
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_1^2n_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_1^2n_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. 
More References and Links
Numerical Aperture of Optical Fibers .Optical Fibers .
Total Internal Reflection of Light Rays at an Interface .