Huygens-Fresnel principle: Analyzing consistency at the photon level

TL;DR

This paper develops a quantum formalism linking classical diffraction and photon propagation, demonstrating that Fresnel diffraction accurately describes photon behavior at the quantum level through experiments and mathematical analysis.

Contribution

It introduces a photon propagation model based on electromagnetic quantization and path integrals, connecting classical Fresnel diffraction with quantum photon detection.

Findings

Fresnel diffraction aligns with photon counting experiments.

The photon propagator can be expressed as a Fresnel diffraction integral.

Quantum and classical diffraction phenomena are closely related at the photon level.

Abstract

Typically the use of the Rayleigh-Sommerfeld diffraction formula as a photon propagator is widely accepted due to the abundant experimental evidence that suggests that it works. However, a direct link between the propagation of the electromagnetic field in classical optics and the propagation of photons where the square of the probability amplitude describes the transverse probability of the photon detection is still an issue to be clarified. We develop a mathematical formulation for the photon propagation using the formalism of electromagnetic field quantization and the path-integral method, whose main feature is its similarity with a fractional Fourier transform (FrFT). Here we show that, because of the close relation existing between the FrFT and the Fresnel diffraction integral, this propagator can be written as a Fresnel diffraction, which brings forward a discussion of the fundamental character of it at the photon level compared to the Huygens-Fresnel principle. Finally, we carry out an experiment of photon counting by a rectangular slit supporting the result that the diffraction phenomenon in the Fresnel approximation behaves as the actual classical limit.