Understanding Refraction, Critical Angle, and Total Internal Reflection

Light Refraction and Reflection from Denser to Rarer Media

• When a light ray traveling in a denser medium falls on the surface separating it from a rarer medium, it undergoes two simultaneous processes:
  • Part of the light is reflected back into the denser medium.
  • Part of the light is refracted into the rarer medium.
• The distribution of light between refraction and reflection depends entirely on the angle of incidence ($i$).
  • If the angle of incidence is small, the refracted ray is more dominant than the reflected ray.
  • As the angle of incidence increases, the proportion of reflected light increases while the refracted light changes behavior based on specific thresholds.

The Definition and Behavior of the Critical Angle

• Definition: The critical angle is the specific angle of incidence in the denser medium corresponding to which the angle of refraction in the rarer medium is exactly $90^\circ$.
• Primary Condition: This phenomenon only occurs when the incident ray originates from the denser medium and travels toward the rarer medium. If the incident ray travels from a rarer medium to a denser medium, the critical angle phenomenon does not occur.
• At the critical angle:
  • The angle of incidence ($i$) is equal to the critical angle ($C$).
  • The angle of refraction ($r$) is exactly $90^\circ$.
  • The refracted ray travels along the glass-air interface (the boundary surface separating the two media).
  • The refracted ray at this point is described as being "very weak."

Three Cases of Media Interaction Based on Angle of Incidence

• Case 1: Angle of incidence is less than the critical angle ($i < C$)
  • Normal refraction takes place.
  • As light moves from a denser to a rarer medium, it moves away from the normal.
  • The reflected ray is less intense than the refracted ray.
• Case 2: Angle of incidence is equal to the critical angle ($i = C$)
  • The incident ray is equal to the critical angle threshold.
  • The refracted light moves along the boundary separating the two media.
  • The angle of refraction is exactly $90^\circ$.
• Case 3: Angle of incidence is greater than the critical angle ($i > C$)
  • Total reflection occurs and no refraction is obtained.
  • The boundary surface separating the two media behaves entirely as a reflecting surface.
  • The whole light ray is reflected back into the denser medium.

Mathematical Relationship Between Critical Angle and Refractive Index

• The relationship can be expressed using the refractive index ($\mu$) of the media.
• Considering the refractive index of air (rarer) with respect to glass (denser):
  • ${}_{g}\mu_{a} = \frac{\sin(C)}{\sin(90^\circ)}$
  • Since $\sin(90^\circ) = 1$, the formula simplifies to: ${}_{g}\mu_{a} = \sin(C)$
• Conversely, the refractive index of the denser medium (glass) with respect to the rarer medium (air) is the reciprocal:
  • ${}_{a}\mu_{g} = \frac{1}{\sin(C)}$

Factors Affecting the Critical Angle

• Dependence on the Color of Light (Wavelength):
  • The refractive index of a transparent medium decreases as the wavelength of light increases.
  • The critical angle for a pair of media increases with the increase in wavelength.
  • Violet Light: Has the shortest wavelength, the highest refractive index, and therefore the least critical angle.
  • Red Light: Has the longest wavelength, the lowest refractive index, and therefore the most (largest) critical angle.
• Dependence on Temperature:
  • Increasing the temperature of a medium causes its refractive index to decrease.
  • Because the refractive index decreases when temperature rises, the critical angle increases with the increase in temperature.

Total Internal Reflection (TIR)

• Definition: When a ray of light passes from a denser medium to a rarer medium at an angle of incidence greater than the critical angle ($i > C$), it is totally reflected back into the denser medium. This phenomenon is known as Total Internal Reflection.
• Contrast with Rare-to-Denser Travel: When light travels from a rarer to a denser medium, reflection and refraction always occur simultaneously at all angles of incidence.
• Mechanism of TIR:
  • The light ray gets entirely reflected back into the same medium, obeying the standard laws of reflection.
  • The light does not suffer any refraction.
• Essential Conditions for Total Internal Reflection:
  1. Light must travel from a denser medium to a rarer medium.
  2. The angle of incidence in the denser medium must be greater than the critical angle for that specific pair of media.
• Efficiency of Reflection:
  • In the process of Total Internal Reflection, 100% of the light energy is reflected back.
  • This distinguishes TIR from reflection by other devices like plane mirrors, which cannot produce 100% reflection due to absorption and partial refraction of light.

Practical Applications of Total Internal Reflection

• Due to the property of 100% energy reflection, total internal reflection is utilized in various optical constructions where high efficiency is required.
• Total reflecting prisms are used to replace traditional plane mirrors in several devices, including:
  • Periscopes.
  • Binoculars.
  • Certain types of cameras.