Diffraction Grating

A diffraction grating is an optical element with a periodic structure that splits and diffracts light into several beams traveling in different directions. The directions of these beams depend on the wavelength and the grating period.

Structure

A grating consists of $N$ equally spaced elements (slits, grooves, or phase steps) with center-to-center separation $\Delta$ (the grating period). Each element has width $Y$ (for a slit grating).

Fraunhofer Diffraction Pattern

The far-field intensity pattern of an $N$-slit amplitude grating is:

$$ I \propto \operatorname{sinc}^2(Yf_Y) \left[\frac{\sin(N\pi\Delta f_Y)}{\sin(\pi\Delta f_Y)}\right]^2 $$

Factor	Origin	Effect
$\operatorname{sinc}^2(Yf_Y)$	Single-element diffraction	Broad envelope
$\left[\frac{\sin(N\pi\Delta f_Y)}{\sin(\pi\Delta f_Y)}\right]^2$	$N$-element interference (array factor)	Sharp principal maxima at $f_Y = m/\Delta$

Key Properties

• Principal maxima occur at angles satisfying the grating equation: $\sin\theta_q = q\lambda/\Delta$
• Peak width scales as $1/N$ — more slits produce sharper peaks
• Peak intensity scales as $N^2$
• Resolving power: $\lambda/\Delta\lambda = qM$, where $q$ is the diffraction order and $M$ is the number of illuminated periods

Amplitude vs. Phase Gratings

Property	Amplitude grating	Phase grating
Transmittance magnitude	Varies ($0$ to $1$)	Constant ($= 1$)
Transmittance phase	Constant	Varies with position
Energy efficiency	Low — absorbs light	High — redistributes all light

Both types obey the same resolving power formula $\lambda/\Delta\lambda = qM$.