Grating Equation

The grating equation relates the diffraction order, wavelength, grating period, and propagation angle for a diffraction grating:

$$ \sin\theta_q = q \lambda f_0 $$

where:

• $\theta_q$ is the diffraction angle of the $q$th order
• $q$ is the integer diffraction order ($0, \pm 1, \pm 2, \ldots$)
• $\lambda$ is the wavelength
• $f_0 = 1/d$ is the spatial frequency of the grating (with period $d$)

Propagating vs. Evanescent Orders

For a diffracted beam to propagate, we require $|\sin\theta_q| \leq 1$:

$$ |q| \leq \frac{1}{\lambda f_0} = \frac{d}{\lambda} $$

Orders exceeding this limit become evanescent waves that decay exponentially with distance from the grating.

Resolving Power

The resolving power of a grating with $M$ illuminated periods, observed in the $q$th order, is:

$$ \frac{\lambda}{\Delta\lambda} = qM $$

This follows from the Rayleigh criterion applied to the sinc-shaped diffraction peaks of a finite grating.

Connection to Fraunhofer Diffraction

In the Fraunhofer regime, each diffraction order produces a peak at position $x_q = q\lambda z f_0$ in the observation plane at distance $z$. The grating equation is the angular form of this relationship.