Calculate Single-Slit Diffraction

Online calculator and formulas for the diffraction condition a·sin(θ) = m·λ

Diffraction Calculator (JavaScript)

Single-slit formula

Intensity minima satisfy: a·sin(θ) = m·λ. Use this to calculate the diffraction angle θ or the slit width a.

m = 1, 2, 3, ...
Result

Example calculations

Example 1: Angle for first minimum

Given: a = 0.20 mm, λ = 550 nm, m = 1

\[sin(θ)=\frac{m·λ}{a}=\frac{1·550\,nm}{0.20\,mm}=0.00275\]

Result: θ ≈ 0.16°

Example 2: Determine slit width

Given: θ = 0.16°, λ = 550 nm, m = 1

\[a=\frac{m·λ}{sin(θ)}\]

Result: a ≈ 0.20 mm

Example 3: Effect of slit width

A smaller slit width leads to larger diffraction angles.

Rule of thumb: The smaller a, the wider the diffraction pattern.

Single-slit diffraction formulas

In single-slit diffraction, intensity minima occur for a·sin(θ) = m·λ with m = 1,2,3, ...

Diffraction condition
\[a·sin(θ)=m·λ\]
Angle
\[θ=arcsin\left(\frac{m·λ}{a}\right)\]
Slit width
\[a=\frac{m·λ}{sin(θ)}\]
Validity range
\[\left|\frac{m·λ}{a}\right|\le 1\]
Note
The calculator accepts angles in degrees. Trigonometric calculations are internally performed in radians.

Description

What is single-slit diffraction?

Diffraction refers to the propagation of waves beyond an obstacle or through a narrow opening into regions that, according to geometric optics, would lie in the shadow. When a plane light wave with wavelength λ strikes a slit of width a, the Huygens-Fresnel principle dictates that every point within the slit acts as the source of an elementary wave. The superposition of these elementary waves creates a characteristic diffraction pattern of bright and dark fringes on a distant screen.

Diffraction condition

Intensity minima (dark fringes) occur where the elementary waves cancel each other out in pairs. This happens precisely when the path difference between the edge rays is an integer multiple of the wavelength:

\[a·sin(θ)=m·λ \qquad m = 1, 2, 3, \dots\]
  • a – Slit width (width of the opening)
  • θ – Diffraction angle relative to the straight-ahead direction
  • m – Order of the minimum (integer, excluding 0)
  • λ – Wavelength of the light used

The central principal maximum is the brightest and approximately twice as wide as the secondary maxima on the sides. The smaller the slit width a is relative to the wavelength, the more strongly the light is spread out and the wider the diffraction pattern appears.

Applications
  • Determination of wavelengths and slit widths in spectroscopy
  • Resolution limit of optical instruments (microscope, telescope, eye)
  • Quality inspection of apertures, wires, and fine structures
  • Basis for understanding diffraction gratings and double-slit experiments
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