Geometric phase accumulation in first-order optical systems

The geometric phase accumulated by the transversal Gaussian mode undergoing a cyclic transformation in parametric space is a possible candidate for information encoding [1]. Therefore, the design of optical systems and appropriate beams suitable for the desirable geometric phase accumulation becomes a challenging task. We show that the eigenvalue analysis of ray transformation matrices permits to identify the first-order optical systems and the appropriate Gaussian-type beams that acquire the geometric phase while propagating through them. Strictly speaking, a beam of light, propagating through an optical system described by the operator R, accumulates a phase shift only if it is an eigenfunction of R with the eigenvalue in the form of a complex exponent. The operator responsible for geometric phase acquirement describes the rotation of the orbital angular momentum of the beam [2]. We demonstrate that if the properly normalized 4x4 ray transformation matrix T has two unimodular complex conjugated eigenvalues, which means that it is similar to the ray transformation matrix of the antisymmetric fractional Fourier transformer (FT), then there exists an orthonormal set of Gaussian-type modes which remain self-reciprocal after propagation through the first-order optical system described by T. These modes are closely related to the Hermite-Gaussian ones Hmn [3]. In particular, they can be obtained from Hmn after propagation through the system associated with the ray transformation matrix M whose definition is based on the eigenvectors of T. The accumulated geometric phase is proportional to the index difference m–n. As an example we mention the helicoidal Laguerre-Gaussian and the Hermite-Gaussian modes, which acquire the geometric phase after propagation through an image rotating system and an antisymmetric fractional FT, respectively. If the eigenvalues of the matrix T are not unimodular complex, we can speak only about the geometric phase accumulation in a wide sense, where an additional scaling and quadratic phase modulation of the beam are present under a loop transformation. In this case we permit the beam to be an eigenfunction for the transform described by the orthogonal matrix in the modified Iwasawa decomposition [2] of T, which matrix is similar to the ray transformation matrix of the antisymmetric fractional FT.