Total Internal Reflection

Fundamentals of Total Internal Reflection

Total internal reflection (TIR) is an optical phenomenon that occurs when light traveling through an optically denser medium encounters a boundary with an optically less dense medium.

• Definition: Total internal reflection (TIR) occurs when light travelling from a denser medium reaches the boundary with a less dense medium at an angle greater than the critical angle ($c$). Instead of refracting out, the light is completely reflected back into the denser medium.
• Conditions Required for TIR:
  • Light must be travelling from a denser medium toward a less dense medium.
  • The angle of incidence ($i$) must be greater than the critical angle ($c$) ($i > c$).

Boundary Behavior Across Angles of Incidence

When light travels from a denser medium (such as water) toward a boundary with a less dense medium (such as air), its behavior varies depending on the angle of incidence ($i$) relative to the critical angle ($c$):

• Sub-Critical Angle ($i < c$):

  • The angle of incidence $i$ is less than the critical angle $c$.
  • The incident ray passes through the boundary and refracts into the less dense medium as a refracted ray at angle $r$, bending away from the normal.

• Critical Angle Boundary ($i = c$):

  • The angle of incidence $i$ equals the critical angle $c$.
  • The refracted ray travels along the boundary surface between the two media, creating an angle of refraction of $r = 90^\boxdot$ (or $r = 90^\top$).

• Super-Critical Angle ($i > c$):

  • The angle of incidence $i$ is strictly greater than the critical angle $c$.
  • Light cannot refract out into the less dense medium. Total internal reflection occurs, completely reflecting the light ray back into the denser medium at an angle of reflection $r$ equal to the angle of incidence $i$ ($r = i$).

Mathematical Formulation of the Critical Angle

The mathematical formula relating the critical angle ($c$) to the refractive index ($n$) of the denser medium is:

$\sin(c) = \frac{1}{n}$

Rearranging the equation allows direct calculation of the critical angle $c$:

$c = \sin^{-1}\left(\frac{1}{n}\right)$

Worked Calculation Example

To find the critical angle $c$ for a medium with a refractive index $n = 1.60$:

$c = \sin^{-1}\left(\frac{1}{1.60}\right)$

$c = 38.7^\circ$

Total Internal Reflection in Prisms

Total internal reflection in right-angled prisms ($45^\circ - 90^\circ - 45^\circ$) is utilized to change the direction of light rays by precise angles.

Light Ray Bends by $90^\circ$

• Mechanism: A light ray enters perpendicular to one of the shorter faces of a right-angled prism and continues straight without bending.
• Internal Reflection: The ray hits the hypotenuse internal surface at an angle of incidence $i = 45^\circ$.
• Condition: Since $45^\circ$ is greater than the critical angle for typical optical medium ($c = 38.7^\circ$ when $n = 1.60$), total internal reflection occurs.
• Result: The ray reflects internally once and exits through the adjacent face, bending the light ray overall by $90^\circ$.

Light Ray Bends by $180^\circ$

• Mechanism: Light rays enter perpendicular to the long hypotenuse face of a $45^\circ - 90^\circ - 45^\circ$ prism without refraction at the entry boundary.
• First Internal Reflection: Rays encounter the first internal surface at an angle of incidence of $45^\circ$. Since $45^\circ > c$, total internal reflection occurs, changing the ray direction by $90^\circ$.
• Second Internal Reflection: The redirected light travels across to strike the second internal surface, again at an angle of incidence of $45^\circ$. Total internal reflection occurs a second time, turning the light ray by an additional $90^\circ$.
• Result: The combined deviation of two successive $90^\circ$ internal reflections results in the light ray bending by a total of $180^\circ$, turning the light completely around back toward its original source direction.