Propagation of electromagnetic surface waves in a radially inhomogeneous optical waveguide

The propagation of electromagnetic surface waves along a radially inhomogeneous dielectric waveguide is investigated. The problem is formulated in terms of differential equations to be satisfied by the radially dependent parts of the electromagnetic field vectors. The dielectric waveguide is assumed to consist of a homogeneous cladding of infinite extent and a radially inhomogeneous core of higher permittivity. Numerical solutions of the differential equations in the core are obtained by two different methods, viz. by direct numerical integration and by substitution of an appropriate power series expansion. In the cladding the field is expressed in terms of modified Bessel functions. Imposing the boundary conditions at the interface of core and cladding, an equation for the unknown propagation coefficient is obtained. From this equation the propagation coefficients for the lower order modes are computed numerically. Numerical results are presented for some permittivity profiles of practical interest in single-mode transmission along optical fibres.Nomenclature a radius of the core - a_q vector coefficient in the power series expansion off() - A_i constants - A square matrix - b_q coefficient in the power series expansion of_r() - B square matrix - C square matrix - c_n unknown constant - d_n unknown constant - D() fundamental matrix - E,E_r,,z electric field vector and components - E,e_r,,z radially dependent parts ofe_r,,z - f solution vector - G_q square matrix - H,H_r,,z magnetic field vector and components - h_r,,z radially dependent parts ofE_r,,z - h reduced wavenumber - i radial mode number - j imaginary unit - k_0,m wave number - K_n modified Bessel function of the second kind and order n - n azimuthal mode number - t time - U normalized propagation constant - Z_m plane wave impedance of the cladding - r, , z cylindrical co-ordinates - p, q, s integers - propagation coefficient - increment - _0,m,r permittivity - normalized radiusr - ₀ wavelength in free space - ₀ permeability - angular frequency - d_r, differentiation with respect tor, Engineering and Professor H. J. Frankena of the Physics Department for their valuable discussions.