Past exam of the physics A course of the University of Cambridge / 2022 / Paper 2 / Section B / B7 / a

For the normally incident plane wave, the incident complex amplitude and phase are uniform across the aperture. Treat the field using scalar diffraction, with the aperture represented by its prescribed amplitude transmittance $h(x,y)$. Let $R$ be the distance from the aperture centre to the observation point and let ${\ell }_x,{\ell }_y$ be the transverse direction cosines of that point.

Across an aperture of characteristic size $a$, the Fraunhofer diffraction assumptions are that $a/R\ll 1$, so $1/r\simeq 1/R$ and the obliquity factor is nearly constant, and that the quadratic contribution to the propagation phase is negligible:The remaining path-length dependence is linear:A coherent monochromatic wave and an aperture large enough relative to wavelength for the scalar aperture model complete the usual physical assumptions. The complex amplitude becomeswhere $C$ includes the common propagation phase, distance factor and nearly constant obliquity factor. For the central part of a planar screen in the paraxial approximation, ${\ell }_x\simeq X/R$ and ${\ell }_y\simeq Y/R$. Small observation angles are useful for that screen-coordinate formula; a general far-field Fourier transform can instead use the exact direction cosines. A lens can also produce this pattern in its focal plane without requiring a physically very distant screen.