- N-slit interferometric equation
Quantum mechanics was first applied to optics, and interference in particular, by Paul Dirac. Feynman, in his lectures, uses Dirac’s notation to describe thought experiments on double-slit interference of electrons. Feynman’s approach was extended to N-slit interferometers using narrow-linewidth laser illumination, that is, illumination by indistinguishable photons, by researchers working on the measurement of complex interference patterns.
Probability amplitudes and the N-slit interferometric equation
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 N-slit interferometer. The Dirac-Duarte 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 N-slit interferometric patterns continuously from the near to the far field. 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.
At a practical level, the N-slit interferometric equation was introduced for imaging applications and is routinely applied to predict N-slit laser interferograms, both in the near and far field. Thus, it has become a valuable tool in the alignment of large, and very large, N-slit laser interferometers used in the study of clear air turbulence and the propagation of interferometric characters for secure free-space optical communications.
Also, the N-slit interferometric equation has been applied to describe interference, diffraction, refraction, and reflection, in a rational and unified approach. For example, the phase term (in parenthesis) can be used to derive
which is also known as the diffraction grating equation. Here, θm is the angle of incidence, ϕm is the angle of diffraction, λ is the wavelength, and M is the order of diffraction. The N-slit interferometric approach is one of several approaches applied to describe basic optical phenomena in a cohesive and unified manner.
Note: given the various terminologies in use, for N-slit interferometry, it should be made explicit that the N-slit interferometric equation applies to two-slit interference, three-slit interference, four-slit interference, etc.
- Beam expander
- Dirac's notation
- Free-space optical communications
- Grating equation
- N-slit interferometer
- ^ 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 N-slit interference phenomena, in Proceedings of the International Conference on Lasers '88, R. C. Sze and F. J. Duarte (Eds.) (STS, McLean, Va, 1989) pp. 42-47.
- ^ F. J. Duarte, Dispersive dye lasers, in High Power Dye Lasers, F. J. Duarte (Ed.) (Springer-Verlag, Berlin, 1991) Chapter 2.
- ^ a b c d e F. J. Duarte, On a generalized interference equation and interferometric measurements, Opt. Commun. 103, 8-14 (1993).
- ^ F. J. Duarte, Comment on "reflection, refraction, and multislit interfernce," Eur. J. Phys. 25, L57-L58 (2004).
- ^ a b c d F. J. Duarte, Tunable Laser Optics (Elsevier-Academic, New York, 2003).
- ^ F. J. Duarte, T. S. Taylor, A. B. Clark, and W. E. Davenport, The N-slit 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, N-slit interferometer for secure free-space 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, 637-640 (1997).
- ^ J. Kurusingal, Law of normal scattering - a comprehensive law for wave propagation at an interface, J. Opt. Soc. Am. A 24, 98-108 (2007).
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