AcademyLight Propagation
Academy
Total Internal Reflection
Level 1 - Physics topic page in Light Propagation.
Principle
Total internal reflection occurs when light in a higher-index medium reaches a boundary with a lower-index medium above the critical angle.
Notation
\(n_1\)
refractive index of incident medium
\(n_2\)
refractive index of transmitted medium
\(\theta_i\)
angle of incidence in the higher-index medium
\(\mathrm{rad}\)
\(\theta_c\)
critical angle
\(\mathrm{rad}\)
\(\theta_2\)
refracted angle
\(\mathrm{rad}\)
Method
Derivation 1: Check the direction of travel
Total internal reflection is only possible when light travels from higher refractive index to lower refractive index.
Required condition
\[n_1>n_2\]
Derivation 2: Find the critical angle
At the critical angle, the refracted ray would skim along the boundary, so \(\theta_2=90^\\circ\).
Snell at critical angle
\[n_1\sin\theta_c=n_2\sin90^\circ\]
Critical angle
\[\sin\theta_c=\frac{n_2}{n_1}\]
Derivation 3: Compare with the incident angle
If \(\theta_i>\theta_c\), no transmitted ray carries energy away in the ideal ray model; the light reflects internally.
Rules
Critical angle
\[\theta_c=\sin^{-1}\left(\frac{n_2}{n_1}\right)\quad(n_1>n_2)\]
Total internal reflection
\[\theta_i>\theta_c\Rightarrow \text{total internal reflection}\]
Critical ray
\[\theta_i=\theta_c\Rightarrow \theta_2=90^\circ\]
Examples
Question
Find the critical angle for glass
\[(n=1.50)\]
to air \[(n=1.00)\]
Answer
\[\theta_c=\sin^{-1}\left(\frac{1.00}{1.50}\right)=41.8^\circ\]
Checks
- Total internal reflection cannot occur from lower index to higher index.
- The critical angle is measured from the normal.
- At the critical angle, the refracted ray is along the boundary.
- Optical fibers use repeated total internal reflection to guide light.