## Abstract

We address the propagation dynamics of the spiraling elliptic beams in nonlocal nonlinear media with losses based on the variational approach. It is found that the spiraling elliptic beams exhibit complicated behaviors, which result from the combined effects of the losses and orbital angular momentum (OAM). The OAM brings in an effective anisotropic diffraction and rotation for the spiraling elliptic beams. However, due to the losses, the rotation of the spiraling beams slows down. Besides, the ellipticity of the spiraling elliptic beams is greatly affected by the lossesand the OAM. When the OAM is not equal to its critical value, a periodic oscillation of the ellipticity is found in the presence of losses. However, when the OAM is equal to the critical one, the ellipticity of the spiraling elliptic beam remains unchanged during propagation regardless of the loss factor. The comparisons between our approximate analytic solutions and numerical simulations confirm our results.

© 2017 Optical Society of America

## 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 spiraling elliptic beams carrying the OAM have attracted quite a lot of interest in recent years [2–7]. It was demonstrated in [2] that the contribution of OAM to the dynamics of elliptic beams in nonlinear self-focusing media is twofold. First, it effectively strengthens the diffraction against self-focusing, which can suppress collapse in Kerr media. Second, it preserves the elliptic profile of stably rotating solitons in optical media with collapse-free nonlinearities. The role of OAM in the formation of the spiraling elliptic beams was then systematically discussed in [3], where it was found that the OAM results in the effective anisotropic diffraction. The OAM enhances the diffraction in the major axis direction and weakens the diffraction in the minor axis direction. The dynamical properties of the spiraling elliptic beams in nonlocal nonlinear media were analytically discussed in [4], and the exact analytical solution was derived in the strongly nonlocal nonlinear limit (called the Snyder-Mitchell model [8]). Very recently, the spiraling elliptic soliton in a finite nematic liquid crystal cell was studied in [5], which shows that a shelf of radiation forms under the spiraling elliptic solitons radiating momentum. In addition, the dynamics of spiraling elliptic beams were discussed in the saturable nonlinear media with linear anisotropy, where the linear anisotropy exhibits an important effect on the dynamics of spiraling elliptic beams. The rotation and the molecule-like libration were predicated for small and large linear anisotropy, respectively.

To observe the spiraling elliptic solitons experimentally, a factor that must be considered is the losses, which more or less affect the propagation properties of spiraling elliptic solitons obtained in lossless situation. To our knowledge the propagation dynamics of the spiraling elliptic beams in lossy nonlinear media have not been reported. In this paper, taking into the loss factor, we discuss the propagation properties of spiraling elliptic beams in nonlocal nonlinear media. Our theoretical analysis will pave the way for the experimental observation of the spiraling elliptic solitons in nonlocal nonlinear media, such as nematic liquid crystal (NLC) [9–12], lead glass [13,14], and thermal nonlinear liquid [15].

## 2. Model

The propagation of the (1+2)-dimensional optical beams in nonlocal nonlinear media with losses is described by the nonlocal nonlinear Schrödinger equation (NNLSE) with damping

**r**and the longitudinal coordinate

*z*are scaled to the input beam width and Rayleigh length, respectively, ${\nabla}_{\perp}^{2}=\frac{{\partial}^{2}}{\partial {x}^{2}}+\frac{{\partial}^{2}}{\partial {y}^{2}}$ is the transverse Laplacian operator,

*∊*is the loss factor,

*R*is the response function of the media, which takes the form of the Gaussian function in the paper, i.e. [16,17] where

*w*is the width of the response function.

_{m}By introducing the transformation [18,19]

NNLSE (1) is changed to*n*. Then the NNLSE is simplified to the Snyder-Mitchell model [8, 20] with a lossy term

*P*= ∬ |

*ψ*(

**r′**)|

^{2}d

^{2}

**r′**is the input optical power. The Snyder-Mitchell model (6) can be stated as a variational problem in terms of the following Lagrangian density [18,19]

## 3. Variational solution

We introduce a trial function [2,3]

*P*= ∫ |

*ψ*|

^{2}d

^{2}

**r**is the optical power,

*b*(

*z*) and

*c*(

*z*) are the semi-axes of the elliptic beam,

*ϕ*=

*B*(

*z*)

*X*

^{2}+ Θ(

*z*)

*XY*+

*Q*(

*z*)

*Y*

^{2}+

*ϑ*(

*z*) is the phase, and the rotating coordinates such as

*X*and

*Y*are defined as

*X*=

*x*cos

*β*(

*z*) +

*y*sin

*β*(

*z*),

*Y*= −

*x*sin

*β*(

*z*) +

*y*cos

*β*(

*z*), where

*β*is angle between the semi-axes of the elliptic beam and the

*x*-axis or the

*y*-axis. When

*β*= 0 or

*π*/2, the beam (8) is of a standard ellipse, whose semi-axes are parallel to the

*x*-axis and

*y*-axis, respectively. But for the general case the shape of the beam (8) is an oblique ellipse. It should be noted that a cross term Θ

*XY*exists in the expression of the phase

*ϕ*. We have shown that the cross term in the phase results in the rotation of the optical beam [3]. The optical beam in the form of Eq. (8) owns the OAM

*z*. Then, following the standard procedures of the variational approach, we obtain Euler-Lagrangian equations for the parameters of the trial-function

*ψ*

*φ*. It can be seen from Eq.(4) that the decay of the optical power for the beam

*φ*is included in the exponential exp(−

*∊z*). However, it can be found in the following that the OAM in lossy media is conserved. Indicated by the Schrödinger equation in quantum mechanics, it is found that the NNLSE (1) can be restated as

*i∂*

_{z}*φ*=

*Ĥφ*, where Hamiltonian operator $\widehat{H}=-(1/2){\nabla}_{\perp}^{2}-\mathrm{\Delta}n-i\u220a$. In addition, the OAM operator can be expressed as [21]

*Ô*= −

*i*(

*x∂*−

_{y}*y∂*). It can be proven that the commutator [

_{x}*Ô*,

*Ĥ*] = 0 in a similar way used in [21], which reveals that the OAM is a conserved quantity. In fact, it is well known that every conserved quantity corresponds to a kind of symmetry [22]. The NNLSE (1) owns the exchange symmetry between

*x*and

*y*in the transverse, which demonstrates the conservation of OAM. The angular velocity of the spiraling elliptic beams is obtained from Eq. (13) Substitutions of Eqs. (13) (14) and (15) into Eqs. (16) and (17) yield

## 4. Discussions

In this section, the effects of the losses on the intensity(= |*φ*|^{2}), the semi-axes, the angular velocity, and the ellipticity of the spiraling elliptic beams will be discussed analytically and numerically. The method of numerical simulation for the NNLSE (1) used here is the split-step Fourier method.

By setting
$\frac{{d}^{2}b}{d{z}^{2}}=\frac{{d}^{2}c}{d{z}^{2}}=0$ and *∊* = 0, we can obtain the lossless soliton, which preserves its beam width unchanged when the input power and the input OAM are equal to their critical values, i.e.

*∊*= 0.01,

*P*=

*P*,

_{c}*σ*=

*σ*,

_{c}*b*= 2,

*c*= 1, and

*w*= 20, we obtain the evolutions of the lossy spiraling elliptic soliton shown in Fig. 1. Although both the input power and the input OAM are equal to their critical values, the beam still diffracts due to the losses, which results in the linear diffraction overcoming nonlinear focusing. As a result of the losses, the optical intensity decays and the semi-axes

_{m}*b*and

*c*expand, as shown in Fig. 2(a)–2(c). From Fig. 2(d), it is found that the angular velocity decreases as the nonlinearity decays due to the losses. The reasons for these phenomena can be stated as follows. Inserting Eqs. (21) and (23) into (18), we have Obviously, the expansion of semi-axes

*b*and

*c*as a result of the losses weakens the angular velocity of the lossy spiraling elliptic soliton. The decrease of the angular velocity is confirmed by the numerical simulations, as shown in Fig. 1.

For the general cases that *P* ≠ *P _{c}* or

*σ*≠

*σ*, we can explore the lossy spiraling elliptic breather by solving Eqs. (19) and (20). Fig. 3 presents the evolutions of the optical intensity, the semi-axes, and the angular velocity of the lossy spiraling elliptic breather, where we keep the input OAM equals to the critical value but the input optical power does not. Comparing Fig. 3 with Fig. 2, it is found that the optical intensity, the semi-axes, and the angular velocity of the lossy spiraling elliptic breather all exhibit more severe vibrations than those of the lossy spiraling elliptic soliton. Besides, when

_{c}*P*>

*P*, the initial nonlinear focusing overcomes the linear diffraction, so the optical intensity increases, the semi-axes decrease, and the angular velocity increases initially. The opposite happens for the case that

_{c}*P*<

*P*. Regardless of the input power, all the spiraling beams finally diffract and the rotation slows down due to the losses. Fig. 4 shows the evolutions of the optical intensity, the semi-axes, and the angular velocity of the lossy spiraling elliptic breather, where we keep the input optical power equals to the critical value but the input OAM does not. It is shown that the OAM introduces an effective diffraction into the spiraling elliptic beams [3]. Therefore, with the increase of the input OAM, the linear diffraction will overcome the nonlinear focusing and the spiraling elliptic beams diffracts as a whole. Conversely, when we decrease the input OAM, the spiraling elliptic beams converges as a whole. Besides, it is noted that the angular velocity increases when the input OAM increases. But regardless of the input OAM, all the spiraling beams finally diffract and the rotation slows down due to the losses.

_{c}We now discuss the ellipticity under the influence of the loss, which is a key parameter of the elliptic beam. It is interesting to find that the ellipticity of the spiraling elliptic beam remains unchanged, regardless of the loss factor *∊*, during propagation when the initial OAM equals to the critical value, as shown in Fig. 5 (a). The reason is stated as follows. It is well known that the larger the beam width, the weaker the diffraction is. Therefore, for the elliptic beams, the diffraction is stronger in the minor axis and weaker in the major axis. We have shown that the OAM weakens the diffraction in the minor-axis direction and enhance the diffraction in the major-axis direction [3]. Then, it is well expected that for appropriate OAM, the diffraction of the elliptic beams equals in all directions, and the ellipticity remains unchanged, where the appropriate OAM is just the critical value mentioned above. When *σ* ≠ *σ _{c}* the ellipticity of the spiraling elliptic beam changes periodically with the period increasing at the first stage of the evolutions for the nonzero loss factor

*∊*, and then it approaches to a saturation, as shown in Fig. 5 (b). Periodic changes of the ellipticity is due to the combined effects of the isotropic nonlinearity and the effective anisotropic diffraction resulting from the OAM. Due to the losses, the nonlinear effect becomes weaker and weaker as the beam propagates in the lossy media. Therefore, the periodic changes of the ellipticity disappear when the nonlinear effect is negligible. Fig. 6 shows the evolutions of the ellipticity of the spiraling elliptic beam before the disappearance of periodicchanges for different loss factors. The loss factor determines the oscillating frequencies of ellipticity. As can be seen from Fig. 6 that the ellipticity oscillates more rapidly for smaller loss factors, which can be explained as follows. The evolutions of

*b*and

*c*are governed by Eqs.(19) and (20), which are similar to the Newton’s second law formally. The last term −

*Pγ*exp(−2

*∊z*) acts as an elastic force (restoring force) of non-harmonic oscillator, and determines the oscillating frequency. The amplitude of the ellipticity is determined by the OAM, which results from the fact that the OAM leads to the effective anisotropic diffraction [3]. When the initial OAM is larger than the critical value, the ellipticity oscillates between the value

*b*(0)/

*c*(0) and a larger one, as shown in Fig. 6(a) and 6(b). When the initial OAM is smaller than the critical value, the ellipticity oscillates between the value

*b*(0)/

*c*(0) and a smaller one, as shown in Fig. 6(c) and 6(d).

## 5. Conclusion

We have discussed the propagation properties of the spiraling elliptic beams in nonlocal nonlinear media with losses. Based on the variational approach, we obtained the lossy spiraling elliptic solitons and breathers, which are the approximate analytical solutions of the nonlocal nonlinear Schrödinger equation in the presence of the losses. The optical intensity, the beam width, and specially the angular velocity are analytically and numerically discussed in detail. Due to the losses, the spiraling beams finally diffract independent of the input power, and the rotation slows down regardless of the input OAM. The numerical simulations agree well with the variational approximate solutions. The theoretical analysis in the paper will pave the way for the experimental observation of the spiraling elliptic solitons in nonlocal nonlinear media.

## Funding

National Natural Science Foundation of China (NSFC) (11604199); Key Research Fund of Higher Education of Henan Province, China (16A140030).

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