Complex Phase Function Research Articles

Complex polynomial phase integration

A previously published integration algorithm applicable to the numerical computation of integrals with rapidly oscillating integrands is generalized. The previous algorithm involved quadratic approximation of the phase function which was assumed to be real. The present generalization concerns approximation of a complex phase function by a polynomial of arbitrary degree. As before, the integrand is then written without approximation as a slowly varying function multiplied by the polynomial phase exponential and the slowly varying factor is approximated by a finite sum of Chebyshev polynomials. The integral is thus expressed as a sum of constituent integrals which are computed recursively via LU decomposition applied to a system of linear equations with a banded coefficients matrix. Examples are presented comparing various degree phase approximants.

ON FOURIER INTEGRAL OPERATORS

On the basis of the stationary phase method for oscillatory integrals with complex phase function, the authors prove the coincidence of Fourier integral operators and Maslov's canonical operator. Bibliography: 17 titles.

Fourier integral operators with complex phase functions and parametrix for an interior boundary value problem

(1976). Fourier integral operators with complex phase functions and parametrix for an interior boundary value problem. Communications in Partial Differential Equations: Vol. 1, No. 4, pp. 313-400.

Application of the boundary-diffraction-wave theory to gaussian beams*

Within the limits of the paraxial approximation used in treating gaussian beams, the ordinary boundary-diffraction-wave theory is also applicable to diffraction problems that involve gaussian incident beams. The total field diffracted by an aperture is thus given by the interference of two component waves: a boundary-diffraction wave and, if allowed by geometrical considerations, the unperturbed wave that would propagate freely to the observation point in the absence of the diffracting aperture. To this end, the gaussian field distribution must be described by properly defined complex amplitude and phase functions. Examples are calculated for gaussian beams with cylindrical symmetry. The general equation for the ray paths associated with the gaussian beams is also derived; it is used to show that the shadow boundary behind the diffracting screen follows a hyperbola.

Analysis of Gaussian beam propagation and diffraction by inhomogeneous wave tracking

Inhomogeneous waves behave locally like A(r) exp[ikS(r)], where A and S are spatially dependent complex amplitude and phase functions, and k is the (large) free-space wavenumber. A previously developed asymptotic theory for high-frequency propagation and scattering of such waves is here applied to the propagation and scattering of paraxial Gaussian beams. Attention is given to Gaussian beams in free space, to beams in a lens-like medium with parabolic variation of the refractive index, and to beam reflection by a cylindrical obstacle. In the latter instance, the obstacle size may be comparable to the incident beamwidth, thereby introducing substantial distortion into the reflected beam. The results obtained from the asymptotic theory are verified by comparison with rigorously derived solutions, thereby confirming the validity of the theory, which can also be applied to more general medium and obstacle configurations.