Multimode vs single mode; NA derivation

Multimode fibre compared with single mode fibre

Advantages of multimode fibre

• Larger core (50–200 µm) and larger NA, so light can be launched from cheap LEDs.
• Easier and cheaper splicing, connectors and alignment.
• Lower total system cost for short links (LANs, buildings, campuses).
• Can be used with low-cost sources at 850 nm.

Disadvantages of multimode fibre

• Large intermodal dispersion, so low bandwidth (MHz·km to about 1 GHz·km).
• Higher attenuation than single mode fibre.
• Short maximum link length; not suitable for long-haul or high-speed WDM systems.

Derivation of NA of a step-index fibre

Consider a step-index fibre with core index $n_1$, cladding index $n_2$ ($n_1 > n_2$), and outside medium index $n_0$ (air, $n_0 = 1$). A meridional ray enters the core end at angle $\theta_0$ to the axis.

        n0  |    cladding n2
   ray \    |__________________
        \th0|   /\phi
 axis ---*--+--/--\----------- core n1
            |th1   \
            |__________________
              cladding n2

Step 1: refraction at the input face. By Snell's law:

$$n_0\sin\theta_0 = n_1\sin\theta_1$$

Step 2: angle at the core–cladding boundary. The refracted ray meets the boundary at angle $\phi$ to the normal. From the right-angled triangle:

$$\phi = 90^\circ - \theta_1 \quad\Rightarrow\quad \sin\theta_1 = \cos\phi$$

Step 3: condition for total internal reflection. The ray is guided only if $\phi \ge \phi_c$, where the critical angle is given by

$$\sin\phi_c = \frac{n_2}{n_1}$$

Step 4: maximum acceptance angle. The largest input angle $\theta_0 = \theta_a$ corresponds to $\phi = \phi_c$:

$$\begin{aligned} n_0\sin\theta_a &= n_1\sin\theta_1 = n_1\cos\phi_c \\ &= n_1\sqrt{1 - \sin^2\phi_c} \\ &= n_1\sqrt{1 - \frac{n_2^2}{n_1^2}} \\ &= \sqrt{n_1^2 - n_2^2} \end{aligned}$$

Step 5: definition of NA.

$$\text{NA} = n_0\sin\theta_a = \sqrt{n_1^2 - n_2^2}$$

For air, $\text{NA} = \sin\theta_a$ and $\theta_a = \sin^{-1}\sqrt{n_1^2 - n_2^2}$.

Step 6: in terms of relative index difference. With $\Delta = \dfrac{n_1^2 - n_2^2}{2n_1^2} \approx \dfrac{n_1 - n_2}{n_1}$ (since $n_1 \approx n_2$):

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

Example: $n_1 = 1.48$, $n_2 = 1.46$:

$$\begin{aligned} \text{NA} &= \sqrt{1.48^2 - 1.46^2} = \sqrt{0.0588} = 0.2425 \\ \theta_a &= \sin^{-1}(0.2425) = 14.03^\circ \end{aligned}$$

NA depends only on the refractive indices, not on the core diameter. The full acceptance cone angle is $2\theta_a$.