Unit 15: Refraction at Plane Surfaces

Introduction

Refraction is the bending of light as it passes from one medium to another with different optical densities. At plane surfaces, the behavior of light can be predicted using simple geometric laws. This chapter covers the laws of refraction, the concept of refractive index, relationships between indices, lateral shift produced by a parallel slab, and the phenomenon of total internal reflection (TIR). Each topic is presented with definitions, mathematical formulations, illustrative examples, and diagrams (described in text).

1. Laws of Refraction and Refractive Index

1.1 First Law of Refraction

The incident ray, the refracted ray, and the normal to the interface at the point of incidence all lie in the same plane.

This law ensures that refraction is a two‑dimensional phenomenon; we can analyse it using a single plane diagram.

1.2 Second Law of Refraction (Snell’s Law)

When a light ray passes from medium 1 to medium 2, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for the pair of media.

where:

• n₁ = absolute refractive index of medium 1
• n₂ = absolute refractive index of medium 2
• θ₁ = angle of incidence (measured from the normal)
• θ₂ = angle of refraction (measured from the normal)

Snell’s law follows from the principle of least time (Fermat’s principle) and can be derived from the wave‑front construction.

1.3 Refractive Index

The refractive index of a medium quantifies how much the speed of light is reduced inside that medium relative to its speed in vacuum.

where:

• c = speed of light in vacuum ≈ 3.00 × 10⁸ m s⁻¹
• v = speed of light in the medium

Since v ≤ c, the refractive index n is always ≥ 1.

1.4 Absolute and Relative Refractive Index

Absolute refractive index (n) is defined with respect to vacuum (or air, approximated as vacuum).

Relative refractive index of medium 2 with respect to medium 1 is the ratio of their absolute indices:

It also equals the ratio of the speeds of light in the two media:

If n₂₁ > 1, medium 2 is optically denser than medium 1; if n₂₁ < 1, medium 2 is rarer.

2. Relation Between Refractive Indices

2.1 Basic Relation

For any two media, the relative refractive index can be expressed as:

2.2 Chain Rule for Three Media

When light travels sequentially from medium 1 → 2 → 3, the overall relative index from medium 1 to 3 is the product of the successive relative indices:

Proof: Using definitions, n₃₁ = n₃ / n₁ = (n₃ / n₂) × (n₂ / n₁) = n₃₂ × n₂₁.

2.3 Principle of Reversibility

If a light ray follows a particular path from point A to point B, reversing the direction of the ray will cause it to retrace the same path. Mathematically, this implies that Snell’s law is symmetric:

Thus, the refractive indices are reciprocal when the direction is reversed:

3. Lateral Shift in a Parallel Glass Slab

When a ray of light passes through a rectangular slab with parallel faces, the emergent ray is parallel to the incident ray but displaced sideways. This sideways displacement is called the lateral shift.

3.1 Derivation of the Shift Formula

Consider a slab of thickness t, refractive index n (relative to surrounding air, nₐᵢᵣ ≈ 1). Let the angle of incidence be i and the angle of refraction inside the slab be r. From geometry:

where:

• d = lateral shift
• t = thickness of the slab
• i = angle of incidence (in air)
• r = angle of refraction (inside the slab)

Using Snell’s law for air–glass interface: sin i = n sin r.

3.2 Independence of Wavelength

For a parallel‑sided slab, the emergent ray is parallel to the incident ray irrespective of the wavelength, because the two refractions (entry and exit) cancel any angular deviation. Hence, the lateral shift does not depend on λ (no dispersion) as long as the slab faces remain parallel and the medium is non‑dispersive (or dispersion is negligible).

3.3 Numerical Example

A glass slab of thickness t = 5.0 cm and refractive index n = 1.50 is placed in air. A light ray strikes the slab at an angle of incidence i = 45°. Find the lateral shift.

1. Apply Snell’s law: sin i = n sin r → sin 45° = 1.50 sin r → sin r = (0.7071)/1.50 = 0.4714 → r ≈ 28.1°.
2. Compute shift: d = t sin(i‑r)/cos r = 5.0 cm × sin(45°‑28.1°)/cos 28.1°.
3. sin(16.9°) = 0.291, cos 28.1° = 0.882.
4. d = 5.0 × 0.291 / 0.882 ≈ 1.65 cm.

Thus the emergent ray is displaced by about 1.65 cm sideways.

4. Total Internal Reflection (TIR)

Total internal reflection occurs when light travelling from a denser medium to a rarer medium strikes the interface at an angle greater than a certain limiting angle called the critical angle. Under this condition, no refracted ray exists; the entire incident intensity is reflected back into the denser medium.

4.1 Condition for TIR

• Light must travel from a medium of higher refractive index (denser) to one of lower refractive index (rarer).
• The angle of incidence i must exceed the critical angle c.

4.2 Critical Angle

From Snell’s law, setting the angle of refraction to 90° (grazing emergence) gives:

where:

• n_denser = refractive index of the denser medium
• n_rarer = refractive index of the rarer medium

If n_rarer / n_denser ≥ 1, no critical angle exists (TIR cannot occur).

4.3 Applications

1. Optical Fibers: Light is guided along the fiber by repeated TIR at the core‑cladding interface, enabling low‑loss communication over long distances.
2. Mirage: On hot days, a layer of hot, less dense air near the ground creates a gradient in refractive index; light from the sky undergoes TIR, producing the illusion of water.
3. Diamond Brilliance: Diamond’s high refractive index (n ≈ 2.42) yields a small critical angle (~24.4°). Facets are cut so that light entering the stone undergoes multiple TIR, returning to the observer as sparkling flashes.
4. Prism Binoculars (Porro Prisms): Right‑angled prisms use TIR to invert and redirect images without loss of brightness, unlike silvered mirrors.

4.4 Example: Critical Angle for Water–Air Interface

Refractive index of water n_w ≈ 1.33, air n_a ≈ 1.00.

Thus, any ray inside water striking the water‑air surface at an angle greater than ≈48.8° will be totally internally reflected.

Summary

This chapter has presented the fundamental laws governing refraction at plane surfaces, including Snell’s law and the concept of refractive index. We explored how indices relate across multiple media and the principle of reversibility. The lateral shift produced by a parallel slab was derived and shown to be wavelength‑independent. Finally, we examined total internal reflection, derived the critical angle, and highlighted its importance in technologies such as optical fibers and everyday phenomena like mirages and diamond sparkle. Mastery of these concepts provides a solid foundation for understanding more complex optical systems involving lenses, prisms, and waveguides.