Ray Optical Solution Research Articles

Off-axial aberrations of a Gaussian beam obliquely passing through a thinradial GRIN lens

We analyze the off-axial aberrations of a slightly inclined Gaussian beam in a thin radial graded-index (GRIN) lens. We assume that the endfaces of the lens are flat and the aperture stop is generally not in contact with the lens surface. First we formulate the diffraction of a complex-source-point spherical wave through a thin GRIN lens, while the terms of up to fourth order in aperture variables are taken into account. Then we find a ray-optical solution to the diffracted beam which is represented in terms of aberration coefficients of the Seidel type. Finally we evaluate the effect of off-axial aberrations on the quality of the diffracted Gaussian beam. The quality of an oblique Gaussian beam is much degraded by increasing the angle of inclination (or the spot size) of the beam and taking the aperture stop out of the lens surface.

Fourth order correction to a Gaussian beam focused with a parabolic index lens

We formulate the fourth order correction to a paraxial Gaussian beam propagated along the axis of symmetry of a parabolic index lens. First we examine the evolution of a complex-source-point spherical wave (equivalent paraxially to a Gaussian beam) through the lens in a two-dimensional xz plane. Taking into account the terms of up to fourth order in aperture variables, we find a ray-optical solution to the exit beam that is represented in terms of aberration function. We also analyze the effect of the lens aberration exerted on the degradation in the quality of a Gaussian beam. The fourth order-corrected wave function derived here may be used to evaluate the quality of a Gaussian beam focused with a parabolic index lens. Further it may be applied to the case of an orthogonal system in which the index variations are different in the xz and yz planes.

A Canonical UTD Solution for Electromagnetic Scattering by an Electrically Large Impedance Circular Cylinder Illuminated by an Obliquely Incident Plane Wave

A uniform geometrical theory of diffraction (UTD) solution is developed for the canonical problem of the electromagnetic (EM) scattering by an electrically large circular cylinder with a uniform impedance boundary condition (IBC), when it is illuminated by an obliquely incident high frequency plane wave. A solution to this canonical problem is first constructed in terms of an exact formulation involving a radially propagating eigenfunction expansion. The latter is converted into a circumferentially propagating eigenfunction expansion suited for large cylinders, via the Watson transform, which is expressed as an integral that is subsequently evaluated asymptotically, for high frequencies, in a uniform manner. The resulting solution is then expressed in the desired UTD ray form. This solution is uniform in the sense that it has the important property that it remains continuous across the transition region on either side of the surface shadow boundary. Outside the shadow boundary transition region it recovers the purely ray optical incident and reflected ray fields on the deep lit side of the shadow boundary and to the modal surface diffracted ray fields on the deep shadow side. The scattered field is seen to have a cross-polarized component due to the coupling between the TEz and TMz waves (where z is the cylinder axis) resulting from the IBC. Such cross-polarization vanishes for normal incidence on the cylinder, and also in the deep lit region for oblique incidence where it properly reduces to the geometrical optics (GO) or ray optical solution. This UTD solution is shown to be very accurate by a numerical comparison with an exact reference solution.

Aberrations of a slightly inclined Gaussian beam in a symmetric optical system

We discuss the Seidel-type aberrations which degrade the quality of the Gaussian beam, while the beam axis is slightly inclined to the axis of an optical system. First we examine the evolution of a complex-source-point spherical wave through multiple spherical boundaries. Then we find a ray-optical solution for the diffracted beam that can be represented as a function of Seidel-type aberrations. We also formulate the characteristic parameters of the aberrated beam. If the beam axis is placed at the yz plane, the x component of beam quality factor is degraded by both spherical aberration and field curvature, and the difference in the x and y components of beam quality factor is caused by linear coma, linear astigmatism, and distortion. The fourth order theory of propagation of the Gaussian beam presented here may be applied to determine the parameters of a slightly inclined Gaussian in a symmetric optical system.

Fourth order correction to a paraxial Gaussian beam in a rotationally symmetric system of non-spherical surfaces

AbstractWe formulate the fourth order correction to a paraxial Gaussian beam propagating through a non-spherical surface system that is rotationally symmetric. First we represent the Gaussian beam by a complex-source-point spherical wave (CSPSW). Next we examine the evolution of the CSPSW through the non-spherical surface system by applying a Fresnel–Kirchhoff diffraction integral. We find a ray-optical solution to the diffraction integral in terms of both the conic constant and the aspheric coefficient of fourth order. Then we numerically evaluate the quality factor of the fourth order-corrected beam propagating through either an aspheric mirror or a thin lens made up of two non-spherical surfaces. The fourth order formulas derived here can be useful in determining the quality factor of the Gaussian beam degraded by a rotationally symmetric system of non-spherical surfaces.

Ultrashort pulse inscription of tailored fiber Bragg gratings with a phase mask and a deformed wavefront [Invited

We report on the inscription of chirped fiber Bragg Gratings (FBGs) with a phase mask and a deformed wavefront using a femtosecond laser. A qualitative model is developed to predict the behavior of the resulting grating period for a deformed wavefront. In addition the quantitative change of the period was simulated based on a ray optical solution of the diffraction behind the phase mask. For deforming the wavefront experimentally a cylindrical tuning lens was used. Tilting of the lens increased the higher order aberrations like coma and spherical aberration, which leads to chirped FBGs. A chirped FBG with a FWHM bandwidth of 2.5 nm could be realized. The change of the resulting fiber Bragg grating period was measured using a side diffraction setup yielding good agreement with the measured spectra.

Evolution of a complex-source-point spherical wave through a symmetric optical system with spherical aberration

Abstract We discuss the evolution of a complex-source-point spherical wave (CSPSW) along the axis of a symmetric optical system with spherical aberration. The CSPSW is equivalent to the sum of all the higher order corrections to a paraxial Gaussian beam. First we formulate the diffracted CSPSW through spherical boundaries, where the terms of up to fourth order in aperture variables are taken into account. We then find a ray-optical solution for the diffracted beam that can be represented in terms of the coefficient of spherical aberration. Next we formulate the beam spot size, the far-field divergence angle, and the beam quality factor which are characteristic of the aberrated beam. For the case of a thick lens, we examine the effect of lens aberration on the beam parameters. The derived formulas can be used to determine the parameters of the Gaussian beam degraded by an arbitrary symmetric optical system with spherical aberration.

Shortwave scattering by a diffraction echelette-grating

The two-dimensional problem of the scattering of a plane wave by a periodic perfectly conducting grating, an echelette with a right angle, is considered in the case of a high-frequency approximation (the wavelength is assumed to be small as compared with the period of the grating). The situation where the incident plane wave glides along one of the faces of a wedge is discussed. A ray-optical solution of the problem (a shortwave asymptotic result) is derived by the method of summing multiple diffracted fields, which is well known in the geometric theory of diffraction. The main result of this paper consists of obtaining simple formulas for the efficiency en of a diffraction order with maximal value of en, derived in the shortwave approximation. Numerical results are presented, and important optical properties obtained by asymptotic analysis are described. Bibliography: 10 titles.

Ray Optical Analysis of a Plane Wave Scattering by An Echelette Diffraction Grating

The two-dimensional problem of scattering of a nearly grazing plane wave by a diffraction grating, echelette with right angle is considered for the case of high-frequency approximation (when period of grating is far greater than the wavelength). The two classical boundary conditions, transverse electric (TE) and transverse magnetic (TM) cases in electromagnetic theory, are implied. The ray optical solution of the problem (short-wave asymptotic result) is derived on the basis of the method of summation of multiple diffracted fields, which is well known in the geometric theory of diffraction. Numerical results are presented and compared with calculations based on the rigorous method of integral equation.

Optical properties of an echelette grating in the short-wave approximation

The problem of plane-wave scattering by an echelette grating with a right angle is considered in the case of high-frequency approximation (the wavelength is assumed to be small compared with the period of grating). We present the results of short-wave asymptotic analysis for a ray optical solution to the problem that was derived on the basis of the method of summation of multiple diffracted fields that is well known in the geometric theory of diffraction. A new effect of perfect blazing for an echelette grating in the high-frequency approximation is also discussed.

Calculation of the wave field scattered by an echelette grating in the short‐wave approximation: Transverse electric case

The two‐dimensional problem of the plane wave scattering by a periodic grating, an echelette with a right angle, is considered in the case of high‐frequency approximation (the wavelength is assumed to be small compared with the period of grating) and where only one face of a wedge is illuminated. The ray optical solution to the problem (short‐wave asymptotic result) derived on the basis of the method of summation of multiple diffracted fields, which is well‐known in the geometric theory of diffraction, is described. Numerical results are presented and compared with calculations based on an integral equation approach.

UTD solution for electromagnetic scattering by a circular cylinder with thin lossy coatings

A uniform geometrical theory of diffraction (UTD) solution is obtained for the field exterior to a two-dimensional circular cylinder with a thin lossy dielectric coating. The solution is convenient for engineering applications due to its simple ray format. In the lit region, the geometrical optics (GO) solution consists of the direct incident ray and the reflected ray. In the shadow region, the geometrical theory of diffraction (GTD) uses the creeping-wave format to calculate the diffracted field. In the transition regions adjacent to the shadow boundaries, where the pure ray optical solution fails, a 'universal' transition integral is used for the UTD solution to calculate the field. Numerical values for the essential transition integral are deduced, by a heuristic approach, from alternative representations of the Green's function for a circular cylinder with coating. Numerical results obtained from the UTD solution show excellent agreement with the eigenfunction results for cylinders with thin dielectric coatings. >

A uniform GTD analysis of the diffraction of electromagnetic waves by a smooth convex surface

The problem of the diffraction of an arbitrary ray optical electromagnetic field by a smooth perfectly conducting convex surface is investigated. A pure ray optical solution to this problem has been developed by Keller within the framework of his geometrical theory of diffraction (GTD). However, the original GTD solution fails in the transition region adjacent to the shadow boundary where the diffracted field plays a significant role. A uniform GTD solution is developed which remains valid within the shadow boundary transition region, and which reduces to the GTD solution outside this transition region where the latter solution is valid. The construction of this uniform solution is based on an asymptotic solution obtained previously for a simpler canonical problem. The present uniform GTD solution can be conveniently and efficiently applied to many practical problems. Numerical results based on this uniform GTD solution are shown to agree very well with experiments.

An asymptotic analysis of the scattering of plane waves by a smooth convex cylinder

An approximate asymptotic high‐frequency result which is convenient for engineering applications is obtained for the field exterior to a smooth perfectly conducting convex cylinder when it is illuminated by a plane wave. This result is uniform in the sense that it remains valid within the transition regions adjacent to the shadow boundaries where the pure ray optical solution based on the geometrical theory of diffraction (GTD) fails, and it automatically reduces to the GTD solution exterior to the transition regions where the latter solution becomes valid. Furthermore, this result is expressed in the simple format of the GTD, and it employs the same ray paths as in the GTD solution. This uniform result is not valid in the close neighborhood of the cylinder; hence a separate asymptotic result is presented for this special case in a form which is also convenient for applications. The asymptotic results presented here are useful for predicting the patterns of antennas radiating in the presence of convex conducting cylindrical structures.

Radiation from a line charge source located near an air-piezoelectric interface

The ray optical solution to the problem of a time harmonic line charge source radiating acoustic energy into a semi-infinite piezo-electric half space is outlined. This ray optical technique differs from the approach employed in recent publications [1], [2], and leads to a better understanding of bulk-wave radiation in SAW devices. PZT ferroceramic materials have been chosen for this study and numerical results are obtained and given.

Ray-optical analysis of fields on shadow boundaries of two parallel plates

The electromagnetic diffraction by two parallel plates of semi-infinite length is treated by ray methods. Two special problems are considered: (i) calculation of the fields in the forward and backward directions due to diffraction of a normally incident plane wave by two nonstaggered parallel plates; (ii) calculation of the field due to a line source in the presence of two staggered parallel plates when the source, the two edges, and the observation point are on a straight line. The crucial step in the ray-optical analysis is the calculation of the interaction between the plates. This calculation is performed by two methods, namely, the uniform asymptotic theory of edge diffraction and the method of modified diffraction coefficient. The relative merits of the two methods are discussed. The ray-optical solution of problem (i) agrees with the asymptotic expansion (plate separation large compared to wavelength) of the exact solution.

Ray-Optical Analysis of Reflection in an Open-Ended Parallel-Plane Waveguide. I: TM Case

The reflection problem for a TM mode traveling toward the open end of a semi-infinite parallel-plane waveguide, is solved by ray methods. Unlike a previous solution due to Yee, Felsen and Kelley, the present ray-optical solution is a rigorous asymptotic result, i.e., it is identical with the asymptotic expansion of the exact solution when the width of the waveguide is large compared to the wavelength. Numerical results for the modal reflection coefficients are presented and are compared with calculations based on the exact solution. It is found that the agreement between ray-optical and exact values is excellent and even better than in the approach of Yee et al., especially in the vicinity of cutoff frequencies of higher order modes.

Edge diffracted caustic fields

The fields near a caustic created by an edge diffraction process are computed using the equivalent current concept [1], [2]. These fields are shown to have the property commonly associated with ray optical analysis or the geometrical theory of diffraction, e.g., a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">90\deg</tex> phase shift as the ray passes through the caustic. This result was predicted by Kay and Keller [3] on the basis of solutions for reflected fields from singly curved surfaces. The present effort is directed toward consideration of the caustic created by an edge diffraction process. Particular attention is focused on electromagnetic excitation. The acoustic excitation for the hard boundary condition is outlined in the Appendix. In addition, our present goal is to establish the extent of the caustic region. This is of particular importance when a ray optical solution involves multiply diffracted terms in that the minimum size of the body that can be analyzed may be restricted by the extent of the caustic, i.e., the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">90\deg</tex> phase shift used in ray optical analysis may be introduced only if the caustic is contained on the surface being studied.

Ray-optical analysis of reflection in an open-ended parallel-plane waveguide: II-TE case

The reflection problem for a TE mode traveling toward the end of a semi-infinite parellel-plane waveguide is solved by ray methods. Unlike a previous solution due to Yee et al., the present ray-optical solution is a rigorous asymptotic result, i.e., it is identical with the asymptotic expansion (width of waveguide large compared to wavelength) of the exact solution. Numerical results for the modal reflection coefficients are presented and are compared with calculations based on the exact solution. It is found that the agreement between ray-optical and exact values is excellent and even better than in the approach of Yee et al., especially in the vicinity of cutoff frequencies of higher order modes.

Multiple scattering of EM waves by spheres part I--Multipole expansion and ray-optical solutions

Solution to the multiple scattering of electromagnetic (EM) waves by two arbitrary spheres has been pursued first by the multipole expansion method. Previous attempts at numerical solution have been thwarted by the complexity of the translational addition theorem. A new recursion relation is derived which reduces the computation effort by several orders of magnitude so that a quantitative analysis for spheres as large as 10\lambda in radius at a spacing as small as two spheres in contact becomes feasible. Simplification and approximation for various cases are also given. With the availability of exact solution, the usefulness of various approximate solutions can be determined quantitatively. For high frequencies, the ray-optical solution is given for two conducting spheres. In addition to the geometric and creeping wave rays pertaining to each sphere alone, there are rays that undergo multiple reflections, multiple creeps, and combinations of both, called the hybrid rays. Numerical results show that the ray-optical solution can be accurate for spheres as small as \lambda/4 in radius is some cases. Despite some shortcomings, this approach provides much physical insight into the multiple scattering phenomena.