S. Chávez-Cerda

Alternative formulation for invariant optical fields: Mathieu beams

Based on the separability of the Helmholtz equation into elliptical cylindrical coordinates, we present another class of invariant optical fields that may have a highly localized distribution along one of the transverse directions and a sharply peaked quasi-periodic structure along the other. These fields are described by the radial and angular Mathieu functions. We identify the corresponding function in the McCutchen sphere that produces this kind of beam and propose an experimental setup for the realization of an invariant optical field.

Mathieu functions, a visual approach

The behavior of the Mathieu functions is illustrated by using a variety of plots with representative examples taken from mechanics. We show how Mathieu functions can be applied to describe standing, traveling, and rotating waves in physical systems. Some background is provided on notation and analogies with other mathematical functions. Our goal is to increase the familiarity with Mathieu functions in the scientific and academic community using visualization. For this purpose we adopt a strategy based on visual recognition rather than only looking at equations and formulas.

Experimental demonstration of optical Mathieu beams

We report the first experimental observation of zero-order Mathieu beams which are fundamental non-diffracting solutions of the wave equation in elliptic cylindrical coordinates.

Holographic generation and orbital angular momentum of high-order mathieu beams

We report the first experimental generation of high-order Mathieu beams and confirm their propagation invariance over a limited range. In our experiment we use a computer-generated phase hologram. The peculiar behaviour of the vortices in Mathieu beams gives rise to questions about their orbital-angular-momentum content, which we calculate by performing a decomposition in terms of Bessel beams.

Elliptic vortices of electromagnetic wave fields.

We demonstrate the existence of elliptic vortices of electromagnetic scalar wave fields. The corresponding intensity profiles are formed by propagation-invariant confocal elliptic rings. We have found that copropagation of this kind of vortex occurs without interaction. The results presented here also apply for physical systems described by the (2+1) -dimensional Schrödinger equation.

Evolution of focused Hankel waves and Bessel beams

The focusing properties of a Bessel beam are analysed and interpreted in terms of its constituent conical waves. While these waves focus to a ring image in the focal plane of the lens, the beam also exhibits an apparent focus on the axis in front of the focal plane. The physical origin of this feature is explained, and it is also shown that, under certain focusing conditions, a Bessel beam can be reconstructed beyond the focal plane.

Focusing evolution of generalized propagation invariant optical fields

The focusing evolution of apertured propagation invariant optical fields (PIOFs) with arbitrary transverse intensity distribution is analysed in detail. A decomposition of the PIOF into its constituent plane waves is applied to find a simple expression of the normalized intensity along the propagation axis which is valid for any PIOF. It is shown that the presence of an apparent focus close to the focal plane is a general property of any focused PIOF and that it is not related to the focal shift due to apertures. By selecting appropriate parameters it is possible to generate a second intense peak beyond the focal plane. Additionally, we define the conditions under which the original PIOF can be reconstructed beyond the focal plane with a desired magnification.

Parabolic propagation-invariant optical beams

We present in this work the propagation-invariant optical fields (PIOFs) which are exact solutions of the Helmholtz equation. These solutions are given in parabolic coordinates and have an inherent parabolic geometry. Our new approach allowed to obtain the proper solutions that describe the whole family of parabolic PIOFs in a straightaway manner. Based on the McCutchen theorem we have also identified the corresponding angular spectrum, which common to all PIOFs, lies on a ring. Another interesting features of these beams is that their order is not quantised as occurs for the Bessel and Mathieu beams and that they have a definite parity.

Scalar representation of paraxial and nonparaxial laser beams

The development of technology of small dimensions requires a different treatment of electromagnetic beams with transverse dimensions of the order of the wavelength. These are the nonparaxial beams either in two or three spatial dimensions. Based on the Helmholtz equation, a theory of nonparaxial beam propagation in two and three dimensions is developed by the use of the Mathieu and oblate spheroidal wave functions, respectively. Mathieu wave functions are the solutions of the Helmholtz equation in planar elliptic coordinates that is a special case of the prolate spheroidal geometry. So we may simply refer to the solutions, either in two or three dimensions, as spheroidal wave functions. Besides the order mode, the spheroidal wave functions are characterized by a parameter that will be referred to as the spheroidal parameter. Divergence of the beam is characterized by choosing the numeric value of this spheroidal parameter, having a perfect control on the nonparaxial properties of the beam under study. When the spheroidal parameter is above a given threshold, the well known paraxial Laguerre-Gauss and Hermite-Gauss beams are recovered, in their respective dimensions. In other words, the spheroidal wave functions represent a unified theory that can describe electromagnetic beams in the nonparaxial regime as well as in the paraxial one.