Huygens–Fresnel principle

Diffraction of a plane wave when the slit width equals the wavelength

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.[2] He assumed that the secondary waves travelled only in the "forward" direction and it is not explained in the theory why this is the case. He was able to provide a qualitative explanation of linear and spherical wave propagation, and to derive the laws of reflection and refraction using this principle, but could not explain the deviations from rectilinear propagation that occur when light encounters edges, apertures and screens, commonly known as diffraction effects.[3] The resolution of this error was finally explained by David A. B. Miller in 1991.[4] The resolution is that the source is a dipole (not the monopole assumed by Huygens), which cancels in the reflected direction.

In 1818, Fresnel[5] 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 experimental results, he had to include additional arbitrary assumptions about the phase and amplitude of the secondary waves, and also 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.[6] 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, Arago, another member of the committee, performed the experiment and showed that the prediction was correct. (Lisle had observed this fifty years earlier.[3]) This was one of the investigations that led to the victory of the wave theory of light over then predominant corpuscular theory.

In antenna theory and engineering, the reformulation of the Huygens–Fresnel principle for radiating current sources is known as surface equivalence principle.[7][8]

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 k = 2π/λ. Within a constant of proportionality, the complex amplitude of the primary wave at the point Q located at a distance r₀ from P₀ is:

${\displaystyle U(r_{0})\propto {\frac {U_{0}e^{ikr_{0}}}{r_{0}}}.}$

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 get an 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 means that the secondary waves oscillate at a quarter of a cycle out of phase with respect to the primary wave and that the magnitude of the secondary waves are 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:[13]

${\displaystyle U(P)=-{\frac {i}{\lambda }}U(r_{0})\int _{S}{\frac {e^{iks}}{s}}K(\chi )\,dS}$

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,[6] which enabled him to make predictions that were in agreement with experimental results. The integral theorem of Kirchhoff includes the basic idea of Huygens–Fresnel principle. Kirchhoff showed that in many cases, the theorem can be approximated to a simpler form that is equivalent to the formation of Fresnel's formulation.[6]

For an aperture illumination consisting of a single expanding spherical wave, if the radius of the curvature of the wave is sufficiently large, Kirchhoff gave the following expression for K(χ):[6]

${\displaystyle ~K(\chi )={\frac {1}{2}}(1+\cos \chi )}$

K has a maximum value at χ = 0 as in the Huygens–Fresnel principle; however, K is not equal to zero at χ = π/2, but at χ = π.

Above derivation of K(χ) assumed that the diffracting aperture is illuminated by a single spherical wave with a sufficiently large radius of curvature. However, the principle holds for more general illuminations.[13] An arbitrary illumination can be decomposed into a collection of point sources, and the linearity of the wave equation can be invoked to apply the principle to each point source individually. K(χ) can be generally expressed as:[13]

${\displaystyle ~K(\chi )=\cos \chi }$

In this case, K satisfies the conditions stated above (maximum value at χ = 0 and zero at χ = π/2).

Many books and references e.g.[14] and [15] refer to the Generalized Huygens' Principle as the one referred by Feynman in this publication.[16]

Feynman defines the generalized principle in the following way:

"Actually Huygens’ principle is not correct in optics. It is replaced by Kirchoff’s [sic] modification which requires that both the amplitude and its derivative must be known on the adjacent surface. This is a consequence of the fact that the wave equation in optics is second order in the time. The wave equation of quantum mechanics is first order in the time; therefore, Huygens’ principle is correct for matter waves, action replacing time."

This clarifies the fact that in this context the generalized principle reflects the linearity of quantum mechanics and the fact that the quantum mechanics equations are first order in time. Finally only in this case the superposition principle fully apply, i.e. the wave function in a point P can be expanded as a superposition of waves on a border surface enclosing P. Wave functions can be interpreted in the usual quantum mechanical sense as probability densities where the formalism of Green's functions and propagators apply. What is note-worthy is that this generalized principle is applicable for "matter waves" and not for light waves any more. The phase factor is now clarified as given by the action and there is no more confusion why the phases of the wavelets are different from the one of the original wave and modified by the additional Fresnel parameters.

As per Greiner [14] the generalized principle can be expressed for ${\displaystyle t'>t}$ in the form:

${\displaystyle \psi '(\mathbf {x} ',t')=i\int d^{3}x\,G(\mathbf {x} ',t';\mathbf {x} ,t)\psi (\mathbf {x} ,t)}$

where G is the usual Green function that propagates in time the wave function ${\displaystyle \psi }$. This description resembles and generalize the initial Fresnel's formula of the classical model.

In 1900, Jacques Hadamard observed that Huygens' principle was broken when the number of spatial dimensions is even.[20][21][22] From this, he developed a set of conjectures that remain an active topic of research.[23][24] In particular, it has been discovered that Huygens' principle holds on a large class of homogeneous spaces derived from the Coxeter group (so, for example, the Weyl groups of simple Lie algebras).[19][25]

The traditional statement of Huygens' principle for the D'Alembertian gives rise to the KdV hierarchy; analogously, the Dirac operator gives rise to the AKNS hierarchy.[26][27]

• Stratton, Julius Adams: Electromagnetic Theory, McGraw-Hill, 1941. (Reissued by Wiley – IEEE Press, ISBN 978-0-470-13153-4).
• B.B. Baker and E.T. Copson, The Mathematical Theory of Huygens' Principle, Oxford, 1939, 1950; AMS Chelsea, 1987.