Editing Huygens–Fresnel principle

{{Short description|Method of analysis}}

The '''Huygens–Fresnel principle''' (named after [[Netherlands|Dutch]] [[physicist]] [[Christiaan Huygens]] and [[France|French]] physicist [[Augustin-Jean Fresnel]]) states that every point on a [[wavefront]] is itself the source of spherical wavelets, and the secondary wavelets emanating from different points mutually [[Wave interference|interfere]].<ref name="MathPages">{{cite web | title=Huygens' Principle | website=MathPages | url=https://www.mathpages.com/home/kmath242/kmath242.htm | access-date=2017-10-03}}</ref> 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 diffraction pattern|far-field limit]] and in near-field [[diffraction]] as well as [[Reflection (physics)|reflection]].

==History==
In 1678, Huygens proposed<ref>Chr. Huygens, ''[[Treatise on Light|Traité de la Lumière]]'' '' (drafted 1678; published in Leyden by Van der Aa, 1690), translated by [[Silvanus P. Thompson]] as ''[[iarchive:treatiseonlight031310mbp|Treatise on Light]]'' (London: Macmillan, 1912; [https://www.gutenberg.org/ebooks/14725 Project Gutenberg edition], 2005), p.19.''</ref> 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'''.<ref name = "Born and Wolf"/>{{rp|132}} 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.<ref name = "Insight into Optics">{{Cite book |last=Heavens |first=Oliver S. |title=Insight into optics |last2=Ditchburn |first2=R. W. |date=1993 |publisher=Wiley |isbn=978-0-471-92901-7 |edition=Reprint with corrections August 1993 |location=Chichester}}</ref> 

In 1818, Fresnel<ref>A. Fresnel, "Mémoire sur la diffraction de la lumière" (deposited 1818, "crowned" 1819), in ''Oeuvres complètes'' (Paris: Imprimerie impériale, 1866&ndash;70), vol.1, pp.&nbsp;247&ndash;363; partly translated as "Fresnel's prize memoir on the diffraction of light", in H.&nbsp;Crew (ed.), ''[https://archive.org/details/wavetheoryofligh00crewrich The Wave Theory of Light: Memoirs by Huygens, Young and Fresnel]'', American Book Co., 1900, pp.&nbsp;81&ndash;144. (Not to be confused with the earlier work of the same title in ''Annales de Chimie et de Physique'', 1:238&ndash;81, 1816.)</ref> showed that Huygens's principle, together with his own principle of [[Interference (wave propagation)|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]].

[[Siméon Denis Poisson|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 [[Arago spot|the prediction was correct]].<ref name = "Born and Wolf">{{cite book |first1=Max |last1=Born |author-link=Max Born |first2=Emil |last2=Wolf |title=[[Principles of Optics]] |year=1999 |publisher=Cambridge University Press |isbn=978-0-521-64222-4 }}</ref> 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.<ref name = "Born and Wolf"/>{{rp|375}} Very few rigorous solutions to diffraction problems are known however, and most problems in optics are adequately treated using the Huygens-Fresnel principle.<ref name = "Born and Wolf"/>{{rp|370}}

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 [[Gravity wave|ocean wave]] or [[Longitudinal wave|sound wave]].<ref name="TheoryOfHuygens">{{cite web | title=TheoryOfHuygens|website=Archive.org|year=1939|url=https://archive.org/details/in.ernet.dli.2015.84565}}</ref><ref>{{cite journal|author=Bleick, Willard Evan|title=Review: ''The Mathematical Theory of Huygens' Principle'' by B. B. Baker and E. T. Copson|journal=Bull. Amer. Math. Soc.|year=1940|volume=46|issue=5|pages=386–388|url=https://www.ams.org/journals/bull/1940-46-05/S0002-9904-1940-07203-9/S0002-9904-1940-07203-9.pdf|doi=10.1090/s0002-9904-1940-07203-9|doi-access=free}}</ref>

In [[antenna (radio)|antenna theory]] and engineering, the reformulation of the Huygens–Fresnel principle for radiating current sources is known as [[surface equivalence principle]].<ref>{{cite book|last=Balanis|first= Constantine A.|author-link=Constantine A. Balanis|title=Advanced Engineering Electromagnetics|date=2012|publisher=John Wiley & Sons|isbn=978-0-470-58948-9|pages=328–331}}</ref><ref>{{cite book |last=Balanis |first=Constantine A.|author-link=Constantine A. Balanis |title=Antenna Theory: Analysis and Design |edition=3rd |publisher=John Wiley and Sons |date=2005|isbn=047166782X|page=333}}</ref>

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.<ref>{{cite journal |first=David A. B. |last=Miller |title=Huygens's wave propagation principle corrected |journal=Optics Letters |volume=16 |issue= 18|pages=1370–1372 |date=1991 |doi=10.1364/OL.16.001370 |pmid=19776972 |bibcode=1991OptL...16.1370M |s2cid=16872264 }}</ref> In 2021, Forrest L. Anderson showed that treating the wavelets as [[Dirac delta function]]s, summing and differentiating the summation is sufficient to cancel reverse propagating waves.<ref>{{Cite journal |last=Anderson |first=Forrest L. |date=2021-10-12 |title=Huygens' Principle geometric derivation and elimination of the wake and backward wave |url=https://www.nature.com/articles/s41598-021-99049-7 |journal=Scientific Reports |language=en |volume=11 |issue=1 |doi=10.1038/s41598-021-99049-7 |issn=2045-2322 |pmc=8511121 |pmid=34642401}}</ref>

==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.<ref name=Wood-1905> 
{{cite book|author=Wood, R. W.|title=Physical Optics|year=1905|isbn=978-1-172-70918-2|publisher=MacMillan|location=New York|url=https://archive.org/details/physicaloptics00wooduoft}}</ref>{{rp|56}}

[[File:Refraction - Huygens-Fresnel principle.svg|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 higher index.<ref name=Wood-1905/>{{rp|68}} 
[[File:HuygensRefractionVariableIndex.svg|thumb|Huygens-Fresnel construction of wave refraction in a medium with variable index of refraction|center]]
{{clear}}

===Diffraction===
[[File:Refraction on an aperture - Huygens-Fresnel principle.svg|thumb|Wave diffraction in the manner of Huygens and Fresnel|center]]
{{clear}}
==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 and far field|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.<ref name="Born and Wolf"/>

[[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.<ref>{{cite book |first1=M. V. |last1=Klein |first2=T. E. |last2=Furtak |title=Optics |year=1986 |publisher=John Wiley & Sons |location=New York |edition=2nd |isbn=0-471-84311-3 }}</ref>

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==
[[File:Huygens-Fresnel BW.svg|thumb|300px|right|Geometric arrangement for Fresnel's calculation]]
Consider the case of a point source located at a point '''P'''<sub>0</sub>, vibrating at a [[frequency]] ''f''. The disturbance may be described by a complex variable ''U''<sub>0</sub> known as the [[complex amplitude]]. It produces a spherical wave with [[wavelength]] λ, [[wavenumber]] {{math|''k'' {{=}} 2''&pi;''/''&lambda;''}}. Within a constant of proportionality, the complex amplitude of the primary wave at the point '''Q''' located at a distance ''r''<sub>0</sub> from '''P'''<sub>0</sub> is:

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

Note that [[amplitude|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 [[superposition principle|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''<sub>0</sub>. 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:<ref name = "Introduction to Fourier Optics">{{cite book|author=J. Goodman|year=2005|title=Introduction to Fourier Optics|edition=3rd|publisher=Roberts & Co Publishers|isbn=978-0-9747077-2-3|url= https://books.google.com/books?id=ow5xs_Rtt9AC}}</ref>

:<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,<ref name="Born and Wolf"/> which enabled him to make predictions that were in agreement with experimental results. The [[Kirchhoff integral theorem|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.<ref name="Born and Wolf"/>

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):<ref name="Born and Wolf"/>
:<math>~K(\chi )= \frac{1}{2}(1+\cos \chi)</math>

''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.<ref name="Introduction to Fourier Optics"/> 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:<ref name="Introduction to Fourier Optics"/>

:<math>~K(\chi )= \cos \chi</math>

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)<ref name="Greiner">{{cite book | author=Greiner W. | title= Quantum Electrodynamics | publisher=Springer, 2002}}</ref> and (Enders, 2009)<ref name="Enders">{{cite journal |last1=Enders |first1=Peter |title=Huygens' Principle as Universal Model of Propagation |journal=Latin-American Journal of Physics Education |date=2009 |volume=3 |issue=1 |pages=19–32 |url=https://lajpe.org/jan09/04_Peter_Enders.pdf }}</ref> - refer to the Generalized Huygens' Principle using the definition in ([[Feynman]], 1948).<ref name="Fey1">{{cite journal |last1=Feynman |first1=R. P. |title=Space-Time Approach to Non-Relativistic Quantum Mechanics |journal=Reviews of Modern Physics |date=1 April 1948 |volume=20 |issue=2 |pages=367–387 |doi=10.1103/RevModPhys.20.367 |bibcode=1948RvMP...20..367F |url=https://resolver.caltech.edu/CaltechAUTHORS:20140731-165931911 }}</ref>

Feynman defines the generalized principle in the following way:
{{bquote| "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 function (many-body theory)|Green's functions]] and [[propagator]]s 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 (physics)|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 <ref name="Greiner"/> the generalized principle can be expressed for <math>t'>t </math> in the form:
:<math>\psi'(\mathbf{x}',t') = i \int d^3x \, G(\mathbf{x}',t';\mathbf{x},t)\psi(\mathbf{x},t)</math>
where ''G'' is the usual Green function that propagates in time the wave function <math>\psi</math>. 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]] 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.<ref>{{cite book|last1=Baggott|first1=Jim|title=The Quantum Story|url=https://archive.org/details/quantumstoryhist00bagg|url-access=limited|date=2011|publisher=Oxford Press|isbn=978-0-19-965597-7|page=[https://archive.org/details/quantumstoryhist00bagg/page/n136 116]}}</ref>  

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.<ref>{{cite web|last1=Peter|first1=Rodgers|title=The double-slit experiment|url=https://physicsworld.com/a/the-double-slit-experiment/|website=www.physicsworld.com|publisher=Physics World|access-date=10 Sep 2018|date=September 2002}}</ref>

==Quantum field theory==
Huygens' principle can be seen as a consequence of the [[homogeneous space|homogeneity]] of space—space is uniform in all locations.<ref name="veselov"/> 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 principle|superposition]] of all the waves results in the observed pattern of wave propagation.

Homogeneity of space is fundamental to [[quantum field theory]] (QFT) where the [[wave function]] of any object propagates along all available unobstructed paths. When [[partition function (quantum field theory)|integrated along all possible paths]], with a [[Phase (waves)|phase]] factor proportional to the [[action (physics)|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]] observed that Huygens' principle was broken when the number of spatial dimensions is even.<ref>{{cite web |first=Alexander P. |last=Veselov |url=http://www.lboro.ac.uk/microsites/maths/research/preprints/papers02/02-49.pdf |title=Huygens' Principle |archive-url=https://web.archive.org/web/20160221215126/http://www.lboro.ac.uk/microsites/maths/research/preprints/papers02/02-49.pdf |archive-date=2016-02-21 |date=2002 }}</ref><ref>{{cite web |url=https://web.stanford.edu/class/math220a/handouts/waveequation3.pdf |title=Wave Equation in Higher Dimensions |publisher=Stanford University |work=Math 220a class notes }}</ref><ref>{{cite journal |first1=M. |last1=Belger |first2=R. |last2=Schimming |first3=V. |last3=Wünsch |title=A Survey on Huygens' Principle |journal=Zeitschrift für Analysis und ihre Anwendungen |volume=16 |issue=1 |date=1997 |pages=9–36 |doi=10.4171/ZAA/747 |doi-access=free }}</ref> From this, he developed a set of conjectures that remain an active topic of research.<ref>{{cite journal |first=Leifur |last=Ásgeirsson |author-link=Leifur Ásgeirsson |title=Some hints on Huygens' principle and Hadamard's conjecture |journal=Communications on Pure and Applied Mathematics |volume=9 |issue=3 |pages=307–326 |date=1956 |doi=10.1002/cpa.3160090304 }}</ref><ref>{{cite journal |first=Paul |last=Günther |title=Huygens' principle and Hadamard's conjecture |journal=The Mathematical Intelligencer |date=1991 |volume=13 |issue=2 |pages=56–63 |doi=10.1007/BF03024088 |s2cid=120446795 }}</ref> In particular, it has been discovered that Huygens' principle holds on a large class of [[homogeneous space]]s derived from the [[Coxeter group]] (so, for example, the [[Weyl group]]s of simple [[Lie algebra]]s).<ref name="veselov">{{cite journal |first=Alexander P. |last=Veselov |title=Huygens' principle and integrable systems |journal=Physica D: Nonlinear Phenomena |volume=87 |issue=1–4 |year=1995 |pages=9–13 |doi=10.1016/0167-2789(95)00166-2 |bibcode=1995PhyD...87....9V }}</ref><ref>{{cite journal |first1=Yu. Yu. |last1=Berest |first2=A. P. |last2=Veselov |title=Hadamard's problem and Coxeter groups: New examples of Huygens' equations |journal=Functional Analysis and Its Applications |date=1994 |volume=28 |issue=1 |pages=3–12 |doi=10.1007/BF01079005 |s2cid=121842251 }}</ref>

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.<ref>{{cite journal |first1=Fabio A. C. C. |last1=Chalub |first2=Jorge P. |last2=Zubelli |title=Huygens' Principle for Hyperbolic Operators and Integrable Hierarchies |journal=Physica D: Nonlinear Phenomena |volume=213 |issue=2 |date=2006 |pages=231–245 |doi=10.1016/j.physd.2005.11.008 |bibcode=2006PhyD..213..231C }}</ref><ref>{{cite journal |first1=Yuri Yu. |last1=Berest |first2=Igor M. |last2=Loutsenko |title=Huygens' Principle in Minkowski Spaces and Soliton Solutions of the Korteweg-de Vries Equation |journal=Communications in Mathematical Physics |date=1997 |volume=190 |issue=1 |pages=113–132 |arxiv=solv-int/9704012 |doi=10.1007/s002200050235 |bibcode=1997CMaPh.190..113B |s2cid=14271642 }}</ref>

==See also==
{{Commons|Huygens' principle}}
{{Div col|colwidth=20em}}
* [[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]]
{{div col end}}

==References==
{{Reflist|30em}}

==Further reading==
* 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.

{{Christiaan Huygens}}
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[[Category:Wave mechanics]]
[[Category:Diffraction]]
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