Numerical Aperture of Optical Fibers

What is the Numerical Aperture of an Optical Fiber?

The optical fiber system shown below has a core of refractive index n1 and a cladding of refractive index n2 such that n1 > n2. For a light ray to be internally reflected at the core - cladding interface, the angle on incidence θ must be greater than the critical angle θc given by
Numerical Aperture of an optical fiber

\theta_c = \sin^{-1} \left(\dfrac{n_2}{n_1} \right)
When a light ray is incident from the outside (left side of the diagram) with refractive index n at an angle α at the outside - core interface, it will be refracted at an angle β inside the core of the fiber and both angles are related by Snell's law as follows
n \sin\alpha = n_1 \sin\beta
But angles θ and β are complementary; hence
\cos\theta = \sin\beta
which gives the equation
n \sin\alpha = n_1 \cos\theta
Solve the above equation for θ
\theta = cos^{-1} \left( \dfrac{n \sin \alpha}{n_1} \right)
To have total internal  internal reflection at the core cladding interface, angle θ must be greater than the critical angle θc given above; hence the inequality
\cos^{-1} \left( \dfrac{n \sin \alpha}{n_1} \right) > \sin^{-1} \left(\dfrac{n_2}{n_1} \right)
Take the cosine of both sides of the above inequality and change the symbol of the inequality because cos(x) is a decreasing function on the interval [0 , π/2]. Hence
\cos \left(\cos^{-1} \left( \dfrac{n \sin \alpha}{n_1} \right) \right) < \cos \left( \sin^{-1} \left(\dfrac{n_2}{n_1} \right) \right)
Use the following properties of trigonometric functions and their inverses in the above inequality
\cos(\cos^{-1} x ) = x \text{ and } \cos(\sin^{-1} x ) = \sqrt{1 - x^2}
to simplify and obtain
\dfrac{n \sin \alpha}{n_1} < \sqrt{1-\dfrac{n_2^2}{n_1^2}}
Multiply both sides of the inequality by n1 / n and simplify
\sin \alpha < \dfrac{1}{n} \sqrt{n_1^2-n_2^2}
Take the sin-1 of both sides to obtain
\alpha < \sin^{-1} \left (\dfrac{1}{n} \sqrt{n_1^2-n_2^2} \right)
The quantity
N.A = \sqrt{n_1^2-n_2^2}
is called the numerical aperture and
\alpha_{max} = \sin^{-1} \left (\dfrac{1}{n} \sqrt{n_1^2-n_2^2} \right)
is called the angle of acceptance which is the largest angle α for which light is totally reflected at the core-cladding interface and hence guided along the fiber.

Numerical Aperture of Optical Fiber Sytems Problems with Solutions

A numerical aperture of optical fibers calculator is included in this site and may be used to check the calculations in the following problems.
Problem 1
let n = 1, n
1 = 1.46 and n2 = 1.45 in the diagram of the optical fiber system above. Find
a) the critical angle θ
c at the core - cladding interface.
b) the numerical aperture N.A. of the optical fiber
c) the angle of acceptance α
max of the the optical fiber system.
Solution to Problem 1
a) θ
c = sin-1 (n2 / n1) = sin-1 (1.45 / 1.46) = 83.29 °
b) N.A. = √(n
21 - n22) = √(1.462 - 1.452) = 0.17
c) α
max = sin-1√(1.462 - 1.452) = 9.82 °

Problem 2
We use the same values for n , n1 and n2 in the diagram of the optical fiber system above as those used in problem 1. Let the angle of incidence of a light ray on the outside - core interface be α = 5°. Find
a) angle of refraction β at the outside - core interface.
b) angle θ
c) and explain why this light ray will be reflected at the core - cladding interface and hence guided along the fiber.

Solution to Problem 2
a) Angle β is found using Snell's law at the outside - core interface as follows
n sin(α) = n1 sin(β)
Substitute the given parameters from problem 1 to obtain
β = sin-1 ( sin(5°) / n1) = 3.42 °
b) Angle θ is complementary to angle β hence
θ = 90 - 3.42 = 86.58 °
c) The angle of incidence θ = 86.58 ° at the core - cladding interface is larger that the critical angle θc = 83.29 ° calculated in problem 1 above and will therefore be totally reflected at this interface and hence guided along the fiber.
Of course we could have answered this question by stating that since angle α = 5° is smaller to αmax = 9.82 ° calculated in problem 1, the given light ray will be reflected at the core - cladding interface, but the idea behind this problem is to gain better understanding of the concept of numerical aperture and angle of acceptance of an optical fiber systems using numerical values and calculations at each interface.

Problem 3
let n = 1 and n1 = 1.48 in the diagram of the optical fiber system above. Find n2 such that light rays incident at an angle α greater than 12 ° are not reflected at the core - cladding and therefore not guided along the optical fiber system.
Solution to Problem 3
In this problem, we are given αmax = 12 ° given by the formula
αmax = sin-1(√(n12 - n22)/ n)
The above is equivalent to
sin(αmax) = √(n12 - n22)/ n
Substitute αmax, n and n1 by the given values and calculate n2
n22 = n12 - sin2max)
n2 = √(1.482 - sin2(12°)) = 1.465

Problem 4
let n = 1 in the diagram of the optical fiber system above. Find n1 and n2 such that the acceptance αmax = 10° and the critical angle at the core - cladding interface θc = 82 ° .
Solution to Problem 4
In this problem, we are given αmax and θc whose formulas are given by

\alpha_{max} = \sin^{-1} \left (\dfrac{1}{n} \sqrt{n_1^2-n_2^2} \right)
and
\theta_c = \sin^{-1} \left(\dfrac{n_2}{n_1} \right)
Take the sine of the first formula, simplify and square both sides to obtain
\sin^2\alpha_{max} = \left (\dfrac {n_1^2-n_2^2} {n^2} \right)
The second equation is equivalent to
\sin \theta_c = \left(\dfrac{n_2}{n_1} \right)
Substitute the known values to obtain the equations
n_1^2-n_2^2 = n^2 \sin^2 \alpha_{max} \quad\quad (equation 1)
\dfrac{n_2}{n_1} = \sin \theta_c \quad\quad (equation (2)
The last equation gives
n_2 = n_1 \sin \theta_c
Substitute the above in equation (1) and solve for n1 and n2
n_1^2- (n_1 \sin \theta_c )^2 = n^2 \sin^2 \alpha_{max}
n_1^2(1 - \sin^2 (\theta_c)) = n^2 \sin^2 \alpha_{max}
n_1^2 = \dfrac{\sin^2 \alpha_{max} }{1 - \sin^2 \theta_c} = n^2 \dfrac{\sin^2 \alpha_{max}}{ \cos^2 \theta_c }
n_1 = \dfrac{n\sin \alpha_{max} }{ \cos (\theta_c)}
n_2 = n_1 \sin \theta_c = \dfrac{n \sin(\alpha_{max})}{ \cos \theta_c} \sin \theta_c = n \sin \alpha_{max} \tan \theta_c
Substitute n , θc and αmax by their values to obtain numerical values for n1 and n2
n_1 = \dfrac{\sin 10^{\circ} }{ \cos 82^{\circ}} = 1.2477
and
n_2 = \tan 82^{\circ} \sin 10^{\circ} = 1.2355

More References and Links

Optical Fibers.
numerical aperture of optical fibers calculator
Total Internal Reflection of Light Rays at an Interface.
Refraction of Light Rays, Examples and Solutions.

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