Diffraction Grating Formula
Calculate diffraction grating angles using d sin(theta) = m lambda.
Used in spectrometers to separate light into wavelengths.
The Formula
The diffraction grating equation describes how a surface with many equally spaced slits or grooves separates incoming light into its component wavelengths. When white light passes through or reflects from a diffraction grating, each wavelength is bent at a different angle, producing a rainbow-like spectrum. This makes diffraction gratings indispensable tools in spectroscopy, the science of analyzing light.
A diffraction grating works by exploiting the wave nature of light. Each slit in the grating acts as a source of secondary waves. These waves spread out and overlap with each other. At certain angles, waves from adjacent slits arrive in phase (their peaks align), creating bright spots of constructive interference. The angle depends on the wavelength of light and the spacing between slits, which is why different colors appear at different angles.
The slit spacing d is the distance between adjacent slits, often specified indirectly as the number of lines per millimeter. For example, a grating with 600 lines/mm has d = 1/600 mm = 1.667 micrometers. Higher line densities produce greater angular separation between wavelengths, which improves the spectral resolution of the grating.
The integer m is called the diffraction order. The zeroth order (m = 0) corresponds to straight-through light with no dispersion. The first order (m = 1) produces the first spectrum on either side of center, the second order (m = 2) produces a wider but dimmer spectrum, and so on. Higher orders provide better wavelength resolution but receive less light intensity.
Diffraction gratings were first developed by Joseph von Fraunhofer in Bavaria in the early 1800s. Today they are manufactured by ruling fine lines on glass or metal, or by holographic methods. They are used in spectrometers, monochromators, wavelength-division multiplexers in fiber optic communications, and even in everyday items like CDs and DVDs, which act as reflection gratings.
Variables
| Symbol | Meaning |
|---|---|
| d | Slit spacing / grating period (meters, m) |
| θ | Diffraction angle (degrees or radians) |
| m | Diffraction order (integer: 0, ±1, ±2, ...) |
| λ | Wavelength of light (meters, m or nanometers, nm) |
Example 1
A grating with 500 lines/mm is illuminated with red light (λ = 650 nm). At what angle does the first-order maximum appear?
d = 1 / 500 lines per mm = 0.002 mm = 2000 nm
d · sin(θ) = mλ → sin(θ) = mλ / d
sin(θ) = (1 × 650) / 2000 = 0.325
θ = arcsin(0.325)
θ ≈ 18.97° (first-order red maximum)
Example 2
Using the same 500 lines/mm grating, what is the maximum diffraction order visible for violet light (λ = 400 nm)?
d = 2000 nm, and sin(θ) cannot exceed 1
Maximum m: sin(θ) = mλ/d ≤ 1 → m ≤ d/λ
m ≤ 2000 / 400 = 5
Maximum order m = 5 (at θ = 90°, the practical limit)
When to Use It
Use the diffraction grating formula when you need to:
- Calculate the angle at which a specific wavelength appears in a spectrometer
- Determine the wavelength of light from measured diffraction angles
- Design spectrometers and monochromators for specific wavelength ranges
- Calculate the resolving power of a grating (R = mN, where N is total slits)
- Separate closely spaced spectral lines in astronomical or chemical analysis
Note that the formula assumes the light hits the grating at normal incidence (perpendicular). For oblique incidence at angle α, the generalized form becomes d(sin(α) + sin(θ)) = mλ.