1. What a diffraction grating is
A diffraction grating is an optical device with a very large number of equally spaced, parallel slits. Instead of just one or two slits, a grating may have hundreds or thousands of slits per millimetre.
Because so many slits produce light waves that interfere together, the resulting diffraction pattern is extremely sharp, bright, and well-defined.
2. Why a grating produces sharp patterns
Each slit acts as a tiny source of light waves. When many such waves overlap, only very specific directions survive constructive interference, while all other directions cancel out. As the number of slits increases:
- bright fringes become narrower and sharper,
- dark regions become darker,
- overall contrast improves greatly.
This makes diffraction gratings useful for analysing wavelengths precisely.
3. Grating element (slit separation)
The distance between two adjacent slits is called the grating element, often denoted by \(d\). It includes one slit and the adjacent opaque region.
If a grating has \(N\) slits per metre, then:
\( d = \dfrac{1}{N} \)
4. Condition for maxima in a diffraction grating
Bright fringes (maxima) occur when waves from all slits add constructively. The condition is:
\( d \sin \theta = n\lambda \)
Here:
- \(d\): grating spacing
- \(\theta\): angle of the bright fringe
- \(\lambda\): wavelength of light
- \(n = 0, 1, 2, ...\): order of the maxima
4.1. Order of maxima
The integer \(n\) represents the order of the bright fringe. The central bright fringe is the zeroth order (\(n = 0\)). The first bright fringe on either side is \(n = 1\), then \(n = 2\), and so on.
5. Order and number of visible spectra
Higher orders occur at larger angles. However, when the argument of \(\sin \theta\) exceeds 1, no more maxima are possible. So the maximum order is:
\( n_{\text{max}} = \left\lfloor \dfrac{d}{\lambda} \right\rfloor \)
6. Why gratings separate colours well
Because the angle \(\theta\) depends on \(\lambda\), different wavelengths satisfy the grating condition at different angles. This spreads the colours out much more effectively than a prism.
Longer wavelengths like red produce maxima at larger angles, while shorter wavelengths like violet appear closer to the centre.
7. Intensity pattern in a grating
As the number of slits increases:
- bright fringes become extremely narrow,
- intensity at the maxima increases,
- dark regions become very wide and nearly zero in intensity.
This makes gratings ideal for precise measurements.
8. Comparison: diffraction grating vs single slit
| Feature | Single Slit | Diffraction Grating |
|---|---|---|
| Number of slits | 1 | Many (hundreds or thousands) |
| Fringe sharpness | Broad, less sharp | Very sharp and well-defined |
| Spectrum separation | Weak | Strong, clear separation of colours |
| Intensity | Moderate | High at maxima |
9. Example: Finding the angle of first order maximum
Suppose a grating has 5000 slits/cm:
5000 \, \text{slits/cm} = 5 \times 10^5 \, \text{slits/m}
d = \dfrac{1}{5 \times 10^5} = 2 \times 10^{-6} \, \text{m}
For light of wavelength \(\lambda = 600 \, \text{nm}\):
d \sin \theta = 1 \times 600 \times 10^{-9}
\sin \theta = \dfrac{600 \times 10^{-9}}{2 \times 10^{-6}} = 0.3
\theta \approx 17.5^\circ
This angle marks the position of the first bright fringe for red light.
10. Where diffraction gratings are used
Diffraction gratings appear in many scientific and everyday applications:
- Spectrometers for analysing light
- Laser tuning and wavelength measurement
- CD/DVD surfaces (which behave like natural gratings)
- Optical communication devices
- Scientific instruments in astronomy
Gratings are essential wherever precise wavelength separation is needed.