A quasi-monochromatic

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Interference and diffraction · Coherence and statistical optics

What an Instrument Can See, Ch Spatial coherence and the van Cittert-Zernike theorem

Problem

A quasi-monochromatic, stationary scalar field

V_0(\boldsymbol{\xi})

in a source plane propagates through free space to two points

\mathbf{r}_1

and

\mathbf{r}_2

via the linear Fresnel propagation integral

V(\mathbf{r}) = \int V_0(\boldsymbol{\xi})\,h(\mathbf{r}-\boldsymbol{\xi})\,d\boldsymbol{\xi}

, where

h

is the free-space impulse response. ((a)) Starting from this integral applied at

\mathbf{r}_1

and at

\mathbf{r}_2

, derive the propagation law for the mutual intensity

J(\mathbf{r}_1,\mathbf{r}_2)=\langle V(\mathbf{r}_1)V^*(\mathbf{r}_2)\rangle

in terms of the source mutual intensity

J_0(\boldsymbol{\xi}_1,\boldsymbol{\xi}_2)=\langle V_0(\boldsymbol{\xi}_1)V_0^*(\boldsymbol{\xi}_2)\rangle

. ((b)) State the two physical assumptions used in moving the ensemble average inside the double integral.