Nonparaxial Gaussian Research Articles

A comparative study of the propagation of vectorial nonparaxial beams in the use of different approaches

Starting from the vectorial Rayleigh diffraction integral formula and using a simple expansion of the nucleus in the Rayleigh formula, an analytical propagation equation of vectorial nonparaxial Gaussian beams in free space is derived, which permits us to perform numerical calculations in comparison with the expression derived by Ciattoni et al. and with the direct numerical integration of the Rayleigh formula. It is found that as usual the use of expansion of vectorial Rayleigh diffraction integral is sufficient to provide satisfactory numerical results as compared with the direct integration of the Rayleigh formula. The above two analytical expressions are valid under certain conditions, however both are applicable in the far field.

Far-field properties of non-paraxial elliptical Gaussian beams

By using the Rayleigh diffraction integral formula, a far-field asymptotic solution of the wave equation is found, which represents non-paraxial elliptical Gaussian beams. The basic properties of such beams are studied, and a comparison with the previous work is also made.

Nonparaxial description of reflection and transmission at the interface between an isotropic medium and a uniaxial crystal.

Angular spectra of reflected and transmitted fields, induced by an arbitrary electromagnetic beam passing through the planar interface between a homogeneous medium and a uniaxially anisotropic medium, are derived and related to the incident medium. By using these formulas, we obtain the expressions for paraxial and slightly nonparaxial fields. The reflected paraxial field is related to the incident one by means of Fresnel relations; the transmitted paraxial field is the superposition of an ordinary and an extraordinary beam, multiplied by the Fresnel coefficient. We find that the nonparaxial corrections, owing to the medium discontinuity, are larger than their free-propagation counterparts and that they are very simply related to the paraxial solutions of the incident beam. The case of two homogeneous media with different refractive indices is also discussed. The general expressions obtained are applied to the case of a nonparaxial Gaussian beam.

Structure of a nonparaxial gaussian beam near the focus: III. Stability, eigenmodes, and vortices

Exact analytical structurally stable solutions of the Maxwell equations for singular mode beams propagating in free space or a uniform isotropic medium are obtained. Approximate boundary conditions are chosen in the form of the requirement that in the paraxial approximation the fields of nonparaxial mode beams in the waist plane are transformed into the fields of eigenmodes and vortices of weakly guiding optical fibers with the axial symmetry of refractive index. It is shown that optical vortices, in spite of a rather complex structure of field distribution, do not experience substantial changes in the beam form and reproduce, in general features, the field of paraxial vortices. Linear perturbations of the characteristic parameters of mode beams do not change the structure of their electromagnetic field. Nonparaxial singular beams have one more important property, in addition to the fact that the structure of these beams in the paraxial approximation is similar to the structure of the fields of eigenmodes in a fiber. The propagation constants of eigenmodes of a fiber exactly coincide (in the first approximation of perturbation theory) with the projection of the wave vector of a mode beam on the optical axis (an analog of the propagation constant). The possibility of the paraxial transition for nonparaxial mode beams with arbitrary values of azimuthal and radial indices is shown. The properties of nonparaxial modes are illustrated by numerous examples. The solutions obtained and the results of their analysis can be used for exact matching optical fibers and laser beams in various applications.

The structure of a nonparaxial Gaussian beam near the focus: II. Optical vortices

Exact analytical solutions of Maxwell’s equations describing the behavior of a nonparaxial optical vortex in the vicinity of a focal waist are obtained using the Whittaker method of scalar potentials, the point complex source method, and approximate Davis boundary conditions. It is shown that nonparaxial optical vortices in free space fall into three large groups: even and odd vortices with preferential circular polarization and azimuthally symmetric TE and TM vortices. The fields of even and odd nonparaxial vortices agree well with the fields of guided homogeneous and inhomogeneous vortices of a weakly guiding fiber. In the paraxial approximation, the expressions obtained for the fields are transformed to the fields of paraxial optical vortices. In the focal region, a nonparaxial beam experiences elliptic deformation of the cross section. This elliptic deformation is shown to result from the asymmetric location of regions with negative energy flows. The reversal of sign of the topological charge and the helicity of a combination of even and odd vortices causes both rotation of the dislocation axis through π/2 and longitudinal displacement of the focal spot, which are the transverse and the longitudinal optical Magnus effects.

Nonparaxial gaussian beams: 2. Splitting of the singularity lines and the optical magnus effect

The Poynting vector field lines for the lowest modes of a nonparaxial Gaussian beam exhibit a number of loops and rings in the vicinity of the phase singularity lines (Airy’s fringes), with negative energy fluxes present inside these loops and rings. The positions of these fluxes are nonsymmetric with respect to rotation about the optical axis. This asymmetry leads to a local splitting of the phase singularity lines, after which the beam cross section transforms from circular to elliptic. The asymmetry of the beam cross section can be eliminated by considering a superposition of circularly polarized even and odd modes. However, this approach only uniformly redistributes the negative energy fluxes in the azimuthal direction of the cross section, rather than completely eliminates these fluxes. Any small perturbation of the resulting symmetric beam gives rise to a unique phenomenon—the optical Magnus effect in the free space, whereby the beam intensity pattern rotates upon changing the circular polarization from right to left. This effect implies the presence of a spin-orbit coupling in the nonparaxial Gaussian beam propagating in the free space.

Nonparaxial gaussian beams: 1. Vector fields

Rigorous analytical solutions were obtained for the vector field of a propagating nonparaxial Gaussian beam in the vicinity of the beam focus. It is demonstrated that a nonparaxial beam of the lowest order has six modes. Upon the limiting transition kz0 ≫ 1, four of these modes exhibit degeneracy and convert into a paraxial Gaussian beam, while the other two modes form the azimuth-symmetric TE and TM fields. A characteristic feature of the nonparaxial modal beam is the absence of symmetry between (still mutually orthogonal) electric and magnetic fields. The deviation from symmetry results in the appearance of negative energy fluxes in the vicinity of the phase singularity manifested by Airy’s fringes, “light drops” (special states of the light field), and “singularity islands”. The obtained results are in good agreement with other relevant data published.

Topological phase in a nonparaxial Gaussian beam

An analysis is made of the structure and evolution of the singularities of a nonparaxial Gaussian beam. It is shown that a Gaussian beam may be represented by a family of straight lines lying on the surface of a hyperboloid and that the wavefront of this beam is a function of a point source situated at a point on the z axis with the imaginary coordinate iz0. The argument of this complex function is the topological phase of the beam which characterizes the rotation of the wavefront. The singularities of a nonparaxial Gaussian beam are located in the focal plane and are annular edge dislocations. Dislocation processes near the constriction of the Gaussian beam only occur as a result of aperture diffraction.

Wave dislocation reactions in non-paraxial gaussian beams

For exact (i.e., non-paraxial) waves ψ representing freely propagating Gaussian beams in two and three dimensions, the patterns of phase singularities, that is zeros of ψ, are studied in detail. The zeros (points in two dimensions, and rings in three) are phase dislocations (optical vortices). The waves depend on a single parameter L, representing the radius of the waist of the beam. As L increases, pairs of dislocations interact and depart from the focal plane. Each such interaction comprises three events where the phase topology of ψ changes; each event is a reaction between the dislocations and associated phase saddles, conserving two topological quantum numbers. The same behaviour was predicted and observed by Karman et al. for beams truncated by apertures. The geometrical sensitivity of the wave to L is astonishing: changes in phase topology can occur when the waist expands by a few thousandths of a wavelength. The integral representing ψ is evaluated asymptotically, leading to a global explanation of the dislocations and topological events in terms of interference between complex and diffracted rays. Locally, all details of the topological changes can be captured by a local model, constructed using a classification devised by Nye.

Evanescent and real waves in quantum billiards and Gaussian beams

A mode in a confined planar region can contain evanescent waves in its plane-wave superposition. It would seem impossible to construct such a mode by continuation of an external scattering superposition. Which must contain only real plane waves. However, evanescent waves can be expressed as the singular limit of an angular superposition of real plane waves. This is surprising because. In the direction perpendicular to that in which it decays. An evanescent wave oscillates faster than the free-space wavenumber; thus the singular superpositions lie in the class of 'superoscillatory' functions, which vary faster than any of their Fourier components. The superposition is the limit of an exact (i.e. non-paraxial and non-singular) Gaussian beam on its evanescent side slopes. Far from the axis, the beam possesses, on each side, a line of phase singularities (nodal points) which organize the global energy current. The three-dimensional generalization provides an explicit elementary construction of a superoscillatory function.

Nonparaxial Gaussian beams

When the waist size of a Gaussian beam becomes of the order of the wavelength of light, the beam does not satisfy the paraxial condition in which it is derived. In this paper, we define the lower bound to the waist size by showing that a Gaussian beam whose waist size is larger than this bound safely satisfies the paraxial condition. A beam which is Gaussian in form but violates the paraxial condition is called a nonparaxial Gaussian beam. We clarify the range of the waist size for which the first-order correction to this beam is effective. It is shown that a distinct value of the waist size exists for which the paraxial approximation completely fails and the first-order correction never works.

Analysis of forward wide-angle light propagation in semiconductor rib waveguides and integrated-optic structures

We analyse longitudinally varying semiconductor rib waveguides with a new nonparaxial wide-angle equation for unidirectional light propagation. We further develop a solution method involving multiplication of the incoming electric field by a series of unitary operators which are evaluated by splitstep fast Fourier transform and finite difference techniques. In the test case of a strongly guiding rib waveguide Yjunction, the estimated losses nearly coincide with those of standard Fresnel equation procedures. We also present a numerical analysis of an integrated-optic lens, demonstrating significant errors in previous results. Finally, we explicitly illustrate the increased accuracy of our new method in comparison to the Fresnel equation for a highly nonparaxial Gaussian beam.