Internal Conical Refraction Research Articles

Group velocity of light in internal conical refraction.

We calculated the group velocity of light in internal conical refraction in a biaxial crystal as a function of the direction of the electric displacement vector, or the vibration direction, of its carrier wave. Our method represents group velocity through the electromagnetic fields of light, rather than its wave normal or ray direction. The travel time of a light pulse traversing a parallel plate biaxial crystal in internal conical refraction is found to vary as a sinusoidal function of twice the vibration angle of the light wave. Our method distinguishes the four directions of the two optic axes in monoclinic and triclinic crystals. Numerical examples are given for K N b O 3 at the wavelength of 400nm, and for S n 2 P 2 S 6 at the wavelength of 550nm.

Theoretical and numerical investigation of internal conical refraction of structured light beams.

We present an ab initio numerical investigation of the internal conical refraction of structured light beams in a biaxial crystal. Starting from the solutions of the Fresnel equation, a theoretical analysis is developed without assuming any analytical approximation, thus obtaining a set of exact equations that can be solved by standard methods of integration for any impinging light beam. As examples of applications, we consider the particular cases of linearly and circularly polarized Gaussian and Bessel beams inside a KTP crystal. Numerical calculations follow the evolution of the refracted beam inside the crystal. It is seen that for realistic boundary conditions, a refraction cone appears in a certain range of distances within the crystal, and its shape is rather sensitive to the initial conditions.

A detailed study of internal conical refraction phenomenon of Flattened Gaussian beams propagating in a biaxial crystal

By means of the internal conical refraction phenomenon, the transformation of input Flattened Gaussian beams (FGBs) through a biaxial crystal has been investigated in this paper. The effect of this phenomenon on these input beams leads to the formation of two bright rings separated by a dark ring in the focal plane. Analytical expressions of electric fields and intensity profiles of the considered light beams propagating in a biaxial crystal have been derived. Some numerical simulations are performed and discussed by studying the influences of some parameters as the input beam order, the ratio between the ring radius and the waist radius of the incident beam, and the propagation distance. The propagation of circularly polarized Gaussian beams in the considered optical system is also deduced as particular case in this work.

Hamilton’s optics

Building on work by Fermat and Huygens, Hamilton transformed the study of geometrical optics in his very first paper, presented when still in his teens. His ‘characteristic function’ was an analytical way to describe wavefronts, and in his hands a powerful tool to look at families of rays rather than isolated ones. His prediction of internal and external conical refraction in some crystals and its spectacular verification in a few months established his reputation among his contemporaries. This formulation of optics uncovered many general properties, not easy to see in the conventional method of tracing individual rays. The deepest outcome of his early optical work was a parallel view of the mechanics of particles, which played a fundamental role in the birth of quantum mechanics and continues to be the standard framework for classical mechanics up to the present time.

Collinear Acousto-Optical Transformation of Bessel Light Beams in Biaxial Gyrotropic Crystals

The collinear acousto-optical transformation of Bessel light beams in biaxial gyrotropic crystals into two annular, internal conical refraction beams with orthogonal elliptical polarization is studied. It is found that the diffraction efficiency is maximal (~50–60%) for low ultrasound intensities and varies slightly with further increases in acoustic power. At high ultrasound intensities, the intensities of the transmitted and diffracted annular beams differ insignificantly. The possible use of this acousto-optical interaction for creating collinear tuneable narrow-band acousto-optical filters at low ultrasonic frequencies is demonstrated.

Experimental study of internal conical refraction in a biaxial crystal with Laguerre–Gauss light beams

The effect of internal conical refraction (CR) in a biaxial crystal was studied using Laguerre–Gauss light beams with and 2, while the lowest-order beam was used as a reference. The transition from ordinary double refraction to CR was examined. It has been shown that double refraction of an beam forms two focal spots containing dark stripes. These stripes evolve into dark rings over an annular focal image when CR is established, and it results in a fine-structure of bright focal rings with different intensities. In a sharp contrast to the lowest-order reference, the multiring focal structure has a distinct asymmetry with respect to the focal image plane. It has been shown that bright off-axis ‘hot spot’ can be formed on the far-field profiles of outgoing light beams when the biaxial crystal is slightly tilted, and a small angle between the propagation axis of the beam and the optic axis of the crystal arises. These off-axis light structures emerge as either a charge-one optical vortex or a zero-charge spot with annihilated vorticity. Polarization selection reveals or Bessel-like profiles of the corresponding ‘hot spots’, and a complex pattern of forked fringes in the dark region near the beam core.

Fourier space method for calculating the propagation of laser radiation in biaxial crystals taking into account the angle between the eigenpolarisations

We have proposed a technique for calculating the propagation of laser radiation in biaxial optical crystals in arbitrary directions. The technique is based on the use of the Fourier space method and takes into account both diffraction and angle beween the eigenpolarisations of the spatial spectrum components, phase shift differences for them with account for all orders of the spatial dispersion and also the features of the boundary conditions at the input and output facets. Using internal conical refraction as an example, we have compared the results of calculations with experimental data.

Wave-vector and polarization dependence of conical refraction

We experimentally address the wave-vector and polarization dependence of the internal conical refraction phenomenon by demonstrating that an input light beam of elliptical transverse profile refracts into two beams after passing along one of the optic axes of a biaxial crystal, i.e. it exhibits double refraction instead of refracting conically. Such double refraction is investigated by the independent rotation of a linear polarizer and a cylindrical lens. Expressions to describe the position and the intensity pattern of the refracted beams are presented and applied to predict the intensity pattern for an axicon beam propagating along the optic axis of a biaxial crystal.

The far-field structure of Gaussian light beams transformed by internal conical refraction in a biaxial crystal

The far-field asymptotic evolution of circularly and linearly polarized Gaussian laser beams transformed by internal conical refraction in a biaxial crystal is considered. The specific intensity and polarization pattern of far-field beams is analyzed. A good agreement between experimental observations and predictions of paraxial theory of conical refraction is shown.

Acoustics and optics of absorptive crystals: a universal formalism for topological effects

A universal form of the equations for acoustical and optical wave fields in absorptive crystals is obtained. On this basis, a unified formalism is constructed and used to describe the effect of absorption on the topology of polarization fields and wave surfaces of elastic and electromagnetic waves close to conical acoustic and optical axes. Unless forbidden by symmetry, both types of axes are split by absorption, with the consequences that the wave surfaces acquire self-intersection lines connecting pairs of split axes and that new singular points with the Poincaré indices arise in the polarization fields. Near the split points, the waves of degenerate branches show an abrupt increase in ellipticity, which transforms the internal conical refraction from a local property along the degeneracy direction into a continuum phenomenon occurring throughout this entire region. For each direction of the wave normal, there is a universal refraction cone, the same as for zero absorption. The ends of ray velocity vectors move along the universal section of this cone. This section is elliptic in acoustics and circular in optics. The kinematics of this precession depend in an essential way on the direction of the wave normal.

The phenomenon of conical refraction in a thin-layer periodic semiconductor-dielectric structure in a magnetic field

The propagation of electromagnetic waves along the optical axes of a thin-layer periodic semiconductor-dielectric structure in an external magnetic field (i.e., under conditions of external and internal conical refraction) has been investigated. It is shown that the conditions for conical refraction can be implemented in certain regions by changing the external magnetic field, wave frequency, and thickness of the layers forming the structure. By varying the above-mentioned characteristics, one can efficiently control the conical refraction parameters; in particular, the opening angles of the cone of internal and external conical refraction and the inclination of the optical axes with respect to the periodicity axis can be varied in a wide range. The results of this study may be useful for designing millimeter-wave, submillimeter-wave, and IR devices.

Conical refraction and formation of multiring focal image with Laguerre–Gauss light beams

For a light beam focused through a biaxial crystal along one of its optical axes, the effect of internal conical refraction in the crystal leads to the formation in the focal image plane of two bright rings separated by a dark ring. It is shown that, with circularly polarized Laguerre-Gauss LG(0)(ℓ) beams entering the crystal, this classical double-ring pattern is transformed into a multiring one consisting of ℓ+2 bright rings.

Frequency doubling with laser beams transformed by conical refraction in a biaxialcrystal

The second-harmonic generation with laser beams transformed by internal conicalrefraction in a biaxial crystal was studied in a cascaded two-crystal arrangement, wherethe biaxial crystal transformed the input Gaussian beam and another nonlinearcrystal was used for frequency doubling of the produced conical refraction pattern.Several specific second-harmonic beam profiles obtained for various positions of thenonlinear crystal along the beam propagation direction are considered and discussed.

Magnetic birefringence and conical refraction of elastic waves in antiferromagnetic α-Fe2O3

This paper reports on the results of an experimental investigation into the influence of linear magnetic birefringence of transverse elastic waves due to magnetoelastic interaction on the conical refraction of sound in antiferromagnetic α-Fe2O3. It has been found that, in a weakly ferromagnetic state, the incoming flux of the sound energy is separated into two fluxes corresponding to two normal modes of transverse vibrations in the sample. They propagate along the C 3 trigonal axis of this crystal with different phase velocities due to the linear magnetic birefringence of sound. It has been established that the angle of deviation (θη) from the C 3 axis for the group velocity of the normal mode of elastic vibrations, which does not interact with the magnetic subsystem, is determined by the internal conical refraction and does not depend on the magnitude of the magnetic field in the basal plane (H ⊥ C 3). It has also been shown that the angle of deviation (θξ) of the group velocity of the interacting mode of normal vibrations depends on the magnitude of the magnetic field and tends to θη at large values of H. The group velocity vectors lie in the same azimuth plane passing through the C 3 axis and deviate in opposite directions. The obtained results confirm the conclusions drawn from the theoretical description of the conical refraction of sound in antiferromagnets.

Biaxial crystal as a versatile mode converter

The extreme versatility of the internal conical refraction in biaxial crystals in thetransformation of Gaussian laser beams is demonstrated and discussed. By means ofsimple variations in the focusing and polarization of the input beam, various beamconfigurations like Bessel–Gauss, Hermite–Gauss, Laguerre–Gauss, and othersare shown to be produced from a lowest-order Gaussian beam passed through abiaxial crystal along one of its optical axes. Further transformations of the beamprofile and formation of more complex light patterns were obtained in a cascadedscheme, when the beam was passed consecutively through two crystals. Theseobservations, together with the known ability of conical refraction to form a variety ofother complex light structures, demonstrate the unique properties of the effectin manipulations with the amplitude, phase, and polarization of light beams.

Antiferromagnetic conical refraction of elastic waves in hematite

Conical refraction, which is due to the renormalization of the elastic moduli by the effective magnetoelastic interaction and depends on the static magnetic field, has been experimentally observed in an α-Fe2O3 trigonal easy-plane antiferromagnet in addition to the usual internal conical refraction of transverse elastic waves propagating along the trigonal C3 axis. It has been shown that the deviation angle θη of the energy flux from the C3 axis at the internal conical refraction is independent of the magnetic field applied in the basal plane (H ⊥ C3) and is a constant determined by the ratio of the elastic moduli C14 and C44. The deviation angle of the energy flux at the antiferromagnetic conical refraction increases with the magnetic field and approaches the value θη at large H values. The results are well described by the theory of this phenomenon developed by E.A. Turov and confirm its basic conclusions.

Analogies between the torque-free motion of a rigid body about a fixed point and light propagation in anisotropic media

An apparently unnoticed analogy between the torque-free motion of a rotating rigid body about a fixed point and the propagation of light in anisotropic media is stated. First, a new plane construction for visualizing this torque-free motion is proposed. This method uses an intrinsic representation alternative to angular momentum and independent of the modulus of angular velocity \U0001d6da. The equivalence between this plane construction and the well-known Poinsot's three-dimensional graphical procedure is also shown. From this equivalence, analogies have been found between the general plane wave equation (relation of dispersion) in anisotropic media and basic equations of torque-free motion of a rigid body about a fixed point. These analogies allow reciprocal transfer of results between optics and mechanics and, as an example, reinterpretation of the internal conical refraction phenomenon in biaxial media is carried out. This paper is intended as an interdisciplinary application of analogies for students and teachers in the context of intermediate physics courses at university level.

The fundamental matrix of the system of linear elastodynamics in hexagonal media. Solution to the problem of conical refraction

An explicit integral representation by single definite integrals of the fundamental matrix (Green's tensor) of the time-dependent system of hexagonal elastic media is derived. Thereby the problem of internal conical refraction in such media is solved.

Polarization-tunable Yb:KGW laser based on internal conical refraction

Using internal conical refraction, we demonstrate an Yb:KGW laser where the polarization state can be arbitrarily altered without any additional cavity components. The extinction ratio can also be altered between 1:1 and 40:1. The maximum output power achieved was 8.6 W at a slope efficiency of 60.5% with respect to incident pump power. This equals the power performance of a standard Yb:KGW laser used for reference.

Conical refraction: Experiments and large-scale demonstrations

Results of experiments on internal and external conical refraction on a rhombic sulfur single crystal are presented. A high-transmission optical scheme is suggested with light focusing on the crystal surface which allows two fringes of internal conical refraction ∼1 m in diameter to be observed in laser radiation. Two fringes of external conical refraction with unique polarization state are first observed, and conditions of light energy concentration along the biradial are discussed. The focusing power of a plane-parallel biaxial crystal plate is measured.