TIR depends on a clean glass-air interface. Reflective surfaces must be free of foreign materials. TIR may also be defeated by decreasing the incidence angle beyond a critical value. For a right angle prism of index n, rays should enter the prism face at an angle θ:

θ < arcsin (((n2-1)1/2-1)/√2)

In the visible range, θ = 5.8° for BK 7 (n = 1.517) and 2.6° for fused silica (n = 1.46). Finally, prisms increase the optical path. Although effects are minimal in laser applications, focus shift and chromatic effects in divergent beams should be considered.

i - incident medium

t - transmitted medium

use Snell’s law to find θ_{t}

r = (n_{i}-n_{t})/(n_{i} + n_{t})

t = 2n_{i}/(n_{i} + n_{t})

θ_{β} = arctan (n_{t}/n_{i})

Only s-polarized light reflected.

θ_{TIR} > arcsin (n_{t}/n_{i})

n_{t} < n_{i} is required for TIR

The field reflection and transmission coefficients are given by:

r = E_{r}/E_{i} t = E_{t}/E_{i}

r_{s} = (n_{i}cosθ_{i} -n_{t}cosθ_{t})/(n_{i}cosθ_{i} + n_{t}cosθ_{t})

r_{p} = (n_{t}cos θ_{i} -n_{i}cosθ_{t})/n_{t}cosθ_{i} + n_{i}cosθ_{t})

t_{s} = 2n_{i}cosθ_{i}/(n_{i}cosθ_{i} + n_{t}cosθ_{t})

t_{p} = 2n_{i}cosθ_{i}/(n_{t}cosθ_{i} + n_{i}cosθ_{t})

The power reflection and transmission coefficients are denoted by capital letters:

R = r2 T = t2(n_{t}cosθ_{t})/(n_{i}cosθ_{i})

The refractive indices account for the different light velocities in the two media; the cosine ratio corrects for the different cross sectional areas of the beams on the two sides of the boundary.

The intensities (watts/area) must also be corrected by this geometric obliquity factor:

I_{t} = T x I_{i}(cosθ_{i}/cosθ_{t})

R + T = 1

This relation holds for p and s components individually and for total power.

To simplify reflection and transmission calculations, the incident electric field is broken into two plane-polarized components. The “wheel” in the pictures below denotes plane of incidence. The normal to the surface and all propagation vectors (**k**_{i}, **k**_{r}, **k**_{t}) lie in this plane.

**E** parallel to the plane of incidence; p-polarized.