Nonreciprocal Birefringence Research Articles

Anisotropic topological magnetoelectric effect in axion insulators

Three-dimensional topological insulators or axion insulators exhibit the topological magnetoelectric effect, which is isotropic with a universal coefficient of proportionality quantized in units of $e^2/2h$. Here we study the finite-size effect of topological magnetoelectric effect, and find the magnetoelectric coefficients are anisotropic, namely $\alpha_{xx}\neq\alpha_{zz}$. Both of them are shown to converge to a quantized value when the thickness of topological insulator film $d$ increases reaching the three-dimensional bulk limit. The nonzero value of $(\alpha_{xx}-\alpha_{zz})\propto1/d$ could be measured by using the gyrotropic or nonreciprocal birefringence of terahertz light. The unique $1/d$ dependence on film thickness of the rotation angle of optical principle axes is the manifestation of topological magnetoelectric effect, which may also serve as a smoking gun signature for axion insulators.

Linear Optic Response of a Noncollinear Magnetic System: Hydrodynamic Theory

A hydrodynamic theory of the linear response of a noncollinearly magnetized medium interacting with electromagnetic radiation has been developed. Linear and quadratic magnetization effects caused by the spatial inhomogeneity of the magnetic moment have been analyzed. Linear magnetization effects include an effect similar to nonreciprocal birefringence, as well as reciprocal and nonreciprocal rotation of the plane of polarization, caused by the inhomogeneity of the magnetic moment. It has been shown that an effect caused by the equilibrium spin current can appear in the considered medium. This effect is determined either by the inhomogeneity of the spin current or by the spatial dispersion of a wave. The effect associated with the spatial dispersion of the wave is linear in its wave vector and is similar to nonreciprocal birefringence. The effect associated with the inhomogeneity of the spin current describes the rotation of the plane of polarization, which, however, can occur in the system with zero average magnetization.

Magnetic field induced nutation of exciton-polariton polarization in (Cd,Zn)Te crystals

We study the polarization dynamics of exciton-polaritons propagating in sub-mm thick (Cd,Zn)Te bulk crystals using polarimetric time-of-flight techniques. The application of a magnetic field in Faraday geometry leads to synchronous temporal oscillations of all Stokes parameters of an initially linearly or circularly polarized, spectrally broad optical pulse of 150 fs duration propagating through the crystal. Strong dispersion for photon energies close to the exciton resonance leads to stretching of the optical pulse to a duration of 200$-$300 ps and enhancement of magneto-optical effects such as the Faraday rotation and the non-reciprocal birefringence. The oscillation frequency of the exciton-polariton polarization increases with magnetic field $B$, reaching 10 GHz at $B\sim 5$T. Surprisingly, the relative contributions of Faraday rotation and non-reciprocal birefringence undergo strong changes with photon energy, which is attributed to a non-trivial spectral dependence of Faraday rotation in the vicinity of the exciton resonance. This leads to polarization nutation of the transmitted optical pulse in the time domain. The results are well explained by a model that accounts for Faraday rotation and magneto-spatial dispersion in zinc-blende crystals. We evaluate the exciton $g$-factor $|g_{\rm exc}|=0.2$ and the magneto-spatial constant $V= 5 \times 10^{-12}$ eVcm$\textup{T}^{-1}$.

Distributed Polarization-Sensitive Reflectometry in Nonreciprocal Single-Mode Optical Fibers

We present a novel theoretical model for the description of round-trip propagation of polarized optical signals in single-mode fibers affected by nonreciprocal birefringence. The model has a general validity, but it has been specialized to the case of Faraday rotation. Building on the theoretical result, we propose also a method to measure both reciprocal and nonreciprocal birefringence of single-mode fibers by means of polarization-sensitive reflectometry. The theoretical model and the measurement technique have been tested in a preliminary experiment, which consisted in measuring the static magnetic field of a 1.5 T magnetic resonance imaging scanner. Experimental results are in agreement with the theory.

Two-Pass Method for Studying Induced Birefringence Effects

We consider a two-pass method for studying the effects of induced reciprocal and nonreciprocal birefringence. In the case of complex anisotropy of the studied sample, methods for isolating reciprocal and nonreciprocal induced effects are proposed. The results of an experimental study of the effect of nonreciprocal magneto-optical linear birefringence (difference between the phase velocities of counterpropagating waves with identical linear polarizations) in crystals of lithium iodate and quartz are presented.

Reciprocal and nonreciprocal magnetic linear birefringence in the γ-Dy2S3 sesquisulfide

A study has been made of the spectral dependence of the Cotton-Mouton effect (CME) quadratic in magnetic field, nonreciprocal birefringence (NB) linear in magnetic field, and the Faraday effect (FE) in the cubic magnetic semiconductor γ-Dy2S3. Unlike the FE, the CME and the NB in this crystal are anisotropic, with the pattern of the anisotropy being dependent on the photon energy. The dependence of the CME and NB dispersion on the direction of the magnetic field B indicates contribution from a variety of electronic transitions and mechanisms to these phenomena. It is shown that the resonant contributions to the CME and NB in the transparency region originate from electronic transitions near E≅3.4 eV (beyond the band edge Eg=2.8 eV), which are likely transitions from the localized ground state of the Dy3+ ion to states derived from mixing of the band and 4fN−1 5d states of the dysprosium ion. The character of the CME anisotropy in the transparency region and near the local electronic transition 6H15/2 → 6F3/2 connecting states of the unfilled 4f shell of the Dy3+ ion suggests the presence of a strong axial component of the crystal field acting on the rare earth ion.

Second Order Magnetoelectric Susceptibility in the Optical Region

The physical phenomena related to manifestation of the second order magnetoelectric susceptibility in optical region are considered. In magnetoelectrics of the type EB 2 this susceptibility contributes to magneto-induced spatial dispersion optical phenomena such as a non-reciprocal birefringence or a non-reciprocal linear dichroism. The analysis of experimental data in boracite and rare-earth sesquisulphide single crystals shows that the magnetoelectric mechanism is the basic one in formation of these phenomena. In magnetic semiconductors type Cd 1 m x Mn x Te the magnetoelectric effect results in the frequency independent part of non-reciprocal birefringence. The results of the theoretical treatment of the second order magnetoelectric susceptibility in the optical region are presented.

Microscopic Theory of Optical Second-Order Magneto-Electric Susceptibility in the Boracite Co 3 B 7 O 13 I

The microscopic theory of second-order magneto-electric susceptibility in optical region is presented for the case of optical transition 4 A 2 ( 4 T 1 ) M 4 E( 4 T 1 ) in the boracite Co 3 B 7 O 13 I. Both the splitting of Kramers doublets of the ground and excited states and the magnetic field dependence of electric and magnetic dipole transition matrix elements contribute to this susceptibility. The experimental data on dispersion, temperature dependence and absolute value of non-reciprocal birefringence in Co 3 B 7 O 13 I can be adequately described in considered model.

Microscopic theory of the second-order magnetoelectric susceptibility in the optical region: Case of boraciteCo3B7O13I

The microscopic theory of second-order magnetoelectric susceptibility in the optical region is presented for the case of optical transitions between states of Co 2 + ions in boracite Co 3 B 7 O 1 3 I. Using many particle wave functions of the Co 2 + ion in the crystal field of tetragonal symmetry D 2 d and accounting for spin-orbit and Zeeman interactions we obtain an expression for the second-order magnetoelectric susceptibility at optical frequencies, which takes into account the population of all Kramers doublets of the ground state and is valid in a wide temperature range. Both the splitting of Kramers doublets of the ground and excited states and the magnetic field dependence of electric and magnetic dipole matrix elements contribute to this susceptibility. The experimental data on dispersion, temperature dependences, and the absolute value of nonreciprocal birefringence in Co 3 B 7 O 1 3 I can be adequately described in the considered model.

Low-frequency behavior of optical spatial dispersion effects

A theoretical explanation is given for the frequency independence of the nonreciprocal birefringence of light, which was recently observed in the semiconductors Cd1−xMnxTe, Zn1−xMnxTe, and GaAs in the frequency range below the frequency corresponding to the interband absorption edge. It is shown that the symmetry of the effect becomes higher at such frequencies if the light-induced excitation energy ℏωn(k) only slightly depends on the photon momentum k. In this case, the nonreciprocal birefringence is completely determined by the second-rank magnetoelectric tensor. It is shown that the nonreciprocal birefringence of light can be observed in magnetic media with a tensor order parameter.

Nonreciprocal light birefringence in the R 3B7O13 X Boracites (R=Co, Cu, Ni; X=I, Br)

The field and angular dependences of nonreciprocal birefringence (NB), which is linear in magnetic field B and is due to magnetic-field-induced spatial dispersion, have been studied in the cubic (symmetry class T d) paraelectric phase of the R 3B7O13 X boracites (R=Co, Cu, Ni; X=I, Br) at a wavelength λ=633 nm. It is shown that the NB in crystals with different 3d and halogen ions exhibits the same anisotropy. The relation between the A and g parameters, A=2g, which determine the NB anisotropy, suggests that the microscopic mechanism of the NB is the manifestation of second-order magnetoelectric susceptibility at optical frequencies.

Second-order magnetoelectric susceptibility in the optical region of the boraciteCo3B7O13I

Nonreciprocal linear birefringence induced by a magnetic field is observed in the cubic phase of ${\mathrm{Co}}_{3}{\mathrm{B}}_{7}{\mathrm{O}}_{13}\mathrm{I}$ single crystals. This effect is one of a class of magneto-induced spatial dispersion phenomena allowed in noncentrosymmetric crystals. It shows S-shaped dispersion and anisotropy in the region of electronic transitions between states of ${\mathrm{Co}}^{2+}$ ions. Microscopic mechanisms of the nonreciprocal birefringence are evaluated on the basis of local optical transitions. The anisotropy of the effect originates in the second-order magnetoelectric susceptibility in optical regions. The S shape of dispersion and the temperature dependence of the effect are explained by the paramagnetic nature of the phenomenon, i.e., by the splitting of Kramers doublets of the ground state of the ${\mathrm{Co}}^{2+}$ ion in the magnetic field.

Manifestation of magnetically induced spatial dispersion in the cubic semiconductors ZnTe, CdTe, and GaAs

Nonreciprocal birefringence due to magnetically induced spatial dispersion was observed in the Td-class cubic semiconductors ZnTe, CdTe, and GaAs near the fundamental absorption edge. The dispersion of the parameters A and g, describing the contributions from terms of the type Bikj to the diagonal and off-diagonal components of the permittivity tensor eij(ω,B,k), is determined for ZnTe and CdTe. Analysis of the dispersion and anisotropy of the nonreciprocal birefringence shows that in ZnTe, CdTe, and GaAs, in contrast to magnetic semiconductors of the type Cd1−xMnxTe, it is due excitonic mechanisms.

Extensions to the Maxwell boundary conditions: Reflection from uniaxial and cubic antiferromagnets

Standard Maxwell boundary conditions do not always apply at the surface of an anisotropic magnetic medium in a vacuum, even when the $\mathbf{D}$ and $\mathbf{H}$ fields that describe the properties of the medium are taken in covariant form. It is shown that discontinuities may arise in these fields due to bound surface current and electric-dipole moment, and the leading multipole forms of these discontinuities are identified. Reflection matrices for normal incidence are derived for all uniaxial and cubic antiferromagnetic crystals. These matrices satisfy reciprocity (time-reversal symmetry) only when the extended boundary conditions are used. Various magnetic point groups are identified that exhibit nonreciprocal effects in reflection, but not in transmission. It is also shown that when nonreciprocal birefringence exists in transmission, nonreciprocal effects are absent in reflection at normal incidence. The magnetic point-group symmetry assigned to ${\mathrm{Nd}}_{2}{\mathrm{CuO}}_{4}$ is questioned on the basis of the theory.

Use of a Sagnac interferometer for measurements of linear nonreciprocal birefringence in a transverse magnetic field

A method of measuring phase nonreciprocity in materials using a Sagnac interferometer is proposed and implemented in which the working point is shifted into the region of maximum slope of the interference pattern by means of an absorbing exit mirror. The principal advantage of this method over the laser technique is that there are no stringent constraints on the losses in the materials. This method was used for the first time at 0.83 μm to measure the nonreciprocal linear birefringence in a LiIO3 crystal in a transverse magnetic field.

Reciprocal reflection interferometer for a fiber-optic Faraday current sensor

A reciprocal fiber-optic reflection interferometer for remote measurement of electrical current through the Faraday effect is described. The effects of polarization cross coupling because of nonideal elements are eliminated with a low-coherence source. Nonreciprocal birefringence phase modulation is employed for detection of the Faraday phase shift. The theoretical predictions are confirmed by measurements with a piece of straight fiber as the sensing element in a 100-turn solenoid. Currents from 0 to 40 A have been measured with a linear response and a noise limit of ~0.015 A/√Hz.

Magnetic effects in antiferromagnetic crystals in the electric quadrupole-magnetic dipole approximation

Abstract A multipole theory of light propagation in matter is used to describe magnetic effects due to electric quadrupoles and magnetic dipoles induced by light wave fields in non-absorbing antiferromagnetic crystals of the uniaxial and cubic systems. A new effect predicted by the theory is linear birefringence, of electric quadrupole origin, for light travelling along the edge directions in certain cubic crystals. Also predicted in many classes of uniaxial crystal, when propagation is along a crystallographic axis perpendicular to the axis of highest symmetry, is the existence of skew or S-rays. These are characterized by a divergence of the Poynting vector from the wave normal. A previously known effect, non-reciprocal or gyrotropic birefringence, is reconsidered and shown to be a property of media which exhibit Jones birefringence. Furthermore, the theory shows that non-reciprocal birefringence is not in general gyrotropic and that, where effects have previously been described, the results obtained by...

Combined reciprocal and non-reciprocal birefringence in optical monomode fibres

We analyse the overall birefringence which arises from the combined effects of Faraday rotation (non-reciprocal) and bending stress (reciprocal) applied to a monomode optical fibre loop. The theory is substantiated by numerical evaluations and appropriate experimental data are reported. These indicate how it is possible to build non-reciprocal birefringent components, i.e. rotators and isolators, by means of monomode coils.

A nonreciprocal birefringence effect in microwave ferrite and its applications

In this paper, a new kind of gyromagnetic effect in square waveguide loaded by ferrite, Nonreprocal Birefringence Effect(i.e. NRB-effect), is discussed here with coupling wave theory. It is different from the reciprocal birefringence effect under dipolemagnetization and the Faraday-Rotation effect under longitutinal magnetization. The NRB-effect is a special nonreciprocal GM-effect in waveguide where the quadrupole magnetizing field is added(Fig.1). The geometry exhibits both birefringence and non-reciprocal phase shift. The values of the three basic GM-effects(Fig.2) show that the NRB-effect is the largest. It is further disscussed here in detail in terms of coulped-mode theory. Finally, some applications of the NRB-effect are presented. A new 4-port circulator's design principle and its behavior are suggested, for a latching variable-polarizer, which is only recommended in brief.

Theory of Gyrotropic Birefringence

A theoretical treatment of the optical effect known as gyrotropic or nonreciprocal birefringence is presented. By suitably renormalizing the electric dipole moment tensor, it is shown, for the case of lossless media, that 10 of the 18 independent quantities in the gyrotropic-birefringence tensor have their origin in electric quadrupole effects. The other eight are shown to be related to the magnetoelectric effect. The general results are applied to the materials ${\mathrm{Cr}}_{2}$${\mathrm{O}}_{3}$ and MnTi${\mathrm{O}}_{3}$. The propagation of a plane wave along one of the crystalline axes of ${\mathrm{Cr}}_{2}$${\mathrm{O}}_{3}$ is then considered. It is shown that the gyrotropic birefringence exhibits itself as a rotation of the principal optic axes, together with a change in the velocity of propagation of the wave in the medium. Next, the modified boundary conditions corresponding to the renormalized field vectors are given, and the case of a plane wave normally incident on a gyrotropically birefringent medium is discussed. It is noted that the field relations at a boundary will be modified even when the quadrupole contribution vanishes and the magnetoelectric tensor is isotropic, a case in which there is no gyrotropic birefringence in the medium itself. Foinally, a quantum-mechanical calculation of the gyrotropic-birefringence tensor at 0\ifmmode^\circ\else\textdegree\fi{}K is given. The expression obtained is applied to the case of ${\mathrm{Cr}}_{2}$${\mathrm{O}}_{3}$, and the electric quadrupole and magnetoelectric contributions are separated. It is roughly estimated that, at optical frequencies, the electric-quadrupole-induced rotation of the principal optic axes of ${\mathrm{Cr}}_{2}$${\mathrm{O}}_{3}$ is of the order of ${10}^{\ensuremath{-}6}$ rad, and the magnetoelectric-induced shift is two orders of magnitude less.