18.2 Diffraction Grating

What is a Diffraction Grating?

A diffraction grating is a glass (or metal) plate ruled with a very large number of close, parallel, equally spaced slits — typically thousands of lines per centimetre. When light passes through (transmission grating) or reflects off (reflection grating) these slits, it undergoes both diffraction and interference, producing a sharp, well-separated spectrum.

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Grating Element ($d$)

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

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

where:

• $L$ = total ruled length of the grating
• $N$ = total number of lines on the grating
• $N^{\prime }$ = number of lines per unit length (e.g., lines per metre)

Example: A grating with 500 lines/mm has $d=\frac{1}{500\times 10^3{\text{ m}}^{-1}}=2\times 10^{-6}\text{ m}=2\mu \text{m}$

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The Grating Equation

When monochromatic light of wavelength $\lambda$ is incident normally on a diffraction grating, constructive interference (principal maxima) occurs at angles $\theta$ satisfying:

$\boxed{d\sin \theta =n\lambda }$

where:

• $d$ = grating element (m)
• $\theta$ = angle of diffraction from the normal
• $n$ = order of the maximum ($n=0,\pm 1,\pm 2,\dots$)
• $\lambda$ = wavelength of light (m)

Orders of Maxima

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Maximum Observable Order

Since $\sin \theta \le 1$, the maximum possible order is:

$n_{\max }\le \frac{d}{\lambda }$

If $n_{\max }$ gives exactly $\theta =90°$, that order travels along the grating surface and is not physically observable. The highest observable order is the largest integer $n$ for which $\theta <90°$.

Example: If $d=2\lambda$, then $n_{\max }=d/\lambda =2$, but at $n=2$, $\theta =90°$ (unobservable). So the highest observable order is $n=1$.

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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).
• The dark regions between maxima become darker.
• The peaks become brighter (more slits contribute constructively).

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Dispersion of White Light

From $d\sin \theta =n\lambda$, since $\sin \theta \propto \lambda$:

• Longer wavelengths (red, ~700 nm) are diffracted through larger angles.
• Shorter wavelengths (violet, ~400 nm) are diffracted through smaller angles.
• White light is dispersed into a spectrum at each order (except $n=0$).

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Using a Diffraction Grating to Determine Wavelength

The diffraction grating is one of the most precise instruments for measuring the wavelength of light (SLO P-12-D-24).

Procedure

1. Set up the grating on a spectrometer table with light incident normally.
2. Measure the angle ${\theta }_n$ of a particular spectral line at order $n$ using the spectrometer's vernier scale.
3. The grating element $d$ is known from the manufacturer's specification ($d=1/N^{\prime }$).
4. Calculate the wavelength using:

$\lambda =\frac{d\sin {\theta }_n}{n}$

Why Gratings are Preferred Over Double Slits

• Gratings produce much sharper maxima → more precise angle measurement.
• Multiple orders allow cross-checking of the calculated $\lambda$.
• Higher resolving power allows separation of closely spaced spectral lines (e.g., sodium doublet).

Applications

• Spectroscopy: Identifying elements from their emission/absorption spectra.
• Astronomy: Analysing starlight to determine composition, temperature, and redshift.
• Laser wavelength measurement: Precise determination of laser output wavelength.

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Worked Example

Problem: A diffraction grating has 600 lines/mm. Light of an unknown wavelength produces a second-order maximum at $\theta =38.2°$. Find the wavelength.

Solution: $d=\frac{1}{600\times 10^3}=1.667\times 10^{-6}\text{ m}$ $\lambda =\frac{d\sin \theta }{n}=\frac{1.667\times 10^{-6}\times \sin 38.2°}{2}$ $\lambda =\frac{1.667\times 10^{-6}\times 0.6170}{2}=5.14\times 10^{-7}\text{ m}\approx 514\text{ nm (green light)}$

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