The Cluster's Huygens-Fresnel Principle

A page about the Huygens-Fresnel principle — the wavefront construction for diffraction.

The Huygens-Fresnel principle

The Huygens-Fresnel principle states that every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront at a later time is the envelope of these wavelets. Fresnel added interference: the amplitude at any point is the superposition of all wavelets, with the correct phase.

The Kirchhoff diffraction formula gives the rigorous version: E(P) = -(i / lambda) integral_S E(Q) (e^{ikr} / r) (cos(theta) / 2) dA where Q is a point on the aperture S, r is the distance from Q to P, theta is the angle between the normal and r, and the factor (cos theta / 2) is the obliquity factor.

In the cluster, the edit Huygens-Fresnel principle states that an edit wavefront is constructed from edit wavelets.

Fraunhofer vs. Fresnel diffraction

• Fresnel diffraction (near-field): The source or screen is at a finite distance. The phase factor e^{ikr} must be expanded fully.
• Fraunhofer diffraction (far-field): Source and screen are at infinity (or lenses are used). The integral reduces to a Fourier transform: E(P) ~ integral E(Q) e^{-ik x sin(theta)} dA.

In the cluster, the edit Fraunhofer diffraction gives an edit Fourier transform.

The single slit

For a single slit of width a, the intensity pattern is: I(theta) = I_0 (sin(beta) / beta)^2 where beta = (pi a sin theta) / lambda. The first minimum is at sin theta = lambda / a. The central maximum has width 2 lambda / a (in sin theta space).

In the cluster, the edit single slit gives an edit intensity pattern.

The double slit

For two slits of width a separated by distance d: I(theta) = I_0 (sin beta / beta)^2 cos^2(gamma) where beta = (pi a sin theta) / lambda and gamma = (pi d sin theta) / lambda. The fine fringes (cos^2) are modulated by the single-slit envelope (sin^2/beta^2).

The fringe spacing: Delta x = lambda L / d (for L >> d). In the cluster, the edit double slit gives an edit fringe pattern.

Applications

• Diffraction gratings: N slits give sharp peaks at d sin theta = m lambda, with angular width Delta theta ~ lambda / (N d cos theta)
• Optical instruments: The Rayleigh criterion (resolution limit): theta_min = 1.22 lambda / D for a circular aperture of diameter D
• Crystallography: Bragg's law n lambda = 2 d sin theta from the same mathematics
• Antenna theory: The radiation pattern of an antenna array is mathematically identical
• Astronomy: Interferometry (Aperture synthesis) uses the same principles

In the cluster, edit applications include:

• edit Diffraction gratings
• edit Optical instruments
• edit Crystallography
• edit Antenna theory
• edit Astronomy

This principle

This page is about the Huygens-Fresnel principle. Every point on a wavefront is a source. Fraunhofer: E(P) ~ integral E(Q) e^{-ik x sin theta} dA. Single slit: I = I_0 (sin beta / beta)^2. The principle is real.