N-Slit Diffraction Grating

I(θ) = sinc²(β/2) × [sin(Nδ/2)/sin(δ/2)]² — from Young's double slit to diffraction grating

INTENSITY PATTERN I(θ)

GRATING DIAGRAM

Slits N 4

Slit width a (×λ) 0.5

Slit spacing d (×λ) 2.0

Wavelength λ (nm) 550

Primary max: m=0,±1,±2...

Resolution λ/Δλ: –

Angular width: –

Key formula: I(θ) = I₀ · sinc²(πa·sinθ/λ) · [sin(Nπd·sinθ/λ)/sin(πd·sinθ/λ)]². N slits narrow the peaks by factor N, increase peak intensity by N², and resolving power = mN. Single envelope (sinc²) modulates the interference pattern.

Diffraction grating: N equally-spaced slits with width a and separation d. The intensity is the product of two factors: the single-slit diffraction envelope sinc²(πa sinθ/λ) and the multi-slit interference factor [sin(Nφ/2)/sin(φ/2)]² where φ = 2πd sinθ/λ. With many slits, the interference peaks become extremely sharp — this is the principle behind spectroscopes and spectrometers. Resolving power R = λ/Δλ = mN where m is the diffraction order. Move from N=2 (Young's experiment) to N=20 to see the dramatic sharpening.