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

The simplest dielectric guide that can be solved exactly is the symmetric slab of Figure 7.6.2: a lossless layer of index $n_{1}$ and thickness $d$ , occupying $∣ y ∣ \leq d /2$ , in a medium of index $n_{2} < n_{1}$ above and below. The slab is unbounded in $x$ , and we look for waves that do not vary along $x$ and travel along $z$ with the same factor $e^{- j τ z}$ in all three regions (phase matching, as in Snell's law).

$The symmetric dielectric slab: index n_1, thickness d, in a medium of index n_2\lt n_1. Inside, the transverse field is a cosine or a sine of hy. Outside, it decays exponentially with the transverse decay constant \alpha. All regions share the factor \mathrm{e}^{-\mathrm{j}\tau z}.$

Figure 7.6.2. The symmetric dielectric slab: index

n_{1}

, thickness

d

, in a medium of index

n_{2} < n_{1}

. Inside, the transverse field is a cosine or a sine of

h y

. Outside, it decays exponentially with the transverse decay constant

α

. All regions share the factor

e^{- j τ z}

Two independent families

With $\partial / \partial x = 0$ and $\partial / \partial z \to - j τ$ the two curl equations $\nabla \times E = - j ω μ_{0} H$ and $\nabla \times H = j ω ε E$ (Section 5.6) read, component by component,

- j τ E_{x} - \frac{\partial E _{x}}{\partial y} \frac{\partial H _{z}}{\partial y} + j τ H_{y} = - j ω μ_{0} H_{y}, = - j ω μ_{0} H_{z}, = j ω ε E_{x}, (7.6.5)

- j τ H_{x} - \frac{\partial H _{x}}{\partial y} \frac{\partial E _{z}}{\partial y} + j τ E_{y} = j ω ε E_{y}, = j ω ε E_{z}, = - j ω μ_{0} H_{x} . (7.6.6)

The first set contains only $E_{x}$ , $H_{y}$ and $H_{z}$ , the second only $H_{x}$ , $E_{y}$ and $E_{z}$ : the fields split into two independent families.

TE waves ( $E_{z} = 0$ ): the only electric component is $E_{x} (y)$ , and $H_{y}$ , $H_{z}$ follow from it.
TM waves ( $H_{z} = 0$ ): the only magnetic component is $H_{x} (y)$ , and $E_{y}$ , $E_{z}$ follow from it.

Eliminating $H_{y}$ and $H_{z}$ from Eq. (7.6.5) (put $H_{y} = τ E_{x} / (ω μ_{0})$ and $H_{z} = - (j / ω μ_{0}) \partial E_{x} / \partial y$ into the third equation and multiply by $ω μ_{0} / j$ ) gives a second-order equation for $E_{x}$ alone, and Eq. (7.6.6) gives the same equation for $H_{x}$ . In region $i$ , with $ω^{2} μ_{0} ε = k_{0}^{2} n_{i}^{2}$ ,

\frac{d ^{2} ψ}{d y ^{2}} + (k_{0}^{2} n_{i}^{2} - τ^{2}) ψ = 0, ψ = E_{x} (TE) or ψ = H_{x} (TM) . (7.6.7)

This is the Helmholtz equation of Section 5.6 with $\partial^{2} / \partial z^{2} \to - τ^{2}$ and nothing depending on $x$ .

Inside and outside

For a guided wave $k_{0} n_{2} < τ < k_{0} n_{1}$ (Eq. (7.6.3)), so the bracket in Eq. (7.6.7) is positive inside and negative outside. Inside we write $h^{2} = k_{0}^{2} n_{1}^{2} - τ^{2} > 0$ , and the solutions are $cos h y$ and $sin h y$ , a standing wave across the slab. Outside we write $α^{2} = τ^{2} - k_{0}^{2} n_{2}^{2} > 0$ , and the solutions are $e^{\pm α y}$ , of which only the one decaying away from the slab is kept. Adding the two definitions eliminates $τ$ :

The transverse wavenumbers of a slab

h^{2} = k_{0}^{2} n_{1}^{2} - τ^{2}, α^{2} = τ^{2} - k_{0}^{2} n_{2}^{2}, h^{2} + α^{2} = k_{0}^{2} (n_{1}^{2} - n_{2}^{2}) . (7.6.8)

In plain words

Inside the slab the field oscillates across the guide with the wavenumber $h$ , outside it dies away with the decay constant $α$ . The two are not independent: they share the fixed budget $k_{0}^{2} (n_{1}^{2} - n_{2}^{2})$ . A mode that oscillates fast inside (large $h$ ) decays slowly outside (small $α$ ), and the other way round.

The relation Eq. (7.6.8) is one equation for the two unknowns $h$ and $α$ (both follow from $τ$ ). The second equation comes from the boundary conditions at $y = \pm d /2$ , and it is different for TE and TM. Because the slab is symmetric about $y = 0$ , every mode is either even ( $cos h y$ inside) or odd ( $sin h y$ inside), and it suffices to match the fields at the upper face $y = d /2$ .

Common mistake · Reading

α

as a loss

$α$ is a decay constant across the guide, not along it. The slab is lossless, and the guided wave travels along $z$ without attenuation, with $τ$ real. $α$ only says how far the field reaches into the surrounding medium: it falls to $1/ e$ at the distance $1/ α$ from the face.

7.6.2 The symmetric dielectric slab

Two independent families

Inside and outside