Calculating Normalized Frequency in Multimode Step-Index Fiber

The question asks us to calculate the normalized frequency (also known as the V-number) for a multimode step-index optical fiber given its physical parameters. The normalized frequency is a crucial parameter in optical fiber characterization, particularly for determining the number of modes a fiber can support and understanding propagation characteristics.

Understanding Normalized Frequency (V-number)

The normalized frequency, or V-number, is a dimensionless quantity that describes the electromagnetic waveguiding properties of an optical fiber. For a step-index fiber, it combines the effects of the core radius, operating wavelength, and the refractive indices of the core and cladding. A higher V-number generally indicates that the fiber can support a larger number of propagating modes.

Key Formulas

The formula for the normalized frequency ($V$) of a step-index fiber is:

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

where:

• $a$ is the radius of the core.
• $\lambda$ is the operating wavelength.
• $\text{NA}$ is the numerical aperture of the fiber.

The numerical aperture (NA) is related to the core refractive index ($n_1$) and the relative index difference ($\Delta$) by the formula:

$$\text{NA} = n_1 \sqrt{2\Delta}$$

Given Parameters

From the question, we are given:

• Core diameter ($2a$) = 80 μm. So, the core radius ($a$) = 80 μm / 2 = 40 μm.
• Relative index difference ($\Delta$) = 1.5% = 0.015.
• Operating wavelength ($\lambda$) = 0.85 μm.
• Core refractive index ($n_1$) = 1.48.

Step-by-Step Calculation of Normalized Frequency

Step 1: Calculate the Numerical Aperture (NA)

Using the formula $\text{NA} = n_1 \sqrt{2\Delta}$:

$$\text{NA} = 1.48 \sqrt{2 \times 0.015}$$ $$\text{NA} = 1.48 \sqrt{0.03}$$ $$\text{NA} \approx 1.48 \times 0.173205$$ $$\text{NA} \approx 0.25634$$

Step 2: Calculate the Normalized Frequency (V)

Using the formula $V = \frac{2\pi a}{\lambda} \text{NA}$:

Substitute the values $a = 40$ μm, $\lambda = 0.85$ μm, and $\text{NA} \approx 0.25634$:

$$V = \frac{2\pi \times 40 \text{ μm}}{0.85 \text{ μm}} \times 0.25634$$ $$V = \frac{80\pi}{0.85} \times 0.25634$$ $$V \approx 295.09 \times 0.25634$$ $$V \approx 75.635$$

Rounding the calculated value to one decimal place gives approximately 75.6. Comparing this with the given options, the closest value is 75.8.

Comparing with Options

Our calculated normalized frequency is approximately 75.6. Let's look at the options provided:

The calculated value of 75.635 is very close to 75.8. Small differences can arise due to rounding during intermediate steps or in the provided data/options. Based on standard calculation methods, 75.8 is the correct choice among the given options.

Conclusion on Normalized Frequency

The normalized frequency for the given multimode step-index fiber operating at 0.85 μm is approximately 75.8. This high V-number confirms that the fiber is indeed multimode, as multimode fibers typically have V-numbers much greater than 2.405 (the cutoff V-number for the fundamental mode).

Revision Table: Fiber Optics Parameters

Additional Information on Multimode Step-Index Fiber and V-number

Multimode Fiber: Multimode fibers have a large core diameter (typically 50 to 100 μm) compared to single-mode fibers (typically 8 to 10 μm). This large core allows multiple modes (paths) for light rays to travel along the fiber. While this makes coupling light into the fiber easier and allows the use of less expensive light sources like LEDs, it also leads to modal dispersion, where different modes arrive at the destination at slightly different times, limiting the fiber's bandwidth and transmission distance.

Step-Index Fiber: In a step-index fiber, the refractive index of the core is constant, and there is an abrupt change (a 'step') to a lower refractive index in the cladding. This simple refractive index profile contributes to modal dispersion in multimode step-index fibers.

Significance of the V-number: The V-number is a key characteristic for classifying optical fibers and predicting their behavior.

• For a step-index fiber, if $V < 2.405$, the fiber supports only the fundamental mode and is considered a single-mode fiber (SMF).
• If $V \ge 2.405$, the fiber is multimode (MMF) and can support multiple modes. The approximate number of guided modes ($M$) in a multimode step-index fiber is given by: $$M \approx \frac{V^2}{2}$$

In our case, with $V \approx 75.8$, the estimated number of modes would be approximately $(75.8)^2 / 2 \approx 5745 / 2 \approx 2872$, confirming it is indeed a multimode fiber supporting a large number of modes.