# Numerical Aperture of Optical Fibers Calculator

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.