Fiber optics: Derivation for acceptance angle or numerical aperture


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

Derive expression for Fractional refractive index change or relative refractive index

Answer:

The light propagates through the fiber only when its angle of incidence at 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 cladding.

Acceptance angle or Numerical aperture

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

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

Now, by applying Snell’s law at launching end

Fractional refractive index change or relative refractive index

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