- Question 1 A Consider A Dispersion Relation Ok Which Describes The Wave Like Propagation In A Uniform Medium Brief 1 (137.58 KiB) Viewed 69 times
Question 1 (a) Consider a dispersion relation, ok), which describes the wave-like propagation in a uniform medium. Briefly describe the main differences between a dispersion relation that is linear and one that is quadratic. Relate your answer to the propagation of photons in a dielectric and electron waves in a semiconductor. [4 marks) (b) πω cos A distributed Bragg reflector (DBR or "quarter wave stack”) comprises an infinite number of alternating layers of dielectric material with thicknesses a, b and refractive indices ng, n, The dispersion relation is given by = (1–r?) cos(kd )+r? . where ris the reflectivity of a single interface, d = a +b is the period and o, is the frequency at the centre of the first band gap. The propagation of a wave packet can be described by its group refractive index n defined by dk n = do do dk do where y = is the group velocity. dk с C =c 8 Show that the group refractive index for the DBR structure considered here can be written as πω πω sin 1 0 1-r2 sin(kd) sin 0. sin (kd) 1 - - Find the expression physical quantity denoted by ñ in the expression above and briefly explain the origin of each of the terms in square brackets. [8 marks] (11) Obtain an approximate expression (valid for small values of r, r«l) for n, at the zone boundary (k = -), and sketch the behaviour of n in the frequency range 0 0 50,- T 8 [8 marks]