Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

An optical fibre is a dielectric rod: a glass core of index $n_{1}$ and radius $a$ , surrounded by a glass cladding of slightly lower index $n_{2}$ and protected by a polymer coating (Figure 7.6.9). In a step-index fibre the index jumps from $n_{1}$ to $n_{2}$ at the core boundary. Light is guided in the core by total internal reflection at that boundary, exactly as in the slab, and the evanescent tail runs in the cladding.

The acceptance cone

Which rays launched from the air into the flat end face are trapped? A ray entering at the angle $θ_{a}$ to the axis is refracted to the angle $θ_{b}$ with $sin θ_{a} = n_{1} sin θ_{b}$ (Snell's law with $n_{0} = 1$ ). It then meets the core boundary at the angle $9 0^{\circ} - θ_{b}$ from the normal, and it is totally reflected if $9 0^{\circ} - θ_{b} \geq θ_{c}$ . The largest accepted angle therefore belongs to $θ_{b} = 9 0^{\circ} - θ_{c}$ :

sin θ_{a} = n_{1} sin (9 0^{\circ} - θ_{c}) = n_{1} cos θ_{c} = n_{1} 1 - \frac{n _{2}^{2}}{n _{1}^{2}} = n_{1}^{2} - n_{2}^{2} .

Numerical aperture of a step-index fibre

NA Δ = sin θ_{a} = n_{1}^{2} - n_{2}^{2} = n_{1} 2Δ, = \frac{n _{1}^{2} - n _{2}^{2}}{2 n _{1}^{2}} \approx \frac{n _{1} - n _{2}}{n _{1}} . (7.6.34)

Rays inside the acceptance cone of half-angle $θ_{a}$ are trapped. The normalised frequency is that of the slab with the core radius in place of the half-thickness:

V = k_{0} a NA = \frac{2 π a}{λ _{0}} NA . (7.6.35)

In plain words

The numerical aperture is the sine of the widest angle at which light can be thrown into the fibre and still be held. It depends only on the two indices. $V$ adds the size of the core in wavelengths and decides how many modes the fibre carries.

The same construction is Example 7.5.5 of Orfanidis (Electromagnetic Waves and Antennas), which also introduces the name numerical aperture.

The step-index fibre. Left: cross-section (not to scale) with the core of radius a, the cladding and the coating. Right: a ray entering the end face within the acceptance cone of half-angle \theta_a (shaded) meets the core boundary beyond the critical angle \theta_c and is trapped. — **Figure 7.6.9.** The step-index fibre. Left: cross-section (not to scale) with the core of radius $a$ , the cladding and the coating. Right: a ray entering the end face within the acceptance cone of half-angle $θ_{a}$ (shaded) meets the core boundary beyond the critical angle $θ_{c}$ and is trapped.

Glass fibres have $Δ ≪ 1$ : the index of the core is raised only slightly, by doping the silica with germanium. The critical angle is then close to $9 0^{\circ}$ and the guided rays run nearly parallel to the axis. Such fibres are called weakly guiding. Table 7.6.1 lists the three common types. All glass fibres have a cladding of $125 μ m$ diameter, many decay lengths of the evanescent tail.

Table 7.6.1. Typical step- and graded-index fibres.

Fibre	core $2 a$ [µm]	$Δ$ , NA	Use
single-mode silica	$8$ – $9$	$0.35 %$ , $0.12$	long-haul and access links
multimode, graded index	$50$	$1 %$ , $0.20$	data centres, buildings
plastic (PMMA)	$980$	$6 %$ , $0.5$	cars, home networks

Quick check

A silica fibre has $n_{1} = 1.45$ and $Δ = 0.35 %$ . Find NA and the acceptance angle.

$NA = n_{1} 2Δ = 1.45 0.0070 = 0.121$ and $θ_{a} = arcsin 0.121 = 6.9 7^{\circ}$ : a narrow cone, which is why light is launched into such fibres with lasers and lenses.

7.6.10 The step-index optical fibre

The acceptance cone