Diffraction Grating Calculator

A diffraction grating splits light into its component wavelengths. The grating equation gives the angles at which constructive interference occurs:

mλ = d sin(θ) d = 1/N (where N = lines per mm)

Where:

• m = Diffraction order (0, ±1, ±2, …)
• λ = Wavelength (m)
• d = Grating spacing = slit-to-slit distance (m)
• θ = Diffraction angle
• N = Number of lines per mm

Angular dispersion: dθ/dλ = m / (d cos(θ)), which is how much the angle changes per unit wavelength

Resolving power: R = mN_total, which is how well the grating separates closely spaced wavelengths

Diffraction gratings vs. prisms:

Applications:

• Spectroscopy: Gratings are the heart of spectrometers in chemistry, astronomy, and environmental monitoring
• Astronomy: Spectrographs on telescopes use gratings to measure stellar composition and redshift
• CDs and DVDs: The closely spaced tracks (d ≈ 1.6 μm) act as a reflection grating, producing rainbow colors
• Laser wavelength selection: Tunable lasers use gratings to select specific wavelengths

How many orders can you actually see

The grating equation has a hard ceiling, because sin(θ) cannot exceed 1. Rearranged, the highest visible order is whatever m keeps mλ below d. A 600 lines/mm grating has d = 1,667 nm, so at 550 nm green you get m = 1, 2 and 3, and there is no fourth order at any angle. Push the line count up to 1,800 per mm and d falls to 556 nm, which is barely more than the wavelength itself: now only the first order exists. That is a real design trade, not a curiosity. Finer gratings disperse more strongly but throw away the higher orders that give a spectrometer its resolving power.

Where the orders overlap

Second-order violet at 400 nm and first-order deep red at 800 nm come off the grating at the same angle, since 2 × 400 = 1 × 800. Any spectrometer working over more than an octave has to deal with this, usually with a coloured blocking filter that kills the short wavelengths before they reach the detector. Miss that and you get a perfectly convincing spectral line that is not there.

A note on the numbers you enter

Gratings are sold by lines per millimetre, typically 300, 600, 1200 or 1800 for laboratory work. The calculator converts to the spacing d for you, because that is what the equation actually uses. Watch the units: 600 lines per mm means d = 1/600 mm, which is 1.667 micrometres, not 600 of anything.