• Numerical Aperture is the ability of fiber to collect the light from the source and save the light inside it by maintaining the condition of total internal reflection.

• Consider a light ray entering from a medium air of refractive index $n_o$ into the fiber with a core of refractive index $n_1$ which is slightly greater than that of the cladding $n_2$.

Applying Snell’s law of reflection at point A,

$$\frac{sin⁡Ө_1 }{sin⁡Ө_2} = \frac{n_1}{n_o} =n_1 \ \ as \ \ n_0= 1$$

$$θ_2 = \frac{π}2 - ∅_c$$

$$sin θ_1= n_1sin (\frac{π}2 - ∅_c) = n_1cos ∅_c$$

$$cos ∅_c = (1-sin⁡∅_c^2)^{1/2}$$

$$sin {θ_1}=n_1 (1-sin⁡∅_c ^2)^{1/2}$$

When the TIR takes place, $\phi_c=\theta_c$ &$ \theta_1$=$\theta_a$

whrer $\theta_c$=critical angle and$ \theta_a$= acceptance angle

therefore,

$$sin\theta_a=n_1(1-sin\theta_{c}^2)^{1/2}$$

we know,

$$sin\theta_c=\frac{n_2}{n_1}$$

$$sin\theta_a=n_1\left[1-\left(\frac{n_2}{n_1}\right)^2\right]^{1/2}$$

$$N.A.=sin\theta_a$$

• sin $θ_a$ represents all the light rays within cone of $θ_a$ , which maintain the condition of TIR inside the fiber.

• The NA is always chosen so as to accept maximum incident light, satisfying other requirements

$$NA = sin θ_a= \sqrt{n_1^2-n_2^2 }$$

• The core transmits the optical signal while the cladding guides the light within the core and if the cladding is removed then there is a loss of signal.

• For a step index fiber with a constant refractive index core, the wave equation is Bessel differential equation and solutions are cylindrical functions, therefore if cladding is removed the step index fiber behaves like a cylindrical circular optical fiber with $n_2$=1 of air.

• Thus, NA increases if the cladding is removed.