PHYS 630: Homework IVdue date:Thursday, November 13th, 2008 at class meeting.You are welcome to work together. If you partially use work from other (e.g.something you might have fo und in a b ook or a journal paper), you should properlycredit the author by citing the material used.1. Refractive index of air (20 pts) The refractive index of air can be pre-cisely measured with the help of a Michelson interferometer and a tunablelight source. At atmospheric pressure and 20otemperature, the atmosphericrefractve index differs from unity by n − 1 = 2.672 × 10−4for λ = 0.76 µm,n−1 = 1.669×1 0−4for λ = 0.81 µm, and n−1 = 2.665×10−4for λ = 0.86 µm.(a) Using a quadratic fit of these data, find the wavelength dependence ofthe group velocity (10 pts).(b) Estimate the value for the dispersion Dλ(10 pts).2. Amplitude modulation using electro-optical effect (30 pts): We con-sider a KDP crystal, of length L, followed by a wave plate sandwiched betweentwo orthogonal polarizers; see Fig. 1. The principal transverse axes of thecrystal are rotated by 45 deg compared to the (x, y) system (see F ig. 1). Thenon-vanishing elements for the electro-optics tensor of the KDP crystal are r63and r41= r32. We take the externally applied electric field Eato be along theˆz-axis (the correspo nding voltage is V = EaL).(a) Write down the dielectric permittivity tensor in presence of the Eafield(5 pts).(b) Find the refractive index asso ciated t o the principal axes (15 pts).(c) Consider a plane wave with wavelength λ propagating along t he ˆz-direction.What is the phase shift between the two transverse (⊥ ˆz) principal po-larization? [you will assume that r63Eais small so that the phase shift islinearly dependent on Ea] (10 pts).(d) Give the ratio of input-to-output optical intensity as a function of Ea(10pts).13. Jones matrix of a rotated linear polarizer (20 pts) Show that the Jonesmatrix of a linear polar izer with transmission axis making an angle theta withthe ˆx-axis is given byT =cos2θ sin θ cos θsin θ cos θ sin2θ.4. Density of mode in a three-dimensional resonator (30 pts): We con-sider a cubic resonator with sides length d.(a) Write down the condition on the components of the wavevector k = 2πν/cimposed by the boundary conditions. Deduce the general form of thewavevector modulus (5 pts).(b) In the k space what is the volume occupied by all the mode (considera lar ge number of mode so that the continuum approximation is valid)(5pts).(c) Estimate the volume associated to one mode, and deduce the number ofmode lying in the frequency interval [0,ν] (10 pts).(d) From the number of mode in [ν, ν +∆ν], deduce the density o f modes, i.e.the number of mode per unit volume of the resonator per unit bandwidthsurrounding the frequency ν (10 pts).Figure 1: Figure associated to Problem 2. x′and y′are the two tra nsverse principalaxes associated to the