Modeling errors in microwave plasma diagnostics employing the reflection coefficient.
Complex reflection coefficient for finite-width boundary used for plasma diagnostics in high electron-density range, discussing modeling errors
ELECTRON DENSITY↗
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The far field diffraction pattern for corner reflectors with complex reflection coefficients
The far field diffraction pattern of a geometrically perfect corner reflector is examined analytically for normally incident monochromatic light. The states of polarization and the complex amplitudes of the emerging light are expressed through transformation matrices in terms of those of the original incident light for each sextant of the face in a single coordinate system. The analytic expression of the total diffraction pattern is obtained for a circular face. This expression consists of three component functions in addition to the basic Airy function. The coefficient of each function is expressed in terms of complex coefficients of reflectance of the reflecting surface. Some numerical results for different reflecting surfaces, including total internal reflection, are presented. The iso-intensity contours of the diffraction pattern evaluated from the analytical expressions for an uncoated solid corner reflector are also presented along with the photographs of the pattern.
Chang, R. F.↗
Far-field diffraction pattern for corner reflectors with complex reflection coefficients
Far field diffraction pattern of corner reflector for normally incident monochromatic light, considering polarization characteristics
Alley, C. O.↗
Ellipsometry
Ellipsometry is a technique often used to measure the thickness and properties of a thin film. This article covers the instrumental, theoretical, and practical aspects of this technique. Notably, different types of instruments including nulling ellipsometer, rotating compensator ellipsometer, and photoelastic modulator ellipsometer designs are presented. Elements of proper data analysis are also included, such as the use of an error-based figure of merit.
Complex reflection coefficients↗
A Computer Aided Broad Band Impedance Matching Technique Using a Comparison Reflectometer
An improved broadband impedance matching technique was developed. The technique is capable of resolving points in the waveguide which generate reflected energy. A version of the comparison reflectometer was developed and fabricated to determine the mean amplitude of the reflection coefficient excited at points in the guide as a function of distance, and the complex reflection coefficient of a specific discontinuity in the guide as a function of frequency. An impedance matching computer program was developed which is capable of impedance matching the characteristics of each disturbance independent of other reflections in the guide. The characteristics of four standard matching elements were compiled, and their associated curves of reflection coefficient and shunt susceptance as a function of frequency are presented. It is concluded that an economical, fast, and reliable impedance matching technique has been established which can provide broadband impedance matches.
Gordy, R. S.↗
Low frequency sound absorption of orifice plates, perforated plates and nozzles
Analyses of impulse time history data from acoustic transmission tests for conical nozzles attached to a pipe show internal reflections from the solid contraction and open area tend to cancel. To gain understanding of the opposing reflections, tests were conducted by replacing the conical nozzles with orifice plates. The primary variable was the open to solid area ratio. Internal reflection coefficient data reveal that, at an area ratio of 10-12%, the low frequency internal reflection is reduced from unity to about 0.2. Based on comparisons of far-field and internal data, acoustic energy is not conserved. Results are presented for complex reflection coefficient and far-field noise for a series of orifice and perforated plate configurations.
Salikuddin, M.↗
Temporally and spatially resolved characterization of microwave induced argon plasmas: Experiment and modeling
Experiments and modeling of the plasma-microwave interaction have been performed in a coaxial microwave plasma source at a field frequency of 2.45 GHz generating argon plasmas at pressures of 20 and 40 millibars and a ratio of flow rate to pressure of 0.125 sccm/Pa. The incident microwave power between 100 W and 300 W is supplied in a regime of a pulse-width modulation with cycle duration of 110 ms and a power-on time of 23 ms. The experiments are based on heterodyne reflectometry and microwave interferometry at 45.75 GHz. They provide the temporal behaviour of the complex reflection coefficient, the microwave power in the plasma, as well as the electron density in the afterglow zone of the discharge. The self-consistent spatially two-dimensional and time-dependent modeling complements the analysis of the plasma-microwave interaction delivering the plasma and electromagnetic field parameters. The consolidating experimental observations and model predictions allow further characterizing the plasma source. The generated plasma has a core occupying the region close to the end of the inner electrode, where maximum electron densities above 10{sup 20} m{sup −3} and electron temperatures of about 1 eV are observed. Due to a longer outer electrode of the coaxial structure, the plasma region is extended and fills the volume comprised by the outer electrode. The electron density reaches values of the order of 10{sup 19} m{sup −3}. The heating of the gas occurs in its great part due to elastic collisions with the plasma electrons. However, the contribution of the convective heating is important especially in the extended plasma region, where the gas temperature reaches its maximum values up to approximately 1400 K. The temporally and spatially resolved modeling enables a thorough investigation of the plasma-microwave interaction which clearly shows that the power in-coupling occurs in the region of the highest electron density during the early stage of the discharge. In the steady state phase, however, the power in-coupling occurs close to the source walls where the electron density is significantly lower than on the discharge axis.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Experimental investigation of microwave interaction with magnetoplasma in miniature multipolar configuration using impedance measurements
A miniature microwave plasma source employing both radial and axial magnetic fields for plasma confinement has been developed for micro-propulsion applications. Plasma is initiated by launching microwaves via a short monopole antenna to circumvent geometrical cutoff limitations. The amplitude and phase of the forward and reflected microwave power is measured to obtain the complex reflection coefficient from which the equivalent impedance of the plasma source is determined. Effect of critical plasma density condition is reflected in the measurements and provides insight into the working of the miniature plasma source. A basic impedance calculation model is developed to help in understanding the experimental observations. From experiment and theory, it is seen that the equivalent impedance magnitude is controlled by the coaxial discharge boundary conditions, and the phase is influenced primarily by the plasma immersed antenna impedance.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Optimization of the imaging response of scanning microwave microscopy measurements
In this work, we present the analytical modeling and preliminary experimental results for the choice of the optimal frequencies when performing amplitude and phase measurements with a scanning microwave microscope. In particular, the analysis is related to the reflection mode operation of the instrument, i.e., the acquisition of the complex reflection coefficient data, usually referred as S{sub 11}. The studied configuration is composed of an atomic force microscope with a microwave matched nanometric cantilever probe tip, connected by a λ/2 coaxial cable resonator to a vector network analyzer. The set-up is provided by Keysight Technologies. As a peculiar result, the optimal frequencies, where the maximum sensitivity is achieved, are different for the amplitude and for the phase signals. The analysis is focused on measurements of dielectric samples, like semiconductor devices, textile pieces, and biological specimens.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Electron density dependence of impedance probe plasma potential measurements
In earlier works, we used spheres of various sizes as impedance probes in demonstrating a method of determining plasma potential, φ{sub p}, when the probe radius is much larger than the Debye length, λ{sub D}. The basis of the method in those works [Walker et al., Phys. Plasmas 13, 032108 (2006); ibid. 15, 123506 (2008); ibid. 17, 113503 (2010)] relies on applying a small amplitude signal of fixed frequency to a probe in a plasma and, through network analyzer-based measurements, determining the complex reflection coefficient, Γ, for varying probe bias, V{sub b}. The frequency range of the applied signal is restricted to avoid sheath resonant effects and ion contributions such that ω{sub pi} ≪ ω ≪ ω{sub pe}, where ω{sub pi} is the ion plasma frequency and ω{sub pe} is the electron plasma frequency. For a given frequency and applied bias, both Re(Z{sub ac}) and Im(Z{sub ac}) are available from Γ. When Re(Z{sub ac}) is plotted versus V{sub b}, a minimum predicted by theory occurs at φ{sub p} [Walker et al., Phys. Plasmas 17, 113503 (2010)]. In addition, Im(Z{sub ac}) appears at, or very near, a maximum at φ{sub p}. As n{sub e} decreases and the sheath expands, the minimum becomes harder to discern. The purpose of this work is to demonstrate that when using network analyzer-based measurements, Γ itself and Im(Z{sub ac}) and their derivatives are useful as accompanying indicators to Re(Z{sub ac}) in these difficult cases. We note the difficulties encountered by the most commonly used plasma diagnostic, the Langmuir probe. Spherical probe data is mainly used in this work, although we present limited data for a cylinder and a disk. To demonstrate the effect of lowered density as a function of probe geometry, we compare the cylinder and disk using only the indicator Re(Z{sub ac})
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Optical Elements: Effect of structural defects of aperiodic multilayer mirrors on the properties of reflected (sub)femtosecond pulses
The effect of structural defects (for example, of interlayer roughness, layer thickness fluctuations and departures of Mo film density from the tabular one) on the amplitude and phase of the complex reflection coefficient as well as on the amplitude and duration of reflected pulses is numerically studied by the example of a model aperiodic Mo/Si multilayer mirror intended for the compression of a chirped pulse with a spectrum lying in a 50 – 80 eV photon energy range. The departures of Mo film density from the tabular values and film thickness fluctuations are shown to exert the strongest effect on the amplitude and duration of the reflected pulses. The interlayer roughness has a comparable effect on the amplitude of the reflection coefficient, but its effect on the duration of reflected pulses is weaker. Even small film thickness fluctuations may give rise to additional reflected pulses of high intensity, which are delayed in time relative to the principal pulse. The Mo-film density in a Mo/Si mirror is shown to vary from 0.77 to 0.97 (in units of the tabular value for massive molybdenum) as the film thickness varies from 1.5 to 5.5 nm. We discuss the key problems that have to be solved in the development of the fabrication technology of multilayer mirrors with desired characteristics. (paper)
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Interpretation of dispersion relations for bounded systems.
Treatment of the problem of constructing normal modes for an arbitrarily bounded system from roots of the linear dispersion relation D(omega, k) = 0 for the corresponding infinite or periodically bounded system. For a system described by continuous macroscopic variables, and of general cylindrical form, each transverse eigenmode gives rise to a set of axial normal modes constructed from a pair of dominant roots of D = 0 satisfying the boundary conditions which are characterized by complex reflection coefficients for the dominant waves. The implications of the results for the interpretation of experiments on plasma waves and instabilities on finite cylinders are discussed, with particular reference to the effects of end-plate damping and axial current on Q-machines.
Rognlien, T. D.↗
Microwave dielectric behavior of vegetation material
The microwave dielectric behavior of vegetation was examined through the development of theoretical models involving dielectric dispersion by both bound and free water and supported by extensive dielectric measurements conducted over a wide range of conditions. The experimental data were acquired using an open-ended coaxial probe that was developed for sensing the dielectric constant of thin layers of materials, such as leaves, from measurements of the complex reflection coefficient using a network analyzer. The probe system was successfully used to record the spectral variation of the dielectric constant over a wide frequency range extending from 0.5 to 20.4 GHz at numerous temperatures between -40 to +40 C. The vegetation samples were measured over a wide range of moisture conditions. To model the dielectric spectrum of the bound water component of the water included in vegetation, dielectric measurements were made for several sucrose-water solutions as analogs for the situation in vegetation. The results were used in conjunction with the experimental data for leaves to determine some of the constant coefficients in the theoretical models. Two models, both of which provide good fit to the data, are proposed.
Elrayes, Mohamed A.↗
Dielectric measurement using an open ended coaxial line with an extended center conductor
Permittivity may be determined by measuring the complex reflection coefficient of an open-ended coaxial line placed in contact with a test sample. This method works best for liquid samples. For the measurement of non-liquid materials problems can arise. A perfect preparation is needed to provide a good contact between the tip and the dielectric medium. At times, the dimension of the sensor cannot be freely chosen, as is the case for the measurement of the dielectric constant of the xylem layer of a living tree. The influence of the installation of the sensor on a tree's water status must be minimized by using a small sensor. In such cases the capacitance cannot be optimized. By extending the center conductor of the open-ended coaxial line, some of these problems can be avoided. This provides an additional tool to tune the capacitance of the sensor by adjusting the length of the extension. Therefore the measurement accuracy can be optimized. The sensor also becomes sensitive to a larger volume. A comparative study of a flush and extended tipped probes shows that the ability to measure the dielectric constant of trees has been notably increased due to the extension of the center conductor.
Wegmueller, Urs↗
An electromagnetic noncontacting sensor for thickness measurement in a dispersive medium
This paper describes a general purpose imaging technology developed by the U.S. Bureau of Mines (USBM) that, when fully implemented, will solve the general problem of 'seeing into the earth.' A first-generation radar coal thickness sensor, the RCTS-1, has been developed and field-tested in both underground and highwall mines. The noncontacting electromagnetic technique uses spatial modulation created by moving a simple sensor antenna in a direction along each axis to be measured while the complex reflection coefficient is measured at multiple frequencies over a two-to-one bandwidth. The antenna motion imparts spatial modulation to the data that enables signal processing to solve the problems of media, target, and antenna dispersion. Knowledge of the dielectric constant of the media is not necessary because the electrical properties of the media are determined automatically along with the distance to the target and thickness of each layer of the target. The sensor was developed as a navigation guidance sensor to accurately detect the coal/noncoal interface required for the USBM computer-assisted mining machine program. Other mining applications include the location of rock fractures, water-filled voids, and abandoned gas wells. These hazards can be detected in advance of the mining operation. This initiating technology is being expanded into a full three-dimensional (3-D) imaging system that will have applications in both the underground and surface environment.
Chufo, Robert L.↗
Implications of Microwave Holography Using Minimum Required Frequency Samples for Weakly- and Strongly-Scattering Indications
Microwave imaging techniques, an integral component of nondestructive testing and evaluation (NDTE), have received significant attention in the past decade. These techniques have included the implementation of synthetic aperture focusing (SAF) algorithms for obtaining high spatial resolution images. The next important step in these developments is the implementation of 3-D holographic imaging algorithms. These are well-known wideband imaging technique requiring a swept-frequency (i.e., wideband), which unlike SAF that is a single frequency technique, are not easily performed on a real-time basis. This is due to the fact that a significant number of data points (in the frequency domain) must be obtained within the frequency band of interest. This not only makes for a complex imaging system design, it also significantly increases the image-production time. Consequently in an attempt to reduce the measurement time and system complexity, an investigation was conducted to determine the minimum required number of frequency samples needed to image a specific object while preserving a desired maximum measurement range and range resolution. To this end the 3-D holographic algorithm was modified to use properlyinterpolated frequency data. Measurements of the complex reflection coefficient for several samples were conducted using a swept-frequency approach. Subsequently, holographical images were generated using data containing a relatively large number of frequency samples and were compared with images generated by the reduced data set data. Quantitative metrics such as average, contrast, and signal-to-noise ratio were used to evaluate the quality of images generated using reduced data sets. Furthermore, this approach was applied to both weakly- and strongly-scattering indications. This paper presents the methods used and the results of this investigation.
Fallahpour, M.↗
PyOECP: A flexible open-source software library for estimating and modeling the complex permittivity based on the open-ended coaxial probe (OECP) technique
Here, we present PyOECP, a Python-based flexible open-source software for estimating and modeling the complex permittivity obtained from the open-ended coaxial probe (OECP) technique. The transformation of the measured reflection coefficient to complex permittivity is performed based on three different methods. The software library contains the dielectric spectra of common reference liquids, which can be used to transform the reflection coefficient into the dielectric spectra. Several Python routines that are commonly employed (e.g., SciPy and NumPy) in the field of science and engineering are required only so that the users can alter the software structure depending on their needs. The modeling algorithm exploits the Markov Chain Monte Carlo method for the data regression. The discrete relaxation models can be built by a proper combination of well-known relaxation models. In addition to these models, electrode polarization, a typical measurement artifact for interpreting dielectric spectra, can be incorporated into the modeling algorithm. A continuous relaxation model, which solves the Fredholm integral equation of the first kind (a mathematically ill-posed problem), is also included. This open-source software enables users to freely adjust the physical parameters to obtain physical insight into their materials under test and will be consistently updated for more accurate measurement and interpretation of dielectric spectra in an automated manner. This work describes the theoretical and mathematical background of the software, lays out the workflow, and validates the software functionality based on both synthetic and empirical data included in the software.
97 MATHEMATICS AND COMPUTING↗
Infrared optical properties of water and ice spheres.
IR absorption coefficient, reflectivity and complex index of refraction of ice and water spheres, calculating single scattering albedo, asymmetry factor and extinction cross section
Irvine, W. M.↗