Vector singularities of Gaussian beams in uniaxial crystals: Optical vortex generation

The propagation of a circularly polarized singular beam through a uniaxial crystal is accompanied by the appearance of additional singularities in the polarization structure of the beam field. These vector singularities combine to form concentric ombilic lines—degenerate ombilic points of the star type, thus significantly changing the entire fine structure of the field. When the beam passes through a birefringent quarter-wave plate and a polarizer, the vector singularities transform into the usual optical vortices. Rotation of the polarizer and/or of the quarter-wave plate drives the vortices to move by preset trajectories, merge with one another, or break into elementary singularities. These processes are studied using theoretical and experimental methods.     

Propagation-invariant beams in uniaxial crystals

Propagation-invariant electromagnetic beams, propagating along the optical axis of a uniaxially anisotropic crystal, are investigated. The beams are a superposition of propagation-invariant ordinary and extraordinary fields, with different polarizations states and Bessel profiles. The analytical expression of the Poynting vector reveals an azimuthal transverse energy flow that becomes pure longitudinal, if the complex amplitudes of the ordinary and extraordinary fields are either in phase or in phase opposition.     

Jones matrices of a quarter-wave plate for Gaussian beams

The Jones matrix of a quarter-wave plate is studied theoretically and experimentally, taking into account internal reflections, the ellipsoid of the indices, geometric defects, the tilt angle, and the characteristics of the incident Gaussian beam. The influence of the different parameters is isolated, and large discrepancies are observed with respect to results obtained from the Jones matrix that are usually given in textbooks. It is shown that the effective Jones matrix of the plate does not depend on the longitudinal position of the plate on the Gaussian beam but only on the beam-waist size. This leads to a method of characterization of the defects of a quarter-wave plate that is more precise than the usual methods. Different procedures to optimize the efficiency of a given plate are discussed, taking the plate defects into account. In all cases, a good agreement between experiments and theory is obtained.     

Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals

A simple correspondence between the paraxial propagation formulas along the optical axis of a uniaxial crystal and inside an isotropic medium is found in the case of beams with linearly polarized circularly symmetric boundary distributions. The electric fields of the ordinary and the extraordinary beams are related to the corresponding expressions in a medium with refractive index n(o) and n(e)2/n(o), where n(o) and n(e) are the ordinary and the extraordinary refractive indexes, respectively. Closed-form expressions for Laguerre-Gauss and Bessel-Gauss beams propagating through an anisotropic crystal are given.     

Application of the Jones calculus for a modulated double-refracted light beam propagating in a homogeneous and nondepolarizing electro-optic uniaxial crystal

The Jones matrix calculus is applied to an electro-optic crystal with uniaxial symmetry when the light beam is incident nearly normally on the crystal face. The approach allows one to treat refracted waves and rays that diverge in the crystal and are modulated by an external low-frequency field. The effect of partial interference of overlapping refracted beams is allowed for and calculated for the case of uniform intensity of the beam over its cross section. The method is employed to analyze optical systems containing an imprecisely cut and aligned electro-optic crystal plate.     

Propagation properties of partially coherent anomalous hollow beams in uniaxial crystals

Analytical propagation formulae for partially coherent anomalous hollow beams in uniaxial crystals are derived. Paraxial propagation of partially coherent anomalous hollow beams in uniaxial crystals orthogonal to the optical axis is investigated based on the beam propagation equations. The propagation properties of partially coherent anomalous hollow beams in uniaxial crystals and in free space are studied numerically and comparatively. It is found that the propagation properties of partially coherent anomalous hollow beams in uniaxial crystals behave very differently from those in free space and are closely determined by the parameters of the uniaxial crystals and the initial coherence width. The uniaxial crystals provide an effective way for generating astigmatic beams.     

Alignment of Gaussian beams

The design of a coupling between a semiconductor laser and a single-mode fiber, or between any two optical or acoustical elements that support Gaussian modes, is presented as a trade-off among coupling efficiency T(a), offset misalignment tolerance d(e), and angular misalignment tolerance theta(e). We show that these three parameters are subject to a trade-off limitation which takes the form 0 < T(a)(1/2)theta(e)d(e) < or = lambda/pi, and we show how to design a coupling so that the upper bound on the alignment product T(a)(1/2)theta(e)d(e) is achieved.     

Reflectivity of uniaxial absorbing crystals

Equations for the reflectances of ordinary and extraordinary waves from the basal plane of strongly absorbing uniaxial crystals are extended to reflectances from planes parallel to the optic axis for configurations likely to be useful and convenient for experimentalists.     

Focusing of spherical Gaussian beams

Simple procedures and formulas for tracing the characteristics of a spherical Gaussian beam through a train of lenses or mirrors are described which are analogous to those used in geometrical optics to trace repeated images through an optical train.     

Grazing reflection of Gaussian beams

The reflectivities of most surfaces are higher for grazing or near-90-deg angles of incidence than for more perpendicular or near-zero-deg angles. Grazing-incidence configurations are especially important in the development of lasers and optical systems that operate in the far-ultraviolet and soft-x-ray regions of the spectrum, where transparent or highly reflecting media are almost unknown. Analytical solutions of the paraxial wave equation are obtained for the grazing reflection and complex interference effects that take place when a Gaussian beam interacts at shallow angles with a reflecting surface.     

Gaussian beams in hollow metal waveguides

Various families of Gaussian beams have been explored previously to represent the propagation of nearly plane electromagnetic waves in media having at most quadratic transverse variations of the index of refraction and the gain or loss in the vicinity of the beam. However, such beams cannot directly represent the wave solutions for propagation in planar or rectangular waveguides, and sinusoidal mode functions are more commonly used for such waveguides. On the other hand, it is also useful to consider the possibility of recurring Gaussian beams that have an approximately Gaussian transverse profile at certain distinct planes along the propagation path. It is shown here that under some conditions recurring Gaussian beams can describe wave propagation in hollow metal waveguides, and they can also lead to efficient coupling between the waveguide fields and free-space beams.     

Acoustooptic interaction in uniaxial gyrotropic paratellurite crystals

An analysis is made of an intermediate mode of light diffraction by ultrasound in a uniaxial gyrotropic paratellurite crystal. A system of coupled wave equations is presented to calculate the polarization and energy characteristics of the diffracted light for the Raman-Nath, intermediate, and Bragg modes of acoustooptic interaction. The diffraction of light propagating at small angles to the crystal optic axis by a slow ultrasonic shear wave propagating along the [110] crystallographic axis is studied. The amplitude-frequency characteristics of a modulator-deflector for optical radiation are investigated. Curves of the diffraction efficiency as a function of the ultrasonic wave intensity are plotted for various acoustooptic interaction lengths. Pis’ma Zh. Tekh. Fiz. 23, 84–89 (January 12, 1997)     

Ray-tracing formulas for refraction and internal reflection in uniaxial crystals

Formulas for the calculation of the direction cosines of refracted and internally reflected rays in anisotropic uniaxial crystals are presented. The method is based on a transformation to a nonorthonormal coordinate system in which the normal surface associated with the extraordinary ray is of spherical shape. A numerical example for the case of refraction and internal reflection in calcite is given.     

Internal reflections of the Gaussian beams in Faraday isolators

The reflection coefficient of the Faraday isolator-mirror system in the presence of internal reflections of the incident Gaussian beam is theoretically and experimentally explored for three different architectures of a Faraday isolator. In every case, these internal reflections are shown to alter widely the behavior of the system. In particular, we propose and test a design using a quarter-wave plate that can, in some experiments, give better isolation ratios than conventional isolators.     

Propagation of cylindrically symmetric fields in uniaxial crystals

We investigate the paraxial propagation along the optical axis of a uniaxially anisotropic crystal of a general paraxial beam whose boundary Cartesian components possess cylindrical symmetry. This property allows us to obtain expressions whose dependence on the azimuth angle phi (in cylindrical coordinates) is fully described and very simple. We also find that the beam loses its boundary cylindrical symmetry during propagation, as a consequence of medium anisotropy. Further, these expressions elucidate the way in which the anisotropy changes the state of polarization. As an example, we discuss the case of a Gaussian beam focused into the crystal by a thin spherical lens.     

Aberration limits for annular Gaussian beams for optical storage

Annularly apodized beams have been suggested for use in optical storage because of their potential to go beyond the conventional spot size and depth-of-focus limits. One concern for such applications is the effects of small aberrations on beams in which the energy is concentrated in a small annular ring. We present calculations and experimental results that show that annular apodization of a Gaussian beam reduces the sensitivity to defocus as well as balanced spherical and coma aberrations. The sensitivity to astigmatism is increased by a small amount.     

Hollow Gaussian beams with the power-exponent-phase vortex

A kind of hollow Gaussian beams with the power-exponent-phase vortex is introduced. Based on the Collins integral, an analytical expression of a hollow Gaussian beam with the power-exponent-phase vortex passing through a paraxial optical system described by the ABCD matrix approach is derived. The analytical expressions for the beam propagation factors and the orbital angular momentum density of such hollow vortex Gaussian beam passing through a paraxial optical system described by the ABCD matrix approach are also derived, respectively. As a numerical example, the propagation properties of a hollow Gaussian beam with the power-exponent-phase vortex are demonstrated in free space. The evolutions of the normalized intensity, the phase and the orbital angular momentum density distributions are investigated, respectively. The influences of the power order and the topological charge on the beam propagation factors in the x- and y-directions are analysed. The introduced hollow Gaussian beam has potential applications in the atom manipulation and the optical trapping.     

Scintillation index of flat-topped Gaussian beams

The scintillation index is formulated for a flat-topped Gaussian beam source in atmospheric turbulence. The variations of the on-axis scintillations at the receiver plane are evaluated versus the link length, the size of the flat-topped Gaussian source, and the wavelength at selected flatness scales. The existing source model that represents the flat-topped Gaussian source as the superposition of Gaussian beams is employed. In the limiting case our solution correctly matches with the known Gaussian beam scintillation index. Our results show that for flat-topped Gaussian beams scintillation is larger than that of the single Gaussian beam scintillation when the source sizes are much smaller than the Fresnel zone. However, this trend is reversed and scintillations become smaller than the Gaussian beam scintillations for flat-topped sources with sizes much larger than the Fresnel zone.