Nonparaxial accelerating waves as a superposition of nondiffracting Bessel-lattice optical fields.

In the first part of this work, we introduce a monochromatic solution to the scalar wave equation in free space, defined by a superposition of monochromatic nondiffracting half Bessel-lattice optical fields, which is determined by two scalar functions; one is defined on frequency space, and the other is a complete integral to the eikonal equation in free space. We obtain expressions for the geometrical wavefronts, the caustic region, and the Poynting vector. We highlight that this solution is stable under small perturbations because it is characterized by a caustic of the hyperbolic umbilical type. In the second part, we introduce the corresponding solution to the Maxwell equations in free space.

Classical characterization of quantum waves: comparison between the caustic and the zeros of the Madelung-Bohm potential.

From a geometric perspective, the caustic is the most classical description of a wave function since its evolution is governed by the Hamilton-Jacobi equation. On the other hand, according to the Madelung-de Broglie-Bohm equations, the most classical description of a solution to the Schrödinger equation is given by the zeros of the Madelung-Bohm potential. In this work, we compare these descriptions, and, by analyzing how the rays are organized over the caustic, we find that the wave functions with fold caustic are the most classical beams because the zeros of the Madelung-Bohm potential coincide with the caustic. For another type of beam, the Madelung-Bohm potential is in general distinct to zero over the caustic. We have verified these results for the one-dimensional Airy and Pearcey beams, which, according to the catastrophe theory, have stable caustics. Similarly, we introduce the optical Madelung-Bohm potential, and we show that if the optical beam has a caustic of the fold type, then its zeros coincide with the caustic. We have verified this fact for the Bessel beams of nonzero order. Finally, we remark that for certain cases, the zeros of the Madelung-Bohm potential are linked with the superoscillation phenomenon.

Refracting and reflecting interfaces transforming a given wavefront into another one.

The aim of this work is threefold. First, following Luneburg and using our own notation, we review the Cartesian ovals. Second, we obtain analytical expressions for the reflecting and refracting surfaces that transform a prescribed smooth two-dimensional wavefront into a spherical one. These results are applied to show that the reflecting surface that connects a plane wavefront to a spherical one is a parabolical surface, and we design a lens, with two freeform surfaces, that transforms a spherical wavefront into another spherical one. These examples show that our equations provide the well-known solution for these problems, which is given by the Cartesian ovals method. Third, we present a procedure to obtain exact expressions for the refracting and reflecting surfaces that connect two given arbitrary wavefronts; that is, by assuming that the optical path length between two points on the prescribed wavefronts is given by the designer the refracting and reflecting surfaces we are looking for are determined by a set of two algebraic equations, which in the general case have to be solved in a numerical way. These general results are applied to compute the analytical expressions for the reflecting and refracting surfaces that transform a parabolical initial wavefront into a plane one.

Wavefronts and caustics associated with Mathieu beams.

In this work we compute the wavefronts and the caustics associated with the solutions to the scalar wave equation introduced by Durnin in elliptical cylindrical coordinates generated by the function A(ϕ)=ceν(ϕ,q)+iseν(ϕ,q), with ν being an integral or nonintegral number. We show that the wavefronts and the caustic are invariant under translations along the direction of evolution of the beam. We remark that the wavefronts of the separable Mathieu beams generated by A(ϕ)=ceν(ϕ,q) and A(ϕ)=seν(ϕ,q) are cones and their caustic is the z axis; thus, they are not structurally stable. However, in general, the Mathieu beam generated by A(ϕ)=ceν(ϕ,q)+iseν(ϕ,q) is stable because locally its caustic has singularities of the fold and cusp types. To show this property, we present the wavefronts and the caustics for the Mathieu beams with characteristic value aν=0 and q=0,0.2,0.3,0.5. For q=0, we obtain the Bessel beam of order zero; in this case, the wavefronts are cones and the caustic coincides with the z axis. For q≠0, the wavefronts are deformations of conical ones, and the caustic surface, for some values of q, has singularities of the cusp ridge type. Furthermore, we remark that the set of Mathieu beams with characteristic value aν=0 and 0≤q<1 has associated a caustic with singularities of the swallowtail type, which is structurally stable. Therefore, we conclude that this type of Mathieu beam is more stable than plane waves, Bessel beams, parabolic beams, and those generated by A(ϕ)=ceν(ϕ,q) and A(ϕ)=seν(ϕ,q). To support this conclusion, we present experimental results showing the pattern obtained after obstructing a plane wave, the Bessel beam of order m=5, and the Mathieu beam of order m=5 and q=50 with complex transversal amplitude given by Ce5(ξ,50)ce5(η,50)+iSe5(ξ,50)se5(η,50), where (ξ, η) are the elliptical coordinates on the plane.

Partially coherent diffraction-free vortex beams with a Bessel-mode structure.

A new family of partially coherent beams carrying optical vortices is introduced. Any member of this family represents an incoherent superposition of fully coherent orthogonal Bessel modes with the same helical wavefront and is notable for its diffraction-free propagation. It is shown analytically and experimentally that such beams can be approximately generated in the Fourier-transforming optical system with a computer-controlled liquid-crystal spatial light modulator.

Simple technique for generating the perfect optical vortex.

We propose an improved technique for generating the perfect optical vortex. This technique is notable for the simplicity of its practical realization and high quality of the results. The efficiency of the proposed technique is illustrated with the results of physical experiments and an example of its application in optical trapping of small particles.

Generation of the "perfect" optical vortex using a liquid-crystal spatial light modulator.

We introduce the concept of the perfect optical vortex whose dark hollow radius does not depend on the topological charge. It is shown analytically and experimentally that such a vortex can be approximately generated in the Fourier transforming optical system with a computer-controlled liquid-crystal spatial light modulator.

Experimental generating the partially coherent and partially polarized electromagnetic source.

The technique for generating the partially coherent and partially polarized source starting from the completely coherent and completely polarized laser source is proposed and analyzed. This technique differs from the known ones by the simplicity of its physical realization. The efficiency of the proposed technique is illustrated with the results of physical experiment in which an original technique for characterizing the coherence and polarization properties of the generated source is employed.

Modulation of coherence and polarization using liquid crystal spatial light modulators.

We propose a method for modulation of coherence and polarization of electromagnetic fields, employing two crossed zero-twisted nematic liquid crystal spatial light modulators. In contrast to a similar technique analyzed by Shirai and Wolf [J. Opt. Soc. Am A, 21, 1907, (2004)] our method provides a wide range simultaneous modulation of coherence and polarization. The dependence of the obtained results on different definitions of electromagnetic coherence is considered.