Numerical aperture refers to the maximum angle at which light incident on the fiber end is very internally reflected and is transmitted properly along the fiber. The cone formed by the rotation of this angle along the axis of the fiber is the cone of acceptance of the fiber. The light ray should strike the fiber end within its cone of acceptance else it is refracted out of the fiber.

From Snell’s law, the minimum angle φ_min that supports total internal reflection for the meridional ray is given by :

sin φ_min  = n₂/n₁                                        ----------------------------------------------- 1

Rays striking the core-cladding interface at angles less than φ_min  will refract out of the core and be lost in the cladding. By applying Snell’s law to the air-fiber face boundary, the condition of equation 1 can be related to the maximum entrance angle θ _0,max through the relationship

n sin θ _0,max = n₁ sin θ _c =     ----------------------------------------2

where θ _c  = π/2 – φ_c

Thus those rays having entrance angles θ ₀ less than θ _0,max will be totally internally reflected at the core-cladding interface.

Equation 2 also defines the numerical aperture (NA) of a step-index fiber for meridional rays:

NA =   n sin θ _0,max =  = n₁√2∆

where n is the refractive index of air, n₁ is the refractive index of the core and n₂ is the refractive index of the cladding.

The approximation on the right hand side is valid when ∆ is much less than 1. The numerical aperture is a dimensionless quantity which is less than unity, with values normally ranging from 0.14 to 0.50.

Fig 1 : Numerical Aperture

Consideration in NA measurement :

It is very important that the LDR should be properly aligned with the cable and the distance from the cable and the LDR be properly selected to ensure that the maximum amount of optical power is transferred to the LDR.

The numerical aperture is computed from the formula given below : θ

NA =  sin θ _max

It is observed that the light intensity at the centre of the screen gradually increases as we move closer to the light detector placed at the screencentre.