Fresnel number

The Fresnel number (F), named after the physicist Augustin-Jean Fresnel, is a dimensionless number occurring in optics, in particular in scalar diffraction theory.

Aperture real amplitude as estimated at focus of a half inch perfect lens having Fresnel number equal to 100. Adopted wavelength for propagation is 1 µm.

Aperture real amplitude as estimated at focus of a half inch perfect lens having Fresnel number equal to 1. Adopted wavelength for propagation is 1 µm.

Aperture real amplitude as estimated at focus of a half inch perfect lens having Fresnel number equal to 0.01. Adopted wavelength for propagation is 1 µm.

For an electromagnetic wave passing through an aperture and hitting a screen, the Fresnel number F is defined as

${\displaystyle F={\frac {a^{2}}{L\lambda }}}$

where

${\displaystyle a}$ is the characteristic size (e.g. radius) of the aperture

${\displaystyle L}$ is the distance of the screen from the aperture

${\displaystyle \lambda }$ is the incident wavelength.

The Fresnel number is a useful concept in physical optics. Conceptually, it is the number of half-period zones in the wavefront amplitude, counted from the center to the edge of the aperture, as seen from the observation point (the center of the imaging screen), where a half-period zone is defined so that the wavefront phase changes by ${\displaystyle \pi }$ when moving from one half-period zone to the next.[1] An equivalent definition is that the Fresnel number is the difference, expressed in half-wavelengths, between the slant distance from the observation point to the edge of the aperture and the orthogonal distance from the observation point to the center of the aperture.

The Fresnel number establishes a coarse criterion to define the near and far field approximations. Essentially, if Fresnel number is small – less than roughly 1 – the beam is said to be in the far field. If Fresnel number is larger than 1, the beam is said to be near field. However this criterion does not depend on any actual measurement of the wavefront properties at the observation point.

Another criterion called Gaussian pilot beam allowing to define far and near field conditions, consists to measure the actual wavefront surface curvature for an unaberrated system. In this case the wavefront is planar at the aperture position, when the beam is collimated, or at its focus when the beam is converging/diverging.[2] In detail, within a certain distance from the aperture – the near field – the amount of wavefront curvature is low. Outside this distance – the far field – the amount of wavefront curvature is high. This concept applies equivalently close to the focus.[3]

According to the user manual for the Zemax optical design software, the correct approximation for propagation in the near field follows the angular spectrum method. This approximation works well when at the observation point the distance to the aperture is of the same order as the aperture size. This propagation regime satisfies ${\displaystyle \ F\gg 1}$.

The correct approximation for the propagation in the near field is Fresnel diffraction. This approximation works well when at the observation point the distance to the aperture is bigger than the aperture size. This propagation regime verifies ${\displaystyle \ F\sim 1}$.

Finally, once at the observation point the distance to the aperture is much bigger than the aperture size, propagation becomes well described by Fraunhofer diffraction. This propagation regime verifies ${\displaystyle \ F\ll 1}$.