Question

Fun with coherent modes! Suppose we have a partially coherent light wave in a slit of width 2a, with a cross-spectral density in the slit of the form, $$W(x_1, x_2) = \lambda_1 \cos(\pi x_1/2a) \cos(\pi x_2/2a) + \lambda_2 \sin(\pi x_1/2a) \sin(\pi x_2/2a).$$ where $\lambda_i$ are the non-negative eigenvalues. (a) Using trigonometric identities, write this cross-spectral density in terms of sum and difference variables $x_1 - x_2$ and $x_1 + x_2$. (b) Calculate the spectral density $S(x) = W(x, x)$ of the field. Describe what happens to the spectral density in the limits $\lambda_1 = \lambda_2 = 1/2$ and $\lambda_1 = 1, \lambda_2 = 0$. (c) Calculate the "cross-correlation" function $\chi(x)$, given by $$\chi(x) = \frac{W(-x, x)}{\sqrt{W(x, x)W(-x, -x)}}.$$ Describe what happens to the cross-correlation in the previously mentioned cases for $\lambda_1, \lambda_2$. (d) Comment on the previous two results. Which case is the least coherent, based on the results you've found?

          Fun with coherent modes! Suppose we have a partially coherent light wave in a slit of width 2a, with a cross-spectral density in the slit of the form,
$$W(x_1, x_2) = \lambda_1 \cos(\pi x_1/2a) \cos(\pi x_2/2a) + \lambda_2 \sin(\pi x_1/2a) \sin(\pi x_2/2a).$$
where $\lambda_i$ are the non-negative eigenvalues.
(a) Using trigonometric identities, write this cross-spectral density in terms of sum and difference variables $x_1 - x_2$ and $x_1 + x_2$.
(b) Calculate the spectral density $S(x) = W(x, x)$ of the field. Describe what happens to the spectral density in the limits $\lambda_1 = \lambda_2 = 1/2$ and $\lambda_1 = 1, \lambda_2 = 0$.
(c) Calculate the "cross-correlation" function $\chi(x)$, given by
$$\chi(x) = \frac{W(-x, x)}{\sqrt{W(x, x)W(-x, -x)}}.$$
Describe what happens to the cross-correlation in the previously mentioned cases for $\lambda_1, \lambda_2$.
(d) Comment on the previous two results. Which case is the least coherent, based on the results you've found?

fun with coherent modes suppose we have a partially coherent light wave in a slit of width 2a with a cross spectral density in the slit of the form wx1 x2 lambda1 cospi x12a cospi x22a 39695

Added by Howard S.

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