Lorentz Oscillator Model Research Articles

Extended coupled Lorentz oscillator model and analogue of electromagnetically induced transparency in coupled plasmonic structures

We report a new extended coupled Lorentz oscillator (ECLO) model by adding a coupled phase factor for the first time, to the best of our knowledge, in contrast to the conventional coupled Lorentz oscillator (CLO) model for describing the analogue of electromagnetically induced transparency (EIT) in a dimer, which is composed of two identically broken eccentric silver nanorings (ESNRs) arranged in an asymmetric manner. Based on the finite element method, the distribution features of the absorption spectra, the charges, and the electric field of the ESNRs are numerically simulated under the action of the applied light field. The ECLO model reveals the physical mechanism of the spectra features and the EIT-like effect. Our findings indicate that the destructive interference effect plays a key role in the generation of the EIT-like effect. The theoretical analyses are in good agreement with the numerical results.

Plasmon-Induced Transparency Based on Triple Arc-Ring Resonators.

This paper presents a plasmon-induced transparency (PIT) using an easy-fabricating metamaterial composed of three pieces of metallic arc-rings on top of a dielectric substrate. The transmission of the transparent peak of 1.32 THz reaches approximately 93%. The utilization of the coupled Lorentzian oscillator model and the distribution of electromagnetic fields together explain the cause of the transparent peak. The simulation results further demonstrate that the bandwidth of the transmission peak can be narrowed by changing the sizes of the arc-rings. Moreover, an on/off effect based on the transparent peak is discussed by introducing photosensitive silicon into the air gaps of the suggested metamaterial structure.

Phase aspect in photon emission and absorption

In the literature one finds several conflicting accounts of the phase difference of stimulated and spontaneous emission, as well as absorption, with respect to an existing (triggering) electromagnetic field. One of these approaches proposes that stimulated emission and absorption occur in phase and out of phase with their driving field, respectively, whereas spontaneous emission occurs under an arbitrary phase difference with respect to an existing field. It has served as a basis for explaining quantum-mechanically the laser linewidth, its narrowing by a factor of 2 around the laser threshold, as well as its broadening due to amplitude–phase coupling, resulting in Henry’s α-factor. Assuming the validity of Maxwell’s equations, all three processes would, thus, violate the law of energy conservation. In semi-classical approaches, we investigate stimulated emission in a Fabry–Perot resonator, analyze the Lorentz oscillator model, apply the Kramers–Kronig relations to the complex susceptibility, understand the summation of quantized electric fields, and quantitatively interpret emission and absorption in the amplitude–phase diagram. In all cases, we derive that the phase of stimulated emission is 90° in lead of the driving field, and the phase of absorption lags 90° behind the transmitted field. Also spontaneous emission must obey energy conservation, hence it occurs with 90° phase in lead of an existing field. These semi-classical findings agree with recent experimental investigations regarding the interaction of attosecond pulses with an atom, thereby questioning the physical explanation of the laser linewidth and its narrowing or broadening.

Electromagnetically induced transparency control in terahertz metasurfaces based on bright-bright mode coupling

We demonstrate a classical analogue of electromagnetically induced transparency (EIT) in a highly flexible planar terahertz metamaterial (MM) comprised of three-gap split ring resonators. The keys to achieve EIT in this system are the frequency detuning and hybridization processes between two bright modes coexisting in the same unit cell as opposed to bright-dark modes. We present experimental verification of two-bright mode coupling for a terahertz EIT-MM in the context of numerical results and theoretical analysis based on a coupled Lorentz oscillator model. In addition, a hybrid variation of the EIT-MM is proposed and implemented numerically in order to dynamically tune the EIT window by incorporating photosensitive silicon pads in the split gap region of the resonators. As a result, this hybrid MM enables the potential active optical control of a transition from the on-state (EIT mode) to the off-state (dipole mode).

Infrared reflectance spectroscopy of MgAl2O4 nanoparticles substituted by K+ ions

The infrared reflectivity spectra for potassium-doped polycrystalline magnesium aluminates Mg[Formula: see text]K[Formula: see text]Al2O4 (x=0, 0.25, 0.50, 0.75, 1) are measured in the frequency range between 10–15, 500 cm[Formula: see text] using FTIR spectrometer at room-temperature. Four optical phonon modes are observed in measured spectra, which are fitted by Lorentz oscillator model for semiconducting behavior and Lorentz–Drude model for metallic behavior. Moreover, optical parameters are also determined for these modes which may attribute to spinel structure for samples Mg[Formula: see text]K[Formula: see text]Al2O4, their reflectivity spectra shows a typical semiconducting nature. To study ionicity and effect of polarization, Born and Szigeti effective charges are calculated from longitudinal optical and transverse optical (LO–TO) splitting of modes for all samples. Optical bandgap has been estimated through optical conductivity ([Formula: see text]([Formula: see text])) and found to be x dependent.

Bifurcation of avoided crossing at an exceptional point in the dispersion of sound and light in locally resonant media

The avoided crossing behavior in the interaction of propagating sound or light waves with resonant inclusions is analyzed using a simple model of an acoustic medium containing damped mass-spring oscillators, which is shown to be equivalent to the Lorentz oscillator model in the elementary dispersion theory in optics. Two classes of experimental situations dictating the choice in the analysis of the dispersion relation are identified. If the wavevector is regarded as the independent variable and frequency as a complex function of the wavevector, then the avoided crossing bifurcates at an exceptional point at a certain value of the parameter γβ−1/2, where γ and β characterize the oscillator damping and interaction strength, respectively. This behavior is not observed if the wavevector is regarded as a complex function of frequency.

Bistable enhanced total reflection in Kretschmann configuration containing a saturable gain medium.

The reflection of a TM-polarized light beam from a Kretschmann configuration with a saturable gain medium is investigated theoretically. Here, the dielectric constant of the gain medium is described by a classical Lorentzian oscillator model. When surface plasmon polaritons are effectively excited in this structure, it is demonstrated that the curves of enhanced total reflection (ETR) show different shaped hysteresis loops associated with optical bistability owing to gain saturation effect. The effects of the angle of incidence, the thickness of metal film, and the value of small-signal gain on bistable ETR are discussed in detail in a homogeneously broadened (HB) gain medium at line center. Analogous results can also be obtained in an inhomogeneously broadened (inHB) gain medium, while the two switch thresholds and the width of optical bistability hysteresis in an inHB gain medium are significantly different from those in a HB gain medium.

Tunable plasmon-induced transparency with graphene-based T-shaped array metasurfaces

The frequency tunable Plasmonic induced transparency (PIT) effect is researched with a periodically patterned T-shaped graphene array in mid-infrared region. We adjust the geometrical parameters to obtain the optimized combination for the realization of the PIT response and use the coupled Lorentz oscillator model to analysis the physical mechanism. Due to the properties of graphene, the PIT effect can be easily and markedly enhanced with the increase of chemical potential and carrier mobility. The frequency of PIT effect is also insensitive with the angle of incident light. In addition, we also propose the π shaped structure to realizing the double-peak PIT effect. The results offer a flexible approach for the development of tunable graphene-based photonic devices.

Fano Resonances in Ultracompact Silicon‐on‐Insulator Compatible Integrated Photonic–Plasmonic Hybrid Circuits

AbstractA novel on‐chip Fano device is experimentally demonstrated operating at telecom wavelengths, which consists of an ultracompact plasmonic nanocavity integrated with two Si waveguides on a silicon‐on‐insulator substrate. It is observed that an Au symmetric split ring with a large linewidth can be coupled to an embedded Au nanorod with a narrow linewidth via excitation of an Si waveguide, contributing to a Fano resonance phenomenon. In addition, the relative precise control of the resonant properties, including the depth, lineshape, and central wavelength, is realized by varying the rotation angle of the Au nanorod. For further investigations, a coupled Lorentz oscillator model is applied to study the transmission peak arising within an absorption region in the nanocavity. Slow‐light effects and sensing characteristics are also verified with finite difference time domain simulations and numerical calculations. This device has achieved Fano resonance in integrated photonic–plasmonic hybrid circuits, which may find utility in optical communications, buffering, and sensing.

Investigation of Cody–Lorentz and Tauc–Lorentz Models in Characterizing Dielectric Function of (HfO2) x (ZrO2)1–x Mixed Thin Film

In the present study, Tauc–Lorentz and Urbach tail assisted Cody–Lorentz models were intuitively derived, explained, and employed to simulate the imaginary dielectric function of the atomic layer deposited (HfO2)x(ZrO2)1–x in the energy range of 0.7–9.3 eV. A fitting procedure is carried out using the Levenberg–Marquardt method for minimizing mean square errors. Both models support the experimental data derived by vacuum ultraviolet spectroscopic ellipsometry. For a zirconium content of 60 and 75%, a weak absorption profile is observed in the visible region and elaborately explained by the Lorentz oscillator model. In order to calculate the band gap of different compositions, Cody plots and Tauc plots are presented. The results show that the Cody–Lorentz model is more accuracy due to its exponential absorption below the band gap energy and modified density of states.

Near-field enhancement of thermoradiative devices

Thermoradiative (TR) device has recently been proposed for noncontact direct photon-electricity energy conversion. We investigate how the near-field effect can boost the performance of a TR device. For a near-field TR device, a heat sink is placed close to the TR cell, with the separation being small compared to the characteristic photon wavelength. It is demonstrated that the TR device, like the thermophotovoltaic device, can be formulated using the transmissivity and the generalized Planck distribution. We quantitatively show that δ-function transmissivity is a very good approximation (capturing up to 90% of total radiative energy transfer) when the radiative energy transfer is governed by resonances. Three practical types of heat sinks are considered, a metallic material described by the Drude model, a polar dielectric material described by the Lorentz oscillator model, and a semiconductor material that is identical to the TR cell. The blackbody heat sink serves as the far-field reference. By properly choosing the resonant frequencies supported by the heat sink, we show that the heat sink made of a Drude or Lorentz material can enhance the output power by about 60 and 20 times, respectively, as compared to the blackbody reference. Even with a heat sink made of the same material as the TR-cell, which does not support any resonant modes, the output power can be enhanced by about 10 times. The mechanisms can be elucidated from the impedance matching condition derived from the coupled-mode theory.

Effect of Co2+ substitution on MgAl2O4 studied by infrared reflectance spectroscopy

Infrared reflectivity spectra are measured in the frequency range between 10 and 15,500cm−1 at room temperature for cobalt substituted nanocrystalline magnesium aluminates MgAl2−xCoxO4 (x=0.0, 0.5, 1.0, 1.5, 2.0). Four optical phonon modes manifesting the spinel structure have been observed and analyzed using the Lorentz oscillator model. The resonant frequency, oscillator strength and the damping factor of these modes were extracted from Lorentz oscillator fit to measured spectra. The observed reflectivity spectra exhibit a typical semiconductor behavior. Born and Szigeti effective charges have been estimated and the results are discussed to understand polarization and ionicity of MgAl2−xCoxO4 system.

Modeling the PbS quantum dots complex dielectric function by adjusting the E-k diagram critical points of bulk PbS

The complex dielectric function ϵ(E)=ϵR(E)+iϵI(E) of a semiconductor is a key parameter that dictates the material's optical and electrical properties. Surprisingly, the ϵ(E) of Lead Sulfide (PbS) quantum dots (QDs) has not been widely studied. In the present work, we develop a new model that aims to simulate the ϵ(E) of QDs. Our model is based on the fact that the quantum confinement in the nano regime affects all the electronic transitions throughout the entire Brillouin zone. Hence, as a first approximation, we attribute an equal contribution of energy, equivalent to the bandgap broadening, to each critical point (CP) in the E-k diagram. This is mathematically realized by adding these energy contributions to the central energy parameters of the Lorentz oscillator model. In order to validate our model, we used the CP parameters of bulk PbS to simulate the ϵ(E) of PbS QDs. Next, we use Maxwell Relations to calculate the refractive index and the extinction coefficient of PbS QDs from ϵE. Our results were compared with those published in the previous literature and showed good agreement. Our findings open a new avenue that may enable the calculation of the ϵE for nanoparticle systems.

Controlled Terahertz Birefringence in Stretched Poly(lactic acid) Films Investigated by Terahertz Time-Domain Spectroscopy and Wide-Angle X-ray Scattering.

We report a correlation between the dielectric property and structure of stretched poly(lactic acid) (PLA) films, revealed by polarization-sensitive terahertz time-domain spectroscopy and two-dimensional (2D) wide-angle X-ray scattering (WAXS). The experiments evidence that the dielectric function of the PLA film becomes more anisotropic with increasing draw ratio (DR). This behavior is explained by a classical Lorentz oscillator model assuming polarization-dependent absorption. The birefringence can be systematically altered from 0 to 0.13 by controlling DR. The combination of terahertz spectroscopy and 2D WAXS measurement reveals a clear correlation between the birefringence in the terahertz frequency domain and the degree of orientation of the PLA molecular chains. These findings imply that the birefringence is a result of the orientation of the PLA chains with anisotropic macromolecular vibration modes. Because of a good controllability of the birefringence, polymer-based materials will provide an attractive materials system for phase retarders in the terahertz frequency range.

Estimated permittivity functions for NIR and SWIR absorbing dyes by inverse analysis of transmission spectra

An inverse analysis of transmission spectra for near-infrared (NIR, 0.7–0.9 μm) and shortwave infrared (SWIR, 0.9–1.7 μm) absorbing dyes in solution is presented. This analysis employs a parametric model of transmission through a sample of finite thickness, where the permittivity function is represented parametrically by a linear combination of Lorentz oscillator models. The results of this analysis provide estimates of permittivity functions, which can be adopted as input data to other types of models, such as those for prediction of transmission and reflectivity spectra for composites containing mixtures of dyes and other materials. In addition, the results of this analysis should contribute to a database of estimated permittivity functions for practical analysis of spectra.

Tunable polarization-independent plasmonically induced transparency based on metal-graphene metasurface.

A tunable polarization-independent dual-band plasmonically induced transparency (PIT) device based on metal-graphene nanostructures is proposed theoretically and numerically at mid-infrared frequencies, which is composed of two kinds of center-symmetric metallic nanostructure array with different sizes and element numbers placed on separate graphene interdigitated finger sets, respectively. The coupled Lorentz oscillator model is used to explain the physical mechanism of PIT effect at multiple frequency domains. The finite-difference time-domain (FDTD) solutions are employed to simulate the characteristics of the polarization-independent metal-graphene PIT device, which is consistent with the theoretical analysis. The PIT peaks, obtained at two frequency domains, are separately and dynamically modulated by varying the Fermi energy of corresponding graphene finger set without changing the geometrical parameter of the metallic nanostructure. By the carefully selected element numbers of nanostructure arrays, the resonance strength of the PIT peaks at two frequency domains are nearly close. And the PIT device has identical response to the various polarized incident field due to the center symmetry of the metallic nanostructure, which have advantages in practical applications with no polarization-dependent loss.

Refractive Index and Absorption Coefficient of Undoped and Mg-Doped Lithium Tantalate in the Terahertz Range

Dielectric material parameters of lithium tantalate (LT) in the terahertz region have been investigated using terahertz time-domain spectroscopy (THz-TDS). Undoped congruent, undoped stoichiometric, and Mg-doped stoichiometric LT crystals were measured. The Mg content was 0.5 and 1.0 mol% for the stoichiometric composition. Index of refraction and absorption coefficient spectra were determined in the 0.3–2.0-THz frequency range for beam polarization both parallel (extraordinary polarization) and perpendicular (ordinary polarization) to the optical axis [001] of the crystal at room temperature. For the calculation of the refractive index and absorption coefficient spectra from the measured data, we used TeraMat software (Menlo System) belonging to the spectrometer. The refractive index and the absorption coefficient for stoichiometric crystals were lower than for the congruent one. In the case of stoichiometric crystals, the Mg dopant caused a slight reduction of both ordinary and extraordinary refractive index compared to the undoped crystal. However, the presence of Mg did not reduce the absorption coefficient either for the ordinary or for the extraordinary polarization. In order to fit the measurement data, a Lorentz oscillator model was used. Good agreement was obtained between the measured data and the fitting curves by using the Lorentz oscillator model containing three terms.

Application of coupled mode theory on radiative heat transfer between layered Lorentz materials

The coupled mode theory (CMT) provides a simple and clear framework to analyze the radiation energy exchange between reservoirs. We apply CMT to analyze the radiative heat transfer between layered Lorentz materials whose dielectric functions can be approximated by the Lorentz oscillator model. By comparing the transmissivity computed by the exact solution to that computed by CMT, we find that CMT generally gives a good approximation for this class of materials. The biggest advantage of CMT analysis, in our opinion, is that only the (complex) resonant energies are needed to obtain the radiation energy transfer; the knowledge of the spatial profile of resonances is not required. Several issues, including how to choose the resonant modes, what these modes represent, and the limitation of this method, are discussed. Finally, we also apply the CMT method to the electronic systems, demonstrating the generality of this formalism.

The dielectric response to the magnetic field of electromagnetic radiation

Light–matter interaction in transparent dielectrics is revisited, including the magnetic force on bound charges in the Lorentz oscillator model. The parameter ranges of incident radiation and the medium on which the magnetic field of the electromagnetic radiation will have a significant effect are traced using Floquet theory. The analysis reveals that the threshold intensity for a significant response of the magnetic field of the radiation at the second harmonic of the incident radiation can be reduced to for off resonant and even lower for resonant interaction. This phenomenon has already been observed indirectly in experiments [, ]. Induced magnetizing current due to the magnetic force is shown to originate from a modified dielectric response, which may be useful in future magneto-optic devices, solar energy harvesting, and studying the ultrafast dynamics in doped dielectrics.

Independently tunable dual-band plasmonically induced transparency based on hybrid metal-graphene metamaterials at mid-infrared frequencies.

A tunable dual-band plasmonically induced transparency (PIT) device based on hybrid metal-graphene nanostructures is proposed theoretically and numerically at mid-infrared frequencies, which is composed of two kinds of gold dolmen-like structures with different sizes placed on separate graphene interdigitated finger sets respectively. The coupled Lorentz oscillator model is used to explain the physical mechanism of the PIT effect at multiple frequency domains. The finite-difference time-domain (FDTD) solutions are employed to simulate the characteristics of the hybrid metal-graphene dual-band PIT device. The simulated spectral locations of multiple transparency peaks are separately and dynamically modulated by varying the Fermi energy of corresponding graphene finger set, which is in good accordance with the theoretical analysis. Distinguished from the conventional metallic PIT devices, multiple PIT resonances in the hybrid metal-graphene PIT device are independently modulated by electrostatically changing bias voltages applied on corresponding graphene fingers, which can be widely applied in optical information processing as tunable sensors, switches, and filters.