Scalar Wave Approximation Research Articles

Polarization effects in the diffraction of electromagnetic waves: the role of disclinations

Three-dimensional diffraction patterns of monochromatic electromagnetic waves contain moving line singularities where the magnitude of the transverse field is zero, and its direction is therefore indeterminate. They are called disclinations, by analogy with the corresponding linear features in liquid crystals. A disclination in a vector wave is a natural generalization of a dislocation in a scalar wave. Where the scalar wave approximation in optics predicts a dislocation, or interference null, the full vector theory reveals a double helix structure: a disclination line in the electric field and another in the magnetic field winding around each other with a spacing of order (wavelength/2π). A perturbing plane wave causes this composite structure itself to become coiled. When there is a ‘polarization effect’ in the diffraction pattern a disclination in the electric field becomes a moving helix or, more generally, a coiled coil. As it moves it sweeps out a surface on which the polarization is everywhere linear. In optical experiments this observable surface is the most significant effect of disclinations. In general, however, the disclinations constitute elements of structure of the electromagnetic field, and their arrangement thus provides an effective way of describing the three-dimensional geometry of even very complicated diffraction fields, for example of microwaves.

Generalized Curvilinear Diffraction Gratings: I. Image diffraction patterns

This paper is the first in a series on the diffraction and imaging properties of generalized curvilinear diffraction gratings. The ‘image diffraction pattern’ (IDP) of a generalized grating is defined as the optical image that is observed by the naked eye focused on the grating when it is illuminated by an arbitrarily extended diffuse source. In this paper, a general theory of the IDP is derived in both the scalar wave and geometric optics approximations. As an application of the theory, it is shown how the IDP of a generalized grating may be simulated by an optical data processing system in which the Fresnel transform of a low line density representation of the grating is filtered by an aperture of the same shape and orientation as the source. The theory is shown to lead to a simple explanation of the moiré effect first observed by Hamann and to a new technique for performing optical map transformations. As a practical test on the theory, the IDP of a circular diffraction grating is calculated and shown to be in good agreement with experiment.

Attenuation of short period seismic waves due to scattering

We point out first the inadequacy of the two widely used approaches in calculating the amplitude attenuation of seismic waves in a random medium, the formulation of mean field attenuation and that of the scattering coefficient under the single‐scattering approximation. Then we calculate the average amplitude attenuation due to scattering in an infinite random slab. We slice the random slab into layers of thickness a correlation length and use the Born approximation for each slice to calculate the scattered field. To include the effect of multiple scattering, we take the back‐halfspace integration of the scattered energy as the energy loss and do energy correction for each successive slice. Taking scalar wave approximation for seismic waves, we get a formula for average amplitude attenuation essentially valid for high frequency range (ka»1). The attenuation depends strongly on the form of correlation function of the random inhomogeneities. We derive the formulas for Gaussian, exponential and Von Kárman correlation functions. The frequency dependence of attenuation by our formulation agrees well with experiments.

Cutoff frequencies of vector wave modes in cladded inhomogeneous optical fibre

Asymptotic vector wave formulas for calculating the cutoff frequencies in optical fibres with graded-index core and uniform cladding are derived. Numerical examples are given in comparison with the scalar wave solutions, in order to show that the mode degeneracy present in the regime of scalar wave approximations is removed and the cutoff values in degenerate states split slightly.

Electromagnetic waves in two-dimensionally inhomogeneous media*

The investigation of electromagnetic waves in two-dimensionally inhomogeneous media leads to an approximation that decouples Maxwell’s equations. The two equations obtained for the longitudinal field components are free from the inconsistencies of the scalar wave approximation. For cylindrically symmetric media the resulting radial equations are basically equivalent to those already derived by Kurtz and Streifer, but a more refined WKB-type approximation of the solutions, using the “method of comparison equations,” leads to the result that all guided modes are split with respect to scalar propagation constants. Calculations carried out for quadratic media indicate that this splitting does not decrease with frequency in the way predicted by previous treatments, using first-order perturbation. The present theory is equally valid for scattered modes, for which an approximate phase shift formula is provided.

Vectorial Wave Analysis of Inhomogeneous Optical Fibers Using Finite Element Method

A vectorial wave analysis of the propagation characteristics of radially inhomogeneous optical fibers is presented. The vectorial wave equation is first translated into a variational problem, and then it is solved by using the finite element method. The results are compared with those of the scalar wave analysis. The error caused by the scalar wave approximation is discussed for wide variety of refractive index profiles. It is shown that the error caused by the scalar wave approximation is about 0.1 percent for eigenvalues and 1 percent for delay time, when the relative index difference between core and cladding is 1 percent.

Pulse dispersion in a lens-like medium

An analysis is given of dispersion in an optical fibre waveguide having a continuous radial variation of refractive index using the scalar wave approximation. Solutions are presented for the particular case of Selfoc fibre taking into account mode dispersion, material dispersion and group delay. It is shown that for a correctly matched input Gaussian beam the pulse dispersion is small although in practice it is likely to be ∼ 1 ns over a length of 1 km.