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Derive an expression for Acceptance angle or Numerical aperture for fiber optics.

Derive an expression for Fractional refractive index change or relative refractive index

Answer:

The light propagates through the fiber only when its angle of incidence at the launching end is less than or equal to a certain value. This maximum possible angle with the axis of the fibre up to which a light ray accepted into the core of the fibre is called acceptance angle. The light rays that enter the fibre beyond acceptance angle refracts into the cladding.

Acceptance angle or Numerical aperture

Let us consider an optical fibre having a core with refractive index n₁ and cladding with refractive index n₂ such that (n₁> n₂). The refractive index of the launching medium is n₀. Let us consider a light ray AO enters the fiber making an angle q_i with its axis. OB is the refracted ray that makes an angle Theta_r with the axis and strikes core-cladding interface at an angle Fi, which is greater than critical angle Fi_c. Thus, it undergoes total internal reflection at the interface.

It is clear from the figure that as the value of angle Theta_i increases, Theta_r will also increases and fi will decreases. However, for light propagation through the fiber, it is compulsory that the value of angle Fi should not be less than critical angle F_c. Thus we may increase the incident angle Theta_i up to a certain value that is acceptance angle Theta_max.

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Now, by applying Snell’s law at the launching end

Fractional refractive index change or relative refractive index

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