Total internal reflection

The two conditions for total internal reflection

First, light must travel from n₁ to a lower n₂. Second, the incidence angle measured from the normal must exceed the critical angle.

Critical angle from Snell's law

At θ = θc, the refracted ray is at 90° to the normal and runs along the boundary. For θ > θc, total internal reflection occurs.

Reflection is still ordinary reflection

The reflected ray obeys the usual law of reflection: reflection angle equals incidence angle. “Total” refers to the absence of a transmitted propagating ray in the ideal lossless boundary model.

An evanescent electromagnetic field can still exist just beyond the boundary, but it decays rapidly and is not the ordinary transmitted ray described by geometric optics.

Optical fibers

A fiber core has higher refractive index than its cladding. Rays launched within an appropriate angular range strike the internal boundary above θc and remain guided through repeated reflection.

Real fibers support electromagnetic modes rather than literal zigzag rays, but the ray picture gives a useful geometric introduction to why the index contrast guides light.

Critical angle is not an incidence angle from the surface

As with Snell's law generally, θc is measured from the normal. A ray that is nearly parallel to the boundary therefore has a large incidence angle and is more likely to satisfy θ > θc when traveling from high n to low n.

The condition depends on both materials. Replacing air with a higher-index cladding increases n₂/n₁ and changes θc. Fiber design deliberately controls this index contrast, along with core size and wavelength, to determine which electromagnetic modes can propagate.

Worked examples

1. Glass to air critical angle

Find θc for glass n₁ = 1.50 to air n₂ = 1.00.

2. Can air-to-glass show TIR?

A ray travels from air (1.00) to glass (1.50). Can total internal reflection occur at that boundary?

3. Classify an incidence angle

For n₁ = 1.60 and n₂ = 1.20, a ray hits at 55°. Does TIR occur?