V-Number Calculation for Optical Fiber

Calculate the V-number for a multimode step-index fiber using the formula and given parameters.

Given Fiber Parameters:

• Core radius, $a$ = 25 $\mu$m = $25 \times 10^{-6}$ m
• Core refractive index, $n_1$ = 1.48
• Index difference, $\Delta$ = 0.01
• Operating wavelength, $\lambda$ = 840 nm = $840 \times 10^{-9}$ m

Calculate Cladding Index ($n_2$):

Using the relation $n_2 = n_1 (1 - \Delta)$:

$n_2 = 1.48 \times (1 - 0.01) = 1.48 \times 0.99 = 1.4652$

V-Number Calculation Steps:

The V-number formula is $V = \frac{2 \pi a}{\lambda} \sqrt{n_1^2 - n_2^2}$.

Substitute the given and calculated values:

$V = \frac{2 \pi (25 \times 10^{-6} \, \text{m})}{840 \times 10^{-9} \, \text{m}} \sqrt{(1.48)^2 - (1.4652)^2}$

Calculate the difference of squares:

$(1.48)^2 - (1.4652)^2 = 2.1904 - 2.14681504 = 0.04358496$

Calculate the V-number:

$V = \frac{2 \pi \times 25}{840} \times 10^3 \sqrt{0.04358496}$

$V = 0.18700 \times 1000 \times 0.20877$

$V \approx 39.08$

The resulting V-number is approximately 39.