Total internal reflection Knowledge-Based Study

Study notes

1. What is Total Internal Reflection?

   When a light ray travels across an interface between two transparent media with different refractive indices, it generally undergoes both reflection and refraction. However, under specific conditions, refraction ceases entirely, and all the incident energy is reflected back into the original medium. This phenomenon is known as total internal reflection (TIR).

   Necessary Conditions for TIR

   • Optically denser to rarer medium: The light ray must be traveling in a medium with a higher refractive index ($n_1$) toward an interface with a medium of lower refractive index ($n_2$), so that $n_1 > n_2$.
   • Angle of incidence exceeding the critical angle: The angle of incidence ($\theta_1$) at the boundary must be strictly greater than a specific threshold value called the critical angle ($\theta_c$).

   When light travels from an optically denser medium to an optically rarer medium, the refracted ray bends away from the normal. As the angle of incidence increases, the angle of refraction increases correspondingly until it reaches $90^\circ$.

   Defining and Calculating the Critical Angle

   The critical angle ($\theta_c$) is defined as the angle of incidence in the denser medium for which the corresponding angle of refraction in the rarer medium is exactly $90^\circ$.

   Mathematical Derivation

   Applying Snell's law at the interface between the two media gives:

   $$n_1 \sin\theta_1 = n_2 \sin\theta_2$$

   At the critical angle, $\theta_1 = \theta_c$ and $\theta_2 = 90^\circ$. Substituting these values yields:

   $$n_1 \sin\theta_c = n_2 \sin(90^\circ)$$

   Since $\sin(90^\circ) = 1$, the expression simplifies to:

   $$\sin\theta_c = \frac{n_2}{n_1}$$

   For the common case where light emerges from a medium of refractive index $n$ into air or vacuum ($n_2 \approx 1.00$), the relationship becomes:

   $$\sin\theta_c = \frac{1}{n} \quad \text{or} \quad \theta_c = \arcsin\left(\frac{1}{n}\right)$$

   Because the sine function cannot exceed $1$, total internal reflection can only occur when $n_1 > n_2$. If $\theta_1 > \theta_c$, no real angle of refraction exists, and complete reflection occurs following the standard law of reflection: the angle of reflection equals the angle of incidence.

   Practical Applications of Total Internal Reflection

   Because total internal reflection reflects $100\%$ of incident light without absorption losses typical of metallic mirrors, it is widely utilized in modern optical technology:

   • Optical Fibres: An optical fibre consists of a central core of high refractive index surrounded by a cladding layer of slightly lower refractive index. Light launched into the core strikes the core-cladding boundary at angles greater than the critical angle, propagating through kilometers of fibre with minimal attenuation. This forms the backbone of modern telecommunications and medical endoscopes.
   • Totally Reflecting Prisms: Right-angle glass or plastic prisms ($45^\circ\text{–}90^\circ\text{–}45^\circ$) are used in periscopes, binoculars, and single-lens reflex (SLR) cameras to turn light rays by $90^\circ$ or $180^\circ$ without the ghost reflections or silver degradation common to surface mirrors.
   • Brilliance of Diamonds: Diamond has a very high refractive index ($n \approx 2.42$), giving it a small critical angle in air of approximately $24.4^\circ$. Expertly faceted gems ensure that light entering the top facets undergoes multiple internal reflections before exiting toward the observer, producing intense brilliance and dispersion.

   Atmospheric Phenomena

   Variations in air temperature create layers of varying density and refractive index. Mirages on hot roads or deserts occur when light rays curving through warmer, optically rarer air near the ground undergo total internal reflection, creating virtual inverted images that resemble reflections from a pool of water.