Geometrical Optics Approach Research Articles

Space wave field produced by a vertical electric dipole above a rough surface

The present diffraction problem is solved by means of a perturbation calculus in the transition conditions and by repeated application of the method of steepest descent to two-dimensional Fourier integrals. We obtain a reflection coefficient for the rough surface resulting in a geometrical-optics approximation for the space wave field strength. In the case of a periodic roughness profile the application of the method of steepest descent in the transform space can be avoided and we get the electromagnetic field through differentiation of the Bromwich potentials. The numerical results of the two methods are discussed in the case of a one-dimensional cosine profile. We show that the influence of the earth's roughness increases with increasing receiver heights and fixed receiver distance. On the other hand, we point out that the geometric optical approach is a rather good approximation for the space wave field strength.

Progress in Radio-Echo Sounding Theory

AbstractThe inverse problem of radio-echo sounding consists in the reconstruction of subglacial relief from the known radio-echo profile, and the path, and speed of the aircraft. The present work shows that in the geometrical optical approach the solution of the inverse problem for a homogeneous, two-dimensional object (valley glacier) exists, and there is a unique solution. An algorithm for interpretation of the experimental data is suggested. It may be considered as the generalization of Harrison’s transformations for any surfaces. paths, and speed.The direct problem of radio-echo sounding consists in the reconstruction of radio-echo profile from the known surface and subsurface relief, path, and aircraft speed. The analysis of traces in the standard radio-echo sounding mode of operation reveals the possibility of introducing a three-index trace classification {K, sign S’(o), K+} where K is the number of real roots characteristic of the equations, K+ the number of positive roots, and S’(o) the position derivative, the argument being equal to zero. The form {o, — o} is optimal for the precision of the calculation of the reflected surface coordinates, as well as for the simplicity of the interpreted picture. By a special choice of the altitude of flight, any form of any surface can be brought to {o, —o}. The decrease in beam width is equivalent to the diminution of roots of the characteristic equations. For a pencil beam the trace degenerates into a point.The attenuation of the reflected signal depends on the glacier geometry, the dielectric parameters of the medium, the altitude and the course of the aircraft, as well as statistical characteristics of the mutual orientation of the interfaces and aerials. For the description of the energetics of radio-echo sounding, equivalent reflecting surfaces are suggested. These surfaces correspond to the Harrison’s equivalent reflecting surface. Exact formulae for the power of coherent and incoherent components of the reflected signal are obtained. Components of the full attenuation such as absorption, depolarization, and spherical divergence, are investigated with respect to refraction and focusing effects.

Progress in Radio-Echo Sounding Theory

Abstract The inverse problem of radio-echo sounding consists in the reconstruction of subglacial relief from the known radio-echo profile, and the path, and speed of the aircraft. The present work shows that in the geometrical optical approach the solution of the inverse problem for a homogeneous, two-dimensional object (valley glacier) exists, and there is a unique solution. An algorithm for interpretation of the experimental data is suggested. It may be considered as the generalization of Harrison’s transformations for any surfaces. paths, and speed. The direct problem of radio-echo sounding consists in the reconstruction of radio-echo profile from the known surface and subsurface relief, path, and aircraft speed. The analysis of traces in the standard radio-echo sounding mode of operation reveals the possibility of introducing a three-index trace classification {K, sign S’(o), K+} where K is the number of real roots characteristic of the equations, K+ the number of positive roots, and S’(o) the position derivative, the argument being equal to zero. The form {o, — o} is optimal for the precision of the calculation of the reflected surface coordinates, as well as for the simplicity of the interpreted picture. By a special choice of the altitude of flight, any form of any surface can be brought to {o, —o}. The decrease in beam width is equivalent to the diminution of roots of the characteristic equations. For a pencil beam the trace degenerates into a point. The attenuation of the reflected signal depends on the glacier geometry, the dielectric parameters of the medium, the altitude and the course of the aircraft, as well as statistical characteristics of the mutual orientation of the interfaces and aerials. For the description of the energetics of radio-echo sounding, equivalent reflecting surfaces are suggested. These surfaces correspond to the Harrison’s equivalent reflecting surface. Exact formulae for the power of coherent and incoherent components of the reflected signal are obtained. Components of the full attenuation such as absorption, depolarization, and spherical divergence, are investigated with respect to refraction and focusing effects.

Geometrical optical approach for electromagnetic wave propagation in rectangular mine tunnels

Electromagnetic wave propagation inside an empty rectangular mine tunnel with imperfect walls is considered. The modal expansion of the fields is complicated by the coupling of the basic modes by the imperfect walls. To avoid this difficulty, and in view of the large guide dimensions relative to the free space wavelength, a geometrical ray approach is proposed. To provide a theoretical foundation for the method, we first consider an idealized waveguide model with two perfectly reflecting side walls. The modal field expansion for this prototype model is fully analyzed to provide a satisfactory comparison between the modal and the geometrical ray sums. The proposed general ray method is then applied to the rectangular waveguide when all four walls are imperfectly conducting. Finally, the influence of wall roughness is considered by a relatively simple method.

Spectral line absorption measurement using optical cavities

A simple technique using a conventional gas laser with spherical mirrors having identical radii of curvature in the nonoscillating regime for spectral line absorption measurements is described and applications for laser work are suggested. The theory of the measurement carried out in the geometrical optical approach for Doppler-broadened lines was checked experimentally and conditions are specified for which measurement inaccuracies of the order of 1% for the peak value of the line absorption coefficient can be obtained. Since the device provides a fine adjustment of the cavity losses, formation of the diffraction modes could be observed in the preoscillating regime of the optical cavity.

An electromagnetic analysis of a cylindrical homogeneous lens

Radially symmetric lenses are ideally suited for use as multiple-beam antennas. In recent years, several investigations using a geometrical optics approach have indicated that simple lens structures such as homogeneous lenses might be suitable antennas. This paper presents an electromagnetic analysis of a cylindrical homogeneous lens as well as examining in detail the effect upon performance of parameter variations such that engineering and design evaluations can be made. It is shown that a lens with a relative dielectric constant of 3.5 gives the best performance. With the use of simple matching techniques, overall performance comparable to the more complex Luneburg lens is obtained.

INFLUENCE OF THE INVERSION INHOMOGENEITY ON THE TRANSVERSE STRUCTURE OF OSCILLATIONS IN SOLID-STATE LASERS

The geometrical-optics approach is used in an analysis of the influence of the inhomogeneity of population inversion, which appears under spiking conditions, on the transverse structure of the radiation field of a laser. It is shown that under certain conditions the inversion inhomogeneity at right angles to the resonator axis has little influence on the properties of the output radiation. The results are given of an experimental investigation of the transverse structure of the radiation emitted by a neodymium-glass laser under conditions corresponding to different degrees of influence of the inversion inhomogeneity.

Relationship between Geometrical-Optical and Full-Wave Solutions to the Problem of Propagation over a Nonparallel Stratified Medium

The problem of propagation over a wedge-shaped overburden has been analyzed using a full-wave-solution approach. The relationship between this solution and an earlier one employing the compensation theorem is discussed in detail. The compensation theorem method makes use of the simplifying concept of of the surface impedance which is determined by a geometrical-optical approach. The applicability and limitations of the latter approach are carefully studied for both dissipative and nondissipative over-burdens.

Calculations of the effect of roughness on microwave emissivity with application to the Moon and Venus

Geometric optics approach to calculating surface roughness effects on microwave emissivity from Moon and Venus

Dynamische Theorie der Röntgenstrahl-Interferenzen an schwach verzerrten Kristallgittern

A geometrical-optical approach to the problem of the propagation of X-rays in weakly deformed crystals is developed in a general form along the lines of the Hamilton-Jakobi theory. It starts from the eikonal equation, which has been derived from Maxwell’s equations in Part I1 and arrives at the equations of Penning and Polder3 and Fermat’s principle by Kato 4. The amplitude equation, which has also been derived in Part I, is interpreted by means of the energy-flow picture, the absorption being taken into account. Application of the theory to the two-beam case gives an equation of rays, which has a remarkable resemblance to the relativistic equation of motion of charged particles in an electromagnetic field. Representation of the lattice distortion by a displacement vector enables one to obtain equations of rays, phases and amplitudes in a form suited to the practical calculation.

Rays and Ray Envelopes within Stable Optical Resonators Containing Focusing Media

In stable resonators any given initially paraxial rays remain close to the axis of the structure and are, in fact, confined within well-defined contours-the envelope of the ray system. Previously, envelopes of rays in empty resonators had been found and their form identified with the variation of the spot size. This geometric optical approach is extended to general resonators, comprising arbitrary arrangements of lenses and convergent or divergent inhomogeneous focusing media. An invariant quadratic form involving parameters descriptive of any of the ray segments that result from a given initial ray segment leads to a differential equation satisfied by the ray segments and their envelope in portions of the resonator. A maximum-minimum problem for the envelope is formulated and solved. In convergent media the envelope function is found to be periodically modulated. The period of the modulation depends only on the properties of the convergent medium; the location of relative maxima and minima, as well as their ratio, depends on both the medium and associated optics. In special cases, results are compared with available solutions of the corresponding electromagnetic problem. A particularly simple resonator is analyzed, and envelope characteristics correlated with the stability limits.

Relationship of geometric optics and autocorrelation approaches to the analysis of lunar and planetary radar

The formal relationship is established between the geometric optic approach and the autocorrelation approach to the analysis of lunar and planetary radar echoes for a Gaussian autocorrelation function when the surface has Gaussian height statistics and introduces deep phase modulation on the incident wave. Similarly it is inferred that such an analogy also holds whenever the surface undulations contain structure no smaller than the wavelength of observation divided by the rms slope of the surface. If an appreciable amount of small-scale structure is present, it is shown that (1) the range of scales responsible for the scattering will include an increasing amount of small-scale structure with increasing angle of incidence and (2) no ample relationship appears to exist between the distribution of apparent surface slopes and the power backscattered as a function of angle of incidence.

Investigation of the Geometrical-Optical Theory of Magnetic Structure Imaging in the Electron Microscope

The imaging of domain walls in thin ferromagnetic films by off-focus transmission electron microscopy is generally considered in terms of a simple geometrical-optical theory in which the electron beam is deviated by the Lorentz force exerted by the internal magnetic flux of the film. Calculated intensity profiles of the wall images are rectangular in form for large off-focus distances where the geometrical-optical approach is valid. The width of the images is simply related to the thickness and internal magnetic flux of the film and to the off-focus distance of the observation plane and the electron beam accelerating potential. An experimental investigation of these relationships for vacuum-condensed films of iron, nickel, and their alloys shows general agreement with the theoretical relationships derived for 180° Néel walls. The observed electron beam deflections within the films are about half of the calculated values. This is largely due to the wall angles being less than 180° and to the film density being below that of the bulk material. Domain wall widths in 80–20 Ni–Fe films about 200 Å thick were found to be approximately 2000 Å, leading to a value for the exchange constant of this material of around 8×10−7 erg·cm−1.

A geometrical optics approach for the radar cross section of thin shells

A geometrical optics approach for the radar cross section of thin shells