The numerical aperture of an optical fiber system as the one shown in the diagram below, has been defined and all important formulas found.
For a core of refractive index n_{1} and a cladding of refractive index n_{2} such that n_{1} > n_{2},
a light ray incident from outside the core at an angle α will be internally reflected at the core - cladding interface, if α is smaller that α_{max}
given by the formula

\alpha_{max} = \sin^{-1} \left (\dfrac{1}{n} \sqrt{n_1^2-n_2^2} \right)

Other formulas related to the critical angle at the core - cladding interface θ _{c}and the numerical aperture N.A. are given by

\theta_c = \sin^{-1} \left(\dfrac{n_2}{n_1} \right)

N.A = \sqrt{n_1^2-n_2^2}

The angle of refraction β at the outside - core interface and angle of incidence α are related by
n \sin\alpha = n_1 \sin\beta

and the angle of incidence at the core - cladding interface θ and angle β are complementary.
\theta = 90^{\circ} - \beta

Two calculators that uses the above formulas are presented below.

##
1 - Calculate Numerical Aperture and α_{max}

Enter the refractive index of the outside n, the refractive index of the core n _{1}, the refractive index of the cladding n

_{2}and the angle of incidence α of the ray coming from the left outside the fiber then press "Calculate".

The outputs are: the angle of refraction β at the outside - core interface, the angle of incidence θ at the core - cladding Interface, the critical angle θ

_{c}at the Core - cladding interface, the numerical aperture N.A. and the maximum angle α

_{max}

_{c}for which total internal refraction inside the fiber occurs which also called the angle of acceptance.

## 2 - Calculate Refractive Indices Given Critical θ_{c} Angle and α_{max}

Enter the refractive index of the outside n, the critical angle θ _{c}at the core - cladding interface, the angle of acceptance α

_{max}then press "Calculate".

The outputs are the refractive indices of the core n

_{1}and of the cladding n

_{2}.

## More References and Links

Numerical Aperture of Optical Fibers .Optical Fibers .

Total Internal Reflection of Light Rays at an Interface .