Electrodynamics Time remaining: 00:09:00 1. Consider the three layer planar dielectric waveguide m. Ixkd/2 n2. else with refractive profile n(x) =- where n >n2 For the TE case, with Ey component the only non-zero element, the general solution can be written as: Acos AY+B sin Ax |xkd/2 E, (x) = { Ce x d/2 Write down the boundary conditions that the TE general solution must satisfy at x=+d. a. b. Derive the transcendental equations that relate u and v for the TE symmetric and anti-symmetric modes. Where d d x = kn - B; a = B - kin ; u = xv = a c. In Ray optics, the mode in a symmetric slab waveguide can be thought of as a plane wave that is either traveling upwards or downwards with the general solution for the E field as E(x,z) = E,e kon,(cos@ x+sin =) State the condition the x-component of the propagation vector of the waveguide mode must satisfy for it to be a guided mode? 3 d. Derive the set of transcendental equations that relates the components k, and k. (ork and B) of the propagation vector. Describe how one should solve these types of equations.