Numerical aperture in GIF.

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Determining Numerical Aperture for graded index fibers is more complex than for step index fibers, since it is a function of position across the core end face.
Graded Index Fiber does not have a constant refractive index in the core. Due to this property they are also called inhomogeneous core fibers.
It has a decreasing core index n(r) with radial distance from a maximum value of $n_1$ at the axis to a constant value $n_2$ beyond the core radius a in the cladding.

The index variation is given by

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Where, r = radial distance from the fiber axis a = core radius
$n_1 $= refractive index at the core axis Δ = relative refractive index difference $Δ = \frac{(n_1 ^2- n_2 ^2)}{(2n_1 ^2 )} ≈ \frac{n_1 - n_2}{n_1} $

$\alpha$ = profile parameter which gives the characteristics

refractive index profile of the fiber core.

Light incident on the fiber core at position r propagates as guided mode if it is within local NA. The local NA is defined as:

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Where the axial NA is defined as:

$$NA(0)= n_{1 } \sqrt{2∆}$$

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Thus, the NA of GIF decreases from NA(0) to 0 as r moves from fiber axis to core-cladding boundary.

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