# 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 n1 and a cladding of refractive index n2 such that n1 > n2, 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 αmax 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 θc 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 αmax

Enter the refractive index of the outside n, the refractive index of the core n1, the refractive index of the cladding n2 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 αmaxc for which total internal refraction inside the fiber occurs which also called the angle of acceptance.

 Refractive Index Outside Fiber: n = 1 Refractive Index of Core: n1 = 1.46 Refractive Index of Cladding: n2 = 1.45 Angle of Incidence Outside Fiber: α = 5 � Decimal Places = 4 Angle of Refraction: β = � Angle of Incidence at Core - Cladding Interface: θ = � Critical Angle Core - Cladding Interface: θc = � Numerical Aperture of Fiber: N.A. = Maximum angle: αmaxc =

## 2 - Calculate Refractive Indices Given Critical θc Angle and αmax

Enter the refractive index of the outside n, the critical angle θc at the core - cladding interface, the angle of acceptance αmax then press "Calculate".
The outputs are the refractive indices of the core n
1 and of the cladding n2.

 Refractive Index Outside Fiber: n = 1 Critical Angle Core - Cladding Interface: θc = 82 � Maximum angle: αmaxc = 10 Decimal Places = 4 Refractive Index of Core: n1 = Refractive Index of Cladding: n2 =