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Numerical Aperture and Acceptance Angle of the Optical Fibre

Angle of Acceptance →
If incident angle of light on the core for which the incident angle on the core-cladding interface equals the critical angle then incident angle of light on the core is called the "Angle of Acceptance.
Transmission of light when the incident angle is equal to the acceptance angle
Transmission of light when the incident angle is equal to the acceptance angle

If the incident angle is greater then the acceptance angle i.e. $\theta_{i}>\theta_{0}$ then the angle of incidence on the core-cladding interface will be less than the critical angle due to which part of incident light is transmitted into cladding as shown in the figure below
Transmission of light when the incident angle is greater than the acceptance angle
Transmission of light when the incident angle is greater than the acceptance angle
If the incident angle is less then the acceptance angle i.e. $\theta_{i}<\theta_{0}$ then the angle of incidence on the core-cladding interface will be greater than the critical angle for which total internal reflection takes place inside the core. As shown in the figure below
Transmission of light when the incident angle is less than the acceptance angle
Transmission of light when the incident angle is less than the acceptance angle
The light entering the core in a cone of semi-vertical angle $\theta_{0}$ is transmitted in the core through total internal reflections. This cone is known as the acceptance cone.
Numerical Aperture
The sine of the angle of acceptance of the optical fibre is known as the numerical aperture of optical fibre.
The numerical aperture determines the light-gathering ability of the fibre. It measures the amount of light that can be accepted by a fibre. The numerical aperture depends upon the refractive index of the core and cladding material and does not depend on the physical dimension of the fibre. It is a dimensionless quantity that is less than unity. The value of the numerical aperture range from $0.13$ to $0.15$. A large numerical aperture implies that a fibre accepts a large amount of light from the source. It varies due to variations of refractive index in the core and it has zero value after the core-cladding boundary. The number of propagation modes to multimode graded-index fibre depends upon the parameter of numerical aperture and hence upon the relative refractive index difference $\Delta n$

Derivation of Angle of Acceptance and Numerical Aperture

Let us consider, step-index optical fibre for which

The incident angle on the axis of core = $\theta_{i}$
The refracted angle on the axis of core = $\theta_{r}$
The refractive index of core = $n_{1}$
The refractive index of cladding = $n_{2}$
The incident angle at core-cladding interface = $\phi$.
Transmission of light when the incident angle is equal to the acceptance angle
Transmission of light when the incident angle is equal to the acceptance angle

When ray incident at point $A$ on the core then According to Snell's law

$\frac{sin \theta_{0}}{sin \theta_{r}}= \frac{n_{1}}{n_{0}} $

Where $n_{0}$ → refractive index of air and vacuum

$sin\theta_{0}=\frac{n_{1}}{n_{0}} sin \theta_{r} \qquad(1)$

Now the refracted ray incident at point $B$ at the interface of core and cladding. So for critical angle condition

$n_{1}\: sin\phi=n_{2} \: sin90^{0} $

$n_{1}\: sin(90-\theta_{r})=n_{2} \: sin90^{0} \qquad (\because \phi=90-\theta_{r})$

$n_{1}\: cos \theta_{r}=n_{2}$

$cos\theta_{r}=\frac{n_{2}}{n_{1}} \qquad(2)$

$sin\theta_{r}=\sqrt{1-cos^{2}\theta_{r}}$

Now substitue the value of $cos\theta_{r}$ from equation$(2)$ to above equation then we get

$sin\theta_{r}=\sqrt{1- \left ( \frac{n_{2}}{n_{1}} \right)^{2}}$

$sin\theta_{r}=\frac{1}{n_{1}}\sqrt{n_{1}^{2}- n_{2}^{2}}$

Now substitue the value of $sin\theta_{r}$ in equation$(1)$ then we get

$sin\theta_{0}=\frac{n_{1}}{n_{0}}\frac{1}{n_{1}}\sqrt{n_{1}^{2}- n_{2}^{2}}$

$sin\theta_{0}=\frac{\sqrt{n_{1}^{2}- n_{2}^{2}}}{n_{0}}\qquad(3)$

This $sin\theta_{0}$ is known as Numerical Aperture. i.e.

$N.A.=\frac{\sqrt{n_{1}^{2}- n_{2}^{2}}}{n_{0}}$

If fibre is in the air then $n_{0}=1$ so the above equation can be written as

$N.A.=\sqrt{n_{1}^{2}- n_{2}^{2}}$

The equation $(3)$ also can be written as

$\theta_{0}=sin^{-1}\frac{\sqrt{n_{1}^{2}- n_{2}^{2}}}{n_{0}}$

The angle $\theta_{0}$ is known as the Angle of Acceptance.

The light is transmitted through the fibre when

$\theta_{i} < \theta_{0}$

i.e. $sin \theta_{i} < sin \theta_{0}$

$sin \theta_{i} < N.A.$

The light will be transmitted through the fibre with multiple total internal reflections when the above condition is satisfied.

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