Mechanism of the formation of static and rotating multilobe structures in a nonlinear Fizeau interferometer

Abstract

The article considers the formation in a nonlinear Fizeau interferometer with two-dimensional feedback. In a previous paper [1] we proposed an approach that could elucidate the mechanisms of formation of simple optical structures for specific values of the parameters of the interferometer and the input optical beam. In the present paper we try to explain the formation of static and rotating multilobe structures.In the present paper we discuss the mechanisms of formation of multilobe structures, using the phenomenological approach that we proposed and substantiated in [1]. The equation of the dynamics of the phase advance u (x, y, t), written asserves as the model of formation.From the results of full-scale experiments [2] we know that a multilobe structure forms when a field rotates in the x, y plane in the feedback loop (A ~ 0) (Fig. I). Depending on the angle of rotation, when it has formed completely the structure rotates clockwise, remains immobile, or rotates counterclockwise. Let us consider how the given phenomenological approach can explain the circumstances of the multilobe structure formation.We use the concept of a transpositional point of the Nth order, which was introduced in [1]. We recall the definition of the concept. If only field rotation occurs in the feedback loop, i.e., A ;e 0, ~ = 1, O = 0, then the field is transformed point by point in the system when the local transverse interactions of the fields are disregarded (D = 0). As a result the fields are bound at a finite number of points (xl, yl), (x2, y2) ..... (xN, yN) of the cross-section of the light beam. If the coordinates of points (xl, yl) and (xN + 1, yN + 1) coincide, it is customary to speak of degenerate two-dimensional feedback [2]. In [1] we proposed calling those points transpositional and the number N, the order of the transpositional point or the order of degeneracy. Since diffusion (diffraction) that lifts the degeneracy is assumed to occur, in the exposition of our approach without risk we can operate with a D-dependent fmite-dimensional neighborhood of an isolated transpositional point rather than with the point itself. From the definition of a transpositional point it follows that the point is of order N, when the angle of rotation of the field has a definite value AT = 27r/N. It is legitimate to ask to what type of multilobe structure (rotating or static) observed in full-scale experiments does the structure formed when a field rotates by A T in the feedback loop belong.In looking for an answer we hypothesize that for an angle of rotation AT a static multilobe structure forms if N is even. That structure forms in a minimal interval of time, when the phase advances u (x, y, t) at the transpositional points (xl, yl), (x3, y3) ..... (x2m -1, y2m -1) ..... (xN -1, yN -1) are the same and close to the lower/upper stable state on the diagram of stationary solutions (Fig. 6.22 in [2]) and the phase advances (x2, y2), (x4, y4) ... .. (x2m, y2m) ... .. (xN, yN) are also the same and close to the upper/lower stable stationary stat...