1. [20 pts] Slab waveguide. Consider the symmetric slab waveguide with a GaAs core (n=3.5) and air cladding at the wavelength of 10 = 980 nm as shown in the following figure. n = 1 y n = 3.5 n = 1 2a (a) What is the fundamental TE mode (the TE mode with the lowest cutoff frequency) of this waveguide? Write down the expressions for the Ex in the core and cladding regions for this mode. (b) Find the maximum waveguide thickness (2a) such that the waveguide is single-mode at 10. (c) Assume 2a = 1 m, how many propagating modes does the waveguide support? (d) Assume 2a = 80 nm and find the propagation constant, the phase index, and the propagation angle of the fundamental mode of the waveguide. Use a program such as MATLAB to find the solution to the two nonlinear equations. (e) Assume 2a = 80 nm and find the propagation constant at the wavelength of 10 = 990 nm. Using the propagation constants at 990 nm and 980 nm estimate the group index of the waveguide. Assume the refractive index of core is wavelength independent.