Waveguides & fibres formulas

$k_c$ is the mode's cut-off wavenumber, set by the guide's shape and size, and $f_c=k_{c}u/(2\pi )$. $\beta$ is real (the mode propagates) only when $f>f_c$.

Same formula for TE and TM. Allowed indices: ${\mathrm{TE}}_{mn}$ with $m,n=0,1,2,\dots$ but not both zero, and ${\mathrm{TM}}_{mn}$ with $m,n\ge 1$. A filling with ${\mu }_r=1$ divides every air-filled cut-off by $\sqrt{{\epsilon }_r}$.

${\lambda }_c$ depends only on the geometry, not on the filling. A mode propagates when the wavelength in the filling, $\lambda =u/f$, is shorter than its ${\lambda }_c$.

When $a>b$, ${\mathrm{TE}}_{10}$ has the lowest cut-off of all modes, so it is the dominant mode. Its cut-off does not depend on $b$. In a square guide ($a=b$), ${\mathrm{TE}}_{01}$ has the same cut-off.

$\lambda =u/f$ is the wavelength in the unbounded filling, and ${\lambda }_g>\lambda$ for every propagating mode. For ${\mathrm{TE}}_{10}$, ${\lambda }_c=2a$, so ${\lambda }_g=\lambda /\sqrt{1-(\lambda /2a{)}^2}$.

$\lambda$ is the wavelength in the filling. This says the same thing as $k^2={\beta }^2+k_c^2$. It is the quickest route when two of the three wavelengths are given.

$v_p>u$ is the speed of the phase pattern and carries no energy or information. $v_g<u$ is the speed of energy in a lossless guide. Also, $v_p=f{\lambda }_g$.

This is the ratio of the mode's transverse $E$ to its transverse $H$ (for ${\mathrm{TE}}_{10}$, $Z_{TE}=E_y/(-H_x)$). Above cut-off, $Z_{TE}>\eta >Z_{TM}$. Equivalently, $Z_{TE}=\eta {\lambda }_g/\lambda$ and $Z_{TM}=\eta \lambda /{\lambda }_g$.

For $f<f_c$ the fields vary as $e^{-\alpha z}$, with no phase shift. $Z_{TE}=j\omega \mu /\alpha$ becomes inductive and $Z_{TM}=-j\alpha /(\omega \epsilon )$ becomes capacitive, so no average power flows. Well below cut-off, $\alpha \approx 2\pi /{\lambda }_c$.

Time-average power in a lossless guide. $E_0$ is the peak phasor amplitude of $E_y$ on the centre line $x=a/2$, and $Z_{TE}$ is the ${\mathrm{TE}}_{10}$ wave impedance.

$a$ is the RADIUS. $p_{nm}$ is the $m$-th non-zero root of $J_n(x)$ (TM), and $p_{nm}^{\prime }$ is the $m$-th non-zero root of $J_n^{\prime }(x)$ (TE). The $\beta$, ${\lambda }_g$, $v_p$, $v_g$ and $Z_{TE}$/$Z_{TM}$ formulas from the previous section apply unchanged with this $f_c$.

Air-filled: $f_{c,11}\approx 8.79/a$ GHz, with $a$ in cm. Only ${\mathrm{TE}}_{11}$ propagates for $1.841<ka<2.405$, that is, up to the ${\mathrm{TM}}_{01}$ cut-off, a frequency ratio of about 1.31.

${\theta }_c$ is measured from the NORMAL to the boundary, inside the core. Rays that meet the boundary at $\varphi >{\theta }_c$ are totally reflected and stay guided; $\varphi ={\theta }_c$ is the limiting ray.

$\Delta$ is the relative index difference. With the first form of $\Delta$, $\mathrm{NA}=n_1\sqrt{2\Delta }$ exactly. The form $(n_1-n_2)/n_1$ is the usual approximation for $\Delta \ll 1$. NA depends only on $n_1$ and $n_2$, not on the launch medium.

${\theta }_a$ is a half-angle measured from the fibre AXIS, outside the fibre. Rays inside this cone are guided. For the same fibre, a denser launch medium (larger $n_0$) gives a smaller ${\theta }_a$. The formula needs $\mathrm{NA}\le n_0$; if $\mathrm{NA}>n_0$, every ray entering the end face is guided. This is the acceptance angle for meridional rays; skew rays, which never cross the axis, can be guided at somewhat larger launch angles.

$a$ is the core RADIUS and ${\lambda }_0$ is the free-space wavelength. $V$ grows with core size and NA and falls as the wavelength rises.

2.405 is the first zero of $J_0$. The fundamental mode (${\mathrm{LP}}_{01}$, also called ${\mathrm{HE}}_{11}$) has no cut-off, and below $V=2.405$ it is the only guided mode (in two orthogonal polarisations). Here ${\lambda }_c$ is the fibre's cut-off wavelength: the fibre is single-mode at every longer wavelength.

These are the usual large-$V$ estimates ($V\gg 1$), counting both polarisations. Use them only for multimode fibres. Both are cases of $M\approx \frac{g}{g+2}\cdot \frac{V^2}{2}$ for the power-law profile above.

The difference in arrival time over length $L$ between the axial ray (path $L$) and the steepest guided ray, at the critical angle (path $L{n}_1/n_2$), both travelling at $c/n_1$. It is proportional to $L$ and, for $\Delta \ll 1$, to $\Delta$.

Losses in dB add along a link: a fibre of ${\alpha }_{\mathrm{dB}}$ dB/km loses ${\alpha }_{\mathrm{dB}}L$ dB over $L$ km. A loss of 3 dB leaves about half the power, and 10 dB leaves one tenth.