Components Of Susceptibility Tensor Research Articles

Emission Dipole and Pressure‐Driven Tunability of Second Harmonic Generation in vdWs Ferroelectric NbOI2

Abstract2D in‐plane ferroelectric NbOI2 exhibits strong second harmonic generation (SHG) and ultrahigh effective susceptibility. To push forward their applications in nonlinear photonics and optoelectronics, it is highly desirable to understand the emission dipole orientation and tunability of SHG, which is not achieved. Here, by integrating tight focusing from parabolic mirror with back focal plane (BFP) imaging technique, for the first time it is demonstrated that SHG emission of NbOI2 presents purely in‐plane dipole orientation in consistent with numerical simulations, suggesting the in‐plane components of the SHG susceptibility tensor in NbOI2 dominate the emission. Moreover, with the aid of ab‐initio calculations, it is found that the hydrostatic pressure can dramatically change the structure and resultant SHG intensity of NbOI2. Explicitly, SHG intensity endures a slight increase due to the distortion of octahedral at low pressure pressure, and then monotonously decreases due to the improvement of structural symmetry with further increasing pressure, and drastically quenching resulting from the ferroelectric to paraelectric phase transition. This work unambiguously demonstrates the dipole emission behavior of SHG and the relationship between structural evolution and SHG intensity, which provides an avenue for tunable nonlinear optics and optoelectronics.

Polarimetric second-harmonic generation microscopy of partially oriented fibers I: Digital modeling

Second-harmonic generation (SHG) in biological tissues originates predominantly from noncentrosymmetric fibrillar structures partially oriented within a focal volume (voxel) of a multiphoton excitation microscope. This study is aimed to elucidate fibrillar organization factors influencing SHG intensity, as well as achiral, R, and chiral, C, nonlinear susceptibility tensor component ratios. SHG response is calculated for various configurations of fibrils in a voxel using the digital nonlinear microscope. The R and C ratios are calculated using linear incident and outgoing polarization states that simulate polarization-in polarization-out polarimetric measurements. The investigation shows strong SHG intensity dependence on parallel/antiparallel fiber organization. The R and C ratios are strongly influenced by the fiber chirality, tilting of the fibers out of the image plane, and crossing of the fibers. The computational modeling provides the basis for the interpretation of polarimetric SHG microscopy images in terms of the ultrastructural organization of fibers in each voxel of the samples. The modeling results are employed in the accompanying paper to investigate the ultrastructures with parallel/antiparallel fibers and two-dimensional and tree-dimensional crossing fibers in biological and biomimetic structures.

Terahertz-induced polar state in quantum paraelectric SrTiO3

The second harmonic generation from strontium titanate crystal under the terahertz excitation has studied at room temperature. The obtained anisotropic dependences show the presence of a terahertz electric field induced polar phase. By analyzing the polarization dependencies of the optical second harmonic intensity, the relative values of the third-order nonlinear susceptibility tensor components were obtained.Graphical abstract

Application of cylindrical IE-GSTC to physical metasurfaces

The cylindrical Integral Equation -- Generalized Sheet Transition Condition (IE-GSTC) is validated by applying it to physical metasurfaces, i.e. metasurfaces defined by material properties and dimensions rather than by susceptibility tensor components. The previously reported IE-GSTC, which was formulated for zero thickness GSTC discontinuities, is extended to handle the finite thickness of physical metasurfaces. A simple analytical approach is used to extract the bianisotropic susceptibility tensor of concentric, multilayered, magneto-dielectric shells. Plane wave scattering by a physical metasurface constructed of four segments of multilayered, magneto-dielectric metasurface scatterers is used as an example problem to validate cylindrical IE-GSTC. A second example considers an opening on the cylindrical metasurface, confirming IE-GSTC can handle metasurfaces with openings. Third example is that of plane wave scattering by an eight segment metasurface. Good agreement is obtained between IE-GSTC results and full wave simulation results for all the cases. The paper concludes with a novel technique to handle PEC segments embedded in a cylindrical metasurface.

Inertial rotational diffusion and magnetic relaxation of the spin system in a strong magnetic field

Analytical expressions for the longitudinal and transverse correlation functions pertaining to inertial magnetic relaxation are obtained by expanding the deterministic magnetization trajectories into a Fourier series and averaging the result over all possible initial conditions applying the stationary Boltzmann distribution function. The longitudinal and transverse components of the magnetic susceptibility tensor are calculated for a system of noninteracting macrospins in a strong uniform external field using Bloch's phenomenological approach, which postulates the exponential evolution of the spin system towards equilibrium, and the Lorentz model of rotational diffusion of the magnetization vector. It is shown that the strength of the external field and the magnitude of the inertia parameters noticeably affect the shape of the susceptibility in the THz (nutation resonance) spectrum region. It is also demonstrated that the simple analytical Lorentz-type expression describes the main features of the complex susceptibility in the ferromagnetic resonance and nutation resonance regions.

Wide-field Stokes polarimetric microscopy for second harmonic generation imaging.

We employ wide-field second harmonic generation (SHG) microscopy together with nonlinear Stokes polarimetry for quick ultrastructural investigation of large sample areas (700 μm × 700 μm) in thin histology sections. The Stokes vector components for SHG are obtained from the polarimetric measurements with incident and outgoing linear and circular polarization states. The Stokes components are used to construct the images of polarimetric parameters and deduce the maps of ultrastructural parameters of achiral and chiral nonlinear susceptibility tensor components ratios and cylindrical axis orientation in fibrillar materials. The large area imaging was employed for lung tumor margin investigations. The imaging shows reduced SHG intensity, increased achiral susceptibility ratio values, and preferential orientation of collagen strands along the boarder of tumor margin. The wide-field Stokes polarimetric SHG microscopy opens a possibility of quick large area imaging of ultrastructural parameters of tissue collagen, which can be used for nonlinear histopathology investigations.

Enhanced Sum Frequency Generation for Monolayers on Au Relative to Silica: Local Field Factors and SPR Effect

Using metals as signal magnified substrates, surface plasmon-enhanced sum frequency generation (SFG) vibrational spectroscopy is a promising technique to probe weak molecular-level signals at surfaces and interfaces. In this study, the vibrational signals of the n-alkane monolayer on the gold (Au) and silica substrates are investigated using the broadband femtosecond SFG. The enhancement factors are discovered to be up to ∼1076 and ∼31 for the methyl symmetric and asymmetric stretching (ss and as) modes of the monolayer, respectively. By systematically analyzing the second-order nonlinear susceptibility tensor components (χijks), the Fresnel coefficients (Fijks), and the surface plasmon resonance (SPR) effect, we find that the interplay between Fijk and χijk terms and the SPR effect dominate the SFG signal enhancement. Our study reveals that the relative contributions of different influencing factors (i.e., Fresnel coefficients and SPR) to the SFG signal enhancement provide an approach to interpreting enhanced SFG vibrational signals detected from probe molecules on distinct substrates and may finally guide the design of the experimental methodology to improve the detection sensitivity and signal-to-noise ratio.

Optical second-harmonic response of axially symmetric media under tightly focused excitation: linear versus radial polarization

We compared the optical second-harmonic (SH) response of axially symmetric media excited by tightly focused radially and linearly polarized beams. The simulation performed showed a significant redistribution of energy between the lobes/cones of the SH radiation pattern, depending on the focusing accuracy in the case of a purely surface/interface nonlinearity. This phenomenon is due to the change in the contributions of second-order susceptibility tensor components to SH generation as we demonstrated via a comparison of SH patterns related to different components of the tensor. Replacement of the interface with a nonlinear layer leads to noticeable broadening of SH radiation patterns, which can allow one to distinguish surface nonlinearity from bulk. The model developed is applicable for SH generation by interfaces, films, and poled glasses.

Study of a bi-axial (KTP) crystal using double Stokes–Mueller polarimetry

In this paper, we report the significance of the double Stokes–Mueller polarimetry (DSMP) technique, to characterize a large size ([Formula: see text][Formula: see text]mm) Potassium titanyl phosphate (KTP) crystal. The crystal undergoes second harmonic generation with type II phase matching. The study of standard KTP crystal using the DSMP technique helps to validate the efficiency of this technique. We were able to extract the crystal’s double Mueller matrix, relative contribution of the susceptibility tensor components, the phase difference between the susceptibility tensor components, etc. We could determine the crystal axes orientation using this optical technique, which was not possible through a single crystal X-Ray diffraction technique for such a large size crystal for which both optic axes and crystallographic axes are the same. Axes direction determined from polarization microscope measurements and Laue diffraction measurements on KTP crystal is compared with those obtained from DSMP measurements.

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.

Identification of a state of persistent spin helix in a parallel magnetic field, and exploration of its transport properties

The tensors of conductivity and spin susceptibility of a two-dimensional electron gas (2DEG) with equal Rashba and Dresselhaus spin-orbit interaction constants [the persistent spin helix (PSH) state] in a parallel magnetic field are calculated. Applying a parallel magnetic field to the PSH state leads to the appearance of a saddle point in the spectrum and, accordingly, to a Van Hove singularity (VHS), whose amplitude increases indefinitely as the magnetic field strength $(\mathbf{b})$ approaches to some critical value ${b}_{cr}$. The presence of a VHS in the density of states is an important factor determining the conductivity and spin susceptibility tensors. When only the lower spin subband is filled, the off-diagonal elements of the conductivity tensor are nonzero; that is, a Hall voltage arises due to the anisotropy of the Fermi surface and scattering. In the region where two spin subbands are filled, the diagonal and off-diagonal components of the spin susceptibility tensor are equal, and the off-diagonal terms of the conductivity tensor vanish. A sharp decrease in the conductivity and spin susceptibility at the beginning of the filling of the second spin subband, the ratio of the diagonal component of the spin susceptibility to the off-diagonal one, and also the number of critical points in the spectrum make it possible to establish the PSH 2DEG state.

Intrinsic symmetry of nonlocal nonlinear optical susceptibilities

The permutation symmetry relations of the nth order nonlocal nonlinear optical susceptibilities emerging in the absence of absorption are obtained from the basic statements of electrodynamics. The values of the nth order nonlocal nonlinear susceptibility tensor components with swapped frequencies and indices corresponding to the generated electric field and the electric field under the operator of differentiation in the constitutive equation only differ by their signs and signs of the interchanged frequencies. Additionally, there exist relations between three components of this tensor with cyclically permuted indices and frequencies corresponding to the generated field, the field under the operator of differentiation in the constitutive equation and one of the remaining fields involved into nonlinear process under consideration.

Microcrystal ordering and second-order optical susceptibilities of zinc oxide films

Second harmonic generation in zinc oxide films deposited by the hydrothermal method was investigated. A correlation between second-order optical susceptibility tensor components d311 and d333 was studied by using the model of noninteracting microcrystals, and the origin of different contributions to the film nonlinearities was discussed. The change of stacking sequences and the inhomogeneity in the distribution of the nonlinearities over the film thickness are the most probable effects hindering the investigations of the thickness dependence of ZnO film nonlinearities. A model of the inhomogeneous distribution of the nonlinearities over the film thickness was used for doing an analysis of the experimental data features, which were accounted for by a sublayer, adjacent to the seed layer, and possess high optical nonlinearity, d333 = −72.6 pm/V.

Third-harmonic Mie scattering from semiconductor nanohelices

Chiroptical spectroscopies provide structural analyses of molecules and nanoparticles but they require sample volumes that are incompatible with generating large chemical libraries. New optical tools are needed to characterize chirality for the ultrasmall (<1 µl) volumes required in the high-throughput synthetic and analytical stations for chiral compounds. Here we show experimentally a novel photonic effect that enables such capabilities—third-harmonic Mie scattering optical activity—observed from suspensions of CdTe nanostructured helices in volumes <<1 µl. Third-harmonic Mie scattering was recorded on illuminating CdTe helices with 1,065, 1,095 and 1,125 nm laser beams and the intensity was around ten-times higher in the forward direction than sideways. The third-harmonic ellipticity was as high as 3° and we attribute this effect to the interference of chiral and achiral effective nonlinear susceptibility tensor components. Third-harmonic Mie scattering on semiconductor helices opens a path for rapid high-throughput chiroptical characterization of sample volumes as small as 10−5 µl.

Second-harmonic generation in the surface layer of a dielectric spheroidal particle: II. Analysis of the solution

The vectors used in the solution of the problem of second-harmonic generation in the surface layer of a dielectric spheroidal particle are explicitly expressed in terms of the basis vectors of the spherical, cylindrical, and Cartesian coordinate systems. Three-dimensional directivity patterns characterizing the spatial distribution of the generated radiation in the far-field region and its polarization are constructed. It is found that for a small size of a spheroidal particle, the directivity pattern due to each of the non-chiral components of the nonlinear dielectric susceptibility tensor has its own individual shape. A proportional increase in the linear dimensions of the particle leads to separation of several directions of predominant radiation with a high directivity in the directivity pattern. If the exciting radiation has a linear polarization, then the generated radiation due to one (any) of the independent components of the tensor is also linearly polarized. Mathematical properties characterizing the spatial distribution of the generated radiation in the far-field region and properties associated with the change of problem parameters are found for the functions used in the solution. A relationship between the symmetries of the directivity patterns of doubled-frequency radiation and the indicated properties is revealed. The conditions under which the generation of radiation does not occur and the conditions under which the generated radiation has a linear polarization are found. The above conditions are related to the features of the spatial distribution of the generated radiation and its polarization, illustrated in the directivity patterns. Methods for estimating the independent components of the nonlinear dielectric susceptibility tensor using these conditions are proposed. Keywords: second-harmonic generation, spheroidal dielectric particle, symmetry of the spatial distribution of radiation, conditions for the absence of generation, conditions for the generation of linearly polarized radiation.

Генерация второй гармоники в поверхностном слое диэлектрической сфероидальной частицы: II. Анализ решения

The vectors used in the solution of the problem of second-harmonic generation in the surface layer of a dielectric spheroidal particle are explicitly expressed in terms of the basis vectors of the spherical, cylindrical, and Cartesian coordinate systems. Three-dimensional directivity patterns characterizing the spatial distribution of the generated radiation in the far-field region and its polarization are constructed. It is found that for a small size of a spheroidal particle, the directivity pattern due to each of the non-chiral components of the nonlinear dielectric susceptibility tensor has its own individual shape. A proportional increase in the linear dimensions of the particle leads to separation of several directions of predominant radiation with a high directivity in the directivity pattern. If the exciting radiation has a linear polarization, then the generated radiation due to one (any) of the independent components of the tensor is also linearly polarized. Mathematical properties characterizing the spatial distribution of the generated radiation in the far-field region and properties associated with the change of problem parameters are found for the functions used in the solution. A relationship between the symmetries of the directivity patterns of doubled-frequency radiation and the indicated properties is revealed. The conditions under which the generation of radiation does not occur and the conditions under which the generated radiation has a linear polarization are found. The above conditions are related to the features of the spatial distribution of the generated radiation and its polarization, illustrated in the directivity patterns. Methods for estimating the independent components of the nonlinear dielectric susceptibility tensor using these conditions are proposed.

Генерация второй гармоники в поверхностном слое диэлектрической сфероидальной частицы. I. Аналитическое решение

The problem of second-harmonic generation by a plane elliptically polarized electromagnetic wave in a thin optically nonlinear surface layer of a dielectric particle shaped as an ellipsoid of revolution is solved. The generalized Rayleigh–Gans–Debye approximation is used for an analytical description with taking into account the difference in refractive indices of the medium corresponding to the frequencies of the exciting and generated radiation. The limiting forms of functions are obtained, with the use of which the electric field strength of the generated radiation is expressed. The order of dependence of these functions on the linear dimensions is found, when the lengths of the semiaxes of the particle are small compared to the wavelength of the exciting radiation and their ratio remains constant. It was found that the power density of the generated radiation in this case is determined to a greater extent by the chiral components of the nonlinear dielectric susceptibility tensor and is proportional to the fourth power of the length of the semiaxis of the particle, if the shape of the spheroidal particle differs significantly from the spherical one. The solution of this problem, obtained by other authors, is supplemented for the possibility of applying to the description of generation in the surface layer of a dielectric particle not only in the form of a prolate, but also in the form of an oblate spheroid. Corrections of inaccuracies and misprints made in similar works by other authors are proposed. The relationships between the formulas used in these works are found, taking into account the corrections and the formulas used in this work.

Second-harmonic generation in the surface layer of a dielectric spheroidal particle: I. Analytical solution

The problem of second-harmonic generation by a plane elliptically polarized electromagnetic wave in a thin optically nonlinear surface layer of a dielectric particle shaped as an ellipsoid of revolution is solved. The generalized Rayleigh-Gans-Debye approximation is used for an analytical description with taking into account the difference in refractive indices of the medium corresponding to the frequencies of the exciting and generated radiation. The limiting forms of functions are obtained, with the use of which the electric field strength of the generated radiation is expressed. The order of dependence of these functions on the linear dimensions is found, when the lengths of the semiaxes of the particle are small compared to the wavelength of the exciting radiation and their ratio remains constant. It was found that the power density of the generated radiation in this case is determined to a greater extent by the chiral components of the nonlinear dielectric susceptibility tensor and is proportional to the fourth power of the length of the semiaxis of the particle, if the shape of the spheroidal particle differs significantly from the spherical one. The solution of this problem, obtained by other authors, is supplemented for the possibility of applying to the description of generation in the surface layer of a dielectric particle not only in the form of a prolate, but also in the form of an oblate spheroid. Corrections of inaccuracies and misprints made in similar works by other authors are proposed. The relationships between the formulas used in these works are found, taking into account the corrections and the formulas used in this work. Keywords: second-harmonic generation, dielectric spheroidal particle, generalized Rayleigh-Gans-Debye model, small particle approximation, chiral component.

Specific features of the conductivity and spin susceptibility tensors of a two-dimensional electron gas with Rashba and Dresselhaus spin-orbit interactions

The two-dimensional electron gas with spin-orbit interactions (SOIs) of Rashba and Dresselhaus types is known to form an anisotropic system with a Van Hove singularity in the density of states. Moreover, the ``amplitude'' and the energy position of this singularity depend on the ratio of the constants of Rashba $\ensuremath{\alpha}$ and Dresselhaus $\ensuremath{\beta}$ SOIs, $\ensuremath{\gamma}=\ensuremath{\beta}/\ensuremath{\alpha}$. Here the dependencies of the conductivity and spin susceptibility tensors on the position of the Fermi level are calculated for a wide range of $\ensuremath{\gamma}$. It is shown that if only the lower spin subband is filled the diagonal elements of the conductivity tensor have dips that appear when the Fermi level passes the singularity point both for $\ensuremath{\gamma}&lt;1$ and $\ensuremath{\gamma}&gt;1$. The amplitude of these features and their energy position depend on $\ensuremath{\gamma}$ and, in particular, for the state of persistent spin helix $\ensuremath{\gamma}=1$ they disappear. When only the lower spin subband is filled, the off-diagonal elements of the conductivity tensor are nonzero, that is, there is a Hall voltage due to the anisotropy of the Fermi surface and the scattering. In the region of filling of two spin subbands the ratio of diagonal and nondiagonal components of the spin susceptibility tensor is equal to $\ensuremath{\gamma}$ and in the conductivity tensor, the off-diagonal terms vanish.

Генерация второй гармоники-суммарной частоты в тонком сферическом слое: III. Свойства решения

Mathematical properties of functions characterizing symmetries of spatial distribution of second harmonic–sum frequency radiation are determined. The conditions at which there occurs no generation and the conditions at which the polarization of the generated radiation is linear are found. The revealed properties are systematized by their manifestations in the directivity patterns of the generated radiation. Methods based on these properties for determining the components of the nonlinear dielectric susceptibility tensor are proposed. The relationship is described between the revealed properties as well as conditions and previously found similar properties and conditions for the phenomena of second–harmonic generation and sum–frequency generation.