Single Slit Diffraction: Definition & Equations

[Greg Bernhardt]

Definition/Summary

This entry describes diffraction of a wave when it passes through a single narrow slit.

Equations

The far-field (Fraunhofer) diffraction pattern has a power per area (irradiance) at an angle $\theta$ from a single slit of width $d$, for wavelength $\lambda$ and wavenumber $k\ =\ 2\pi/\lambda$ of:

$$I(\theta)\ =\ \left( \frac{\sin \beta}{\beta}\right)^2\,I(0)$$

where:
$$\beta\ \equiv\ \frac{\pi d}{\lambda} \ \sin\theta\ =\ \frac{k d}{2} \ \sin\theta$$

which for very small angles is approximately:
$$\beta \ \approx \ \frac{\pi d}{\lambda} \ \theta \ = \ \frac{kd}{2} \ \theta$$The diffraction minima (dark fringes) occur when

$$\beta \ = \ n \pi, \ \ n \ = \ \pm 1, \ \pm 2, \ \pm 3, \ ...$$

or, for small angles,

$$\theta \ \approx \ n \lambda / d, \ \ n \ = \ \pm 1, \ \pm 2, \ \pm 3, \ ...$$

Note that n=0 corresponds to the central maximum, not a minimum.

Extended explanation

Definition of terms

I = irradiance of the wave, with SI units of W/m²
I(0) = the irradiance at θ=0
d = the slit width
λ = the wavelength of the wave
k = 2π/λ
θ = the angle at which the irradiance is evaluated
​

* This entry is from our old Library feature, and was originally created by Redbelly98

 

[APhysicist]

It discusses wave diffraction and provides equations for the far-field (Fraunhofer) diffraction pattern of a wave that passes through a single narrow slit. The equation provided gives the irradiance at an angle θ, with the minima (dark fringes) occurring at regular intervals, determined by the ratio of the slit width and the wavelength of the wave. The terms used in the equation and their SI units are also provided.