Waveguides and optical fibres

Conventions on this page. SI units throughout. Time-harmonic fields are phasors with $e^{j\omega t}$. The guide axis is $z$, and a propagating mode varies as $e^{-j\beta z}$ (as $e^{-\alpha z}$ below cut-off). Metal guides have perfectly conducting walls and a lossless, homogeneous filling with $\mu ={\mu }_r{\mu }_0$ and $\epsilon ={\epsilon }_r{\epsilon }_0$, where ${\mu }_0=4\pi \times 10^{-7}$ H/m and ${\epsilon }_0=8.854\times 10^{-12}$ F/m. In the filling, $u=1/\sqrt{\mu \epsilon }$ is the speed of light, $\eta =\sqrt{\mu /\epsilon }$ is the intrinsic impedance, $\lambda =u/f$ is the wavelength and $k=2\pi /\lambda =\omega \sqrt{\mu \epsilon }$ is the wavenumber. Air is treated as free space: $u=c\approx 3\times 10^8$ m/s and $\eta ={\eta }_0=\sqrt{{\mu }_0/{\epsilon }_0}\approx 376.7$ Ω, which is usually rounded to $120\pi \approx 377$ Ω.

A hollow pipe has only one conductor, so it cannot carry a TEM wave. TEM needs at least two separate conductors, as in a coaxial line. Instead, the fields form discrete modes. In a TE (transverse electric, or H) mode, $E_z=0$ and $H_z\ne 0$. In a TM (transverse magnetic, or E) mode, $H_z=0$ and $E_z\ne 0$. Each mode has a cut-off frequency $f_c$ set by the cross-section. Above $f_c$ the mode propagates. Below $f_c$ it is evanescent: its fields decay exponentially along $z$ and it carries no average power. A metal waveguide is therefore a high-pass structure.

A useful picture: a propagating mode is a superposition of plane waves bouncing between the walls, each travelling at an angle $\theta$ to the axis, with $\sin \theta =f_c/f$. For a ${\mathrm{TE}}_{m0}$ mode this is exactly one pair of plane waves; for $m,n\ge 1$ there are four. The pattern moves along $z$ at $v_p=u/\cos \theta$, while energy moves at $v_g=u\cos \theta$. As $f$ falls to $f_c$, the waves bounce straight across ($\theta \to 90^{\circ }$) and stop moving forward. Far above cut-off they travel almost along the axis.

Take inner dimensions $a$ along $x$ and $b$ along $y$, with $a\ge b$ ($a$ is the broad wall). In ${\mathrm{TE}}_{mn}$ and ${\mathrm{TM}}_{mn}$, $m$ counts the half-cycle field variations across the broad dimension (along $x$) and $n$ counts them across the narrow dimension (along $y$).

In a circular guide of inner radius $a$, the fields vary with radius as Bessel functions $J_n(k_c\rho )$ and around the axis as $\cos n\varphi$ or $\sin n\varphi$. In ${\mathrm{TE}}_{nm}$ and ${\mathrm{TM}}_{nm}$, the first index $n=0,1,2,\dots$ counts the full-cycle variations around the circumference, and the second index $m=1,2,\dots$ numbers the root (the radial variations). The wall conditions fix $k_c$. For TM, $E_z\propto J_n(k_c\rho )$ must vanish at $\rho =a$, so $k_{c}a=p_{nm}$, the $m$-th non-zero root of $J_n$. For TE, the tangential field $E_{\varphi }\propto J_n^{\prime }(k_c\rho )$ must vanish, so $k_{c}a=p_{nm}^{\prime }$, the $m$-th non-zero root of $J_n^{\prime }$. The root $x=0$ is never counted: it would give $k_c=0$ and no field.

Two non-dominant circular modes are worth recognising. ${\mathrm{TM}}_{01}$ has a field pattern symmetric about the axis ($n=0$), which suits rotating joints. ${\mathrm{TE}}_{01}$ has only circumferential wall currents, and its conductor loss falls as frequency rises, which makes it attractive for long low-loss runs. At any frequency where ${\mathrm{TE}}_{01}$ propagates, the lower modes (${\mathrm{TE}}_{11}$, ${\mathrm{TM}}_{01}$, ${\mathrm{TE}}_{21}$) and the degenerate ${\mathrm{TM}}_{11}$ propagate too, so the guide must be kept from converting ${\mathrm{TE}}_{01}$ into them.

A step-index fibre has a core of refractive index $n_1$ and radius $a$, surrounded by a cladding of lower index $n_2<n_1$. Light is launched from a medium of index $n_0$ ($n_0=1$ for air). A refractive index is $n=c/v$, the speed of light in vacuum divided by the speed in the medium, and ${\lambda }_0$ denotes the free-space wavelength. Rays are trapped by total internal reflection at the core–cladding boundary. This happens only when light travels from the higher index to the lower one and hits the boundary at more than the critical angle. The ray picture below (meridional rays, which pass through the axis) holds for multimode fibres whose core is many wavelengths wide.

Modal dispersion spreads a short input pulse by $\Delta \tau$, so it limits the bit rate of multimode step-index fibre. A graded-index core, whose index falls from the axis outward, removes much of this spread. Rays on longer, steeper paths spend more time in lower-index (faster) glass. For a near-parabolic profile, the delay difference scales as ${\Delta }^2$ instead of $\Delta$, which makes it smaller by a factor of the order of $\Delta$. A single-mode fibre has no intermodal dispersion. The spreading that remains is mainly chromatic dispersion (material and waveguide), with a smaller contribution from polarisation-mode dispersion.