Launching and Controlling Gaussian Beams from Point Sources via Planar Transformation Media

TL;DR

This paper demonstrates how complex transformation optics can be used to generate and control Gaussian beams from point sources using anisotropic gain media slabs, relaxing material constraints and enabling beam steering.

Contribution

It extends CTO analysis to non-symmetric transformations, enabling Gaussian beam generation and control with simpler, homogeneous gain media slabs and outside sources.

Findings

Gaussian beams can be produced from point sources outside the slab.

Beam properties like amplitude and waist location are controllable via CTO Jacobian.

Negative Jacobian entries enable bidirectional waist movement.

Abstract

Based on operations prescribed under the paradigm of Complex Transformation Optics (CTO) [1-5], it was recently shown in [5] that a complex source point (CSP) can be mimicked by a parity-time ($\mathcal{PT}$) transformation media. Such coordinate transformation has a mirror symmetry for the imaginary part, and results in a balanced loss/gain metamaterial slab. A CSP produces a Gaussian beam and, consequently, a point source placed at the center of such metamaterial slab produces a Gaussian beam propagating away from the slab. Here, we extend the CTO analysis to non-symmetric complex coordinate transformations as put forth in [6] and verify that, by using simply a (homogeneous) doubly anisotropic gain-media metamaterial slab, one can still mimic a CSP and produce Gaussian beam. In addition, we show that a Gaussian-like beams can be produced by point sources placed {\it outside} the slab as well [6]. By making use of the extra degrees of freedom (real and imaginary part of the coordinate transformation) provided by CTO, the near-zero requirement on the real part of the resulting constitutive parameters can be relaxed to facilitate potential realization of Gaussian-like beams. We illustrate how beam properties such as peak amplitude and waist location can be controlled by a proper choice of (complex-valued) CTO Jacobian elements. In particular, the beam waist location may be moved bidirectionally by allowing for negative entries in the Jacobian (equivalent to inducing negative refraction effects). These results are then interpreted in light of the ensuing CSP location.