Gaussian Beam Approximation Research Articles

The Influence of PE Initial Field Construction Method on Radio Wave Propagation Loss and Tropospheric Duct Inversion

Parabolic equations (PE) are commonly employed for calculating the spatial propagation loss of wireless signals. The initial field is a crucial factor. To investigate the impact of construction accuracy on the calculation of radio wave propagation loss, we selected the half-wave dipole antenna and its Gaussian approximation to examine the influence of wide-angle PE modeling. We analyzed the disparities between the actual antenna pattern and the Gaussian beam approximation, as well as the discrepancies in the corresponding initial field and the calculation of radio wave propagation loss in PE modeling. The simulation results indicate that the error of the Gaussian approximation increases as the angle of departure from the antenna main beam increases, with a relative error of approximately 30% in the initial field. A comparison between the experimental test of the broadcast signal and the simulation calculation reveals that the model based on the actual antenna aligns more closely with the measured value on a flat underlying surface. However, in mountainous areas with significant fluctuations, the simulation results are consistent with each other and higher than the measured value. The inversion results obtained through the particle swarm optimization algorithm demonstrate that the model based on the actual antenna exhibits superior inversion accuracy for the tropospheric atmospheric duct structure.

Non-Hermitian Mode Cleaning in Periodically Modulated Multimode Fibers.

We show that the simultaneous modulation of the propagation constant and of the gain/loss coefficient along the graded index multimode fibers results in unidirectional coupling among the modes, which leads to either the enhancement or the reduction of the excitation of higher order transverse modes, depending on the modulation parameters. In the latter case, effective mode cleaning is predicted, ideally resulting in single-mode spatially coherent output. The effect is semi-analytically predicted on a simplified Gaussian beam approximation and numerically proven by solving the wave propagation equation introducing the non-Hermitian modulated potential.

Theoretical Analysis of Continuous-Wave Mid-Infrared Optical Vortex Source Generated by Singly Resonant Optical Parametric Oscillator

Due to the important application in the study of vibrational circular dichroism and helical dichroism of chiral molecules, the tunable vortex beam at mid-infrared region has attracted increasing attention. Based on orbital angular momentum (OAM) conservation in nonlinear interactions, the vortex pumped singly resonant optical parametric oscillator (SRO) is recognized as a versatile source of coherent vortex radiation providing high power and broad wavelength coverage from a single device. However, the low parametric gain and high oscillation threshold under continuous wave (cw) pumping has so far been the most challenging factor in generating cw tunable vortex beams. To predict the output characteristic of vortex pumped SRO, a theoretical model describing the vortex pumped SRO is needed. In this study, the theoretical model describing the vortex pumped SRO is set up under collimated Gaussian beam approximation. Output characteristics of different SROs are simulated numerically. By proper selection of pump scheme (such as double-pass pumping scheme), the vortex pumped mid-infrared SRO can oscillate at a relatively low pump power. By controlling the gain (mode overlap ratio between the pump and resonant wave in the nonlinear crystal) and loss (employing a spot-defect mirror with different defect size as the output coupler) of the resonant signal mode in the SRO, the OAM of the pump beam can be directionally transferred to a specific down converted beam. The transfer mechanism of the OAM among the pump light and the down-converted beams and factors affecting the transfer are studied. Our study provides the guidelines for the design and optimization of vortex pumped SRO under cw operation.

Wave focusing by submerged islands and gravitational analogues

We study water waves propagating over a smooth obstacle in a fluid of varying depth, motivated by the observation that submerged islands in the ocean act as effective lenses that increase the amplitude and destructive power of tsunami waves near focal points. We show that islands of substantial height (compared to the water depth) lead to strong focusing in their immediate vicinity, and generate caustics of either cusp or butterfly type. We highlight similarities and differences with focusing of (high-frequency) gravitational waves by a neutron star. In the linear regime, the comparison is made precise through an effective-spacetime description of the island-fluid system. This description is then put to practical use: We identify caustics by solving the Raychaudhuri equation (a transport equation) along rays of the effective metric. Next, the island-fluid scattering processes are examined in detail (i.e., deflection angle, phase shifts, scattering amplitudes) using numerical simulations and analytical techniques, including the eikonal approximation and its generalisation in the form of the Gaussian beam approximation. We show that the techniques capture the key features of the simulations. Finally, we extend the eikonal approximation to the dispersive regime, demonstrating that the essential features are robust in dispersive settings. This paves the way for future exploration in a controlled laboratory set-up.

Gravity Wave Focusing on the Antarctic Polar Vortex Using Gaussian Beam Approximation in Horizontally Nonuniform Flows

AbstractRay path theory is an asymptotic approximation to the wave equations. It represents efficiently gravity wave propagation in nonuniform background flows so that it is useful to develop schemes of gravity wave effects in general circulation models. One of the main limitations of ray path theory to be applied in realistic flows is in caustics where rays intersect and the ray solution has a singularity. Gaussian beam approximation is a higher-order asymptotic ray path approximation that considers neighboring rays to the central one, and thus, it is free of the singularities produced by caustics. A previous implementation of the Gaussian beam approximation assumes a horizontally uniform flow. In this work, we extend the Gaussian beam approximation to include horizontally nonuniform flows. Under these conditions, the wave packet can undergo horizontal wave refraction producing changes in the horizontal wavenumber, which affects the ray path as well as the ray tube cross-sectional area and so the wave amplitude via wave action conservation. As an evaluation of the Gaussian beam approximation in horizontally nonuniform flows, a series of proof-of-concept experiments is conducted comparing the approximation with the linear wave solution given by the WRF Model. A very good agreement in the wave field is found. An evaluation is conducted with conditions that mimic the Antarctic polar vortex and the orography of the southern flank of South America. The Gaussian beam approximation nicely reproduces the expected asymmetry of the wave field. A much stronger disturbance propagates toward higher latitudes (polar vortex) compared to lower latitudes.

Gaussian Beam Shaping and Multivariate Analysis in Plasmonic Sensing.

This work demonstrates a novel strategy to improve the sensing performance of a prism-coupled surface plasmon resonance system by Gaussian beam shaping and multivariate data analysis. The propagation of the beam along the optical system has been studied using the Gaussian beam approximation to design the incident beam such that the beam waist is aligned precisely and that stability is assured at the metal-dielectric interface. This renders a collimated incident beam, hence least angular dispersion, yielding a stronger and sharper plasmonic resonance. Moreover, we use the multivariate analysis method partial least squares that combines multiple features of the surface plasmon resonance curve and allows for a more precise analysis of the plasmonic response. Compared to univariate analysis, partial least squares improves typical sensing performance parameters remarkably. The combination of both aspects, beam shaping and multivariate analysis, overcomes current limitations of plasmonic detection systems. Thereby, we improve analytical sensitivity by a factor of 16, decrease the prediction error of the concentration of an unknown analyte by a factor of 11, and enhance resolution to the order of 5 × 10-7 RIU in angular interrogation.

Optimization of sum frequency output from intra-cavity sum frequency generation in singly resonant optical parametric oscillator

Theoretical investigation and optimization of output characteristics of intra-cavity sum frequency generation between pump and the signal waves in singly resonant optical parametric oscillators (IC-SFG-SROs) are presented. Under the collimated Gaussian beam approximation, simple models considering different spatial orderings of SRO and SFG processes of IC-SFG-SROs are set up. The calculations based on experimental parameters are in good agreement with experimental results. Output characteristics of IC-SFG-SROs operated at other wavelengths are optimized and predicted. For given experimental conditions (pump wavelength, pump power and SRO crystal length), the optimum length of SFG crystal and SF power are calculated, respectively. Furthermore, when a periodically poled crystal with a cascaded pattern is used as the SRO crystal and SFG crystal, the optimum SFG crystal length and the SF power are also calculated. These calculations and models provide guidelines for designing and optimizing continuous-wave tunable IC-SFG-SROs.

Asymptotic Properties of Linear Field Equations in Anti-de Sitter Space

We study the global dynamics of the wave equation, Maxwell’s equation and the linearized Bianchi equations on a fixed anti-de Sitter (AdS) background. Provided dissipative boundary conditions are imposed on the dynamical fields we prove uniform boundedness of the natural energy as well as both degenerate (near the AdS boundary) and non-degenerate integrated decay estimates. Remarkably, the non-degenerate estimates “lose a derivative”. We relate this loss to a trapping phenomenon near the AdS boundary, which itself originates from the properties of (approximately) gliding rays near the boundary. Using the Gaussian beam approximation we prove that non-degenerate energy decay without loss of derivatives does not hold. As a consequence of the non-degenerate integrated decay estimates, we also obtain pointwise-in-time decay estimates for the energy. Our paper provides the key estimates for a proof of the non-linear stability of the anti-de Sitter spacetime under dissipative boundary conditions. Finally, we contrast our results with the case of reflecting boundary conditions.

Gaussian beam sphere optics in condensed matter research

Transparent spheres in which the radius R of the sphere is larger than or equal to the wavelength λ of light have currently become of particular interest for quite a number of physical problems (2D colloidal crystal structures composed of transparent spheres, photonic crystals and meta-materials) and applications (laser lithography, fiber-optic systems, super resolution microscopy, biomedical applications, and others). The optical properties of a “full” sphere, when the light aperture r is of order R and the sphere is a natural spherical aberration, have not been investigated in ample detail yet. Strong spherical aberration makes focusing nontrivial. Usually, the exact solution of sphere optics is obtained using the Mie theory, the generalized Lorenz–Mie theory (GLMT), DFT, DDFT codes, which do not give much of a physical insight, as it requires summation of a large number of terms in a multipole expansion even for moderate sphere sizes. In this work we present an algorithm for describing the focusing properties of a transparent sphere using the Gaussian beam approximation that gives a good description of the field in the region of lens caustic. The algorithm developed for the spherical systems was used to create a code for calculating based on standard computation systems spheres with diameters ranging from ∼λ to thousands of λ. The results obtained were compared with the data of some authors obtained earlier by more sophisticated methods.

Quantum motion with trajectories: beyond the Gaussian beam approximation

A quantum model based on the Euler–Lagrange variational approach is proposed. Similarly to classical transport, our approach maintains the description of particle motion in terms of trajectories in a configuration space. Our method is designed to provide some corrections to the classical motion of nearly localized particles due to quantum phenomena. We describe the spatial localization of the particles by extending the usual ansatz which is at the basis of the Gaussian beam method. We discuss the completeness of our ansatz and the connection of our results with the Bohm trajectories approach. Our method finds application in the simulation of the motion of light nuclei in ab initio calculations.

Error estimates for Gaussian beam methods applied to symmetric strictly hyperbolic systems

In this work we construct Gaussian beam approximations to solutions of linear symmetric hyperbolic systems with highly oscillatory initial data, including both strictly and non-strictly hyperbolic systems as long as they are diagonalizable. The evolution equations for each Gaussian beam component are derived. Under some regularity assumptions of the data we obtain an error estimate between the exact solution and the first order Gaussian beam superposition in terms of the high frequency parameter ε−1. The main result is that the relative local error measured in energy norm in the beam approximation decays as ε12 independent of dimension and presence of caustics, for first order beams. This result is shown to be valid when the gradient of the initial phase is bounded away from zero. Applications to some non-strictly hyperbolic systems including both the acoustic equation and the system of Maxwell equations are discussed.

Quantifying the Effect of Horizontal Propagation of Three-Dimensional Mountain Waves on the Wave Momentum Flux Using Gaussian Beam Approximation

Abstract This work examines the influence of horizontal propagation of three-dimensional (3D) mountain waves on the wave momentum flux (WMF) within finite domains (e.g., the grid cell of general circulation models). Under the Wentzel–Kramers–Brillouin (WKB) approximation, analytical solutions are derived for hydrostatic nonrotating mountain waves using the Gaussian beam approximation (GBA), which incorporates both the wind vertical curvature effect and the height variation of stratification. The GBA solutions are validated against numerical simulations conducted using the Advanced Regional Prediction System (ARPS). In the situation of idealized terrain, wind, and stratification, the WMF obtained from the GBA shows a good agreement with the numerical simulation. The effect of wind curvature in enhancing the WMF is captured, although the WKB-based GBA solution tends to overestimate the WMF, especially at small Richardson numbers of order unity. For realistic terrain and/or atmospheric conditions, there are some biases between the WKB GBA and simulated WMFs, arising from, for example, the missing physics of wave reflection. Nonetheless, the decreasing trend of finite-domain WMF with height, because of the horizontal propagation of 3D mountain waves, can be represented fairly well. Using the GBA, a new scheme is proposed to parameterize the orographic gravity wave drag (OGWD) in numerical models. Comparison with the traditional OGWD parameterization scheme reveals that the GBA-based scheme tends to produce OGWD at higher altitudes, as the horizontal propagation of mountain waves can reduce the wave amplitude and thus inhibit wave breaking.

Collimated thermal radiation transfer via half Maxwell's fish-eye lens for thermophotovoltaics

Thermophotovoltaics (TPV) convert heat into electricity by capturing thermal radiation with a photovoltaic (PV) cell, ideally at efficiencies of 50% or more. However, excess heating of the PV cell from close proximity to the emitter substantially reduces the system efficiency. In this work, we theoretically develop and numerically demonstrate an approach to fundamentally improving TPV systems that allow for a much greater separation of an emitter and a receiver. Thus, we solve the excess heating dilemma, required for achieving theoretically high efficiencies. It consists of a spherically graded index lens known as Maxwell's Fish-Eye (MFE) structure, capable of collimating hemispherical emission into a much narrower range of angles, close to the normal direction. To fully characterize the power radiation profile of the MFE, we perform finite-difference time-domain simulations for a quarter MFE and then map it onto a Gaussian beam approximation. The modeled beam properties are subsequently used to study a half MFE. In an optimized half MFE design, 90% of all thermal photons reach a receiver at a distance of 100 λ; by comparison, only 15.6% of a blackbody emitter reach a receiver in the same geometry. It is also shown that the emission achieved by a half MFE can lead to a photon recycling rate above 95% for below bandgap photons at an emitter-receiver separation of 100 λ. By applying a half MFE, the absolute TPV efficiency can be improved from 5.74% to 37.15%, which represents a significant step forward in realizing high-efficiency TPV systems.

A Gaussian Beam Approximation Approach for Embedding Antennas Into Vector Parabolic Equation-Based Wireless Channel Propagation Models

Vector parabolic equation (VPE) methods have been widely applied to the modeling of radio-wave propagation in tunnel environments, offering high computational efficiency and fidelity. While the propagation environment can be discretized and represented in detail, the representation of radiating sources (such as transmitting antennas) requires the calculation, analytical if possible or numerical via another method such as ray-tracing (RT), of the fields that the sources generate on the initial plane of the VPE model. These initial conditions are necessary for subsequently applying VPE. However, the solutions offered so far compromise either the accuracy or the efficiency of VPE. For example, generating the initial conditions for VPE through RT adds significant computational overhead to the typically fast VPE solver. To address this significant limitation of VPE methods, we introduce a technique that allows one to directly embed antennas into a VPE mesh, via a Gaussian beam approximation of their radiated fields. Hence, the initial conditions for VPE are generated for practical antenna patterns, without invoking other techniques and with no compromise on the inherent efficiency of VPE. Concrete guidelines on how to choose parameters for Gaussian beams are provided. Numerical results are compared to experimental measurements in various tunnel scenarios, demonstrating the validity and usefulness of the technique.

Sobolev and max norm error estimates for Gaussian beam superpositions

This work is concerned with the accuracy of Gaussian beam superpositions, which are asymptotically valid high frequency solutions to linear hyperbolic partial differential equations and the Schr\"odinger equation. We derive Sobolev and max norms estimates for the difference between an exact solution and the corresponding Gaussian beam approximation, in terms of the short wavelength $\varepsilon$. The estimates are performed for the scalar wave equation and the Schr\"odinger equation. Our result demonstrates that a Gaussian beam superposition with $k$-th order beams converges to the exact solution as $O(\varepsilon^{k/2-s})$ in order $s$ Sobolev norms. This result is valid in any number of spatial dimensions and it is unaffected by the presence of caustics in the solution. In max norm, we show that away from caustics the convergence rate is $O(\varepsilon^{\lceil k/2\rceil})$ and away from the essential support of the solution, the convergence is spectral in $\varepsilon$. However, in the neighborhood of a caustic point we are only able to show the slower, and dimensional dependent, rate $O(\varepsilon^{(k-n)/2})$ in $n$ spatial dimensions.

Coupling of a laser diode to a quadric interface lensed monomode elliptic-core step index fiber: efficiency computation with the ABCD matrix

We report what to our knowledge is the first theoretical prediction of the coupling efficiency of a laser diode emitting two wavelengths either 1.3 or 1.5 μm, to three elliptic core step index fibers with different aspect ratios via quadric interface microlens of three different focal lengths on the fiber tip. Here, quadric interface microlens contains elliptical (hemispherical included), hyperbolic and parabolic microlenses. Instead of considering special ABCD matrix for individual microlens, a simple, accurate, and popular unified transfer ABCD matrix for refraction of quadric microlens under paraxial approximation is utilized to analyze the theoretical coupling efficiency, based on Gaussian beam approximation. The coupling efficiency can reach as high as 98.63 % with optimizing structure parameters of specific lensed fiber. The analysis can give some guidance for fabricating quadric interface lensed elliptic core fiber which are mainly useful in polarisation-maintaining fiber optic sensors and coherent fiber optic communication systems.

Characterisation of the energy of Gaussian beams on Lorentzian manifolds: with applications to black hole spacetimes

It is known that using the Gaussian beam approximation one can show that there exist solutions of the wave equation on a general globally hyperbolic Lorentzian manifold whose energy is localised along a given null geodesic for a finite, but arbitrarily long time. In this paper, we show that the energy of such a localised solution is determined by the energy of the underlying null geodesic. This result opens the door to various applications of Gaussian beams on Lorentzian manifolds that do not admit a globally timelike Killing vector field. In particular we show that trapping in the exterior of Kerr or at the horizon of an extremal Reissner-Nordstr\"om black hole necessarily leads to a `loss of derivative' in a local energy decay statement. We also demonstrate the obstruction formed by the red-shift effect at the event horizon of a Schwarzschild black hole to scattering constructions from the future (where the red-shift turns into a blue-shift): we construct solutions to the backwards problem whose energies grow exponentially for a finite, but arbitrarily long time. Finally, we give a simple mathematical realisation of the heuristics for the blue-shift effect near the Cauchy horizon of sub-extremal and extremal black holes: we construct a sequence of solutions to the wave equation whose initial energies are uniformly bounded, whereas the energy near the Cauchy horizon goes to infinity.

Semi-analytical model of arrayed waveguide grating in SOI using Gaussian beam approximation.

The arrayed waveguide grating structure can be used as an important component in high-speed CMOS optical interconnects in silicon-on-insulator (SOI) platform. However, the performance of such device is found to be extremely sensitive to the fabrication-related errors in defining the critical features. In the absence of an appropriate analytical model, one needs to rely on numerical computation to analyze the device characteristics and fabrication tolerances. Because compact design of such a device structure has foot-print ∼mm2 and the smallest features can be as small as ∼500 nm×220 nm (waveguide cross section), it demands a huge computational budget to optimize the design parameters. A semi-analytical model using Gaussian beam approximation of guided mode profiles has been developed to analyze the output spectrum of arrayed waveguide grating and to estimate the phase errors due to waveguide inhomogeneities. This model has been validated with existing numerical methods and published experimental results. It has been observed that a probabilistic waveguide width variations of ΔW∼5 nm can cause a cross-talk degradation of about 40 dB (25 dB) for a device (operating at λ∼1550 nm) fabricated on SOI substrate with 220 nm (2 μm) device layer thickness.

Modulation of the interstitial fluid pressure by high intensity focused ultrasound as a way to alter local fluid and solute movement: insights from a mathematical model

High intensity focused ultrasound (HIFU) operated in thermal mode has been reported to reduce interstitial fluid pressure and improve the penetration of large macromolecules and nanoparticles in tumor and normal tissue. Little is understood about how the interstitial fluid pressure and velocity as well as the interstitial macromolecule transport are affected by HIFU exposure. A mathematical model is presented here which sheds light on the initial biophysical changes brought about HIFU. Our continuum model treats tissue as an effective poro-elastic material that reacts to elevated temperatures with a rapid drop in interstitial elastic modulus. Using parameters from the literature, the model is extrapolated to derive information on the effect in tumors, and to predict its impact on the convective and diffusive transport of macromolecular drugs. The model is first solved using an analytical approximation with step-wise changes at each boundary, and then solved numerically starting from a Gaussian beam approximation of the ultrasound treatment. Our results indicate that HIFU causes a rapid drop in interstitial fluid pressure that may be exploited to facilitate convection of macromolecules from vasculature to the exposed region. However, following a short recovery period in which the interstitial fluid pressure is normalized, transport returns to normal and the advantages disappear over time. The results indicate that this effect is strongest for the delivery of large molecules and nanoparticles that are in the circulation at the time of treatment. The model may be easily applied to more complex situations involving effects on vascular permeability and diffusion.

Optimal Error Estimates for First-Order Gaussian Beam Approximations to the Schrödinger Equation

Gaussian beams are generally local asymptotic solutions to the linear wave equations in the high-frequency regime. Each Gaussian beam is concentrated around a specific ray path determined by the underlying Hamiltonian system. Expressed as some superposition of Gaussian beams, Gaussian beam approximation is expected to be a high-frequency asymptotic solution which remains globally valid even around caustics. We derive optimal first-order error estimates for first-order Gaussian beam approximations, in both continuous and discrete levels, to the Schrodinger equation equipped with a WKB initial data. Our error estimates are valid for any spatial dimension and unaffected by the presence of caustics. Some numerical tests are presented to validate the theoretical results.