# N-slit interferometric equation

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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

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 $\langle x | s \rangle = \sum_{j=1}^\N\, \langle x | j \rangle \langle j | s \rangle$

Using a wavefunction representation for probability amplitudes, after some algebra, the corresponding probability becomes $|\langle x | s \rangle|^2\ = \sum_{j=1}^\N\,\Psi(r_j)^2\ +2 \sum_{j=1}^\N\,\Psi(r_j)\bigg(\sum_{m=j+1}^\N\,\Psi(r_m)cos(\Omega_m-\Omega_j)\bigg)$

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.

## Applications

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.  Interferogram for N = 3 slits with diffraction pattern superimposed on the right outer wing.

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 $d_m \left( \sin{\theta_m} + \sin{\phi_m} \right) = M \lambda$

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.

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