Diffraction Grating Calculator

Diffraction grating calculator analyzes what happens when a light ray meets a surface with multiple apertures.

Angle of order 1 maximum15.962°d sin θ = mλ

sin θ0.275

Wavelength550 nm550 nm

Spacing2 µm3.6364 wavelengths

Highest order available3beyond this sin θ would exceed 1, so the order simply does not exist — which is why a grating must have a spacing comparable to the wavelength

Position on a screen 2 m away572.0561 mm572.056 mm from the centre

Rayleigh angular resolution1.342e-5 rad2.7681 arcseconds — 1.22λ/D, the diffraction limit no amount of magnification can beat

Smallest detail at 1 km13.42 mm

At twice the aperture1.384 arcsecondstwice as fine — which is the whole reason telescopes are built large

The formula

d sin θ = mλ; Bragg 2d sin θ = nλ; Rayleigh θ = 1.22λ ÷ D

Why a grating needs fine spacing

Diffraction maxima occur where the path difference between adjacent slits is a whole number of wavelengths, giving d sin θ = mλ. Since sin θ cannot exceed 1, an order only exists if mλ ≤ d — so a grating must have a line spacing comparable to the wavelength it is meant to disperse. A grating with spacing below one wavelength produces no orders at all beyond the central one.

Bragg reflection carries an extra factor of two because the ray must travel down to the next plane and back, doubling the path difference. Forgetting it halves every spacing derived from an X-ray pattern, which is the classic error in crystallography arithmetic.

The Rayleigh criterion sets a hard limit: an aperture of diameter D cannot resolve detail finer than about 1.22λ/D, no matter how good the optics or how much you magnify. This is diffraction, not imperfection, and it is why telescope apertures grow rather than their eyepieces — and why electron microscopes exist, since electrons have far shorter wavelengths than light.