### Abstract

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.

Original language | English |
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Title of host publication | ICO-21 Congress Proceedings 2008, 21th Congress of the International Commission for optics |

Pages | 298- |

Publication status | Published - 2008 |

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## Cite this

Alieva, T., & Bastiaans, M. J. (2008). Geometric phase accumulation in first-order optical systems. In

*ICO-21 Congress Proceedings 2008, 21th Congress of the International Commission for optics*(pp. 298-)