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Suppose there is a (one dimensional quasi monochromatic) spatially incoherent source with intensity distribution $I_0(x')$. What would the intensity distribution $I_z(x)$ be a distance $z$ from it (in the far field paraxial regime)?

Here is what I have tried. The field distribution at $z$ according to Fraunhofer diffraction is $$E_z(x)=\frac{e^{ikz}}{\sqrt{i\lambda z}}\int_{-\infty}^{\infty}E_0(x')e^{-ik\frac{x}{z}x'}\mathrm{d}x'.$$ This means that the field from a small section $\mathrm{d}x'$ of the source is $$\mathrm{d}E_z(x)=\frac{e^{ikz}}{\sqrt{i\lambda z}}E_0(x')e^{-ik\frac{x}{z}x'}\mathrm{d}x'$$ which gives rise to an intensity $$\mathrm{d}I_z(x)=\frac{I_0(x')\mathrm{d}x'^2}{\lambda z}.$$ Because the source is incoherent, we must add all these intensity contributions, i.e. integrate the above expression for $\mathrm{d}I_z(x)$ to find $I_z(x)$. The problem is that this expression includes a $\mathrm{d}x'^2$ so the integral tends to zero, i.e. apparently $$I_z(x)=0.$$

I highly doubt the intensity is zero everywhere, but I cannot find my mistake.

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This is because a perfectly incoherent source is an idealization which requires infinite intensity. In actuality, the minimum distance over which coherence can exist is of the order of one wavelength (see e.g. page 135 of Goodman) which gives a finite answer.

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