When a light ray travels from a denser to a rarer medium, it bends away from the normal path. As the angle of incidence in denser medium increases, the angle of refraction in the rarer medium also increases. At a particular angle of incidence, the angle of refraction becomes 90°. Then the refracted ray grazes the surface at the interface between the two media. This angle of incidence is called the critical angle (θ_c).

When the angle of incidence exceeds the critical angle, the light ray reflects back to the denser medium following the laws of reflection. This phenomenon is known as Total internal reflection (TIR). It is commonly observed in optics when light travels through total reflecting glass prisms, mirages, optical fibers, etc.

Necessary conditions for total internal reflection:

• The light ray must travel from a denser medium to a rarer medium.
• The angle of incidence (θ_i) must be greater than the critical angle (θ_c). 
• There must be a significant difference in the refractive indices of the two media.
• Interface between the media, where the transition from one medium to another occurs.

Relation between the Critical angle and Refractive Index

The relation between the refractive index and the critical angle is given by the Snell's law. The refractive index of any medium is reciprocal to the sine of the critical angle.

n - Refractive index

θ_c - Critical angle

• The sine of the critical angle depends upon the wavelength as given by the equation  

• Critical angle increases with the rise in temperature as the refractive index of the material decreases. As the temperature of a medium rises, the speed at which light propagates within it generally increases. This results in lower refractive index.

Examples of Total Internal Reflection

• Prism

The right-angled isosceles glass prism is a totally reflecting prism used to change the path of light without losing light energy. The light ray gets totally reflected when it is incident perpendicularly on the prism. The ray makes 45 deg angle at one face of the prism, which is greater than the critical angle for glass, 42 deg.

• Mirage

Mirage is an optical illusion seen in deserts or metallic roads. It occurs when the ray of light travels from the top of a distant tree, gets refracted, and thus bends away from normal. At a particular layer when θ_i > θ_c , the light ray is totally reflected and appears to be the inverted image. This creates the impression that such a distant object is on the bank of the water.

• Total Internal Reflection in Optical fiber

Optical fiber is a flexible, transparent material used to transmit light. It consists of two main parts, the core and cladding region, where the refractive index of the core region n₁ is higher than that of the cladding region n₂.

The light ray undergoes its first refraction at the air-core interface in the optical fiber.  It is important to know the angle at which this refraction takes place since it determines whether or not the following internal reflections will follow the principle of Total Internal Reflection.

Not all rays entering the fiber core will continue propagating down its length. The acceptance angle, θ_a is the maximum angle to the fiber axis at which the light ray incident on the fiber core is propagated. 

• θ_i = θ_a: Any ray entering the fiber at an angle, θ_a, is refracted at the air-core interface and is transmitted to the core-cladding interface. The incident ray strikes the core-cladding interface at critical angle and it passes through the interface.

• θ_i > θ_a: If the incident ray enters the fiber at an angle greater than θ_a, it strikes the core-cladding interface at an angle less than the critical angle. These rays will be refracted into the cladding and are eventually lost to radiation.

• θ_i < θ_a: Only with an angle less than the acceptance angle strikes the core-cladding interface at an angle greater than the critical angle of the core region. These rays undergo repeated total internal reflections along the fiber.

The light ray incident on the fiber core must be within the acceptance cone, a cone of semi-vertical angle, defined by the θ_a to be propagated along an optical fiber through the core-cladding interface.

The numerical aperture (N.A.) measures the ability of an optical fiber to capture light. The sine of the angle of acceptance of the optical fiber is N.A.

It depends on fiber properties such as refractive indices of the core, cladding, and the surrounding medium & transmission conditions.

A large numerical aperture (NA) implies that the fiber can accept light from a wide range of angles, which leads to a larger amount of light being collected by the fiber from the source.

Angle of acceptance and numerical aperture

By using Snell's law and basic trigonometric relationships, the N.A. of the fiber is given by: 

θ_a is the acceptance angle 

n₁ is the index of refraction of the fiber core

n₂ is the index of refraction of the fiber cladding

n₀ is the index of refraction of the surrounding medium

Since the surrounding medium is air, n₀ is 1, N.A. would be sin θ_a.

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