Optical Fibres

An optical fibre consists of a core (refractive index $n_1$) surrounded by a cladding ($n_2 \lt n_1$). Light is guided by total internal reflection.

The numerical aperture:

$\mathrm{NA} = \sin\theta_{\mathrm{max}} = \sqrt{n_1^2 - n_2^2}$

Where $\theta_{\mathrm{max}}$ is the maximum acceptance angle for light entering the fibre.

The number of modes supported depends on the V-number:

$V = \frac{2\pi a}{\lambda}\mathrm{NA}$

Where $a$ is the core radius.

• Single-mode fibre: $V \lt 2.405$. Only the fundamental HE$_{11}$ mode propagates.
• Multimode fibre: $V \gt 2.405$. Multiple modes propagate, causing modal dispersion.

Fibre attenuation is dominated by Rayleigh scattering ($\propto \lambda^{-4}$) and infrared absorption peaks. The minimum attenuation for silica fibre is $\sim 0.2$ dB/km at $\lambda \approx 1550$ nm.

Dispersion broadens optical pulses as they travel, limiting the bit rate.

Modal dispersion occurs in multimode fibres because different modes travel at different speeds. This is the dominant dispersion mechanism in multimode fibres.

Chromatic dispersion arises from the wavelength dependence of the refractive index. It has two components: material dispersion (due to intrinsic glass properties) and waveguide dispersion (due to mode confinement). Standard silica fibre has zero chromatic dispersion at $\lambda \approx 1300$ nm.

Polarisation mode dispersion (PMD) results from birefringence in the fibre core, causing the two orthogonal polarisation components to travel at slightly different speeds.

Optical fibres are made by the Modified Chemical Vapour Deposition (MCVD) process. A thin layer of pure silica is deposited inside a rotating silica tube by passing SiCl$_4$ and O$_2$ through it. The tube is then collapsed into a solid preform rod at approximately 2000 $^\circ$C.

The preform is then placed in a drawing tower, heated to its melting point, and pulled into a thin fibre under tension. The fibre diameter is monitored precisely to maintain a 125 $\mu$m outer diameter.

Doping with GeO$_2$ or P$_2$O$_5$ increases the core refractive index, while doping with F or B$_2$O$_3$ decreases it for the cladding.

Step-index fibre has a uniform core refractive index with an abrupt step at the core-cladding boundary. Graded-index fibre has a core index that decreases parabolically from the centre, reducing modal dispersion significantly.

Photonic crystal fibres (PCFs) use a periodic array of air holes running along the fibre length to guide light via photonic bandgap effects. They can achieve single-mode operation over an extremely wide wavelength range and offer very high nonlinearity for supercontinuum generation.

Dispersion-shifted fibre moves the zero-dispersion wavelength to 1550 nm to coincide with the minimum attenuation window. Dispersion-flattened fibre maintains low dispersion across a broad wavelength range for WDM systems.

Telecommunications is the dominant application. Fibre links form the backbone of the internet, using wavelength-division multiplexing (WDM) to send multiple channels at different wavelengths on a single fibre, achieving Tb/s data rates.

Fibre optic sensors exploit changes in intensity, phase, polarisation, or wavelength caused by external stimuli. Fibre Bragg gratings measure strain and temperature in structural health monitoring of bridges, dams, and pipelines.

Medical applications include endoscopy (imaging via fibre bundles) and laser surgery, where high-power laser light is delivered through thin fibres.

A step-index fibre has $n_1 = 1.48$, $n_2 = 1.46$, and core radius $a = 25\ \mu$m. Calculate the NA, acceptance angle, and V-number at $\lambda = 850$ nm.

$\mathrm{NA} = \sqrt{1.48^2 - 1.46^2} = \sqrt{2.1904 - 2.1316} = \sqrt{0.0588} \approx 0.242$

$\theta_{\mathrm{max}} = \arcsin(0.242) \approx 14.0^\circ$