Welcome to Refraction and Optical Fibres
Have you ever noticed how a straw looks bent or broken when you place it in a glass of water? Or wondered how high-speed internet cables send vast amounts of data across the world in the blink of an eye? Both of these everyday wonders happen because of a single wave phenomenon: refraction.
In this chapter of AS 2: Waves, Photons and Astronomy, we will explore how light behaves when it travels from one material to another, uncover the maths that describes this bending, and discover how total internal reflection powers modern fibre optics. Don't worry if physics equations sometimes seem daunting; we will break down every concept step-by-step!
1. The Fundamentals of Refraction
What is Refraction?
Refraction is the change in direction of a wave when it crosses a boundary between two different materials (media), caused by a change in the wave's speed.
The Wave Model: Why Does Light Bend?
Imagine pushing a toy car across a smooth wooden floor onto a thick carpet at an angle. The wheel that hits the carpet first slows down before the other wheel. Because one side is moving slower than the other, the car pivots and changes direction. Light behaves in the exact same way!
When light enters an optically denser medium (such as going from air into glass), it slows down. If it enters at an angle, the part of the wavefront that enters first slows down first, causing the light ray to bend towards the normal.
What Happens to Frequency, Speed, and Wavelength?
When a light wave passes from one medium to another:
• Frequency (\(f\)): Stays constant. The frequency is set by the source of the wave and cannot change across a boundary.
• Speed (\(v\)): Changes depending on the optical density of the medium.
• Wavelength (\(\lambda\)): Changes in direct proportion to the speed, according to the wave equation: \(v = f\lambda\).
Because \(f\) is constant, if the speed \(v\) decreases, the wavelength \(\lambda\) must also decrease proportionally: \(\frac{v_1}{v_2} = \frac{\lambda_1}{\lambda_2}\).
Did You Know?
Because the frequency does not change when light travels from air into water, the perceived colour of light remains the same, even though its wavelength shrinks inside the water!
Key Takeaway
Refraction occurs because waves change speed at a boundary. Frequency remains constant, so when speed decreases, wavelength also decreases.
2. The Refractive Index and Snell's Law
The Absolute Refractive Index (\(n\))
The refractive index (\(n\)) of a material is a measure of how much it slows down light compared to the speed of light in a vacuum (\(c = 3.00 \times 10^8\text{ m s}^{-1}\)).
The formula for absolute refractive index is:
\(n = \frac{c}{v}\)
Where:
• \(n\) = absolute refractive index (has no units because it is a ratio)
• \(c\) = speed of light in a vacuum (\(3.00 \times 10^8\text{ m s}^{-1}\))
• \(v\) = speed of light in the material (\(\text{m s}^{-1}\))
Note: For air or a vacuum, \(n \approx 1.00\). For all other transparent media, \(n > 1.00\) because light always travels slower in a material than in a vacuum.
The Direction of Bending: A Handy Memory Aid
To remember which way a light ray bends, use the mnemonic FAST:
• Faster \(\rightarrow\) Away from the normal (when moving to a lower \(n\))
• Slower \(\rightarrow\) Towards the normal (when moving to a higher \(n\))
Snell's Law
Snell's Law describes the mathematical relationship between the angles of incidence and refraction and the refractive indices of the two media:
\(n_1 \sin\theta_1 = n_2 \sin\theta_2\)
Where:
• \(n_1\) = refractive index of the first medium
• \(\theta_1\) = angle of incidence (measured between the incident ray and the normal)
• \(n_2\) = refractive index of the second medium
• \(\theta_2\) = angle of refraction (measured between the refracted ray and the normal)
Step-by-Step Worked Example
Question: A ray of light travels from air (\(n_1 = 1.00\)) into a glass block (\(n_2 = 1.52\)) at an angle of incidence of \(35.0^\circ\). Calculate the angle of refraction inside the glass.
Step 1: Write down the known quantities.
\(n_1 = 1.00\), \(\theta_1 = 35.0^\circ\), \(n_2 = 1.52\)
Step 2: State Snell's Law.
\(n_1 \sin\theta_1 = n_2 \sin\theta_2\)
Step 3: Substitute the values.
\(1.00 \times \sin(35.0^\circ) = 1.52 \times \sin\theta_2\)
Step 4: Rearrange to find \(\sin\theta_2\).
\(\sin\theta_2 = \frac{1.00 \times \sin(35.0^\circ)}{1.52} = \frac{0.5736}{1.52} = 0.3774\)
Step 5: Take the inverse sine (\(\sin^{-1}\)).
\(\theta_2 = \sin^{-1}(0.3774) = 22.2^\circ\)
Common Pitfall to Avoid
Always measure angles relative to the normal line (the imaginary line perpendicular to the surface at \(90^\circ\)), never relative to the surface itself! If an exam question gives the angle to the surface, subtract it from \(90^\circ\) first.
Key Takeaway
Snell's Law links angles and refractive indices: \(n_1 \sin\theta_1 = n_2 \sin\theta_2\). Light bends towards the normal when slowing down and away from the normal when speeding up.
3. Total Internal Reflection (TIR) and the Critical Angle
What Happens as the Angle of Incidence Increases?
Consider light travelling from an optically denser medium to a less dense medium (e.g., from glass into air, where \(n_1 > n_2\)):
1. At small angles of incidence, most light refracts away from the normal, with a weak reflected ray.
2. As the angle of incidence increases, the angle of refraction eventually reaches \(90^\circ\). The light ray skims along the boundary. The angle of incidence at which this happens is called the critical angle (\(\theta_c\) or \(c\)).
3. If the angle of incidence is increased beyond the critical angle, no light can refract out. All light is reflected back into the denser medium. This is Total Internal Reflection (TIR).
Two Essential Conditions for TIR
For total internal reflection to take place, two conditions must be met:
• Condition 1: The wave must be travelling from a medium with a higher refractive index to a medium with a lower refractive index (\(n_1 > n_2\)).
• Condition 2: The angle of incidence must be greater than the critical angle (\(\theta_1 > \theta_c\)).
Deriving the Critical Angle Formula
At the critical angle, the angle of incidence is \(\theta_1 = \theta_c\) and the angle of refraction is \(\theta_2 = 90^\circ\).
Applying Snell's Law:
\(n_1 \sin\theta_c = n_2 \sin(90^\circ)\)
Since \(\sin(90^\circ) = 1\), this simplifies to:
\(\sin\theta_c = \frac{n_2}{n_1}\)
If the second medium is air or a vacuum (\(n_2 = 1.00\)), the equation becomes:
\(\sin\theta_c = \frac{1}{n}\)
Worked Example: Calculating Critical Angle
Question: Calculate the critical angle for a water-to-air boundary, given that the refractive index of water is \(n = 1.33\).
Step 1: Use the critical angle formula.
\(\sin\theta_c = \frac{1}{n} = \frac{1}{1.33} = 0.7519\)
Step 2: Calculate the inverse sine.
\(\theta_c = \sin^{-1}(0.7519) = 48.8^\circ\)
Key Takeaway
TIR occurs only when light moves from a denser to a less dense medium and the angle of incidence exceeds the critical angle (\(\sin\theta_c = \frac{n_2}{n_1}\)).
4. Step-Index Optical Fibres
Structure of an Optical Fibre
Optical fibres are thin, flexible strands of highly transparent glass or plastic used to transmit signals as pulses of light. A standard step-index optical fibre consists of three layers:
• Core: The central glass strand where light travels. It has a high refractive index (\(n_{\text{core}}\)).
• Cladding: A protective layer surrounding the core with a slightly lower refractive index (\(n_{\text{cladding}} < n_{\text{core}}\)).
• Protective Outer Jacket / Buffer: A plastic coating that protects the fibre from moisture and physical damage.
Why is the Cladding Necessary?
Students often ask: "Why do we need cladding if air has an even lower refractive index?" Cladding is vital for several reasons:
• Maintains a Clean Boundary: Scratches, dirt, or moisture on the surface of the core would scatter light and reduce internal reflection.
• Prevents Crosstalk / Light Leakage: In bundles of fibres, cladding prevents light from leaking from one core into an adjacent core.
• Increases Critical Angle: A small difference between \(n_{\text{core}}\) and \(n_{\text{cladding}}\) creates a large critical angle. This limits the range of reflection angles, reducing multi-path dispersion.
Signal Degradation: Dispersion and Pulse Broadening
In digital communications, information is sent as rapid light pulses (representing 1s and 0s). As a pulse travels along an optical fibre, it can spread out over time—a problem known as pulse broadening. If pulses broaden too much, they overlap, leading to data corruption and limiting transmission speed.
Pulse broadening is caused by two main types of dispersion:
1. Modal Dispersion (Multipath Dispersion):
Light rays take different paths down the fibre. A ray travelling straight along the central axis takes the shortest path and arrives first. A ray bouncing repeatedly at angles takes a longer zigzag path and arrives later. Because different paths take different times, the pulse spreads out.
• How to reduce modal dispersion: Use a monomode (single-mode) fibre with an extremely narrow core (a few micrometres across). This forces light to travel only along a single path.
2. Material Dispersion (Spectral Dispersion):
White light is made of different wavelengths (colours). The refractive index of glass varies slightly with wavelength (e.g., blue light travels slightly slower than red light in glass). Different wavelengths in the same pulse travel at different speeds and arrive at different times.
• How to reduce material dispersion: Use monochromatic light (a single wavelength), such as light from a laser diode.
Signal Loss: Attenuation and Absorption
Light pulses also lose energy as they travel down a fibre due to:
• Absorption: Small impurities in the glass absorb light energy and convert it into heat.
• Scattering: Microscopic variations in glass density cause Rayleigh scattering, directing light out of the core.
• Solution: Optical amplifiers/repeaters are placed at regular intervals along long-distance lines to boost signal strength.
Key Takeaway
Step-index fibres rely on TIR with \(n_{\text{core}} > n_{\text{cladding}}\). Modal and material dispersion cause pulse broadening, which is reduced by using very thin (single-mode) cores and monochromatic light sources.
5. Quick Summary and Revision Checklist
Make sure you can comfortably recall and apply the following core equations and definitions:
• Wave Equation: \(v = f\lambda\) (Frequency \(f\) is unchanged during refraction)
• Refractive Index: \(n = \frac{c}{v}\)
• Snell's Law: \(n_1 \sin\theta_1 = n_2 \sin\theta_2\)
• Critical Angle: \(\sin\theta_c = \frac{n_2}{n_1}\) (or \(\sin\theta_c = \frac{1}{n}\) for a boundary with air)
• Conditions for TIR: Light travels from higher \(n\) to lower \(n\), and \(\theta_1 > \theta_c\)
• Fibre Structure: \(n_{\text{core}} > n_{\text{cladding}}\)
• Reducing Dispersion: Narrow core (single-mode) prevents modal dispersion; monochromatic light prevents material dispersion.