Spiraling elliptic beam in nonlocal nonlinear media

Guo Liang; Qi Guo; Wenjing Cheng; Naiqiang Yin; Ping Wu; Hongmin Cao

doi:10.1364/OE.23.024612

1. Introduction

The technologies associated with orbital angular momentum (OAM), including spatial light modulators and hologram design, have found their own applications ranging from optical tweezers to microscopy [1]. A kind of the spiraling elliptic solitons carrying the OAM was introduced in saturable nonlinear media [2], and was systematically discussed in media of the nonlocal nonlinear media without anisotropy [3], where it was claimed that the OAM can result in the effective anisotropic diffraction for the spiraling elliptic beams. In the nonlinear media with linear anisotropy, the OAM is not conserved. Depending on the linear anisotropy of the media, two kinds of evolution behaviors for the dynamic breathers, rotations and molecule-like librations were predicated analytically and confirmed in numerical simulations [4]. Very recently, the polarized vector spiraling elliptic solitons were reported to stably exist in nonlocal nonlinear media [5]. It was shown that the spiraling elliptic solitons can be formed and can propagate stably in the waveguide induced by orthogonally polarized Gaussian beams.

It has been claimed that [3] only when the input optical power is equal to the critical power and the input OAM is equal to the critical OAM the spiraling elliptic beams can evolve as the spiraling elliptic solitons. In fact, external perturbations will easily make the input power and the input OAM deviate from their critical values, in this general case the evolution of the spiraling elliptic beams is still an open question. For the special case of the strongly nonlocal nonlinear (SNN) media in which the characteristic length of the material response function is much larger than the beam width, the propagation equation can be linearized to the well-known Snyder-Mitchell model (SMM) [6]. The propagation of optical beams in SNN media can be simply regarded as a self-induced fractional Fourier transform [7], then there exists a one-to-one correspondence between free propagation and propagation in SNN media [8]. In this paper we use the established one-to-one correspondence to analytically investigate the dynamic properties of the spiraling elliptic beams. We find that both the deviations from the critical power and the deviations from the critical OAM can make the spiraling elliptic beams breathe. The decrease (increase) of the OAM or the increase (decrease) of the power can both make the spiraling elliptic breathers contract (diffract), however, there still exist differences between them. In addition, the rotating speed can be changed by the input optical power and the input OAM, it may have potential applications in the controlling of the optical beams.

2. Theoretical model and analytical solution

The propagation of the (1+2)-dimensional optical beam in nonlocal nonlinear media is described by the nonlocal nonlinear Schrödinger equation (NNLSE) [6, 9, 10]

i \frac{\partial ψ}{\partial z} + \frac{1}{2} \nabla_{⊥}^{2} ψ + Δ n ψ = 0,

where ψ(r, z) is the complex amplitude envelope, Δn = ∬R(r − r′)|ψ(r′)|²d²r′ is the light-induced nonlinear refractive index, z is the longitudinal coordinate, r is the transverse coordinate vector with r = xe_x + ye_y,

\nabla_{⊥}^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}

is the transverse Laplacian operator, and R is normalized symmetrical real spatial response of the media such that Δn = ∬R(r′)d²r′ = 1. In the case of strongly nonlocal nonlinear (SNN) media we need only keep the first two terms of the expansion of Δn and the NNLSE is simplified to the Snyder-Mitchell model [6, 11]

i \frac{\partial ψ}{\partial z} + \frac{1}{2} \nabla_{⊥}^{2} ψ - \frac{1}{2} γ^{2} P_{0} (x^{2} + y^{2}) ψ = 0,

where

γ^{2} = - \frac{1}{2} \partial_{x}^{2} {R (x, y) |}_{x = 0, y = 0} = - \frac{1}{2} \partial_{y}^{2} {R (x, y) |}_{x = 0, y = 0}

, and P₀ = ∬|ψ(r′)|²d²r′ is the input optical power.

With an established one-to-one correspondence between free and SNN propagation [7, 8, 12], it would be easy to deal with the propagation of an arbitrary field in SNN media by using the well-developed theory of free propagation. The expression of paraxial optical beams at any propagation distance z in SNN media can be obtained by the following integral formula [7, 12]:

\begin{array}{l} ψ (x, y, z) = & - \frac{i γ \sqrt{P_{0}}}{2 π sin (γ \sqrt{P_{0}} z)} exp [\frac{i γ \sqrt{P_{0}} (x^{2} + y^{2})}{2 tan (γ \sqrt{P_{0}} z)}] \iint ψ (x_{0}, y_{0}, 0) \\ \times exp [\frac{i γ \sqrt{P_{0}} (x_{0}^{2} + y_{0}^{2})}{2 tan (γ \sqrt{P_{0}} z)} - \frac{i γ \sqrt{P_{0}} (x x_{0} + y y_{0})}{sin (γ \sqrt{P_{0}} z)}] d x_{0} d y_{0}, \end{array}

where ψ(x₀, y₀, 0) is the input optical beam. The input optical beam considered here has the form of

ψ (x_{0}, y_{0}, 0) = \sqrt{\frac{P_{0}}{π b c}} exp (- \frac{x_{0}^{2}}{2 b^{2}} - \frac{y_{0}^{2}}{2 c^{2}}) exp (i Θ x_{0} y_{0}),

which has an elliptic shape with b and c being the major semiaxis and the minor semiaxis, and has a cross term Θx₀y₀ in the phase. We have shown that the cross term in the phase will result in the rotation of the optical beam [3]. The optical beam expressed as Eq. (4) owns the OAM

M = Im \int \int ψ^{*} (r \times \nabla ψ) d^{2} r = \frac{P_{0}}{2} (b^{2} = c^{2}) Θ .

It is known that the NNLSE, i.e. Eq. (1) conserves the OAM [13]. Substituting Eq. (4) into Eq. (3) yields the expression of the optical beam at any propagation distance z:

\begin{array}{l} ψ (x, y, z) & = & ψ_{0} exp (- \frac{x^{2}}{2 w_{x}^{2}} - \frac{y^{2}}{2 w_{y}^{2}} - \frac{x y}{2 w_{x y}}) \\ \times exp (i c_{x} x^{2} + i c_{y} y^{2} + i c_{x y} x y + i \frac{arctan λ - π}{2}), \end{array}

where

\begin{array}{l} ψ_{0} = & {[{(1 + b^{2} c^{2} Θ^{2} - b^{2} c^{2} P_{0} γ^{2} {cot}^{2} β)}^{2} + {(b^{2} + c^{2})}^{2} P_{0} γ^{2} {cot}^{2} β]}^{- 1 / 4} \\ \times \frac{P_{0} \sqrt{b c} γ}{\sqrt{π}} csc β, \end{array}

w_{x} = \frac{1}{2 b γ} \sqrt{\frac{α_{r}}{P_{0} [1 + c^{4} P_{0} γ^{2} + b^{2} c^{2} Θ^{2} + (c^{4} P_{0} γ^{2} - b^{2} c^{2} Θ^{2} - 1) cos 2 β]}},

w_{y} = \frac{1}{2 c γ} \sqrt{\frac{α_{r}}{P_{0} [1 + b^{4} P_{0} γ^{2} + b^{2} c^{2} Θ^{2} + (b^{4} P_{0} γ^{2} - b^{2} c^{2} Θ^{2} - 1) cos 2 β]}},

w_{x y} = \frac{α_{r}}{8 b^{2} c^{2} (b^{2} + c^{2}) P_{0}^{3 / 2} γ^{3} Θ},

\begin{array}{l} c_{x} = & \frac{8 \sqrt{P_{0}} γ cos β {sin}^{7} β}{α_{i}} [P_{0} γ^{2} (b^{4} - c^{4} + b^{4} c^{4} P_{0} γ^{2}) - 2 b^{2} c^{2} Θ^{2} - b^{4} c^{4} Θ^{4} \\ - 1 + (1 - c^{4} P_{0} γ^{2} + 2 b^{2} c^{2} Θ^{2} + b^{4} c^{4} P_{0}^{2} γ^{4} + b^{4} c^{4} Θ^{4} \\ - b^{4} P_{0} γ^{2} + 2 b^{4} c^{4} P_{0} γ^{2} Θ^{2}) cos 2 β], \end{array}

\begin{array}{l} c_{y} = & \frac{8 \sqrt{P_{0}} γ cos β {sin}^{7} β}{α_{i}} [P_{0} γ^{2} (c^{4} - b^{4} + b^{4} c^{4} P_{0} γ^{2}) - 2 b^{2} c^{2} Θ^{2} - b^{4} c^{4} Θ^{4} \\ - 1 + (1 - c^{4} P_{0} γ^{2} + 2 b^{2} c^{2} Θ^{2} + b^{4} c^{4} P_{0}^{2} γ^{4} + b^{4} c^{4} Θ^{4} \\ - b^{4} P_{0} γ^{2} + 2 b^{4} c^{4} P_{0} γ^{2} Θ^{2}) cos 2 β], \end{array}

\begin{array}{l} c_{x y} = & - \frac{16 b^{2} c^{2} P_{0} γ^{2} Θ {sin}^{6} β}{α_{i}} [b^{2} c^{2} (P_{0} γ^{2} - Θ^{2}) - 1 \\ + (b^{2} c^{2} P_{0} γ^{2} + b^{2} c^{2} Θ^{2} + 1) cos 2 β], \end{array}

λ = \frac{(b^{2} + c^{2}) \sqrt{P_{0}} γ cot β}{1 + b^{2} c^{2} Θ^{2} - b^{2} c^{2} P_{0} γ^{2} {cot}^{2} β},

with

\begin{array}{l} α_{r} = & 3 + P_{0} γ^{2} (b^{4} + c^{4} - 2 b^{4} c^{4} Θ^{2}) + 4 [b^{4} c^{4} (P_{0}^{2} γ^{4} - Θ^{4}) - 2 b^{2} c^{2} Θ^{2} - 1] cos 2 β \\ + [b^{4} c^{4} (P_{0}^{2} γ^{4} + Θ^{4} + 2 P_{0} γ^{2} Θ^{2}) - P_{0} γ^{2} (b^{4} + c^{4}) + 2 b^{2} c^{2} Θ^{2} + 1] cos 4 β \\ + 6 b^{2} c^{2} Θ^{2} + 3 b^{4} c^{4} Θ^{4} + 3 b^{4} c^{4} P_{0}^{2} γ^{4}, \end{array}

\begin{array}{l} α_{i} = & [{(b^{2} c^{2} Θ^{2} + 1)}^{2} - P_{0} γ^{2} (2 b^{4} c^{4} Θ^{2} - b^{4} - c^{4}) {cot}^{2} β + b^{4} c^{4} P_{0}^{2} γ^{4} {cot}^{4} β] \\ \times {(cos 2 β - 1)}^{5}, \end{array}

β = \sqrt{P_{0}} γ z .

The analytic solution, i.e. Eq. (6), the key result of the paper, is an exact one to the Snyder-Mitchell model, i.e. Eq. (2). The spiraling elliptic beam expressed as Eq. (6) carries the OAM and rotates on the cross section perpendicular to the propagation direction z during the propagations.

3. Discussion of solutions

From Eq. (6), it can be easily shown that the shape of the spiraling elliptic beam is an inclined ellipse, whose major axis and minor axis are not parallel to x-axis and y-axis. The elliptic optical beam, i.e. Eq. (6), can be transformed to its standard elliptic form by a coordinate rotation through angle ϑ, which satisfies the following equation

tan (2 ϑ) = \frac{γ_{x y}}{γ_{x x} - γ_{y y}},

where

γ_{x x} = \frac{1}{w_{x}^{2}}, γ_{y y} = \frac{1}{w_{y}^{2}}, γ_{x y} = \frac{1}{w_{x y}} .

One of the semi-axes of the standard elliptic optical spot is

w_{b} = \sqrt{\frac{1}{γ_{x x} {cos}^{2} ϑ + γ_{y y} {sin}^{2} ϑ + γ_{x y} sin ϑ cos ϑ}},

the other is

w_{c} = \sqrt{\frac{1}{γ_{x x} {sin}^{2} ϑ + γ_{y y} {cos}^{2} ϑ - γ_{x y} sin ϑ cos ϑ}} .

The angular velocity of the optical beam in the xyz-coordinate frame can be obtained from Eq. (18)

ω \equiv \frac{d ϑ}{d z} = \frac{1}{2 {(1 + tan 2 ϑ)}^{2}} \frac{d}{d x} (\frac{γ_{x y}}{γ_{x x} - γ_{y y}}) .

3.1. Spiraling elliptic solitons

Generally speaking, the semi-axes w_b and w_c are the functions of the propagation distance z. As the special case, the spiraling elliptic solitons can be obtained by letting $\frac{\partial w_{b}}{\partial z} = 0$ and $\frac{\partial w_{c}}{\partial z} = 0$ . For the spiraling elliptic solitons, from Eqs. (20) and (21) we have

P_{c} = \frac{{(b^{2} + c^{2})}^{2}}{4 b^{4} c^{4} γ^{2}},

Θ_{c} = \frac{b^{2} - c^{2}}{2 b^{2} c^{2}} .

The critical OAM can be obtained by combining Eq. (5) and Eq. (24) as

σ_{c} \equiv \frac{M_{c}}{P_{c}} = \frac{{(b^{2} - c^{2})}^{2}}{4 b^{2} c^{2}} .

Substitution of Eqs. (23) and (25) into Eq. (22) yields

ω_{c} = \frac{b^{2} + c^{2}}{2 b^{2} c^{2}},

which shows that the spiraling elliptic solitons make constant-angular rotations. Then the period of rotation can be obtained as

T = \frac{2 π b^{2} c^{2}}{b^{2} + c^{2}}

.

We suppose the material response to be the Gaussian function [14, 15]

R (x, y) = \frac{1}{2 π w_{m}^{2}} exp (- \frac{x^{2} + y^{2}}{2 w_{m}^{2}}),

where w_m is the width of the response function. The ratio of the width of the response function to the scale in the transverse dimension occupied by the optical beam determines the degree of nonlocality, which reads

δ \equiv \frac{w_{m}}{max (b, c)}

. The larger is δ, the stronger is the degree of nonlocality. For the response function of Eq. (27), we obtain that

γ = \frac{1}{\sqrt{2 π} w_{m}^{2}}

. Figs. 1 present the propagation dynamics of the spiraling elliptic solitons. The excellent agreement between the exact analytical spiraling elliptic soliton solutions of the Snyder-Mitchell model, i.e. Eq. (2), and the numerical simulation of the NNLSE for the case of strong nonlocality δ = 10 is shown in Figs. 1. When the degree of nonlocality becomes weaker (δ = 5), the approximations of the analytical results of the Snyder-Mitchell model to the exact ones of the NNLSE are a little bit worse. When the degree of nonlocality is weak enough (δ = 1), the dynamics of the spiraling elliptic solitons will be unstable.

Fig. 1 Propagation dynamics of the spiraling elliptic solitons in the Gaussian-shaped response material. (a)–(d) Transverse normalized intensity distributions from the analytical solution, (e)–(h) distributions from the numerical simulation (δ = 10), (i)–(l) distributions from the numerical simulation (δ = 5), (m)–(p) distributions from the numerical simulation (δ = 1). The different columns represent the different propagation distances given in the top of the figure. The parameters are chosen as P₀ = P_c, b = 2 and c = 1.

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3.2. Spiraling elliptic breathers

In general, for the case that either the input optical power or the input OAM is not equal to the critical power and the critical OAM, all the spiraling elliptic beams will evolve as the spiraling elliptic breathers, following Eq. (6). In the following, we will consider the breathing behavior of the spiraling elliptic breathers, and discuss the evolutions of the optical intensity, the beam width, and the angular velocity in details.

3.2.1. Breathing behavior resulting from input optical power

It is well known that [6] for P₀ > P_c, self-focusing initially overcomes diffraction and the beam initially contracts shown in Figs. 2, whereas for P₀ < P_c, the reverse happens and the beam initially expands shown in Figs. 3. The excellent agreement between the exact analytical spiraling elliptic breathers of the Snyder-Mitchell model and the numerical simulation of the NNLSE for the case of strong nonlocality δ = 10 are also shown in Figs. 2 and Figs. 3, where we have assumed that the input optical power P₀ = 1.5P_c and P₀ = 0.5P_c, respectively, but meanwhile kept the input OAM equal to the critical OAM.

Fig. 2 (Color online) Same as Fig. 1 but with plots corresponding to the spiraling elliptic breathers. The parameters are chosen as P₀ = 1.5P_c, b = 2 and c = 1.

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Fig. 3 Same as Fig. 2. The parameters are chosen as P₀ = 0.5P_c, b = 2 and c = 1.

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We start our considerations with the evolution of the on-axis optical intensity for the spiraling elliptic breathers. Note, the intensities of the spiraling elliptic beams are periodic functions of the propagation distance z, the period can be obtained from Eq. (7) as $T_{b} = \frac{π}{\sqrt{P_{0}} γ}$ , which can be rewritten as

T_{b} = \frac{\sqrt{2 π} π w_{m}^{2}}{\sqrt{P_{0}}}

for the Gaussian response function of Eq. (27). It is shown from Eq. (28) that the evolutional periods of the spiraling elliptic breathers decrease as the input optical powers increase. Taking parameters as P₀ = 1.5P_c and P₀ = 0.5P_c, the evolutional periods can be calculated to be 4.1 and 7.1, respectively. The analytical results are confirmed by the numerical simulation shown in Fig. 4. Under the assumption that the input OAM equals to the critical OAM, the on-axis optical intensity will increase initially and then decrease if P₀ > P_c, and will decrease initially and then increase if P₀ < P_c. Both the maximum on-axis optical intensity for the case that P₀ > P_c and the minimum on-axis optical intensity for the case that P₀ < P_c can be obtained from Eq. (7) as

I_{m} = \frac{P_{0}}{π b c}

, the ratio of which to the critical optical intensity is I_m/I_c = P₀/P_c, with the critical optical intensity being

I_{c} = \frac{P_{c}}{π b c}

. Therefore, for the spiraling elliptic breathers the ratio of the on-axis optical intensity to the input optical intensity vibrates between 1 and P₀/P_c, which also agrees well with the numerical simulation shown in Fig. 4.

Fig. 4 Comparison of analytical solution (solid red curves) with numerical simulation (dotted blue curves) for the intensity of the spiraling elliptic breathers propagated in the Gaussian-shaped response material, where the lines above the dashed blue line are plotted for P₀ = 1.5P_c and the lines below the dashed blue line are plotted for P₀ = 0.5P_c, respectively, but keeping the input OAM equal to the critical OAM. The parameters are chosen as b = 2, c = 1 and w_m = 20. Dashed blue line denotes the intensity of the spiraling elliptic solitons.

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And then, we investigate the evolution of the major semiaxis w_b and the minor semiaxis w_c of the spiraling elliptic breathers. By using Eqs. (20) and (21), after somewhat complicated calculations we obtain that the ratio of the major semiaxis w_b to b and the ratio of the minor semiaxis w_c to c vibrate between 1 and $\sqrt{P_{c} / P_{0}}$ . Fig. 5 shows the evolution of the semi-axes w_b and w_c of the spiraling elliptic breathers, and the analytical solution agrees well with the numerical simulation. When we take P₀ = 1.5P_c, the major semiaxis w_b vibrates between 1.6 (i.e., $2 / \sqrt{1.5}$ ) and 2, and the minor semiaxis w_c vibrates between 0.8 (i.e., $1 / \sqrt{1.5}$ ) and 1. And when we take P₀ = 0.5P_c, the major semiaxis w_b vibrates between 2 and 2.8 (i.e., $2 / \sqrt{0.5}$ ), and the minor semiaxis w_c vibrates between 1 and 1.4 (i.e., $1 / \sqrt{0.5}$ ).

Fig. 5 Comparison of analytical solution (dashed red curves) with numerical simulation (dashed blue curves) for the semi-axes w_b and w_c of the spiraling elliptic breathers for P₀ = 1.5P_c in (a) and for P₀ = 0.5P_c in (b), respectively, but keeping the input OAM equal to the critical OAM. In numerical simulations, the parameters are chosen as b = 2, c = 1 and w_m = 20.

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In the end, we discuss the angular velocity of the spiraling elliptic breathers. For the case that the input optical power is not equal to the critical power but the input OAM is equal to the critical OAM, the angular velocity can be obtained by using Eqs. (22) and (24)

ω = \frac{2 b^{2} c^{2} (b^{2} + c^{2}) P_{0}}{2 b^{4} c^{4} P_{0} + {(b^{2} + c^{2})}^{2} π w_{m}^{4} + [2 b^{4} c^{4} P_{0} - {(b^{2} + c^{2})}^{2} π w_{m}^{4}] cos (2 \sqrt{P_{0} / P_{c}} ω_{c} z)},

where the critical optical power P_c and the critical angular velocity ω_c are expressed as Eqs. (23) and (26), respectively. It can be found that the spiraling elliptic breathers perform nonuniform rotations, the angular velocity is not a constant but is a periodic function of the propagation distance z. From Eq. (29) with the aid of Eqs. (23) and (26), after some simple calculations, the angular velocity can be found vibrating between ω_c and

\frac{P_{0}}{P_{c}} ω_{c}

, as shown in Fig. 6. It is interesting to note that the angular velocity of the spiraling elliptic breathers depends on the optical power P₀. Consequently, the angular velocity of the spiraling elliptic beams can be controlled by the input optical power.

Fig. 6 Angular velocity of the spiraling elliptic breathers for P₀ = 1.5P_c (dashed red line) and P₀ = 0.5P_c (solid blue line), respectively, but keeping the input OAM equal to the critical OAM.

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3.2.2. Breathing behavior resulting from input OAM

In this section, we consider the evolutions of the spiraling elliptic breathers for the case that the input OAM is not equal to the critical OAM but keeping P₀ = P_c. It is shown that [2, 3] the OAM can introduce an effective diffraction into the spiraling elliptic beams. Therefore, when we increase the input OAM, the spiraling elliptic beams will diffract in the whole, as shown in Fig. 7. Conversely, when we decrease the input OAM, the spiraling elliptic beams will contract in the whole, as shown in Fig. 8. Comparing Figs. 2 and Figs. 8 (or comparing Figs. 3 and Figs. 7), it seems that both the decrease (increase) of the OAM and the increase (decrease) of the power have the similar effects on the evolutions of the spiraling elliptic breathers in the sense that they both make the spiraling elliptic breathers contract (diffract). But we will show that there still exist differences between the effects of the decrease (increase) of the OAM and the increase (decrease) of the power on the evolutions of the spiraling elliptic breathers.

Fig. 7 Same as Fig. 2. The parameters are chosen as P₀ = P_c, σ = 1.5σ_c, b = 2 and c = 1.

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Fig. 8 Same as Fig. 7. The parameters are chosen as P₀ = P_c, σ = 0.5σ_c, b = 2 and c = 1.

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Firstly, we investigate the effect of the OAM on the evolution of the on-axis optical intensity for the spiraling elliptic breathers. To this end, we assume that the input OAM departs from the critical OAM but the input optical power is equal to the critical power, in this case we can obtain from Eq. (7) that the ratio of the on-axis optical intensity I(z) to the input optical intensity I(0) has such three extrema as

κ_{1} = 1,

κ_{2} = \frac{{(b^{2} + c^{2})}^{2}}{4 b^{2} c^{2} + {(b^{2} - c^{2})}^{2} {(σ / σ_{c})}^{2}},

κ_{3} = \frac{\sqrt{3 (b^{4} + c^{4}) + 10 b^{2} c^{2} + {(b^{2} - c^{2})}^{2} {(σ / σ_{c})}^{2}}}{2 (b^{2} + c^{2})},

with κ denoting the extrema of I(z)/I(0). Fig. 9 gives the evolution of the on-axis optical intensity when σ ≠ σ_c and P₀ = P_c. From Eqs. (30), (31) and (32), the three extrema of the intensity can be obtained as 1, 1.05 and 0.69 when σ = 1.5σ_c, and obtained as 1, 0.97 and 1.37 when σ = 0.5σ_c, which are confirmed by the numerical simulation in Fig. 9. Comparing Fig. 9 with Fig. 4, it is observed that although both the decrease (increase) of the OAM and the increase (decrease) of the power can make the spiraling elliptic breathers contract (diffract), the effect of the OAM is obviously more complicated. The decrease (increase) of the OAM makes spiraling elliptic beams contract (diffract) on the whole, but in some small propagation ranges the spiraling elliptic beams diffract (contract).

Fig. 9 Evolution of the on-axis optical intensity for the spiraling elliptic breathers when σ ≠ σ_c and P₀ = P_c. The solid red line denotes the case that σ = 0.5σ_c and the green solid line denotes the case that σ = 1.5σ_c. All the solid lines are plotted for the analytic solution, and the dashed lines are plotted for the numerical simulation under strong nonlocality δ = 10.

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Subsequently, we consider the effects of the OAM on the evolutions of the semi-axes for spiraling elliptic beams. When we increases the input OAM, the OAM will enhance the diffraction, and the semi-axes will be increased to their maxima

w_{bm} = \frac{c}{b^{2} + c^{2}} \sqrt{4 b^{2} c^{2} + {(b^{2} - c^{2})}^{2} {(σ / σ_{c})}^{2}},

w_{c m} = \frac{b}{b^{2} + c^{2}} \sqrt{4 b^{2} c^{2} + {(b^{2} - c^{2})}^{2} {(σ / σ_{c})}^{2}},

respectively, as shown in Fig. 10. Differently form the case in Fig. 5 that it can be observed that the minor semiaxis initially contracts and the major semiaxis initially expands in Fig. 10. On the contrary, when the input OAM decreases, the effective diffraction of the spiraling elliptic beams will be weakened, then the semi-axes will be decreased to their minimums w_bm and w_cm, respectively, but the major semiaxis will initially contract and the minor semiaxis will initially expand, as shown in Fig. 10.

Fig. 10 Evolution of the semi-axes w_b and w_c for the spiraling elliptic breathers when σ ≠ σ_c and P₀ = P_c. The solid red line denotes the analytical solution and the blue solid red line denotes the numerical simulation under strong nonlocality δ = 10.

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In addition, the angular velocity of the spiraling elliptic breathers can be obtained from Eq. (22) when σ ≠ σ_c and P₀ = P_c, as shown in Fig. 11. It is shown from Fig. 11 that the angular velocity of the spiraling elliptic breathers also depends on the input OAM. It was claimed that [2] when the elliptic beams are focused by a cylindrical lens, and the major axis of the elliptically shaped beam makes an angle with the cylindrical lens, the elliptic beams will carry the OAM. Therefore, the angular velocity of the spiraling elliptic beams can also be changed by the angle made between the major axis of the elliptically shaped beam makes and the cylindrical lens.

Fig. 11 Angular velocity of the spiraling elliptic breathers for σ = 1.5σ_c (solid blue line) and σ = 0.5σ_c (solid blue line), respectively, but keeping P₀ = P_c.

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Before concluding, we give a short discussion of the applicability of the Snyder-Mitchell model of Eq. (2), and perform a comparison between the Snyder-Mitchell model and the actual physical systems. The Snyder-Mitchell model [6] is a phenomenological model, and is also the simplified model of the NNLSE for the case of the strong nonlocality (i.e. the the degree of nonlocality δ ≫ 1) and the condition that the response function R(x, y) is symmetrical and regular (or at least twice differentiable) at x = y = 0, such as the extremely instructive Gaussian shaped response function of Eq. (27). The derivation from the NNLSE to the Snyder-Mitchell model is shown for the strong nonlocality in [11]. On the other hand, the response functions of the actual physical systems, such as the SNN media nematic liquid crystal (NLC) and the lead glass, are always singular [11]. The singularity in the response functions results in the actual physical systems such as the NLC and the lead glass can not be simplified to the Snyder-Mitchell model [11]. However, the Snyder-Mitchell model keeps the main features of the SNN media, and the physical properties do not depend strongly on the shapes of the response functions for the SNN media [16, 17]. For example, the theoretical predictions by the Snyder-Mitchell model [6], such as the accessible solitons and the attraction of spatial solitons, have been observed in experiments in the NLC [18, 19, 20] and the lead glass [21]. It is worth mentioning that [7] the fractional Fourier transform existing the SNN media was predicted by the Snyder-Mitchell model, and was also observed in the lead glass. Therefore, the dynamics of elliptic beams in highly nonlocal media discussed in the paper by the Snyder-Mitchell model pave the way to the observation in the NLC and the lead glass in experiments.

4. Conclusion

We analytically discussed the dynamical properties of the spiraling elliptic beams in nonlocal nonlinear media. This class of spiraling elliptic beams carry the orbital angular momentum (OAM), and can rotate on the cross section perpendicular to the propagation direction during the propagations. We obtained the analytical solutions for the evolution period,the beam width, and specially the angular velocity of the spiraling elliptic beams in strongly nonlocal nonlinear media. Although both the deviations from the critical power and the deviations from the critical OAM can make the spiraling elliptic beams breathe, there still exist differences between them. The breathing behaviors resulting from the OAM are more complicated. The rotating speed can be changed by the input optical power and the input OAM, which may have potential applications in the controlling of the optical beams.

Acknowledgments

This research was supported by the National Natural Science Foundation of China, Grant Nos. 11274125 and 11474109.

References and links

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Spiraling elliptic beam in nonlocal nonlinear media

Abstract

1. Introduction

2. Theoretical model and analytical solution

3. Discussion of solutions

3.1. Spiraling elliptic solitons

3.2. Spiraling elliptic breathers

3.2.1. Breathing behavior resulting from input optical power

3.2.2. Breathing behavior resulting from input OAM

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (11)

Equations (34)

Optics Express