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

In a multimode fibre each mode, in the ray picture each ray angle, has its own delay. All rays travel at the same speed $c / n_{1}$ in a step-index core, but along paths of different length. The axial ray covers the length $L$ of fibre in $L n_{1} / c$ . The ray at the critical angle runs the path $L / sin θ_{c} = L n_{1} / n_{2}$ and needs $L n_{1}^{2} / (c n_{2})$ . A short pulse that excites all modes is therefore spread over

Modal dispersion of a step-index fibre

Δ T = \frac{L n _{1}}{c} (\frac{1}{sin θ _{c}} - 1) = \frac{L n _{1}}{c} \frac{n _{1} - n _{2}}{n _{2}} \approx \frac{L n _{1} Δ}{c} . (7.6.39)

Consecutive bits stay apart if $Δ T < 1/ B$ ( $B$ the bit rate), which limits the product of the bit rate and the length:

B L < \frac{c n _{2}}{n _{1}^{2} Δ} (step index) . (7.6.40)

In plain words

In a step-index fibre the steep rays take the long way round and arrive late. The spread grows with the length and with the index step, so a multimode fibre is either short or slow.

Common mistake · The slow ray is not slower

Saying that the higher modes "travel more slowly". In the ray picture of a step-index core every ray travels at $c / n_{1}$ . The steep ray is late only because its path is longer by the factor $1/ sin θ$ . In the mode picture the same fact appears as a group velocity that differs from mode to mode.

Graded index

The remedy for multimode fibres is a core whose index falls gradually from $n_{1}$ on the axis to $n_{2}$ at the core boundary (Figure 7.6.11). The long, steep paths now run through the outer core, where the index is low and the light is fast, and the short axial path runs where the light is slow. With the optimum profile, very nearly parabolic, the delays become almost equal:

Δ T \approx \frac{L n _{1} Δ ^{2}}{8 c} (optimum graded index), (7.6.41)

which is $8/Δ$ times smaller than Eq. (7.6.39): for $Δ = 1 %$ , eight hundred times.

Chromatic dispersion

A single-mode fibre has no modal dispersion, but the source is never perfectly monochromatic. Its spectral width $Δ λ$ contains components that travel at slightly different group velocities, because the index of silica, and with it the mode, depends on the wavelength. The pulse spreads by

Δ τ = ∣ D ∣ L Δ λ, (7.6.42)

where the dispersion parameter $D$ of standard fibre is close to zero at $1310 nm$ and about $17 ps/ (nm km)$ at $1550 nm$ .

For the step-index fibre of Problem 7.6.4 ( $50 μ m$ core, $n_{1} = 1.48$ , $n_{2} = 1.46$ ), Eq. (7.6.39) gives $Δ T = 68 ns/km$ and $B L \approx 15 Mb/s km$ . A graded-index fibre with the same indices has $Δ T = 0.11 ns/km$ and $B L \approx 9 Gb/s km$ .

Modal dispersion. Top: in a step-index core the axial ray (green) and a steep ray (dark red) travel at the same speed along paths of different length, and a short input pulse comes out spread by \Delta T (right). Bottom: in a graded-index core the rays bend back smoothly, and the long paths run where the index is low and the light is fast, so all rays arrive almost together. — **Figure 7.6.11.** Modal dispersion. Top: in a step-index core the axial ray (green) and a steep ray (dark red) travel at the same speed along paths of different length, and a short input pulse comes out spread by $Δ T$ (right). Bottom: in a graded-index core the rays bend back smoothly, and the long paths run where the index is low and the light is fast, so all rays arrive almost together.

Application 7.6.1 · Optical-fibre communication

Nearly all long-distance data, the internet backbone, undersea cables and mobile backhaul, travel as light in single-mode silica fibre. Two properties of the glass decide where in the spectrum. Silica scatters light by Rayleigh scattering on frozen density fluctuations, which falls as $λ_{0}^{- 4}$ , and it absorbs in the infrared, beyond about $1.6 μ m$ , by vibrations of the Si–O bond. Between the two lie the low-loss windows (Figure 7.6.12 and Table 7.6.2).

Table 7.6.2. The three windows of silica fibre (typical values).

Window	loss [dB/km]	$D$ [ps/(nm km)]	Use
$850$ nm	$\approx 2$	large	buildings (multimode)
$1310$ nm	$0.33$	$\approx 0$	access and metro links
$1550$ nm	$0.19$	$17$	long haul

Single-mode against multimode. Single-mode fibre ( $2 a \approx 9 μ m$ , $V < 2.405$ ) has no modal dispersion. It carries $10$ to $400 Gb/s$ per wavelength, dozens of wavelengths per fibre (wavelength-division multiplexing, WDM), and erbium-doped fibre amplifiers (EDFA, $1530$ to $1565 nm$ ) restore the signal every $80$ to $100 km$ . Multimode graded-index fibre ( $50 μ m$ core) accepts cheap $850 nm$ surface-emitting lasers and easy connectors. Its modal dispersion limits it to a few hundred metres at $10 Gb/s$ , which is enough inside a data centre. The link budget is written in decibels, where every loss is subtracted:

P_{rx} [dBm] = P_{tx} [dBm] - α L - splices - connectors \geq receiver sensitivity . (7.6.43)

(b) A two-span link. — (a) Loss of silica fibre (model).

Worked example 7.6.1 puts numbers on an $80 km$ span. It shows the trade-off that shapes every link: the loss favours $1550 nm$ , the dispersion favours $1310 nm$ , and long-haul systems take $1550 nm$ and manage the dispersion.

Application 7.6.2 · Dielectric resonators

A short cylinder, a "puck", of low-loss ceramic with $ε_{r} = 30$ to $80$ is a piece of dielectric waveguide that reflects back on itself. At its faces the index ratio is $n_{1} / n_{2} = ε_{r} \approx 6$ , so a wave inside meets them far beyond the critical angle ( $θ_{c} \approx 1 0^{\circ}$ ) and is trapped, as in the slab, with an evanescent field outside. The puck becomes a resonator, like the metal cavities of Section 7.8, but without walls. Its lowest mode, TE $_{01 δ}$ , has circular $E$ lines in the puck, and an $H$ field that loops through the puck and closes outside, like the field of a small magnetic dipole (Figure 7.6.13).

Application 7.6.2. A dielectric resonator in its TE_{01\delta} mode beside a microstrip line. Left (side view): the magnetic field (blue, dashed) loops through the puck and closes outside it, and this evanescent field couples the puck to the line. Right (top view): the electric field lines (orange) are circles inside the ceramic. — **Figure 7.6.13.** Application 7.6.2. A dielectric resonator in its TE $_{01 δ}$ mode beside a microstrip line. Left (side view): the magnetic field (blue, dashed) loops through the puck and closes outside it, and this evanescent field couples the puck to the line. Right (top view): the electric field lines (orange) are circles inside the ceramic.

The resonance frequency of the TE $_{01 δ}$ mode of an isolated puck of radius $a$ and height $H$ is well described by the empirical formula of Kajfez (accuracy about $2 %$ for $0.5 < a / H < 2$ and $30 < ε_{r} < 50$ , $a$ and $H$ in millimetres):

f [GHz] Q_{d} = \frac{34}{a ε _{r}} (\frac{a}{H} + 3.45), \approx \frac{1}{tan δ} . (7.6.44)

Engineers like dielectric resonators for four reasons.

Size. All dimensions scale with the wavelength in the ceramic, $λ_{0} / ε_{r}$ , so the volume falls as $ε_{r}^{3/2}$ : about $200$ times smaller than an air-filled metal cavity at the same frequency (Worked example 7.6.2).
Quality factor. Most of the energy is in the low-loss ceramic, and $Q \approx 1/ tan δ \approx 1 0^{4}$ .
Stability. The temperature coefficient of $f$ is tailored by the ceramic recipe to $0 \pm 2 ppm/K$ .
No walls. The field is held by the dielectric, so the puck can sit on a circuit board.

In a dielectric-resonator oscillator (DRO) the puck sits beside a microstrip line, couples to it through its evanescent magnetic field, and fixes the frequency of a transistor oscillator: the $9.75$ and $10.6 GHz$ local oscillators of satellite-TV receivers are the classic example. Several pucks in a metal box form the filters of mobile base stations, and a puck without the box radiates: the dielectric resonator antenna.

7.6.12 Modal and chromatic dispersion

Modal dispersion

Graded index

Chromatic dispersion