Fresnel diffraction and near fields

Define the Fresnel number using an explicitly chosen transverse aperture scale $a$, taken here as a radius or half-width. The aperture-dependent quadratic phase reaches approximately $\pi N_F$ at that radius on axis. Large or order-one Fresnel number warns against discarding that phase. A small Fresnel number supports a Fraunhofer approximation, but the required smallness depends on the permissible field or intensity error, rather than a universal boundary exactly at one.

Fresnel propagation itself still neglects higher terms of the distance expansion and assumes small propagation angles. It can remain accurate for a large Fresnel number when those higher phase terms are small. Conversely, a label such as far away in metres cannot establish a far-field regime without the aperture scale and wavelength. Doubling an aperture dimension requires four times the distance to keep the same Fresnel number, which often makes an apparently minor experimental change consequential.

On axis, the excess path from radius $r$ is approximately $r^2/(2z)$. Boundaries where this excess increases by half a wavelength define half-period Fresnel zones. Neighbouring zones then contribute roughly opposite phases. Their areas are approximately equal because squared radius, rather than radius, increases by a constant increment. Slowly varying amplitudes allow strong cancellation between neighbouring annuli; stopping the aperture at a different zone can therefore increase or decrease the central brightness.

This reasoning explains why more open area need not increase irradiance at a particular screen point. Energy conservation constrains power integrated over an appropriate whole surface, not monotonic brightness at every pixel. Zone arguments also require care at edges: the contributions form a curved path in the complex plane, and treating each zone as one exactly equal arrow is a qualitative approximation. The actual Fresnel integral remains the quantitative statement.

A periodic aperture field can be expanded into transverse Fourier orders of pitch $d$. Under paraxial propagation, order $m$ acquires a relative phase $-\pi\lambda z m^2/d^2$, after removing the common carrier phase. Unlike a far-field picture that spatially separates orders, a near-field plane lets them overlap. Their changing relative phases can reconstruct, shift, or reshape the original periodic field as distance changes.

At the paraxial Talbot distance $2d^2/\lambda$, every integer order gains an integer multiple of $2\pi$, so an ideal infinite periodic field reproduces itself up to a common phase. At half that distance, the factor is $(-1)^m$, equivalent to a half-period lateral translation. Real finite gratings have edge diffraction and limited useful repetitions. High orders also violate the paraxial approximation, so the simple distance is a model prediction with explicit angular limits.