Dynamic and geometric phase accumulation by Gaussian-type modes in first-order optical systems

Abstract

Based on the ray transformation matrix formalism, we propose a simple method for the identification of the dynamic and geometric parts of the Gouy phase, acquired by an appropriate Gaussian-type beam while propagating through a first-order optical system. © 2008 Optical Society of America OCIS codes: 070.2575, 070.2580, 070.2590, 070.3185, 070.4690, 080.2468 A transversal mode with a Gaussian envelope, undergoing a cycle of transformations while propagating through a paraxial optical system, accumulates Gouy phase, usually divided into two parts: A dynamic part and a geometric part [1][2][3][4]. The identification of the Gouy phase is important in resonator theory [5], in optical trapping [6], and in possible applications of its geometric part for quantum computation [7]. The physical origin of the geometric phase, its calculation in parametric space [2,3], and its experimental measurement [4] have been discussed in a series of publications.In this Letter we propose a simple method for the determination of the Gouy phase-and in particular its dynamic and geometric parts-accumulated by an appropriate Gaussian-type mode during its propagation through a first-order optical system. It is based on the analysis of the eigenvalues of the ray transformation matrix associated with the first-order optical system.Strictly speaking, a coherent beam of light ⌿͑r͒, propagating through an optical system described by an operator R, accumulates a phase shift only if it is an eigenfunction of R with a unimodular eigenvalue exp͑i͒: R͓⌿͑r i ͔͒͑r o ͒ = exp͑i͒⌿͑r o ͒. Therefore we need to identify a system where phase accumulation is possible and determine the corresponding beams.Beam propagation through a lossless first-order optical system is described by the canonical integral transformation R T [8,9], whose kernel is parameterized by the real symplectic ray transformation matrix T, which relates the (properly normalized) position r = ͑x , y͒ t and direction p = ͑p x , p y ͒ t of an incoming ray to those of the outgoing ray. Using the modified Iwasawa decomposition [10] the (normalized) ray transformation matrix T can be written as a product of three symplectic matrices,where the first matrix T L represents a lens described by the symmetric matrix G =−͑CA t + DB t ͒͑AA t + BB t ͒ −1 , the second matrix T S corresponds to a scaler described by the positive definite symmetric matrix S = ͑AA t + BB t ͒ 1/2 , and the third matrix T O is orthogonal and can be described by the unitary matrix U = X + iY = ͑AA t + BB t ͒ −1/2 ͑A + iB͒. A simple example of Gouy phase accumulation is the propagation of the well-known HermiteGaussian (HG) modes H m,n ͑r͒ = H m ͑x͒H n ͑y͒, where H n ͑x͒ =2

Ray-wave dual descriptions of classical light and birefringence: a path integral story

“…Thus we have a representation of the Faddeev-Popov determinant that can be used directly to remove the unphysical degrees of freedom from path integral. Inserting Equation ( 20) directly into the path integral Equation (19) gives…”

“…It is a natural extension of our previous results, 7 since there the focal points were not explored. Of note here is also the work, 19 where the Gouy phase was considered in the context of the fractional Fourier transform.…”

Ray-wave duality of electromagnetic fields: a Feynman path integral approach to classical vectorial imaging

“…It is a natural extension of our previous results [7], since there the focal points were not explored. Of note here is also the work [21], where the Gouy phase was considered in the context of the fractional Fourier transform.…”

The Phase-Space Approach to Optical System Theory

“…It is noted that the WDF can also be used to obtain the Gouy phase from an eigenvalue analysis of the ABCD system matrix. 15 …”