Welcome to Superposition, Interference and Diffraction
Have you ever wondered why you can hear someone talking from around a corner even when you cannot see them? Or why the surface of a soap bubble or the back of a DVD shimmers with vibrant rainbow colours? The answer lies in the fascinating wave behaviours we are going to explore in this chapter: superposition, interference, and diffraction.
These topics form the heart of wave optics in your AS 2 Physics journey. Don't worry if wave calculations and phase angles seem a bit daunting at first — we will break every concept down into simple, manageable steps with plenty of real-world analogies and exam tips!
1. The Principle of Superposition
What happens when waves meet?
Unlike solid objects (which bounce off one another when they collide), waves pass right through each other! At the exact point where they cross paths, their displacements combine. This behaviour is governed by the Principle of Superposition.
Definition (Memorise this!):
The Principle of Superposition states that when two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements of the waves at that point.
Constructive vs Destructive Interference
When waves superpose, they produce interference. There are two main types:
1. Constructive Interference:
This happens when two waves arrive in phase (crest meets crest, or trough meets trough).
• The displacements add together to create a wave with maximum amplitude.
• For light, this means a bright spot or fringe.
• For sound, this means a louder volume.
2. Destructive Interference:
This happens when two waves arrive in antiphase (a crest meets a trough).
• If two waves have equal amplitude and meet in exact antiphase (phase difference of \(180^\circ\) or \(\pi\text{ rad}\)), they completely cancel each other out.
• Resultant displacement = \(0\).
• For light, this creates darkness (a dark fringe).
• For sound, this creates silence (which is how noise-cancelling headphones work!).
Analogy: Imagine two people jumping together on a trampoline. If you both land and bounce upwards at the exact same moment (in phase), you launch much higher (constructive interference). If one lands just as the other pushes up (out of phase), your motions cancel out and neither of you bounces high (destructive interference)!
Key Takeaway for Superposition: Waves add their displacements together when they overlap. Peak + Peak = Super Peak (Constructive); Peak + Trough = Flat line / Zero (Destructive).
2. Coherence and Path Difference
To see a clear, steady interference pattern on a screen (instead of a messy, constantly flickering blur), our wave sources must be coherent.
What is Coherence?
Definition: Two wave sources are coherent if they have a constant phase difference and the same frequency (or wavelength).
Did you know? Two separate light bulbs can never produce a steady interference pattern because they emit light in random, chaotic bursts that constantly change phase. A laser, however, emits light that is highly coherent!
Path Difference vs Phase Difference
When waves travel from two different sources to reach the same point on a screen, one wave usually has to travel further than the other. The difference in distance travelled is called the path difference.
Let \(\lambda\) represent the wavelength of the waves, and let \(n\) be an integer (\(n = 0, 1, 2, 3\dots\)):
Condition for Constructive Interference (Bright fringe):
\(\text{Path Difference} = n\lambda\)
The path difference is a whole number of wavelengths.
Condition for Destructive Interference (Dark fringe):
\(\text{Path Difference} = \left(n + \frac{1}{2}\right)\lambda\)
The path difference is an odd number of half-wavelengths.
Phase Difference Formula:
You can convert path difference to phase difference (\(\Delta\phi\)) using the relationship:
\(\Delta\phi = \frac{2\pi}{\lambda} \times \text{Path Difference}\) (in radians)
or
\(\Delta\phi = \frac{360^\circ}{\lambda} \times \text{Path Difference}\) (in degrees)
Common Mistake to Avoid: Don't mix up path difference and phase difference in exam questions! Path difference is measured in metres (\(\text{m}\)), while phase difference is measured in degrees (\(^\circ\)) or radians (\(\text{rad}\)).
3. Young's Double-Slit Experiment
The Setup
In 1801, Thomas Young performed a landmark experiment showing that light produces an interference pattern, proving light behaves as a wave.
• A monochromatic light source (single wavelength) illuminates two very narrow, closely spaced parallel slits.
• The slits act as two coherent wave sources because both originate from the same initial wavefront.
• On a distant screen, an alternating pattern of bright and dark equally spaced bands (fringes) is observed.
The Double-Slit Equation
The spacing of the fringes on the screen is given by:
\(\lambda = \frac{a y}{D}\)
Which can also be rearranged as:
\(y = \frac{\lambda D}{a}\)
Where:
• \(\lambda\) = wavelength of light in metres (\(\text{m}\))
• \(a\) = slit separation (distance between the centres of the two slits) in metres (\(\text{m}\))
• \(y\) = fringe separation (distance between adjacent bright fringes or adjacent dark fringes) in metres (\(\text{m}\))
• \(D\) = distance from the slits to the screen in metres (\(\text{m}\))
How to Increase Fringe Separation (\(y\)):
Looking at \(y = \frac{\lambda D}{a}\), you can make the fringes wider and easier to measure by:
1. Increasing \(\lambda\): Use red light instead of blue light (red has a longer wavelength).
2. Increasing \(D\): Move the screen further away from the slits.
3. Decreasing \(a\): Move the two slits closer together.
Top Exam Tip on Units: Double-slit questions are notorious for tricky unit conversions! Always convert everything to standard SI units (\(\text{m}\)) before calculating:
• Nanometres (\(\text{nm}\)): multiply by \(10^{-9}\)
• Micrometres (\(\mu\text{m}\)): multiply by \(10^{-6}\)
• Millimetres (\(\text{mm}\)): multiply by \(10^{-3}\)
Key Takeaway for Young's Experiment: Fringes are equally spaced. The central fringe (\(n = 0\)) is bright because path difference is zero. Measuring \(a\), \(y\), and \(D\) gives an experimental method to measure the tiny wavelength of light \(\lambda\).
4. Diffraction
What is Diffraction?
Definition: Diffraction is the spreading out of waves as they pass through a gap or around the edge of an obstacle.
The Golden Rule of Diffraction
The extent of diffraction depends on the ratio of the wavelength (\(\lambda\)) to the width of the gap (\(d\)):
• Maximum Diffraction occurs when: \(\lambda \approx d\) (the gap size is roughly equal to the wavelength). The waves spread out into semicircular wavefronts.
• Little or No Diffraction occurs when: \(d \gg \lambda\) (the gap is much wider than the wavelength). The waves pass straight through with only slight spreading at the edges.
Everyday Example: Why can sound bend around an open doorway, but light cannot?
• Sound waves have wavelengths of around \(0.1\text{ m}\) to \(1\text{ m}\), which is very close to the width of a standard door frame (\(\sim 0.8\text{ m}\)). Hence, sound diffracts strongly.
• Visible light has a tiny wavelength (\(\sim 5 \times 10^{-7}\text{ m}\)), which is millions of times smaller than a doorway. Light barely diffracts at all, creating sharp shadows!
5. The Diffraction Grating
What is a Diffraction Grating?
A diffraction grating is a flat glass or plastic slide with thousands of closely spaced, parallel, microscopic lines etched onto it. When light hits the grating, each gap acts as a separate coherent source of diffracted light.
The Diffraction Grating Equation
The positions of the bright maxima (lines of constructive interference) are given by:
\(d \sin \theta = n \lambda\)
Where:
• \(d\) = grating spacing (distance between adjacent slit centres) in metres (\(\text{m}\))
• \(\theta\) = diffraction angle (angle between the straight-through zero-order beam and the \(n\)-th order maximum) in degrees (\(^\circ\))
• \(n\) = order of the maximum (\(n = 0\) is the central beam, \(n = 1\) is first order, \(n = 2\) is second order, etc.)
• \(\lambda\) = wavelength of light in metres (\(\text{m}\))
Finding Slit Spacing \(d\) from "Lines per mm"
Exam papers often tell you the grating has "\(N\) lines per mm" or "\(N\) lines per metre".
To find \(d\):
\(d = \frac{1}{\text{number of lines per metre}}\)
Example: If a grating has \(500\text{ lines per mm}\):
First convert to lines per metre: \(500 \times 1000 = 5 \times 10^5\text{ lines m}^{-1}\).
Then: \(d = \frac{1}{5 \times 10^5} = 2.0 \times 10^{-6}\text{ m}\).
Finding the Maximum Number of Orders Visible
The maximum angle light can be diffracted to is \(\theta = 90^\circ\). Since \(\sin(90^\circ) = 1\):
\(n_{\text{max}} \le \frac{d}{\lambda}\)
Step-by-step method:
1. Calculate \(\frac{d}{\lambda}\).
2. Always round down to the nearest whole integer (you cannot have part of an order).
3. If an exam asks for the total number of bright spots/beams on a screen, remember to count both sides AND the central zero order: \(\text{Total maxima} = 2n_{\text{max}} + 1\).
Why is a Grating Better than Double Slits for Measuring Wavelength?
• Sharper, narrower maxima: Because light from thousands of slits superposes, destructive interference happens almost everywhere except at precise angles. This creates extremely sharp lines.
• Larger angles (\(\theta\)): Because \(d\) is very small, \(\theta\) is large, which drastically reduces percentage uncertainty when measuring angles with a spectrometer.
Diffraction of White Light
If white light passes through a diffraction grating:
• The central maximum (\(n = 0\)) is white because all wavelengths have zero path difference (\(\theta = 0^\circ\)) and recombine.
• For higher orders (\(n \ge 1\)), the white light splits into a continuous rainbow spectrum.
• Since \(d \sin \theta = n \lambda\), \(\sin \theta \propto \lambda\). Red light has the longest visible wavelength, so red is diffracted the most (furthest from the centre). Violet light has the shortest wavelength, so violet is diffracted the least (closest to the centre).
Quick Summary and Chapter Checklist
• Superposition: Resultant displacement is the sum of individual wave displacements.
• Constructive: Path diff = \(n\lambda\) (in phase).
• Destructive: Path diff = \(\left(n + \frac{1}{2}\right)\lambda\) (in antiphase).
• Coherence: Constant phase difference and same frequency.
• Young's Double-Slit Formula: \(\lambda = \frac{a y}{D}\)
• Diffraction: Spreading of waves through a gap; maximum when \(\lambda \approx d\).
• Diffraction Grating Formula: \(d \sin \theta = n \lambda\)
• Max Orders: \(n_{\text{max}} = \text{integer floor of } \frac{d}{\lambda}\); total beams = \(2n_{\text{max}} + 1\).