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I have an aperture of length a and width b (sizes are comparable) How do I calculate the irradiance for a point on the screen? I know that I should calculate the electric field due to an element of area, then integrate over the whole area of the aperture, and then use the amplitude to calculate the irradiance. But I don't know how to express the optical path difference $\Delta$ between any element and the element at the center of the aperture. This is all I've been able to write down:

$dE = \frac{dE_{0}}{r_{0}+\Delta } e^{i[k(r_{0}+\Delta )-\omega t]}\simeq \frac{E_{A}dA}{r_{0}}e^{i[k(r_{0}+\Delta )-\omega t]}$

Where $dE_{0}$ is the amplitude of element dA at unit distance away, $E_{A}$ is amplitude per unit area at unit distance away, $r_{0}$ is the optical path of the central element, and $dA=dxdy$.

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    $\begingroup$ We have the MathJax rendering engine active on the site explicitly to allow you to write clear and readable math in a LaTeX math-mode-alike language.. A picture or scan of a handwritten document is neither searchable nor something that a page reader can parse. This is something that you'll want to fix. $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Nov 23, 2016 at 20:04

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As you may be able to see from your expression, the Fraunhofer (far field) diffraction pattern is the Fourier transform of the aperture function.

The FT of an infinite slit is a sinc function; the FT of the product of two slits (at right angles: that is what a rectangular aperture is) is the convolution of two sinc functions at right angles.

You should be able to do the math from here.

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