Calculate Total Internal Reflection

Total internal reflection can only occur when light travels from an optically denser medium (n₁) to an optically less dense medium (n₂).

What is Total Internal Reflection?

Total internal reflection (or total reflection) is an optical phenomenon that occurs when light travels from an optically denser medium (higher refractive index) to an optically less dense medium (lower refractive index). Instead of the light refracting and entering the less dense medium, the light is completely reflected at the interface. This occurs only when the angle of incidence exceeds the critical angle.

Historical Background

Total internal reflection is a phenomenon known since antiquity. As early as 300 BC, Euclid described the reflection of light at water and glass surfaces. Mathematical description came later with Snellius (Willebrord Snell van Royen, 1580–1626), who formulated Snell's law. Later, total internal reflection became crucial in telecommunications (fiber optics) and is today indispensable for fiber optic technology.

Snell's Law and the Critical Angle

Snell's law describes light refraction at an interface between two media:

• n₁ – Refractive index of the first (denser) medium
• n₂ – Refractive index of the second (less dense) medium
• θ₁ – Angle of incidence (to the normal)
• θ₂ – Angle of refraction (to the normal)

The critical angle (θc) is the angle of incidence at which the angle of refraction becomes exactly 90°. At this angle, the refracted ray runs along the interface. For larger angles of incidence, total internal reflection occurs.

• The critical angle depends only on the ratio of refractive indices
• For glass (n=1.5) → air (n=1.0): θc ≈ 41.81°
• For water (n=1.33) → air (n=1.0): θc ≈ 48.75°

Conditions for Total Internal Reflection

Total internal reflection occurs only when two conditions are met:

1. n₁ > n₂: Light must travel from optically denser to optically less dense medium
2. θ₁ ≥ θc: The angle of incidence must be greater than or equal to the critical angle

If only the first condition is met, but the angle of incidence is less than θc, the light is partially reflected and partially refracted.

Refractive Indices of Materials

Critical Angles for Common Materials

Applications of Total Internal Reflection

1. Fiber Optic Cables (Optical Fibers)

The most important modern application of total internal reflection is fiber optic cables. The light ray is trapped in the fiber and propagates through repeated total internal reflection. This enables:

• Fast data transmission over long distances (internet, telecommunications)
• Minimal signal loss through loss reduction
• Resistance to electromagnetic interference
• High bandwidth (Tbit/s)

2. Prisms and Roof Prisms

Total internal reflection prisms use the critical angle to redirect light by 45° or 90°:

• Binoculars and telescopes (roof prisms)
• Cameras and projectors
• Optical instruments for light redirection
• Simple, reliable, no moving parts

3. Diamonds and Gemstones

The high refractive index of diamonds (n ≈ 2.42) results in a very small critical angle (≈24.4°):

• Much of the incident light is reflected back through total internal reflection
• This creates the characteristic sparkle ("fire")
• Cutting is designed to maximize these total internal reflection effects

4. Underwater Phenomena

Looking up from underwater can encounter a remarkable phenomenon:

• The critical angle for water → air is approximately 48.75°
• Above this angle, a fish sees only the reflection of the underwater world
• This is called "Snell's window"

5. Optical Coatings

Anti-reflective coatings on cameras and eyeglasses function through control of total internal reflection and interference at interfaces.

Graphical Representation: Total Internal Reflection vs. Refraction

Simplified diagram:

• θ₁ < θc: Light is partially reflected and partially refracted (normal Snell's law)
• θ₁ = θc: Angle of refraction becomes 90° (critical case)
• θ₁ > θc: Light is completely reflected (total internal reflection)

Mathematical Condition

From Snell's law follows for the critical angle:

This explains why total internal reflection is only possible when n₁ > n₂ (otherwise sin(θc) > 1, which is impossible).

Intensity at Total Internal Reflection (Fresnel Equations)

The Fresnel equations describe the proportion of reflected and transmitted light:

• Below critical angle: Reflection coefficient < 1 (partial reflection)
• At critical angle: Reflection coefficient ≈ 1 (transition)
• Above critical angle: Reflection coefficient = 1 (complete reflection)

Numerical Examples

Example 1: Glass → Air (n₁=1.5, n₂=1.0)

• Critical angle: θc = arcsin(1.0/1.5) = arcsin(0.6667) ≈ 41.81°
• At θ₁ = 40°: Refraction, no TIR
• At θ₁ = 45°: Total internal reflection (45° > 41.81°)

Example 2: Water → Air (n₁=1.33, n₂=1.0)

• Critical angle: θc = arcsin(1.0/1.33) ≈ 48.75°
• This explains why a diver underwater sees a flattened view of the water surface
• Everything above the viewing angle appears as a reflection from below

Example 3: Diamond → Air (n₁=2.42, n₂=1.0)

• Critical angle: θc = arcsin(1.0/2.42) ≈ 24.42°
• Much smaller angle than glass, hence much more total internal reflection
• This explains the sparkle and brilliance of diamonds

Frequently Asked Questions

Q: Can total internal reflection occur from optically less dense to optically denser medium?

A: No. The condition requires n₁ > n₂. Light from optically less dense to denser medium is always refracted, never totally reflected.

Q: Is total internal reflection loss-free?

A: Theoretically yes – the Fresnel equations show reflection coefficient = 1 at total internal reflection. In practice, there are tiny losses due to material absorption.

Q: Why is light not simply absorbed at high angles?

A: Total internal reflection is a property of Maxwell's equations and Snell's law. At incidence angles greater than θc, the wave mathematically cannot enter the second medium – it must be reflected.

Q: How does total internal reflection differ from mirroring?

A: A mirror is material with high reflivity across a large angle range. Total internal reflection is an effect that only occurs above the critical angle, but works without special coatings (only the refractive index difference is needed).

Summary