Physics (9702) | The diffraction grating

The Diffraction Grating: Peeking into the Wavelengths of Light

Hello future Physicists! This chapter is where we take the principles of superposition and interference (which you learned about with the double slit) and turbocharge them!
The diffraction grating is one of the most powerful tools in physics, allowing us to accurately measure the tiny wavelengths of light and even analyze the composition of stars.
Don't worry if this topic feels a little complex; we will break down the key equation and its application step-by-step. Let's get started!

Quick Review: Coherence and Interference

Remember from the Interference section (Syllabus 8.3) that to observe a stable interference pattern (fringes), the two sources must be coherent. This means they must have:

• The same frequency and wavelength.
• A constant phase relationship.

The double-slit experiment produces interference fringes, but they are often quite dim and spread out. The diffraction grating solves this problem beautifully!

1. What is a Diffraction Grating?

A Diffraction Grating is essentially an optical component with a very large number of equally spaced parallel slits (or lines) close together.

• A typical grating might have 300 to 600 lines per millimeter.
• The lines are opaque, and the gaps between them act as coherent sources of light when illuminated by a single source.

The Grating Spacing (\(d\))

The most crucial property of the grating is the distance between the centres of adjacent slits, known as the grating spacing, \(d\).

If a grating has \(N\) lines per unit length (e.g., lines per metre or lines per millimeter), the spacing \(d\) is calculated simply as the reciprocal of \(N\):

$$d = \frac{1}{\text{Number of lines per unit length}}$$

Example: If a grating has 500 lines per millimeter (mm), you must convert this to lines per metre (m) before finding \(d\).

500 lines/mm = 500,000 lines/m.
$$d = \frac{1}{500,000 \text{ m}^{-1}} = 2.0 \times 10^{-6} \text{ m}$$

Key Takeaway (Diffraction Gratings)

Diffraction gratings have many slits, leading to interference patterns that are much sharper and brighter than those from a double slit. The essential parameter is the grating spacing \(d\).

2. The Diffraction Grating Equation

Constructive Interference Condition

When monochromatic light (light of a single wavelength, \(\lambda\)) passes through the grating, every single slit acts as a source. For the light waves from all these slits to constructively interfere (i.e., produce a bright spot or maximum) at a specific angle \(\theta\), the path difference between light waves from adjacent slits must be an integer multiple of the wavelength \(\lambda\).

Let's consider two adjacent slits separated by distance \(d\).

If the light travels at an angle \(\theta\) relative to the original direction (the normal to the grating), the path difference, \(\Delta x\), between the rays is:

$$\Delta x = d \sin \theta$$

For a bright maximum (constructive interference) to occur, this path difference must equal \(n \lambda\):

$$\Delta x = n \lambda$$

Combining these gives us the fundamental equation for the diffraction grating:

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

Understanding the Variables

\(\mathbf{d}\) (Grating Spacing)

This is the distance between adjacent slits, calculated from the number of lines per metre. It must be measured in metres (m).

\(\mathbf{\theta}\) (Diffraction Angle)

This is the angle between the central maximum (the zero order, where \(\theta = 0\)) and the specific bright maximum being observed. It is measured in degrees or radians.

\(\mathbf{n}\) (Order Number)

The order number is an integer (\(n = 0, 1, 2, 3, \ldots\)) representing which bright fringe (maximum) you are looking at.

• \(n = 0\): This is the central maximum. \(\sin \theta = 0\), so \(\theta = 0\). This is always present and is usually the brightest.
• \(n = 1\): The first order maximum.
• \(n = 2\): The second order maximum, and so on.

\(\mathbf{\lambda}\) (Wavelength)

This is the wavelength of the light passing through the grating. Measured in metres (m).

Common Application: Maximum Order

Since \(\sin \theta\) can never be greater than 1, there is a maximum possible order number, \(n_{max}\), that can be observed.

To find the theoretical maximum order, set \(\sin \theta = 1\) (which corresponds to an angle of \(90^{\circ}\), the limit of observation):

$$d(1) = n_{max} \lambda$$
$$n_{max} = \frac{d}{\lambda}$$

Since the order must be an integer, you always round this value down to the nearest whole number.

Key Takeaway (The Grating Equation)

The equation \( d \sin \theta = n \lambda \) links the physical properties of the grating (\(d\)) and the light (\(\lambda\)) to the observed angles (\(\theta\)) of the bright maxima (\(n\)).

3. Using a Diffraction Grating to Determine Wavelength (\(\lambda\))

One of the most important practical uses of the diffraction grating is precisely measuring the wavelength of unknown light sources, like laser beams or light from gases.

Experimental Procedure (Syllabus Requirement 8.4.2)

We need to measure the grating spacing \(d\) and the angle \(\theta\) for a known order \(n\).

Step 1: Determine the Grating Spacing (\(d\)).

If the grating is labelled (e.g., 500 lines per mm), calculate \(d\) using \(d = 1/N\), ensuring \(d\) is in metres.

Step 2: Set up the Equipment.
Use a source of monochromatic light (like a laser or a lamp emitting light passed through a single colour filter) and shine it perpendicularly onto the diffraction grating.

Step 3: Locate and Measure the Central Maximum (\(n=0\)).
This is the brightest spot, directly ahead, corresponding to \(\theta = 0\). This acts as our baseline.

Step 4: Locate and Measure Higher Order Maxima (\(n=1, 2, \ldots\)).
Identify the first order maximum (\(n=1\)) on either side of the central maximum. Measure the angle of this spot, \(\theta_1\), relative to the central maximum.

(Did you know? Because the grating creates sharp lines instead of diffuse fringes, the measurement of \(\theta\) is highly accurate, making this experiment better than using a double-slit.)

Step 5: Apply the Equation.
Substitute the known values \(d\), the measured angle \(\theta_1\), and the order number \(n=1\) into the equation:

$$d \sin \theta_1 = 1 \times \lambda$$

This allows you to calculate the unknown wavelength \(\lambda\).

Increasing Accuracy (The 2\(\theta\) Method)

To reduce percentage uncertainty, especially when measuring small angles:

Instead of measuring \(\theta\) from the centre (\(n=0\)) to one side (\(n=1\)), it is often more accurate to measure the total angle between the two corresponding maxima, say \(\theta_{L}\) (left) and \(\theta_{R}\) (right).

If the total angle measured between the \(n=1\) maximum on the left and the \(n=1\) maximum on the right is \(2\theta\), then the angle \(\theta\) used in the formula is simply:

$$\theta = \frac{\text{Angle between maxima } (\theta_R \text{ to } \theta_L)}{2}$$

4. Comparing Grating Spectra to Double-Slit Fringes

Why do physicists prefer gratings over double slits for precise measurements? The resulting patterns are very different:

Grating Spectrum Characteristics:

• Sharp and Narrow: The maxima (bright lines) are very narrow and sharp. This is because tiny changes in angle cause huge changes in the phase relationship across thousands of slits, quickly leading to destructive interference immediately outside the peak angle.
• Brighter: Because light from many thousands of slits contributes to the maximum, the bright lines are much more intense and easier to measure.
• Dispersion: The grating causes greater angular separation (dispersion) between different orders, making measurements clearer.

Did You Know?

This principle isn't just for visible light! Gratings are crucial components in spectroscopy, where physicists analyze light emitted by chemical elements (like hydrogen or iron) to identify them. Each element produces its own unique set of wavelengths, which show up as distinct lines in the grating spectrum. This is how we know what stars are made of!