d (spacing)—
θ for m = −2—
θ for m = −1—
θ for m = 00.00°
θ for m = +1—
θ for m = +2—
Resolving power R—
Angular disp. dθ/dλ—
FSR Δλ (m=1)—
Max visible order—
🌈 Diffraction Grating Physics
A transmission grating with N slits of spacing d and slit width a produces a far-field (Fraunhofer) intensity pattern:
I(θ) = I₀ · [sin(β/2)/(β/2)]² · [sin(Nδ/2) / (N·sin(δ/2))]²
where β = 2πa·sin(θ)/λ (single-slit envelope phase) and δ = 2πd·sin(θ)/λ (inter-slit phase difference).
- Principal maxima (grating equation): d · (sin θᵢ + sin θₘ) = m · λ for m = 0, ±1, ±2, …
- Resolving power: R = mN — smallest resolvable wavelength difference Δλ = λ / (mN)
- Angular dispersion: dθ/dλ = m / (d · cos θₘ) — higher orders disperse more
- Free spectral range: Δλ_FSR = λ / m — range before overlap of adjacent orders
- Missing orders: when a/d = 1/k (integer k), grating maxima at those orders are suppressed by the single-slit zero
White light mode overlays all visible wavelengths (380–740 nm), revealing the characteristic rainbow spectrum in each diffraction order — the principle behind spectrographs and CDs.