Calculate Grating Equation

Online calculator and formulas for diffraction by optical gratings

Grating Calculator (JavaScript)

Grating equation

Diffraction maxima satisfy: d·sin(θ)=m·λ. Calculate diffraction angle θ, wavelength λ, or grating constant d.

Result

Example calculations

Example 1: Calculate angle

Given: d = 1 mm, λ = 550 nm, m = 1

\[sin(θ)=\frac{m·λ}{d}=\frac{1·550\,nm}{1\,mm}=0.00055\]

Result: θ ≈ 0.03°

Example 2: Determine wavelength

Given: d = 1 mm, θ = 0.03°, m = 1

\[λ=\frac{d·sin(θ)}{m}\]

Result: λ ≈ 523.60 nm

Example 3: Orders

Higher order m gives larger diffraction angles.

Note: Solutions exist only if |m·λ/d| ≤ 1.

Grating equation formulas

Diffraction by a grating produces discrete maxima for integer orders m.

Grating equation
\[d·sin(θ)=m·λ\]
Angle
\[θ=arcsin\left(\frac{m·λ}{d}\right)\]
Wavelength
\[λ=\frac{d·sin(θ)}{m}\]
Grating constant
\[d=\frac{m·λ}{sin(θ)}\]

Description

What is an Optical Grating?

An optical grating is a periodic structure with many parallel, closely spaced slits or lines. When light strikes a grating, it is diffracted at many points. The resulting diffraction pattern shows discrete bright and dark stripes created by constructive and destructive interference. Optical gratings are fundamental to spectroscopy and enable the analysis of light by wavelength.

The Grating Equation

The grating equation describes the condition for diffraction maxima (bright orders) at a grating:

\[d \cdot \sin(θ) = m \cdot λ\]
  • d – grating constant (spacing between adjacent slits/lines) in meters
  • θ – diffraction angle from the optical axis in degrees
  • m – order of the maximum (m = 0, ±1, ±2, ...)
  • λ – wavelength of light in meters
Meaning of Orders
  • m = 0 (Central maximum): Undeviated light, passes directly through the grating, θ = 0°
  • m = ±1 (First order): First visible diffraction order on both sides
  • m = ±2, ±3, ... (Higher orders): Further weaker maxima at larger angles
  • Maximum order: Solutions only exist when |m·λ/d| ≤ 1
Typical Grating Constants
Grating Type Grating Constant d Lines per mm Application
Coarse grating 10 µm 100 Visible light, demonstrations
Standard grating 1 µm 1000 Laboratory spectroscopy, visible range
Fine grating 500 nm 2000 High-resolution spectroscopy
Very fine grating 200 nm 5000 UV spectroscopy
High resolution 100 nm 10000 Extreme resolution, specialized applications
Important Properties of Diffraction Gratings
  • Dispersion characteristic: Different wavelengths are deflected at different angles
  • Spectral resolution: A fine grating with large d can generate multiple orders and provide better resolution
  • Wavelength range: The smaller the grating constant d, the higher orders are possible
  • Diffraction efficiency: The intensity of various orders depends on the grating design
Practical Applications
  • Spectroscopy: Decomposition of light into its spectrum to determine wavelengths and chemical composition
  • Monochromator: Selection of specific wavelengths from white light
  • Laser applications: Wavelength-selective components in laser systems
  • CD/DVD: The iridescent colors result from the periodic structure (similar to a grating)
  • Holographic gratings: Used in modern spectrometers and optical systems
Note
The resolution of a grating improves with the number of grating lines and the order. A grating with N lines and light of order m can separate wavelengths differing by approximately λ/(N·m).
Is this page helpful?            
Thank you for your feedback!

Sorry about that

How can we improve it?