N-slit interferometric equation

[1] Richard Feynman, in his Lectures on Physics, uses Dirac's notation to describe thought experiments on double-slit interference of electrons.

Although originally derived to reproduce and predict N-slit interferograms,[3][4] 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[3] This equation represents the probability amplitude of a photon propagating from s to x via an array of j slits.

Thus, the overall probability amplitude can be rewritten as where and 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 parentheses represents the phase that is directly related to the exact path differences derived from the geometry of the N-slit array (j), the intra interferometric distance, and the interferometric plane x.

The N-slit interferometric equation has been applied to describe classical phenomena such as interference, diffraction, refraction (Snell's law), and reflection, in a rational and unified approach, using quantum mechanics principles.

Top view schematics of the $N$ -slit interferometer indicating the position of the planes $s$ , $j$ , and $x$ . The $N$ -slit array, or grating, is positioned at $j$ . The intra interferometric distance can be several-hundred meters long. TBE is a telescopic beam expander, MPBE is a multiple-prism beam expander .

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

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