 Nslit interferometric equation

Quantum mechanics was first applied to optics, and interference in particular, by Paul Dirac.^{[1]} Feynman, in his lectures, uses Dirac’s notation to describe thought experiments on doubleslit interference of electrons.^{[2]} Feynman’s approach was extended to Nslit interferometers using narrowlinewidth laser illumination, that is, illumination by indistinguishable photons, by researchers working on the measurement of complex interference patterns.^{[3]}
Contents
Probability amplitudes and the Nslit interferometric equation
In this approach the probability amplitude for propagation from a source (s) to an interference plane (x) via an array of slits (j) is given, using Dirac’s notation, as^{[3]}
Using a wavefunction representation for probability amplitudes,^{[1]} after some algebra, the corresponding probability becomes^{[3]}^{[4]}^{[5]}
where N is the total number of slits in the array, or transmission grating, and the term in parenthesis represents the phase that is directly related to the exact geometry of the Nslit interferometer. The DiracDuarte interferometric equation applies to the propagation of a single photon, or the propagation of an ensemble of indistinguishable photons, and enables the accurate prediction of measured Nslit interferometric patterns continuously from the near to the far field.^{[5]}^{[6]} Interferograms generated with this equation have been shown to compare well with measured interferograms for both even (N = 2, 4, 6...) and odd (N = 3, 5, 7...) values of N from 2 to 1600.^{[5]}^{[7]}
Applications
At a practical level, the Nslit interferometric equation was introduced for imaging applications^{[5]} and is routinely applied to predict Nslit laser interferograms, both in the near and far field. Thus, it has become a valuable tool in the alignment of large, and very large, Nslit laser interferometers^{[8]}^{[9]} used in the study of clear air turbulence and the propagation of interferometric characters for secure freespace optical communications.
Also, the Nslit interferometric equation has been applied to describe interference, diffraction, refraction, and reflection, in a rational and unified approach.^{[7]}^{[10]} For example, the phase term (in parenthesis) can be used to derive^{[7]}^{[10]}
which is also known as the diffraction grating equation. Here, θ_{m} is the angle of incidence, ϕ_{m} is the angle of diffraction, λ is the wavelength, and M is the order of diffraction. The Nslit interferometric approach^{[5]}^{[7]}^{[10]} is one of several approaches applied to describe basic optical phenomena in a cohesive and unified manner.^{[11]}Note: given the various terminologies in use, for Nslit interferometry, it should be made explicit that the Nslit interferometric equation applies to twoslit interference, threeslit interference, fourslit interference, etc.
See also
 Beam expander
 Dirac's notation
 Freespace optical communications
 Grating equation
 Nslit interferometer
References
 ^ ^{a} ^{b} P. A. M. Dirac, The Principles of Quantum Mechanics, 4th Ed. (Oxford, London, 1978).
 ^ R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. III (Addison Wesley, Reading, 1965).
 ^ ^{a} ^{b} ^{c} F. J. Duarte and D. J. Paine, Quantum mechanical description of Nslit interference phenomena, in Proceedings of the International Conference on Lasers '88, R. C. Sze and F. J. Duarte (Eds.) (STS, McLean, Va, 1989) pp. 4247.
 ^ F. J. Duarte, Dispersive dye lasers, in High Power Dye Lasers, F. J. Duarte (Ed.) (SpringerVerlag, Berlin, 1991) Chapter 2.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} F. J. Duarte, On a generalized interference equation and interferometric measurements, Opt. Commun. 103, 814 (1993).
 ^ F. J. Duarte, Comment on "reflection, refraction, and multislit interfernce," Eur. J. Phys. 25, L57L58 (2004).
 ^ ^{a} ^{b} ^{c} ^{d} F. J. Duarte, Tunable Laser Optics (ElsevierAcademic, New York, 2003).
 ^ F. J. Duarte, T. S. Taylor, A. B. Clark, and W. E. Davenport, The Nslit interferometer: an extended configuration, J. Opt. 12, 015705 (2010).
 ^ ^{a} ^{b} F. J. Duarte, T. S. Taylor, A. M. Black, W. E. Davenport, and P. G. Varmette, Nslit interferometer for secure freespace optical communications: 527 m intra interferometric path length , J. Opt. 13, 035710 (2011).
 ^ ^{a} ^{b} ^{c} F. J. Duarte, Interference, diffraction, and refraction, via Dirac's notation, Am. J. Phys. 65, 637640 (1997).
 ^ J. Kurusingal, Law of normal scattering  a comprehensive law for wave propagation at an interface, J. Opt. Soc. Am. A 24, 98108 (2007).
External links
Categories: Optics
 Interference
 Interferometry
 Interferometers
 Quantum mechanics
 Wave mechanics
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