Welcome to Refraction (AS 2: Topic 2.2)

Have you ever noticed how a drinking straw looks bent or broken when you place it in a glass of water? Or wondered how high-speed internet data travels through thin glass cables across the ocean? Both of these everyday wonders happen because of a single optical phenomenon: refraction.

In this chapter, we explore how light behaves when moving between different materials, how to calculate its path using Snell's Law, and how to harness Total Internal Reflection in modern optical technology. Don't worry if physics equations sometimes seem intimidating—we will break down each concept step by step!

1. What is Refraction?

Refraction is the change in direction (bending) of a wave caused by a change in its wave speed as it passes across a boundary from one optical medium into another of different optical density.

What Happens to Wave Properties?

When light crosses an optical boundary, its physical properties do not all change in the same way:

  • Frequency (\(f\)): Remains constant. The frequency of a wave is determined entirely by the original source that created it. Crossing into glass, water, or air never alters its frequency.
  • Wave Speed (\(v\)): Changes. Light travels fastest in a vacuum (\(c \approx 3.00 \times 10^8\text{ m s}^{-1}\)) and slows down when entering denser substances like water, perspex, or glass.
  • Wavelength (\(\lambda\)): Changes proportionally to speed. From the wave equation \(v = f\lambda\), because \(f\) is fixed, if the wave speed \(v\) decreases, the wavelength \(\lambda\) must also decrease by the same factor.

The Direction of Bending

To predict how a ray bends, always draw an imaginary line perpendicular (\(90^\circ\)) to the interface surface at the point where the light hits. This reference line is called the normal.

  • Entering a more optically dense medium (e.g., Air \(\to\) Glass): The ray slows down and bends towards the normal.
  • Entering a less optically dense medium (e.g., Glass \(\to\) Air): The ray speeds up and bends away from the normal.

Helpful Analogy: Imagine pushing a lawnmower from smooth concrete onto thick grass at an angle. The wheel that hits the grass first slows down immediately, while the other wheel stays fast on the concrete. This mismatch swings the entire lawnmower towards the slower side!

Key Takeaway: Refraction is caused by a change in speed. Frequency never changes across a boundary; speed and wavelength change together.

2. Absolute Refractive Index (\(n\))

The absolute refractive index (\(n\)) of a material measures how much the medium reduces the speed of light compared to the speed of light in a vacuum.

Formula:

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

Where:

  • \(n\) = Absolute refractive index (a dimensionless ratio with no units)
  • \(c\) = Speed of light in a vacuum (\(3.00 \times 10^8\text{ m s}^{-1}\))
  • \(v\) = Speed of light in the medium (\(\text{m s}^{-1}\))

Because light travels at its maximum possible speed in a vacuum, \(c \ge v\) for any material medium. Therefore, the refractive index of any medium is always greater than or equal to 1 (\(n \ge 1\)). For air or a vacuum, \(n \approx 1.00\).

Key Takeaway: A higher refractive index means light travels slower in that material, indicating greater optical density.

3. Snell's Law

Snell's Law describes the mathematical relationship between the angle of incidence and the angle of refraction when light passes across a boundary between two media.

The Basic Form (From Air/Vacuum into a Medium)

When light enters a substance of refractive index \(n\) from air or vacuum:

\(\frac{\sin i}{\sin r} = n\)

Where:

  • \(i\) = Angle of incidence (angle between the incident ray and the normal)
  • \(r\) = Angle of refraction (angle between the refracted ray and the normal)

The General Form (Between Any Two Media)

When light passes from medium 1 into medium 2:

\(n_1 \sin \theta_1 = n_2 \sin \theta_2\)

Where:

  • \(n_1\) = Refractive index of the first medium
  • \(\theta_1\) = Angle of the ray in the first medium (measured relative to the normal)
  • \(n_2\) = Refractive index of the second medium
  • \(\theta_2\) = Angle of the ray in the second medium (measured relative to the normal)

Common Examiner Pitfall: Measuring from the Surface

Examiner Warning: Never measure angles from the flat glass surface! All angles (\(i\), \(r\), \(\theta_1\), \(\theta_2\)) must strictly be measured from the normal line (\(90^\circ\) perpendicular to the boundary interface). If an exam question gives an angle of \(30^\circ\) to the glass surface, the angle of incidence is \(90^\circ - 30^\circ = 60^\circ\).

Key Takeaway: Snell's Law connects angles to refractive indices via sines: \(n_1 \sin \theta_1 = n_2 \sin \theta_2\).

4. Total Internal Reflection (TIR) and the Critical Angle

When light travels from a denser medium into a less dense medium (e.g., from glass into air), it speeds up and refracts away from the normal (\(r > i\)). As you increase the angle of incidence \(i\), the angle of refraction \(r\) reaches its theoretical limit of \(90^\circ\).

The Critical Angle (\(C\))

The critical angle (\(C\)) is defined as the angle of incidence in the more optically dense medium for which the angle of refraction in the less dense medium is exactly \(90^\circ\).

Applying Snell's Law for light going from a medium of index \(n\) to air (\(n_{\text{air}} = 1.00\)) at the critical angle:

\(n \sin C = 1.00 \times \sin(90^\circ)\)

Since \(\sin(90^\circ) = 1\):

\(\sin C = \frac{1}{n} \implies C = \sin^{-1}\left(\frac{1}{n}\right)\)

General form between two media where \(n_1 > n_2\): \(\sin C = \frac{n_2}{n_1}\)

The Two Essential Conditions for Total Internal Reflection (TIR)

CCEA exam mark schemes require both conditions when asked to state how TIR occurs:

  1. The light ray must be travelling in a more optically dense medium towards a less optically dense medium (\(n_1 > n_2\)).
  2. The angle of incidence at the interface must exceed the critical angle (\(i > C\)).

When both conditions are met, no light refracts out; instead, \(100\%\) of the light energy reflects back inside the denser medium according to the law of reflection (angle of incidence = angle of reflection).

Practical Applications of TIR

  • Step-Index Optical Fibres: These fibres consist of a cylindrical glass or plastic core with a high refractive index (\(n_{\text{core}}\)), surrounded by an outer cladding of slightly lower refractive index (\(n_{\text{cladding}} < n_{\text{core}}\)). Light injected into the core strikes the core-cladding boundary at angles greater than the critical angle, undergoing repeated total internal reflections to carry data signals over long distances with minimal loss.
  • Reflecting Prisms: Right-angled triangular glass prisms (\(45^\circ\text{–}90^\circ\text{–}45^\circ\)) use TIR internally to redirect light by \(90^\circ\) or invert it by \(180^\circ\). These are widely used in binoculars and periscopes instead of standard silvered mirrors because TIR provides highly efficient reflection without double-image reflections.

Key Takeaway: TIR traps light completely inside a denser medium whenever \(i > C\). To calculate the critical angle for glass to air, use \(\sin C = \frac{1}{n}\).

5. Prescribed Practical: Determining the Refractive Index (\(n\))

In Unit AS 2 and practical Unit AS 3, you are required to understand how to measure the refractive index of a rectangular transparent block (glass or perspex).

Experimental Procedure:

  1. Place a rectangular glass or perspex block flat on a sheet of plain white paper and draw around its outline.
  2. Direct a single narrow beam of light from a ray box (or use optical pins) toward the long edge of the block at a chosen angle of incidence.
  3. Mark the path of the incident ray, the exit point where the ray emerges from the opposite side, and the path of the emergent ray.
  4. Remove the block and use a ruler to join the entry point to the exit point. This line represents the refracted ray inside the block.
  5. Draw a normal line (\(90^\circ\)) at the exact point of entry. Use a protractor to measure the angle of incidence \(i\) and the angle of refraction \(r\).
  6. Repeat the procedure for multiple different incident angles (e.g., \(20^\circ, 30^\circ, 40^\circ, 50^\circ, 60^\circ, 70^\circ\)).

Graphical Analysis:

  • Calculate \(\sin i\) and \(\sin r\) for each pair of measurements.
  • Plot a graph with \(\sin i\) on the y-axis against \(\sin r\) on the x-axis.
  • According to Snell's Law (\(\sin i = n \sin r\)), comparing this to the equation of a straight line (\(y = mx + c\)) confirms:
    • The graph produces a straight line passing through the origin.
    • The gradient of the line is equal to the refractive index (\(n\)) of the block.
  • Note: A plot of \(i\) against \(r\) yields a curve (non-linear relationship), demonstrating that \(i\) and \(r\) are not directly proportional, but their sines are!

Quick Review: Summary of Key Formulae & Reminders

  • Absolute Refractive Index: \(n = \frac{c}{v}\)
  • Snell's Law (Air \(\to\) Medium): \(\frac{\sin i}{\sin r} = n\)
  • General Snell's Law: \(n_1 \sin \theta_1 = n_2 \sin \theta_2\)
  • Critical Angle to Air: \(\sin C = \frac{1}{n} \implies C = \sin^{-1}\left(\frac{1}{n}\right)\)
  • TIR Checklist: Must travel from more dense \(\to\) less dense medium AND angle of incidence must be greater than critical angle (\(i > C\)).
  • Ray Diagram Rule: Always draw arrows on rays to show the direction of light travel, and measure all angles from the normal.