Classical Lorentz Oscillator Research Articles

Optical Spectra of Plasmon–Exciton Core–Shell Nanoparticles: A Heuristic Quantum Approach

Light–matter coupling in plasmonic nanocavities has been widely studied in the past years. Yet, for core–shell particles, popular electromagnetic models that use the classical Lorentz oscillator to describe the shell predict extinction spectra with three maxima, if the plasmon and the shell absorption are in resonance. In contrast, experiments exhibit only two peaks, as also expected from simple quantum models of hybrid states. In order to reconcile the convenient and widely used classical electromagnetic description with experimental data, we connect it to the quantum world by conceiving a heuristic quantum model. Our model is based on the permittivity of a two-level system in a classical electric field derived from the optical Bloch equations. The light–matter coupling is included via the collective vacuum Rabi frequency Ω0. Using our model, we obtain excellent agreement with a series of experimental extinction spectra of particles with various coupling strengths due to a systematic size variation. The suppression of the third maximum, which mainly stems from the absorption in the shell, can be interpreted as a vacuum induced power broadening, which may occur in lossy (plasmonic) cavities below the strong-coupling regime.

STRUCTURE, PHONON VIBRATIONAND ELECTRICALPROPERTIESOFBa2+ DOPEDBISMUTH FERRITE


 
 
 Multiferroic Bi1-xBaxFeO3 (BBFO) (x = 0.0, 0.1, 0.2, 0.25, 0.3) nanoparticles were successfully synthesized utilizing sol–gel process. The structure of the prepared samples was characterized by employing x-ray diffraction technique. The results confirmed the transformation of crystal structure from rhombohedral to tetragonal at x ≥ 0.25. Infrared (IR) reflectivity spectra measured at ambient temperature in the frequency range 10-7500 cm-1. The resonant frequency, oscillator strength and the damping factor of all modes were computed by fitting the IR reflectivity spectra to the classical Lorentz oscillator model. From the temperature dependent electrical resistivity, Ba2+ doped BFO exhibited metallic behavior at low temperature, which transform to semiconducting behavior on increasing the temperature. The temperature of this metal to semiconductor (MS) transition depends on doping concentration of Barium.
 
 

Optical constants of amorphous Sb2Se3 thin films in the spectral range 2.5 to 15 µm

Sb2Se3 plays an important role as high-index layers in mid-infrared edge filters used in thermophotovoltaics (TPVs) to enable the spectral control performance. However, the information on its mid-infrared optical constants is still insufficient. In our investigation, Sb2Se3 thin films were deposited on Ge wafers at different substrate temperatures using electron beam evaporation. The compositions, surface morphologies and mid-infrared spectral transmittance were characterized using the energy dispersive X-ray analysis (EDX), scanning electron microscopy (SEM) and Fourier-transform infrared (FTIR) spectrometer, respectively. The crystallographic structure was also identified using X-ray diffraction (XRD). The mid-infrared optical constants were obtained using the classical Lorentz oscillator models as the dispersion equation in the fitting to the measured spectral transmittance. It was revealed that thin films were amorphous deposited at a low substrate temperature less than 150 °C, which has an approximately ideal stoichiometric ratio and the lowest free carrier absorption, being preferential for achieving the optimal optical constants. In the spectral region from 2.5–15 µm, the amorphous thin films have a refractive index in the range of 3.00–3.18, and an extinction coefficient less than 0.04.

Quantum impedance Lorentz oscillator and its 1- and 2-photon-absorption applications

In this paper, a classical Lorentz oscillator is quantized via Bohr–Sommerfeld quantum theory and 1- and 2-photon absorption (1PA and 2PA) selection rules of quantum mechanics. Based on the Bohr–Sommerfeld model of a hydrogen-like atom in the adiabatic approximation, the computational formulas of the linear and nonlinear parameters and the damping coefficient of the quantized oscillator are derived and further expressed in terms of microphysical quantities, such as electronic charge and mass, Bohr radius, and effective quantum number. In accordance with Boltzmann thermal equilibrium distribution, here, the atom number density in general electric susceptibility is changed to the energy level transition one from the initial to the final state at equilibrium between atomic emission and absorption under light field. A new relationship is proposed to determine the transition eigenfrequency according to the peak frequency and full width at half maximum of an absorption spectrum. Our theoretical simulations of the 1PA spectra of atomic hydrogen and lithium and 1PA and 2PA spectra of two kinds of organic molecules turn out to be in good agreement with the experimental ones. These results suggest that our advancement in the quantization of the Lorentz oscillator is likely successful to make it available for use in the quantitative description of atomic or molecular 1PA and 2PA processes. Generally, the improved Lorentz oscillator may also be more suitable for approximating both linear and nonlinear properties of many dielectric or optoelectronic materials due to its relative simplicity.

Precise control of the size and gap between gold nanocubes by surface-based synthesis for high SERS performance.

The optical properties of a monolayer of nanocomposite film (PMMA/gold nanocubes) were provided by fitting a proposed theoretical model to spectroscopic ellipsometry (SE) measurements. For such a thin film, these features cannot be successfully determined by means of experimental and conventional effective medium theory such as Maxwell-Garnett or Bruggeman. To make it possible, we developed a model of two classical Lorentz oscillators; one for a PMMA layer and the other for GNCs, revealing one homogeneous layer and rapid analysis without the need for large computational resources. Additionally, we tailored both the size and number of GNCs in the PMMA layer by tuning the synthesis parameters as seen in scanning electron microscopy (SEM) images. In parallel, SE measurements clearly highlighted the change in the optical properties of GNCs as a function of their density on the substrate and dimensions. Our findings demonstrate that SE is an alternative method to characterize layered GNCs on opaque substrates efficiently, which has potential implications for designing other morphologies in the future.

Extracting the differential inverse inelastic mean free path and differential surface excitation probability of Tungsten from X-ray photoelectron spectra and electron energy loss spectra

Precise knowledge of the differential inverse inelastic mean free path (DIIMFP) and differential surface excitation probability (DSEP) of Tungsten is essential for many fields of material science. In this paper, a fitting algorithm is applied for extracting DIIMFP and DSEP from X-ray photoelectron spectra and electron energy loss spectra. The algorithm uses the partial intensity approach as a forward model, in which a spectrum is given as a weighted sum of cross-convolved DIIMFPs and DSEPs. The weights are obtained as solutions of the Riccati and Lyapunov equations derived from the invariant imbedding principle. The inversion algorithm utilizes the parametrization of DIIMFPs and DSEPs on the base of a classical Lorentz oscillator. Unknown parameters of the model are found by using the fitting procedure, which minimizes the residual between measured spectra and forward simulations. It is found that the surface layer of Tungsten contains several sublayers with corresponding Langmuir resonances. The thicknesses of these sublayers are proportional to the periods of corresponding Langmuir oscillations, as predicted by the theory of R.H. Ritchie.

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.

Differential inverse inelastic mean free path and differential surface excitation probability retrieval from electron energy loss spectra

Quantitative interpretation of the electron spectroscopy data requires the information on differential inverse inelastic mean free paths (DIIMFP) and differential surface excitation probabilities (DSEP). In this paper, we test an algorithm of extracting DIIMFP and DSEP from reflected electron energy loss spectra (REELS) and photo-electron spectra (PES) in which the desired functions are parametrized on the base of a classical Lorentz oscillator. Unknown parameters are found by using the fitting procedure. To account for surface excitations, the investigated samples are considered as multi-layer systems. Simulations of REELS and PES are performed by making use of the partial intensity approach. The partial intensities for the reflection function and the photo-electron density flux are computed on the base of the invariant imbedding method. Extracted DIIMFPs and DSEPs are compared with those obtained by other authors. Finally, REELS and PES spectra for Be, Mg, Al, Si, Nb and W are computed using the retrieved DIIMFPs and DSEPs, and compared with the experimental spectra. All comparisons show good agreement.

Analysis of transition from Lorentz resonance to Fano resonance in plasmon and metamaterial systems

Fano resonance, viewed as a quantum interference between continuum and discrete states, demonstrates an asymmetric spectral line shape. Fano resonance is also observed in metamaterials and plasmonic nanostructures and could be explained by the coupling of two classical Lorentz oscillators. A unambiguous connection between Fano resonance and Lorentz resonance plays a vital role in the design and optimization of metamaterials and plasmonic materials for both mechanism understanding and applications. In this paper, a numerical method is employed to schematically investigate the parameters connection between Fano and Lorentz resonance. Both Fano resonant frequency and line-width are parabolic dependent with the coupling coefficient of the Lorentz oscillators. A distinct transition effect can be observed in the Fano parameters when the eigen-frequencies of Lorentz oscillators are tuned. Our results suggest that the controlling of Fano resonance can be done by designing of Lorentz resonance in metamaterials and plasmonic materials artificially, which sustains mimicking the quantum effect by classic systems.

Electromagnetic energy storage and power dissipation in nanostructures

The processes of storage and dissipation of electromagnetic energy in nanostructures depend on both the material properties and the geometry. In this paper, the distributions of local energy density and power dissipation in nanogratings are investigated using the rigorous coupled-wave analysis. It is demonstrated that the enhancement of absorption is accompanied by the enhancement of energy storage both for material at the resonance of its dielectric function described by the classical Lorentz oscillator and for nanostructures at the resonance induced by its geometric arrangement. The appearance of strong local electric field in nanogratings at the geometry-induced resonance is directly related to the maximum electric energy storage. Analysis of the local energy storage and dissipation can also help gain a better understanding of the global energy storage and dissipation in nanostructures for photovoltaic and heat transfer applications.

Infrared optical properties of amorphous and nanocrystalline Ta2O5 thin films

The optical constants of tantalum pentoxide (Ta2O5) are determined in a broad spectral region from the visible to the far infrared. Ta2O5 films of various thicknesses from approximately 170 to 1600 nm are deposited using reactive magnetron sputtering on Si substrates. X-ray diffraction shows that the as-deposited films are amorphous, and annealing in air at 800 °C results in the formation of nanocrystalline Ta2O5. Ellipsometry is used to obtain the dispersion in the visible and near-infrared. Two Fourier-transform infrared spectrometers are used to measure the transmittance and reflectance at wavelengths from 1 to 1000 μm. The surface topography and microstructure of the samples are examined using atomic force microscopy, confocal microscopy, and scanning electron microscopy. Classical Lorentz oscillators are employed to model the absorption bands due to phonons and impurities. A simple model is introduced to account for light scattering in the annealed films, which contain micro-cracks. For the unannealed samples, an effective-medium approximation is used to take into account the adsorbed moisture in the film and a Drude free-electron term is also added to model the broad background absorption.

Tuning the effective dielectric function of thin film metal-dielectric composites by controlling the deposition temperature

The influence of the substrate temperature on the effective optical behavior of Ag-SiO2 composites obtained by electron beam evaporation was studied. Optical characterization of the composites was performed by means of spectroscopic ellipsometry measurements. The effective dielectric function of the composites, modeled using a multiple oscillator approach, could be widely tuned by controlling the deposition temperature. The spectral dependence of the composite absorption appeared to be better described with a Gaussian line shape than with the classical Lorentz oscillator model. The description of the effective dielectric function using standard effective medium theories failed and the experimental results could be explained only in the general framework of the Bergman spectral density theory.

Light-induced dynamics in the Lorentz oscillator model with magnetic forces

The classical Lorentz oscillator model of bound electron motion ordinarily excludes magnetic forces at nonrelativistic intensities for the simple reason that their magnitude is small. However, perturbative and numerical results show that when the $\mathrm{vP\vec}\ifmmode\times\else\texttimes\fi{}\mathrm{BP\vec}$ term is retained, dynamically enhanced terms give rise to large amplitude, magnetically induced charge displacements at zero frequency and at twice the driving frequency in the Cartesian laboratory frame. Numerical simulations of electron motion are in accord with the predictions of perturbative theory for steady-state motion in the classical picture. Direct integration shows that magnetic response which is comparable to electric dipole response can arise in transparent dielectrics at optical frequencies. Parametric instability in the equations of motion is implicated as the source of rapid energy transfer from electric to magnetic motions by reduction of the equations to a complex Mathieu equation.

Time‐domain simulation of electromagnetic wave propagation in two‐level dielectrics

AbstractA method for the time‐domain simulation of electromagnetic wave propagation in two‐level dielectric media is developed. The technique involves using transmission‐line modelling (TLM) and 𝒵‐transform methods to develop an algorithm to simulate the Maxwell–Bloch equations. The approach is accurate and free from numerical instabilities. The scheme was validated for the small‐signal case by demonstration of the correspondence principle with the classical Lorentz oscillator. Results obtained for propagation in the two‐level medium are in agreement with previous studies in the literature. In addition, results are shown demonstrating gain, self‐induced transparency (SIT), Rabi flopping and laser oscillations. Copyright © 2008 John Wiley & Sons, Ltd.

The determination of infrared optical constants of rare earth fluorides by classical Lorentz oscillator model

Rare earth fluoride (REF3, RE = La, Ce, Pr, Sm, Er, Yb, Y) films were deposited on Ge(1 1 1) and silicon wafers in order to determine optical constants from the near infrared up to the high frequency tail of the reststrahlen band. The FT-IR transmission spectrum and the reflection spectrum were used to examine the infrared optical properties of the fluorides. The optical constants of the films in the infrared spectrum from 2 to 20 µm were calculated using the classical Lorentz oscillator model by fitting the transmission spectrum. The reflective spectrum of the fluorides on silicon was used to demonstrate the absorption difference in this region. It is found that the extinction coefficients of rare earth fluorides films with Pnma structure (REF3, RE = Y, Yb, Er) are larger than the other fluorides films with structure (REF3, RE = La, Ce, Pr, Sm) from 15 to 20 µm.

Composition dependence of the infrared dielectric functions in Si-doped hexagonalAlxGa1−xNfilms onc-plane sapphire substrates

The optical properties of hexagonal ${\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{N}$ ($x$ from 0.05 to 0.42) epitaxial films with Si doping concentrations up to ${10}^{18}\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$ grown on $c$-plane sapphire substrates by metal-organic chemical vapor deposition have been investigated using infrared reflectance spectra. The dielectric functions $(\ensuremath{\epsilon}={\ensuremath{\epsilon}}_{1}+i{\ensuremath{\epsilon}}_{2})$ of the ${\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{N}$ films are determined in the wavelength region of $1.54--50\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{m}$ at room temperature. The experimental reflectance spectra are analyzed using classical harmonic Lorentz oscillators and Drude model in the transparent, reststrahlen, and free carrier absorption regions. GaN-like ${E}_{1}$ transverse-optical (TO) phonon frequency linearly decreases and its strength increases with decreasing Al composition. However, AlN-like ${E}_{1}(\mathrm{TO})$ phonon frequency shows a relatively weak composition dependence and its strength increases with increasing composition. At a wavelength of $1.54\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{m}$, ${\ensuremath{\epsilon}}_{1}$ varies between 4.86 and 5.2 when the composition $x$ changes from 0.05 to 0.42 and ${\ensuremath{\epsilon}}_{2}$ is close to zero. The longitudinal-optical phonon plasmon (LPP) coupled modes of $n$-type hexagonal ${\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{N}$ films are also discussed. For samples with higher concentrations beyond ${10}^{18}\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$ the upper LPP coupled mode frequencies increase with increasing carrier concentration indicating the transition from phononlike to plasmonlike behavior.

Zn-doped GaSb epitaxial film absorption coefficients at terahertz frequencies and detector applications

The reflectance measurements of p-type GaSb:Zn epitaxial films with different hole concentrations, grown by metalorganic vapor-phase epitaxy, have been investigated in the 3–30-THz frequency region. The experimental spectra were fitted using a classical harmonic Lorentz oscillator and the Drude model, illustrating that the hole effective mass and the mobility change with the carrier concentration. The hole effective mass was found to vary from 0.22m0 to 0.41m0 as the carrier concentration changed from 3.5×1017 to 3.8×1018cm−3. The mobility values derived from the reflectance measurements were slightly smaller than the values obtained from Hall-effect measurements. A sublinear relationship between the absorption coefficient and the hole concentration was found at a frequency of 3 THz. Those results can be used for designing GaSb-based terahertz detectors.

Optical functions of InGaP/GaAs epitaxial layers from 0.01 to 5.5 eV

In0.49Ga0.51P films, both undoped and doped n- and p-type (up to 1018 cm-3), were grown lattice matched on GaAs substrates, with different miscut angles, by Metal-Organic Vapour Phase Epitaxy (MOVPE) at different temperatures. The shift of the fundamental gap E0, caused by “ordering effect” was measured as a function of temperature by photoluminescence. The complex refractive index = n + ik and the dielectric function = ɛ 1 + iɛ 2 at room temperature were determined from 0.01 to 5.5 eV by using complementary data from fast-Fourier-transform far-infrared (FFT-FIR), dispersive, and ellipsometric spectroscopies. The effect of the native oxide was accounted for and the self-consistency of the optical functions was checked in the framework of the Kramers-Kronig causality relations. In the restrahlen region the dielectric function was well fitted by classical Lorentz oscillators; in the transparent region below E0, the refractive index was modelled by a Sellmeier dispersion relation; in the interband region the dielectric function was well reproduced by analytical lineshapes associated to seven critical points. Thus parametrized analytical expressions were obtained for the optical functions all over the spectral range, without discontinuities, to be used in the modelling and characterization of multi-layer structures, also on opaque substrates.

Crystalline to amorphous transition and band structure evolution in ion-damaged silicon studied by spectroscopic ellipsometry

Crystalline to amorphous transition and subsequent microstructural evolution in silicon induced by Ar+-ion implantation over a wide range of ion fluences (6×1013–1×1017 cm−2) have been investigated by spectroscopic ellipsometry. In the evaluation of the optical and microstructural properties of the damaged layer, the contribution of the surface overlayer to the measured dielectric spectra was separated by fitting a multilayer model with an effective medium approximation. The best fit to the dielectric spectra for disordered silicon could be obtained by taking our highest-fluence implanted (fluence=1×1017 ions/cm2) amorphous silicon (a-Si) data as reference data instead of a-Si data available in the handbook. The derivative spectra as a function of fluence show a distinct and sharp transition from the crystalline to amorphous phase. The threshold fluence for this transition is derived from fitting. Evaluation of standard sum rules and optical moments for imaginary part of the pseudodielectric function reveals no substantial change in various physical parameters below the transition indicating their insensitivity to point defects, while it shows a large change with fluence above the threshold for amorphization. The disorder induced changes in the effective dielectric constant, number of valence electrons per atom participating in optical transition, Penn gap energy, average bond length, coordination number, effective dispersion oscillator energy, an average strength of the interband optical transition with fluence is discussed on the basis of microstructural evolution and corresponding band structure modification. It is also shown that the dielectric functions of damaged silicon are well represented by a sum of six classical Lorentz oscillators. With increasing fluences, each of the oscillator amplitude decreases and linewidth increases except for the 3.3 eV transition which shows increasing amplitude with fluence. These results are discussed in the context of short-range order/disorder and effective band gap reduction along with flattening of the bands with increasing fluence above the amorphization threshold.

Determination of optical properties of nitrogen-doped hydrogenated amorphous carbon films by spectroscopic ellipsometry

Nitrogen-doped hydrogenated amorphous carbon films have been deposited on silicon substrates by radio-frequency plasma-enhanced chemical vapor deposition using different N2/CH4 gas ratios from 0 to 3. The real and imaginary parts, n and k, of the complex index of refraction of these films have been determined for wavelengths between 300 and 830 nm by spectroscopic ellipsometry. Excellent agreement has been found between measured and modeled spectra, in which an empirical dielectric function based on classical Lorentz oscillator and Tauc joint density of states, and a linear void distribution along the thickness of the films have been assumed. Decrease in the optical energy gap and increase in the extinction coefficient, k, with increase in nitrogen concentration have been observed. Refractive index, n, increases rapidly with increase in nitrogen concentration up to 6.8 at. % (∼7.0 at. %) and then increases slowly with further increase in nitrogen concentration. For all the samples, n is found to be highest at the film-substrate interface which gradually decreases towards the film surface.