What is Snell's Law in TIR?

Snell's Law, when applied to Total Internal Reflection (TIR), describes the specific conditions under which light traveling from a denser medium to a rarer medium does not refract, but instead reflects entirely back into the denser medium.

Understanding Snell's Law

Snell's Law is a fundamental principle in optics that describes the relationship between the angles of incidence and refraction for a light wave passing through an interface between two different isotropic media, such as air and water. It mathematically reveals the relationship between the directions of travel in the two media.

The general form of Snell's Law is:

n1 sinθ1 = n2 sinθ2

Where:

• n1 is the refractive index of the first medium (where light originates).
• θ1 is the angle of incidence (the angle between the incident ray and the normal to the surface).
• n2 is the refractive index of the second medium.
• θ2 is the angle of refraction (the angle between the refracted ray and the normal to the surface).

From this, the angle of refraction can be expressed as: sinθ2 = (n1/n2)sinθ1.

Snell's Law and Total Internal Reflection (TIR)

Total Internal Reflection occurs when light attempts to pass from a medium of higher refractive index (denser medium) to a medium of lower refractive index (rarer medium) at an angle greater than a specific value called the critical angle. In such cases, there is no refracted ray, and all the light is reflected back into the denser medium.

Let's consider an example where the ratio of refractive indices n1/n2 = 1.5, so Snell's Law becomes sinθ2 = 1.5 sinθ1.

• The Critical Angle (θc): As the angle of incidence (θ1) increases, the angle of refraction (θ2) also increases. At a specific angle of incidence, known as the critical angle (θc), the refracted ray grazes along the interface between the two media, meaning θ2 = 90° and thus sinθ2 = 1.
  According to Snell's Law, for θ2 = 90°:
  n1 sinθc = n2 sin90°
n1 sinθc = n2
  Therefore, the critical angle θc is given by:
  sinθc = n2/n1

• Condition for TIR: When the angle of incidence (θ1) exceeds the critical angle (θc), sinθ1 becomes greater than sinθc (which is n2/n1). Consequently, when applying Snell's Law, sinθ2 = (n1/n2)sinθ1 becomes greater than 1. Mathematically, there is no real solution for θ2 if sinθ2 > 1, indicating that refraction is not possible. Instead, the light undergoes Total Internal Reflection.

  For instance, as provided in the reference, if sinθ1 is greater than (1/1.5) = 2/3, or θ1 is greater than approximately 41.8°, then sinθ2 will be greater than 1, and there is no solution for θ2. This means that at angles of incidence greater than 41.8° (which is θc for n1/n2 = 1.5), Total Internal Reflection occurs.

Conditions for Total Internal Reflection

For TIR to occur, two crucial conditions must be met:

1. Light must travel from a denser medium to a rarer medium: The refractive index of the first medium (n1) must be greater than the refractive index of the second medium (n2).
2. The angle of incidence must be greater than the critical angle: θ1 > θc, where sinθc = n2/n1.

Practical Applications of TIR

Total Internal Reflection is a phenomenon with numerous practical applications in various fields:

• Fiber Optics: Optical fibers transmit data through light pulses by continuously reflecting them off the inner walls of the fiber. This is achieved because the light travels from the denser core to the rarer cladding, and the angle of incidence is always kept above the critical angle.
• Prisms in Binoculars/Periscopes: Prisms are used to redirect light without significant loss of intensity, providing a brighter image compared to mirrors.
• Endoscopes: Medical instruments used for internal examination of the body utilize optical fibers based on TIR to transmit images from inside the body.
• Diamonds: The brilliant sparkle of a diamond is largely due to total internal reflection. Diamonds are cut in such a way that light entering them undergoes multiple total internal reflections before exiting, enhancing their brilliance.

Summary of Key Terms