Question

Optical fibers are constructed with a cylindrical core surrounded by a sheath of cladding material. Common materials used are pure silica $n_2=1.4502$ for the cladding and silica doped with germanium $n_1=1.4652$ for the core. (a) What is the critical angle ${\theta }_{crit}$ for light traveling in the core and reflecting at the interface with the cladding material? (b) The numerical aperture (NA) is defined as the angle of incidence $thet{a}_i$ at the flat end of the cable for which light is incident on the core-cladding interface at angle ${\theta }_{crit}$ ($\mathbf{\text{Fig. P33.46}}$). Show that sin ${\theta }_i$ =$\sqrt{n_1^2-n_2^2}$ . (c) What is the value of ${\theta }_i$ for $n_1$ = 1.465 and $n_2$ = 1.450?

Short answer

(a) ${\theta }_{crit}\approx {80.57}^{\circ }$; (b) $\sin ({\theta }_i)=\sqrt{n_1^2-n_2^2}$; (c) ${\theta }_i\approx {14.02}^{\circ }$.

Step-by-step solution

Understand Refraction and Critical Angle

To find the critical angle ${\theta }_{crit}$, we use the concept of total internal reflection that occurs when light moves from a medium with a higher refractive index to one with a lower refractive index. The critical angle is the angle of incidence in the denser medium (core) for which the angle of refraction is 90 degrees in the less dense medium (cladding).

Apply Snell's Law

Snell's Law is given by $n_1\sin ({\theta }_1)=n_2\sin ({\theta }_2)$. For the critical angle, we set ${\theta }_2={90}^{\circ }$, making $\sin ({\theta }_2)=1$, then the formula becomes $n_1\sin ({\theta }_{crit})=n_2\cdot 1$.

Solve for the Critical Angle

Rearrange Snell's Law to solve for ${\theta }_{crit}$: $$\sin ({\theta }_{crit})=\frac{n_2}{n_1}$$. Substituting the given values $n_1=1.4652$ and $n_2=1.4502$, we have $$\sin ({\theta }_{crit})=\frac{1.4502}{1.4652}\approx 0.9897$$. Thus, ${\theta }_{crit}=\arcsin (0.9897)\approx {80.57}^{\circ }$.

Understand Numerical Aperture (NA)

The Numerical Aperture (NA) of an optical fiber describes the range of angles at which the fiber can accept light. It relates to the critical angle by connecting angles of incidence outside and inside the fiber.

Derive the Formula for Numerical Aperture

By definition, NA is $\sin ({\theta }_i)$ for which light refracts to ${\theta }_{crit}$ at the core-cladding interface. Using our understanding of $n_1\sin ({\theta }_{crit})=n_2$, the largest ${\theta }_i$ occurs when $\sin ({\theta }_i)=\sqrt{n_1^2-n_2^2}$.

Calculate the Incidence Angle ${\theta }_i$

Using the formula $\sin ({\theta }_i)=\sqrt{n_1^2-n_2^2}$, we plug in the given $n_1=1.465$ and $n_2=1.450$. The calculation yields $$\sin ({\theta }_i)=\sqrt{{1.465}^2-{1.450}^2}\approx 0.242$$. Therefore, ${\theta }_i=\arcsin (0.242)\approx {14.02}^{\circ }$.

Key concepts

Total Internal Reflection

Total internal reflection is a fascinating phenomenon that occurs in optical fibers. It happens when light travels from a medium with a higher refractive index, like the fiber core, to a medium with a lower refractive index, such as the cladding.
The light is completely reflected back into the core if the angle of incidence exceeds a certain value, the critical angle. This is important because it allows light to be transmitted over long distances with minimal loss. Optical fibers take advantage of this property to efficiently guide light along their length.
Effects of total internal reflection are extensively used in telecommunications and medical instruments, among other applications. These include transmitting internet signals or creating endoscopes that help doctors look inside the human body.

Critical Angle

The critical angle is the minimum angle of incidence at which light is entirely reflected within a denser medium. When light reaches this angle, it transitions from traveling through the core to being confined within it. This means no refraction occurs beyond the critical angle.
The critical angle depends on the refractive indices of the two materials involved. For the fiber, using Snell's Law, it can be calculated as $\arcsin \left(\frac{n_2}{n_1}\right)$.
For instance, in an optical fiber where the refractive index of the core $n_1=1.4652$ and the cladding $n_2=1.4502$, the critical angle is about ${80.57}^{\circ }$.
This precise condition helps in ensuring that light continues to propagate within the fiber, which is crucial for efficient signal transmission.

Numerical Aperture

Numerical Aperture (NA) is a measure of how much light can be captured and transmitted through an optical fiber. It indicates the size of the cone of light that enters the fiber.
The NA is based upon the critical angle and is calculated using the indices of refraction of both the core and the cladding. The NA formula is $\sin ({\theta }_i)=\sqrt{n_1^2-n_2^2}$, where ${\theta }_i$ is the maximum angle of light entering the fiber that results in total internal reflection.
For example, with core and cladding refractive indices of 1.465 and 1.450, the NA gives us insight into how effectively the fiber can gather light, and we find $\sin ({\theta }_i)$ to be approximately 0.242.
This value is vital for ensuring that the fiber captures the maximum amount of light, maintaining strong signal integrity.

Snell's Law

Snell's Law is fundamental in understanding how light travels through different media. It provides the relationship between the angle of incidence and the angle of refraction when light passes from one medium to another.
The formula is $n_1\sin ({\theta }_1)=n_2\sin ({\theta }_2)$, where $n_1$ and $n_2$ are the refractive indices of the two media, ${\theta }_1$ is the angle of incidence, and ${\theta }_2$ is the angle of refraction.
When applied to finding the critical angle, Snell's Law helps determine the angle beyond which no refraction occurs. This reflects the principle of total internal reflection, a vital aspect of optical fiber technology.
Understanding Snell's Law allows us to predict and manipulate how light behaves when entering different materials, which is essential for designing efficient fiber optics systems.