Q21Engineering Physics

Optical fibers are the backbone of the modern global telecommunications infrastructure, capable of transmitting vast amounts of digital data using pulses of light over transoceanic distances with near-zero loss. To utilize them effectively, engineers must understand exactly how they trap light and how much light they can physically accept from a source.

1. The Principle of Propagation (Total Internal Reflection)

An optical fiber is fundamentally constructed as a highly pure, solid cylindrical glass 'core' (with a specific refractive index $n_1$) entirely encased by an outer 'cladding' layer of slightly different glass (with a strictly lower refractive index $n_2$). The entire operational principle relies on the optical phenomenon of Total Internal Reflection (TIR).

When light is injected into the core and travels down the fiber, it eventually hits the boundary interface between the denser core and the rarer cladding. According to Snell's Law, light traveling from a denser to a rarer medium bends away from the normal. If the light strikes the interface at a shallow angle—specifically, an angle of incidence greater than a critical threshold known as the critical angle (${\theta }_c={\sin }^{-1}(n_2/n_1)$)—it cannot refract into the cladding. The light is perfectly, $100\%$ reflected back into the core, acting as if the interface were a perfect mirror. This continuous bouncing confines the light entirely within the core, allowing it to propagate for hundreds of kilometers around bends and curves without escaping.

2. Numerical Aperture and Acceptance Angle Theory

To ensure TIR occurs inside the fiber, the light must be injected into the front face of the fiber at an appropriate angle. The Acceptance Angle (${\theta }_a$) defines the maximum possible angle (measured relative to the central longitudinal axis of the fiber) at which an incoming ray of light can hit the fiber face and still successfully undergo TIR inside. Any light entering at an angle larger than ${\theta }_a$ will strike the internal boundary at less than the critical angle, refract into the cladding, and be permanently lost (radiation loss).

The light-gathering capacity of the fiber is quantified by its Numerical Aperture (NA), which is an intrinsic design property of the core/cladding indices ($\text{NA}=\sqrt{n_1^2-n_2^2}$). The absolute relationship between the acceptance angle, the Numerical Aperture, and the refractive index of the surrounding external medium ($n_0$) in which the fiber is placed is given by the equation:

3. Step-by-Step Numerical Calculation

We are tasked with finding the exact acceptance angle of a specific fiber when it is completely submerged underwater. We must carefully identify the parameters:

• Numerical Aperture of the fiber ($\text{NA}$) = $0.20$
• Refractive Index of the external medium (water, $n_0$) = $1.33$

We rearrange the acceptance angle formula to solve for the angle ${\theta }_a$:

Substitute the given numerical values into the equation:

Perform the division:

To find the physical angle in degrees, we must take the inverse sine ($\arcsin$) of this value using a scientific calculator:

This calculation proves that immersing an optical fiber in a denser fluid like water (compared to air where $n_0=1$) drastically reduces its acceptance angle. In air, the angle would be $\sim 11.5^{\circ }$, but in water, the "cone of acceptance" shrinks to a mere $8.65^{\circ }$, requiring the light source to be aligned much more precisely.