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. He assumed that the secondary waves traveled only in the "forward" direction, but 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. 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, François 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. Very few rigorous solutions to diffraction problems are known, however, and most problems in optics are adequately treated using the Huygens–Fresnel principle. In 1939 Edward Copson, extended Huygens's original principle to consider the polarization of light, which requires a vector potential, in contrast to the scalar potential of a simple ocean wave or sound wave. 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.
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.
Diffraction
Huygens's principle as a microscopic model The Huygens–Fresnel principle provides a reasonable basis for understanding and predicting the classical wave propagation of light. However, there are limitations to the principle, namely the same approximations made for deriving the Kirchhoff's diffraction formula and the approximations of near field due to Fresnel. These can be summarized in the fact that the wavelength of light is much smaller than the dimensions of any optical components encountered. Kirchhoff's diffraction formula provides a rigorous mathematical foundation for diffraction, based on the wave equation. The arbitrary assumptions made by Fresnel to arrive at the Huygens–Fresnel equation emerge automatically from the mathematics in this derivation. 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
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:
U ( r 0 ) ∝ U 0 e i k r 0 r 0 . {\displaystyle U(r_{0})\propto {\frac {U_{0}e^{ikr_{0}}}{r_{0}}}.}
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