Measuring wavelength with a grating

The main symbols used in this lesson:

You need:

• a source of monochromatic light — a laser is ideal, or a lamp with a coloured filter,
• a diffraction grating of known slit spacing,
• a screen (or a wall) a measured distance away,
• a metre rule to measure distances.

Shine the light at normal incidence on the grating, so the beam arrives perpendicular to its surface. The central maximum then goes straight through, and the orders appear symmetrically on either side.

Let the screen be a distance $D$ from the grating. Choose an order $n$ — the first order is usually bright and easy to find.

Measure the distance from the central maximum to the $n$-th order maximum. Call it $y$.

A better method is to measure the distance between the $+n$ and $-n$ maxima and halve it. This avoids having to locate the centre to the nearest millimetre, and it makes the measured distance twice as large, which reduces the percentage uncertainty.

Then:

$\tan\theta = \frac{y}{D}.$

From $\tan\theta$ you get $\theta$, and hence $\sin\theta$. Finally

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

On a screen you measure two sides of a right-angled triangle: $y$ opposite the angle and $D$ adjacent to it. So the ratio $\dfrac{y}{D}$ is $\tan\theta$, not $\sin\theta$.

For small angles the two are nearly equal, which is why the double-slit formula uses $\dfrac{ax}{D}$ as an approximation. But grating angles are often large — $30^\circ$ or more — so the difference matters.

For example, if $\tan\theta = 0.75$ then $\theta = 36.9^\circ$ and $\sin\theta = 0.60$. Using $0.75$ instead of $0.60$ would make the wavelength $25\%$ too large.

If you know $\tan\theta$, you can find $\sin\theta$ from $\sin\theta = \dfrac{\tan\theta}{\sqrt{1 + \tan^2\theta}}$.

With a double slit the maxima are broad, so locating their centres is imprecise. A grating produces narrow, sharp maxima, so the angle can be measured accurately.

The angles are also large, and you can use higher orders, where a small error in distance corresponds to a small error in angle.

Practical points:

• Measure across several orders and average, or use the largest order you can see clearly.
• Measure $D$ from the grating to the screen with the same rule, and keep the screen perpendicular to the beam.
• Use a dark room so faint orders are visible.
• Never look directly into a laser beam. Use a screen to view the pattern.