Diffraction Grating Angle
Diffraction Grating Angle
Diffraction gratings — surfaces ruled with thousands of closely spaced lines — split light into its component wavelengths by diffraction, the same principle behind the rainbow sheen on a CD or DVD. Spectrometers used in chemistry, astronomy, and materials science rely on precisely this relationship to separate and identify wavelengths.Because the diffraction angle depends on both wavelength and grating spacing, a grating with finer line spacing spreads colors out over a wider angular range, giving higher spectral resolution — which is why grating spacing is chosen carefully when designing a spectrometer for a target wavelength range.
The grating equation is d·sin θ = m·λ, so the diffraction angle is θ = arcsin(mλ/d). where lambda_d is the light wavelength, d_grat is the grating line spacing, m_ord is the diffraction order, and theta_d is the resulting diffraction angle.
Inputs
lambda_d := 550 nm=5.5e-04 mm
d_grat := 2000 nm=2e-03 mm
m_ord := 1=1
Solve the grating equation for the angle: the ratio of order times wavelength to line spacing gives the sine of the diffraction angle directly.
theta_d := asin(m_ord·lambda_d/d_grat)=0.28 rad
Results
For green light on a grating with 2000 nm line spacing, the first-order beam diffracts at a moderate angle of roughly 16° from straight through. Higher orders (m = 2, 3, …) diffract at progressively larger angles until mλ/d exceeds 1, at which point that order no longer exists for that wavelength — which is why only a limited number of orders are visible for a given grating and wavelength.
Assumptions & limitations
- Monochromatic, normally incident light.
- Grating spacing much larger than the wavelength (many lines illuminated).
- Far-field (Fraunhofer) diffraction regime.