Total Internal Reflection: Critical Angle and Light Trapped in Materials

What “total internal reflection” really means

Total internal reflection (TIR) is a special boundary behavior where no transmitted (refracted) ray propagates into the second medium. Instead, the light reflects back into the first medium as if the boundary were a perfect mirror (often with very high efficiency).

TIR is not “extra strong reflection.” It is a regime that happens only under specific conditions involving the direction of travel (which side you start from) and the angle of incidence.

Two required conditions

• Light must travel from a higher refractive index to a lower refractive index (e.g., glass → air, water → air, diamond → air). If you go from lower to higher (air → glass), TIR cannot occur.
• The incidence angle must exceed a threshold called the critical angle. Below that threshold, you get partial reflection and partial refraction; above it, you get TIR.

Critical angle: the “skimming” refracted ray idea

Conceptually, increase the incidence angle inside the higher-index material while aiming at the boundary. The refracted ray in the lower-index material bends farther away from the normal as the incidence angle increases. At one particular incidence angle, the refracted ray would travel exactly along the boundary (i.e., at 90° to the normal). That incidence angle is the critical angle θc.

For incidence angles larger than θc, a propagating refracted ray would require an impossible geometry (it would need to refract beyond 90°), so the transmitted ray disappears and the energy stays in the first medium as TIR.

Critical angle relationship (for quick predictions)

When light goes from n1 (higher) to n2 (lower), the critical angle satisfies:

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sin(θc) = n2 / n1   (only valid when n1 > n2)

Key implications you can use without heavy math:

• If n1 ≤ n2, there is no critical angle (and no TIR).
• The bigger the index contrast (large n1 compared to n2), the smaller the critical angle, so TIR is easier to achieve.

Diagrams: partial refraction vs total internal reflection

Case A: Below the critical angle (partial refraction)

Lower index (n2)  (e.g., air)          refracted ray exists (transmitted)  ↗ (θ2 < 90°)  | boundary ---------------------------------------------------------------  | normal (perpendicular)                         |                         |                         |                         | incident ray in higher index (n1)          ↘ (θ1 < θc)  Higher index (n1) (e.g., glass)

What to notice: you get two rays: a reflected ray in the first medium and a refracted ray in the second medium.

Case B: At the critical angle (refracted ray skims the boundary)

Lower index (n2)                         refracted ray along boundary  →→→  | boundary ---------------------------------------------------------------  | normal                         |                         | incident ray in higher index (n1)          ↘ (θ1 = θc)  Higher index (n1)

This is the conceptual “edge” of refraction: the transmitted ray is just barely propagating.

Case C: Above the critical angle (total internal reflection)

Lower index (n2)                         no propagating refracted ray  (none)  | boundary ---------------------------------------------------------------  | normal                         |                         | incident ray in higher index (n1)          ↘ (θ1 > θc)                         ↗ reflected ray (TIR)  Higher index (n1)

What to notice: the light stays in the higher-index medium; the boundary behaves mirror-like for that ray.

Common misconceptions to avoid

• “TIR happens when light goes from air into glass at a steep angle.” No. TIR requires going from higher index to lower index. Air → glass is lower → higher, so you can get refraction and some reflection, but not TIR.
• “If the angle is big enough, any interface will totally reflect.” No. Even at grazing incidence, if you start in the lower-index medium, there is still a refracted ray into the higher-index medium.
• “TIR means 100% of light reflects in real life.” TIR can be extremely efficient, but real systems may lose light due to absorption, surface roughness, imperfect geometry, or coupling into other modes. The key idea is: no propagating transmitted ray in the second medium.
• “The critical angle is measured from the surface.” In standard optics geometry, angles are measured from the normal (perpendicular to the surface). Mixing these up flips the interpretation.

Step-by-step: deciding if TIR can occur

Step 1: Identify the direction of travel

Determine which medium the ray is currently in (n1) and which medium it is trying to enter (n2).

Step 2: Check the index condition

If n1 > n2, TIR is possible in principle. If n1 ≤ n2, TIR is impossible.

Step 3: Estimate the critical angle

Use sin(θc) = n2/n1. You can do rough estimates with common values.

Step 4: Compare your incidence angle to the critical angle

• If θ1 > θc → TIR.
• If θ1 = θc → refracted ray skims the boundary.
• If θ1 < θc → partial refraction.

Quick reference: approximate refractive indices (for predictions)

Applications: where light gets “trapped” on purpose

Fiber optics: guiding light by repeated TIR

An optical fiber has a core with slightly higher refractive index than the surrounding cladding. Light launched into the core at suitable angles hits the core–cladding boundary above the critical angle and undergoes repeated TIR, staying confined over long distances.

• Design idea: make n_core > n_clad so a critical angle exists at the core–cladding boundary.
• Practical consequence: not every launch angle is guided. Rays that hit the boundary below the critical angle leak into the cladding and are lost.

Light pipes and acrylic rods: “bending” light around corners

Clear acrylic or glass rods can guide light similarly to fibers (often with air outside). If the rod is surrounded by air, the index contrast is large, making TIR easier. This is used in instrument panels, decorative lighting, and illumination guides where a source is placed at one end and light is delivered elsewhere.

Practical tip: fingerprints, scratches, or intentional roughening locally disrupt TIR and let light escape—useful when you want the rod to glow along its length.

Diamond sparkle: strong index contrast and small critical angle

Diamond’s high refractive index means the critical angle at a diamond–air boundary is relatively small, so many rays inside a well-cut diamond meet facets above the critical angle and undergo multiple TIR bounces. This increases the chance that light exits through the top in concentrated directions, contributing to brilliance.

Cut geometry matters because it controls which internal rays meet facets above the critical angle and which leak out through the sides or bottom.

Endoscopy: flexible imaging and illumination

Endoscopes use bundles of optical fibers (or coherent fiber bundles for imaging) to deliver illumination and return images from inside the body. The guiding principle is the same: light is confined by TIR at the core–cladding boundary, allowing transmission through bends with manageable losses.

Prediction exercises (use the index table)

Exercise 1: Can TIR occur at an air → glass boundary?

Given: n1 = 1.00 (air), n2 = 1.50 (glass). Light travels from air into glass.

• Check condition: is n1 > n2? No (1.00 is not greater than 1.50).
• Prediction: TIR cannot occur for air → glass, at any angle.

Exercise 2: Glass → air: estimate the critical angle

Given: n1 = 1.50 (glass), n2 = 1.00 (air).

sin(θc) = 1.00 / 1.50 ≈ 0.667  →  θc ≈ 42°

Prediction checks:

• If θ1 = 30° (from the normal): 30° < 42° → partial refraction occurs.
• If θ1 = 50°: 50° > 42° → TIR occurs.

Exercise 3: Water → air: is TIR easier or harder than glass → air?

Given: n1 = 1.33 (water), n2 = 1.00 (air).

sin(θc) = 1.00 / 1.33 ≈ 0.75  →  θc ≈ 49°

Because the critical angle is larger (~49° vs ~42° for glass → air), you need a more “glancing” incidence in water to get TIR. Prediction: TIR is harder to achieve in water → air than in glass → air.

Exercise 4: Acrylic light pipe in air

Given: n1 = 1.49 (acrylic), n2 = 1.00 (air).

sin(θc) = 1.00 / 1.49 ≈ 0.67  →  θc ≈ 42°

Prediction: Many rays inside an acrylic rod will TIR at the acrylic–air boundary if they hit the surface at angles greater than ~42° from the normal, making acrylic effective for light piping.

Exercise 5: Core–cladding in a fiber (small index difference)

Given: n_core = 1.48, n_clad = 1.46. Light travels in the core toward the cladding.

sin(θc) = 1.46 / 1.48 ≈ 0.986  →  θc ≈ 80°

Prediction: The critical angle is very large (~80° from the normal), meaning the ray must strike the boundary very close to grazing incidence to get TIR. This is why fiber acceptance angles are limited and why launch conditions matter.

Exercise 6: Diamond → air: why sparkle is enhanced

Given: n1 = 2.42 (diamond), n2 = 1.00 (air).

sin(θc) = 1.00 / 2.42 ≈ 0.41  →  θc ≈ 24°

Prediction: With such a small critical angle, many internal rays exceed it at facets, so TIR is common—supporting strong internal bouncing and bright return of light for suitable cuts.