Starting from the propagation law for an incoherent source of intensity distribution

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

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

Problem

Starting from the propagation law J(r1,r2)=∬J0(ξ1,ξ2) h(r1−ξ1)h∗(r2−ξ2) dξ1dξ2J(\mathbf{r}_1,\mathbf{r}_2) = \iint J_0(\boldsymbol{\xi}_1,\boldsymbol{\xi}_2)\,h(\mathbf{r}_1-\boldsymbol{\xi}_1)h^*(\mathbf{r}_2-\boldsymbol{\xi}_2)\,d\boldsymbol{\xi}_1 d\boldsymbol{\xi}_2 for an incoherent source of intensity distribution I0(ξ)I_0(\boldsymbol{\xi}), for which J0(ξ1,ξ2)=κI0(ξ1)δ(ξ1−ξ2)J_0(\boldsymbol{\xi}_1,\boldsymbol{\xi}_2) = \kappa I_0(\boldsymbol{\xi}_1)\delta(\boldsymbol{\xi}_1-\boldsymbol{\xi}_2) with κ\kappa a constant: ((a)) Collapse the double integral using the delta function, and insert the paraxial Fresnel kernel h(r−ξ)∝exp⁡ ⁣[ik∣r−ξ∣2/2z]h(\mathbf{r}-\boldsymbol{\xi}) \propto \exp\!\big[ik|\mathbf{r}-\boldsymbol{\xi}|^2/2z\big] for propagation over a distance zz, to show that μ12\mu_{12} is, apart from a phase factor independent of the source, the normalised Fourier transform of I0I_0 evaluated at spatial frequency (r1−r2)/(λz)(\mathbf{r}_1-\mathbf{r}_2)/(\lambda z). ((b)) Verify that ∣μ12∣=1|\mu_{12}| = 1 when r1=r2\mathbf{r}_1=\mathbf{r}_2. ((c)) State the two conditions this derivation requires beyond incoherence of the source.

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