The formula for single-slit diffraction is derived using Huygens’ principle and the concept of path difference, which are essential to understanding wavefronts and rays.

To derive the formula for single-slit diffraction, we begin with Huygens’ principle. This principle posits that each point on a wavefront acts as a source of secondary wavelets that propagate forward at the same speed as the original wave. The new wavefront can be visualized as the tangential surface that touches all these secondary wavelets.

Consider a single slit with a width of $a$ and a screen positioned at a distance $D$ from the slit. When monochromatic light of wavelength $\lambda$ strikes the slit, diffraction occurs, resulting in a pattern of alternating bright and dark fringes on the screen. The central maximum is the brightest part of the pattern, and as we move away from the center, the intensity of light diminishes. This phenomenon exemplifies how interference patterns are formed.

To determine the condition for the minima (dark fringes), we analyze two rays of light: one originating from the top of the slit and another from the middle of the slit. The path difference between these two rays is given by $\frac{a}{2} \sin \theta$, where $\theta$ is the angle that the ray makes with the line perpendicular to the slit. For destructive interference to occur, this path difference must equal half the wavelength, or $\frac{\lambda}{2}$. Thus, the condition for the minima can be expressed as:

where $m$ is an integer (1, 2, 3, …). This derivation underscores the principle of interference, which is a fundamental aspect of wave behavior, albeit in a different context.

The angle $\theta$ can be related to the position $y$ on the screen using the small angle approximation, where $\tan \theta \approx \sin \theta \approx \frac{y}{D}$. Therefore, the position of the $m$-th minima on the screen is given by:

This derivation operates under the assumption that the width of the slit $a$ is on the order of the wavelength of light $\lambda$, and that the distance to the screen $D$ is significantly larger than $a$. Additionally, it is assumed that the light is incident normally on the slit, and the screen is perpendicular to the direction of the incoming light.