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## Waveguide Modes Edit

### Wave Equations Edit

From Maxwell's equations, the field equation of a plane wave propagating along a waveguide can be derived, in terms of the spatial dimensions $x$, $y$ and $z$, and the temporal dimension $t$, as

$\mathbf{E}\left(x,y;z,t\right)=\mathbf{e}\left(x,y\right)\exp{\left(j\beta z\right)}\exp{\left(-j\omega t\right)}$

where $\mathbf{e}\left(x,y\right)$ is the 2-dimensional cross-section of the distribution of the electromagnetic field, $\beta$ is the propagation constant, and $\omega=2\pi f$ is the angular frequency of the wave excitation in units of radians (with $f$ being the frequency in hertz). Technical names exist for each term in the equation:

• $\mathbf{e}\left(x,y\right)$ is called the transverse term
• $\exp{\left(j\beta z\right)}$ is known as the phase term
• $\exp{\left(-j\omega t\right)}$ is the harmonic term

Note that the wave propagation is in the $z$-direction, and since the cross-sectional field pattern is normally assumed constant, it is therefore customary to express the wave equation without $x$ and $y$ dependence as

$\mathbf{E}\left(z,t\right)=\mathbf{e}\cdot\exp{\left(j\left(\beta z-\omega t\right)\right)}$

where only the phase changes along the waveguide axis ($z$-direction). By comparing this with the general equation for wave propagation $f\left(z-vt\right)$, where $v$ is the velocity of the propagation, it is simple to define the propagation velocity of the wave as

$v_p=\frac{\omega}{\beta} \text{.}$

Recalling that the speed of light in a specific material (i.e. the waveguide) of refractive index $n$ is given by

$v_p=\frac{c}{n}=\frac{\omega}{kn}$

where $c\approx 3\times 10^{8}$ m/s is the speed of light in free space, and $k=\frac{2\pi}{\lambda}$ is the free space wavenumber (with $\lambda$ being the wavelength), we see that $\beta=kn$, or put another way, the effective refractive index of the waveguide is

$n_{\text{eff}}=\frac{\beta}{k}$

which always satisfies $n_{\text{cl}}$ < $n_{\text{eff}}$ < $n_{\text{co}}$, where $n_{\text{cl}}$ and $n_{\text{co}}$ are the refractive indices of the waveguide cladding and core, respectively.

### Degree of Guidance Edit

The degree of guidance of a waveguide gives an indication on the total number of individual modes that the waveguide can support. It is given by

$\mathrm{V}=\frac{2\pi}{\lambda}\rho\sqrt{n_{\text{co}}^2-n_{\text{cl}}^2}$

where $\rho$ is the characteristic dimension of the waveguide (e.g. it is the half-thickness of a slab/square waveguide, the radius for a circular waveguide or optical fibre, and the e-folding distance of a graded-index waveguide), and $n_{\text{co}}^2-n_{\text{cl}}^2$ is known as the index contrast. In practical systems, we are usually interested in number of supportable modes for a given waveguide, at different wavelengths (frequencies). To determine this, the degree of guidance is first calculated for a particular wavelength. The result is then compared with the eigenvalue U(V) graph for the particular dimensions and index contrast of the waveguide. An example is shown in Figure 1 for a step index fibre. For instance, if the degree of guidance is calculated to be 2.4, then from looking at the graph, we see that only one mode (i.e. the $\sf LP_{01}$ or fundamental mode) is supported, since no other modes have an intersection point before V = 2.4. On the other hand, if the degree of guidance is calculated as say, 3.5, then two modes (i.e. the $\sf LP_{01}$ and $\sf LP_{11}$ modes) will be supported.

These results are important in telecommunication systems that employ optical fibre links. Having more than one supportable mode in a waveguide (optical fibre) means that inter-mode interference will occur, hence degrading communication performance. Telecommunications engineers thus seek to reduce the degree of guidance to achieve single-moded fibres. This can be done by increasing the wavelength of communication. However, in bandlimited systems, this may not be feasible, and so the physical charateristics of the fibre (i.e. size or index contrast) may be varied. Of course, waveguide modes are important in contexts other than telecommunications.