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The Huygens–Fresnel principle (named after Dutch
physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
Christiaan Huygens Christiaan Huygens, Halen, Lord of Zeelhem, ( , ; ; also spelled Huyghens; ; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor who is regarded as a key figure in the Scientific Revolution ...
and French physicist
Augustin-Jean Fresnel Augustin-Jean Fresnel (10 May 1788 – 14 July 1827) was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Isaac Newton, Newton's c ...
) states that every point on a
wavefront In physics, the wavefront of a time-varying ''wave field (physics), field'' is the set (locus (mathematics), locus) of all point (geometry), points having the same ''phase (waves), phase''. The term is generally meaningful only for fields that, a ...
is itself the source of spherical wavelets, and 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 In physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. '' Periodic waves'' oscillate repeatedly about an equilibrium (resting) value at some f ...
both in the far-field limit and in near-field
diffraction Diffraction is the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the Wave propagation ...
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' construction. 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 Diffraction is the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the Wave propagation ...
effects. In 1818, Fresnel showed that Huygens's principle, together with his own principle of
interference Interference is the act of interfering, invading, or poaching. Interference may also refer to: Communications * Interference (communication), anything which alters, modifies, or disrupts a message * Adjacent-channel interference, caused by extra ...
, 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. 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 Dominique François Jean Arago (), known simply as François Arago (; Catalan: , ; 26 February 17862 October 1853), was a French mathematician, physicist, astronomer, freemason, supporter of the Carbonari revolutionaries and politician. Early l ...
, 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 Gustav Robert Kirchhoff (; 12 March 1824 – 17 October 1887) was a German chemist, mathematician, physicist, and spectroscopist who contributed to the fundamental understanding of electrical circuits, spectroscopy and the emission of black-body ...
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 the Huygens' 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 physics, sound is a vibration that propagates as an acoustic wave through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
. 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' construction quantitatively correct. In 2021, Forrest L. Anderson showed that treating the wavelets as
Dirac delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
s, 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 angle can be understood by the Huygens 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 higher index.


Diffraction


Huygens' 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 done 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 seen 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 Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
''f''. The disturbance may be described by a complex variable ''U''0 known as the complex amplitude. It produces a spherical wave with
wavelength In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same ''phase (waves ...
λ,
wavenumber In the physical sciences, the wavenumber (or wave number), also known as repetency, is the spatial frequency of a wave. Ordinary wavenumber is defined as the number of wave cycles divided by length; it is a physical quantity with dimension of ...
. Within a constant of proportionality, the complex amplitude of the primary wave at the point Q located at a distance ''r''0 from P0 is: :U(r_0) \propto \frac . 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 ''r''0. 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: : U(P) = -\frac U(r_0) \int_ \frac 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, 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. 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): :~K(\chi )= \frac(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. 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: :~K(\chi )= \cos \chi In this case, ''K'' satisfies the conditions stated above (maximum value at χ = 0 and zero at χ = π/2).


Generalized Huygens' principle

Many books and references – e.g. (Greiner, 2002) and (Enders, 2009) - refer to the Generalized Huygens' Principle using the definition in ( Feynman, 1948). Feynman defines the generalized principle in the following way: 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 those of the original wave and modified by the additional Fresnel parameters. As per Greiner the generalized principle can be expressed for t'>t in the form: :\psi'(\mathbf',t') = i \int d^3x \, G(\mathbf',t';\mathbf,t)\psi(\mathbf,t) where ''G'' is the usual Green function that propagates in time the wave function \psi. This description resembles and generalize the initial Fresnel's formula of the classical model.


Feynman's path integral and the modern photon wave function

Huygens' 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 In modern physics, the double-slit experiment demonstrates that light and matter can exhibit behavior of both classical particles and classical waves. This type of experiment was first performed by Thomas Young in 1801, as a demonstration of ...
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 which 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' principle can be seen as a consequence of the
homogeneity Homogeneity and heterogeneity are concepts relating to the Uniformity (chemistry), uniformity of a Chemical substance, substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, ...
of space—space is uniform in all locations. Any disturbance created in a sufficiently small region of homogeneous space (or in a homogeneous medium) propagates from that region in all geodesic directions. The waves produced by this disturbance, in turn, create disturbances in other regions, and so on. The superposition of all the waves results in the observed pattern of wave propagation. Homogeneity of space is fundamental to
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
(QFT) where the
wave function In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
of any object propagates along all available unobstructed paths. When integrated along all possible paths, with a phase factor proportional to the action, the interference of the wave-functions correctly predicts observable phenomena. Every point on the wavefront acts as the source of secondary wavelets that spread out in the light cone with the same speed as the wave. The new wavefront is found by constructing the surface tangent to the secondary wavelets.


In other spatial dimensions

In 1900,
Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry, and partial differential equations. Biography The son of a tea ...
observed that Huygens' principle was broken when the number of spatial dimensions is even. From this, he developed a set of conjectures that remain an active topic of research. In particular, it has been discovered that Huygens' principle holds on a large class of
homogeneous space In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and ...
s derived from the
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
(so, for example, the
Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections t ...
s of simple
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
s). The traditional statement of Huygens' principle for the D'Alembertian gives rise to the KdV hierarchy; analogously, the
Dirac operator In mathematics and in quantum mechanics, a Dirac operator is a first-order differential operator that is a formal square root, or half-iterate, of a second-order differential operator such as a Laplacian. It was introduced in 1847 by William Ham ...
gives rise to the AKNS hierarchy.


See also

* Fraunhofer diffraction * Kirchhoff's diffraction formula *
Green's function In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if L is a linear dif ...
*
Green's theorem In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region (surface in \R^2) bounded by . It is the two-dimensional special case of Stokes' theorem (surface in \R^3) ...
* Green's identities * Near-field diffraction pattern *
Double-slit experiment In modern physics, the double-slit experiment demonstrates that light and matter can exhibit behavior of both classical particles and classical waves. This type of experiment was first performed by Thomas Young in 1801, as a demonstration of ...
* Knife-edge effect *
Fermat's principle Fermat's principle, also known as the principle of least time, is the link between geometrical optics, ray optics and physical optics, wave optics. Fermat's principle states that the path taken by a Ray (optics), ray between two given ...
* Fourier optics * Surface equivalence principle * Wave field synthesis * Kirchhoff integral theorem


References


Further reading

* Stratton, Julius Adams: ''Electromagnetic Theory'', McGraw-Hill, 1941. (Reissued by Wiley – IEEE Press, ). * B.B. Baker and E.T. Copson, ''The Mathematical Theory of Huygens' Principle'', Oxford, 1939, 1950; AMS Chelsea, 1987. {{DEFAULTSORT:Huygens-Fresnel Principle Wave mechanics Diffraction Christiaan Huygens