Numerical Aperture of Optical Fiber & Its Derivation

Optical fiber is a plastic or transparent fiber that is used to propagate light. The working principle of this is the total internal reflection from completely different walls. So light can be transmitted for long distances because the flexibility of fiber optics is sufficient. So this is used in microscopes which are in micro size, data communication, in fine endoscopes design, etc. An optical fiber cable includes three layers like core, cladding, and jacket. A core layer is enclosed through a cladding. Here cladding layer is normally designed with plastic or silica. The main function of the core within the optical fiber is to transmit an optical signal while the cladding directs the light in the core. As the optical signal is guided throughout the fiber, then it is called an optical waveguide. This article discusses an overview of the numerical aperture of optical fiber.

What is the Numerical Aperture of Optical Fiber?

Definition: The measurement of an optical fiber ability to collect the occurrence light ray in it is known as the numerical aperture. The short form of this is NA that illustrates the efficiency with the light which is collected within the fiber to get propagated. We know that when the light is propagated through an optical fiber during total internal reflection. So multiple total internal reflections take place within the fiber to transmit from one end to another.

Optical Fiber Cable with Internal Reflection
Optical Fiber Cable with Internal Reflection

Once the light ray is produced from the source of an optical fiber, then the optical fiber should be very efficient to get the maximum emitted radiation in it. So we can say that the efficiency of a light which is getting from the optical fiber is the main character once transmitting a signal throughout an optical fiber.

The numerical aperture is connected to the acceptance angle because the acceptance angle is the maximum angle during light travels through the fiber. Therefore the NA & acceptance angle is associated with each other.

Numerical Aperture of Optical Fiber Experiment

The diagram of the optical fiber experiment is shown below. In the following image, a light ray that is transmitted into fiber optic is denoted with ‘XA’. Here ‘ƞ1’ is the refractive index of the core and ‘ƞ2’ is the cladding.

The following image illustrates the light ray is focused on an optical fiber. Here, the light ray travels from denser to rarer medium with an angle ‘α’ through the fiber axis. The ‘α’ angle is called the acceptance angle in the fiber optic cable.


This incident ray travels within the fiber cable to get reflected totally through the interface of core-cladding. However, the incident angle must be more when contrasted to the critical angle or else, if the incident angle is low compare with the critical angle, then the ray gets refracted instead of reflected.
Based on the law of Snell, the refracted ray & the incident angle will transmit within the same angle.

Numerical Aperture of Optical Fiber
Numerical Aperture of Optical Fiber

Therefore, by applying this law at medium 1 (air) & core interface, then the equation will be

Ƞ sin α = Ƞ1 sin θ

The ‘θ’ value can be written from the above image like the following.

Θ = π/2- θc

By substituting the value of ‘θ’ in the above equation

Ƞ sin α = Ƞ1 sin (π/2- θc)

Ƞ sin α = Ƞ1* sin (π/2)- sin(θc)

From the trigonometry, we know that sin θ = cosθ and sin π/2 = 1

Ƞ sin α = Ƞ1cos(θc)

sin α = Ƞ1/ Ƞ cos(θc)

We know that, cos θc = √1-sin2θc

By applying snell’s law at the interface of core-cladding, then we can get

Ƞ1 sin θc = Ƞ2 sin π/2

Ƞ1 sin θc = Ƞ2

Here sin π/2 value is ‘1’ according to standard trigonometry values

sin θc = Ƞ2/ Ƞ1

Substitute the sin θc value in cos θc equation, then

cos θc = √1- cos θc = √1- (Ƞ2/ Ƞ1) 2

Substitute the cos θc value in sin α equation, then

sin α = Ƞ1/ Ƞ√1- (Ƞ2/ Ƞ1) 2

sin α = √( Ƞ12- Ƞ22)/ Ƞ

We have already discussed that medium 1 is nothing but air, so the refractive index (ƞ) will be 1. So more especially we can say

sin α = √( Ƞ12- Ƞ22)

NA= √( Ƞ12- Ƞ22)

The numerical aperture of the optical fiber formula is derived above. So this is the formula for NA, where ‘ƞ1’is the refractive index for core & ‘ƞ2’ is the refractive index for the cladding.

Applications of Numerical Aperture

The applications of NA include the following

  • Fiber Optics
  • Lens
  • Microscope Objective
  • Photographic Objective


1). What is the numerical aperture (NA)?

Numerical aperture is the ability to gather light otherwise an optical fiber capacity.

2). What is the application of the numerical aperture of optical fiber?

In fiber optics, it describes the angles range where light is occurring on the fiber optic will be broadcasted along with it.

3). What is the application of numerical aperture?

NA is generally used in microscopy for describing the acceptance cone

4).What is the acceptance angle in fiber optic cable?

The maximum angle completed through the light ray using the fiber axis to propagate the light via the fiber after the whole internal reflection is known as the acceptance angle.

5). What is the formula for the numerical aperture?

The main formula for numerical aperture (NA) is = √( Ƞ12- Ƞ22)

6). How to select an optical fiber?

There are various parameters that should be taken into reflection to select the suitable optical fiber in signal propagation.

7).What is the working principle of a fiber optic cable?

The working principle of a fiber optic cable is total internal reflection where the light signals can be broadcasted from one position to another through a small loss of energy.

Thus, this is all about what is a numerical aperture in optical fiber, the derivation of the numerical aperture of optical fiber, and its applications From the above information finally, we can conclude that the light-collecting ability is known as NA. So the value of NA should be high that can be attained simply once the dissimilarity between the two refractive indexes is high. For this, ƞ1 must be high otherwise ƞ2 must below. Here is a question for you, what is the value of NA?