Welcome to Refraction at a Plane Surface

Ever noticed how a straw looks broken when you put it in a glass of water? Or how a swimming pool looks shallower than it actually is? That is refraction in action! In this chapter, we will explore why light bends when it moves from one material to another and how we use this property to send high-speed internet data around the world using optical fibres.

Don't worry if the math looks a bit intimidating at first; we will break it down step-by-step so you can master it for your AS exams.

1. What is Refractive Index?

Light travels at different speeds in different materials. The refractive index (\(n\)) is a number that tells us how "optically dense" a material is. It is a ratio of the speed of light in a vacuum to the speed of light in the substance.

The formula for the refractive index of a substance is:

\(n = \frac{c}{v}\)

Where:
\(c\) = the speed of light in a vacuum (\(3.00 \times 10^8 \, \text{m s}^{-1}\))
\(v\) = the speed of light in the substance (\(\text{m s}^{-1}\))

Important things to remember:

  • The refractive index \(n\) has no units (it is a ratio).
  • For air, we usually assume \(n = 1\) because light travels almost as fast in air as it does in a vacuum.
  • The higher the value of \(n\), the slower light travels in that material.

Quick Review: If light slows down, it bends towards the normal. If light speeds up, it bends away from the normal.

2. Snell's Law

When light crosses a boundary between two materials (like air to glass), it changes direction. We use Snell's Law to calculate exactly how much it bends.

The Formula:
\(n_1 \sin \theta_1 = n_2 \sin \theta_2\)

Where:
\(n_1\) = refractive index of the first material.
\(\theta_1\) = angle of incidence (the incoming ray).
\(n_2\) = refractive index of the second material.
\(\theta_2\) = angle of refraction (the outgoing ray).

Common Mistake Alert! Always measure your angles (\(\theta\)) from the normal (an imaginary line 90 degrees to the surface). Never measure from the surface of the glass itself!

3. Total Internal Reflection (TIR)

Sometimes, light doesn't refract at all. Instead, it acts like it has hit a mirror and reflects back into the original material. This is called Total Internal Reflection.

This only happens when two conditions are met:

  1. The light is travelling from a higher refractive index to a lower refractive index (e.g., from glass to air).
  2. The angle of incidence is greater than a specific angle called the critical angle (\(\theta_c\)).

The Critical Angle (\(\theta_c\))

The critical angle is the angle of incidence that causes the light to refract at exactly \(90^{\circ}\) along the boundary. We can find it using this formula:

\(\sin \theta_c = \frac{n_2}{n_1}\)

(Note: In this formula, \(n_1\) is the material the light is starting in, and \(n_2\) is the material it is heading towards.)

Summary of outcomes:
- If \(\theta < \theta_c\): Most light refracts out, some reflects internally.
- If \(\theta = \theta_c\): Light travels along the boundary.
- If \(\theta > \theta_c\): Total Internal Reflection occurs.

4. Optical Fibres

Optical fibres are thin strands of glass or plastic that carry data as pulses of light. They rely entirely on Total Internal Reflection to keep the light trapped inside the fibre.

For your AQA AS level, you only need to know about step-index optical fibres. These consist of two main parts:

  1. The Core: The central part of the fibre where the light travels. It has a high refractive index.
  2. The Cladding: A layer surrounding the core. It has a lower refractive index than the core.

Why is the Cladding important?
- It allows Total Internal Reflection to occur (because \(n_{\text{core}} > n_{\text{cladding}}\)).
- It protects the core from scratches or moisture, which would allow light to leak out.
- It prevents "crosstalk" (light leaking between adjacent fibres in a cable).

5. Signal Degradation: Absorption and Dispersion

In a perfect world, a light pulse would stay exactly the same as it travels through a fibre. In reality, signals can be distorted. You need to know these three terms:

1. Absorption

Some of the light's energy is absorbed by the glass material. This makes the signal weaker (reduced amplitude) as it travels. To fix this, we use optical regenerators (boosters) at intervals along the cable.

2. Modal Dispersion

This happens because light rays enter the fibre at different angles. Some rays take a "straight" path, while others bounce back and forth many times. The rays taking the "zig-zag" path travel a longer distance and take longer to reach the end.
Result: The pulse spreads out over time, known as pulse broadening.

3. Material Dispersion

Light is often made of different wavelengths (colours). Different wavelengths travel at slightly different speeds through glass.
Result: Just like modal dispersion, this causes pulse broadening because different colours arrive at different times.

Why is Pulse Broadening bad?
If pulses broaden too much, they start to overlap. If the pulses overlap, the computer at the other end cannot distinguish between the individual "bits" of data, leading to errors in the signal.

Key Takeaway: To reduce dispersion, we use very thin cores (monomode fibres) and monochromatic light (lasers of a single wavelength).

Summary Checklist

  • Can you define refractive index \(n\)?
  • Can you use \(n_1 \sin \theta_1 = n_2 \sin \theta_2\) to find an angle?
  • Do you know the two conditions for Total Internal Reflection?
  • Can you explain why the cladding has a lower refractive index than the core?
  • Can you distinguish between absorption and dispersion?

Memory Tip: Remember that Absorption affects Amplitude (how tall the pulse is), while Dispersion affects Duration (how wide the pulse is).