A semiconductor slab waveguide is formed by a two different semiconductor layers, with the core of thickness d being GaAs and both cladding layers being AlGaAs, as shown in the figure. The index of refraction of the ternary layer (Al, Ga-As) depends upon the fractional aluminium content x, and has a lower index of refraction than GaAs. At an optical wavelength = 850 nm, the refractive index of GaAs is n GaAs = 3.59, while that of Al, GaAs is n = 3.59 - 0.710x + 0.091x Rectangular Ship AlGaAs n GaAs nGaAs = 3.59 AlGaAs n With these parameters, design a symmetric single-mode waveguide for TE polarization by properly choosing the waveguide core thickness & and the material composition x. (a) [3 marks] If d is fixed at 1.5 m, what is the range of x that allows only the fundamental mode? (b) [3 marks] If x is fixed at 0.3 m, what is the range of a that allows only the first 3 TE modes? (c) [3 marks] Using the upper bound of the values found in part (b), calculate the effective index of the first 3 TE modes. (d) [3 marks] Plot the first 3 modes in the waveguide - your plot should include the fields in the Al Ga-As and GaAs layers. (e) [3 marks] Design a single-mode waveguide (by varying x, d, or x and d) that has a confinement factor of TTE = 0.5 for the TE, mode. d