Total Internal Reflection

• The mathematical foundation of TIR is derived from Snell's Law, which relates the angles and refractive indices of two media: $n_1 \\sin(\theta_1) = n_2 \\sin(\theta_2)$.

• As the angle of incidence $\\theta_1$ increases in the denser medium, the angle of refraction $\\theta_2$ in the rarer medium increases even faster because $n_1 > n_2$.

• When the refracted angle $\\theta_2$ reaches its physical limit of $90^\circ$, the sine of the angle becomes $1$, leading to the derivation of the critical angle formula.

Critical Angle Formula: $\\sin(\theta_c) = \\frac{n_2}{n_1}$, where $n_1$ is the refractive index of the incident (denser) medium and $n_2$ is the refractive index of the exit (rarer) medium.

• Step 1: Identify the Media: Determine which medium has the higher refractive index ($n_{high}$) and which has the lower ($n_{low}$). TIR can only occur if the light starts in $n_{high}$.

• Step 2: Calculate the Critical Angle: Use the inverse sine function: $\\theta_c = \\sin^{-1}(\\frac{n_{low}}{n_{high}})$. Ensure your calculator is in degree mode if the answer is required in degrees.

• Step 3: Compare Angles: Measure the angle of incidence ($\\theta_i$) relative to the normal (the line perpendicular to the surface).

• Step 4: Determine Outcome: If $\\theta_i < \\theta_c$, refraction occurs. If $\\theta_i = \\theta_c$, the ray skims the surface. If $\\theta_i > \\theta_c$, Total Internal Reflection occurs.

• Check the Direction: Always verify that light is moving from a higher $n$ to a lower $n$. If a question asks for the critical angle of light moving from air to glass, it is a trick question; TIR cannot occur in that direction.

• Normal vs. Surface: Examiners often provide the angle between the ray and the surface. You must subtract this from $90^\circ$ to find the angle of incidence relative to the normal before using Snell's Law.

• The Sine Limit: If you accidentally calculate $\\frac{n_{high}}{n_{low}}$ (e.g., $1.5/1.0$), your calculator will give an error because $\\sin(\\theta)$ cannot exceed $1$. This is a physical reminder that the light must be moving toward a less dense medium.

• Sanity Check: The critical angle for glass ($n \\approx 1.5$) to air is roughly $42^\circ$. If your calculated angle is very small (e.g., $5^\circ$) or very large (e.g., $85^\circ$), re-check your refractive index values.

• Fiber Optics: Thin strands of glass use TIR to transmit data as light pulses over vast distances. The light is trapped in the 'core' because it hits the 'cladding' (a lower-index material) at angles greater than the critical angle.

• Optical Instruments: Binoculars and periscopes use 45-90-45 degree prisms. Because the critical angle for glass is $\\approx 42^\circ$, light hitting the internal face at $45^\circ$ undergoes TIR, allowing for efficient image flipping without the silvering required for mirrors.

• Gemology: Diamonds are cut with specific facet angles to ensure that light entering the top undergoes multiple TIR events before exiting. This 'trapping' of light creates the characteristic sparkle or 'fire' of the gem.