Ray Optics Method Research Articles

A modified ray-optic method for planar waveguides with graded-index profiles and with truncated-index profiles

A modified ray-optic (MRO) method is proposed for evaluating the propagation characteristics of planar waveguides with arbitrary refractive index profiles. We correct the error caused by assuming a constant phase shift for any mode at the turning point on the substrate side. Very simple formulas are given for calculating the dispersion characteristics of planar optical waveguides. The results are much more accurate than those of the ray-optic method, especially in the near cutoff regions, and agree well with those of exact numerical methods. A comparison with other analytical and exact numerical methods shows that our method is much simpler than any other methods.

D-shaped optical fibre design for methane gas sensing

The authors demonstrate how the sensitivity of D-fibre based sensors depends on its parameters (refractive index difference, core diameter and flat surface-core distance). The optimal fibre parameters are predicted using the authors' finite difference/perturbation theory model, and similar results were obtained using a ray optics method for the planar waveguide to model the D fibre. It is concluded that the D fibre within its optimum architecture exhibits relatively low sensitivity. This sensitivity can however be improved using a suitable overcoating.

A modified ray-optic method for arbitrary dielectric waveguides

A modified ray-optic method (MRO) is proposed for evaluating the propagation characteristics of dielectric waveguides with arbitrary refractive index profile. The authors correct the error caused by assuming a constant phase shift for any mode at the turning point on the substrate side. Very simple formulas are given for calculating the dispersion characteristics including those for planar optical waveguides and single-mode fibers. The results are more accurate than those of the ordinary ray-optic method, especially in the near cutoff regions, and agree well with those of the exact numerical methods. A comparison with other analytical and the exact numerical methods shows that the proposed method is simpler than other methods. This method can be extended to other more complicated cases.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

RADIATION FROM STAGGERED PARALLEL-PLATE WAVEGUIDES: RAY-OPTICAL ANALYSIS AND EXPERIMENTAL RESULTS

ABSTRACT The radiation patterns of a three-element array of staggered parallel-plate waveguides is calculated by ray-optical methods. Experiments with a long H-plane sectoral horn at about 12 GHz show good agreement with the theory.

Evaluation Method Of Gradient-Index Rod Lenses

The ray optic evaluation method of gradient-index (GRIN) rod lenses is described. The feature of this method is to utilize the fact that the ray path in an inhomogeneous refractive index medium is determined uniquely by the initial condition given externally. Some numerical examples of the ray tracing on one-quarter pitch and one-half pitch GRIN lenses and simple expressions corresponding to the refractive index profile are shown.

Coherent states and nonparaxial propagation of light in graded-index media

A study is made of the trajectories and widths of coherent states in nonparaxial propagation of light in a medium with a refractive index which differs slightly from a quadratic profile. Explicit expressions are obtained for the trajectories and widths and it is shown that they generally depend on the wavelength of light. In the limit of wavelengths tending to zero the expressions for the trajectories reduce to the corresponding expressions in ray optics and the ray widths decrease to zero. The coherent states thus make it possible to generalize the concept of a geometric ray and to allow for the wave properties of light in solutions obtained by ray optics methods.

Waveguide loss evaluation by the ray-optics method

Attenuation constants of bound or leaky modes in a general class of straight planar waveguides evaluated by the ray-optics method are analytically shown to be identical in the first-order of approximation with those obtained from the characteristic equation when one takes the Goos–Hänchen shift into account. It is also pointed out that significant errors will result if the Goos–Hänchen shift is ignored.

High-frequency propagation in an elevated tropospheric duct

High-frequency propagation in an elevated tropospheric duct above the earth is analyzed conventionally by normal mode summation, which may be inefficient and often of questionable accuracy because of the large number of modes required. Ray-optical methods, which have been employed, are likewise subject to limitations. The present study explores a new method that combines a judiciously chosen mixture of ray-optical and normal mode fields plus a remainder field that is usually small. The modes may be guided along the concave side of the duct boundary or between this boundary and the earth's surface. By retaining the advantages of both the modal and ray formulations, the hybrid scheme generally requires far fewer rays and modes than the number of modes in a conventional normal mode representation. The hybrid method is applied here to the Green's function problem for a two-dimensional model of the duct formed by a refractive index discontinuity above a perfectly conducting earth. Field values for various locations of source and observation points are computed from the normal mode formulation, which is used as a reference solution, and from the hybrid formulation. The features of these alternative computational methods are discussed in detail and reveal the advantages of the hybrid scheme. Physical interpretation of the hybrid ray-mode field is emphasized throughout and grants new insight into various wave processes.

Ray-optical calculation of eigenmode behavior of unstable laser resonators with rounded edges

The eigenmode resonance equation, previously developed by the ray-optical method of analysis for unstable symmetric bare resonators with strip and spherical mirrors, is employed here to establish the basic properties of resonators with rounded mirror edges. The numerical results obtained in this manner are far more comprehensive than those presented earlier from a waveguide model approach to the resonator problem. As in this earlier study, edge rounding is found to improve mode separation and to shift the onset of low-loss mode detachment to lower Fresnel numbers. It is also found that a simple closed form expression predicts accurately the low-loss mode behavior after it has detached from the remaining mode spectrum.

Quasi-optics of the waves guided by a slab of uniaxially anisotropic dielectric

A ray optical method is given for deducing the dispersion relation and the group velocity of the TM wave guided by a slab of uniaxially anisotropic dielectric with particular emphasis on confirming the correctness of the reverse lateral displacement of the ray on total reflection at an interface separating the dielectric from the free space.

Ray method for unstable resonators

The previously developed ray-optical method for unstable, symmetric, bare resonators with sharp-edged stop and circular mirrors is reviewed here. A deductive stepwise procedure is presented, with emphasis on the physical implications. It is shown how the method can accommodate other edge configurations such as those produced by rounding, and also more complicated nonaxial structures such as the half-symmetric resonator with internal axicon. For the latter, the ray approach categorizes those rays that must be eliminated from the equivalent aligned unfolded symmetric resonator, and it identifies the canonical diffraction problems that must be addressed to account for shadowing and scattering due to the axicon tip. Effects due to shielding or truncation of the axicon tip are also considered. Approximate calculations of the eigenvalues for the lowest-loss modes illustrate the effects due to various tip shielding lengths and spacings of the axicon from the output mirror.

Multipath Delay Characteristics on Line-of-Sight Microwave Radio System

Multipath properties on microwave line-of-sight links for the purpose of applying to microwave systems design are discussed. The theoretical analyses of the fading mechanism and the relation between multipath delay and radio duct parameters are carried out using the ray-optical method. Multipath delay have been measured with the frequency-sweep method and the three-frequency method on several paths in Japan since 1954. These paths are distributed in length from 20 km to 80 km, and include horizontal and slant paths. Through these theoretical and experimental analyses, a number of wave components in multipath propagation, the probability density function of multipath delay, the maximum delay and the relation between multipath delay and radio duct parameters are examined. The estimation method for path-length difference statistics, needed for evaluation of propagation distortion, is quantitatively given.

Ray-optical analysis of unstable resonators with spherical mirrors

A previously developed ray-optical method for determining the eigenvalues and eigenmodes of unstable resonators with circular strip mirrors is here applied to spherical mirrors. The results obtained in this manner agree with those of Butts and Avizonis who extended to the three-dimensional case the postulative procedure employed by Horwitz for the strip configuration. The individual and collective treatment of rays in the analysis provides a physical basis that should be applicable also to more complicated resonator structures.

Transform approach to electromagnetic scattering

In this paper, we present a comprehensive review of the Fourier transform technique as applied to the problem of high-frequency scattering and introduce the concepts of the spectral theory of diffraction (STD). In contrast to the more commonly employed ray-optical method for high-frequency scattering, viz. the geometrical theory of diffraction (GTD), the STD approach interprets the scattered field as the spectrum, or the Fourier transform of the induced current on the scatterer. Such an interpretation offers several important advantages: uniform nature of representation, capacity to improve and extend the ray-optical formulas in a systematic manner, and convenient accuracy tests for the results. Methods for combining integral equation methods with the Galerkin procedure and asymptotic techniques in the transform domain are described, and representative examples illustrating the application of the spectral approach are included.

Multiply Reflected Gaussian Beams in a Circular Cross Section

A well-collimated beam reflected repeatedly within a circular cross section undergoes periodic focusing and defocusing. This behavior is of interest for tracking of beams around a type of acoustic surface wave disk delay line, and it also relates to beam monitoring after oblique injection into the endface of a multimode optical fiber. The problem is analyzed by considering first the field excited by an isotropic line source inside a dielectric cylinder, and then converting this to Gaussian beam excitation by assigning a complex value to the source coordinate location. Because the wavelength is small compared to the cylinder radius, ray-optical methods are employed to construct the solution, with inclusion of such novel ingredients as the lateral ray shift on a curved boundary. Results are obtained for the amplitude and phase of the ray and beam field and for such beam parameters as the location of the focus the minimum beam width and the rate of beam divergence between successive reflections.

Comparison of Wave and Ray Techniques for Solution of Graded Index Optical Waveguide Problems

Transit times and eigenvalue equations found using ray-optics methods are compared with the exact wave-equation solution results. Geometric optics gives excellent accuracy for commonly used values of the dimensionless waveguide parameter V. Even when the effects of dispersion are included, the WKB condition gives exactly the same transit times as the more physically intuitive ray-optics technique.

Impulse response of a self-focusing optical fiber

By using wave optics method, analytical expressions are derived for the impulse response of a self-focusing optical fiber with a core whose index profile is described by the second- and the fourth-order terms of the radial distance. Two cases are considered: (1) where all the possible propagating modes are equally excited; (2) where only the modes with no azimuthal variations are equally or unequally excited. It is shown that the shape and the rms width of the impulse response depend strongly on the fourth-order coefficient of the index distribution and on the excitation conditions. Comparison is also made between the results obtained from the wave optics method and those obtained from the ray optics method, and both results are found to be identical.

Dispersion in optical waveguides with graded refractive index

Dispersion in graded-index optical fibres can be calculated by ray-optical or wave-optical methods. This letter reports a seeming discrepancy between the results of both methods: in a fibre with a refractive profile that has been optimised by the WKB wave-optical method, helical rays may occur with a much larger transit-time spread than the modes computed with the WKB method. The question is what transit-time spread is relevant for the bandwidth of the fibre.

Antennas employing conical-dielectric horns. Part 2: The Cassegrain antenna

A design theory is presented for Cassegrain antennas employing dielectric-cone feeds. A modal approach is used to describe the aperture field over the subreflector, while a ray-optics method is employed to determine the sub-reflector profile and main-reflector aperture distribution. Experiments have been made with antennas designed for operation over bands centred near 11 and 18 GHz, and results are in generally good agreement with theory. These antennas are shown to have high efficiency, good main-beam symmetry and low mutual coupling.

Intensity and Polarization for Single Scattering by Polydisperse Spheres: A Comparison of Ray Optics and Mie Theory

The intensity and polarization for single scattering by large spherical particles are computed using both the exact Mie theory and the approximation of ray optics. It is found that the ray-tracing method can yield accurate results for particle size parameters in the range of interest for some meteorological applications, where the size parameter is the ratio of the particle circumference to the wavelength of the incident light. Since this method is practical for application to nonspherical particles, it should be of use in studies of cloud microstructure. The ray-optics method is also useful in the case of spherical particles because it provides a physical explanation for features which occur in the exact theory. The ray-optics calculations include Fraunhofer diffraction as well as geometrical reflection and refraction; rays undergoing one or two internal reflections, which give rise to the observable rainbows, are also included. Calculations are made for non-absorbing and absorbing spheres for several refractive indices in the range 1.1 ≤ nr ≤ 2.0. Comparisons between the ray-optics approximation and the exact Mie theory are made for nr = 1.33 and 1.50. It is found that the two methods are in close agreement, if the particle size parameter is ≳400. It is also shown that, to a good approximation, the ray-optics solution may often be used to obtain the entire phase matrix for single scattering.