3.3.2.2 - Diffraction

What diffraction means

Diffraction is a wave effect, so light shows it just as sound and water waves do. The spreading is most noticeable when the size of the gap or obstacle is similar to the wavelength of the wave. If the gap is much wider than the wavelength, the wave still travels mostly straight ahead and the diffraction is much less obvious.

For visible light, the wavelength is very small, so a wide opening produces very little spreading. A narrow slit produces much more. This is why single-slit diffraction becomes easier to observe as the slit is reduced in width.

Single-slit diffraction patterns

With monochromatic light, a single narrow slit produces a broad bright central maximum with weaker bright fringes on either side. These outer fringes are separated by dark regions where destructive interference occurs. In the usual small-angle screen pattern, the central maximum is approximately twice the width of a side maximum when widths are measured between neighbouring dark minima. The side maxima have approximately equal widths and rapidly decreasing intensity.

The figure below shows the pattern you are expected to recognise; notice the broad central maximum, the narrower outer bright fringes, and the way the intensity falls away from the centre.
[DIAGRAM: asset_name: 3.2.2 - Diffraction - Diagram 1; asset_slug: 3.2.2 - Diffraction - Diagram 1; caption: A single slit gives a broad central maximum and weaker side maxima; description: Calculated monochromatic single-slit screen pattern with a bright central maximum between the first two minima and diminishing side maxima. In the small-angle approximation the central width is 2a and each adjacent side width between minima is a.]

For this specification, you need the appearance of the pattern and the qualitative changes in its width. You do not need an intensity-against-angle graph.

If white light is used instead, each wavelength diffracts by a slightly different amount. The central maximum is white because all wavelengths overlap there. On each side of the centre, the fringes are coloured spectra: violet is closest to the central maximum and red is furthest away.

The width of the central maximum changes in a simple qualitative way:

• a longer wavelength gives a wider central maximum
• a narrower slit gives a wider central maximum

Diffraction gratings

A diffraction grating contains a very large number of equally spaced parallel slits. Light is incident normally on the grating, and each slit diffracts the light that passes through it. The diffracted waves from adjacent slits only reinforce strongly in certain directions, so the transmitted light appears as a series of sharp maxima.

The central beam is the zero order maximum, where the light continues straight through. Maxima on either side are called first order, second order, and so on. Because a grating has many slits, the bright maxima are much sharper than those from a double-slit arrangement, which makes their angles easier to measure accurately.

At a principal maximum, path differences between adjacent slits are whole wavelengths, so the contributions from all slits reinforce. Slightly away from that direction, phase differences accumulate across the many slits, so their contributions largely cancel. This explains why the maxima are narrow.

A longer wavelength is diffracted to a larger angle for the same grating and the same order. A grating with smaller slit spacing also gives larger diffraction angles.

Deriving the grating equation

To derive the diffraction grating equation, consider two neighbouring slits separated by a distance $d$. Look at rays from these two slits travelling to the same bright maximum at an angle $\theta$ to the normal.

In the diagram, A and B are adjacent slits. Draw AC perpendicular to the parallel outgoing rays, meeting the lower ray at C. A and C lie on the same outgoing wavefront, so the lower ray has travelled an extra BC. The angle between AB and AC equals $\theta$, giving $BC=d\sin\theta$. Normal incidence is essential: the incident wavefront reaches A and B together.
[DIAGRAM: asset_name: 3.2.2 - Diffraction - Diagram 2; asset_slug: 3.2.2 - Diffraction - Diagram 2; caption: The grating path difference is the segment BC; description: Normal incidence at two adjacent grating slits A and B separated by d. Outgoing parallel rays make angle theta to the horizontal normal. AC is perpendicular to both rays and BC is the extra path d sin theta. At a principal maximum BC equals an integer number of wavelengths.]

For a maximum to form, the waves from adjacent slits must arrive in phase. That happens when their path difference is a whole number of wavelengths:

• first order maximum: path difference = $\lambda$
• second order maximum: path difference = $2\lambda$
• nth order maximum: path difference = $n\lambda$

From the right-angled triangle in the diagram,
$\sin \theta = \frac{\text{path difference}}{d}$

So for the nth order maximum,
$\sin \theta = \frac{n\lambda}{d}$

Rearranging gives the required result.

Here, $d$ is the slit spacing, $\theta$ is the angle from the normal, $n$ is the order number, and $\lambda$ is the wavelength. If a grating is labelled by the number of lines per metre $N$, then the slit spacing is $d = 1/N$.

A physical order requires $|n\lambda/d|\leq1$. Test the greatest whole number no larger than $d/\lambda$; do not round this limit to the nearest integer. At equality the ray is at $90^\circ$ and will not reach an ordinary forward screen. A finite screen may capture still fewer orders.

For example, $500\,\mathrm{lines\,mm^{-1}}$ means $N=5.00\times10^5\,\mathrm{m^{-1}}$ and $d=2.00\times10^{-6}\,\mathrm{m}$. With $\lambda=600\,\mathrm{nm}$, the first-order angle is $\sin^{-1}(0.300)=17.5^\circ$. The third order exists because $3\lambda/d=0.900$; a fourth would require a sine of $1.20$, which is impossible.

Applications of diffraction gratings

Diffraction gratings are useful because they separate different wavelengths very clearly. That lets physicists identify substances and measure very small spacings.

This is why gratings are central to spectroscopy. A sharp pattern makes it easier to distinguish wavelengths that are close together.

A crystal is a three-dimensional array rather than a plane transmission grating. Its diffraction also depends on path differences, but the transmission-grating formula should not simply be substituted for the crystal geometry.