Ellipsoidal Wave Research Articles

Uniform Asymptotic Expansions for Ellipsoidal Wave Functions I. High Frequency Solutions of the Ellipsoidal Wave Equation

High frequency solutions of the ellipsoidal wave equation have been known for some time. However these solutions were non-uniform. A method for second order turning points of differential equations is used to obtain uniform solutions of the equation. On account of the nature of the turning point these solutions are necessarily related to parabolic cylinder functions.

Common Aspects of Gaussian Beams, Ellipsoidal Waves and Multipole Radiation

A close connection is found among spheroidal waves, light beams with the Gaussian amplitude profile or the resonant mode of Fabry-Perot resonators and the multipole expansion of electro-magnetic radiation. For usual applications where aberrations are ignored, the dipole radiation term aR-1exp jγ0nR is sufficient to represent those beam waves, where R is defined through R2=( r- r0+j q0), 2 in which r is the radius vector and r0, q0, a are constant vectors specifying the position of waist, direction of propagation as well as the waist size, and polarization, respectively. Two beams, each consisting of electric dipole, magnetic dipole and electric quadrupole radiation terms, are found necessary and sufficient to match the magnetic field and tangential electric field on both sides of a spherical surface up to the fourth order.

Theory of Ellipsoidal Waves and Seidel Aberrations of Gaussian Beams

Hermite-Gaussian beam type solutions of the Helmholtz' equation are obtained by the method of separation of variables using the ellipsoidal coordinates. Reflection and refraction of ellipsoidal electromagnetic waves are investigated as vector boundary value problems, and first and third order theories are established in accordance with first and third order geometrical optics. the first order theory is formally identical to the conventional theory of mode transformation of Gaussian beams. In the third order theory spherical aberration is interpreted as a superposition of two Gaussian beams. The third order (Seidel) aberrations obtained in this paper include those of geometrical optics (Schwarzschild's perturbation eikonal) as a special limiting case.

The Angular Spectrum Representation of Electromagnetic Fields in Crystals. I. Uniaxial Crystals

The electromagnetic field in a linear, nonmagnetic, nonabsorbing uniaxial crystal which fills the entire half-space z ≥ 0 and whose optic axis is perpendicular to the plane z = 0 is represented as an angular spectrum of plane waves. The angular spectrum representation consists of a superposition of plane electromagnetic waves expressed as the sum of two integrals. In general both homogeneous and evanescent plane waves are required in each integral. Each plane wave of the spectrum satisfies the identical equations obeyed by the entire field. The homogeneous plane waves of the first integral are all ordinary waves and those of the second integral all extraordinary waves. The spectral amplitudes of the field are explicitly expressed in terms of the Fourier transform of the field in the place z = 0. The method of stationary phase is applied to the integral representation and it is thereby shown that in the far zone the field may be expressed as the sum of an outgoing (nonuniform) spherical wave and an outgoing (nonuniform) ellipsoidal wave. The amplitude of these waves, at each point on the wave surface, is expressed in terms of the Fourier transform of the field in the plane z = 0.

High‐frequency approximations to ellipsoidal wave functions

Abstract : The ellipsoidal wave equation is the ordinary differential equation which arises when the reduced wave equation 2V+ 2V = 0 is separated in ellipsoidal coordinates; doubly-periodic solutions of this equation are known as ellipsoidal wave functions. Approximations to the latter, in the form of asymptotic series valid for 2 large positive or large negative, were given but the analysis was incomplete in that the values of two integral parameters were left undetermined. This gap is filled, the complete results re-calculated and the connections established between the asymptotic series and the standard solutions in various parts of the plane. As a by-product, some unpublished transformation formulae are given, linking ellipsoidal wave functions with negative 2 to those with positive 2. (Author)

A comparative study of Ellipsoidal gaussians: H 2+

A series of calculation is carried out for H 2 + using ellipsoidal Gaussian wave functions. Comparison to spherical Gaussians shows that the additional variational freedom results in lower energy expectation values.

Integral equations and relations for Lamé functions and ellipsoidal wave functions

AbstractNon-linear integral equations and relations, whose nuclei in all cases is the ‘potential’ Green's function, satisfied by Lamé polynomials and Lamé functions of the second kind are discussed. For these functions certain techniques of analysis are described and these find their natural generalization in ellipsoidal wave-function theory. Here similar integral equations are constructed for ellipsoidal wave functions of the first and third kinds, the nucleus in each case now being the ‘free space’ Green's function. The presence of ellipsoidal wave functions of the second kind is noted for the first time. Certain possible generalizations of the techniques and ideas involved in this paper are also discussed.

Asymptotic Expansions of Ellipsoidal Wave Functions and their Characteristic Numbers

AbstractIt is shown that a simple pair of asymptotic expansions exists for solutions of the ellipsoidal wave equation. An asymptotic expansion for the characteristic numbers is also obtained.

On Asymptotic Expansions of Ellipsoidal Wave Functions

AbstractIn two previous papers it was shown that three pairs of asymptotic expansions exist for the solutions of the ellipsoidal wave equation: One pair in terms of Jacobian elliptic functions and two pairs in terms of Hermite functions. In the present investigation we show that the expansions can be joined in regions of common validity. We then identify these solutions with known functions, and finally derive asymptotic expansions for their normalization constants.

Asymptotic Expansions of Ellipsoidal Wave Functions in Terms of Hermite Functions

AbstractTwo simple pairs of asymptotic expansions in terms of Hermite functions are obtained for the solutions of the ellipsoidal wave equation.

Electric Dipole Fields in an Axially Symmetric Anisotropic Plasma

Electric dipole fields in an axially symmetric anisotropic medium are calculated by means of the cylindrical coordinates. Comparing with the spherical waves in an isotropic medium, it becomes clear that the anisotropy deforms the spherical waves into ellipsoidal waves, then it splits into two kinds of waves with different phase velocities. Far fields and near fields are also discussed.

VI.—Neumann-Series Solutions of the Ellipsoidal Wave Equation

SynopsisThe ellipsoidal wave equation is the name given to the ordinary differential equation which arises when the wave equation (Helmholtz equation) is separated in ellipsoidal co-ordinates. In this paper, solutions of the equation are expressed as Neumann series (series of Bessel functions of increasing order).

A New Treatment of the Ellipsoidal Wave Equation

A New Treatment of the Ellipsoidal Wave Equation