All electric field solutions to the scalar wave equation can be expressed as a weighted sum of plane wave solutions of different amplitudes A(k) and spatial frequencies, k, expressed as: E(*) = S A(K)e"jë*ak. For a one dimensional problem, the complex amplitude spectrum A(k,) can be found from the Fourier transform integral A(k.) = ± S E(x)e/hor dæ. Note that there are various conventions about which of the two integrals is assigned the normalization factor of 2. The pair of equations above is self consistent, but if you use a table of Fourier transforms or a mathematics package to perform a Fourier transform, your answer may not aggree to within a constant. Use this to express the following one dimensional Gaussian profile, E(z) = e as a function of one dimensional plane waves. Give an analytical solution to the amplitude A(k,), in terms of k, (k_x), Wo (w_0), and x. If you need an exponential function, use eæp(2), not e*. Hint: The result must have an amplitude which depends on Wg. You can use unit analysis or variable substitution to determine this dependency.