Huygens–Fresnel principle

TheHuygens–Fresnel principle(named afterDutchphysicistChristiaan HuygensandFrenchphysicistAugustin-Jean Fresnel) states that every point on awavefrontis 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 luminouswave propagationboth in thefar-field limitand in near-fielddiffractionas well asreflection.

Waverefractionin the manner of Huygens

Wave diffraction in the manner of Huygens and Fresnel

History

Diffractionof 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.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 asdiffractioneffects.The resolution of this error was finally explained byDavid A. B. Millerin 1991.The resolution is that the source is a dipole (not the monopole assumed by Huygens), which cancels in the reflected direction.

In 1818, Fresnelshowed that Huygens's principle, together with his own principle ofinterference,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 thePoisson spot.

Poissonwas 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, Arago, another member of the committee, performed the experiment and showed thatthe prediction was correct.(Lisle had observed this fifty years earlier.) This was one of the investigations that led to the victory of the wave theory of light over then predominantcorpuscular theory.

Inantenna theoryand engineering, the reformulation of the Huygens–Fresnel principle for radiating current sources is known assurface equivalence principle.

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 theKirchhoff's diffraction formulaand the approximations ofnear fielddue 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 formulaprovides 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.

Modern physics interpretations

Not all experts agree that the Huygens' principle is an accurate microscopic representation of reality. For instance,Melvin Schwartzargued that "Huygens' principle actually does give the right answer but for the wrong reasons".

This can be reflected in the following facts:

• The microscopic mechanics of photon creation and emission is, in general, essentially acceleration of electrons.
• The original analysis by Huygensconsidered only wavefronts of uniform frequency, phase, and propagation speed, and therefore cannot properly account for effects such asinterferenceordispersion.
• Huygens' original principle also does not consider the polarization of light, which would require a vector potential, in contrast to the scalar potential of a simpleocean waveorsound wave,and therefore cannot account for effects such asbirefringence.
• In theHuygensdescription, there is no explanation of why we choose only the forward-going (retarded waveor forward envelope of wave fronts) versus the backward-propagatingadvanced wave(backward envelope).
• In the Fresnel approximation there is a concept of non-local behavior due to the sum of spherical waves with different phases that comes from the different points of the wave front, and non local theories are subject of many debates (e.g., not beingLorentz covariant) and of active research.
• The Fresnel approximation can be interpreted in a quantum probabilistic manner but is unclear how much this sum of states (i.e., wavelets on the wavefront) is a completelist of statesthat are meaningful physically or represents more of an approximation on a genericbasislike in thelinear combination of atomic orbitals(LCAO) method.

The Huygens' principle is essentially compatible with quantum field theory in thefar field approximation,consideringeffective fieldsin the center of scattering, consideringsmall perturbations,and in the same sense thatquantum opticsis compatible withclassical optics,other interpretations are subject of debates and active research.

The Feynman model where every point in an imaginary wave front as large as the room is generating a wavelet, shall also be interpreted in these approximationsand in a probabilistic context, in this context remote points can only contribute minimally to the overall probability amplitude.

Quantum field theory does not include any microscopic model for photon creation and the concept of single photon is also put under scrutiny on a theoretical level.

Mathematical expression of the principle

Geometric arrangement for Fresnel's calculation

Consider the case of a point source located at a pointP₀,vibrating at afrequencyf.The disturbance may be described by a complex variableU₀known as thecomplex amplitude.It produces a spherical wave withwavelengthλ,wavenumberk= 2π/λ.Within a constant of proportionality, the complex amplitude of the primary wave at the pointQlocated at a distancer₀fromP₀is:

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

Note thatmagnitudedecreases in inverse proportion to the distance traveled, and the phase changes asktimes the distance traveled.

Using Huygens's theory and theprinciple of superpositionof waves, the complex amplitude at a further pointPis found by summing the contribution from each point on the sphere of radiusr₀.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 thatK(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 atP,due to the contribution of secondary waves, is then given by:

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

whereSdescribes the surface of the sphere, andsis the distance betweenQandP.

Fresnel used a zone construction method to find approximate values ofKfor the different zones,which enabled him to make predictions that were in agreement with experimental results. Theintegral theorem of Kirchhoffincludes 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 forK(χ):

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

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

Above derivation ofK(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,Ksatisfies 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:

"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 ofGreen's functionsandpropagatorsapply. 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 theactionand 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 Greinerthe generalized principle can be expressed for$t'>t$in the form:

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

whereGis 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.

Huygens' theory, 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-intensitydouble-slit experimentfirst 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 BrusselsSolvay Conference,whereLouis de Broglieproposed 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.

Huygens' principle and quantum field theory

Huygens' principle can be seen as a consequence of thehomogeneityof 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. Thesuperpositionof all the waves results in the observed pattern of wave propagation.

Homogeneity of space is fundamental toquantum field theory(QFT) where thewave functionof any object propagates along all available unobstructed paths. Whenintegrated along all possible paths,with aphasefactor proportional to theaction,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 Hadamardobserved 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 ofhomogeneous spacesderived from theCoxeter group(so, for example, theWeyl groupsof simpleLie algebras).

The traditional statement of Huygens' principle for theD'Alembertiangives rise to theKdV hierarchy;analogously, theDirac operatorgives rise to theAKNShierarchy.

See also

• Fraunhofer diffraction
• Kirchhoff's diffraction formula
• Green's function
• Green's theorem
• Green's identities
• Near-field diffraction pattern
• Double-slit experiment
• Knife-edge effect
• Fermat's principle
• Fourier optics
• Surface equivalence principle
• Wave field synthesis
• Kirchhoff integral theorem

Further reading

• Stratton, Julius Adams:Electromagnetic Theory,McGraw-Hill, 1941. (Reissued by Wiley – IEEE Press,ISBN978-0-470-13153-4).
• B.B. Baker and E.T. Copson,The Mathematical Theory of Huygens' Principle,Oxford, 1939, 1950; AMS Chelsea, 1987.