Huygens–Fresnel principle

The Huygens–Fresnel principle (named after Dutch physicist Christiaan Huygens and French physicist Augustin-Jean Fresnel) states that every point on a wavefront is itself the source of spherical wavelets and that the secondary wavelets emanating from different points mutually interfere. The sum of these spherical wavelets forms a new wavefront. As such, the Huygens–Fresnel principle is a method of analysis applied to problems of luminous wave propagation both in the far-field limit and in near-field diffraction as well as reflection.

History

In 1678, Huygens proposed that every point reached by a luminous disturbance becomes a source of a spherical wave. The sum of these secondary waves determines the form of the wave at any subsequent time; the overall procedure is referred to as Huygens's construction.

In 1818, Fresnel showed that Huygens's principle, together with his own principle of interference, could explain both the rectilinear propagation of light and also diffraction effects. To obtain agreement with the experimental results, he had to include additional arbitrary assumptions about the phase and amplitude of the secondary waves, as well as an obliquity factor. These assumptions have no obvious physical foundation, but led to predictions that agreed with many experimental observations, including the Poisson spot.

Poisson was a member of the French Academy, which reviewed Fresnel's work. He used Fresnel's theory to predict that a bright spot ought to appear in the center of the shadow of a small disc, and deduced from this that the theory was incorrect. However, Fran%C3%A7ois Arago, another member of the committee, performed the experiment and showed that the prediction was correct. This success was important evidence in favor of the wave theory of light over then predominant corpuscular theory.

In 1882, Gustav Kirchhoff analyzed Fresnel's theory in a rigorous mathematical formulation, as an approximate form of an integral theorem.

In antenna theory and engineering, the reformulation of the Huygens–Fresnel principle for radiating current sources is known as surface equivalence principle.

Issues in Huygens–Fresnel theory continue to be of interest. In 1991, David A. B. Miller suggested that treating the source as a dipole (not the monopole assumed by Huygens) will cancel waves propagating in the reverse direction, making Huygens's construction quantitatively correct. In 2021, Forrest L. Anderson showed that treating the wavelets as Dirac delta functions, summing and differentiating the summation is sufficient to cancel reverse propagating waves.

Examples

Refraction

The apparent change in direction of a light ray as it enters a sheet of glass at an angle can be understood by the Huygens's construction. Each point on the surface of the glass gives a secondary wavelet. These wavelets propagate at a slower velocity in the glass, making less forward progress than their counterparts in air. When the wavelets are summed, the resulting wavefront propagates at an angle to the direction of the wavefront in air.

thumb|Wave [[refraction in the manner of Huygens|center]]

In an inhomogeneous medium with a variable index of refraction, different parts of the wavefront propagate at different speeds. Consequently, the wavefront bends around in the direction of the higher index.

A simple example of the operation of the principle can be observed when an open doorway connects two rooms and a sound is produced in a remote corner of one of them. A person in the other room will hear the sound as if it originated at the doorway. As far as the second room is concerned, the vibrating air in the doorway is the source of the sound.

Mathematical expression of the principle

thumb|300px|right|Geometric arrangement for Fresnel's calculation

Consider the case of a point source located at a point P0, vibrating at a frequency f. The disturbance may be described by a complex variable U0 known as the complex amplitude. It produces a spherical wave with wavelength λ, wavenumber . Within a constant of proportionality, the complex amplitude of the primary wave at the point Q located at a distance r0 from P0 is:

:<math>U(r_0) \propto \frac {U_0 e^{ikr_0{r_0}. </math>

Note that magnitude decreases in inverse proportion to the distance traveled, and the phase changes as k times the distance traveled.

Using Huygens's theory and the principle of superposition of waves, the complex amplitude at a further point P is found by summing the contribution from each point on the sphere of radius r0. In order to obtain agreement with experimental results, Fresnel found that the individual contributions from the secondary waves on the sphere had to be multiplied by a constant, −i/λ, and by an additional inclination factor, K(χ). The first assumption implies that the secondary waves oscillate at a quarter of a cycle out of phase with respect to the primary wave and the second implies that the magnitude of the secondary waves is in a ratio of 1:λ to the primary wave. He also assumed that K(χ) had a maximum value when χ = 0, and was equal to zero when χ = π/2, where χ is the angle between the normal of the primary wavefront and the normal of the secondary wavefront. The complex amplitude at P, due to the contribution of secondary waves, is then given by:

:<math> U(P) = -\frac{i}{\lambda} U(r_0) \int_{S} \frac {e^{iks{s} K(\chi)\,dS </math>

where S describes the surface of the sphere, and s is the distance between Q and P.

Fresnel used a zone construction method to find approximate values of K for the different zones,

:<math>\psi(\mathbf{x}',t') = i \int d^3x \, G(\mathbf{x}',t';\mathbf{x},t)\psi(\mathbf{x},t)</math>

where G is known as the Green's function or propagator, which propagates the wave function <math>\psi</math> in time.

This generalized principle is the basis for Feynman's approach to quantum electrodynamics.

Feynman's path integral and the modern photon wave function

Huygens's theory served as a fundamental explanation of the wave nature of light interference and was further developed by Fresnel and Young, but did not fully resolve all observations, such as the low-intensity double-slit experiment first performed by G. I. Taylor in 1909. It was not until the early and mid-1900s that quantum theory discussions, particularly the early discussions at the 1927 Brussels Solvay Conference, where Louis de Broglie proposed his de Broglie hypothesis that the photon is guided by a wave function.

The wave function presents a much different explanation of the observed light and dark bands in a double slit experiment. In this conception, the photon follows a path that is a probabilistic choice of one of many possible paths in the electromagnetic field. These probable paths form the pattern: in dark areas, no photons are landing, and in bright areas, many photons are landing. The set of possible photon paths is consistent with Richard Feynman's path integral theory, the paths determined by the surroundings: the photon's originating point (atom), the slit, and the screen and by tracking and summing phases. The wave function is a solution to this geometry. The wave function approach was further supported by additional double-slit experiments in Italy and Japan in the 1970s and 1980s with electrons.

Quantum field theory

Huygens's principle can be seen as a consequence of the homogeneity of space—space is uniform in all locations. From this, he developed a set of conjectures that remain an active topic of research. In particular, it has been discovered that Huygens's principle holds on a large class of homogeneous spaces derived from the Coxeter group (so, for example, the Weyl groups of simple Lie algebras).

The traditional statement of Huygens's principle for the d'Alembertian gives rise to the KdV hierarchy; analogously, the Dirac operator gives rise to the AKNS hierarchy.