A dark hollow beam (DHB) is defined, in general, as a ring-shaped light beam with a null intensity center on the beam axis. Some known examples of DHBs are the Laguerre-Gaussian beams, high-order Bessel and Bessel-Gauss beams, hollow Gaussian beams, helical Mathieu and Mathieu-Gauss beams, high-order linearly polarized output fiber modes, among others. DHBs have interesting physical properties, such as a helical wavefront, a center vortex singularity, doughnut-shaped transverse intensity distribution. They may carry and transfer orbital and spin angular momentum, and may also exhibit a nondiffracting behavior upon propagation. All these characteristics make DHBs particularly useful in optical tweezers, micromanipulation, optical trapping and cooling, Bose-Einstein condensation, optical metrology, computer-generated holography, and so on. A variety of methods have been proposed to generate DHBs, including passive optical methods, mode selection in laser resonators, passive mode conversion, holographic methods, and liquid crystal displays.
Most of the known theoretical models to describe DHBs consider axially symmetric transverse intensity distributions. Recently, several works have been devoted to explore the properties and applications of DHBs for which the circular symmetry no longer exists, with particular emphasis in the rectangular and elliptical geometry. For example, Cai and Zhang and Cai and Ge based on a suitable superposition of a finite series of fundamental Gaussian beams, proposed a model of a rectangular DHB that seems to be more suitable than a circular DHB to guide atom laser beams with a particular mode. If the beam width in one direction of a rectangular DHB is much larger than that in the other direction, the rectangular DHB can be visualized as a one-dimensional beam, which may be applied to achieve one-dimensional Bose-Einstein condensates.
DHBs with elliptic symmetry can be regarded as transition beams between circular and rectangular DHBs. For example, the high-order modes emitted from resonators with neither completely rectangular nor completely circular symmetry, but in between them, cannot be described by the known Hermite-Gaussian or Laguerre-Gaussian beams. In recent years, there has been an increasing interest in developing models to describe DHBs with elliptic symmetry. An important characteristic of the waves exhibiting elliptic geometries is the possibility of independently choosing the propagation parameters along the two orthogonal transverse directions. For example, in the case of the waveguides with an elliptic cross section, the fundamental propagating mode splits into two independent orthogonally polarized modes. Such geometrical birefringence makes the elliptical fibers particularly useful in applications where the polarization plays a major role. This splitting feature is not restricted to propagation in waveguides, but is also present in free-space propagation of waves with elliptic symmetry.
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