Question

A silicon optical fiber with a care diameter large enough has a core refractive index of \(1.50\) and cladding refractive index 1.47. The critical angle at the core cladding interface is (A) \(87.5^{\circ}\) (B) \(78.5^{\circ}\) (C) \(38.5^{\circ}\) (D) \(45^{\circ}\)

Short answer

The critical angle at the core-cladding interface of a silicon optical fiber with a core refractive index of 1.50 and cladding refractive index of 1.47 is approximately \(78.5^{\circ}\) (B).

Step-by-step solution

Snell's Law and Total Internal Reflection

Snell's law relates the angles of incidence and refraction to the refractive indices of two mediums. Assuming the angle of incidence is the critical angle (θc), we can write Snell's law as: \[n1 \cdot \sin{\theta_{c}} = n2\] Where \(n1\) is the refractive index of the core, \(n2\) is the refractive index of the cladding, and \(\theta_{c}\) is the critical angle. For total internal reflection to occur, the angle of refraction (∅) should be 90 degrees or π/2 radians. In this case, Snell's law can be rewritten as: \[\sin{\theta_{c}} = \frac{n2}{n1}\] Now we have the formula to find the critical angle. We will use the given refractive indices of the core and cladding to find the critical angle.

Calculate the sin of the critical angle

Using the given values of the refractive indices, \(n1 = 1.50\) and \(n2 = 1.47\), we can find the sin of the critical angle: \[\sin{\theta_{c}} = \frac{1.47}{1.50} \approx 0.98\]

Calculate the critical angle

To find the critical angle, we will use the inverse sine function on the sin value of the critical angle: \[\theta_{c} = \arcsin{0.98} \approx 78.5^{\circ}\] Thus, the critical angle at the core-cladding interface is: (A) \(87.5^{\circ}\) (B) \(78.5^{\circ}\) (C) \(38.5^{\circ}\) (D) \(45^{\circ}\)

Identify the answer

Comparing our calculated critical angle to the answer choices given, we find that the correct answer is: (B) \(78.5^{\circ}\)

Key concepts

Snell's Law

Snell's Law is a fundamental principle governing the behavior of light as it travels across the boundary between two different media. It helps in calculating how much light bends or refracts. The law states that the relationship between the angle of incidence \( \theta_{i} \) and the angle of refraction \( \theta_{r} \) is given by:\[ n_1 \cdot \sin{\theta_{i}} = n_2 \cdot \sin{\theta_{r}} \]where \( n_1 \) and \( n_2 \) are the refractive indices of the two media respectively. In the context of optical fibers, Snell's Law explains how light enters the fiber and is guided along it. Understanding this law is crucial when examining how light interacts with fiber optic materials and why it gets confined within the core.

Refractive Index

The refractive index is a measure of how much a medium can bend light rays passing through it. It is denoted by \(n\) and indicates how much the speed of light is reduced inside the material.

• If the refractive index is higher, the medium is denser, and light slows down more.
• Conversely, a lower refractive index means the medium is less dense, and light travels faster.

In optical fibers, the core and cladding have different refractive indices (for example, 1.50 for core and 1.47 for cladding in the given problem). This difference leads to the bending of light at the interface, making the conditions for total internal reflection possible. The difference in refractive indices is essential for the functioning of optical fibers.

Critical Angle

The critical angle is the specific angle of incidence above which light is not refracted out but instead reflected back into the medium. It is crucial for the phenomenon of total internal reflection to occur. Mathematically, it is represented by:\[ \sin{\theta_{c}} = \frac{n_2}{n_1} \]where \( n_1 \) is the refractive index of the core and \( n_2 \) is the refractive index of the cladding. To calculate the critical angle \( \theta_{c} \), we take the inverse sine (\(\arcsin\)) of the result. This ensures that light remains within the optical fiber, continually bouncing back within the core rather than refracting out of the fiber. In the provided exercise, this calculation results in \( 78.5^{\circ} \), meaning this is the angle at which all the light reflects internally.

Total Internal Reflection

Total internal reflection is a phenomenon where all the light striking the boundary between two media is reflected back into the original medium, rather than passing through the boundary. This happens when:

• The light is in the denser medium and hits the boundary to a less dense medium.
• The angle of incidence is greater than the critical angle.

In the operation of optical fibers, total internal reflection is essential for guiding light along lengths of fiber without significant loss of signal. Light that stays inside continually reflects at the core-cladding boundary, following the path of the fiber. This allows optical signals to travel over long distances with minimal attenuation, making fiber-optic cables perfect for high-speed data transmission.