Numerical aperture and dispersion

Numerical aperture (NA)

Numerical aperture is the light-gathering ability of a fibre. It is the sine of the acceptance angle $\theta_a$ (the largest angle to the axis at which an incoming ray is still guided by total internal reflection) multiplied by the index of the launching medium:

$$\text{NA} = n_0\sin\theta_a = \sqrt{n_1^2 - n_2^2} \approx n_1\sqrt{2\Delta}, \qquad \Delta = \frac{n_1 - n_2}{n_1}$$

      acceptance cone
   \            cladding n2
    \  ____________________
  ---*<)th_a   core n1 -->
    /  ____________________
   /

• Rays inside the cone of half-angle $\theta_a$ are guided; rays outside it are lost into the cladding.
• A large NA makes it easy to couple light from an LED, but also increases modal dispersion.
• Typical NA: 0.2–0.3 for multimode, about 0.1 for single mode fibre.
• Example: $n_1 = 1.48$, $n_2 = 1.46$ gives NA $= 0.2425$ and $\theta_a = 14.03^\circ$.

Dispersion

Dispersion is the spreading of a light pulse in time as it travels along the fibre. Spread pulses overlap with neighbours (inter-symbol interference), which limits the bit rate and the bandwidth–distance product.

 input  _|^|_          output  _/^^\_  (wider, lower)

Types:

1. Intermodal (modal) dispersion: in multimode fibre, different modes travel different path lengths and arrive at different times. For step-index fibre $\Delta T \approx \dfrac{L n_1 \Delta}{c}$. Reduced by graded-index fibre; absent in single mode fibre.
2. Intramodal (chromatic) dispersion: the source emits a range of wavelengths that travel at different speeds.
   • Material dispersion: refractive index of silica changes with wavelength.
   • Waveguide dispersion: the mode's speed depends on core size relative to wavelength.
   • Chromatic dispersion is near zero at about 1310 nm in standard fibre.
3. Polarization mode dispersion: the two polarizations travel at slightly different speeds due to fibre asymmetry; important at very high bit rates.

Dispersion is reduced by using single mode fibre, narrow-linewidth lasers, operation near 1310 nm, or dispersion-shifted fibre at 1550 nm.