l. Hermite-Gauss solutions for the parabolic index waveguide Following the solution for the HG eigenfunctions forthe quantum simple harmonic oscillator, calculate the guidedmode solutions for the parabolic index waveguide. Use the index profile shown in the notes' a. find the separable wave equation For the symmetric waveguide, with the refractive index profile, n2 (x, y) = of [1 % (X2 +y2)} with k0 = no 9 n c write the wave equation for the electric field let E = u{x,y) ellk'z'w'l and find an equation for LI Rearrange the equation to Show that the equation is separable, so you can let u (x, y) = f(x) g(y). Hint: there will be separate eigenvalues for the k, and ky components of the wavevector. b. work with the equation for fix) and write it in dimensionless units, lettingx = ax' c. by comparing to the quantum problem and the Hermite equation, find an expression for the quantized transverse wavenumbers kx. and km, the propagation constant km and the fu nctions forthe guided modes. Express all of these in terms of the radius of the guided mode, we. d. Suppose a Gaussian beam with a radius w1!= we enters the waveguide. Calculate the modal expansion of this input beam in terms of the guided modes and show that the beam preserves its Gaussian shape as it propagates