Rotation controlling of spiraling elliptic beams in inhomogeneous nonlocal media

The dynamics of spiraling elliptic beams in longitudinally-inhomogeneous nonlocal media with z-varied characteristic length (CL) were discussed. When the response CL of nonlocal nonlinear media gradually varies, the spiraling elliptic beams at the critical powers can evolve as rotating solitons with CL-dependent rotating velocity. This kind of dynamical evolution for spiraling elliptic beam is confirmed to exist in the actual nonlinear media, the nematic liquid crystals, when the CL varies with z. We also investigate the gradient force exerted by an elliptic beam upon micro-particles from the application point of view in optical tweezers, and find that the gradient force also can be conveniently controlled by the varied CL.


Introduction
Currently, one of the major scientific endeavors is to modulate spatial beams exhibiting novel properties in various applications. Among the concerned optical patterns, optical beams carrying the orbital angular momentum (OAM) can exert forces and torques on the microparticles, which make them rotate [1]. The technologies associated with OAM, including spatial light modulators and hologram design, have wide applications in optical tweezers [2], optical trapping [3], imaging [4], and information processing [5,6]. The optical patterns with the OAM are usually associated with optical vortices and related ring-shaped beams with the phase-singularity, including the Laguerre-Gauss beams [7] and the Bessel beams [8], as well as the other hollow beams [9]. However, there is another kind of OAM-carrying patterns, namely the astigmatic (elliptical Gaussian) beam [10], which exhibits the cross phase and is absolutely different from the corkscrew-like phase of the vortices. The OAM makes the elliptic beams rotate during propagations, therefore the astigmatic elliptic beams are also called the spiraling elliptic beams [11,12]. The propagation of elliptical beam is more complicated than that of circular beams [13][14][15]. An elliptic beam will periodically oscillate during its propagation in isotropic media [16,17]. In 2010, the OAM was proposed to stabilize the elliptical solitary wave [11] due to the OAM-induced anisotropic diffraction (AD), and makes the elliptic Gaussian mode exist in isotropic media [12,18], where only the circular eigenmode is supposed to exist for a beam without OAM.
The spiraling elliptic beams exhibit novel properties during their linear and nonlinear propagations. In linear anisotropy media, the rotating velocity of spiraling elliptic beams can be readily controlled by both initial OAM and the anisotropy parameter of media [19]. When a Hermite-Gaussian beam is modulated with a cross-phase, the inter-conversion between the Hermite-Gaussian modes and the Laguerre-Gaussian modes can be achieved [20]. As for the nonlinear case, the spiraling elliptic solitons are theoretically predicated when the optical power and the initial OAM are both equal to their corresponding critical values in media with the saturable nonlinearity [11] and the nonlocal nonlinearity [12]. Recently, the spiraling elliptic soliton is observed for the first time in cylindrical lead glass [21].
In nonlocal nonlinear media, such as nematic liquid crystal [22] and lead glass [23], the evolution of spatial optical beam is governed by the nonlocal nonlinear Schrödinger equation (NNLSE) [24][25][26], in which the nonlinear term is expressed by a convolution between optical intensity and the response function. The characteristic length (CL) of the response function is in fact quite handy to be modulated. For the nematic liquid crystal, its CL depends closely on the bias voltage [27]. While, the CL of the lead glass is determined by sample sizes [23]. Therefore, it is possible to make the CL vary with z in the two actual nonlinear media [28]. In this paper, we will explore the dynamics of spiraling elliptic beams in such longitudinally-inhomogeneous nonlocal media.

Rotation of spiraling elliptic beams and transitions of soliton states
The propagation of spatial optical beam in nonlocal nonlinear media is modeled by the following NNLSE [24][25][26] i ∂φ ∂z + 1 2 where φ(x, y, z) is the complex amplitude, x and y are the transverse coordinates, z is the longitudinal coordinate, and R is the response function of the media. We take the Gaussian response for analytical discussions in this section where σ is the CL. As mentioned in the introduction, it is possible to realize the z-dependent CL in actual nonlocal nonlinear media such as the nematic liquid crystal and the lead glass. In the following, we will study the evolution of a spiral elliptic beams in longitudinal nonhomogeneous nonlocal medium [12,29] φ(x, y, 0) = P 0 πbc where P 0 = |Φ| 2 dx dy is the optical power, b and c are the semi axes of the elliptic beam, and the last term exp(iΘxy) is called the cross phase [12,[18][19][20][21], which contributes to the OAM [12] Θ is the cross phase coefficient, which can be adjusted by the angle between the major axis of the elliptically shaped beam and the cylindrical lens in experiments [11]. If the CL is a constant, the spiraling elliptic beams can evolve as solitons when the optical power and the initial OAM both equal to their critical valves under the strong nonlocality [12,28] with τ c = M c /P c , where P c and M c are the critical power and OAM, respectively. We have demonstrated that the ellipticity of the spiraling elliptic beam during propagations is only determined by the OAM, and is independent of the nonlinearity [21], which can be also confirmed by the equation (23) of reference [18] In reference [12], it was discovered that the decaying rate of the OAM is extremely low for the spiraling elliptic solitons in nonlocal nonlinear media. Furthermore, the larger is the degree of nonlocality, the lower is the decay of the OAM. In this paper, the nonlocality is quite strong. Therefore, under the critical OAM the elliptic beam will keep its ellipticity ρ invariable during propagations. Combining equations (5) and (6), the beam width can be analytically obtained  The critical angular velocity of soliton rotation is ω c = (b 2 +c 2 )/2b 2 c 2 [12,28], which can be rewritten with the aid of equation (7) Then the total rotation angles of the spiraling elliptical soliton during propagations can be obtained Obviously, the beam widths, angular velocities and rotation angles all can be controlled by the CL. Then we will simulate the evolutions of spiraling elliptic beam given by equation (3) in longitudinally-inhomogeneous nonlocal media by using split-step Fourier method. The gradually-decreasing-CL is shown in figure 1(a), and the evolution characteristics of the spiraling elliptic beams are summarized in figures 1(b) and (c). When P 0 = P c and M 0 = M c , the semi axes b and c also decrease in a nearly linear manner like the CL (solid green lines in figure 1(b)). While, if P 0 = P c the two semi axes will both oscillate around the linear lines (pink and blue lines in figure 1(b)). During propagations, the beam rotates, and the rotating velocity can smoothly speeds up when the CL decreases, which is shown in figure 1(c). The beam's contraction-induced acceleration can be explained in the following. By an analogy between the rotating beam and a rigid body, we calculate the pattern's moment of inertia J and the rotating velocity ω The inertia of a rigid body here corresponds to the beam width. Therefore, when the optical beam contracts, the so-called moment of inertia J decreases, then the rotation will become quicker.
Furthermore, we numerically found that the beam power keeps invariable, and the OAM are nearly constant during propagations when the CL decreases as shown in figures 2(a) and (d), respectively. The invariability of the ellipticity for the elliptic beam when M 0 = M c is also numerically confirmed as shown in figure 2(c). By combining equation (7) and δ = σ/b, we can rewrite the degree of nonlocality as Obviously, the degree of nonlocality also remains the same when the CL change with propagation distance z, which confirmed by the numerical solution as seen in figure 2(d).
The results above provide a way to control the rotation of an optical beam by changing the external conditions. A key point should be pointed out that the spiraling elliptic beam still can keep soliton states even when its rotating velocities change, which is shown in figure 3. When the CL changes from one value to the other, the solitons can smoothly transit to the different ones of different widths ( figure 3(b)), and exhibit a stable rotation with different velocities (figure 3(c)). The transitions among soliton states with different widths and rotating velocities when the CL gradually changes can be clearly confirmed by figure 4, where three dimensional evolutions of the spiraling elliptic beams under different cases of varying CL, slowly increasing in (a), keeping constant in (b), and slowly increasing in (c) are given. In addition, different rotating velocities of the beam will result in the different total rotation angles at the output plane of nonlinear media, which is shown in figure 3(d), and offers a method to engineer beams in optical signal processing. As numerical results shown in figure 5, the orientation at the exit surface can be controlled by changing the CL in nonlocal nonlinear media with fixed length.

Gradient force exerted by spiraling elliptic beams
The spiraling elliptic beams carrying the OAM, have potential applications in optical tweezers. In this section, we will discuss the induced variation of the gradient force exerted by spiraling elliptic beam when the CL gradually changes.
In the presence of an optical field, a particle experiences the gradient force [30] where k is a dimensionless quantity related to the refractive index, the radius, and the effective polarizability of the particle. When the refractive index of particle is larger than that of medium, k > 0, and the particle will be subjected to the gradient force directed at the maximum value of intensity. In the following calculations, we assume that k = 1 for simplicity. Substitution of equation (3) into equation (12) at the critical power yields the gradient force along principal axes of ellipse And the maxima of the gradient force can be obtained from equation (13), which reveals that the gradient force is anisotropic, Obviously, as the CL changes, the anisotropy of the gradient force remains and the gradient force can be controlled as shown in figure 6. The analytical solutions were well confirmed by the numerical results. Although the trapping mechanisms is the same as those mentioned in [30], which are both resulting from the gradient force. However, due to the structured beam considered in the manuscript, the gradient force is anisotropic, that is, the strength of the force is different along different directions as shown in figure 6. Furthermore, because the beam is rotating as shown in figure 7, the orientation of the maximal gradient force will also rotate. We think the direction-controllable optical tweezers can be also used to rotate the rod-shaped micro-particles.

Theoretical extension in the nematic liquid crystal
The discussions above are in the nonlocal nonlinear media with the Gaussian response, which can be analytically analysed. However, the actual response functions in physical nonlinear system always exhibit some singularities. For example, the response function of the nematic liquid crystals, which are typical nonlocal nonlinear media [31,32], can be expressed by [33] R(x, y) = 1 2πσ 2 K 0 where K 0 is the zeroth-order modified Bessel function. In fact, this response function is obtained from the simplified model rather than the complete one for the nematic liquid crystal [27,28,34]. However, this model can be used to reveal the main phenomena of spiraling soliton transition in nematic liquid crystal.
On the other hand, the results obtained here can be used as a reference for the manipulation of widths and velocities of spiraling solitons in other real nonlocal nonlinear media which also have singular response functions. First, we iterate the spiraling elliptic solitons by using the imaginary-time evolution method [35,36] in nematic liquid crystal. Since the iterated solution is spiraling in the Cartesian coordinate system (x, y, z), we must transform the NNLSE to its corresponding form in the rotating coordinate system (X, Y, Z). The relations between the two coordinate systems are X = x cos(ωz) + y sin(ωz), Then equation (1) will be rewritten in the rotating coordinate system Stationary solutions of equation (1) are assumed as ψ = u(X, Y)e iβZ , where u(X, Y) is a complex function and β is a real propagation constant. Inserting ψ = u(X, Y)e iβZ into equation (17), we can obtain − βu + 1 2 Introducing an operator L 00 Then equation (18) can be rewritten as −βu + L 00 u = 0. And the evolution equation in imaginary time is [35,36] ∂u where M = c − ∇ 2 , M is the acceleration operator, and c = 3 in general [35]. Equation (20) can be solved by Euler's methodû where u n is the solution of iteration n times, β n = L 00 u n , M −1 u n / u n , M −1 u n . L 00 u n , M −1 u n is the inner product of L 00 u n and M −1 u n . Whenû n+1 was obtained, it can be adjusted by [35] u n+1 = P û n+1 ,û n+1 where P is the given power, which is fixed [35]. In addition, the σ and ω should be given first before iterations. Substituting a trial solution of elliptic Gaussian-shaped beam into equations (20)- (22), a stable soliton can be iteratively obtained, and it will rotate under the Cartesian coordinate system.  The evolutions of the iterative solitons in nematic liquid crystal are simulated by the split-step Fourier method. In addition to linear gradually change of CL, Gaussian-shaped (as shown in figure 8(a)) or even periodic varying CL also can be designed. The numerical results in figures 8(b)-(e) demonstrate that the beam widths, angular velocity and gradient force all can be controlled by modulating the CL. The transition evolutions of the iterative spiraling elliptic soliton corresponding to figure 8(a) are shown in figure 9. Solitons at z = 0 are the numerically found by means of the imaginary-time evolution method. The soliton stability can be confirmed by the direct propagations from z = 0 to z = 100. The CL begins to change at z = 100, and gradually changes along different curves to another values. Accordingly, the solitons transit to another ones with different widths at z = 300. The new solitons also can stably propagate from z = 300 to z = 400. The comparison of figures 9(a) and (c) shows that the soliton widths can be easy controlled by the varied CL. Figures 10(a) and (b) demonstrate that the powers of spiraling elliptic solitons in nematic liquid crystal with varied CL are constant, and the OAM exhibits a small decay. Of course, the comparison of blue, green and prink lines in figure 10(b) shows that the OAM radiation when the CL changes slowly is slightly larger  It is shown in figure 11 that all the predicated phenomena can be periodically controlled for the 'sin-' varying CL. Which means that the only requirement of the CL for the solitons transition is that the variation of the CL must be slow enough.

Conclusion
In longitudinally-inhomogeneous nonlocal media, we have investigated the propagation properties of the spiraling elliptic beams, which exhibit the cross phase and carry the OAM. We have demonstrated that, when the CL gradually changes, a spiraling elliptic soliton can transit to another different one of the different width. During the transition, the rotating velocity, total rotation angles, and gradient force are all changes, and their values are well controlled by the CL. This kind of rotating pattern carrying the OAM may find its potential applications in optical tweezers. Besides, the CL-dependent rotating is also useful in the field of light controlling.