Question

A silicon optical fiber has a core refractive index of \(1.50\) and a cladding refractive index \(1.47\) what is the numerical aperture of the fiber? (A) \(0.90\) (B) \(0.60\) (C) \(0.45\) (D) \(0.30\)

Short answer

The numerical aperture of the silicon optical fiber with core refractive index 1.50 and cladding refractive index 1.47 is approximately \(0.30\).

Step-by-step solution

Write down the given values

We are given the core refractive index (n1) and the cladding refractive index (n2) of a silicon optical fiber. \[n1 = 1.50\] \[n2 = 1.47\]

Calculate the squared difference between n1 and n2

We first need to find the squared difference between the core refractive index and the cladding refractive index. \[(n1^2 - n2^2) = (1.50^2 - 1.47^2) = (2.25 - 2.1609)\]

Calculate the numerical aperture (NA)

Now, apply the formula for the numerical aperture (NA) \[NA = \sqrt{n1^2 - n2^2} \] Use the squared difference calculated in Step 2: \[NA = \sqrt{2.25 - 2.1609} = \sqrt{0.0891}\] Now, we find the square root of 0.0891: \[NA \approx 0.2985\]

Choose the correct answer

The calculated numerical aperture (NA) is approximately 0.2985, which is closest to the given option (D) \(0.30\). Hence, the correct answer is: (D) \(0.30\)

Key concepts

Refractive Index

The refractive index, often symbolized as \(n\), is a fundamental concept in optics. It represents how much light bends, or refracts, when entering a material. The refractive index is calculated as the ratio of the speed of light in a vacuum to the speed of light in a medium. For example, if light travels slower in a material compared to a vacuum, the refractive index is greater than one.
Understanding the refractive index is crucial when studying optical fibers because it determines how efficiently light can travel through the fiber core. In the exercise, the refractive index values for the core and cladding were given as 1.50 and 1.47, respectively. This slight difference in values is what allows light to be guided along the fiber via a phenomenon known as total internal reflection.

Optical Fiber

Optical fibers are thin, flexible strands of glass or plastic that are used to transmit light signals over long distances. They are essential in telecommunications, medicine, and many other fields because they can carry large amounts of data swiftly and without electromagnetic interference.
Optical fibers consist of a core and a cladding. The core is where the light travels, and it is made of a material with a higher refractive index than the cladding. This configuration facilitates the reflection of light within the core, allowing it to travel long distances with minimal loss.
The performance of an optical fiber depends significantly on its numerical aperture (NA), which is a measure of the fiber's ability to collect light. Understanding these components is crucial for utilizing optical fibers in many technological applications.

Silicon Fiber

Silicon fibers are a type of optical fiber known for their robustness and ability to withstand harsh environments. These fibers are particularly suitable for applications that involve high temperatures or corrosive environments. Silicon has a high melting point and excellent mechanical strength, making it ideal for specific industrial applications.
The refractive index of silicon is typically higher than other materials, which is advantageous for maintaining a strong numerical aperture (NA). This feature allows silicon fibers to transmit light signals with high efficiency, ensuring that data can be relayed quickly and with little loss over long distances.
Given their unique properties, silicon fibers play a critical role in environments where traditional optical fibers might fail.

Core and Cladding

The core and cladding are two essential components of an optical fiber. The core is the central part through which light travels. It usually has a higher refractive index than the cladding, which surrounds the core. This difference in refractive index values is critical for achieving total internal reflection.
Total internal reflection occurs when light traveling through the core hits the core-cladding boundary at an angle greater than the critical angle. Because of the refractive index contrast, the light is reflected back into the core rather than escaping through the cladding.
Our original exercise highlighted the importance of this concept with a core refractive index of 1.50 and a cladding refractive index of 1.47. This slight discrepancy ensures light travels efficiently through the fiber. By understanding the purposes and variations of the core and cladding, one can optimize fiber optic designs for specific applications.