Huygens' principle

Huygens' principle states that every point on a wavefront can be treated as the source of a small spherical wavelet travelling at the local wave speed, and that the wavefront at a later time is the surface tangent to all these wavelets. Christiaan Huygens set it out in his Traité de la Lumière (1690) and used it to explain reflection, refraction and the double refraction of calcite. In 1818 Augustin Fresnel added that the wavelets carry phase and interfere with each other; the resulting Huygens-Fresnel principle predicts diffraction patterns quantitatively and underlies the diffraction integrals used today. As a scale for the effects it describes: at 500 nm, an observer 1 m behind an aperture sees the aperture divided into Fresnel zones whose first has a radius of 0.71 mm.

Refraction from wavelets

At a boundary between media of refractive index $n_1$ and $n_2$, wavelets travel at $c/n_1$ above and $c/n_2$ below. A plane wavefront meeting the boundary obliquely starts its wavelets at different times along the surface, and the envelope of the slower wavelets in the second medium is a plane tilted toward the normal. Equating the time the incident front needs to sweep along the interface with the time the refracted wavelets need to reach the new front gives Snell's law, $n_1 \sin\theta_1 = n_2 \sin\theta_2$. For light entering glass of index 1.5 at 30° the construction gives a refraction angle of 19.5°. Huygens extended the argument to calcite by letting the wavelets for one polarization be ellipsoids, which accounts for the extraordinary ray in birefringence.

The Huygens-Fresnel integral

In its quantitative form, the field at a point $P$ behind an aperture $\Sigma$ is a sum of wavelets from every point $Q$ in the aperture:

$$U(P) = \frac{1}{i\lambda}\iint_{\Sigma} U(Q)\, \frac{e^{ikr}}{r}\,\cos\theta\,dS,$$

where $r$ is the distance from $Q$ to $P$, $k = 2\pi/\lambda$, and $\theta$ is the angle between $r$ and the aperture normal. This is the first Rayleigh-Sommerfeld solution. Kirchhoff's 1882 derivation from the wave equation gives a similar form with an obliquity factor $(1 + \cos\theta)/2$; the two agree near the axis and differ only at large angles. Two features of the formula are absent from Huygens' original picture: the factor $1/(i\lambda)$, which makes the secondary sources a quarter-cycle ahead in phase and scales their strength with $1/\lambda$, and the obliquity factor, which suppresses the backward wave that the simple construction would otherwise produce. In the paraxial limit the integral becomes the Fresnel diffraction integral, and in the far field the Fraunhofer integral, the starting points of Fourier optics.

Fresnel zones

Fresnel evaluated the integral by dividing the wavefront into annular zones whose distances to $P$ differ by successive half wavelengths. For an observation point at distance $z$ on axis, the outer radius of zone $n$ is approximately

$$r_n = \sqrt{n \lambda z}.$$

At 500 nm and $z = 1$ m, $r_1 = 0.71$ mm and $r_{10} = 2.24$ mm. Adjacent zones contribute with opposite sign and nearly cancel, so the on-axis intensity behind a circular aperture oscillates as the aperture opens through successive zones. The number of zones an aperture of radius $a$ contains is the Fresnel number $a^2/(\lambda z)$: 1.58 for a 1 mm radius at 632.8 nm and 1 m.

Two classic results follow. A Fresnel zone plate, which blocks alternate zones, focuses light like a lens with focal length $f = r_1^2/\lambda$; a plate with $r_1 = 0.5$ mm has $f = 0.40$ m at 632.8 nm, and zone plates with much finer zones are the standard focusing optic for soft X-rays. And behind an opaque disk, the unobstructed zones around the edge add in phase on the axis, giving a bright spot at the centre of the shadow. Poisson raised this prediction in 1818 as an objection to Fresnel's theory, and Arago observed the spot shortly after.

Where it is used

The Huygens-Fresnel integral is the basis for computing diffraction from apertures and obstacles, the Airy disk of a circular lens, the patterns of gratings, and the near fields of antennas and phased arrays. Beam-propagation codes and ultrasound and seismic imaging sum secondary sources in the same way. The construction is also used qualitatively to explain interference in slit experiments and the bending of waves around edges.

Limitations and pitfalls

The principle is a mathematical construction; the secondary sources are not physical emitters. The scalar form ignores polarization, so it is inaccurate within a few wavelengths of an aperture edge, for apertures comparable to the wavelength, and for high-NA focusing, where vector diffraction theory is needed. Kirchhoff's boundary conditions are mathematically inconsistent, though accurate for apertures many wavelengths across. When applying $r_n = \sqrt{n\lambda z}$, the illumination must be a plane wave; for a point source at a finite distance, $z$ is replaced by the reduced distance $z_1 z_2/(z_1 + z_2)$.

Common questions

Why do Huygens wavelets not produce a backward wave?

In the simple construction they would. Kirchhoff's obliquity factor $(1 + \cos\theta)/2$ falls to zero in the backward direction, and the Rayleigh-Sommerfeld formulation describes only the half-space beyond the aperture; both follow from solving the wave equation with boundary conditions on the aperture plane.

How does Huygens' principle explain diffraction?

When part of a wavefront is blocked, the wavelets at its edge lose the neighbours that cancelled their sideways spread, so their sum extends into the geometric shadow and forms fringes through interference.

Who formulated the Huygens-Fresnel principle?

Huygens proposed the wavelet construction in 1690; Fresnel added interference in his 1818 memoir on diffraction; Kirchhoff (1882), and later Rayleigh and Sommerfeld, derived it from the wave equation.

References: C. Huygens, Traité de la Lumière (Leiden, 1690); M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 8; J. W. Goodman, Introduction to Fourier Optics, 4th ed. (W. H. Freeman, 2017), Ch. 3 and 4; E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 4 and 10.