# Intensity distribution of spatially incoherent source

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.