Welcome to Refraction at a Plane Surface!
Hey future physicist! This chapter is all about what happens when light moves from one material into another—for example, from air into water or glass. You've seen this phenomenon every time a straw in a drink looks bent, or when light bounces off a mirror (but wait, that’s reflection!). Refraction is the bending of light. It’s a crucial concept for understanding amazing technologies like fibre optics and even how your glasses work.
Don't worry if the formulas seem a bit intimidating; we will break them down piece by piece. Let's dive in!
1. What is Refraction and the Refractive Index?
The Big Idea: Why does light bend?
When a wave travels from one medium (like air) to another (like glass), its speed changes. This change in speed causes the wave to change direction—this process is called refraction.
Analogy: Imagine pushing a shopping trolley across smooth tiles (fast medium) and then hitting a patch of sticky tar at an angle (slow medium). The wheel that hits the tar first slows down, causing the trolley to swerve or "bend" toward the slower direction.
- When light moves from a fast medium (less dense, e.g., air) to a slow medium (more dense, e.g., glass), it bends towards the normal.
- When light moves from a slow medium to a fast medium, it bends away from the normal.
- The normal is an imaginary line drawn perpendicular (90°) to the surface where the two media meet.
Defining the Refractive Index (\(n\))
The refractive index (\(n\)) of a substance tells us exactly how much light slows down when it enters that substance. It is a ratio comparing the speed of light in a vacuum to the speed of light in the medium.
The formula is:
\[n = \frac{c}{c_s}\]
Where:
- \(c\) is the speed of light in a vacuum (approximately \(3.00 \times 10^8 \text{ m s}^{-1}\)).
- \(c_s\) is the speed of light in the substance (the medium).
Key Points about \(n\):
- Since \(c\) is the fastest speed possible, \(n\) is always greater than or equal to 1.
- It has no units (it's a ratio of speeds).
- The refractive index of air is approximately \(1.00\), so for most calculations, air is treated the same as a vacuum.
Quick Review: Refraction Basics
Refraction happens because the speed of light changes. The refractive index (\(n\)) measures this change, and it always relates to the speed of light in a vacuum.
2. Snell’s Law: The Rule of Refraction
Understanding the Angles
To use Snell's Law, we need to carefully define the angles involved. All angles are measured relative to the normal line.
- \(\theta_1\): The angle of incidence (in the first medium, \(n_1\)).
- \(\theta_2\): The angle of refraction (in the second medium, \(n_2\)).
Important Caution! If you measure the angle to the surface instead of the normal, your calculations will be completely wrong. Always measure to the normal!
Applying Snell's Law
Snell's Law mathematically relates the refractive indices of the two materials to the angles of incidence and refraction.
\[n_1 \sin \theta_1 = n_2 \sin \theta_2\]
Where:
- \(n_1\) is the refractive index of the first medium (where the light starts).
- \(\theta_1\) is the angle of incidence.
- \(n_2\) is the refractive index of the second medium (where the light bends).
- \(\theta_2\) is the angle of refraction.
Memory Aid: "N-one-sin-theta-one equals N-two-sin-theta-two." Keep the medium's refractive index and its angle paired together.
Step-by-Step Example of Bending:
If light moves from glass (\(n_1 = 1.5\)) into air (\(n_2 = 1.0\)):
- Since \(n_1 > n_2\), the light is moving from a slow (denser) medium to a fast (less dense) medium.
- Therefore, \(\theta_2\) must be greater than \(\theta_1\). The light bends away from the normal.
Key Takeaway: Snell’s Law
Snell’s Law provides the quantitative relationship for refraction: \(n_1 \sin \theta_1 = n_2 \sin \theta_2\). Remember to always measure angles from the normal.
3. Total Internal Reflection (TIR)
What happens when light tries to leave a very dense material and enter a less dense one, like shining a laser beam from the bottom of a swimming pool upwards?
Prerequisite: TIR can only occur when light travels from a more dense medium (\(n_1\)) to a less dense medium (\(n_2\)). This means \(n_1\) must be greater than \(n_2\).
The Critical Angle (\(\theta_c\))
As the angle of incidence (\(\theta_1\)) in the denser medium increases, the angle of refraction (\(\theta_2\)) increases even faster, bending further away from the normal.
The Critical Angle (\(\theta_c\)) is the specific angle of incidence in the denser medium for which the angle of refraction in the less dense medium is exactly 90°.
At this point, the refracted ray travels along the boundary surface.
We find the critical angle by setting \(\theta_2 = 90^\circ\) in Snell's Law (\(\sin 90^\circ = 1\)):
\[n_1 \sin \theta_c = n_2 \sin 90^\circ\]
This simplifies to the critical angle formula:
\[\sin \theta_c = \frac{n_2}{n_1}\]
Where \(n_1\) is the index of the dense medium and \(n_2\) is the index of the less dense medium.
Conditions for Total Internal Reflection (TIR)
TIR occurs when two conditions are met:
- Light must be traveling from a denser medium to a less dense medium (\(n_1 > n_2\)).
- The angle of incidence (\(\theta_1\)) must be greater than the critical angle (\(\theta_1 > \theta_c\)).
When TIR happens, no light is refracted; all the light is reflected back into the denser medium. This is why the surface of water looks like a perfect mirror when viewed from underneath at a shallow angle.
Key Takeaway: TIR
TIR requires movement from dense to less dense, and the incident angle must exceed the critical angle \(\theta_c\). Calculate \(\theta_c\) using \(\sin \theta_c = \frac{n_2}{n_1}\).
4. Application: Optical Fibres and Data Transfer
One of the most powerful applications of TIR is in fibre optics, which are thin strands of glass or plastic used to transmit light (and therefore information, like internet data) over long distances.
Simple Fibre Optic Structure
An optical fibre consists of two main parts:
- The Core: A central strand made of glass or plastic, which is the path for the light. This material has a high refractive index (\(n_1\)).
- The Cladding: A layer wrapped around the core, made of material with a lower refractive index (\(n_2\)).
The purpose of the cladding is critical. By ensuring \(n_1 > n_2\), the light signals entering the core strike the boundary at an angle greater than \(\theta_c\) and undergo continuous Total Internal Reflection, keeping the light trapped inside the core until it reaches the end.
Syllabus Note: You only need to consider step index fibres, where the refractive index changes abruptly (in a 'step') between the core and the cladding.
Challenges in Optical Fibre Communication
When light pulses travel through long optical fibres, the signals can weaken and become distorted. We study two main challenges:
A. Absorption (Signal Loss)
This occurs when the light energy is converted into heat within the glass material of the core. This reduces the signal strength (attenuation), requiring signal boosters along the fibre route.
B. Dispersion (Pulse Broadening)
Dispersion means that a sharp, distinct pulse of light spreads out as it travels, becoming wider and less distinct. This phenomenon is called pulse broadening. If pulses broaden too much, they overlap with adjacent pulses, making it impossible to read the information. This limits the rate at which data can be sent.
There are two types of dispersion we must consider:
- Modal Dispersion:
- In wider cores, different light rays travel along different paths ("modes").
- A ray bouncing straight down the middle travels a shorter path than a ray that bounces many times near the critical angle.
- Since they travel different distances, the rays arrive at the detector at slightly different times, causing the pulse to spread out.
- Solution: Use very thin fibres (single-mode fibres) so all rays travel nearly the same path length.
- Material Dispersion:
- Even if the light starts as a single pulse, if it is made up of slightly different wavelengths (colours) of light, those wavelengths travel at different speeds within the glass core.
- This is because the refractive index (\(n\)) varies slightly depending on the wavelength.
- The different speeds mean the colours arrive at different times, also causing pulse broadening.
- Solution: Use highly monochromatic light sources, like lasers, which produce only a single wavelength.
Key Takeaway: Fibre Optics
Fibre optics rely on TIR, made possible by a high-index core and a lower-index cladding. Signal degradation is caused by absorption (loss of strength) and dispersion (pulse broadening, which limits data rate).