Showing papers on "Dispersion relation published in 1989"

Optical dispersion relations for GaP, GaAs, GaSb, InP, InAs, InSb, AlxGa1−xAs, and In1−xGaxAsyP1−y

Abstract: A method is described for calculation of the optical constants (the refractive index, extinction coefficient, and absorption coefficient) of some III‐V binaries (GaP, GaAs, GaSb, InP, InAs, and InSb), ternaries (AlxGa1−xAs), and quaternaries (In1−xGaxAsyP1−y) in the entire range of photon energies (0–6.0 eV). The imaginary part of the dielectric function [e2(ω)] is derived first from the joint density‐of‐states functions at energies of various critical points (CPs) in the Brillouin zone; then its real part [e1(ω)] is obtained analytically using the Kramers–Kronig relation. The indirect band‐gap transitions are also assumed to provide a gradually increasing e2 spectrum expressed by a power law of (ℏω−EIDg)2, where ℏω is the photon energy and EIDg is the indirect band‐gap energy. The optical dispersion relations are expressed in terms of these model dielectric functions. The present model reveals distinct structures in the optical constants at energies of the E0, E0+Δ0 [three‐dimensional (3‐D) M0 CP], E1, E...

Surface waves on vertically sheared flows: Approximate dispersion relations

Abstract: Assuming linear wave theory for waves riding on a weak current of 0(e) compared to the wave phase speed, an approximate dispersion relation is developed to 0(e2) for arbitrary current U(z) in water of finite depth. The 0(e2) approximation is shown to be a significant improvement over the 0(e) result, in comparison with numerical and analytic results for the full problem. Various current profiles in the full range of water depths are considered. Comments on approximate action conservation and application to depth-averaged wave models are included.

Oscillations and Waves: in Linear and Nonlinear Systems

Abstract: One Oscillations and Waves in Linear Systems- 1 Linear Oscillators- 11 General Notes- 12 Two Examples The Phase Plane Diagram of an Oscillator- 13 Resonance The Effect of an Aperiodic External Force on an Oscillator- 14 Normal Oscillations Analogy with Quantum Mechanics Production and Extinction Operators- 2 Oscillations in a System with Two Linked Oscillators- 21 Initial Equations- 22 The Fundamental Oscillations of Two Linked Oscillators- 23 Disturbance of Two Linked Oscillators by an External Force The Reciprocity Principle- 3 Oscillations in an Ensemble of Non-Interacting Oscillators- 31 Classical Theory of Dispersion- 32 Oscillations in an Ensemble of Dissimilar Noninteracting Oscillators with a Given Distribution Function- 4 Oscillations in Ordered Structures Limit for a Continuous medium Waves Dispersion- 41 General Remarks- 42 Oscillations in Ordered Structures (Chains of Linked Particles and Identical Linked Oscillators)- 43 Limiting Transition from an Ordered Structure to a One-dimensional Medium Temporal and Spatial Dispersion Physical Nature of Dispersion- 44 Typical Dispersion Characteristics for Medium Models- 45 Formal Method for Obtaining the Dispersion Equation Waves in a One-Dimensional Resonator Resonance in Wave Systems- 5 Properties of Waves with Small Amplitudes in Continuous media- 51 General Remarks- 52 Equations of Hydrodynamics Dispersion for Sound Waves For Sound Waves- 53 A Stratified Fluid Sound in an Ocean- 54 Gravity Waves in an Incompressible Liquid Internal Waves Rossby Waves- 55 Waves in a Superfluid Liquid- 56 Waves in a Plasma Hydrodynamic Description- 6 Stability and Instability of Linear Systems with Discrete Spectra- 61 General Notes and Definitions- 62 The Raus-Gurvits Criterion and Three-Dimensional Systems- 63 The D-Partition Method- 64 Stability of Non-Autonomous Systems- 65 Instability Mechanisms- 7 Stability of Distributed Systems with Continuous Spectra- 71 General Comments- 72 Examples of Instability- 73 Absolute and Convective Instability The Characteristics Method- 74 Waves in Flows Electron Beams Helmholtz Instability- 75 Amplification and Filtering Separation Criteria- 8 Propagation Velocity of Waves- 81 Various Introductions to the Concept of Group Velocity- 82 Group Velocity of Waves in Some Continuous Media- 9 Energy and Momentum of Waves- 91 Equation for the Transport of the Average Energy Density by Wave Packets in Dispersing Media- 92 Density of the Energy of an Electromagnetic Wave in a Medium with Dispersion- 93 Momentum of a Wave Packet- 10 Waves with Negative Energy Linked Waves- 101 General Notes- 102 Waves with Positive and Negative Energies- 103 Coupled Waves Synchronicity Normal and Anomalous Doppler Effects- 11 Parametric Systems and Parametric Instability- 111 General Comments- 112 Parametric Resonance Floquet's (Blokh's) Theorem Mathieu's Equation- 113 Waves in Periodic Structures The Mathieu Zone and the Brillouin Diagram- 114 Motion in a Rapidly Oscillating Field Kapitsa's Pendulum Free Electron Lasers- 12 Adiabatic Invariants Propagation of Waves in Inhomogeneous Media- 121 The Wentsel-Kramers-Brillouin (VCB) Approximation and Adiabatic Invariants- 122 Equivalence Between a Rotor and an Oscillator- 123 Propagation of Waves in Inhomogeneous Media The Approximation of Geometric Optics- 124 The Propagation of Waves in a Plane-Layer Medium in the Geometric Optics Approximation- 125 Linear Wave Interaction in an Inhomogeneous Medium- Two Oscillations and Waves in Nonlinear Systems- 13 The Nonlinear Oscillator- 131 Initial remarks- 132 Qualitative and Analytical Description Examples of Nonlinear Systems- 133 Nonlinear Resonance- 134 Overlap between Nonlinear Resonances- 14 Periodic Self-Excited Oscillations- 141 Definitions- 142 The Van der Pol Generator Self-Excited Oscillations as a Function of System Parameters- 143 Relaxational Self-Excited Oscillations Fast and Slow Motions- 15 General Properties of Nonlinear Dynamic Systems in Phase Space- 151 Basic Types of Trajectory The Fundamentals of Dynamic Systems (Structural Stability)- 152 Basic Bifurcations on a Plane Poincare Indices- 153 Point Transformations- 154 Bifurcation of Periodic Motions- 155 Homoclinic Structures- 16 Self-Excited Oscillations in Multifrequency Systems- 161 Forced Synchronization- 162 Competition- 163 Mutual Mode Synchronization- 17 Resonance Interactions between Oscillators- 171 Interaction Between Three Coupled Oscillators in a System with Quadratic Nonlinearity- 172 Resonance Interactions Between Waves in Weakly Nonlinear Media with Dispersion- 173 Explosive Instability- 18 Simple Waves and the Formation of Discontinuities- 181 Kinematic Waves- 182 Travelling Waves in a Nonlinear Medium Without Dispersion- 183 Determining the Discontinuity Coordinates- 184 Weak Shock Waves Boundary Conditions at a Discontinuity- 19 Stationary Shock Waves and Solitons- 191 Structure of a Discontinuity- 192 Solitary Waves - Solitons- 193 Solitons as Particles- 194 Higher-Dimensional Solitons- 20 Modulated Waves in Nonlinear Media- 201 General Remarks- 202 Self-Modulation Reversibility- 203 Self-Focusing- 20 4 Interaction Between Wave Beams and Packets- 205 Interactions Between Waves Having Randomly Modulated Phases Wave Kinetics- 21 Self-Excited Oscillations in Distributed Systems- 211 General Remarks- 212 Medium Without Dispersion Discontinuous Waves- 213 Stationary Waves- 214 The Existence and Role of Limiting Cycles- 215 Competition Between Stationary Waves in an Active Medium- 216 Periodic Self-Excited Oscillations in Hydrodynamic Flows- 22 Stochastic Dynamics in Simple Systems- 221 How Randomness Appears in a Dynamic System- 222 The Stochastic Dynamics of One-Dimensional Mappings- 223 Noise Generator Qualitative Description and Experiment- 224 Statistical Description of a Simple Noise Generator- 225 Ways in which Strange Attractors Arise- 226 Dimensionality of Stochastic Sets- 23 The Onset of Turbulence- 231 General Remarks- 232 The Occurrence of Stochastic Self-Excited Oscillations in Experimental Fluid Mechanics- 233 Stochastic Modulation- 234 Ideal Flow and Turbulence- 24 Self-Organization- 241 Main Phenomena, Models, and Mathematical Forms- 242 Travelling Pulsations- 243 Spiral and Cylindrical Waves Travelling Centers- 244 Concerning Self-Organization Mechanisms- References

Leaky Lamb waves in an anisotropic plate. I: An exact solution and experiments

Abstract: The propagation of leaky Lamb waves in a plate consisting of a general balanced symmetric composite material is considered. The problem has been examined both analytically as well as experimentally. An exact solution for the dispersion equation was obtained. Numerical results for complex‐valued wavenumber were obtained for an isotropic material (aluminum) and a (0/903)s graphite/epoxy laminate. Excellent agreement for the isotropic case and a satisfactory agreement for the anisotropic case between the theory and experiment were observed.

Sensitivity of Near-Surface Shear-Wave Velocity Determination From Rayleigh And Love Waves

Abstract: Rayleigh and Love waves recorded on seismic-shot gathers can be used to determine the thickness and shear-wave velocity of shallow subsurface layers. After the data are transformed into the k-f domain, the dispersion curve for each of the phases can be picked from maxima on the contour plot. This dispersion curve is then inverted for the velocities and depths. Different frequencies in the dispersion curve yield information about different depths. The fundamental mode has proven to be of greater use than higher modes. Both Rayleigh and Love waves are easily inverted. However, the Love waves seem to yield information in a lower portion of the spectrum than the Rayleigh modes. Three examples are given from field experiments conducted near Canton, Texas.

Surface-plasmon dispersion in simple metals.

Abstract: Angle-resolved reflection inelastic electron scattering has been used to measure the dispersion of the surface plasmon for thick films of Na and K. The measured dispersion is negative at small momentum parallel to the surface as predicted by quantum-mechanical calculations of the dynamic response of the electrons at the surface of an interacting electron gas. A simple physical picture is presented to explain the origin of the negative surface-plasmon dispersion in simple metals.

Stoneley-wave attenuation and dispersion in permeable formations

Abstract: The tube wave, or low‐frequency manifestation of the Stoneley wave, has been modeled previously using the quasi‐static approximation; I extend this method to include the effect of the formation matrix compressibility, which tends to marginally increase the tube‐wave attenuation. Using the Biot theory of poroelasticity, I develop a fully dynamic description of the Stoneley wave. The dispersion relation derived from Biot’s equations reduces in the low‐frequency limit to the quasi‐static dispersion relation. Comparisons of the quasi‐static and dynamic theories for typical sandstones indicate the former to be a good approximation to at least 1 kHz for oil and water infiltration. At higher frequencies, usually between 5 and 20 kHz for the formations considered, a maximum in the Stoneley Q is predicted by the dynamic theory. This phenomenon cannot be explained by the quasi‐static approximation, which predicts a constantly increasing Q with frequency. Instead, the peak in Q may be understood as a transition from...

Nondestructive evaluation of adhesive joints by guided waves

Abstract: Guided waves in an adhesive layer between two adherend half‐spaces are shown to be uniquely sensitive to most types of bond defects. A simple experimental technique based on ultrasonic transmission mesurement is introduced to detect leaky guided modes in the interface layer. Experimental results for these dispersive guided modes are shown to be in good quantitative agreement with theoretical predictions. The suggested technique might find numerous applications in nondestructive evaluation of different bonds of layered structure such as adhesive and brazed joints.

Beyond the two-fluid model: Transition from linear behavior to a velocity-independent force on a moving object in 3He-B.

Abstract: Using a simple one-dimensional model, we show that the existence of the energy gap for excitations in an isotropic BCS superfluid leads to strongly nonlinear mechanical behavior of the liquid in the ballistic quasiparticle limit. The nonlinear damping of a vibrating wire in $^{3}\mathit{B}$ below 200 \ensuremath{\mu}K is explained, both in its velocity dependence and magnitude. At modest velocities (\ensuremath{\upsilon}gkT/${\mathrm{p}}_{\mathrm{F}}$), the damping force on an object moving through the superfluid becomes independent of velocity, an unexpected result with several interesting implications.

Tube diameter and wave frequency limitations when using the electro magnetic surface wave in the m=1 (dipolar) mode to sustain a plasma column

Abstract: An exhaustive experimental investigation of the conditions required to sustain a plasma column through the propagation of the m=1 mode surface wave has been conducted. It reveals that, given a discharge tube radius a, there corresponds a minimum frequency value fm below which the discharge cannot be achieved; conversely, for a given operating frequency f, the tube radius must exceed some minimum value am for the plasma to be sustained. These minimum conditions required to obtain the discharge are observed to obey a scaling law of the form (fa)m≂const., where the constant is independent of the gas nature and pressure. Theoretically, the dispersion equation of the m=1 mode wave shows no low‐frequency cutoff. However, it is found that the specific dependence of the wave attenuation coefficient on the frequency and on the tube diameter can ultimately account for the observed limitations when the wave is used to sustain a plasma. A discharge stability criterion is proposed that recovers the observed scaling la...

Internal kink modes in the ion‐kinetic regime

Abstract: The m=1 kink mode is investigated in the high temperature regime where the width of the singular layer is determined by the mean ion gyroradius. This regime is reached in a number of present‐day fusion experiments with strong auxiliary heating. A dispersion relation that contains the full kinetic response of the ions is derived and analyzed. The growth rates are larger than the corresponding ones obtained from fluid theory. Diamagnetic stabilization is weaker than in the fluid case. Ion temperature gradients are shown to be stabilizing at low values of the diamagnetic frequency and destabilizing at large values.

Optical dispersion relations for Si and Ge

Abstract: A method is described for calculation of the optical constants (the refractive index, extinction coefficient, absorption coefficient, and normal‐incidence reflectivity) of Si and Ge in the entire range of photon energies (0–0.6 eV). The imaginary part of the dielectric function [e2(ω)] is derived first from the joint‐density‐of‐states functions at energies of various critical points (CPs) in the Brillouin zone; then its real part [e1(ω)] is obtained analytically using the Kramers–Kronig relation. The indirect‐band‐gap transitions are also assumed to provide a gradually increasing absorption spectrum expressed by a power law of (ℏω−EIDg)2, where ℏω is the photon energy and EIDg is the indirect‐band‐gap energy. The optical dispersion relations are expressed in terms of these model dielectric functions. The present model reveals distinct structures in the optical data at energies of the E0, E0+Δ0 [three‐dimensional (3D) M0 critical point (CP)], E1, E1+Δ1 [3D M1 or two‐dimensional (2D) M0 CP], E2 [a mixture of damped harmonic oscillator (DHO) and 2D M2 CP], E’0 triplet (DHO), and E1 (DHO). Calculated optical spectra are in satisfactory agreement with the experimental data over a wide range of the photon energies.

Collective excitation spectra of one-dimensional electron systems

Abstract: We calculate, within the random-phase approximation, the elementary excitation spectrum of quasi-one-dimensional electron systems as occurring, for example, in semiconductor microstructures. Using multisubband models, we derive and discuss the dispersion relations for both intrasubband and intersubband excitations and consider the mode-coupling effect between them. We show that the depolarization shift correction for the intersubband excitation could be very large, increasing the intersubband collective mode energy substantially above the single-particle intersubband separation, and, thus explaining a puzzling recent far-infrared spectroscopic experimental observation.

Optical phonon modes in a double heterostructure of polar crystals

Abstract: The equation of motion for the polarization vector for a double heterostructure of polar crystals is solved exactly within the framework of the continuum model. There exist only two types of phonon modes, the interface modes and the confined bulk modes, whose eigenvectors are obtained explicitly. Dispersion relations are derived analytically for the interface modes, while the confined bulk modes are dispersionless, a fact consistent with the model. It is also found that in Raman scattering experiments the symmetric interface modes are predominantly longitudinal optical (LO) and the antisymmetric interface modes transverse optical (TO). In the central region of the Brillouin zone, however, they both split into two branches oscillating at LO and TO frequencies, respectively. Possible reinterpretation of various experiments is briefly discussed.

Guided‐wave optical wavelength demultiplexer using an asymmetric Y junction

Abstract: An optical wavelength demultiplexer using an asymmetric Y junction is numerically analyzed and the experimental results are reported. This demultiplexer utilizes both the mode splitting characteristic of an asymmetric Y junction and the waveguide dispersion of channel waveguides. Analysis shows that the device with step‐index profile has more than 20 dB isolation for wavelengths with 6.5% separation from the center wavelength. The fabricated device composed of two‐step ion exchanged soda‐lime glass waveguides coated asymmetrically with Corning 7059 glass thin film separated the lights of 0.63 and 0.84 μm wavelengths.

Neutrinos in a medium.

Abstract: The method of covariant field theory at finite temperature is applied to the propagation of neutrinos in a medium. The implications due to the discrete space-time symmetries on the dispersion relations are derived. In the calculation of rates for processes involving neutrinos, the neutrino wave functions that must be used differ from those in a vacuum by some normalization factors that depend on the medium. The calculation of these normalization factors is described and general formulas are obtained in terms of the neutrino self-energy in the medium. As an example of the formalism the Wolfenstein formula for the index of refraction of a neutrino propagating through a gas of electrons is rederived. Also, the dispersion relations and wave functions are calculated for the case of a neutrino background, in the presence of new interactions mediated by a light scalar.

Molecular cloud formation by gravitational instabilities in a clumpy interstellar medium

Abstract: A dispersion relation is derived for gravitational instabilities in a medium with cloud collisional cooling, using a time-dependent energy equation instead of the adiabatic equation of state. The instability extends to much smaller length scales than in the conventional Jeans analysis, and, in regions temporarily without cloud stirring, it has a large growth rate down to the cloud collision mean free path. These results suggests that gravitational instabilities in a variety of environments, such as galactic density wave shocks, swept-up shells, and extended, quiescent regions of the interstellar medium, can form molecular clouds with masses much less than the conventional Jeans mass, e.g., from 100 to 10 million solar masses for the ambient medium, and they can do this even when the unperturbed velocity dispersion remains high. Similar processes operating inside existing clouds might promote gravitationally driven fragmentation. 29 refs.

Kinetic-theory approach to quark-gluon plasma oscillations

Abstract: The difficulties in the unique definition of oscillations of a plasma with a non-Abelian interaction are considered. The recently formulated kinetic theory of the quark-gluon plasma in the semiclassical limit is presented and discussed with particular attention to the gluon sector of the theory. The transport equations are linearized around the global equilibrium and the chromoelectric permeability tensor is found. The dispersion relations of the plasma oscillations are discussed and the rate of oscillation damping is estimated.

Local kinetic stability analysis of the ion temperature gradient mode

Abstract: A local dispersion relation is derived for the toroidal ion temperature gradient mode, retaining full drift resonance effects but assuming small ion Larmor radius and fluid-like parallel ion response. The dispersion relation is analysed for marginal stability, in the limit of flat density profiles, to obtain the critical ion temperature gradient length. The relation between these results and anomalous transport in tokamaks in the L-and H-modes is discussed

Drop oscillations in liquid-liquid systems

Abstract: When the ratio of the drop radius to the distance separating any two drops and the relative importance of gravitational to surface forces are both small, the small amplitude oscillations of a drop of one viscous fluid immersed in another fluid are governed by the nonlinear dispersion relation derived by Miller and Scriven. The dispersion relation has been solved numerically to determine the character of oscillations for arbitrary values of drop size, physical properties of the two fluids, and interfacial tension. The new theoretical results determine the range of validity of the low-viscosity approximation, and are also shown to be essential for proper interpretation of many previously reported experimental results. New experimental measurements of natural frequencies of oscillation of water drops falling in 2-ethyl-1-hexanol, a system having properties characteristic of many others in solvent extraction, agree well with the theoretical predictions when drop radius is smaller than a critical size.

A photon accelerator

Abstract: A novel method of frequency upshifting short ({le} 100{mu}s) pulses of laser light, which makes use of relativistic plasma waves, is described. As is well known, photons in a plasma can be thought of as particles possessing and effective mass of m{sub {gamma}} = {Dirac h}{omega}{sub pe}/c{sup 2} and moving with the group velocity of the laser pocket. It is also known that these photons experience a force acting on them when in the presence of a gradient in the plasma density. By using a relativistic plasma wave traveling with the photons, the energy of the photons (thus the frequency) can be continuously increased. Similarities with another laser-plasma interaction, Raman scattering, are discussed. The physical mechanism and results of particle-in-cell computer simulations will be described. 9 refs., 3 figs.

Analytical study of the relativistic dispersion: application to the generation of the auroral kilometric radiation

Abstract: The measurements recently performed by the Viking spacecraft have shown that, in addition to being cold plasma depleted, the source regions of the Auroral Kilometric Radiation (A.K.R.) are characterized by a relatively denser, more energetic electron component (a few perticle by cm−3, 〈E〉 ≈ 5 KeV). In order to properly study the Cyclotron Maser Instability (C.M.I.) which is thought to be responsible for the A.K.R. generation, it is thus necessary to include relativistic corrections in both the hermitian and the antihermitian parts of the dielectric tensor characterizing the linear properties of the plasma. Here one presents an analytical study of the corresponding dispersion equation which aims to describe stable and unstable waves having frequencies lying very close to the electronic gyrofrequency and propagating across the geomagnetic field with a perpendicular refractive index less than a few units (n⊥<5). The electronic population of the auroral plasma is supposed to have two components. The distribution function of the non-thermal, energetic component is modelled by a Dory-Guest-Harris (DGH) distribution function and those of the tenuous thermal one by either a Dirac's function or a maxwellian. The dispersion equation thus depends on four parameters: i) the normalized peak energy of the DGH distribution: δ = 1/2 (v0/c)² which is a small number and scales both the temporal growth rate and the bandwidth of the CMI. ii) a parameter characterising the density of the energetic population: P = 1/δ(ωph/ωc)²(ωph being the plasma frequency of the energetic component). iii) the proportion of thermal plasma : χ = nc/nh where nc and nh are the density of the thermal and the hot plasma components respectively. iv) the ratio τ between the peak energy of the DGH and the mean energy of the maxwellian population. Our calculations allow us to simply perform a parametric study of the various regimes presented by the CMI. For the range of parameters inferred from measurements of the Swedish spacecraft Viking: (P>1 and χ small), the growth rate could maximize at the cut-off frequency of the relativistic X mode. It can be quite large: Im(ω)max ≈ δωc (10−2 ωc for electrons of 5 Kev). Moreover, for small χ, the relativistic X mode is connected to freely propagating modes which guarantees an easy access of the electromagnetic energy to free space.

Electrostatic wave generation above and below the plasma frequency by electron beams

Abstract: The unmagnetized electrostatic dispersion equation is solved numerically to study the growth of electrostatic waves near the plasma frequency resulting from a unstable electron beam. An attempt is made to find the dispersion relations, frequencies of maximum growth, and the resonant or nonresonant character of the waves for particular beam parameters. It is found that the unstable waves do not have the Langmuir dispersion relation except in the limit of a very dilute beam with growth on the connected mode of O'Neil and Malmberg. The properties of the unstable mode depend strongly on beam parameters such as beam density, speed, and temperature.

Nonlinear guided waves in multilayer systems

Abstract: A nonlinear matrix formalism that allows the calculation of the optical field within a nonlinear multilayer system is presented. Taking advantage of this technique, the field profiles are calculated and the dispersion relation of nonlinear guided waves in such configurations is derived. The numerical results reveal that several field profiles occur, each attached to a certain unit cell of the multilayer system. The various field profiles apply to the corresponding branches of rich structured nonlinear dispersion curves. A comparison of the results of those yielded by an effective medium model is carried out. Appreciable differences are found and are discussed in detail. >

Relaxation process in deeply supercooled water by Mandelstam-Brillouin scattering

Abstract: Polarized Mandelstam-Brillouin scattering data in bulk supercooled water are presented. The hypersonic velocity and absorption are measured as a function of the scattering angle {theta} at temperatures ranging from +20 to {minus}27.4{degree}C. The experimental results indicate the existence of a relaxation process in the gigahertz region. The authors discuss the observed dispersion in the frame of the relaxation theory for moderately supercooled liquids. The obtained relaxation times and relaxation strengths are interpreted in terms of man-body effects involved to explain the behavior of supercooled water.

Semiconductor quantum wells as electron wave slab waveguides

Abstract: A quantum well in a semiconductor can act as a slab waveguide for electron waves in a manner analogous to the way a layered dielectric can act as a slab waveguide for electromagnetic waves (e.g., as commonly employed in integrated optics). In this work, the case of a general electron asymmetric slab waveguide (a quantum well comprised of three materials each with a different potential energy and a different effective mass) is analyzed and the conditions for electron waveguiding are quantified. Electron waveguide modes exist for electron energies in the well and for electron energies above one or both of the potential energy barriers. Furthermore, due to dispersion, each electron waveguide mode has an upper‐energy cutoff as well as a lower‐energy cutoff. This is in contrast to electromagnetic guided modes which typically have only lower‐energy (low‐frequency) cutoffs. At the upper‐energy cutoff the electron wave is refracted into the substrate and/or cover. An example quantum well waveguide consisting of Ga0.80 Al0.20 As (substrate), GaAs (film), Ga0.55 Al0.45 As (cover) is analyzed. This structure is a single‐mode electron waveguide for GaAs thicknesses of from 5 (1.413 nm) to 26 monolayers (7.349 nm).