27.4 Multiple Slit Diffraction

Diffraction Grating

A diffraction grating takes the same interference idea from double-slit experiments and scales it up to hundreds or thousands of slits. The result is a much sharper, more precise pattern of bright spots, which is why gratings are the tool of choice for analyzing light in spectroscopy.

Diffraction Grating vs. Double Slit

Both setups produce bright fringes (maxima) and dark fringes (minima) through interference, but the patterns look noticeably different.

Diffraction grating (many slits):

• Produces sharp, narrow maxima that are evenly spaced and symmetric around the central maximum ($m = 0$)
• Intensity of maxima decreases as you move to higher orders ($m = \pm 1, \pm 2$, etc.)
• The dark regions between maxima are wide and very dark, making each bright spot easy to identify

Double slit (two slits):

• Produces broader, more gradually fading maxima
• Fringe spacing increases slightly at higher orders
• The overall pattern is less distinct because only two slits contribute to the interference

The key reason for the difference is the number of slits. With more slits, waves must line up more precisely to produce constructive interference. This makes each maximum narrower and the dark regions between them darker. The underlying physics is the same: the principle of superposition governs how waves from all the slits combine at each point on the screen.

Slit Spacing in Diffraction Gratings

Slit spacing ($d$) is the center-to-center distance between adjacent slits. It controls how spread out the maxima are.

• Smaller $d$ → maxima are spaced farther apart (larger angles from center)
• Larger $d$ → maxima are spaced closer together (smaller angles from center)

Why does this happen? The slit spacing determines the path difference between light waves arriving from neighboring slits. A smaller $d$ means a given path difference (say, one full wavelength) is reached at a larger angle. So the bright spots get pushed farther from the center.

Gratings are often described by their line density, such as "500 lines/mm." To find $d$, just take the reciprocal: $d = \frac{1}{500 \text{ lines/mm}} = 2 \times 10^{-3} \text{ mm} = 2 \text{ μm}$.

For the interference pattern to be clean and consistent, the light source needs to be coherent (waves maintaining a constant phase relationship), which is why lasers work so well in these experiments.

Calculating Constructive Interference Angles

The grating equation relates slit spacing, wavelength, and the angle at which bright maxima appear:

$d \sin \theta = m \lambda$

• $d$ = slit spacing
• $\theta$ = angle of the maximum measured from the central axis
• $m$ = order number (0, ±1, ±2, ...)
• $\lambda$ = wavelength of light

Steps to find the angle for a given order:

1. Identify $d$, $\lambda$, and the order $m$ you're solving for.
2. Plug into the grating equation and isolate $\sin \theta$: $\sin \theta = \frac{m \lambda}{d}$
3. Take the inverse sine to get $\theta$.
4. Check that $\sin \theta \leq 1$. If it's greater than 1, that order doesn't exist for this grating and wavelength.

Example: Find the first-order maximum angle for light with $\lambda = 500 \text{ nm}$ passing through a grating with $d = 2 \text{ μm}$.

1. $\sin \theta = \frac{m \lambda}{d} = \frac{(1)(500 \times 10^{-9})}{2 \times 10^{-6}}$
2. $\sin \theta = 0.25$
3. $\theta = \sin^{-1}(0.25) \approx 14.5°$

To find the maximum possible order, set $\sin \theta = 1$ and solve for $m$: $m_{\text{max}} = \frac{d}{\lambda}$. For this example, $m_{\text{max}} = \frac{2 \times 10^{-6}}{500 \times 10^{-9}} = 4$, so orders up to $m = 4$ are observable.

Applications

• Spectroscopy: Diffraction gratings separate light into its component wavelengths. Each wavelength diffracts at a slightly different angle, so a grating can spread white light into a full spectrum. This is how scientists identify elements in stars and gas samples.
• Resolution: A grating's ability to distinguish two closely spaced wavelengths improves with the total number of slits. More slits mean narrower maxima, making it easier to tell apart wavelengths that are nearly the same.