18.2.1 Diffraction of Light Through a Diffraction Grating | Diffraction And Interference | Physics 12

What is a Diffraction Grating?

A diffraction grating is an optical device consisting of a glass or metal plate ruled with a very large number of close, parallel, and equally spaced slits (typically thousands per centimetre). When light passes through (transmission grating) or reflects off (reflection grating) these slits, it undergoes diffraction and the diffracted beams from all slits interfere with each other, producing a sharp, bright spectrum.

Example: A CD or DVD surface acts as a reflection diffraction grating — the closely spaced tracks disperse white light into its constituent colours.

Grating Element ( $d$ )

The grating element (also called the slit spacing) $d$ is the distance between the centres of two adjacent slits.

$d = \frac{L}{N} = \frac{1}{N ^{'}}$

where:

$L$ = total ruled length of the grating
$N$ = total number of slits
$N^{'}$ = number of slits per unit length (lines per metre or per mm)

Example: A grating with 500 lines/mm has $d = \frac{1}{500 lines/mm} = \frac{1}{500 \times 1 0 ^{3} lines/m} = 2 \times 1 0^{- 6} m$

The Grating Equation

When monochromatic light of wavelength $λ$ is incident normally on a grating with element $d$ , constructive interference (a principal maximum) occurs at angles $θ$ satisfying:

$d sin θ = nλ$

Symbol	Meaning
$d$	Grating element (m)
$θ$	Angle of diffraction from the normal
$n$	Order of maximum ( $n = 0, \pm 1, \pm 2, \dots$ )
$λ$	Wavelength of light (m)

$n = 0$ : zeroth order (straight-through beam, no dispersion)
$n = 1$ : first order, $n = 2$ : second order, etc.

Effect of Wavelength

Since $sin θ = \frac{nλ}{d}$ , a longer wavelength gives a larger diffraction angle. For white light, red (longest $λ \approx 700$ nm) is diffracted most and violet (shortest $λ \approx 400$ nm) least, producing a visible spectrum at each order.

Maximum Observable Order

Because $sin θ \leq 1$ , the maximum possible order is:

$n_{m a x} \leq \frac{d}{λ}$

If this gives $θ = 90°$ exactly, that order travels along the grating surface and is not physically observable. The highest observable order is the largest integer $n$ for which $θ < 90°$ .

Example: If $d = 2 λ$ , then $n_{m a x} = d / λ = 2$ , but at $n = 2$ , $θ = 90°$ (unobservable). So the highest observable order is $n = 1$ .

Effect of Number of Slits

Compared to a double-slit with the same spacing $d$ :

The positions of principal maxima are unchanged (same grating equation).
Increasing the number of slits $N$ makes the maxima narrower and sharper (higher resolving power) and the regions between them darker.
This allows the grating to resolve closely spaced wavelengths that a double-slit cannot separate.

Using a Diffraction Grating to Determine Wavelength

A diffraction grating is one of the most precise instruments for measuring the wavelength of light:

Direct a monochromatic light source normally onto the grating.
Use a spectrometer to measure the angle $θ_{n}$ of the $n$ -th order maximum.
Apply the grating equation: $λ = \frac{d s i n θ _{n}}{n}$
Since $d$ is known (from the grating specification) and $θ_{n}$ is measured, $λ$ is determined with high precision.

Gratings are preferred over prisms for spectroscopy because they produce a linear dispersion and can achieve much higher resolution.

Summary Table

Quantity	Formula
Grating element	$d = 1/ N^{'}$
Principal maxima	$d sin θ = nλ$
Maximum order	$n_{m a x} = ⌊ d / λ ⌋$ (with $θ < 90°$ )
Wavelength from grating	$λ = d sin θ / n$

What is a Diffraction Grating?

Example: A CD or DVD surface acts as a reflection diffraction grating — the closely spaced tracks disperse white light into its constituent colours.

Grating Element ( $d$ )

The grating element (also called the slit spacing) $d$ is the distance between the centres of two adjacent slits.

$d = \frac{L}{N} = \frac{1}{N ^{'}}$

where:

$L$ = total ruled length of the grating
$N$ = total number of slits
$N^{'}$ = number of slits per unit length (lines per metre or per mm)

Example: A grating with 500 lines/mm has $d = \frac{1}{500 lines/mm} = \frac{1}{500 \times 1 0 ^{3} lines/m} = 2 \times 1 0^{- 6} m$

The Grating Equation

When monochromatic light of wavelength $λ$ is incident normally on a grating with element $d$ , constructive interference (a principal maximum) occurs at angles $θ$ satisfying:

$d sin θ = nλ$

Symbol	Meaning
$d$	Grating element (m)
$θ$	Angle of diffraction from the normal
$n$	Order of maximum ( $n = 0, \pm 1, \pm 2, \dots$ )
$λ$	Wavelength of light (m)

$n = 0$ : zeroth order (straight-through beam, no dispersion)
$n = 1$ : first order, $n = 2$ : second order, etc.

Effect of Wavelength

Maximum Observable Order

Because $sin θ \leq 1$ , the maximum possible order is:

$n_{m a x} \leq \frac{d}{λ}$

If this gives $θ = 90°$ exactly, that order travels along the grating surface and is not physically observable. The highest observable order is the largest integer $n$ for which $θ < 90°$ .

Example: If $d = 2 λ$ , then $n_{m a x} = d / λ = 2$ , but at $n = 2$ , $θ = 90°$ (unobservable). So the highest observable order is $n = 1$ .

Effect of Number of Slits

Compared to a double-slit with the same spacing $d$ :

The positions of principal maxima are unchanged (same grating equation).
Increasing the number of slits $N$ makes the maxima narrower and sharper (higher resolving power) and the regions between them darker.
This allows the grating to resolve closely spaced wavelengths that a double-slit cannot separate.

Using a Diffraction Grating to Determine Wavelength

A diffraction grating is one of the most precise instruments for measuring the wavelength of light:

Direct a monochromatic light source normally onto the grating.
Use a spectrometer to measure the angle $θ_{n}$ of the $n$ -th order maximum.
Apply the grating equation: $λ = \frac{d s i n θ _{n}}{n}$
Since $d$ is known (from the grating specification) and $θ_{n}$ is measured, $λ$ is determined with high precision.

Gratings are preferred over prisms for spectroscopy because they produce a linear dispersion and can achieve much higher resolution.

Summary Table

Quantity	Formula
Grating element	$d = 1/ N^{'}$
Principal maxima	$d sin θ = nλ$
Maximum order	$n_{m a x} = ⌊ d / λ ⌋$ (with $θ < 90°$ )
Wavelength from grating	$λ = d sin θ / n$