Internal Reflection in Uniaxial Crystals,III: Transmission and Reflection Coefficients for an Extraordinary Incident Wave

Abstract

The boundary conditions have been solved and the reflection and transmission coefficients calculated for an interface between a uniaxial crystal and an isotropic medium when the incident wave is extraordinary. The existence of a pseudo-Brewster angle for each reflected wave has been verified, and the polarizations of the reflected refracted waves have been determined.     

A Simple Optical Procedure for Characterizing Uniaxial Media

Abstract

Angle-dependent reflectivities, at the interface between an isotropic medium and a uniaxial crystal, have been analysed for the situation where the index of the isotropic medium lies between the ordinary and extraordinary indices of the uniaxial crystal. The analytic results reveal simple relationships between the parameters specifying the optical tensor of the uniaxial medium and the angles of incidence for which a suitable combination of the input and output polarization angles gives zero reflected signal. As a consequence, by using a prism together with a matching fluid of the same index and determining these incidence angles, a procedure for complete optical characterization of a single crystal of calcite with one polished face has been demonstrated. The experimental results are in good agreement with the analytic predictions.     

Guided modes in a uniaxial multilayer

An algorithm is presented for simulation of guided modes in a multilayer uniaxial structure with each layer characterized by its own ellipsoid of refractive indices and direction of optical axis. The proposed approach is based on presenting an electromagnetic field in each layer as a linear combination of ordinary and extraordinary waves coupled through the boundary conditions. The problem is reduced to two dimensions by considering the waves with a given projection of the wave vector on the plane of the waveguide. No a priori assumption about the guided-mode polarization is required in this method. Hybrid polarized modes appear naturally as solutions of a system of linear equations with respect to the amplitudes of the ordinary and extraordinary waves. The proposed approach covers a wide variety of important practical cases including isotropic waveguides, surface waves at the boundary between positive uniaxial crystal and isotropic medium, surface plasmons at metallic interfaces, uniaxial multilayers in a very general form, and leaky modes in such structures.     

Implementation of ordinary and extraordinary beams interference by application of diffractive optical elements

Abstract

We apply diffractive optical elements in problems of transformation of Bessel beams in a birefringent crystal. Using plane waves expansion we show a significant interference between the ordinary and extraordinary beams due to the energy transfer in the orthogonal transverse components in the nonparaxial mode. A comparative analysis of the merits and lack of diffractive and refractive axicons in problems of formation non-paraxial Bessel beams has shown the preferability of diffractive optics application in crystal optics. The transformation of uniformly polarised Bessel beams in the crystal of Iceland spar in the nonparaxial mode by application of a diffractive axicon is investigated numerically and experimentally.     

Field and Poynting vectors of homogeneous waves in uniaxial and absorbing dielectric media

The propagation of a homogeneous plane wave in uniaxial and absorbing dielectric media is described in detail. Complete expressions for electric displacement field, electric field, magnetic field and Poynting vector are derived using a new formalism that simplifies the interpretation of polarization. These expressions are analysed and verified in some particular situations. Whilst absorption is directly determined by the absorption coeffcient k, the field geometry depends exclusively on the absorption index X for given ordinary and extraordinary refractive indices. When ordinary and extraordinary absorption indices are equal, the field geometry coincides with the field geometry in a transparent medium with the same refractive index as the absorbing medium.     

Complex unitary vectors for the direction of propagation and for the polarization of electromagnetic waves in uniaxial and absorbing dielectric media

The propagation of inhomogeneous and elliptically polarized plane waves in absorbing uniaxial anisotropic media is described using complex unitary vectors to represent the direction of propagation and the direction of polarization. Detailed expressions for electric displacement, electric field, and magnetic field vectors are obtained for the ordinary and extraordinary waves, and their geometry is discussed. According to the complex direction of propagation, three particular cases are studied: the real case (homogeneous wave), the case perpendicular to the optical axis, and the case coplanar with the optic axis. The case of isotropic media is also analyzed.     

Behavior of total internal reflection in optical crystals

It is shown that a crystal can be cut so that one incident beam undergoing reflected from an inclined face inside the crystal excites four beams, two ordinary and two extraordinary, propagating in different directions. Pis’ma Zh. Tekh. Fiz. 25, 46–51 (January 12, 1999)     

Interface between transparent isotropic and uniaxial absorbing dielectric media: equations for ray tracing and for the direction of propagation

In this work we analyze the propagation of a plane wave that passes from an isotropic transparent medium to a uniaxial absorbing medium. Detailed expressions that give the real directions of propagation of the wave and the energy of the reflected and refracted ordinary and extraordinary waves are obtained. These expressions are valid for every orientation of the optic axis of the uniaxial medium and for every direction of propagation of the incident wave. Expressions are tested in the case of an interface between a transparent and an absorbing isotropic media and for the air-rutile (TiO2) interface. The effect of absorption has been evaluated by comparing the results obtained in rutile with the results obtained in a transparent uniaxial medium with the same real refractive indices. Results are presented for different values of the angle of incidence and the orientation of the plane of incidence.     

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.     

Propagating and evanescent waves associated with inhibited reflection in uniaxial crystals: Extraordinary incidence

Abstract

When an extraordinary wave under inhibited reflection conditions is incident on an interface between a uniaxial crystal and an isotropic medium, the reflected extraordinary wave is evanescent and the polarization of the refracted wave changes from linear to elliptical. In the present paper it is shown that the refracted ray undergoes a shift which is not only longitudinal (as the Goos—Hänchen effect in total reflection in isotropic interfaces) but also transversal. The structure of the evanescent reflected wave is studied and the polarization of the transmitted wave is analysed.     

Multidimensional calculation of ray path in a twisted nematic liquid crystal cell

In general, light propagating an inhomogeneous liquid crystal (LC) cell can be modeled as ‘bundle rays’ because the LC cell consists of many birefringence layers. In order to calculate the optical path of the propagating light in the inhomogeneous LC cell, we multidimensionally calculated the wavevector, k, and the Poynting vector, S, of an ordinary and an extraordinary ray at LC grid interfaces, which are isotropic to a uniaxial medium and a uniaxial-to-uniaxial medium, by using the phase matching method. Furthermore, we also investigated the transmission coefficients and transmittance of the ordinary and the extraordinary rays as a function of difference of the optical axes of the facing birefringence medium at the interface to obtain the significant rays in the LC cell. Finally, we could calculate the exact path of the significant rays in the inhomogeneous LC cell, and compared the ray path in an electrically controlled birefringence (ECB) mode and a twisted nematic (TN) LC mode.     

Propagation of the Airy beam along the optical axis of a uniaxial medium

We investigate the propagation of an Airy beam along the optical axis of a uniaxial medium, and we find that the propagation property of the Airy beam is determined by the ordinary refractive index of uniaxial crystals and is independent of the ratio of the extraordinary to ordinary refractive index. We also know that the polarization state of linearly polarized Airy beams changes gradually during the propagation. This shows that the propagation properties of the Airy beam in uniaxial crystals along the optical axis is distinctly different from that orthogonal to the optical axis.     

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.     

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.     

Electric polarization and the extinction of the extraordinary reflected ray

Abstract

When light is incident from an isotropic medium on to an interface with a uniaxial crystal, the idea of an effective electric polarization is useful. We show that the same idea may be extended to the reverse situation where the light is incident from the uniaxial crystal. In this case, when the optical axis lies in the plane of incidence, there is a Brewster angle associated with the extinction of the reflected ray, because the waves have a definite polarization. We analyse the relation between the Brewster angle and the effective electric polarization at the interface.     

Light propagation in chiral media with large pitch

Light propagation in uniaxial chiral media with large pitch is studied. In these systems there are forbidden zones for extraordinary beams, which lead to effective reflection on zone boundaries and to wave damping inside the forbidden zone. We analyze the vicinities of the turning points and the transition of an extraordinary wave through the forbidden zone. Narrow forbidden zones with merging turning points are studied in detail. The transition through the forbidden zone is studied experimentally in nematic liquid crystal doped with a chiral addition. There is a good agreement between experimental results and theoretical calculations.     

Dispersion of magnetostatic waves in a tangentially magnetized ferrite wafer with normal uniaxial anisotropy

A dispersion law is obtained and analyzed for the first time for magnetostatic waves in a tangentially magnetized ferrite wafer with uniaxial anisotropy where the axis is perpendicular to the plane of the wafer and the applied static magnetic field is weaker than the anisotropy field. This model qualitatively describes the dispersion of magnetostatic waves in hexa-and orthoferrite wafers and also in an unsaturated ferrite wafer. Pis’ma Zh. Tekh. Fiz. 25, 73–78 (November 26, 1999)     

Polarimetric Study of the Phase Conjugate Wave in an Isotropic Medium

Abstract

We derive expressions for the third-order nonlinearities responsible for the optical phase conjugation via backward degenerate four-wave mixing using various combinations of left and right circularly polarized beams in isotropic media. The effects of polarization conjugation and the nulling effect of phase conjugate waves are also discussed in an isotropic medium.     

Transformation of a Gaussian Beam into a Uniform Beam by a Lenslike Medium with Gain or Loss

Abstract

The propagation of Gaussian beams through lenslike media with gain or loss is analysed in order to discuss the transformation of these beams into uniform beams. The results obtained are applied to a lenslike medium with gain increasing along the radial direction.     

Non-absorbing isotropic–uniaxial interfaces: refraction in ordinary and extraordinary total reflection

We study characteristics of refraction of plane waves in a non-absorbing isotropic–uniaxial interface in conditions of total and partial reflection when the incident wave has arbitrary direction with regard to the optical axis of the crystal and polarization. We analyse the ordinary and extraordinary fields and a phase shift that appears in the propagating wave which subsists in ordinary or extraordinary total reflection. We obtain that, unlike what happens in isotropic interfaces, under conditions of ordinary total reflection, the ordinary time-averaged Poynting vector changes its direction with the angle of incidence which does not depend on the existence of extraordinary total reflection. Extraordinary total reflection gives place to an analogous behaviour of the extraordinary time-averaged Poynting vector. In addition, we define and calculate in an explicit way complex coefficients of transmission for geometries with and without symmetry in partial and total reflection.