- #1

cpburris

Gold Member

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- Homework Statement
- This is problem 5-9 in Goodman's "Statistical Optics".

In Young's interference experiment (diagram below), the normalized power spectral density ##\hat G(\nu)## of the light is measured at a point Q by a spectrometer. The mutual coherence function of the light is known to be separable, $$ \tilde \Gamma (P_1,P_2,\tau)=\tilde \mu (P_1,P_2) \tilde \Gamma (\tau).$$ Show that under the condition ##(r_2-r_1)/c\gg\tau_c##, when no interference fringes are observed, and assuming the intensities of the light from pinholes ##P_1## and ##P_2## at point Q are the same, show that ##\tilde \mu(P_1,P_2)## can be measured by examining the fringes that exist in the normalized spectrum ##\hat G_Q(\nu)## of the light at Q. Specify how both the modulus and the phase of ## \tilde \mu(P_1,P_2)## can be determined.

- Relevant Equations
- Given as hints:

$$ \tilde u (Q,t) = \tilde K _1 \tilde u (P_1, t- \frac {r_1} c )+ \tilde K _2 \tilde u (P_2,t- \frac {r_2} c ) ~~~~~~~~~~~~~~~~(1)$$

$$ \tilde \mu (P_1,P_2) = \tilde \gamma (P_1,P_2,0)= \frac {\tilde J (P_1,P_2)} {\left[ I(P_1)I(P_2) \right] ^\frac 1 2}~~~~~~~~~~~~~~~~~~~~~~~(2) $$

$$G(\nu)=\lim_{n \rightarrow \infty} \frac {\left| \tilde \upsilon _T \right| ^2} T~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(3) $$

## \tilde \upsilon _T## is the Fourier transform of the truncated form of ##u(t)##.

I wasn't sure what to do, so I started with equation (1) and used it to derive the power spectral density at point Q, $$ \begin{align} \tilde G (Q,\nu) = ~ & K_1^2 G(P_1,\nu)+K_2 ^2 G(P_2,\nu) \nonumber \\ & + 2 K_1 K_2 \left| \tilde G (P_1,P_2,\nu) \right| \cos \left[ 2 \pi \nu \frac {r_2-r_1} c - \psi (P_1,P_2,\nu) \right] \nonumber \end{align} $$

Since ##(r_2-r_1)/c\gg\tau_c##, the intensity should be (approximately) constant, $$I(Q)=~2I^{(1)}(Q)=~2I^{(2)}(Q)$$ where ##I^{(1)}(Q)## is the intensity at point Q from ##P_1## and similarly for ##I^{(2)}(Q)##. By this and the integral of the power spectral density over all frequencies is equal to the intensity, I should be able to reduce the power spectral density to $$ \tilde G (Q,\nu) = ~ K_1^2 G(P_1,\nu)+K_2 ^2 G(P_2,\nu) $$.

Then $$ \hat G (Q,\nu) = \frac 1 2 \left[ \frac {G^{(1)}(Q,\nu)} {I^{(1)}(Q)} + \frac {G^{(2)}(Q,\nu)} {I^{(2)}(Q)} \right]$$

I'm not sure if that is useful or even correct, but I couldn't figure out anything else to do.

Other equations I know which may possibly be useful:

$$ G(P_1,P_2,\nu)=~ \sqrt {I(P_1)I(P_2)} \tilde \mu (P_1,P_2) \hat G (\nu) $$

$$ \hat G (\nu) = \frac {G^{(r,r)}(\nu)} {\int_0 ^\infty G^{(r,r)}(\nu) \, d\nu} $$

$$G(\nu)=4G^{(r,r)}(\nu)$$

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