Double and Square Bessel-Gaussian Beams.

We obtain a transform that relates the standard Bessel-Gaussian (BG) beams with BG beams described by the Bessel function of a half-integer order and quadratic radial dependence in the argument. We also study square vortex BG beams, described by the square of the Bessel function, and the products of two vortex BG beams (double-BG beams), described by a product of two different integer-order Bessel functions. To describe the propagation of these beams in free space, we derive expressions as series of products of three Bessel functions. In addition, a vortex-free power-function BG beam of the mth order is obtained, which upon propagation in free space becomes a finite superposition of similar vortex-free power-function BG beams of the orders from 0 to m. Extending the set of finite-energy vortex beams with an orbital angular momentum is useful in searching for stable light beams for probing the turbulent atmosphere and for wireless optical communications. Such beams can be used in micromachines for controlling the movements of particles simultaneously along several light rings.

Multiple-twisted spiral beams.

We propose a method to construct paraxial light fields whose transverse intensity rotates as a whole during the field propagation (spiral beams), and the total rotation angle is an integer multiple of π/2 (multiple-twisted beams). We derive analytical expressions for the field complex amplitude by utilizing an integral description of Laguerre-Gaussian beams. This method provides a straightforward way to obtain multiple-twisted spiral beams of various intensity shapes and rotation rates.

Closed-form expression for mutual intensity evolution of Hermite-Laguerre-Gaussian Schell-model beams.

We derive a comprehensive closed-form expression for the evolution of the mutual intensity (MI) of Hermite-Laguerre-Gaussian Schell-model beams (HLG-SMBs) during propagation through rotationally symmetric optical systems. We demonstrate that the MI of the beam associated with a given HLG mode at any transverse plane can be presented as a linear superposition of the MIs of the SMBs associated with the equal and lower index modes of the same type, but of complex argument. The obtained expression allows easy analysis of the evolution of the intensity distribution and the CCF of such beams and, in particular, an understanding of the coherence singularity formation and modification during the beam propagation.

Definition of the waist plane for general astigmatic Gaussian beams.

We propose a possible generalization of the waist plane for a two-dimensional astigmatic Gaussian beam as a plane of minimal spot area of the beam during propagation. It is shown that the defocusing component of the beam phase vanishes in this plane. Some examples of astigmatic Gaussian beams and corresponding area waist planes are presented and discussed.

Shaping of light beams along curves in three dimensions.

We present a method for efficient and versatile generation of beams whose intensity and phase are prescribed along arbitrary 3D curves. It comprises a non-iterative beam shaping technique that does not require solving inversion problems of light propagation. The generated beams have diffraction-limited focusing with high intensity and controlled phase gradients useful for applications such as laser micro-machining and optical trapping. Its performance and feasibility are experimentally demonstrated on several examples including multiple trapping of micron-sized particles.

Product of three Airy beams.

A two-dimensional field that is a product of three Airy beams is proposed and investigated. It is shown that the Fourier image of this field has a cubic phase and a radially symmetric intensity with a super-Gaussian decrease. Propagation of the product of three Airy beams in a Fresnel zone is investigated numerically.

General astigmatic transform of Hermite-Laguerre-Gaussian beams.

The general astigmatic transform, or two-dimensional non-separable linear canonical transform of a Hermite-Laguerre-Gaussian beam, is investigated by theoretical means. Some corollaries that apply to Hermite-Gaussian and Laguerre-Gaussian beam propagation are presented and discussed.

Rotating beams in isotropic optical system.

Based on the ray transformation matrix formalism, we propose a simple method for generation of paraxial beams performing anisotropic rotation in the phase space during their propagation through isotropic optical systems. The widely discussed spiral beams are the particular case of these beams. The propagation of these beams through the symmetric fractional Fourier transformer is demonstrated by numerical simulations.