= Solution
The far-field <complex amplitude> is the two-dimensional <Fourier transform> of the aperture <amplitude transmittance>, evaluated at transverse <spatial frequencies> $q_x=k\ell_x$, $q_y=k\ell_y$. The <light intensity> is proportional to the squared magnitude of this transform.
Use <linearity> to add the transforms of simpler aperture components, preserving their relative <phases>. A translated component acquires a <complex exponential> factor. Repeated identical openings can be expressed as the <convolution> of one opening with a sum of <Dirac delta functions>; the <convolution theorem> then factors the field into a single-opening transform and an array factor. Multiplying aperture functions instead convolves their transforms. These operations simplify complex apertures without repeatedly evaluating the full <diffraction> integral. It is the <complex amplitudes>, rather than their <wave intensities>, that are added for coherent illumination.
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