Mathematical expression of double-slit interference and intensity distribution

Introduction

Double-slit interference is a cornerstone of wave theory, demonstrating the principles of superposition and coherence in quantum mechanics. When light passes through two closely spaced slits, it creates an interference pattern characterized by alternating bright and dark fringes. This phenomenon arises from the constructive and destructive interference of waves emanating from each slit. The mathematical formulation of this behavior is essential for understanding both classical wave phenomena and the foundational principles of quantum mechanics. This article explores the mathematical framework governing double-slit interference, focusing on the intensity distribution and the factors that shape it.

Mathematical Framework

The double-slit interference arises from the superposition of waves originating from two slits. Let the electric field from each slit be represented by the wave functions:
$$ E_1(\vec{r}) = \frac{1}{\sqrt{2\pi}} \frac{E_0}{\lambda} e^{i(k \vec{r} - \omega t)}
$$
$$ E_2(\vec{r}) = \frac{1}{\sqrt{2\pi}} \frac{E_0}{\lambda} e^{i(k (\vec{r} - \vec{d}) - \omega t)}
$$
where $ \vec{d} $ is the vector from the second slit to the first,