Numerical aperture of an optical fibre (w.r.t air) having core and cladding refractive indices `n_(1) and n_(2)` respectively is

Options

A. `sqrt(n_(1)^(2) - n_(2)^(2))`
B. `sin^(-1) sqrt(n_(1)^(2) - n_(2)^(2))`
C. `cos^(-1) sqrt(n_(1)^(2) - n_(2)^(2))`
D. `tan^(-1) sqrt(n_(1)^(2)-n_(2)^(2))`

Answer

To find the numerical aperture (NA) of an optical fiber with respect to air, we can follow these steps:

### Step 1: Understand the Concept of Numerical Aperture
The numerical aperture (NA) of an optical fiber is a dimensionless number that characterizes the range of angles over which the fiber can accept or emit light. It is defined as:
\[ NA = \sin(\theta) \]
where \(\theta\) is the maximum half-angle of the cone of light that can enter or exit the fiber.

### Step 2: Apply Snell's Law
For the optical fiber, we need to consider the refractive indices of the core (\(n_1\)) and cladding (\(n_2\)). When light travels from air (refractive index \(n_0 = 1\)) into the core of the fiber, we can apply Snell's Law:
\[ n_0 \cdot \sin(i) = n_1 \cdot \sin(r) \]
where \(i\) is the angle of incidence and \(r\) is the angle of refraction.

### Step 3: Total Internal Reflection Condition
For total internal reflection to occur at the core-cladding boundary, the angle of refraction \(r\) must reach \(90^\circ\). Thus, we have:
\[ \sin(r) = 1 \quad \text{(since } r = 90^\circ\text{)} \]
This leads to:
\[ n_1 \cdot \sin(90^\circ) = n_2 \cdot \sin(90^\circ - r) \]
From this, we can derive:
\[ n_2 \cdot \cos(r) = n_1 \]
This implies:
\[ \cos(r) = \frac{n_1}{n_2} \]

### Step 4: Relate Sine and Cosine
Using the identity \(\sin^2(r) + \cos^2(r) = 1\), we can express \(\sin(r)\):
\[ \sin^2(r) = 1 - \cos^2(r) = 1 - \left(\frac{n_1}{n_2}\right)^2 \]
Thus,
\[ \sin(r) = \sqrt{1 - \left(\frac{n_1}{n_2}\right)^2} \]

### Step 5: Substitute into the NA Formula
Now, substituting \(\sin(r)\) into the equation for the numerical aperture:
\[ NA = n_1 \cdot \sin(r) = n_1 \cdot \sqrt{1 - \left(\frac{n_1}{n_2}\right)^2} \]
This gives us the expression for the numerical aperture in terms of the refractive indices.

### Step 6: Final Expression
The final expression for the numerical aperture (NA) of the optical fiber with respect to air is:
\[ NA = \sqrt{n_1^2 - n_2^2} \]

### Conclusion
Thus, the numerical aperture of an optical fiber having core and cladding refractive indices \(n_1\) and \(n_2\) respectively is:
\[ NA = \sqrt{n_1^2 - n_2^2} \]