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

Consider a uniform plane wave travelling inside a layer of refractive index $n_{1}$ , bounded above and below by a medium of lower index $n_{2} < n_{1}$ . It meets a face at the angle $θ$ measured from the normal. Section 6.4 showed that beyond the critical angle

θ_{c} = arcsin \frac{n _{2}}{n _{1}} (7.6.1)

the wave is totally reflected: the reflection coefficient has magnitude one and only its phase depends on the angle. A wave launched at $θ > θ_{c}$ is therefore reflected without loss at the upper face, travels down to the lower face, is reflected again, and zig-zags along the layer indefinitely (Figure 7.6.1). This is the principle of every dielectric waveguide.

$Guiding by total internal reflection. A plane wave in the layer of index n_1 meets the faces at \theta>\theta_c and is totally reflected at both: it zig-zags along z. Its wave vector has the component \tau along the guide and h across it. Outside, the field is evanescent (shaded). Right: the resulting standing-wave profile across the layer (the TE_0 mode of a slab with V=3.5, see Section 7.6.5), a cosine inside and decaying exponentials outside.$

Figure 7.6.1. Guiding by total internal reflection. A plane wave in the layer of index

n_{1}

meets the faces at

θ > θ_{c}

and is totally reflected at both: it zig-zags along

z

. Its wave vector has the component

τ

along the guide and

h

across it. Outside, the field is evanescent (shaded). Right: the resulting standing-wave profile across the layer (the TE

_{0}

mode of a slab with

V = 3.5

, see Section 7.6.5), a cosine inside and decaying exponentials outside.

The wave vector of the zig-zagging wave has the magnitude $k_{0} n_{1}$ . Split it into a component along the guide and a component across it:

τ τ^{2} + h^{2} = k_{0} n_{1} sin θ, h = k_{0} n_{1} cos θ, = k_{0}^{2} n_{1}^{2} . (7.6.2)

Total internal reflection requires $sin θ > n_{2} / n_{1}$ , that is $τ > k_{0} n_{2}$ , and of course $sin θ < 1$ gives $τ < k_{0} n_{1}$ . It is convenient to measure $τ$ in units of $k_{0}$ :

Effective index of a guided wave

n_{eff} = \frac{τ}{k _{0}}, n_{2} < n_{eff} < n_{1} . (7.6.3)

The guided wave travels along $z$ with the phase velocity $c / n_{eff}$ and the guide wavelength $λ_{g} = λ_{0} / n_{eff}$ .

In plain words

A guided wave is slower than a plane wave in the surrounding medium and faster than a plane wave in the core. It travels as if it were in a uniform medium of an intermediate index, $n_{eff}$ , which lies closer to $n_{1}$ the more of the field is inside the core.

The field outside

Phase matching at the faces (Snell's law of Section 6.4) requires the same variation $e^{- j τ z}$ outside the layer. A wave of wavenumber $k_{0} n_{2}$ with the longitudinal component $τ > k_{0} n_{2}$ must have a transverse component $k_{y}$ with $k_{y}^{2} = k_{0}^{2} n_{2}^{2} - τ^{2} < 0$ , an imaginary $k_{y} = - j α$ with

α = τ^{2} - k_{0}^{2} n_{2}^{2} = k_{0} n_{1}^{2} sin^{2} θ - n_{2}^{2} [Np/m] . (7.6.4)

Outside the layer the field therefore decays as $e^{- α ∣ y ∣}$ away from the faces: it is the evanescent wave of total internal reflection. It is a real field, which can be measured and which couples to anything brought close, but its electric and magnetic tangential components are $9 0^{\circ}$ out of phase, so no time-average power crosses the faces. All the power flows along $z$ .

Not every angle

The upward and the downward plane waves overlap in the layer. As they travel along $z$ they must reproduce the same transverse pattern after each round trip, otherwise they would cancel by interference. Only a discrete set of angles $θ_{m}$ satisfies this self-consistency condition. Each of them defines a mode of the guide, with its own $n_{eff}$ . Sections 7.6.2–7.6.5 find these angles from Maxwell's equations, and Section 7.6.8 finds them again from the ray picture.

Quick check

A glass layer, $n_{1} = 1.5$ , is surrounded by air. What is the critical angle, and what is the effective index of a wave that zig-zags at $θ = 6 0^{\circ}$ ?

$θ_{c} = arcsin (1/1.5) = 41. 8^{\circ}$ . At $6 0^{\circ} > θ_{c}$ the wave is totally reflected and $n_{eff} = n_{1} sin θ = 1.5 \times 0.866 = 1.299$ , between $1$ and $1.5$ as Eq. (7.6.3) requires. (Whether $6 0^{\circ}$ is one of the allowed mode angles depends on the thickness and the frequency.)

7.6.1 Guiding by total internal reflection

The field outside

Not every angle