On a Mechanism for the Formation of Spatially Inhomogeneous Structures of Light Waves in Optical Information Transmission Systems

Spatially inhomogeneous structures of light waves are used as a mechanism of compacting information in optical and fiber-optic communication systems. In this paper, we consider a mathematical model of an optical radiation generator with a nonlinear delayed feedback loop and a stretching (compression) operator of the spatial coordinates of the light wave in a plane orthogonal to the radiation direction. It is shown that the presence of a delay in the feedback loop can lead to the generation of stable periodic spatially inhomogeneous oscillations. In the space of the main parameters of the generator, the spaces of generation of stable spatially nonuniform oscillations are constructed, the mechanism of their occurrence is studied, and approximate asymptotic formulas are constructed.


INTRODUCTION
In [1][2][3], experimental results on the formation of spatially inhomogeneous waves in laser beams of an optical radiation generator with a special nonlinear two-dimensional feedback loop are presented. Such structures arise in a plane orthogonal to the direction of propagation of the light wave. Their occurrence is due to the nonlinearity of the system, which is provided by a thin layer of a nonlinear conducting medium and a two-dimensional feedback loop with the operator of the spatial transformation of the light wave in the radiation plane of the optical generator. In [1], a mathematical model for describing this phenomenon is also proposed and the results of its numerical analysis in the case of the rotation operator of the plane of the light wave are presented. The mathematical model is an initial boundary value problem for a nonlinear differential equation of parabolic type with a spatial argument transformation operator, which is considered in the region determined by the aperture of light radiation. This initial boundary value problem and its various generalizations have been studied in numerous works, where spatially inhomogeneous solutions are constructed using various analytical and numerical methods (for an overview, see, for example, [4]). In these works, the rotation operator of a spatial argument is mainly considered, since such an operator is created by the mechanism of generating self-oscillating solutions. The model of [1] does not take into account the time delay factor in the nonlinear feedback loop. The time delay can also serve as a mechanism for the excitation of self-oscillations [4,5], including with a simpler, from a constructive point of view, operator for the transformation of spatial coordinates. In this paper, a mathematical model of [1] is considered in a circular domain with a stretching (compression) operator of the spatial argument and a time delay in the feedback loop, for which the conditions and the nature of the stability loss of equilibrium states are studied depending on the gain and the delay. The possibility of oscillatory loss of stability of equilibrium states and the possibility of bifurcation of stable spatially inhomogeneous periodic solutions are shown. Such solutions can be used as information carriers in optical and fiber-optic communication systems.
1. MATHEMATICAL FORMULATION OF THE PROBLEM For a functional differential equation with a delayed argument with respect to the function determined in polar and , in which Δ ρϕ is the Laplace operator in polar coordinates; Q α is the spatial coordinate transformation operator; D and K are the positive constants; , in the domain where the circle , , an initial boundary value problem of the form (2) is considered. In (1), at (3) is the operator of compression of spatial coordinates, and at , is the operator of stretching of spatial coordinates; in (2), the space of initial conditions where the space of functions is and is obtained by the closure of the set of functions in the metric of the function space . In the following, is the space of real-valued functions defined in , for which here and below and are the spaces of continuous and twice-continuously-differentiable in functions for which the norm is defined, The phase space of the initial boundary value problem (1) and (2) is the space the norm in which we define as The domain of definition of the right side of equation (1) is the space . The norm in is defined as As the solution of the initial boundary value problem (1) and (2), defined at , we will consider a function (for each ) that is continuously differentiable with respect to t at and converts equation (1) into an identity in phase space and satisfies the initial conditions (2).
In this paper, we study the conditions and nature of the loss of stability of equilibrium states and the bifurcations of spatially inhomogeneous self-oscillatory solutions of the initial boundary value problem (1) and (2), as well as their stability.

ANALYZING THE STABILITY OF THE EQUILIBRIUM STATES OF THE INITIAL BOUNDARY VALUE PROBLEM (1) AND (2)
The equilibrium states of the initial boundary value problem (1) and (2) are determined by the solutions of the nonlinear operator equation (5) in . Let us choose one of the solutions of Eq. (5) and write the initial boundary value problem (1) and (2) in its neighborhood, replacing . As a result, we get the initial boundary value problem where the dots indicate the terms with higher order of smallness with respect to in the norm . Consider the linear part of (6)-(7) Defining solutions of (9) and (10) of the form (Euler's solutions), we obtain a bundle of operators (11) acting in with the definition domain , the points of the spectrum of which determine the stability of solutions of the initial boundary value problem (9) and (10), and the corresponding eigenfunctions determine the solutions of the form being sought. Here and in the future, we will use the tilde sign to denote a complex extension of the corresponding functional space, a scalar product and a norm in which are generalized in the common way.   Consider the stretching operator (4). In this case, the initial boundary value problem (1) and (2) can have homogeneous equilibrium states . Homogeneous equilibrium states of the initial boundary value problem (1) and (2) are defined as solutions of equation (15) Depending on K and γ, Eq. (15) can have several solutions, including multiple ones. Note that . With this in mind, in (13) and (14) can be presented in the form . Let us study the conditions at which the equilibrium state loses its stability. For this purpose, we will use the D-splitting method [6]. Assume in (13) and consider for each n a sequence of truncated finite-dimensional matrices in which at n = 0 and at n > 0. Let us first consider the case n = 0. Equating the determinant of the matrix to zero and expressing the element from this equality, we get the expression     (((  ( , , , )) (1 ( , , , ))))), , , )) (1 ( , , , ))), 1,2, .  (9) and (10). In the case , we will build an iterative process in (17) and (18), As , , we choose the solution of (17) and (18) at , . The iterative process (19) and (20) converges quite quickly both in q and in m, since the coefficients of the matrices (13) fairly rapidly tend to zero. Figures 1 and 2 show the D-partitioning of the plane (b, T) at , , . D j indicate the regions of the parameter values at which the bundle of operators (11) has j spectrum points belonging to the right complex half-plane, and the boundaries of these regions correspond to the spectrum points lying on the imaginary axis. It can be seen from the figures that there are parameter values at the boundary of the stability region (region D 0 ), at which a pair of complex conjugate points of the spectrum (11) can pass into the right halfplane (right border), as well as parameter values corresponding to the zero point of the spectrum (11) (the left border is a vertical straight line). The numbers on the border indicate the values of the parameters n and j in the functions that define the corresponding border (see (17) and (18)). Note that the eigenfunctions corresponding to the specified points of the spectrum have a spatially inhomogeneous form.
In the case of the compression operator (3), the scheme for finding the stability regions of solutions of the initial boundary value problem (1) and (2) remains the same, but, in this case, the dependence of b on K and T is more complex.

BIFURCATION OF SPATIALLY INHOMOGENEOUS SOLUTIONS OF THE INITIAL BOUNDARY VALUE PROBLEM (1) AND (2)
Consider the case of stretching . Let us at given and γ, choose the parameters in such a way that they would correspond to a point at the boundary of the stability domain of solutions of the initial  ( ), , ))))), Let us choose and which for given satisfy equalities (8) and (15). Note that such a choice may be ambiguous. Now, we assume that , where is a small parameter and will study the possibility of a bifurcation of the spatially inhomogeneous periodic solutions of the initial boundary value problem (1) and (2) from the equilibrium state when the parameter m is changed. For the analysis, we will use the method of invariant (central) manifolds [7] and the theory of normal forms of ordinary differential equations [8].
Note that now in (8), (12), (13), and (21) Accordingly, in what follows the bundle of operators will be denoted as . Denote the point of the spectrum of the operator , that satisfies the condition , as and the corresponding eigenfunction as ; and  Defining now as a series of functions , we obtain a linear algebraic system in for determining the expansion coefficients, which has a unique solution belonging to . are determined in the same way. Now we equate the coefficients at on the left and right sides of (29)-(31). As a result, for determining we obtain a boundary value problem of the form The solution of (33) and (34)  , the boundary value problem (35) and (36) is not solvable. We achieve solvability by a choice of which is uniquely determined in this case. The continuous in solution of (35) and (36) is also defined uniquely in the form of a expansion in . In this case, the expansion coefficients are uniquely determined by analogy with the boundary value problem (33) and (34). In what follows, only the coefficient is used, so we give the expression for it which is obtained from the solvability condition of (35) and (36) at taking into account Eqs. (22).