Fraunhofer diffraction and aperture transforms

Start with the Fresnel integral and expand the transverse squared distance into an observation-only term, a cross term, and an aperture-only term. For an aperture of radius or half-width $a$, the aperture phase has scale $\pi a^2/(\lambda z)$. When this is much smaller than one radian, replace its exponential by unity. The observation-only quadratic phase may be retained: it has unit modulus and does not alter the intensity pattern.

Define the Fourier transform with a negative exponential and spatial frequencies in cycles per metre. The physical observation coordinates select frequencies $f_x=x/(\lambda z)$ and $f_y=y/(\lambda z)$ in the paraxial limit. Frequencies specified as angular wavenumbers instead require factors of $2\pi$ in a different place. Mixing those conventions shifts every predicted zero, even when the qualitative pattern still looks convincing.

For constant incident phasor $U_a$ across a centred rectangular opening of full widths $b$ and $h$, integrate each coordinate independently. Each integral is its width multiplied by the normalized sinc function. At zero frequency both sinc factors approach one, so the on-axis field magnitude is $|U_a|bh/(\lambda z)$. This provides a useful dimensional and limiting check on the transform calculation.

Dividing the intensity by its on-axis value removes the area factor and yields a shape function. It does not imply equal throughput for different aperture widths. With fixed incident irradiance, widening the aperture increases transmitted power, narrows the angular pattern, and raises the central intensity. A calculation that rescales every pattern to the same peak can hide two of those three effects, so always state which quantity has been normalized.

Translating an aperture field by a fixed amount multiplies its Fourier transform by a phase ramp. Its isolated far-field intensity is therefore unchanged when the illumination translates with it. Tilting the incident plane wave multiplies the aperture field by a phase ramp instead; the shift theorem then moves the transform in spatial frequency. These are reciprocal operations, not two descriptions of the same displacement.

A lens can produce a Fourier-transform plane at its back focal plane within paraxial optics, because its quadratic phase cancels the relevant propagation phase. Thus a laboratory need not put a screen kilometres away to inspect the far-field angular distribution. Finite lens aperture, aberrations, and polarization still matter. The statement concerns a suitably arranged coherent optical system, not every camera image or every plane behind a lens.