Electric Susceptibility Tensor Research Articles

Nonlinear Optics Through the Field Tensor Formalism

AbstractThe “field tensor” is the tensor product of the electric fields of the interacting waves during a sum‐ or difference‐frequency generation nonlinear optical interaction. It is therefore a tensor describing light interacting with matter, the latter being characterized by the “electric susceptibility tensor.” The contracted product of these two tensors of equal rank gives the light‐matter interaction energy, whether or not propagation occurs. This notion having been explicitly or implicitly present from the early pioneering studies in nonlinear optics, its practical use has led to original developments in many highly topical theoretical or experimental situations, at the microscopic as well macroscopic level throughout a variety of coherent or non‐coherent processes. The aim of this review article is to rigorously explain the field tensor formalism in the context of tensor algebra and nonlinear optics in terms of a general time‐space multi‐convolutional development, using spherical tensors, with components expressed in the frame of a common basis set of irreducible tensors, or Cartesian tensors. A wide variety of media are considered, including biological tissues and their imaging, artificially engineered by various combinations of optical and static electric fields, with the two extremes of all‐optical and purely electric poling, and also bulk single crystals.

Raman activity of the longitudinal optical phonons of the LiNbO3 crystal: Experimental determination and quantum mechanical simulation

AbstractIn this study, the and the two equivalent / polarized Raman spectra of a LiNbO3 single crystal have been recorded and used as a benchmark test for the density functional theory (DFT) calculation of the longitudinal modes and of their Raman activity. The theoretical approach, based on periodic boundary conditions and a linear combination of atomic orbitals (LCAO), provides excellent predictions of phonon wavenumbers and relative bands intensities for both A1 and E Longitudinal Optical (LO) modes and complements a previous paper limited to the study of Transverse Optical (TO) modes. Overall, the present investigation demonstrates that the LCAO approach, as implemented in the CRYSTAL software, gives results of similar accuracy for the TO and the LO phonons features. By means of a band deconvolution scheme applied to the experimental spectra, we present, for the first time, a quantitative comparison between experimental and theoretically predicted polarized Raman band intensities of LiNbO3 LO modes. This analysis highlights the role of the suitable determination of the static and high‐frequency dielectric matrices that are needed for the prediction of the TO/LO frequency split but also the first nonlinear electric susceptibility tensor for an accurate description of the Raman intensity pattern.

Dimethylaniline-Based Hybrid Compounds of Cadmium Diiodide: Synthesis, Crystal Structure, and Physical Properties

In this work, synthesis, crystal structure, and selected properties of two new organic–inorganic compounds based on cadmium iodide with 2,5- and 2,6-dimethylaniline: CdI2(2,5-DMA)2 1 and CdI2(2,6-DMA)2 2, respectively, were investigated. Both compounds were obtained using the same methodology, and their crystal structures were determined from powder (1) and single-crystal (2) X-ray diffraction experiments. The examined materials crystallize in the noncentrosymmetric space group Fdd2 and contain large polarizable iodine ions and electron-rich aromatic ligands. Second harmonic generation (SHG) tests performed with urea as a reference revealed high SHG efficiency of powdered samples 1 and 2. SHG signals measured for samples 1 and 2 are, respectively, around 10 and 8 times larger than the signal for urea. The experimental part was supplemented with theoretical calculations of the second-order electric susceptibility tensor components, χijk(2). Calculated values of the χ113(2) tensor component around 1.2–1.4 pm·V–1 suggest that both materials are relatively efficient second harmonic generators. Fibrous morphology of the materials was revealed by scanning electron microscopy (SEM), and the observed bending of the fibers suggested their significant flexibility. Samples 1 and 2 were also characterized by spectroscopic techniques (IR, RS, UV–Vis-DRS) and their thermal stability and decomposition rates were researched using XRPD and TG/DSC, revealing that both investigated materials are thermally stable up to around 100 °C.

Fast computation of scattering by isolated defects in periodic dielectric media

Scattering by an isolated defect embedded in a dielectric medium of two-dimensional periodicity is of interest in many sub-fields of electrodynamics. Present approaches to compute this scattering rely on either the Born approximation and its quasi-analytic extensions or ab initio computation that requires large domain sizes to reduce the effects of boundary conditions. The Born approximation and its extensions are limited in scope, while the ab initio approach suffers from its high numerical cost. In this paper, I introduce a hybrid scheme in which an effective local electric susceptibility tensor of a defect is estimated by solving an inverse problem efficiently. The estimated tensor is embedded into an S-matrix formula based on the reciprocity theorem. With this embedding, the computation of the S-matrix of the defect requires field solutions only in the unit cell of the background. In practice, this scheme reduces the computational cost by almost two orders of magnitude, while sacrificing little in accuracy. The scheme demonstrates that statistical estimation can capture sufficient information from cheap calculations to compute quantities in the far field. I outline the fundamental theory and algorithms to carry out the computations in high dielectric contrast materials, including metals. I demonstrate the capabilities of this approach with examples from optical inspection of nano-electronic circuitry where the Born approximation fails and the existing methods for its extension are also inapplicable.

Measuring linear electric quadrupole susceptibility of non-centrosymmetric crystals with reflectivity difference.

I report experimental methods for measuring real and imaginary parts of the linear electric quadrupole susceptibility tensor of nonlinear optical crystals that lack inversion centers. The third-ranked tensor is related to the corresponding second-order nonlinear susceptibility tensor. For opaque materials such as GaAs, the methods involve normal-incidence reflectivity difference detection schemes. For transparent materials, quadrupole susceptibility tensor elements may be measured with similarly construed transmission difference detection schemes.

An approach for the calculation of local dielectric properties in nanosized systems on the basis of coupled dipoles method

Despite interest in a description of nanoscale systems properties, there is still a lack of methods that can grasp size dependence and variation across the interface of the static dielectric properties. We propose an approach for the calculation of local nanoscopic electric susceptibility and dielectric permittivity tensors in nanosized systems on the basis of the coupled dipoles method. Requiring only the polarizability of individual molecules, this approach allows one to determine the local dielectric properties of arbitrarily shaped nanoparticles as well as (semi)infinite bodies such as bulk material and infinite films, expanding our ability to predict and govern the behavior of nanoscale systems.

Canonical Quantization of Electromagnetic Field in the Presence of Nonlinear Anisotropic Magnetodielectric Medium with Spatial-Temporal Dispersion

Modeling a nonlinear anisotropic magnetodielectric medium with spatial-temporal dispersion by two continuum collections of three dimensional harmonic oscillators, a fully canonical quantization of the electromagnetic field is demonstrated in the presence of such a medium. Some coupling tensors of various ranks are introduced that couple the magnetodielectric medium with the electromagnetic field. The polarization and magnetization fields of the medium are defined in terms of the coupling tensors and the oscillators modeling the medium. The electric and magnetic susceptibility tensors of the medium are obtained in terms of the coupling tensors. It is shown that the electric field satisfy an integral equation in frequency domain. The integral equation is solved by an iteration method and the electric field is found up to an arbitrary accuracy.

On the use of structured light in nonlinear optics studies of the symmetry group of a crystal.

We put forward a technique that allows to extract information about the symmetry group to which certain nonlinear crystals belong using a single illuminating beam. It provides such information by considering the outcome of a nonlinear optics process characterized by the electric nonlinear susceptibility tensor, whose structure is dictated by such symmetry group. As an example, we consider the process of spontaneous parametric down-conversion, when it is pumped with a special type of Bessel beam. The observation of the spatial angular dependence of the lower-frequency generated light provides direct information about the symmetry group of the crystal. We should stress that the choice of the appropriate illumination is of paramount importance for unveiling the sought-after information.

Anisotropy analysis of third-harmonic generation in a germanium-doped silica optical fiber.

We performed an intermodal third-harmonic generation around 516nm in a germanium-doped silica optical fiber. The analysis of the complex polarization behavior that was observed allowed us to determine the orientation symmetry group of the fiber and the relative values of the independent coefficients of the third-order electric susceptibility tensor.

Modeling of the optical properties of molecular crystals

In this contribution we present our current findings in the calculations of the linear and second-order nonlinear electric susceptibility tensor components of organic crystals. The methodology used for this purpose is based on a combination of the electrostatic interaction scheme developed by Hurst and Munn (Hurst & Munn, 1986) with electronic structure calculations for the isolated molecules. Our modification of the method consists in i) running periodic boundary condition (PBC) calculations for an adequate chromophore geometry (either experimental or optimized) to obtain atomic charges and in ii) performing the calculations of the molecular properties within a non-uniform embedding field generated by point charges located spherically around the reference molecule. Using this approach good accuracy is achieved on the electric susceptibility tensor components in comparison with the uniform dipole electric field (Seidler et al., 2013). We extend here the application of this method to other molecular crystals as well as we present the first attempt to predict the chi(1) and chi(2) components of two-component organic crystals (Gryl et al., 2014).

Refractive indices, phase-matching directions and third order nonlinear coefficients of rutile TiO_2 from third harmonic generation

Experiments of third harmonic generation in rutile TiO2 allowed us to determine the phase-matching angles and the refractive indices of the crystal up to 4500 nm. We also showed that χ16 and χ18 coefficients of the third order electric susceptibility tensor exhibit opposite signs, and that |χ18(616.7nm)| = 9.7 × 10−20 m2V−2.

Theoretical and experimental study of second harmonic generation from the surface of the topological insulator Bi2Se3

We develop a theoretical model that describes the second harmonic generation of light from the surface of the topological insulator Bi${}_{2}$Se${}_{3}$ and experimentally demonstrate that the technique is sensitive to the surface electrons. By performing a crystal symmetry analysis of Bi${}_{2}$Se${}_{3}$(111) we determine the nonlinear electric susceptibility tensor elements that give rise to second harmonic generation. Using these results, we present a phenomenological model that shows that the relative magnitudes of these tensor elements can be determined by measuring the polarization and intensity of the radiated second harmonic light as a function of the in-plane crystal orientation and incident laser polarization. We describe optical techniques capable of isolating second harmonic light and, using these techniques, we measure the first-order linear optical and second-order nonlinear optical responses as a function of crystal orientation and laser polarization on bulk single crystals of Bi${}_{2}$Se${}_{3}$(111). The experimental results are consistent with our theoretical description. By comparing the data to our theoretical model we determine that a portion of the measured second harmonic light originates from the accumulation region of Bi${}_{2}$Se${}_{3}$(111), which we confirm by performing surface doping-dependent studies. Our results show that second harmonic generation is a promising tool for spectroscopic studies of topological surfaces and buried interfaces.

The First and Second Static Electronic Hyperpolarizabilities of Zigzag Boron Nitride Nanotubes. An ab Initio Approach through the Coupled Perturbed Kohn–Sham Scheme

The coupled perturbed Kohn-Sham (CPKS) computational scheme for the evaluation of electric susceptibility tensors in periodic systems, recently implemented in the CRYSTAL code, has been extended to third-order. It is, then, used to obtain static electronic hyperpolarizabilities of zigzag BN nanotubes for the first time. This procedure, which is fully analytic in all key steps, requires a double self-consistent treatment for taking into account the first- and second-order response of the system to the applied field. The performance of different functionals is compared and the B3LYP hybrid is ultimately chosen for calculations on nanotubes having radii as large as R = 20 Å (6-200 atoms in the unit cell). Such large radii are sufficient to give the pure longitudinal component of the (hyper)polarizability tensors to within 1% of the "exact" hexagonal BN monolayer limit. Other tensor components involving the transverse direction converge more slowly. They can, however, be extrapolated to the monolayer limit to within 4% accuracy except for the pure transverse second hyperpolarizability, which has an error of 13% in that limit.

Electric susceptibility of a magnetized plasma under electromagnetically induced transparency

This study derives the electric susceptibility tensor of a cold magnetized plasma under electromagnetically induced transparency (EIT) regime (Litvak and Tokman 2002 Phys. Rev. Lett. 88 095003, Shvets and Wurtele 2002 Phys. Rev. Lett. 89 115003) in which an intense right-hand circularly polarized pump wave is injected parallel to the background magnetic field. A dispersion relation of the wave in the electron cyclotron frequency range for an arbitrary propagation angle is obtained from this susceptibility tensor. In the case of purely parallel propagation of the probe wave, the dispersion relation obtained by Litvak, Shvets and others is recaptured. A new finding is that a stop band emerges between left-hand cutoff and upper hybrid frequencies, in which originally an extraordinary-mode (X) branch exists, in the case of perpendicular propagation to the background magnetic field under the EIT. The bandwidth of the stop band expands as the pump wave is intensified. For the situation of launching the probe wave from the high-field side in a tokamak, the accessibility of the probe wave to the region where the EIT effect appears is investigated. The EIT region which is a resonance layer created by the EIT is accessible to the probe wave, indicating the possibility of the application of EIT to control the spatial position of wave power deposition.

New perspective on the reciprocity theorem of classical electrodynamics

We provide a simple physical proof of the reciprocity theorem of classical electrodynamics in the general case of material media that contain linearly polarizable as well as linearly magnetizable substances. The excitation source is taken to be a point-dipole, either electric or magnetic, and the monitored field at the observation point can be electric or magnetic, regardless of the nature of the source dipole. The electric and magnetic susceptibility tensors of the material system may vary from point to point in space, but they cannot be functions of time. In the case of spatially non-dispersive media, the only other constraint on the local susceptibility tensors is that they be symmetric at each and every point. The proof is readily extended to media that exhibit spatial dispersion: For reciprocity to hold, the electric susceptibility tensor χ E_ mn that relates the complex-valued magnitude of the electric dipole at location r m to the strength of the electric field at r n must be the transpose of χ E_ nm . Similarly, the necessary and sufficient condition for the magnetic susceptibility tensor is χ M_ mn = χ T M_ nm .

Calculation of the first static hyperpolarizability tensor of three-dimensional periodic compounds with a local basis set: A comparison of LDA, PBE, PBE0, B3LYP, and HF results

The computational scheme for the evaluation of the second-order electric susceptibility tensor in periodic systems, recently implemented in the CRYSTAL code within the coupled perturbed Hartree-Fock (HF) scheme, has been extended to local-density, gradient-corrected, and hybrid density functionals (coupled-perturbed Kohn-Sham) and applied to a set of cubic and hexagonal semiconductors. The method is based on the use of local basis sets and analytical calculation of derivatives. The high-frequency dielectric tensor (epsilon(infinity)) and second-harmonic generation susceptibility (d) have been calculated with hybrid functionals (PBE0 and B3LYP) and the HF approximation. Results are compared with the values of epsilon(infinity) and d obtained from previous plane-wave local density approximation or generalized gradient approximation calculations and from experiment. The agreement is in general good, although comparison with experiment is affected by a certain degree of uncertainty implicit in the experimental techniques.

The dynamic ionic charge of zincblende type crystals

Abstract : The ionic charges of zincblende type crystals are discussed. Since the static ionic charges are, in general, quite small in these crystals, the magnitudes of the dynamic ionic charge, (e* sub T), are attributed largely to the effects of charge redistribution. The fairly sizeable values of (e* sub T) in Se and Te trigonal crystals must, in fact, be attributed entirely to charge redistribution effects since their static ionic charges are identically zero. Efforts to account for the (e* sub T) values in ZnS type crystals using either the shell model or the deformation dipole model have not been too successful. It is urged that theorists turn their attention to ab-initio energy band calculations of long wavelength dynamical properties of zincblende and diamond type crystals, namely: the first order electric moment (infrared) tensor; the first order electric susceptibility (Raman) tensor; and the change in the electric susceptibility with interatomic distance (piezo-optic) tensor. (Author)

ELECTROMAGNETIC FIELD QUANTIZATION IN AN ANISOTROPIC AND INHOMOGENEOUS MAGNETODIELECTRIC MEDIUM

The electromagnetic field in an anisotropic and inhomogeneous magnetodielectric medium is quantized by modelling the medium with two independent quantum fields. Maxwell and constitutive equations of the medium are obtained using a minimal coupling scheme. The electric and magnetic susceptibility tensors of the medium are calculated. Finally the efficiency of the approach is elucidated by some examples.

Electromagnetic field quantization in an anisotropic magnetodielectric medium with spatial–temporal dispersion

By modeling a linear, anisotropic and inhomogeneous magnetodielectric medium with two independent sets of harmonic oscillators, the electromagnetic field is quantized in such a medium. The electric and magnetic polarizations of the medium are expressed as linear combinations of the ladder operators of the harmonic oscillators modeling the magnetodielectric medium. Maxwell and the constitutive equations of the medium are obtained as the Heisenberg equations of the total system. The electric and magnetic susceptibility tensors of the medium are obtained in terms of the tensors coupling the medium with the electromagnetic field. The explicit forms of the electromagnetic field operators are obtained for a translationally invariant medium.

On relationships between electro-optic coefficients for impermeability and nonlinear electric susceptibilities

Electro-optic coefficients can be defined in terms of the changes, which result from an applied electric field, to components of either the impermeability or electric susceptibility tensors. In this paper exact relationships are derived between electro-optic coefficients defined in these two ways for powers of the field up to four. Previously known approximate relationships are also discussed.