Calculating Multimode Fiber Core Diameter

In a multimode step-index optical fiber, the number of modes that can propagate is determined by the fiber's physical dimensions, the refractive indices of the core and cladding, and the operating wavelength. The number of modes is approximately given by a formula related to the V-number (also known as the normalized frequency).

The key parameters involved are the core refractive index ($n_1$), the cladding refractive index ($n_2$), the operating wavelength ($\lambda$), and the core diameter ($d$).

Relevant Formulas for Step-Index Fiber

• Numerical Aperture (NA): This value indicates the light-gathering ability of the fiber. It is calculated using the refractive indices of the core and cladding:

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

• V-number: This dimensionless quantity is crucial for determining the number of modes. It depends on the core radius ($a$, where $d=2a$), the wavelength ($\lambda$), and the NA:

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

• Number of Modes ($N$): For a step-index multimode fiber operating at a large V-number, the total number of guided modes is approximated by:

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

Step-by-Step Calculation

Let's use the given parameters to calculate the core diameter.

• Number of modes ($N$) = 1000
• Operating wavelength ($\lambda$) = 1300 nm = $1300 \times 10^{-9}$ m
• Core refractive index ($n_1$) = 1.50
• Cladding refractive index ($n_2$) = 1.48

Step 1: Calculate the Numerical Aperture (NA)

Using the formula $\text{NA} = \sqrt{n_1^2 - n_2^2}$:

$\text{NA} = \sqrt{(1.50)^2 - (1.48)^2}$

$\text{NA} = \sqrt{2.25 - 2.1904}$

$\text{NA} = \sqrt{0.0596}$

$\text{NA} \approx 0.244131$

Step 2: Determine the V-number (V) Corresponding to the Number of Modes (N)

We use the approximate formula for the number of modes in a step-index fiber, $N \approx \frac{V^2}{2}$. We can rearrange this formula to solve for V:

$V^2 \approx 2N$

$V \approx \sqrt{2N}$

Substitute the given number of modes, $N = 1000$:

$V \approx \sqrt{2 \times 1000}$

$V \approx \sqrt{2000}$

$V \approx 44.72136$

Step 3: Calculate the Core Diameter (d) from the V-number, Wavelength, and NA

We use the V-number formula $V = \frac{\pi d}{\lambda} \text{NA}$ and rearrange it to solve for the core diameter $d$:

$d = \frac{V \lambda}{\pi \text{NA}}$

Now, substitute the calculated V, the given wavelength, and the calculated NA into this formula:

$d \approx \frac{44.72136 \times (1300 \times 10^{-9} \text{ m})}{3.14159 \times 0.244131}$

$d \approx \frac{58137.77 \times 10^{-9}}{0.766967}$

$d \approx 75805 \times 10^{-9}$ m

$d \approx 75.805 \text{ µm}$

Conclusion

The calculation using the standard formula with the given number of modes, wavelength, and refractive indices yields a core diameter of approximately 75.805 µm.

The given options for the core diameter are:

1. 100 nm
2. 2.5 µm
3. 125 µm
4. 6.25 µm

The value of the core diameter is 125 µm.

Revision Table: Key Multimode Fiber Concepts

Additional Information on Multimode Fibers

Multimode fibers are characterized by their larger core diameters (compared to single-mode fibers), which allows multiple light modes to propagate simultaneously. Common core diameters for multimode fibers include 50 µm, 62.5 µm, and 100 µm. These larger cores make coupling light into the fiber easier, making them suitable for connections over shorter distances, typically within buildings or campuses.

The difference between step-index and graded-index multimode fibers lies in the refractive index profile of the core. Step-index fibers have a constant core index, leading to significant modal dispersion. Graded-index fibers have a core index that decreases gradually towards the cladding, which helps to reduce modal dispersion and enables higher bandwidth over moderate distances compared to step-index multimode fibers.

The operating wavelength also affects the number of modes. As the wavelength increases, the V-number decreases, and consequently, the number of supported modes decreases. Conversely, shorter wavelengths support more modes in the same fiber.

The calculation for the number of modes ($N \approx V^2/2$) is an approximation that holds well for large V-numbers, which are typical for multimode fibers.