Fundamental Electric Field Research Articles

High spectral contrast chiral second-harmonic generation from multilayer gallium selenide coupled to silicon holey disk metasurfaces

We present electromagnetic design and analysis of hybrid metasurfaces composed of multilayer gallium selenide (GaSe) coupled to silicon holey disk arrays for achieving high spectral contrast chiral second-harmonic generation (SHG). The silicon holey disk structures are designed to support electromagnetically induced transparency (EIT)-like optical resonances in the 1.5–1.8 µm wavelength range in the vicinity of electric and toroidal dipole scattering modes. The fundamental electric field enhancement above the silicon structures shows right-circularly polarized (RCP)- and left-circularly polarized (LCP)-like characteristics at distinct wavelengths resulting in two prominent peaks for the LCP and RCP resolved SHG from the GaSe layer above the holey disks. This results in high contrast SHG degree of circular polarization spanning − 1 to + 1 over the fundamental excitation spectral range considered. The chiral SHG response is also found to be strongly influenced by the unit-cell lattice arrangement with square or hexagonal lattice exhibiting very different chiral SHG response. This observation is consistent with the circular polarization state selection rules for the harmonic generation process considering the rotational symmetry of the underlying metasurface lattice arrangement. Two-dimensional (2D) material–dielectric resonant metasurface hybrid systems exhibit nonlinear optical polarization selection rules, which differ from the native 2D material, making them an interesting platform for realizing engineered chiral nonlinear photonic devices.

Observation of a ferro-rotational order coupled with second-order nonlinear optical fields

Ferroic orders can be classified by the symmetry of their order parameters, and ferroelectric, ferromagnetic and ferro-toroidal orders have already been observed. The ferro-rotational order1–3, whose order parameter is an axial vector invariant under both time-reversal and spatial-inversion operations, is the final ferroic to be identified and has a vector order parameter. This order is closely related to a number of phenomena such as polar vortices4, giant magnetoelectric coupling5 and spin-helicity-driven ferroelectricity6, but it has received little attention so far. Here, using high-sensitivity rotational-anisotropy second-harmonic generation, we have exploited the electric quadrupole contribution to the second-harmonic generation to directly couple to this centrosymmetric ferro-rotational order in an archetype of type-II multiferroics, RbFe(MoO4)2. We found that two domain states with opposite ferro-rotational vectors emerge with distinct populations at the critical temperature Tc ≈ 195 K and gradually evolve to reach an even ratio at lower temperatures. Moreover, we have identified the ferro-rotational order phase transition as weakly first order and have revealed its coupling field as a unique combination of the induced electric quadrupole second-harmonic generation and the incident fundamental electric fields. The authors use optical spectroscopy to show that RbFe(MoO4)2 hosts a ferro-rotational phase. This is the final form of ferroic order to be observed.

Silencing the second harmonic generation from plasmonic nanodimers: A comprehensive discussion.

The silencing of the second harmonic generation process from plasmonic nanostructures corresponds to the limited far-field second harmonic radiation despite the huge fundamental electric field enhancement in the interstice between two plasmonic nanoparticles forming a nanodimer. In this article, we report a comprehensive investigation of this effect using a surface integral equation method. Various geometries are considered, including nanoantennas with cylindrical and rectangular arms as well as nanodimers with surface defects. The existence of the silencing of the second harmonic generation from plasmonic nanogaps is first confirmed, and the problem of the origin of the second harmonic light from these plasmonic nanostructures is addressed in detail. Our results show that the distribution of the second harmonic sources, especially on the arm sides, plays a non-negligible role in the overall second harmonic emission. This contribution is induced by retardation effects at the pump wavelength and results in a dipolar second harmonic emission.

Analytical Evaluation of Partial Elements Using a Retarded Taylor Series Expansion of the Green’s Function

The computation of the fundamental magnetic and electric field coupling terms for the full-wave partial element equivalent circuit method is time consuming for large problems. Extensive numerical integration is required for the computation of four to six fold integration of the partial element coefficients. In this paper, we present new full-wave analytical formulas for their computation, which are based on several different Taylor series expansion approaches for the case of rectangular elementary volumes and surfaces.

Geometric effect on second harmonic generation from gold grating

We numerically investigate second harmonic generation from gold gratings of an ideal rectangular and ladder-shaped cross-section. The SHG efficiency from the gold gratings of the ladder-shaped cross-section is significantly enhanced compared with that from the ideal rectangular cross-section with a maximum enhancement factor of around two. The enhancement is ascribe to the nanostructure dependent local fundamental electric field, the nonlinear sources and thus the far field radiation. Our results have a practical meaning in the explanation of experimental SHG measurement, and the modulation of SHG response in the metallic nanostructure.

Phase relationship between ULF waves and drift‐bounce resonant ions: A statistical study

AbstractWe explore the phase relationship between the poloidal mode ULF wave electric field oscillations and drift‐bounce resonant oxygen ions under the resonant condition of N = 2 at the magnetic equator. Using Cluster data from 2001 to 2004, we identify 55 fundamental poloidal mode wave events, among which 42 show “negative slope” pitch angle dispersion signatures in the southern hemisphere, 11 show “positive slope” dispersion in the northern hemisphere, and two near‐equatorial events are associated with in‐phase field‐aligned signatures. For each event, the off‐equatorial resonant ions are traced along their bouncing trajectories to determine the last time they moved across the equator. The resulting time series of the resonant oxygen ion fluxes at the equator are found to be statistically in antiphase with the wave electric fields. The resonant ion flux variation depends on both ion energy change and radial transportation. This antiphase relationship in statistics suggests two possibilities: (1) the fundamental poloidal mode wave electric fields are generally characterized by electric field intensity peaking near the magnetic equator if the flux variation is mainly caused by energy change and (2) the radial gradient of phase space density is positive if the flux variation caused by radial transportation is dominant.

Depth-Dependent Three-Layer Model for the Surface Second-Harmonic Generation Yield

We present a generalization of the three-layer model to calculate the surface second harmonic generation (SSGH) yield, that includes the depth dependence of the surface nonlinear second order susceptibility tensor $\boldsymbol{\chi}(-2\omega;\omega,\omega)$. This model considers that the surface is represented by three regions or layers. The first layer is a semi-infinite vacuum region with a dielectric function $\epsilon_{v}(\omega)=1$, from where the fundamental electric field impinges on the material. The second layer is a thin layer ($\ell$) of thickness $d$ characterized by a dielectric function $\epsilon_{\ell}(\omega)$, and it is in this layer where the SSHG takes place. We consider the position of $\boldsymbol{\chi}(-2\omega;\omega,\omega)$ within this surface layer. The third layer is the bulk region denoted by $b$ and characterized by $\epsilon_{b}(\omega)$. We include the effects caused by the multiple reflections of both the fundamental and the second-harmonic (SH) fields that take place within the thin layer $\ell$. {\color{red} As a test case, we calculate $\boldsymbol{\chi}(-2\omega;\omega,\omega)$ for the Si(111)(1$\times$1):H surface, and present a layer-by-layer study of the susceptibility to elucidate the depth dependence of the SHG spectrum. We then use the depth dependent three-layer model to calculate the SSHG yield, and contrast the calculated spectra with experimental data. We produce improved results over previous published work, as this treatment can reproduce key spectral features, is computationally viable for many systems, and most importantly remains completely \emph{ab initio}.

Three-layer model for the surface second-harmonic generation yield including multiple reflections

We present the three-layer model to calculate the surface second-harmonic generation (SSHG) yield. This model considers that the surface is represented by three regions or layers. The first layer is the vacuum region with a dielectric function ${\ensuremath{\epsilon}}_{v}(\ensuremath{\omega})=1$ from where the fundamental electric field impinges on the material. The second layer is a thin layer $(\ensuremath{\ell})$ of thickness $d$ characterized by a dielectric function ${\ensuremath{\epsilon}}_{\ensuremath{\ell}}(\ensuremath{\omega})$, and it is in this layer where the SSHG takes place. The third layer is the bulk region denoted by $b$ and characterized by ${\ensuremath{\epsilon}}_{b}(\ensuremath{\omega})$. Both the vacuum and bulk layers are semi-infinite. The model includes the multiple reflections of both the fundamental and the second-harmonic (SH) fields that take place at the thin layer $\ensuremath{\ell}$. We obtain explicit expressions for the SSHG yield for the commonly used $s$ and $p$ polarizations of the incoming $1\ensuremath{\omega}$ and outgoing $2\ensuremath{\omega}$ electric fields, where no assumptions for the symmetry of the surface are made. These symmetry assumptions ultimately determine which components of the surface nonlinear second-order susceptibility tensor $\mathbf{\ensuremath{\chi}}(\ensuremath{-}2\ensuremath{\omega};\ensuremath{\omega},\ensuremath{\omega})$ are different from zero, and thus contribute to the SSHG yield. Then, we particularize the results for the most commonly investigated surfaces, the (001), (110), and (111) crystallographic faces, taking their symmetries into account. We use the three-layer model and compare it against the experimental results of a Si(111)$(1\ifmmode\times\else\texttimes\fi{}1)$:H surface, as a test case, and use it to predict the SSHG yield of a Si(001)$(2\ifmmode\times\else\texttimes\fi{}1)$ surface.

Maximizing Nonlinear Optical Conversion in Plasmonic Nanoparticles through Ideal Absorption of Light

The optimization of nonlinear optical processes in plasmonic structures is important for the design of efficient nonlinear nanosources of light. Considering the simple case of spherical nanoparticles, we clearly identify the most efficient channel for second-harmonic generation, thanks to physical insights provided by the generalized Mie theory. This channel corresponds to the excitation of electric dipolar modes at the fundamental wavelength and a quadrupolar second-harmonic emission. Interestingly, it is demonstrated that the second-harmonic generation intensity is directly related to the square of the absorbed power, which reproduces both the electric field enhancement and the specific size dependence of second-harmonic generation in the small-particle limit. Additionally, the absorbed power can be optimized by controlling the nanoparticle size. These results demonstrate that the optimization of the fundamental electric field is not sufficient for reaching the highest nonlinear conversion in plasmonic ...

Numerical Analysis of Carrier-Depletion Strained SiGe Optical Modulators With Vertical p-n Junction

The modulation characteristics of carrier-depletion strained SiGe optical modulators with a vertical p-n junction are numerically analyzed by technology computer-aided design simulation and finite-difference optical mode analysis. In addition to the strong optical confinement in the vertical direction for the fundamental transverse electric field mode, the vertical p-n junction effectively depletes the strained SiGe layer in which the plasma dispersion effect is enhanced owing to the mass modulation of holes by strain. We predict that a Si <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0.7</sub> Ge <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0.3</sub> optical modulator exhibits V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">π</sub> L at 1.55 μm of as small as 0.31 V-cm at a bias voltage of -2 V, which is ~1.8 times smaller than that of a Si optical modulator. The product of V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">π</sub> L and the phase-shifter loss (αV <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">π</sub> L) is also expected to be as low as 18.3 V-dB at -2 V, enabling optical modulation with a 5-dB extinction ratio in a symmetric Mach-Zehnder modulator.

Nonlinear regime of electrostatic waves propagation in presence of electron-electron collisions

The effects are presented of including electron-electron collisions in self-consistent Eulerian simulations of electrostatic wave propagation in nonlinear regime. The electron-electron collisions are approximately modeled through the full three-dimensional Dougherty collisional operator [J. P. Dougherty, Phys. Fluids 7, 1788 (1964)]; this allows the elimination of unphysical byproducts due to reduced dimensionality in velocity space. The effects of non-zero collisionality are discussed in the nonlinear regime of the symmetric bump-on-tail instability and in the propagation of the so-called kinetic electrostatic electron nonlinear (KEEN) waves [T. W. Johnston et al., Phys. Plasmas 16, 042105 (2009)]. For both cases, it is shown how collisions work to destroy the phase-space structures created by particle trapping effects and to damp the wave amplitude, as the system returns to the thermal equilibrium. In particular, for the case of the KEEN waves, once collisions have smoothed out the trapped particle population which sustains the KEEN fluctuations, additional oscillations at the Langmuir frequency are observed on the fundamental electric field spectral component, whose amplitude decays in time at the usual collisionless linear Landau damping rate.

High‐frequency radar cross‐sections of swell‐contaminated seas for a pulsed waveform

The first- and second-order monostatic cross-sections of swell-contaminated seas for high-frequency ground wave pulsed radar operation are derived from the fundamental electric field equations. In this case, the ocean surface, a mixture of both swell and wind wave components, is first represented with a Fourier series. In one of the two approaches used, the total Fourier coefficient is expanded as a linear sum of the contribution from swell and wind waves without considering the possible interaction between the two regimes; whereas in a second, more general situation, the analysis includes coupling effects between the wind waves and swell. Then, the corresponding electric fields received from the swell-contaminated sea surface are obtained. A Fourier transform of the autocorrelation of the electric fields gives the Doppler power spectral density (PSD), and a comparison between this PSD and monostatic radar range equation yields the radar cross-sections. Simulation shows that the assumption incorporating the coupling effects leads to results that better agree with historical field observations. Overall, the model proposed here not only lays a solid foundation for the development of future swell inversion algorithms, but also reveals the existence of non-linear energy transfer between swell and wind waves from a new perspective

High Absorption and Second‐Harmonic Generation in Split Ring Resonator Multilayer Nanostructure

Second‐harmonic generation in split ring resonator multilayer nanostructure is studied with the finite‐difference time‐domain (FDTD) method. The fundamental frequency wave and the second‐harmonic generation at the resonant absorption wavelength are highly localized in the dielectric layer, and the absorption peak is sensitive to dielectric constant of the dielectric layer. Under the excitation of the plasmon resonances mode, the strong local field induces an expected increase of the second‐harmonic generation with conversion efficiencies 10−6‐10−7. The distributions of fundamental frequency electric field and second‐harmonic electric field inside the central dielectric layer region are also shown.

Shaping of attosecond pulses by phase-stabilized polarization gating

We demonstrate that the characteristics of the high-order harmonic spectra generated by few-cycle carrier-envelope phase-stabilized pulses can be finely adjusted by controlling the time-dependent ellipticity. The experimental measurements show evidence for the generation of single, pairs, and trains of attosecond pulses by controlling the time window of linear polarization of the driving pulses. The influence of the carrier-envelope phase on the generation process in different confinement configurations is interpreted and analyzed using a nonadiabatic stationary phase model. We show that the xuv emission depends critically on particular aspects of the fundamental electric field that allows us to steer the electron trajectories on the time scale of tens of attoseconds.

Calculation of second-harmonic wave pattern generated by focused cylindrical vector beams

We calculated the second-harmonic wave pattern induced by focused cylindrically symmetric, polarized vector beams. The second-order nonlinear polarization was expressed for fundamental electric field components passed through a dielectric interface based on vector diffraction theory. Furthermore, the second-harmonic wave pattern was represented on the basis of the far-field approximate expression derived from the formulation of higher-order harmonic generation including a Green’s function. For a (110) zinc selenide crystal, the calculated forward emission patterns of the second-harmonic wave were eight-figure shaped as observed in experiment.

SCALAR AND ELECTROMAGNETIC CONFIGURATIONS IN k = 1 FRIEDMANN–ROBERTSON–WALKER UNIVERSES

The present work focuses on the scalar and electromagnetic fields in a spatially closed FRW Universe with incoherent dust. First, as the complex scalar field is under consideration, we derive the general solutions of the Gordon equation, in terms of the positive-frequency-like parity modes, and the corresponding propagator. Secondly, we turn to the source-free Maxwell equations in order to find a (1 + 4)-parametric class of closed form solutions representing a non-propagating fundamental electric field and a burst of electromagnetic radiation.

Magneto-optical characterization of ferromagnetic ultrathin multilayers in terms of surface susceptibility tensors

Two phenomenological models for considering linear magneto-optical (MO) effects in a ferromagnetically ordered multilayer, which necessarily comprises ultrathin layers, are described assuming that its response can be represented in terms of electric polarizations. As an alternative to a commonly used finite thickness model, an exceptionally simple analytical model is proposed to avoid dealing with thicknesses of the ultrathin layers explicitly, since these parameters might be ill defined. Incident light is assumed to be a plane monochromatic wave, the ultrathin layers (embedded equidistantly into a nonmagnetic optically isotropic medium) are treated as ideal surfaces and corresponding surface polarizations as perturbations. The problem of relating the polarizations to the fundamental electric field through surface susceptibility tensors is addressed, since intrinsic symmetry of the tensors may not be consistent with that given by the Onsager principle. To solve this problem, the general symmetry principle for reflection matrices, due to Shelankov and Pikus, is slightly adjusted for the use within both models. Symmetry analysis based on the Neumann principle is outlined to reveal the form of the tensors through their invariance under magnetic and, alternatively, ordinary point groups (when the linear approximation with respect to magnetization is pursued). New analytical results presented here for the reflection matrix of the multilayer are valid for any relevant ordinary point group and therefore take into account the in-plane (rotational) anisotropy that may be encountered when observing both optical and MO effects. Although both models are equivalent (this is shown merely for the groups $4mm,$ $3m,$ $6mm,$ and $\ensuremath{\infty}m),$ that proposed here can become more preferable in magneto-optics. Interesting and helpful analytical expressions for characterizing the conventional MO Kerr effects are provided along with their numerical illustrations.

Features of the simplest model of second-harmonic magneto-optical Kerr effects

A generalised formalism of characteristic matrices is presented to analyse second-harmonic magneto-optical Kerr effects in an optically anisotropic centrosymmetric ferromagnetically ordered multilayer if its response can be described in terms of electric polarisation. Features of the model associated with ideal (infinitely thin) interfaces are highlighted. These are due to both the existence of two versions of unconventional boundary conditions and an inevitable conventional approach to defining the surface polarisations through the fundamental electric field and surface-susceptibility tensors. New analytical results for linear and second-harmonic Kerr effects are shown to be advantageous for developing an effective algorithm for their numerical simulation. The linear approximation with respect to magnetisation is pursued, thereby also making our results suitable for investigating a great variety of magneto-optical effects and (in the second-harmonic case) effects related to anisotropy.

Boundary conditions in the simplest model of linear and second harmonic magneto-optical effects

This paper is concerned with linear and nonlinear magneto-optical effects in multilayered magnetic systems when treated by the simplest phenomenological model that allows their response to be represented in terms of electric polarization. The problem is addressed by formulating a set of boundary conditions at infinitely thin interfaces, taking into account the existence of surface polarizations. Essential details are given that describe how the formalism of distributions (generalized functions) allows these conditions to be derived directly from the differential form of Maxwell's equations. Using the same formalism we show the origin of alternative boundary conditions that exist in the literature. The boundary value problem for the wave equation is formulated, with an emphasis on the analysis of second harmonic magneto-optical effects in ferromagnetically ordered multilayers. An associated problem of conventions in setting up relationships between the nonlinear surface polarization and the fundamental electric field at the interfaces separating anisotropic layers through surface susceptibility tensors is discussed. A problem of self-consistency of the model is highlighted, relating to the existence of rescaling procedures connecting the different conventions. The linear approximation with respect to magnetization is pursued, allowing rotational anisotropy of magneto-optical effects to be easily analyzed owing to the invariance of the corresponding polar and axial tensors under ordinary point groups. Required representations of the tensors are given for the groups $\ensuremath{\infty}m,$ $4mm,$ $\mathrm{mm}2,$ and $3m.$ With regard to centrosymmetric multilayers, nonlinear volume polarization is also considered. A concise expression is given for its magnetic part, governed by an axial fifth-rank susceptibility tensor being invariant under the Curie group $\ensuremath{\infty}\ensuremath{\infty}m.$

The electromagnetic environment: implications for bodywork

Abstract Rhythmically changing electric, magnetic and electromagnetic fields are ubiquitous in our environment. Some of these fields are natural, others are produced by household appliances and technologies (see Part 1). Many people are adversely affected by natural and/or artificial energy fields (clinically termed weather or electromagnetic sensitivity). Often affected individuals. do not recognize the sources of their ailments. Disturbances in geomagnetic fields (e.g. caused by solar and terrestial magnetic storms) have been correlated with the onset of a variety of disorders, including heart attacks, seizures and strokes. Practitioners of bodywork, energetic, and movement therapies who appreciate the subtleties of energetic interactions can educate their clients about the possible health effects of environmental fields and appropriate precautions. At a fundamental level, electric and magnetic fields arise from quantum phenomena called scalar and vector potentials. Knowledge of these generally unappreciated effects is leading to new hypotheses to help explain biological interactions and phenomena taking place in bodywork and movement therapies that have been puzzling for a long time. In particular, scalar waves may mediate so-called non-local interactions. Natural and artificial scalar energy may help people protect themselves from harmful energies in the environment.