Understanding Numerical Aperture of Optical Fibre

The numerical aperture (NA) of an optical fibre is a measure of its ability to collect light. It is related to the maximum angle at which light rays can enter the fibre and still be guided through it by total internal reflection. A higher numerical aperture means the fibre can accept light over a wider angle, making it easier to couple light into the fibre.

The numerical aperture of a step-index optical fibre is determined by the refractive indices of its core and cladding materials. The core has a higher refractive index ($n_1$) than the cladding ($n_2$).

Calculating Optical Fibre Numerical Aperture

The formula to calculate the numerical aperture (NA) of a step-index optical fibre is given by:

$$NA = \sqrt{n_1^2 - n_2^2}$$

Where:

• $n_1$ is the refractive index of the core material.
• $n_2$ is the refractive index of the cladding material.

Applying Given Values to Numerical Aperture Formula

In this problem, we are given the following values:

• Refractive index of the core material, $n_1 = 1.55$
• Refractive index of the cladding material, $n_2 = 1.50$

Now, we can substitute these values into the numerical aperture formula:

$$NA = \sqrt{(1.55)^2 - (1.50)^2}$$

Step-by-Step Calculation

Let's perform the calculation step by step:

1. Square the refractive index of the core ($n_1$):
   $$(1.55)^2 = 1.55 \times 1.55 = 2.4025$$
2. Square the refractive index of the cladding ($n_2$):
   $$(1.50)^2 = 1.50 \times 1.50 = 2.2500$$
3. Subtract the square of the cladding refractive index from the square of the core refractive index:
   $$n_1^2 - n_2^2 = 2.4025 - 2.2500 = 0.1525$$
4. Take the square root of the result to find the numerical aperture:
   $$NA = \sqrt{0.1525}$$
5. Calculating the square root:
   $$NA \approx 0.3905$$

Resulting Numerical Aperture

The calculated numerical aperture (NA) of the optical fibre is approximately 0.3905.

Rounding to two decimal places, the numerical aperture is 0.39.

Summary of Calculation

Therefore, the numerical aperture of the optical fibre is approximately 0.39.