Solving for Guided Modes in Step-Index Optical Fiber

This question asks us to determine the total number of guided modes that can travel through a multi-mode step-index fiber, given its normalized frequency.

Understanding Guided Modes and Normalized Frequency

In optical fibers, guided modes refer to the distinct pathways or patterns in which light can propagate along the fiber core. A multi-mode fiber allows multiple modes to propagate simultaneously, unlike single-mode fibers.

The number of modes a fiber can support is determined by its physical characteristics and the wavelength of light. A key parameter used to characterize this is the normalized frequency (V), also known as the V-number. It is defined as:

$V = \frac{2 \pi a \cdot \text{NA}}{\lambda}$

Where:

• a is the radius of the fiber core
• NA is the numerical aperture of the fiber
• \(\lambda\) is the wavelength of the light

While the core diameter is given as 80 mm, it's important to note that the calculation for the number of modes directly uses the provided normalized frequency (V).

Calculating the Number of Guided Modes

For a step-index fiber (where the refractive index changes abruptly at the core-cladding boundary), the total number of guided modes (N) can be approximated by the following formula, especially for large values of V (typical for multi-mode fibers):

$N \approx \frac{V^2}{2}$

Step-by-Step Calculation

1. Identify Given Values: The normalized frequency is provided as $V = 72$.
2. Apply the Mode Formula: Input the value of V into the approximation formula:

   $N \approx \frac{(72)^2}{2}$

3. Calculate \(V^2\): First, calculate the square of the normalized frequency:

   $72^2 = 72 \times 72 = 5184$

4. Calculate N: Now, divide the result by 2 to find the approximate number of modes:

   $N \approx \frac{5184}{2}$

   $N \approx 2592$

Therefore, the total number of guided modes in the specified multi-mode step-index fiber is approximately 2592.

Conclusion

The calculation based on the given normalized frequency ($V = 72$) yields 2592 guided modes. This matches one of the provided options.