Consider an incoherent source radiating with spatial intensity distribution \(I(\xi, \eta)\).

(a) Using the Van Cittert-Zernike theorem, show that the coherence area of the light (mean wavelength \(\bar{\lambda}\) ) at distance \(z\) from the source can be expressed as

\[ A_{c}=(\bar{\lambda} z)^{2} \frac{\iint_{-\infty}^{\infty} I^{2}(\xi, \eta) d \xi d \eta}{\left[\iint_{-\infty}^{\infty} I(\xi, \eta) d \xi d \eta\right]^{2}} \]

(b) Show that if an incoherent source has an intensity distribution describable as

\[ I(\xi, \eta)=I_{0} P(\xi, \eta) \]

where \(P(\xi, \eta)\) is a function with values 1 or 0 , then

\[ A_{c}=\frac{(\bar{\lambda} z)^{2}}{A_{s}} \]

where \(A_{s}\) is the area of the source.

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Point source 0 -2f. Figure 5-14p 2f-