Fiber Optics and Numerical Aperture

Optical fibers transmit light over long distances with minimal loss by exploiting total internal reflection. They are the backbone of modern telecommunications, carrying internet traffic across oceans.

Structure of an Optical Fiber

An optical fiber has two concentric layers:

• Core: higher refractive index $n_1$. Light propagates here.
• Cladding: lower refractive index $n_2 < n_1$. Surrounds the core.

Because $n_1 > n_2$, light hitting the core-cladding interface beyond the critical angle undergoes total internal reflection and remains confined in the core.

Numerical Aperture (NA)

The numerical aperture characterizes how much light a fiber can accept. It depends on the difference in refractive indices:

$\text{NA} = \sqrt{n_1^2 - n_2^2}$

A larger NA means the fiber accepts light from a wider cone of angles.

Example: Core $n_1 = 1.5$, cladding $n_2 = 1.4$:

$\text{NA} = \sqrt{1.5^2 - 1.4^2} = \sqrt{2.25 - 1.96} = \sqrt{0.29} \approx 0.539$

Acceptance Angle

The acceptance angle $\theta_a$ is the half-angle of the cone of light that will be accepted into the fiber (measured in air outside the fiber):

$\sin\theta_a = \text{NA} \quad \Rightarrow \quad \theta_a = \arcsin(\text{NA})$

The full acceptance cone has a full angle of $2\theta_a$.

Example: For the fiber above, $\theta_a = \arcsin(0.539) \approx 32.6°$.

Single-Mode vs Multi-Mode Fibers

• Single-mode fiber: very thin core (~9 µm), $n_1 \approx n_2$, NA ≈ 0.12, carries one mode of light. Used for long-distance telecommunications.
• Multi-mode fiber: thicker core (~50–62.5 µm), NA ≈ 0.2–0.5, carries many modes. Used for short distances (data centers, buildings).

Your Task

Implement the numerical aperture and acceptance angle using math.sqrt and math.asin.