Wave diffraction in the manner of Huygens and Fresnel
The Huygens–Fresnel principle (named after Dutch physicist Christiaan Huygens and French physicist AugustinJean Fresnel) is a method of analysis applied to problems of wave propagation both in the farfield limit and in nearfield diffraction.
History
Diffraction of a plane wave at a slit whose width is several times the wavelength
Diffraction of a plane wave when the slit width equals the wavelength
In 1678, Huygens^{[1]} proposed that every point which a luminous disturbance reaches 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 as diffraction effects.^{[2]}
In 1816, Fresnel^{[3]} showed that Huygens' 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 Arago spot.
Poisson was a member of the French Academy, which reviewed Fresnel's work.^{[4]} He used Fresnel's theory to predict that a bright spot will 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 actually observed this fifty years earlier.^{[2]}) This was one of the investigations that led to the victory of the wave theory of light over the then predominant corpuscular theory.
The Huygens–Fresnel principle provides a good basis for understanding and predicting the wave propagation of light. However, this article^{[5]} provides an interesting discussion of the limitations of the principle and also of different scientists' views as to whether it is an accurate representation of reality or whether "Huygens' principle actually does give the right answer but for the wrong reasons".
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.^{[6]}
A simple example of the operation of the principle can be seen when two rooms are connected by an open doorway 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
Geometric arrangement for Fresnel's calculation
Consider the case of a point source located at a point
P_{0}, vibrating at a
frequency f. The disturbance may be described by a complex variable
U_{0} known as the
complex amplitude. It produces a spherical wave with
wavelength λ,
wavenumber k = 2π/λ. The complex amplitude of the primary wave at the point
Q located at a distance
r_{0} from
P_{0} is given by

U(r_0) = \frac {U_0 e^{ikr_0}}{r_0}
since the magnitude decreases in inverse proportion to the distance travelled, and the phase changes as k times the distance travelled.
Using Huygens' theory and the principle of superposition of waves, the complex amplitude at a further point P is found by summing the contributions from each point on the sphere of radius r_{0}. In order to get 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. The complex amplitude at P is then given by:

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,^{[4]} which enabled him to make predictions that were in agreement with experimental results.
The various assumptions made by Fresnel emerge automatically in Kirchhoff's diffraction formula,^{[4]} to which the Huygens–Fresnel principle can be considered to be an approximation. Kirchhoff gave the following expression for K(χ):

~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.
Huygens' principle and quantum electrodynamics
Huygens' principle can be seen as a consequence of the isotropy of space—all directions in space are equal. Any disturbance created in a sufficiently small region of isotropic space (or in an isotropic medium) propagates from that region in all radial directions. The waves created 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.
Isotropy of space is fundamental to quantum electrodynamics (QED) where the wave function of any object propagates along all available unobstructed paths. When integrated along all possible paths, with a phase factor proportional to the path length, the interference of the wavefunctions correctly predicts observable phenomena. Every point on the wave front acts as the source of secondary wavelets that spread out in the forward direction with the same speed as the wave. The new wave front is found by constructing the surface tangent to the secondary wavelets.
See also
References

^ Chr. Huygens, Traitė de la Lumiere (completed in 1678, published in Leyden in 1690)

^ ^{a} ^{b} OS Heavens and RW Ditchburn, Insight into Optics, 1987, Wiley & Sons, Chichester ISBN 0471927694

^ A. Fresnel, Ann Chim et Phys, (2), 1 (1816), Oeuvres, Vol. 1, 89, 129

^ ^{a} ^{b} ^{c} Max Born and Emil Wolf, Principles of Optics, 1999, Cambridge University Press ISBN 9780521642224

^ Huygens' Principle

^ MV Klein & TE Furtak, Optics, 1986, John Wiley & Sons, New York
Further reading
Stratton, Julius Adams: Electromagnetic Theory, McGrawHill, 1941. (Reissued by Wiley – IEEE Press, ISBN 9780470131534).
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.