25 qσ2 , b = 23 qσb Ib , c1 = q′σ′1, c2 = q′σ′2, d = q′σ′bIb + γ, αj is a constant relative with the material and the light. The cis-trans transition is faster than the trans-cis transition, namely, d N b, c1 N a1, c2 N a2. All the parameters are positive. It should be pointed that the presence of background light can greatly influence the nonlinear effects, or even change the sign of nonlinearity. As a matter of fact, Δñj b 0 when Ib = 0 [and then b = 0 in Eq. (4)]; Δñj N 0 when Ib ≠ 0, b N ac11 and bd N ac22 in Eq. (4). That is to say, when the background is d absent, the signal lights induce negative index change; when the background light is present, and the proportion of trans-cis transition velocity to cis-trans transition velocity of background light is larger than that of signal lights, the signal lights induce positive nonlinearity. The homogeneous index does not influence the propagation of light, we only need to consider the index change relative with signal lights that is Δnj ðI1 ; I2 Þ = mj
a 1 I1 + a 2 I2 + b ; c 1 I1 + c 2 I2 + d
ð5Þ
where mj = − 2παjN. Suppose that the one-dimensional beams propagate along z axis, and diffract along the x axis direction. Under the paraxial approximation, the slowly varying amplitudes of the coupled beams are determined by the coupled nonlinear Schrödinger equation (NLSE): ∂Aj ðx; zÞ ∂z
2
−
i ∂ Aj ðx; zÞ ikj Δnj − Aj ðx; zÞ = 0: ð j = 1; 2Þ: 2kj ∂x2 nj
ð6Þ
1 = 2 2k2 m , we obtain Introducing the antatz: ζ = − k1nm1 1 z, ξ = − n11 1 the normalized NLSEs: −i
∂A1 ðξ; ζÞ ∂2 A1 ðξ; ζÞ a1 jA1 j2 + a2 jA2 j2 + b − A1 ðξ; ζÞ; = ∂ζ ∂ξ2 c1 jA1 j2 + c2 jA2 j2 + d
ð7aÞ
−i
∂A2 ðξ; ζÞ ∂2 A2 ðξ; ζÞ a jA j2 + a2 jA2 j2 + b −ρη 1 1 2 =ρ A2 ðξ; ζÞ; 2 ∂ζ ∂ξ c1 jA1 j + c2 jA2 j2 + d
ð7bÞ
where ρ =
k1 , k2
η=
k22 m2 n1 . k21 m1 n2
Now we search for the vector soliton solution of the following form A1 ðξ; ζÞ = u1 ðξÞ exp ð−iδ1 ζÞ; A2 ðξ; ζÞ = u2 ðξÞ exp ð−iρηδ2 ζÞ: The amplitude functions satisfy the following coupled differential equations: " # ∂2 u1 ðξÞ a1 u21 + a2 u22 + b = −u ð ξ Þ δ − ; 1 1 ∂ξ2 c1 u21 + c2 u22 + d
ð8aÞ
" # ∂2 u2 ðξÞ a1 u21 + a2 u22 + b = −ηu2 ðξÞ δ2 − : ∂ξ2 c1 u21 + c2 u22 + d
ð8bÞ
The boundary conditions of bright-bright vector soliton are u1 ð∞Þ = u2 ð∞Þ = 0;
ð9aÞ
u ′1 ð0Þ = u ′2 ð0Þ = 0:
ð9bÞ
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Substituting Eq. (9a) into Eqs. (8a) and (8b), we obtain the asymptotic solutions (ξ → ∞) of Eqs. (8a) and (8b): rffiffiffiffiffiffiffiffiffiffiffiffiffi ! b u1 ðξÞ = λ1 exp − −δ1 ξ ; d sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! b u2 ðξÞ = λ2 exp − η −δ2 ξ : d
ð10aÞ
ð10bÞ
ψ″þ +
ψ″− +
ρηδ2 −σ ψþ −ηVψþ −ηV22 ψþ + ψ− −ηV21 ϕþ + ϕ− = 0; ρ ð12cÞ ρηδ2 + σ ψ− −ηVψ− −ηV22 ψþ + ψ− −ηV21 ϕþ + ϕ− = 0: ρ ð12dÞ
where 2
According to the general experimental conditions, we set the following parameter relations: b = a41 , c1 = 8a1, aa12 = σσ12 = 1:6, cc12 = σ1′ = 1:2, d = 1.2a2, η = 0.70, and ρ = 1.195. The integral constants σ2′ are set as λ1 = 1 and λ2 = 1.2. Then we can compute the eigenvalues (δ1, δ2) with the shooting method. Solving ordinary differential Eqs. (8a) and (8b), combining with the initial conditions Eqs. (10a) and (10b), and imposing the first-order derivative of u1 and u2 to be zero, we get δ1 = 0.2459, and δ2 = 0.2530. We further obtain the intensity distribution of the vector soliton which is shown in Fig. 1. 3. Internal modes of VIOSSs Since single-humped vector solitons in saturable nonlinear media have been known to be stable, we now investigate their internal modes. To determine these internal modes, we write the perturbed soliton solutions as
* A1 ðξ; ζÞ = u1 ðξÞ + ϕþ ðξÞ expðiσζÞ + ϕ− ðξÞ exp ð−iσ*ζÞ expð−iδ1 ζÞ;
ð11aÞ
* A2 ðξ; ζÞ = u2 ðξÞ + ψþ ðξÞ expðiσζÞ + ψ− ðξÞ exp ð−iσ*ζÞ expð−iρηδ2 ζÞ
ð11bÞ where u1(ξ) and u2(ξ) are the envelope amplitudes of the optical fields without perturbation, (ϕ+, ϕ−, ψ+, ψ−) are small perturbations, σ is the eigenvalue, and the asterisk represents complex conjugation. Substituting Eqs. (11a) and (11b) into Eq. (6) and dropping higherorder terms in (ϕ+, ϕ−, ψ+, ψ−), we can obtain the following linearized equations: ϕ″þ + ðδ1 −σ Þϕþ −Vϕþ −V11 ϕþ + ϕ− −V12 ψþ + ψ− = 0;
ð12aÞ
ϕ″− + ðδ1 + σ Þϕ− −Vϕ− −V11 ϕþ + ϕ− −V12 ψþ + ψ− = 0;ð12bÞ
2
a1 u1 + a2 u2 + b ; c1 u21 + c2 u22 + d h i 2 2 ða1 d−bc1 Þ + ða1 c2 −a2 c1 Þu2 u1 V11 = ; ðc1 u21 + c2 u22 + dÞ2 h i 2 ða2 d−bc2 Þ + ða2 c1 −a1 c2 Þu1 u1 u2 ; V12 = ðc1 u21 + c2 u22 + dÞ2 h i ða1 d−bc1 Þ + ða1 c2 −a2 c1 Þu22 u1 u2 ; V21 = ðc1 u21 + c2 u22 + dÞ2 h i ða2 d−bc2 Þ + ða2 c1 −a1 c2 Þu21 u22 : V22 = ðc1 u21 + c2 u22 + dÞ2 V=
Eqs. (12a), (12b), (12c) and (12d) are an eigenvalue problem. If σ is real, the vector soliton is linearly stable with respect to internal mode. In contrast, if σ is purely imaginary, the vector soliton is linearly unstable. According to the boundary conditions [ϕ±(ξ → ± ∞) = 0 and ψ±(ξ → ± ∞) = 0] of bright solitons, we can obtain the asymptotic solutions of Eqs. (12a), (12b), (12c) and (12d) at infinity: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! b −ðδ1 −σ Þξ ; ϕþ ðξ→ + ∞Þ = C1 exp − d rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! b −ðδ1 −σ Þξ ; ϕþ ðξ→−∞Þ = C2 exp d rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! b −ðδ1 + σ Þξ ; ϕ− ðξ→ + ∞Þ = C3 exp − d rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! b −ðδ1 + σ Þξ ; ϕ− ðξ→−∞Þ = C4 exp d sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ! ηb σ − ηδ2 − ξ ; ψþ ðξ→ + ∞Þ = C5 exp − d ρ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ! ηb σ − ηδ2 − ξ ; ψþ ðξ→−∞Þ = C6 exp d ρ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ! ηb σ − ηδ2 + ξ ; ψ− ðξ→ + ∞Þ = C7 exp − d ρ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ! ηb σ − ηδ2 + ξ : ψ− ðξ→−∞Þ = C8 exp d ρ where Ci(i = 1, 2, ···, 8) are integral constants. Making use of the above initial conditions and according to that the values of all the internal mode at ξ = 0 or the first-order derivatives of all the internal mode at ξ = 0 should be zero, we obtain σ = 0.0019, and the internal modes are obtained which are shown in Fig. 2. 4. Internal oscillation of VIOSSs
Fig. 1. Numerical solution of the VIOSS. Different color implies different wavelength of each component of the vector soliton.
We study numerically the dynamics of the VIOSS under the perturbation of the internal modes. Firstly, the propagation of the VIOSS without perturbation is plotted in Fig. 3. One can see that each component of the VIOSS does not change their amplitude shape
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Fig. 5. Propagation of each component of the VIOSS under the perturbation of internal modes.
Fig. 2. Internal modes of the VIOSS.
during the propagation. Next, a white noise with variance of ε2 = 0.02 is added to each component of the vector soliton, unsmoothness emerges, however no collapse comes forth, which implies the stability of the VIOSS (See Fig. 4). Finally, we add the perturbation of the internal modes to the vector soliton; their propagation is shown in Fig. 5. Although the shapes of internal modes are distinct with that of the solitons, they do not give rise to the instability of the vector soliton. In the long-distance propagating process, the shape and intensity of VIOSS under the perturbation of the internal modes show the characteristic of quasi-periodical oscillation. 5. Necklace solitons in Bessel lattices and their stability We assume that an optical beam propagates along the z axis in a bulk medium with defocusing Kerr nonlinearity and imprinted transverse modulation of refractive index. The dynamics of soliton is modeled by the 2-D normalized NLSE for the complex field amplitude q: 2
2
∂q 1 ∂ q ∂ q i + =− 2 ∂x2 ∂z ∂y2
pffiffiffi J1 ½ 2r expð−iz + iφÞ, where r2 = x2 + y2, φ is the azimuthal angle. Accordingly the lattice profile pffiffiffi is determined by the beam intensity in the form of Rðx; yÞ = pJ12 ½ 2r, where p = 40 is the modulation depth of the Bessel lattice. We search for the ten-pearled necklace soliton solutions of Eq. (13) numerically in the form of q(x, y, z) = w(x, y)exp (ibz). Here b is the propagation constant, and w(x, y) is real. Substitution of this expression in Eq. (13) leads to
−bw = −
! 2 2 1 ∂ w ∂ w 3 + + w −Rðx; yÞw: 2 ∂x2 ∂y2
ð14Þ
We solve Eq. (14) numerically using the finite difference method and the standard relaxation method. The initial iterative guess solution of w(x, y) is written as α sec h[(r − L)/W]cos(5θ), where α = 1, is the field amplitude, W = 0.5 is the approximation of the width of the first ring of the Bessel lattice. The first-order Bessel function approaches the first peak value when r = L ≈ 1.3. When b = 2, the resulting contour of the ten-pearled necklace soliton are
! 2
+ qjqj −Rðx; yÞq;
ð13Þ
where x and y are transverse coordinates. We suppose that optical lattice is created by a first-order Bessel beam, whose field is given by
Fig. 3. Propagation of each component of the VIOSS without perturbation.
Fig. 4. Propagation of each component of the VIOSS under the perturbation of white noise with variance of ε2 = 0.02.
Fig. 6. (a) Contour of ten-pearled necklace solitons in Bessel lattices for b = 2. (b) The instability parameter [maximum of Re(δ)] vs propagation constant b. The regions where [Re(δ)]max = 0 correspond to linearly stable solitons.
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shown in Fig. 6 (a). The pearls reside homogeneously on the first ring of the first-order Bessel lattice. As internal modes of solitons are related with stable solitons, we first analyze the stability of the necklace solitons in Bessel lattices prior to analyzing internal modes, taking a ten-pearled necklace for example. As is well known that multi-humped solitons in homogeneous materials are unstable. A photonic lattice may (not definitely) suppress the instability and result in the stable soliton propagation. Now we determine the stability of the necklace solitons in Bessel lattice. To perform the linear stability analysis of necklace solitons, we take the perturbed solutions to Eq. (13) as
! 1 ∂2 v ∂2 v 2 2 iδ*v−bv + −w u*−2w v + Rv = 0: + 2 ∂x2 ∂y2
ð16bÞ
from which the eigenvalue δ and eigenmodes u and v are obtained numerically. For a given b, there are many eigenvalues and related eigenmodes. The soliton is stable if all the real parts of δ equal zero [32]. Fig. 6 (b) gives information on the stability of the necklace solitons which shows the maximum of real parts of δ [Re(δ)] vs b. The solitons are unstable when b b 2.65, while they are stable when b N 5. There are three stable windows between these two points. 6. Internal modes of necklace solitons in Bessel lattices
q = ½w + εu expðδzÞ + εv expðδ*zÞ expðibzÞ;
ð15Þ
with an infinitesimal amplitude ε, complex perturbation eigenmodes u and v, and complex instability growth rate δ (* stands for the complex conjugation). Substitution of this expression in Eq. (13) and linearization around the stationary solution results in the following eigenvalue problem: ! 1 ∂ u ∂ u 2 2 iδu−bu + + −w v*−2w u + Ru = 0; 2 ∂x2 ∂y2 2
2
ð16aÞ
As internal modes are related with stable solitons, we search internal modes in stable region, taking the necklace soliton at b = 3 for example. The boundary conditions for internal modes are u ′r ;i →0; v ′r ;j →0; when r→0; ur ;i →0; vr ;j →0; when r→∞:
ð17Þ
In addition, the internal modes (real and imaginary part) should have similar symmetry as the corresponding soliton, and should have ten symmetric peaks and valleys. According to these conditions, the internal modes are obtained as shown in Fig. 7. The related
Fig. 7. Ten-pearled necklace solitons [(a) Intensity profile; and (b) contour] in Bessel lattices with b = 3, and the field distribution of its internal modes. (c) Real part of eigenmode u; (d) Imaginary part of eigenmode u; (e) Real part of eigenmode v; and (f) Imaginary part of eigenmode v.
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perturbation eigenvalue is δ = −1.0515i. As is well known, internal modes of 1-D solitons have only real part [3,7,8]. Differ from the cases of 1-D solitons, internal modes of 2-D necklace solitons have both real part and imaginary part.
in photonic lattices are investigated for the first time, to the best of knowledge.
7. Propagation of perturbed necklace solitons in Bessel lattices
B.Z. Zhang and H.C. Wang are acknowledged for their helpful discussions.
Now we confirm the stability and investigate the dynamics of necklace solitons under perturbation of internal modes. First we simulate the propagation of solitons that is claimed to be unstable. The soliton shown in Fig. 6(a) is selected. This is an unstable soliton according to the results in Section 5, as the parameters b = 2 correspond to nonzero real instability rate [Re(δ)] from Fig. 6(b). The initial condition is chosen as q(x, y, 0) = w(x, y)(1 + ρ), where ρ is a white noise with variance of σ2 = 0.05. The numerical computation soft ComSol is used based on the finite element method. The resulting Video 1 shows that the soliton collapses under the perturbation of the white noise after a propagation distance, by which the instability of the soliton is confirmed. Then we simulate the propagation of the stable soliton under the perturbation of the internal modes which is shown in Fig. 7. The initial field is denoted by Eq. (15), where z = 0, and ε = 3. The Video 2 shows that the necklace soliton oscillates semi-periodically under the perturbation of the internal modes; no collapse occurs after the propagation distance, and no apparent radiation and damping is observed. The robust stability of the soliton is thus confirmed. 8. Conclusions As a summary, internal modes and internal oscillation of vector solitons associated with photoisomerization and necklace solitons in Bessel lattices are researched. We show that the perturbation of internal modes of the vector solitons results in the long-distance quasi-oscillation of soliton shape while the presence of white noise gives rise to the unsmoothness of the vector soliton. After investigate the stability of necklace solitons in Bessel photonic lattices, we obtain the internal modes of these solitons in stable region. Differing from the internal modes of one-dimensional solitons, which have only real part, internal modes of two-dimensional necklace solitons in Bessel lattices have both real and imaginary parts. The necklace solitons oscillate semi-periodically under the perturbation of their internal modes, without apparent radiation and damping. Internal modes of vector solitons in organic materials and two-dimensional solitons
Acknowledgement
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.optcom.2010.03.019. References [1] D.K. Campell, J.F. Schonfeld, L.E. Guerrero, Physica D 9 (1983) 1. [2] D.E. Pelinovsky, Y.S. Kivshar, V.V. Afannasjev, Physica D 116 (1997) 121. [3] Y.S. Kivshar, D.E. Pelinovsky, T. Cretegny, M. Peyrard, Phys. Rev. Lett. 80 (1998) 5032. [4] P. Anninos, S. Oliveira, R.A. Matzner, Phys. Rev. D 44 (1991) 1147. [5] J. Yang, Y. Tan, Phys. Rev. Lett. 85 (2000) 3624. [6] J.A. González, A. Bellorín, L.E. Guerrero, Phys. Rev. E 65 (2002) 065601. [7] J.K. Yang, Phys. Rev. E 66 (2002) 026601. [8] L.W. Dong, F.W. Ye, J.D. Wang, T. Cai, Y.P. Li, Physica D 194 (2004) 219. [9] I. Shadrivov, A.A. Zharov, Opt. Commun. 216 (2003) 47. [10] J. Yang, D.E. Pelinovsky, Phys. Rev. E 67 (2003) 016608. [11] G.D. Montesinos, V.M. Pérez-García, H. Michinel, Phys. Rev. Lett. 92 (2004) 133901. [12] M. Shen, X. Chen, J. Shi, Q. Wang, W. Krolikowski, Opt. Commun. 282 (2009) 4805. [13] M.I. Carvalho, S.R. Singh, D.N. Christodoulides, R.I. Joseph, Phys. Rev. E 53 (1996) R53. [14] J. Yang, Phys. Rev. E 64 (2001) 026607. [15] Y.V. Kartashov, A.S. Zelenina, V.A. Vysloukh, L. Torner, Phys. Rev. E 70 (2004) 066623. [16] Z.Y. Sun, Y.T. Gao, X. Yu, W.J. Liu, Y. Liu, Phys. Rev. E 80 (2009) 066608. [17] B.A. Malomed, R.S. Tasgal, Phys. Rev. E 58 (1998) 2564. [18] J. Liang, X. Zhou, J. Opt. Soc. Am. B 22 (2005) 2468. [19] J.C. Liang, H. Zhao, X. Zhou, H. Wang, J. Appl. Phys. 101 (2007) 013106. [20] X.S. Wang, W.L. She, W.K. Lee, Opt. Lett. 29 (2004) 277. [21] X.S. Wang, W.L. She, Phys. Rev. E 71 (2005) 026601. [22] H. Wang, B. Zhang, L. Yan, Y. Li, G. Zheng, W. She, Opt. Commun. 275 (2007) 234. [23] B.Z. Zhang, H.C. Wang, W.L. She, J. Opt. A: Pure Appl. Opt. 9 (2007) 395. [24] B.Z. Zhang, H. Cui, W.L. She, Chin. Phys. B 18 (2009) 209. [25] X.S. Wang, W.L. She, S.Z. Wu, F. Zeng, Opt. Lett. 30 (2005) 863. [26] S. Bian, M.G. Kuzyk, Appl. Phys. Lett. 85 (2004) 1104. [27] X.S. Wang, W.L. She, W.K. Lee, J. Opt. Soc. Am. B 23 (2006) 2127. [28] J.C. Liang, Z.B. Cai, Y.Z. Sun, S.L. Xu, L. Yi, J. Opt. Soc. Am. B 26 (2009) 36. [29] J.C. Liang, Z.B. Cai, Y.Z. Sun, L. Yi, H.C. Wang, Eur. Phys. J. D 55 (2009) 685. [30] J. Arlt, K. Dholakia, Opt. Commun. 177 (2000) 297. [31] A. Vasara, J. Truunen, A. Fribert, J. Opt. Soc. Am. A 6 (1989) 1748. [32] Y.V. Kartashov, V.A. Vysloukh, L. Torner, Phys. Rev. Lett. 94 (2005) 043902. [33] J.C. Liang, Z.B. Cai, L. Yi, H.C. Wang, Opt. Commun. 283 (2010) 386.
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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
Internal oscillation of vector solitons and necklace solitons Zebin Cai a,b,⁎, Jianchu Liang a, Lin Yi a, Lei Deng b a b
Department of Physics, Huazhong University of Science and Technology, Wuhan (430074), China Department of Basis, Air Force Radar Academy, Wuhan (430019), China
a r t i c l e
i n f o
Article history: Received 9 December 2009 Received in revised form 9 March 2010 Accepted 9 March 2010 Keywords: Internal mode Photoisomerization Internal oscillation Vector soliton Necklace solitons
a b s t r a c t Internal modes and internal oscillation of vector solitons associated with photoisomerization and necklace solitons in Bessel lattices are researched. While white noise gives rise to the unsmoothness of the vector solitons, the perturbation of internal modes results in the long-distance quasi-periodic oscillation of soliton shape. Internal modes of two-dimensional necklace solitons in Bessel lattices have both real and imaginary parts, which is different with the internal modes of one-dimensional solitons which have only real part. © 2010 Elsevier B.V. All rights reserved.
1. Introduction One of the methods for investigating solitons is to research their internal modes [1,2]. While integrable solitons, such as onedimensional (1-D) Kerr solitons do not have internal mode, stable non-integrable solitons, such as saturable photorefractive solitons have internal modes. Internal modes are responsible for long-lived quasi-periodic oscillation of soliton shape. A small perturbation to such solitons may turn into its internal mode and create internal oscillation [3]. Internal oscillation of stable solitons is helpful for understanding of soliton dynamics under perturbation and collision dynamics of such solitons. If the internal modes exist, OSSs can temporarily store some of the translational energy in the internal oscillation during collision and retrieve this energy after the collision. This energy exchange mechanism can induce very complex collision structures such as window sequences and even fractal structures [4,5]. A better understanding of such collision process requires a good knowledge of internal oscillation in these solitons. Internal modes and internal oscillation have been analyzed for Klein-Gordon solitons [1], Sine-Gordon solitons [6], fundamental solitons in saturable nonlinear media [7], and bright vortex solitons in cubic–quintic nonlinear media [8]. The internal modes mentioned above are internal modes of scalar solitons. The existence, stability, collision of vector solitons [9–16] have been investigated in detail. As for the internal modes of vector solitons, Kaup et al. investigated the internal modes of vector tem-
⁎ Corresponding author. Department of Physics, Huazhong University of Science and Technology, Wuhan (430074), China. E-mail address: [email protected] (Z. Cai). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.03.019
poral solitons in optical fiber with an arbitrary polarization. They found that internal oscillation eigenmode gave rise to antisymmetric oscillations of the symmetric solitons. Malomed and Tasgal [17] studied internal vibrations of a vector soliton in optical fiber via the variational approximation. On the other hand, all these investigations on internal modes are restricted to 1-D solitons in homogeneous material. Refs. [7, 8] can be regarded as the case of 1-D solitons, as the solitons are azimuth-independent. The investigation of internal modes of multi-pole solitons in photonic lattices remains a challenge. The aim of this paper is to investigate internal modes of 1-D vector solitons and 2-D multi-pole solitons in optical lattices. While vector solitons are multi-component solitons in which all the components interact, 2-D multi-pole solitons in optical lattices are higherdimensional solitons in which the solitons are affected by the linear optical lattices. Both the cases attracted much interest recently and the investigation of internal modes of these solitons will help comprehending the essentiality of such solitons. For the first case, we take the solitons in organic material associated with photoisomerization for example. The photorefractive effects in inorganic crystal and photoisomerization nonlinear effects in organic polymer are ideal nonlinear effects for soliton formation, as they can support optical solitons in power of mW/cm2. Photoisomerization nonlinearity was researched recently [18,19], and then photoisomerization spatial solitons were predicted [20–24] and realized [25,26]. Photoisomerization results in negative nonlinearity spontaneously, and thus only dark or gray OSSs can be formed by photoisomerization when no other light is present. Later it was demonstrated that bright isomerization optical spatial solitons (IOSSs) could be obtained by adding a background light [27–29]. The IOSSs mentioned above are scalar solitons, and no vector isomerization optical spatial solitons (VIOSSs) was researched except that the solution of spontaneous bright-
Z. Cai et al. / Optics Communications 283 (2010) 3342–3347
dark soliton pair was given [28]. We will obtain the solution of brightbright vector soliton, and investigate the initial modes and internal oscillation of VIOSSs. We will see that similar to the case of scalar solitons, the perturbation of internal modes of VIOSS results in the long-distance quasi-periodic oscillation. For the second case, we will investigate internal modes of 2-D multi-pole solitons in Bessel photonic crystal. Such solitons are usually called necklace solitons which are constituted of many pearls. As multi-humped solitons in homogeneous materials are unstable, an optical lattice, such as a Bessel photonic lattice should be presented to stabilize the propagation of the solitons. The study of wave propagation in optical periodic structures such as photonic crystals has attracted great interest in recent years. Periodic systems exhibit unique behavior that differs fundamentally from that of their homogeneous counterparts. In these periodic nonlinear lattices, the underlying dynamics are dominated by the interplay between linear coupling effects among adjacent optical potential and nonlinearity; an exact balance between these two processes results in a self-localized state, namely lattice soliton (LS) or discrete soliton. As eigenmodes of linear Schrödinger equation, Bessel beams can propagate in the free space without diffraction; in addition, Bessel lattices can be induced by nondiffracting Bessel beams which are created in experiments by a conical prism (axicon) [30] or computer-generated hologram [31]. So Bessel lattices become an important concentric optical photonic lattice supporting lattice solitons and attracted much interest [32,33]. We will see, internal modes of necklace solitons in Bessel optical lattices have both real part and imaginary part, which is different with the case of one-dimensional solitons. 2. Bright–bright VIOSSs When the bulk material is homogeneously illuminated by an arbitrarily polarized background light, and two linearly polarized light beams with the same polarization direction propagate collinearly through the material, the mechanic equations governing photoisomerization can be written as [30] h i dNT 2 = −NT q ðσ1 I1 + σ2 I2 Þ cos θ + σb Ib dt + NC q′ σ ′1 I1 + σ ′2 I2 + σ ′b Ib + γNC ; N = NT ðθÞ + NC ðθÞ:
ð1Þ
ð2Þ
where θ is the angle between the orientation of the trans molecule and the polarization direction of the signal laser, I is intensity, subscripts 1, 2 and b denote beam 1, beam 2 and background light; N is the initial distribution of trans molecules, NT(θ) is the density of the trans molecules and NC(θ) is the density of the cis molecules at the angle θ, σ1(σ2) is the absorption cross section of the signal light, σb is the absorption cross section of the background light by trans molecules parallel to the electric field. σ′1(σ′2) is the absorption cross section of the signal light by cis molecules, σ′b is the absorption cross section of the background light by cis molecules. q(q′) is called the quantum yields for the trans-cis (cis-trans) transition. γ is the thermal relaxation rate from cis to trans. To achieve low-loss solitons we choose the wavelength of the signal light in the long wavelength region that is far away from the absorption band. Thus the molecules isomerized to be cis ones are much less than the original trans molecules, and the stationary-state distribution of the concentration of trans molecules NT, given by Eqs. (1) and (2) can be expressed as " NT ðθÞ≈N 1−
# qðσ1 I1 + σ2 I2 Þ cos2 θ + qσb Ib : q′ σ ′1 I1 + σ ′2 I2 + σ ′b Ib + γ
ð3Þ
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The change of refractive index can be further derived [30]: 2
Δ n˜ j = αj ∫½NT −NT ðI1 = I2 = 0Þ cos θdΩ = 2παj N
b a1 I1 + a2 I2 + b ; − d c1 I1 + c2 I2 + d
ð4Þ where a1 = 25 qσ1 , a2 = 25 qσ2 , b = 23 qσb Ib , c1 = q′σ′1, c2 = q′σ′2, d = q′σ′bIb + γ, αj is a constant relative with the material and the light. The cis-trans transition is faster than the trans-cis transition, namely, d N b, c1 N a1, c2 N a2. All the parameters are positive. It should be pointed that the presence of background light can greatly influence the nonlinear effects, or even change the sign of nonlinearity. As a matter of fact, Δñj b 0 when Ib = 0 [and then b = 0 in Eq. (4)]; Δñj N 0 when Ib ≠ 0, b N ac11 and bd N ac22 in Eq. (4). That is to say, when the background is d absent, the signal lights induce negative index change; when the background light is present, and the proportion of trans-cis transition velocity to cis-trans transition velocity of background light is larger than that of signal lights, the signal lights induce positive nonlinearity. The homogeneous index does not influence the propagation of light, we only need to consider the index change relative with signal lights that is Δnj ðI1 ; I2 Þ = mj
a 1 I1 + a 2 I2 + b ; c 1 I1 + c 2 I2 + d
ð5Þ
where mj = − 2παjN. Suppose that the one-dimensional beams propagate along z axis, and diffract along the x axis direction. Under the paraxial approximation, the slowly varying amplitudes of the coupled beams are determined by the coupled nonlinear Schrödinger equation (NLSE): ∂Aj ðx; zÞ ∂z
2
−
i ∂ Aj ðx; zÞ ikj Δnj − Aj ðx; zÞ = 0: ð j = 1; 2Þ: 2kj ∂x2 nj
ð6Þ
1 = 2 2k2 m , we obtain Introducing the antatz: ζ = − k1nm1 1 z, ξ = − n11 1 the normalized NLSEs: −i
∂A1 ðξ; ζÞ ∂2 A1 ðξ; ζÞ a1 jA1 j2 + a2 jA2 j2 + b − A1 ðξ; ζÞ; = ∂ζ ∂ξ2 c1 jA1 j2 + c2 jA2 j2 + d
ð7aÞ
−i
∂A2 ðξ; ζÞ ∂2 A2 ðξ; ζÞ a jA j2 + a2 jA2 j2 + b −ρη 1 1 2 =ρ A2 ðξ; ζÞ; 2 ∂ζ ∂ξ c1 jA1 j + c2 jA2 j2 + d
ð7bÞ
where ρ =
k1 , k2
η=
k22 m2 n1 . k21 m1 n2
Now we search for the vector soliton solution of the following form A1 ðξ; ζÞ = u1 ðξÞ exp ð−iδ1 ζÞ; A2 ðξ; ζÞ = u2 ðξÞ exp ð−iρηδ2 ζÞ: The amplitude functions satisfy the following coupled differential equations: " # ∂2 u1 ðξÞ a1 u21 + a2 u22 + b = −u ð ξ Þ δ − ; 1 1 ∂ξ2 c1 u21 + c2 u22 + d
ð8aÞ
" # ∂2 u2 ðξÞ a1 u21 + a2 u22 + b = −ηu2 ðξÞ δ2 − : ∂ξ2 c1 u21 + c2 u22 + d
ð8bÞ
The boundary conditions of bright-bright vector soliton are u1 ð∞Þ = u2 ð∞Þ = 0;
ð9aÞ
u ′1 ð0Þ = u ′2 ð0Þ = 0:
ð9bÞ
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Substituting Eq. (9a) into Eqs. (8a) and (8b), we obtain the asymptotic solutions (ξ → ∞) of Eqs. (8a) and (8b): rffiffiffiffiffiffiffiffiffiffiffiffiffi ! b u1 ðξÞ = λ1 exp − −δ1 ξ ; d sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! b u2 ðξÞ = λ2 exp − η −δ2 ξ : d
ð10aÞ
ð10bÞ
ψ″þ +
ψ″− +
ρηδ2 −σ ψþ −ηVψþ −ηV22 ψþ + ψ− −ηV21 ϕþ + ϕ− = 0; ρ ð12cÞ ρηδ2 + σ ψ− −ηVψ− −ηV22 ψþ + ψ− −ηV21 ϕþ + ϕ− = 0: ρ ð12dÞ
where 2
According to the general experimental conditions, we set the following parameter relations: b = a41 , c1 = 8a1, aa12 = σσ12 = 1:6, cc12 = σ1′ = 1:2, d = 1.2a2, η = 0.70, and ρ = 1.195. The integral constants σ2′ are set as λ1 = 1 and λ2 = 1.2. Then we can compute the eigenvalues (δ1, δ2) with the shooting method. Solving ordinary differential Eqs. (8a) and (8b), combining with the initial conditions Eqs. (10a) and (10b), and imposing the first-order derivative of u1 and u2 to be zero, we get δ1 = 0.2459, and δ2 = 0.2530. We further obtain the intensity distribution of the vector soliton which is shown in Fig. 1. 3. Internal modes of VIOSSs Since single-humped vector solitons in saturable nonlinear media have been known to be stable, we now investigate their internal modes. To determine these internal modes, we write the perturbed soliton solutions as
* A1 ðξ; ζÞ = u1 ðξÞ + ϕþ ðξÞ expðiσζÞ + ϕ− ðξÞ exp ð−iσ*ζÞ expð−iδ1 ζÞ;
ð11aÞ
* A2 ðξ; ζÞ = u2 ðξÞ + ψþ ðξÞ expðiσζÞ + ψ− ðξÞ exp ð−iσ*ζÞ expð−iρηδ2 ζÞ
ð11bÞ where u1(ξ) and u2(ξ) are the envelope amplitudes of the optical fields without perturbation, (ϕ+, ϕ−, ψ+, ψ−) are small perturbations, σ is the eigenvalue, and the asterisk represents complex conjugation. Substituting Eqs. (11a) and (11b) into Eq. (6) and dropping higherorder terms in (ϕ+, ϕ−, ψ+, ψ−), we can obtain the following linearized equations: ϕ″þ + ðδ1 −σ Þϕþ −Vϕþ −V11 ϕþ + ϕ− −V12 ψþ + ψ− = 0;
ð12aÞ
ϕ″− + ðδ1 + σ Þϕ− −Vϕ− −V11 ϕþ + ϕ− −V12 ψþ + ψ− = 0;ð12bÞ
2
a1 u1 + a2 u2 + b ; c1 u21 + c2 u22 + d h i 2 2 ða1 d−bc1 Þ + ða1 c2 −a2 c1 Þu2 u1 V11 = ; ðc1 u21 + c2 u22 + dÞ2 h i 2 ða2 d−bc2 Þ + ða2 c1 −a1 c2 Þu1 u1 u2 ; V12 = ðc1 u21 + c2 u22 + dÞ2 h i ða1 d−bc1 Þ + ða1 c2 −a2 c1 Þu22 u1 u2 ; V21 = ðc1 u21 + c2 u22 + dÞ2 h i ða2 d−bc2 Þ + ða2 c1 −a1 c2 Þu21 u22 : V22 = ðc1 u21 + c2 u22 + dÞ2 V=
Eqs. (12a), (12b), (12c) and (12d) are an eigenvalue problem. If σ is real, the vector soliton is linearly stable with respect to internal mode. In contrast, if σ is purely imaginary, the vector soliton is linearly unstable. According to the boundary conditions [ϕ±(ξ → ± ∞) = 0 and ψ±(ξ → ± ∞) = 0] of bright solitons, we can obtain the asymptotic solutions of Eqs. (12a), (12b), (12c) and (12d) at infinity: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! b −ðδ1 −σ Þξ ; ϕþ ðξ→ + ∞Þ = C1 exp − d rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! b −ðδ1 −σ Þξ ; ϕþ ðξ→−∞Þ = C2 exp d rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! b −ðδ1 + σ Þξ ; ϕ− ðξ→ + ∞Þ = C3 exp − d rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! b −ðδ1 + σ Þξ ; ϕ− ðξ→−∞Þ = C4 exp d sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ! ηb σ − ηδ2 − ξ ; ψþ ðξ→ + ∞Þ = C5 exp − d ρ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ! ηb σ − ηδ2 − ξ ; ψþ ðξ→−∞Þ = C6 exp d ρ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ! ηb σ − ηδ2 + ξ ; ψ− ðξ→ + ∞Þ = C7 exp − d ρ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ! ηb σ − ηδ2 + ξ : ψ− ðξ→−∞Þ = C8 exp d ρ where Ci(i = 1, 2, ···, 8) are integral constants. Making use of the above initial conditions and according to that the values of all the internal mode at ξ = 0 or the first-order derivatives of all the internal mode at ξ = 0 should be zero, we obtain σ = 0.0019, and the internal modes are obtained which are shown in Fig. 2. 4. Internal oscillation of VIOSSs
Fig. 1. Numerical solution of the VIOSS. Different color implies different wavelength of each component of the vector soliton.
We study numerically the dynamics of the VIOSS under the perturbation of the internal modes. Firstly, the propagation of the VIOSS without perturbation is plotted in Fig. 3. One can see that each component of the VIOSS does not change their amplitude shape
Z. Cai et al. / Optics Communications 283 (2010) 3342–3347
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Fig. 5. Propagation of each component of the VIOSS under the perturbation of internal modes.
Fig. 2. Internal modes of the VIOSS.
during the propagation. Next, a white noise with variance of ε2 = 0.02 is added to each component of the vector soliton, unsmoothness emerges, however no collapse comes forth, which implies the stability of the VIOSS (See Fig. 4). Finally, we add the perturbation of the internal modes to the vector soliton; their propagation is shown in Fig. 5. Although the shapes of internal modes are distinct with that of the solitons, they do not give rise to the instability of the vector soliton. In the long-distance propagating process, the shape and intensity of VIOSS under the perturbation of the internal modes show the characteristic of quasi-periodical oscillation. 5. Necklace solitons in Bessel lattices and their stability We assume that an optical beam propagates along the z axis in a bulk medium with defocusing Kerr nonlinearity and imprinted transverse modulation of refractive index. The dynamics of soliton is modeled by the 2-D normalized NLSE for the complex field amplitude q: 2
2
∂q 1 ∂ q ∂ q i + =− 2 ∂x2 ∂z ∂y2
pffiffiffi J1 ½ 2r expð−iz + iφÞ, where r2 = x2 + y2, φ is the azimuthal angle. Accordingly the lattice profile pffiffiffi is determined by the beam intensity in the form of Rðx; yÞ = pJ12 ½ 2r, where p = 40 is the modulation depth of the Bessel lattice. We search for the ten-pearled necklace soliton solutions of Eq. (13) numerically in the form of q(x, y, z) = w(x, y)exp (ibz). Here b is the propagation constant, and w(x, y) is real. Substitution of this expression in Eq. (13) leads to
−bw = −
! 2 2 1 ∂ w ∂ w 3 + + w −Rðx; yÞw: 2 ∂x2 ∂y2
ð14Þ
We solve Eq. (14) numerically using the finite difference method and the standard relaxation method. The initial iterative guess solution of w(x, y) is written as α sec h[(r − L)/W]cos(5θ), where α = 1, is the field amplitude, W = 0.5 is the approximation of the width of the first ring of the Bessel lattice. The first-order Bessel function approaches the first peak value when r = L ≈ 1.3. When b = 2, the resulting contour of the ten-pearled necklace soliton are
! 2
+ qjqj −Rðx; yÞq;
ð13Þ
where x and y are transverse coordinates. We suppose that optical lattice is created by a first-order Bessel beam, whose field is given by
Fig. 3. Propagation of each component of the VIOSS without perturbation.
Fig. 4. Propagation of each component of the VIOSS under the perturbation of white noise with variance of ε2 = 0.02.
Fig. 6. (a) Contour of ten-pearled necklace solitons in Bessel lattices for b = 2. (b) The instability parameter [maximum of Re(δ)] vs propagation constant b. The regions where [Re(δ)]max = 0 correspond to linearly stable solitons.
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shown in Fig. 6 (a). The pearls reside homogeneously on the first ring of the first-order Bessel lattice. As internal modes of solitons are related with stable solitons, we first analyze the stability of the necklace solitons in Bessel lattices prior to analyzing internal modes, taking a ten-pearled necklace for example. As is well known that multi-humped solitons in homogeneous materials are unstable. A photonic lattice may (not definitely) suppress the instability and result in the stable soliton propagation. Now we determine the stability of the necklace solitons in Bessel lattice. To perform the linear stability analysis of necklace solitons, we take the perturbed solutions to Eq. (13) as
! 1 ∂2 v ∂2 v 2 2 iδ*v−bv + −w u*−2w v + Rv = 0: + 2 ∂x2 ∂y2
ð16bÞ
from which the eigenvalue δ and eigenmodes u and v are obtained numerically. For a given b, there are many eigenvalues and related eigenmodes. The soliton is stable if all the real parts of δ equal zero [32]. Fig. 6 (b) gives information on the stability of the necklace solitons which shows the maximum of real parts of δ [Re(δ)] vs b. The solitons are unstable when b b 2.65, while they are stable when b N 5. There are three stable windows between these two points. 6. Internal modes of necklace solitons in Bessel lattices
q = ½w + εu expðδzÞ + εv expðδ*zÞ expðibzÞ;
ð15Þ
with an infinitesimal amplitude ε, complex perturbation eigenmodes u and v, and complex instability growth rate δ (* stands for the complex conjugation). Substitution of this expression in Eq. (13) and linearization around the stationary solution results in the following eigenvalue problem: ! 1 ∂ u ∂ u 2 2 iδu−bu + + −w v*−2w u + Ru = 0; 2 ∂x2 ∂y2 2
2
ð16aÞ
As internal modes are related with stable solitons, we search internal modes in stable region, taking the necklace soliton at b = 3 for example. The boundary conditions for internal modes are u ′r ;i →0; v ′r ;j →0; when r→0; ur ;i →0; vr ;j →0; when r→∞:
ð17Þ
In addition, the internal modes (real and imaginary part) should have similar symmetry as the corresponding soliton, and should have ten symmetric peaks and valleys. According to these conditions, the internal modes are obtained as shown in Fig. 7. The related
Fig. 7. Ten-pearled necklace solitons [(a) Intensity profile; and (b) contour] in Bessel lattices with b = 3, and the field distribution of its internal modes. (c) Real part of eigenmode u; (d) Imaginary part of eigenmode u; (e) Real part of eigenmode v; and (f) Imaginary part of eigenmode v.
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perturbation eigenvalue is δ = −1.0515i. As is well known, internal modes of 1-D solitons have only real part [3,7,8]. Differ from the cases of 1-D solitons, internal modes of 2-D necklace solitons have both real part and imaginary part.
in photonic lattices are investigated for the first time, to the best of knowledge.
7. Propagation of perturbed necklace solitons in Bessel lattices
B.Z. Zhang and H.C. Wang are acknowledged for their helpful discussions.
Now we confirm the stability and investigate the dynamics of necklace solitons under perturbation of internal modes. First we simulate the propagation of solitons that is claimed to be unstable. The soliton shown in Fig. 6(a) is selected. This is an unstable soliton according to the results in Section 5, as the parameters b = 2 correspond to nonzero real instability rate [Re(δ)] from Fig. 6(b). The initial condition is chosen as q(x, y, 0) = w(x, y)(1 + ρ), where ρ is a white noise with variance of σ2 = 0.05. The numerical computation soft ComSol is used based on the finite element method. The resulting Video 1 shows that the soliton collapses under the perturbation of the white noise after a propagation distance, by which the instability of the soliton is confirmed. Then we simulate the propagation of the stable soliton under the perturbation of the internal modes which is shown in Fig. 7. The initial field is denoted by Eq. (15), where z = 0, and ε = 3. The Video 2 shows that the necklace soliton oscillates semi-periodically under the perturbation of the internal modes; no collapse occurs after the propagation distance, and no apparent radiation and damping is observed. The robust stability of the soliton is thus confirmed. 8. Conclusions As a summary, internal modes and internal oscillation of vector solitons associated with photoisomerization and necklace solitons in Bessel lattices are researched. We show that the perturbation of internal modes of the vector solitons results in the long-distance quasi-oscillation of soliton shape while the presence of white noise gives rise to the unsmoothness of the vector soliton. After investigate the stability of necklace solitons in Bessel photonic lattices, we obtain the internal modes of these solitons in stable region. Differing from the internal modes of one-dimensional solitons, which have only real part, internal modes of two-dimensional necklace solitons in Bessel lattices have both real and imaginary parts. The necklace solitons oscillate semi-periodically under the perturbation of their internal modes, without apparent radiation and damping. Internal modes of vector solitons in organic materials and two-dimensional solitons
Acknowledgement
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