Goos-Hänchen effect of an extraordinary refracted beam

As known, when there is total reflection on an isotropic or anisotropic interface, the reflected ray undergoes a displacement on the interface that has been studied by a great number of authors. However, if an isotropic-uniaxial interface is considered, the condition of total reflection for one of the refracted rays can be fulfilled whereas the other subsists as a propagating wave. This leads to the existence of a complex displacement of the propagating refracted beam maximum that can never take place in linear isotropic interfaces. We analyse this displacement up to second order and its relationship with the phase shifts which the waves that synthesize the extraordinary propagating beam suffer under conditions of ordinary total reflection. We compare these first order non-geometric effects with those which an ordinary transmitted beam undergoes in conditions of extraordinary total reflection.     

Measurement of Birefringence,Dispersion and Line Splitting for Biaxial Crystals by Double-layer Interferometer

Abstract

The fringes of a double-layer interferometer containing a birefringent plate, cut from a biaxial crystal, are split into two separate components belonging to the ordinary and extraordinary waves whose refractive indices have a difference determining the component separation. The dependence of the intensity ratio of the resolved components upon the angle of ray propagation in the crystal is used to identify the axes of a biaxial crystal like muscovite mica. A fringe count with a rotated sample accurately determines its optical thickness from which the absolute values of the refractive indices are deduced. The sensitivity of the double-layer interferometer to dispersion effects is so high that a birefringence dispersion is measurable and either the anisotropy of the atomic number density or the splitting of the characteristic spectral line is detectable.     

Topological birefringence and combined Rytov-Magnus effect

It is shown that the symmetry properties of the locally isotropic inhomogeneous medium of an optical fiber cause circular and linear topological birefringence. The circular birefringence δn _C in graded-index fibers is ∼(λ/ρ)² (where λ is the wavelength and ρ is the core radius), while the linear birefringence is δn _L ∼(λ/ρ)³. This topological birefringence is characterized not only by the polarization basis (as in crystals for example) but also by the magnitude and sign of the topological charge of the guided vortex. This topological birefringence forms the basis of the instability of the fiber IV vortex and is observed experimentally as the combined Rytov-Magnus effect. Pis’ma Zh. Tekh. Fiz. 25, 41–46 (March 12, 1999)     

Structure and Properties of Inhomogeneous Waves

Abstract

The structure and properties of optical radiation are considered in terms of inhomogeneous waves propagating in isotropic absorbing media. Within the framework of the proposed representation, it is shown that the optical properties of inhomogeneous waves can be completely defined by three independent variables related to the degree of inhomogeneity of the wave, together with the ellipticity of the E and H components of the wave field. The values of these variables cannot be chosen completely arbitrarily. In a geometrical interpretation we consider these variables to be restricted to a three-dimensional region demarked by a ‘boundary wave surface’. The topology of this boundary wave surface and the possible types of inhomogeneous wave are investigated, and invariant relations are revealed. Simple and compact relations describing inhomogeneous waves are proposed.     

Photonic Bands

Abstract

Photonic band structure has been computed using ellipsoidal grains in f.c.c. lattice. Bandgaps have been found and the conditions for the appearance of such gaps are discussed. The effective long-wavelength dielectric constants for the ordinary and the extraordinary rays are calculated and compared with the predictions of effective medium and Maxwell-Garnett theories.     

Huygens's principle and rays in uniaxial anisotropic media. I. Crystal axis normal to refracting surface

Huygens's principle is used to derive equations for tracing the extraordinary ray in a uniaxial crystal when the crystal axis is normal to the refracting surface. Snell's law is used to trace ordinary rays that lead to an array of ordinary spherical wavelets centered on the refracting surface. Each spherical wavelet is then replaced by an extraordinary wavelet in the shape of a rotationally symmetric ellipsoid whose major axis is in the direction of the crystal axis. The envelope of these is the extraordinary wave front; the extraordinary ray is a vector from the center of the wavelet to its point of contact with the extraordinary wave front. The inverse problem is also solved, yielding expressions for the ordinary ray in terms of the extraordinary ray, making it possible to trace the extraordinary ray out of the system by using a fictive ordinary ray. We found the geometrical invariant for the uniaxial crystals.     

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.     

Paraxial theory for birefringent lenses

The generalized ray tracing for the extraordinary ray through uniaxial crystals developed by Avenda?o-Alejo and Stavroudis [J. Opt. Soc. Am. A 19, 1674 (2002)] has been applied to derive paraxial refracting equations. Paraxial equations are derived for three cases where the incident, ordinary, and extraordinary rays lie in the incident plane: (a) the crystal axis is parallel to the optical axis, (b) the crystal axis is orthogonal to the optical axis and lies in the plane of incidence, and (c) the crystal axis is orthogonal to both the optical axis and the incident plane. The paraxial ray-tracing equations for the extraordinary ray are represented by matrix operators. The elements of the matrix system give all the information of the focal points and of the principal points. Gaussian formulas are derived, and some examples are presented.     

Refractive index of nematic liquid crystals in the submillimeter wave region

Large electro-optic effects of liquid-crystal materials are attractive in applications to various optical devices in a wider wavelength region. Fundamental optical properties in the submillimeter wave region, such as refractive indices and transmission losses for some cyanobiphenyl nematic liquid crystals, have been investigated for the first time, to our knowledge, with a submillimeter laser. Refractive indices of the liquid crystal materials for ordinary and extraordinary rays are a little larger than those in the visible region, and a larger birefringence comparable with the visible region can also be obtained. Although the loss level is larger by ~2 orders of magnitude than that of quartz plate, which is an excellent window in the submillimeter wave region, the transmission of the liquid crystal cell is high enough.     

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.     

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.     

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)     

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.     

Time Reversal for the Polarization State in Optical Systems

Abstract

The time-reversal operator for the polarization state can be successfully implemented in any optical system where a beam retraces its path. A Faraday rotator followed by a mirror realizes a device whose representative matrix is similar to the quantum mechanics time-reversal operator for the spin. Any effect of the medium birefringence is cancelled and, for linear polarization, the beam always returns opposite polarized with respect to the entrance state. Analogies with the operation of a phase-conjugation mirror are pointed out and suggested consequences of the novel optical configuration are given.     

Recursive exact ray trace equations through the foci of the tilted off-axis confocal prolate spheroids

Abstract

Simple, exact ray trace equations are derived for a general ray propagating through the foci of tilted, coupled, confocal spheroids. In this off-axis optical system, there does not necessarily exist a (real) ray for which the paraxial approximation applies.     

Optical path difference in a plane-parallel uniaxial plate

The flux of energy given by the Poynting vector Se and the kt-wave vector normal to the geometrical wavefront for the extraordinary ray propagating through uniaxial crystals can be evaluated by using the theory developed by Avenda?o-Alejo et al. [J. Opt. Soc. Am. A 19, 1668 (2002)] and Avenda?o-Alejo and Stavroudis [J. Opt. Soc. Am. A 19, 1674 (2002)]. We give here the equations necessary to evaluate the general dispersion angle Se x kt. Additionally we define two new dispersion angles, Se x A and kt x A, where A is the crystal axis vector. With these new dispersion angles we evaluate the optical path length traversed by the extraordinary ray in a plane-parallel uniaxial plate when the crystal axis lies in the plane of incidence.     

Three-dimensional polarization ray-tracing calculus II: retardance

The concept of retardance is critically analyzed for ray paths through optical systems described by a three-by-three polarization ray-tracing matrix. Algorithms are presented to separate the effects of retardance from geometric transformations. The geometric transformation described by a "parallel transport matrix" characterizes nonpolarizing propagation through an optical system, and also provides a proper relationship between sets of local coordinates along the ray path. The proper retardance is calculated by removing this geometric transformation from the three-by-three polarization ray-tracing matrix. Two rays with different ray paths through an optical system can have the same polarization ray-tracing matrix but different retardances. The retardance and diattenuation of an aluminum-coated three fold-mirror system are analyzed as an example.     

Propagation modes and regimes of intense laser beam in magnetized plasma

This paper presents an analytical and numerical investigation of a circularly polarized beam propagating along the static magnetic field parallel to oscillating magnetic field in plasma at relativistic intensities. In the high intensity regime, such a magnetic field is created by the pulse itself. The authors have identified three regimes of propagation taking into account the relativistic mass correction. Based on WKB and paraxial ray theory, an appropriate expression for a dielectric tensor has been evaluated in the presence of an externally applied magnetic field. The natural electromagnetic modes are circularly polarized. Consequently, extraordinary and ordinary modes propagate, which are significantly affected due to the relativistic mechanism. The regimes are characterized by dimensionless power and beamwidth, characterizing the nature of propagation as steady divergence, oscillatory divergence and self-focusing. Numerical computations are presented and discussed for typical parameters of laser plasma interaction; defined through critical parameters, namely cyclotron-to-beam frequency (Ω _c ), plasma-to-beam frequency (Ω _p ) and beam power for arbitrary large intensities.     

Diffractive patterns superimposed over propagating N-slit interferograms

Abstract

Transparent, microscopic spider web silk fibers were used to softly intrude into the propagating path of N-slit interferograms. The resulting interferograms, with superimposed diffractive signals, were recorded using digital means and reproduced using N-slit interferometric calculations. We also show, for the first time, very slight and subtle alterations of the propagating interferograms via the soft insertion of spider web silk fibers into the intra-interferometric path. The experiments were performed at an overall intra-interferometric propagation path length of 7.235 m.