A symmetric, multimode slab waveguide has a core width 2ρ, a uniform cladding index n_cl and a linear, graded-index profile n(x) in the core that is continuous and symmetric about the z-axis. The variation in the x-direction is defined by 

where nco is the maximum index on the z-axis, Δ is the relative index difference with the usual definition and the x-axis is orthogonal to the z-axis of symmetry of the waveguide.

(a)   Substitute the above profile into the second-order differential equation as given in  lectures and solve the equation to determine the curved bound ray paths for x as a  function of z. Assume that all ray paths are periodic and start at x=z=0. The initial angle the ray path makes with the z-axis at x=0 is θz, i.e. dx/dz=tanθz at z=0.

(b)   Sketch approximately the shape of a bound ray path along the waveguide. 

(c) Use (a) to determine an expression for the bound ray half-period zp, i.e. the distance between successive crossing points on the z-axis.

(d) Determine the range of values of θz for all bound rays within the core, i.e. those ray paths that do not reach the core-cladding interface and therefore satisfy |x|<ρ.