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 for either single-photon illumination, or narrow-linewidth laser illumination, that is, illumination by indistinguishable photons, by Duarte. The N-slit interferometer was first applied in the generation and measurement of complex interference patterns.

In this article the generalized N-slit interferometric equation, derived via Dirac's notation, is described. Although originally derived to reproduce and predict N-slit interferograms, this equation also has applications to other areas of optics.

In this approach the probability amplitude for the propagation of a photon from a source (s) to an interference plane (x), via an array of slits (j), is given using Dirac's bra–ket notation as

This equation represents the probability amplitude of a photon propagating from s to x via an array of j slits. Using a wavefunction representation for probability amplitudes, and defining the probability amplitudes as

where ${\displaystyle \theta _{j}}$ and ${\displaystyle \phi _{j}}$ are the incidence and diffraction phase angles, respectively. Thus, the overall probability amplitude can be rewritten as

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