Hermite-Gaussian Vector soliton in strong nonlocal media

Optics Communications 333 (2014) 253–260

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Hermite-Gaussian Vector soliton in strong nonlocal media Qing Wang a,b, JingZhen Li a,n a Shenzhen Key Laboratory of Micro-Nano Photonic Information Technology, College of Electronic Science and Technology, Shenzhen University, Guangdong, 518060, China b College of Optoelectronic Engineering, Shenzhen University, Guangdong, 518060, China

art ic l e i nf o

a b s t r a c t

Article history: Received 12 June 2014 Received in revised form 26 July 2014 Accepted 28 July 2014 Available online 11 August 2014

The propagation of two mutually incoherent Hermite-Gaussian (HG) beams in strong nonlocal media was studied. We obtained the evolution equations for the parameters of the two beams and found the condition of forming a HG Vector soliton by variational approach. The numerical result, which accords with the analytical solution very well, shows that a series of vector solitons which consisted of differentorder HG beam pairs can be formed in strong nonlocal media. In addition, we found that the phase shifts are not only related to the total incident power, but also related to the orders of the two HG beams. & 2014 Elsevier B.V. All rights reserved.

Keywords: Nonlinear optics Nonlocal media Vector soliton Hermite-Gaussian beam

1. Introduction Spatial Optical soliton, which is a self-trapped optical beam, exists in that the diffraction is exactly balanced by nonlinearity. Early in 1997, Snyder and Mitchell simplified the nonlocal nonlinear Schrodinger equation (NNLSE) to a simple linear model, namely the Snyder–Mitchell model, and found the accessible soliton [1]. This work stimulates people with a strong interest to study the propagation of optical beam in nonlocal media. So far, a great variety of nonlocal solitons have been studied and a series of achievements have been gained. For instance, D. M. Deng obtained the exact analytical nonlocal Laguerre–Gaussian solutions in a cylindrical coordinate system [2]. The propagation of four-petal Gaussian beams in strongly nonlocal media has been discussed by Z. J. Yang [3]. Families of fundamental and DM solitons are discovered in two-dimensional media with anisotropic semilocal nonlinearity [4]. A class of spiraling elliptic solitons in nonlocal nonlinear media without both linear and nonlinear anisotropy was analyzed by G. Liang [5]. The instability suppression of vectornecklace-ring soliton clusters in different degree of nonlocal media was investigated by M. Shen [6]. Z. Y. Bai demonstrated the dynamics of elegant Ince–Gaussian beams in quadratic-index medium [7] and strongly nonlocal nonlinear media [8]. J. C. Liang found that the Bessel–Gaussian beams only have breather state in strong nonlocal media [9]. The analytical solution of nonlocal Hermite–Gaussian breathers and solitons based on the Snyder–Mitchell model has been

n

investigated [10]. Furthermore, S. W. Zhang found the HG solitons in strong nonlocal media with rectangular boundaries [11]. Buccoliero studied the Laguerre and HG soliton clusters in nonlocal media [12]. Hutsebaut demonstrated that the single-component multihump spatial solitons can travel stably in NLC [13]. The experimental observation of scalar multipole solitons was presented [14] and the stability of multipole-mode solitons in nonlocal nonlinear media was addressed [15]. However, to the best of our knowledge, the HG vector soliton, comprising two incoherent orthogonally polarized beams [16,17], has remained unexplored. In addition, as is widely known, multimodal structure has potential applications value in all-optical control technology [18]. Hence it is necessary to derive the exact analytical solution of HG vector soliton and compare it by direct numerical simulation.

2. Theoretical model and variational approach The propagation of two incoherent orthogonally polarized HG beams in nonlocal nonlinear media can be described by the coupled nonlocal nonlinear Schrodinger equations (NNLSE) [7,17,19–21]: ∂ψ j ∂2 ∂2 i þ μð 2 þ 2 Þψ j ∂z ∂x ∂y Z þ1 Z þ1 þ ρψ j Rðx  x0 ; y y0 Þ½jψ j ðx0 ; y0 ; zÞj2 1

Corresponding author. E-mail address: [email protected] (J. Li).

http://dx.doi.org/10.1016/j.optcom.2014.07.079 0030-4018/& 2014 Elsevier B.V. All rights reserved.

1

þjψ 3  j ðx ; y ; zÞj2 dx0 dy0 ¼ 0 0

0

ð1Þ

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Q. Wang, J. Li / Optics Communications 333 (2014) 253–260

where ψj (j¼ 1, 2) are the paraxial optical beams, μ ¼1/2k, ρ ¼kη, k is the wave number in the media without nonlinearity, η represents the material constant. The Lagrange density equation, which corresponds to Eq. (1), is given as follows:  2  2 ∂ψ n ∂ψ ∂ψ  ∂ψ  L ¼ ∑ 2i ðψ nj ∂z j  ψ j ∂zj Þ  μð ∂xj  þ  ∂yj  Þ j ¼ 1;2

 Z þ1 Z þ1  1   þ ρψ j j2 Rðx  x0 ; y y0 Þ½ψ j ðx0 ; y0 ; zÞj2 2 1 1 ð2Þ

Here we look for the trial solution to Eq. (1) in HG-shaped " # x y x2 þy2 H mj ½ exp iθj ðzÞ þ icj ðzÞðx2 þ y2 Þ  2 ψ j ðx; y; zÞ ¼ Aj ðzÞHnj ½ aj ðzÞ aj ðzÞ 2aj ðzÞ ð3Þ where Aj(z) (j¼1, 2) are the amplitudes, θj(z) represent the phases of complex amplitude, aj(z) are the widths and cj(z) are the phasefront curvatures of the two beams. The characteristic length of the strong nonlocal media is larger than the beam width, therefore the response function can be expanded twice and reduced as follow [19,20] 1 1 Rðx  x0 ; y  y0 Þ  R0  γ x ðx  x0 Þ2  γ y ðy  y0 Þ2 2 2

ð4Þ

where R0 ¼R(0,0), γx ¼ R (0,0), γy ¼  R (0,0) (R (0,0) ¼ d2R(x,y)/dx2|(0.0) ¼0 and R(0,2)(0,0)¼ d2R(x,y)/dy2|(0.0) ¼0). Assuming the response function is circular symmetrical, γx ¼ γy ¼ γ [20]. Inserting the trial function into Eq. (2) and integrating Lagrange density over x, y, we obtain the average Lagrange L ¼  2n1 þ m1 n1 !m1 !A21 π ½ðn1 þ m1 þ 1Þa41 þa21

(0,2)

(2,0)

dc1 dz

ð5Þ

The evolution equations for the parameters of the optical beams can be obtained based on the variational approach A2j a2j ¼ A2j0 a2j0 ¼

P j0 2nj þ mj nj !mj !π

ð6aÞ

daj  4μcj aj ¼ 0 dz

ð6bÞ

dcj μ 1 1 ¼  4c2j μ  ργ P j0  ργ P ð3  jÞ0 2 2 dz a4j

ð6cÞ

dθ j 2μðnj þ mj þ 1Þ 1 ¼ þ ρR0 P j0  ργ P j0 a2j ðnj þ mj þ 1Þ 2 dz a2j 1 þ ρR0 P ð3  jÞ0  ργ P ð3  jÞ0 a2j ðn3  j þ m3  j þ 1Þ 2

ð7Þ

where P0 ¼ P10 þP20 is the total incident power. So the evolution of such HG beam pairs depends only on the total initial power. The critical power of stationary soliton can be obtained by setting d2aj/ dz2j |z ¼ 0 ¼ 0 1 k γ a4j0 η 2

ð8Þ

It is obvious that when the total initial power is equal to the two critical powers, i.e., P0 ¼Pc1 ¼Pc2, the two HG beams will both preserve their widths as they travel in the straight path along the z-axis. Namely the stable HG Vector soliton is formed. Furthermore, we can obtain a10 ¼ a20 by Pc1 ¼Pc2. This is another necessary condition of forming a HG Vector soliton. We normalize Eq. (7) by making wj ¼

aj z ; Z¼ 2 a10 ka10

ð9Þ

where wj is the normalized beam width, Z¼z/ka210 is the normalized Propagation Distance. Eq. (7) can be deduced to 2

d wj dZ 2

¼

1 P0  wj P cj w3j

ð10Þ

Assuming wj(0)¼ 1, dwj/dZ|Z ¼ 0 ¼0, we can obtain the analytical solution by Eq. (10) and then compare the solution with numerical simulation.

3. Numerical results

dθ 1 þ μðn1 þ m1 þ1Þð1 þ 4c21 a41 Þ dz

1 þ ρπ 2 22ðn1 þ m1 Þ ðn1 !m1 !Þ2 A41 ½R0 a41  γ a61 ðn1 þ m1 þ 1Þ 2 dc2 dθ 2 þ a22 2n2 þ m2 n2 !m2 !A22 π ½ðn2 þ m2 þ 1Þa42 dz dz þ μðn2 þ m2 þ 1Þð1 þ4c22 a42 Þ 1 þ ρπ 2 22ðn2 þ m2 Þ ðn2 !m2 !Þ2 A42 ½R0 a42  γ a62 ðn2 þ m2 þ 1Þ 2 þ ρπ 2 2n1 þ m1 n1 !m1 !2n2 þ m2 n2 !m2 !A21 A22 a21 a22 ½R0 1 1  γ ðn1 þ m1 þ 1Þa21  γ ðn2 þ m2 þ 1Þa22  2 2

2

P cj ¼

þ jψ 3  j ðx0 ; y0 ; zÞj2 dx0 dy

(2,0)

d aj 4μ2 ¼ 3  2μργ aj ðP j0 þ P ð3  jÞ0 Þ dz2 aj

ð6dÞ

where Pj0 (j ¼1, 2) are the initial powers, aj0 and Aj0 are the initial beam widths and amplitudes, respectively. The evolution equations of the beam widths can be obtained by combining Eqs. (6b) and (6c)

For confirming the predictions of the variational approach, the split-step Fourier transform method will be employed to numerically simulate the propagation of HG beam pairs in strong nonlocal media. We use the approximate solution resulting from the variational approach as the initial condition and assume that the response function of the material is Gaussian-shaped, i.e., R(x, y) ¼ (1/πσ2)exp[  (x2 þy2)/σ2], where σ is the characteristic length of the material response function. Therefore α ¼ σ/a10 represents the degree of nonlocal. For the strong nonlocal case, we have α Z10. The normalized variables are given as: Y¼ y/a10, X ¼x/a10 are the normalized coordinates in the transverse direction, and Wj0 ¼aj0/ a10 (j¼1, 2) are the normalized initial beam widths.

3.1. The HG vector breather The top and second row of Fig. 1 show that when the total initial power is smaller than the critical power, i.e., P0 oPc, the two beams both expand initially which means that the diffraction effect initially overcomes the nonlinear effect. Therefore, the HG vector breather is formed as seen from the third row of Fig. 1. Likewise, the propagation of coupled HG beam pairs in the case of P0 4Pc can be demonstrated numerically in Fig. 2, and the numerical results indicate that the two beams both contracted initially, which means that the nonlinear effect is stronger than diffraction effect. The Fig. 3 displays the evolution lines of two beam widths. The comparison of analytical solution (solid lines) with numerical solution (dashed lines) shows that the analytical HG vector breathers are in good agreement with the numerical simulation in the case of strong nonlocality.

Q. Wang, J. Li / Optics Communications 333 (2014) 253–260

255

Fig. 1. Propagation dynamics of the HG vector breather in the strong nonlocal media with Gaussian-shaped response function. The parameters are chosen as n1 ¼ 0, m1 ¼0, n2 ¼0, m2 ¼1, W10 ¼W20 ¼ 1, P0 o Pc1 ¼Pc2.

3.2. The HG vector soliton

4. The evolution equation for the parameter of the HG beam pairs

Figs. 4–6 display that the two beams can keep their intensity distributions and widths unchanged during propagation when P0 ¼Pc1 ¼ Pc2. Obviously, the numerical results confirm the analytical results that the HG vector soliton can be formed in strong nonlocal media when the total initial power is equal to the critical power. In addition, by comparing Figs. 4 and 5, we find that the HG vector soliton can be formed with arbitrary value of P1/P2 (concretely, P1 ¼1/ 2P2 in Fig. 4 and P1 ¼ 2P2 in Fig. 5). This is because the nonlinear refractive index, which is only dependent on the total incident power duo to the strong nonlocal [1,20], is the same in the two cases as seen from Fig. 7(a) and (b). It is worth noting that the nonlinear refractive index has been enlarged 100 times for convenient comparing.

It is worth noting that, according to definition of the secondorder moment for beam width, the initial HG beam widths are aj0(2nj þ1)1/2 and aj0(2mj þ 1)1/2 in x-direction and y-direction, respectively [10]. The phases and phase-front curvatures of the two beams can be obtained by substitution of Eq. (11) into Eqs. (6c) and (6d)

3.3. Different-order HG beam pairs

cj ¼

In this article, we are interested in the propagation of vector soliton which consisted of different-order HG beam pairs in strong nonlocal media. Concretely, we numerically simulate the propagation of (00)-mode and (11)-mode HG beam pairs in Fig. 4, (10)-mode and (11)-mode HG beam pairs in Fig. 5, (10)-mode and (01)-mode HG beam pairs in Fig. 6. In fact, there are lots of different-order HG beam pairs cannot be listed completely. The numerical results in Figs. 8–10 all illustrate that when the total initial power is equal to the critical power, the two beams can keep their intensity distributions unchanged during propagation. The analytical solution and numerical solution in Fig. 11 both display that the different-order HG beam pairs can preserve their widths as they travel in the straight path along the z-axis in that the diffraction effect is exactly balanced by nonlinear effect, namely, a series of stationary HG vector solitons are formed.

sffiffiffiffiffiffi P θj ðzÞ ¼ ρR0 P 0 z  ðnj þ mj þ 1Þarctan½ cj tan ðβ0 zÞ P0

By solving Eq. (7), we obtain the evolution equation of beam width a2j ¼ a2j0 ½ cos 2 ðβ0 zÞ þ

P cj 2 sin ðβ 0 zÞ P0

ð11Þ

β0 kðP cj =P 0 1Þ sin ð2β0 zÞ

ð12Þ

4½ cos 2 ðβ 0 zÞ þ ðP cj =P 0 Þ sin ðβ 0 zÞ 2

þ ðnj þ mj þ 1Þ½ 

ðP cj =P 0  1Þργ a2j0 P j0 8β 0

ðP cj =P 0 þ 1Þργ a2j0 P j0 z 4

sin ð2β 0 zÞ



ðP cð3  jÞ =P 0 1Þργ a2ð3  jÞ0 P ð3  jÞ0 sin ð2β 0 zÞ þ ðn3  j þm3  j þ 1Þ½ 8β 0 

ðP cð3  jÞ =P 0 þ 1Þργ a2ð3  jÞ0 P ð3  jÞ0 z 4



ð13Þ

where β0 ¼(γη P0)1/2 ¼[γη (P10 þP20)]1/2. From Eq. (13), we can find that the phase shifts are not only related to the total incident

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Fig. 2. Propagation dynamics of the HG vector breather in the strong nonlocal media with Gaussian-shaped response function. The parameters are chosen as n1 ¼ 0, m1 ¼ 0, n2 ¼ 0, m2 ¼1, W10 ¼ W20 ¼ 1, P0 4Pc1 ¼ Pc2.

Fig. 3. Comparison of analytical solution (solid lines) with numerical solution (dashed lines). The parameters are chosen as n1 ¼ 0, m1 ¼ 0, n2 ¼ 0, m2 ¼1, W10 ¼ W20 ¼ 1. The initial conditions are P0 o Pc1 ¼ Pc2 for (a) and P0 4Pc1 ¼ Pc2 for (b).

power, but also related to the orders of the two HG beams. However, according to the theory of large phase shift which deems that the phase shift of strong nonlocal soliton is mainly dependent on the incident power (namely θj ¼ ρR0P0z) [20], we find that the phase shifts of the two beams are nearly the same.

5. Summary We study the propagation of two incoherent orthogonally polarized HG beams in strong nonlocal media by variational approach and numerical simulation. The evolution equations for

the parameters and the critical powers of forming a HG vector soliton are obtained. The numerical result, which is in good agreement with the analytical solution, shows that the HG vector soliton can be formed in strong nonlocal media when the total initial power is equal to the critical power with arbitrary value of P1/P2. In addition, it demonstrates that a series of vector solitons which consisted of different-order HG beam pairs can propagate stably in strong nonlocal media. The analytical solution shows that the phase shifts are not only related to the total incident power, but also related to the orders of the two HG beams. However, we find that the phase shifts of the two beams are nearly the same in strong nonlocal media according to the theory of large phase shift.

Q. Wang, J. Li / Optics Communications 333 (2014) 253–260

257

Fig. 4. Propagation dynamics of the HG vector soliton in the strong nonlocal media with Gaussian-shaped response function. The parameters are chosen as n1 ¼ 0, m1 ¼0, n2 ¼0, m2 ¼1, W10 ¼W20 ¼ 1, P0 ¼ Pc1 ¼Pc2 and P1 ¼ 1/2P2.

Fig. 5. Propagation dynamics of the HG vector soliton in the strong nonlocal media with Gaussian-shaped response function. The parameters are chosen as n1 ¼ 0, m1 ¼0, n2 ¼0, m2 ¼1, W10 ¼W20 ¼ 1, P0 ¼ Pc1 ¼Pc2 and P1 ¼ 2P2.

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Fig. 6. Comparison of analytical solution (solid lines) with numerical solution (dashed lines). The parameters are chosen as n1 ¼ 0, m1 ¼ 0, n2 ¼ 0, m2 ¼ 1, W10 ¼ W20 ¼ 1, P0 ¼ Pc1 ¼ Pc2. The initial conditions are P1 ¼1/2P2 for (a) and P1 ¼ 2P2 for (b).

Fig. 7. Normalized intensity distribution and nonlinear refractive index of HG vector soliton. The green dashed line is the nonlinear refractive index. The orange and purple lines are the Normalized intensity of ψ1 and ψ2. The parameters are chosen as n1 ¼0, m1 ¼0, n2 ¼0, m2 ¼1, W10 ¼ W20 ¼ 1, P0 ¼Pc1 ¼ Pc2. The difference is P1 ¼1/2P2 for (a) and P1 ¼2P2 for (b).

Fig. 8. Propagation dynamics of the HG vector soliton in the strong nonlocal media with Gaussian-shaped response function. The parameters are chosen as n1 ¼ 0, m1 ¼0, n2 ¼1, m2 ¼ 1, W10 ¼ W20 ¼ 1, P0 ¼Pc1 ¼ Pc2.

Q. Wang, J. Li / Optics Communications 333 (2014) 253–260

259

Fig. 9. Propagation dynamics of the HG vector soliton in the strong nonlocal media with Gaussian-shaped response function. The parameters are chosen as n1 ¼1, m1 ¼0, n2 ¼1, m2 ¼ 1, W10 ¼W20 ¼1, P0 ¼ Pc1 ¼ Pc2.

Fig. 10. Propagation dynamics of the HG vector soliton in the strong nonlocal media with Gaussian-shaped response function. The parameters are chosen as n1 ¼1, m1 ¼0, n2 ¼0, m2 ¼1, W10 ¼W20 ¼ 1, P0 ¼ Pc1 ¼Pc2.

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Fig. 11. Comparison of analytical solution (solid lines) with numerical solution (dashed lines). The initial conditions are W10 ¼ W20 ¼ 1, P0 ¼ Pc1 ¼Pc2. The parameters are chosen as (a) n1 ¼ 0, m1 ¼0, n2 ¼1, m2 ¼ 1, (b) n1 ¼1, m1 ¼ 0, n2 ¼ 1, m2 ¼1, (c) n1 ¼1, m1 ¼ 0, n2 ¼0, m2 ¼1.

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