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Magazine Chapter 5: Problem 99
A multimode stepped-index glass fiber has a core index of 1.50 and a cladding index of 1.48 . Given that the core has a radius of \(50.0 \mu \mathrm{m}\) and operates at a vacuum wavelength of \(1300 \mathrm{nm}\), find the number of modes it sustains.
Short Answer
Expert verified
The fiber sustains approximately 1300 modes.
Step by step solution
01
Understanding the Problem
We have a stepped-index multimode fiber with given core and cladding refractive indices, and we need to find how many modes it sustains at a specific wavelength. This involves calculating the V-number (or normalized frequency) of the fiber.
02
Recall the Formula
The formula for the V-number, which determines the number of modes a multimode fiber can support, is given by \( V = \frac{2\pi a}{\lambda} \sqrt{n_1^2 - n_2^2} \), where \( a \) is the core radius, \( \lambda \) is the wavelength in vacuum, \( n_1 \) is the core refractive index, and \( n_2 \) is the cladding refractive index.
03
Convert Units Appropriately
Ensure all units are consistent. Convert the core radius from micrometers to meters: \( a = 50.0 \, \mu \text{m} = 50.0 \times 10^{-6} \, \text{m} \) and the wavelength from nanometers to meters: \( \lambda = 1300 \, \text{nm} = 1300 \times 10^{-9} \, \text{m} \).
04
Calculate the Numerical Aperture
Calculate the numerical aperture (NA) using \( \text{NA} = \sqrt{n_1^2 - n_2^2} \). Substitute \( n_1 = 1.50 \) and \( n_2 = 1.48 \) into the equation: \( \text{NA} = \sqrt{1.50^2 - 1.48^2} = \sqrt{0.0448} \approx 0.2117 \).
05
Calculate the V-number
Substitute the known values into the V-number equation: \[ V = \frac{2\pi \times 50.0 \times 10^{-6}}{1300 \times 10^{-9}} \times 0.2117 \]. Simplify and calculate \( V \approx 51 \).
06
Determine the Number of Modes
For a step-index multimode fiber, the number of modes \( M \) it supports can be approximated by the formula \( M \approx \frac{V^2}{2} \). Substitute the calculated V-number: \( M \approx \frac{51^2}{2} \approx 1300 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
V-number
The V-number, also known as the normalized frequency, is a crucial parameter in fiber optics, especially when analyzing multimode fibers. It essentially determines how many modes a fiber can support.
The V-number is calculated using the formula \( V = \frac{2\pi a}{\lambda} \sqrt{n_1^2 - n_2^2} \), where \( a \) is the core radius, \( \lambda \) is the operating wavelength, \( n_1 \) is the core refractive index, and \( n_2 \) is the cladding refractive index.
The V-number is calculated using the formula \( V = \frac{2\pi a}{\lambda} \sqrt{n_1^2 - n_2^2} \), where \( a \) is the core radius, \( \lambda \) is the operating wavelength, \( n_1 \) is the core refractive index, and \( n_2 \) is the cladding refractive index.
- The V-number provides insight into how light propagates through the fiber.
- For multimode fibers, a higher V-number indicates more modes can be supported.
- If the V-number decreases below a certain threshold, the fiber behaves like a single-mode fiber instead.
numerical aperture
Numerical aperture (NA) is another essential concept in understanding fiber optics. It measures the light-gathering ability of a fiber and influences the number of modes the fiber can support.
The formula for numerical aperture is \( \text{NA} = \sqrt{n_1^2 - n_2^2} \), where \( n_1 \) and \( n_2 \) are the refractive indices of the core and cladding, respectively.
The formula for numerical aperture is \( \text{NA} = \sqrt{n_1^2 - n_2^2} \), where \( n_1 \) and \( n_2 \) are the refractive indices of the core and cladding, respectively.
- A higher NA indicates the fiber can capture more light from a source.
- This property also influences the fiber鈥檚 ability to sustain modes.
- In this exercise, we found the NA to be approximately 0.2117, influencing the calculation of the V-number.
multimode fiber
A multimode fiber is an optical fiber primarily used to carry multiple light signals over relatively short distances due to its larger core size compared to single-mode fibers. This type of fiber allows multiple modes or paths for light to travel.
- The larger core diameter (such as the 50 \( \mu \)m core mentioned in this exercise) makes it easier to couple more light into the fiber.
- It supports a higher number of modes, which is beneficial for applications requiring less precision in light source alignment.
- However, multimode fibers suffer from modal dispersion, where different modes travel at different speeds causing signal distortion.
refractive index
The refractive index is a fundamental property of materials that indicates how light propagates through it. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the material.
In the context of optical fibers:
In the context of optical fibers:
- The core refractive index (e.g., 1.50 in this exercise) influences the light-carrying capacity of the fiber.
- The cladding refractive index (e.g., 1.48) is slightly lower, helping to maintain the light within the core through total internal reflection.
- Refractive index contrast (difference between core and cladding indices) is crucial for determining the fiber's numerical aperture and V-number.
