A diffraction grating contains many equally spaced parallel slits or grooves. Each illuminated opening contributes a coherent wave, and their superposition produces very sharp intensity maxima at particular directions.

Let the spacing between neighboring slits or grooves be $d$. For light observed at angle $\theta$ from the normal, the path difference between waves from adjacent openings is

$$d\sin\theta.$$

Constructive interference occurs when that path difference is an integer number of wavelengths:

$$\boxed{d\sin\theta=m\lambda},$$

where

$$m=0,\pm1,\pm2,\ldots$$

is the diffraction order.

Which orders can exist?

Because

$$|\sin\theta|\le1,$$

a physical order must satisfy

$$|m|\lambda\le d.$$

The central maximum is the zeroth order $m=0$. Positive and negative orders appear symmetrically on opposite sides when the geometry is symmetric.

Worked example

A grating has

$$600\ \text{lines/mm}.$$

Its spacing is

$$d=\frac{1,\mathrm{mm}}{600} =1.67\times10^{-6},\mathrm m.$$

For green light of wavelength

$$\lambda=500,\mathrm{nm}=5.00\times10^{-7},\mathrm m,$$

the first-order maximum satisfies

$$\sin\theta_1=\frac{\lambda}{d} \approx0.300.$$

Thus

$$\boxed{\theta_1\approx17.5^\circ}.$$

For the second order,

$$\sin\theta_2=\frac{2\lambda}{d}\approx0.600,$$

so

$$\boxed{\theta_2\approx36.9^\circ}.$$

A fourth order would require $\sin\theta>1$, so it cannot occur for this wavelength and grating spacing.

Why a grating separates wavelengths

At fixed order $m$, the angle depends on wavelength:

$$\sin\theta=\frac{m\lambda}{d}.$$

Different wavelengths therefore produce maxima at different angles. A grating can spread light into a spectrum and measure unknown wavelengths from their angular positions.

Higher diffraction orders generally produce greater angular separation between nearby wavelengths, although only orders satisfying the geometric condition can exist.

Why many slits give sharp peaks

With only two slits, constructive interference produces broad fringes. When $N$ equally spaced openings contribute, all $N$ waves are exactly in phase at the grating maxima. Slightly away from those angles, their phases progressively spread around the cycle and cancel strongly.

As $N$ increases, the principal maxima become narrower. This makes a grating capable of distinguishing wavelengths that would overlap in a broad two-slit pattern.

For an ideal grating illuminated across $N$ grooves, a commonly used estimate of spectral resolving power in order $m$ is

$$\boxed{R\equiv\frac{\lambda}{\Delta\lambda}\approx |m|N}.$$

Thus using more illuminated grooves or a higher usable order can separate more closely spaced spectral lines.

Diffraction gratings turn the interference of many coherent contributions into a practical wavelength analyzer: the grating equation sets the peak positions, while the number of illuminated periods controls how sharply those peaks are defined.