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Complex variable function Gaussian beam in strongly nonlocal nonlinear media

We introduce a novel class of spatial complex variable function Gaussian (CVF-Gaussian) beam, which is the product of an arbitrary analytic complex variable function and a Gaussian function, in strongly nonlocal nonlinear media (SNNM). The CVF-Gaussian beam rotates generally during propagation. By choosing the input power of the beam, we can obtain the CVF-Gaussian breather or the CVF-Gaussian soliton. We reveal that stable CVF-Gaussian beam can exist in SNNM with different forms, including rotating dipole, and rotating elliptic donut. A distribution factor which is the parameter for the description of the transverse distribution of the CVF-Gaussian beam is discussed.

In this letter, we introduce a novel class of spatial self-trapped optical beam in the SNNM, the so-called spatial complex variable function Gaussian (CVF-Gaussian) beam which is the product of an arbitrary analytic complex variable function and a Gaussian function.
Depending on the input power of the beam, the CVF-Gaussian beam can be either a soliton state or a breather state.
Our analysis is based on the nonlocal nonlinear Schrödinger equation (NNLSE) for the amplitude Φ(r, z) of a paraxial laser beam propagation in a focusing cubic media with a transverse nonlocality [17,18]: where z is the longitudinal coordinate, µ = 1/(2k), k = ωn 0 /c is the wave number in the media without nonlinearity, n 0 is the linear refractive index of the media, △ ⊥ is the two dimensional transverse Laplacian operator, ∆n = n 2 ∞ −∞ R(r − r 1 )|Φ(r 1 , z)| 2 d 2 r 1 is the nonlinear perturbation of refraction index, n 2 is the nonlinear index coefficient, r and r 1 are the two dimensional transverse coordinate vectors, and R is the normalized symmetrical real nonlinear response function of the media.
In the case of the strong nonlocality [3,5,12], if the response function R(r) is twice differentiable at r = 0, Eq. (1) can be reduced into [4,17,22] the Snyder-Mitchell model [6] i∂Ψ/∂z + µ△ ⊥ Ψ − n 2 /(2n 0 )kγP 0 r 2 Ψ = 0, (2) and the functions Φ in Eq. (1) and Ψ in Eq. (2) are related by [17] where r = |r| is the radial coordinate, γ is the material parameter associated with the response function R(r) [4], is the beam width of the Gaussian beam, c(z) represents the phase-front curvature of the beam, θ(z) is the phase of the complex amplitude. They are given by [5], respectively, w(z) = w 0 P c /P 0 sin 2 β 0 z + cos 2 β 0 z 1/2 , c(z) = kβ 0 w 2 0 P c /P 0 − 1 sin 2β 0 z/ 4w 2 (z) , and where w 0 is the initial beam width at z = 0, P c = n 0 /(k 2 γn 2 w 4 0 ) is the critical power for the soliton propagation, β 0 = n 2 γP 0 /n 0 . To obtain CVF-Gaussian beam solution of Eq. (2), we suppose its solution has a form, Substituting Eq. (4) into Eq. (2), one can obtain, In the transverse plane perpendicular to z, we introduce a rotating coordinate system (x ′ , y ′ , z), x ′ = w 0 [x cos(ϑ(z)) + y sin(ϑ(z))]/w(z), y ′ = w 0 [y cos(ϑ(z)) − x sin(ϑ(z))]/w(z), where ϑ(z) describes the rotation angle of the rotating coordinate, dϑ(z)/dz denotes the angular velocity (AV, the angle rotated per unit propagation distance). In the rotating coordinate system, Eq. (5) can be written as, where is the z component of the angular momentum operator, equivalent to one in quantum mechanics [23]. The solution of Eq. (6) is found to be where f (·) denotes an arbitrary analytical function, b = 0 is an arbitrary real parameter whose physical meaning will be discussed later, θ(z) is given by Eq. (3), and " + (−)" presents the rotation direction which accords to the right-hand (left-hand) rule relative to the propagation direction. From Eqs. (4) and (7), one can obtain the solution of Eq. (2) in the laboratory frame: where C 0 is a constant which can be determined by The solution is called CVF-Gaussian beam due to the fact that f (ζ) is an arbitrary analytical complex variable function. The structure of the beam is determined by the product of the function f (ζ) and a Gaussian function. Solution (8) represents an optical beam that rotates with an AV, dϑ(z)/dz, in the laboratory frame (x, y, z) and is stationary in the rotation frame (x ′ , y ′ , z). Observed in the laboratory frame, the trajectories of the corresponding points of the beam during propagation rotate generally.
To discuss the physical meaning of the parameter b, we define the mean radius of an optical beam as [24] From the axis-symmetrical solution (8), it can be obtained r = 0, which means that the position of the CVF-Gaussian beam center is always located on the propagation axis (z -axis). By use of the Taylor expansion of f (ζ), then, the mean radius of the CVF-Gaussian beam is found to be where A m = d m f (ζ)/dζ m | ζ=0 /m! (m = 0, 1, 2, · · ·) are the coefficients of the Taylor se- and m! is the factorial of an integer m. W characterizes the distribution of the CVF-Gaussian beam from the beam center and provides a way to capture the beam's scale or degree of being spread out. The derivative of W 2 with respect to b can be written as A 1 = 0 (It means that the function f (ζ) is a constant or direct proportional to ζ.), otherwise dW 2 /db < 0. This means that the mean radius of the CVF-Gaussian beam is independent of b for the former case, and is a monotonically descent function of b for the latter case. Therefore, we can find that b, which is nominated a distribution factor, is the parameter for the description of the transverse distribution of the CVF-Gaussian beam, and conclude that the distribution is independent of b if the function f (ζ) is a constant or direct proportional to ζ, otherwise generally for another given f (ζ), the distribution is farther from the beam center for a smaller b and is clustered more closely around the beam center for a larger b. When (4)]. Ψ will, then, equal the Gaussian function or the null function depending upon whether f (0) = 0 or not. Thus we have the following limit: Figure 1 gives an example of the transverse normalized intensity distribution of the CVF-Gaussian beam when f (ζ) = cos(ζ) or f (ζ) = H 4 (ζ) for different b, where H n (ζ) is the nth-order Hermite polynomial. Such a cosine CVF-Gaussian beam or fourth-order Hermite CVF-Gaussian beam is clustered more closely around the beam center as b increases, and will approach to a Gaussian distribution when b → ∞ since cos(ζ)| ζ=0 = 1 and H 4 (ζ)| ζ=0 = 12. In fact, the cosine CVF-Gaussian beam becomes almost the Gaussian function when b = 10, as shown in the inset of Fig. 1(d).
The CVF-Gaussian beam can exhibit the different properties for different input power P 0 . Equation (9) shows that the mean radius of the CVF-Gaussian beam is proportional to the beam width of the linked Gaussian beam when the function f (ζ) and the distribution factor b are given. The CVF-Gaussian beam, therefore, evolves in the similar way the Gaussian beam does [4]. When P 0 < P c [the expression of P c was given under Eq. (3)], the pattern of the CVF-Gaussian beam initially broadens and then narrows, undergoing a periodic evolution; whereas the reverse happens and the pattern initially narrows when P 0 > P c . These two cases are the CVF-Gaussian breather states. When P 0 = P c , Eq. (8) can be reduced as where ζ (s) ± = (x ± iy)/(b w 0 ) exp(±iβ 0 z). This is a CVF-Gaussian soliton, and diffraction is exactly balanced by nonlinearity in this case. Its transverse distribution preserves but rotates as the CVF-Gaussian soliton travels along the propagation axis.
From Eqs. (7) and (3), the AV of the CVF-Gaussian beam can be obtained Equation (11) shows that Ω is only dependent upon the material parameters and input power, and independent of the complex-variable function f and the distribution factor b, that is, independent of the concrete form of the CVF-Gaussian beam. Ω is inversely proportional to w 2 , the squared beam width of the linked Gaussian beam, so Ω behaves periodically as w does [4]. If P 0 < P c , Ω decreases initially, vibrating between a minimum P 0 /(P c z R ) and a maximum 1/z R ; whereas if P 0 > P c , the reverse happens, and Ω increases first, vibrating between a maximum P 0 /(P c z R ) and a minimum 1/z R , where z R = kw 2 0 is the Rayleigh range. When P 0 = P c , Ω of the CVF-Gaussian soliton is a constant 1/z R .
The analytical prediction above is tested by comparison with the numerical solution of Eq. (1). The material response is supposed to be the Gaussian function [3,4], i.e., where w m is the characteristic length of the material response function, α = w 0 /w m denotes the degree of nonlocality for the beam in nonlocal nonlinear media [4]. The less α is, the stronger the nonlocality is. The propagation dynamics of the cosine CVF-Gaussian soliton, and the fourth-order Hermite CVF-Gaussian soliton are presented in Fig. 2, respectively.  CVF-Gaussian soliton consists of a bright elliptic ring with a dark elliptic spot, thus it seems appropriate to refer to this kind of ring soliton as "elliptic donut mode" [25].
A better understanding of this phenomenon of the visible rotation for the optical beam would be provided if an analogy between the rotating beam and the mechanically rotating object [26] was drawn. Being similar to mechanical attributes, the optical beam possesses the AV, the moment of inertia that is proportional to the total beam equivalent mass, which equals the beam power, and to the squared mean radius, and the angular momentum (AM) that is the product of the AV and the moment of inertia. Just like the movement of the rigid body in mechanics, this shows that if the beam is narrowed or broadened while keeping the total "mass", its AV increases or decreases. In this way, the result of Eq. (11) could be clarified physically. From the definition of AM [14], A = Im ∞ −∞ Φ * (r × ∇Φ)d 2 r, one can obtain the AM of the CVF-Gaussian beam is ∞ m=0 (|A m | 2 mm!/b 2m ). The inverse proportion between the AM and distribution factor b could also be explained qualitatively via this analogy. Since the mean radius of the CVF-Gaussian beam is inverse proportional to b and the AV is independent of b, as discussed above, it is easily understood why the AM decreases while b increases with the total "mass" preserving.
Although the visible rotation comes from the non-zero AM, not all the beams with nonzero AM will rotate. It can be observed that the vortex LGS [9] and the vortex IGS [10], which are also the solutions of the SNNM (2), do not rotate during propagation, but both of their AMs are non-zero. The physics behind the rotation phenomenon is the inhomogeneity of the energy flow within the beam cross section. Extending the linear analysis [27] to the nonlinear case, we can have ∂|Φ| 2 /∂z = −∇ ⊥ · S ⊥ , which follows directly from the Eq. (1), where the quantity Ξ S ⊥ · dO is the scalar flux of the field through the closed surface Ξ that surrounds a point (x, y, z) and encloses a domain D with volume V . When the energy flow is homogeneous, the energy outflow will equal to energy inflow through the closed surface, then Ξ S ⊥ · dO = 0 and the intensity pattern is constant during the propagation. If the energy outflow is less than the energy inflow, Ξ S ⊥ · dO < 0, and ∂|Φ| 2 /∂z > 0, hence the intensity grows in a point, otherwise the intensity falls down if the energy outflow is more than the energy inflow. It can be found that S ⊥ = 0 but (1) when the initial beam is perturbed by noises [28] have proved the stability of the solution (8). The detail of the simulation will not discussed here because of the length limitation.
In conclusion, we have provided analytical and numerical evidence for the existence of a novel class of the CVF-Gaussian beams, the CVF-Gaussian breather and the CVF-Gaussian soliton, stabilized by strongly nonlocal nonlinearity. The CVF-Gaussian beams will rotate during propagation, and the angular velocity is generally a function of the material parameters and the input power, independent of the concrete form of the CVF-Gaussian beams.
The distributions of the CVF-Gaussian beams are generally dependent upon the distribution factor b, and the distribution is farther from the beam center for a smaller b and is clustered more closely around the beam center for a larger b. The stable CVF-Gaussian beam can exist in SNNM with different forms, including rotating dipole, and rotating elliptic donut. The physics behind the rotation phenomenon is the inhomogeneity of the energy flow within the beam cross section, which results in the energy redistribution over the beam cross section.