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 core refractive index \( n_1 \) and cladding refractive index \( n_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Concept**:
The numerical aperture is a measure of the light-gathering ability of the optical fiber. It is defined in terms of the refractive indices of the core and cladding.

2. **Using Snell's Law**:
According to Snell's Law, when light travels from one medium to another, the relationship between the angles of incidence and refraction is given by:
\[ n_1 \sin I = n_2 \sin R \]
where \( I \) is the angle of incidence and \( R \) is the angle of refraction.

3. **Critical Angle**:
The critical angle \( \theta_C \) occurs when light travels from the core to the cladding and is refracted at \( 90^\circ \). At this point:
\[ n_1 \sin I_{max} = n_2 \sin 90^\circ \]
This simplifies to:
\[ n_1 \sin I_{max} = n_2 \]
Therefore, we can express \( \sin I_{max} \) as:
\[ \sin I_{max} = \frac{n_2}{n_1} \]

4. **Finding the Numerical Aperture**:
The numerical aperture (NA) is defined as:
\[ NA = n_1 \sin I_{max} \]
Substituting \( \sin I_{max} \) into this equation gives:
\[ NA = n_1 \left(\frac{n_2}{n_1}\right) = \sqrt{n_1^2 - n_2^2} \]

5. **Final Expression**:
Therefore, the numerical aperture of the optical fiber is given by:
\[ NA = \sqrt{n_1^2 - n_2^2} \]

### Final Answer:
The numerical aperture of an optical fiber with core refractive index \( n_1 \) and cladding refractive index \( n_2 \) is:
\[ NA = \sqrt{n_1^2 - n_2^2} \]