Complex Transmission Coefficients Research Articles

Attenuation and dispersion of Rayleigh waves propagating on a cracked surface: an effective field approach

A new effective field approach describing the attenuation and dispersion of a Rayleigh wave propagating on a surface containing a distribution of one-dimensional, surface-breaking cracks is presented. Limited by the validity of the independent scattering approximation, the model utilizes the complex transmission coefficient of a single crack to build expressions for the phase velocity and attenuation coefficient of an effective Rayleigh wave. The model is shown to be able to accommodate the effect of compressive residual stresses that tend to close the distributed cracks at their mouth, and therefore substantially reduce the attenuation and the velocity change caused by an equivalent distribution of open cracks. Wherever possible, the predictions of the new model are compared with those of other approaches and critical remarks are offered which discuss the advantages of the effective field approach over the others. Finally, an extension of this approach to distributions of two-dimensional surface-breaking cracks is outlined.

Microwave properties of metallic nanowires

We report on the microwave properties of arrays of parallel magnetic nanowires constituted of nickel, cobalt, or Ni/Fe alloy embedded in nanoporous track-etched polymer membranes. The experiments consist of transmission measurements carried out on microwave stripline structures using a magnetically loaded membrane as the substrate. Measurements were performed at frequencies ranging from 100 MHz to 40 GHz and under static magnetic fields up to 5.6 kOe applied along the wires axis. Resonance phenomena have been observed in the magnitude of the complex transmission coefficient at frequencies which depend on the nature of the material and applied static magnetic field. Results are consistent with those expected for a ferromagnetic resonance (FMR) experiment and the observed behaviors are analyzed in the framework of the classical FMR theory.

Design of semiconductor heterostructures with preset electron reflectance by inverse scattering techniques

We present the application of the inverse scattering method to the design of semiconductor heterostructures having a preset dependence of the (conduction) electrons’ reflectance on the energy. The electron dynamics are described by either the effective mass Schrödinger equation or by the (variable mass) BenDaniel and Duke equations. The problem of phase (re)construction for the complex transmission and reflection coefficients is solved by a combination of Padé approximation techniques, obtaining reference solutions with simple analytic properties. Reflectance-preserving transformations allow bound state and reflection resonance management. The inverse scattering problem for the Schrödinger equation is solved using an algebraic approach due to Sabatier. This solution can be mapped unitarily onto a family of BenDaniel and Duke type equations. The boundary value problem for the nonlinear equation which determines the mapping is discussed in some detail. The chemical concentration profile of heterostructures whose self consistent potential yields the desired reflectance is solved completely in the case of Schrödinger dynamics and approximately for BenDaniel and Duke dynamics. The Appendix contains a brief digest of results from the scattering and inverse scattering theory for the one-dimensional Schrödinger equation which is used in the article.

Precision of quasi-optical null-balanced bridge techniques for transmission and reflection coefficient measurements

The precision of quasioptical null-balanced bridge instruments for transmission and reflection coefficient measurements at millimeter and submillimeter wavelengths is analyzed. A Jones matrix analysis is used to describe the amount of power reaching the detector as a function of grid angle orientation, sample transmittance/reflectance and phase delay. An analysis is performed of the errors involved in determining the complex transmission and reflection coefficient after taking into account the quantization error in the grid angle and micrometer readings, the transmission or reflection coefficient of the sample, the noise equivalent power of the detector, the source power and the post-detection bandwidth. For a system fitted with a rotating grid with resolution of 0.017 rad and a micrometer quantization error of 1 μm, a 1 mW source, and a detector with a noise equivalent power 5×10−9 W Hz−1/2, the maximum errors at an amplitude transmission or reflection coefficient of 0.5 are below ±0.025.

Theory of phase-sensitive measurement of photon-assisted tunneling through a quantum dot

Recent double-slit interference experiments have demonstrated the possibility of probing the phase of the complex transmission coefficient of a quantum dot via the Aharonov-Bohm effect. We propose an extension of these experiments: an ac voltage imposed on the side gate with the concomitant photonic sidebands leads to additional structure both in the amplitude and in the phase of the Aharonov-Bohm signal. Observation of these effects would be a definitive proof of coherent absorption and reemission of photons from the ac source.

Measurement of resonant frequency and quality factor of microwave resonators: Comparison of methods

Precise microwave measurements of sample conductivity, dielectric, and magnetic properties are routinely performed with cavity perturbation measurements. These methods require the accurate determination of quality factor and resonant frequency of microwave resonators. Seven different methods to determine the resonant frequency and quality factor from complex transmission coefficient data are discussed and compared to find which is most accurate and precise when tested using identical data. We find that the nonlinear least-squares fit to the phase versus frequency is the most accurate and precise when the signal-to-noise ratio is greater than 65. For noisier data, the nonlinear least-squares fit to a Lorentzian curve is more accurate and precise. The results are general and can be applied to the analysis of many kinds of resonant phenomena.

Applications of the causality condition to acoustic scattering

The causality condition states that the response of a passive system cannot precede the cause. Under certain conditions, the causality condition leads to a Hilbert transform relation between the magnitude and phase of the complex Fourier transform of a system’s response. This relation has profound implications for those attempting to design passive structures whose desired scattering characteristics are expressed in the frequency domain. Unless the causality condition is satisfied in the frequency domain, the structure is not physically realizable. In this presentation, some novel applications of this relation are developed for a one-dimensional fluid-loaded structure which scatters incident sound in the backward and forward directions. In each application the reflection and transmission coefficients, which are the complex Fourier transforms of the reflected and transmitted pressures due to an impulsive incident pressure wave, are subject to the causality condition. The Weiner–Lee transform, which is derived from the Hilbert transform but is more easily implemented numerically, is used to find complex reflection and transmission coefficients given only their frequency-dependent magnitudes. By using this informa- tion and structural reciprocity, one can find an impedance matrix of a structure which scatters sound in a specified way.

A broadband free-space dielectric properties measurement system at millimeter wavelengths

A free-space system operating in the 75 GHz to 95 GHz range for measurement of the complex permittivity of matter is presented. The complex transmission coefficient of a parallel polarized quasiplane wave through a planar sample is determined for different incidence angles. The quasiplane wave is realized by a collimated Gaussian beam. The complex permittivity is subsequently calculated from the measured data by applying a nonlinear least-squares method. Errors induced by multiple reflections between the transmitting and receiving horn antennas are removed by a time-domain gating technique. The design of the quasioptical measurement setup is based on Gaussian beam optics. A dielectric lens collimates the beam of the transmitting horn antenna for minimization of diffraction effects at the edges of the sample. In order to verify the Gaussian beam and thin lens approximations at millimeter wavelengths, the far-field pattern of the horn antenna and the dielectric lens system has been measured. The performance of the measurement setup has been tested using glass samples.

Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures.

We derive an exact expression for the electromagnetic mode density, and hence the group velocity, for a finite, N-period, one-dimensional, photonic band-gap structure. We begin by deriving a general formula for the mode density in terms of the complex transmission coefficient of an arbitrary index profile. Then we develop a specific formula that gives the N-period mode density in terms of the complex transmission coefficient of the unit cell. The special cases of mode-density enhancement and suppression at the photonic band edge and also at midgap, respectively, are derived. The specific example of a quarter-wave stack is analyzed, and applications to three-dimensional structures, spontaneous emission control, delay lines, band-edge lasers, and superluminal tunneling times are discussed. \textcopyright{} 1996 The American Physical Society.

Optical sensor based on coated light guides

A new type of intensity-modulated sensors based on a coated light guide is presented. These sensors operate preferentially in the regime of total internal reflection (TIR), rather than relying on violation of TIR as in conventional intensity-modulated sensors. Using a physical model of the new sensor structure, the output light intensity is computed as a function of the complex reflection and transmission coefficients at the substrate-film and film-ambient interfaces. Based on the theoretical results, an experimental setup has been constructed and evaluated for measurements on various liquids. The results confirm the predicted dependence of the output signal on the characteristics of the substrate-film-ambient system. Of particular importance is the novel feature that the output signal increases as the refractive index of the ambient medium increases, at certain incidence angles.

Terahertz transmission of a Ba1-xKxBiO3 film probed by coherent time-domain spectroscopy.

The complex transmission coefficient for millimeter and submillimeter waves incident on a ${\mathrm{Ba}}_{0.6}$${\mathrm{K}}_{0.4}$${\mathrm{BiO}}_{3}$ thin film (82 nm) has been measured over a frequency range of 200--1200 GHz at temperatures above and below ${\mathit{T}}_{\mathit{c}}$ using coherent time-domain spectroscopy. We observe a dramatic change in both the magnitude and phase of the terahertz transmission in the superconducting state caused by a rapid carrier condensation. Both the real (${\mathrm{\ensuremath{\sigma}}}_{1}$) and imaginary (${\mathrm{\ensuremath{\sigma}}}_{2}$) parts of the complex conductivity are determined directly from the amplitude and phase of the transmitted electric field without the need for a Kramers-Kr\"onig analysis. By fitting ${\mathrm{\ensuremath{\sigma}}}_{2}$ in the framework of BCS theory, a superconducting gap 2\ensuremath{\Delta}(0)=6.9 meV=3.8${\mathit{k}}_{\mathit{B}}$${\mathit{T}}_{\mathit{c}}$ is obtained. Below ${\mathit{T}}_{\mathit{c}}$, the ${\mathrm{\ensuremath{\sigma}}}_{1}$ is rapidly enhanced for \ensuremath{\omega}/2\ensuremath{\pi}500 GHz, which is attributed to the BCS coherence effects. However, the conductivity exhibits monotonic temperature dependence and no clear ${\mathrm{\ensuremath{\sigma}}}_{1}$(T) peak is observed throughout the frequency range measured. The high-frequency penetration depth (\ensuremath{\sim}600 nm) is also extracted and discussed. Our results are consistent with a picture of BCS moderate coupling superconductivity in an intermediate to dirty limit.

Method for measuring the acoustical characteristics of waveduct elements with unknown gasdynamic parameters of the internal medium

A method is discussed for measuring the acoustical characteristics (the input impedance and the complex transmission coefficient) of an element of a waveduct (such as a muffler, a section of conduit with an abrupt change of cross section, ect.) during the flow of a medium with unknow velocity and temperature in the waveduct. The errors of the proposed method and existing waveguide methods are compared.

Ray tracing analysis of the transmission performance of curved FSS

Finite frequency selective surfaces (FSS) embedded in a curved dielectric are analysed using the ray-tracing technique. The FSS is treated as an infinitesimally thin interface with known complex reflection and transmission coefficients subjected to multiple interactions with rays internally reflected within the dielectric. Results for flat surfaces agree well with those computed using a full modal analysis. The transmission properties of such arrays are studied as the curvature is steadily increased for the case of paraboloids illuminated by plane waves, and also when enclosing a finite aperture-type antenna. The results obtained are validated by measurement.

A new method for complex spectral measurement in millimeter and submillimeter wave range

New method for measuring complex transmission or reflection coefficients has been proposed and demonstrated using a simple setup. High measurement quality was achieved: phase sensitivity ∼ 0.005 rad., amplitude reproduceability ∼ 1%, dynamic range to ∼ 107 ÷ 108 (previously 104), phase shift range to be measured 105. It is effective for BWT-Spectroscopy due to the absence of any mechanical adjustments during electronic frequency scan.

Compression of pulses spectrally broadened by self-phase modulation using a fiber-grating: a theoretical study of the compression efficiency

We analyse theoretically the potential of short fiber-gratings with constant grating period for the compression of optical pulses spectrally broadened by self phase modulation. A simple analytical model is developed which allows to analyse pulse compression in any dispersive element which has a constant dispersion over a certain bandwidth. For fiber-gratings with constant grating period the model predicts that the maximum achievable pulse compression factor is practically independent of the grating parameters and is typically in the order of 2. Numerical simulations which take into account the exact frequency dependence of the complex transmission and reflection coefficients of the fiber-grating confirm these predictions.

Full-wave analysis of radiation effect of microstrip transmission lines

A full-wave analysis of radiation effects produced by discontinuities in microstrip and buried microstrip transmission lines is presented. Beginning with the dyadic Green's function for a dielectric slab, with an embedded source, an integral equation is formulated. This equation is then solved by the method of moments to obtain the current distributions along the transmission line, in particular, near the discontinuities. Employing these results, the near- and far-zone fields, as well as radiation patterns are computed. The results from our method showed good agreement with those of previous publications in complex reflection and transmission coefficients, and equivalent capacitance values. It is found that under resonance conditions the radiation efficiency of a simple structure can exceed 41%, which may cause a potential problem in electromagnetic compatibility. Our analytic result also shows that the maximum radiation occurs when the source is located at the height of $$\lambda _0 /4\sqrt {\varepsilon _r } $$ from the bottom ground plane, which should be prevented.

Theory of microwave transmission and reflection in type-II superconductors in the mixed state.

We develop a phenomenological theory for microwave transmission and reflection in type-II superconducting films that self-consistently includes the effects of vortices. The theory includes the effects of vortex inertia, pinning, flux flow, and flux creep in a unified manner by using a complex dynamic vortex mobility. The treatment of vortex dynamics in the linear-response regime includes nonlocal vortex interactions in addition to two-fluid effects. Our results, including the complex effective wave number, rf conductivity, and transmission and reflection coefficients, are expected to be valid over a wide range of magnetic induction, frequency, and temperature. Numerical results illustrate the temperature and field dependences of these quantities for typical experimental conditions.

A national intercomparison between automatic network analyzers

In order to estimate the accuracy of heterodyne automatic network analyzers, LCIE organized an intercomparison in France. Measurements were performed on six different passive microwave components at 2, 10, and 18 GHz. The measured results are compared to those given by the Calibration Center of LCIE and those obtained with the dual six-port reflectometer realized at BNM/LCIE. The measured results, in terms of complex reflection and transmission coefficients (amplitude, phase), obtained at different frequencies are reported. >

X-ray transmission phenomenon analysis in a periodic multilayer

The X-ray transmission phenomenon in a periodic perfect multilayer model is analyzed by introducing a self-consistent scattering approach. Defining a complex transmission coefficient, precise calculations have been done for the transmissivity and the transmission phase considering a regular sequence of layers. The results obtained for GaAs and Si layers appear as an optical characteristic of the structure.

Modulated barrier approach to the interaction time in tunneling for arbitrary potentials.

The time that a transmitted particle interacts with a tunneling barrier is evaluated in one of the principal existing approaches by considering a small oscillatory perturbation added to the static barrier. The modulation frequency, at which a serious deviation from quasistatic behavior appears, is indicative of the traversal time. This approach is extended here, to a long region including the barrier, and utilizing a modulation potential which is spatially uniform throughout this region, except in the incident region ahead of the actual barrier, where it is made sufficiently smooth to avoid reflections. The resulting traversal time is \ensuremath{\Elzxh}\ensuremath{\partial} ln[\ensuremath{\partial}t/\ensuremath{\partial}E]/\ensuremath{\partial}E, where the complex transmission coefficient t(E) is defined as the ratio of the wave function at the far end to that at the point of incidence. This answer applies to arbitrary potentials and to all energies, including both propagation and tunneling in a single framework. This result is closely related to, but not identical to, that obtained by another earlier approach in which the potential is fixed, but the incident amplitude is modulated.