Fraunhofer and Fresnel Diffraction

Fraunhofer diffraction describes the far-field pattern of an aperture, while Fresnel diffraction describes the near field where curvature of the wavefront matters.

Definition

The analysis of the wave field diffracted by an aperture, divided into the far-field Fraunhofer regime, where the pattern is the Fourier transform of the aperture, and the near-field Fresnel regime, where wavefront curvature produces more complex patterns.

Scope

This topic covers the two principal regimes of scalar diffraction theory. Fraunhofer (far-field) diffraction applies when source and observation are effectively at infinity, giving patterns that are Fourier transforms of the aperture, including the single-slit sinc pattern and the circular-aperture Airy pattern. Fresnel (near-field) diffraction applies closer to the aperture, where the quadratic phase of the wavefront cannot be neglected, and is analysed with Fresnel zones and the Cornu spiral. The topic treats the Huygens-Fresnel and Kirchhoff formulations and the criteria separating the two regimes.

Core questions

• What distinguishes the near-field and far-field diffraction regimes?
• What is the intensity pattern of a single slit and of a circular aperture?
• How are Fresnel zones used to analyse near-field diffraction?
• How does the Kirchhoff formulation justify the Huygens-Fresnel construction?

Key concepts

• Fraunhofer regime
• Fresnel regime
• single-slit pattern
• Airy disc
• Fresnel zones
• Cornu spiral
• Kirchhoff diffraction integral
• Huygens-Fresnel principle

Key theories

Fraunhofer diffraction as a Fourier transform

In the far field the diffracted amplitude is proportional to the Fourier transform of the aperture transmission, giving the sinc pattern for a slit and the Airy disc for a circular aperture.

Fresnel diffraction and zone construction

Closer to the aperture the wavefront's quadratic phase must be retained; dividing the wavefront into Fresnel zones, or summing along the Cornu spiral, yields the near-field pattern, including the bright spot behind a circular obstacle.

Clinical relevance

Far-field diffraction by the pupil produces the Airy pattern that limits the resolution of the human eye and of ophthalmic and microscopic imaging, so understanding these patterns is essential to interpreting and optimizing medical optical instruments.

History

Fresnel's prize-winning 1818 memoir on diffraction introduced the zone construction and led Poisson to point out, and Arago to confirm, the bright spot at the centre of a circular shadow. Kirchhoff later gave a more rigorous mathematical formulation, while Fraunhofer's far-field studies underpinned spectroscopy.

Key figures

• Augustin-Jean Fresnel
• Joseph von Fraunhofer
• Gustav Kirchhoff
• François Arago

Seminal works

• hecht2017
• bornwolf1999

Frequently asked questions

When can I use the simpler Fraunhofer formula instead of Fresnel?

The Fraunhofer approximation holds when the observation distance is large compared with the aperture size relative to the wavelength, or equivalently when a lens places the aperture and screen effectively at infinity; otherwise the Fresnel treatment with wavefront curvature is required.

What is the Airy disc?

It is the central bright spot, surrounded by faint rings, in the Fraunhofer diffraction pattern of a circular aperture; its size sets the smallest point to which a circular lens can focus light.