This paper presents a theoretical investigation of the propagation and focusing of an electromagnetic wave in an inhomogeneous dielectric whose dielectric constant is given bywhere Z-axis is the direction of propagation of the beam. The quantity ~(r, z) may either be due to non-linear effects or due to inhomogeneity in the actual fabrication of the guide.Numerical calculations show that for a cylindrically symmetrical Gaussian beam, even a small axial inhomogeneity has appreciable effect on self-focusing; if the real part of d~o(z)/dz is larger than zero the focusing effect increases with z, and vice-versa.157

The possibility of focusing a laser beam by the optical inhomogeneities induced by a second strong coaxial laser beam has been investigated. In a particular case in which the power of the strong beam equals the critical power, the medium provides an oscillatory waveguide for the weak beam. When both beams are strong the nonlinearities induced by them are additive, and both beams become focused even when their powers are below the critical powers.

The nonlinear propagation of a cylindrically symmetric microwave beam (having the radial distribution of intensity Gaussian in nature) in n-InSb has been analyzed. The analysis is phenomenological and is based on the first-order perturbation approximation. The focusing effects are mainly due to the nonparabolicity of energy bands and are seen to be important at commonly employed fields. The absorption suppresses the self-focusing effect.

The waveguide propagation of Gaussian laser beams in dielectric media with saturating non-linearity has been investigated analytically, using two profiles for the dependence of the dielectric constant on intensity. An expression for the radius of the uniform wave-guide, corresponding to a given power of the beam, has been obtained and the existence of a minimum radius, corresponding to an optimum power, has been predicted. For high power laser beams, oscillatory waveguides have been obtained; for very high powers the oscillations of the radius of the waveguide are small.

Expressions are derived for some transport coefficients in a simple model degenerate semiconductor in the presence of a strong d.c. electric field and a weak magnetic field using a distribution function of carrier velocities corresponding to an elevated carrier temperature. T, has been calculated following Shockley's treatment. The variation of these transport coefficients with electric field strength, the impurity parameter ]l&, and the chemical potential 7 has been studied. The results are presented in the form of graphs.