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

The modes were found from Maxwell's equations. The same eigenvalue equations follow from the ray picture of Section 7.6.1, and this second derivation explains what the modes are.

The phase of total reflection

The upgoing wave meets the upper face at the angle $θ$ from the normal. For a TE mode the wave is perpendicularly polarised, and Section 6.4 gives

Γ_{⊥} n_{1} sin θ = \frac{η _{2} cos θ - η _{1} cos θ _{t}}{η _{2} cos θ + η _{1} cos θ _{t}}, = n_{2} sin θ_{t} . (7.6.26)

Beyond the critical angle $sin θ_{t} = (n_{1} / n_{2}) sin θ > 1$ , and $cos θ_{t}$ is imaginary. The transmitted wave $e^{- j k_{0} n_{2} (z s i n θ_{t} + y c o s θ_{t})}$ must decay into the upper medium, which fixes $cos θ_{t} = - j ∣ cos θ_{t} ∣$ with $∣ cos θ_{t} ∣ = sin^{2} θ_{t} - 1$ . Then

Γ_{⊥} tan ϕ = \frac{η _{2} cos θ + j η _{1} ∣ cos θ _{t} ∣}{η _{2} cos θ - j η _{1} ∣ cos θ _{t} ∣} = e^{j 2 ϕ}, = \frac{η _{1} ∣ cos θ _{t} ∣}{η _{2} cos θ}, (7.6.27)

since numerator and denominator are complex conjugates: $∣ Γ_{⊥} ∣ = 1$ , and only the phase $2 ϕ$ is left. With $η_{i} = η_{0} / n_{i}$ , $n_{2} ∣ cos θ_{t} ∣ = n_{1}^{2} sin^{2} θ - n_{2}^{2} = α / k_{0}$ by Eq. (7.6.4), and $n_{1} cos θ = h / k_{0}$ by Eq. (7.6.2),

tan ϕ = \frac{n _{2} ∣ cos θ _{t} ∣}{n _{1} cos θ} = \frac{α}{h} . (7.6.28)

(The transmitted wave decays as $e^{- k_{0} n_{2} ∣ c o s θ_{t} ∣ y} = e^{- α y}$ : the evanescent field of the ray picture is exactly the tail of the mode.)

The round trip

Follow a wavefront of the upgoing wave once around: it crosses the slab upwards (phase $- h d$ ), is reflected (phase $+ 2 ϕ$ ), crosses downwards ( $- h d$ ) and is reflected again ( $+ 2 ϕ$ ), Figure 7.6.7. The wave reproduces itself, and the up- and down-going waves reinforce instead of cancelling, only if the total phase is a multiple of $2 π$ :

- 2 h d + 4 ϕ = - 2 mπ ⟹ \frac{h d}{2} = ϕ + \frac{mπ}{2}, m = 0, 1, 2, \dots (7.6.29)

Take the tangent of $ϕ = u - mπ /2$ and use Eq. (7.6.28):

Transverse resonance

tan (u - \frac{mπ}{2}) = \frac{w}{u} ⟺ w = u tan (u - \frac{mπ}{2}), (7.6.30)

the TE equations Eq. (7.6.20) again. With $Γ_{∥}$ of the parallel polarisation the same steps give the TM equations.

Transverse resonance. A wavefront crossing the slab twice collects the phase -2hd, and the two total reflections add 2\times2\phi. The mode exists when the sum is a multiple of 2\pi, which is again the eigenvalue equation Eq. (7.6.30). — **Figure 7.6.7.** Transverse resonance. A wavefront crossing the slab twice collects the phase $- 2 h d$ , and the two total reflections add $2 \times 2 ϕ$ . The mode exists when the sum is a multiple of $2 π$ , which is again the eigenvalue equation Eq. (7.6.30).

Same result, two pictures

In the field picture a mode is a standing wave across the guide and a travelling wave along it. In the ray picture it is a pair of plane waves zig-zagging at $\pm θ_{m}$ , with

sin θ_{m} = \frac{τ _{m}}{k _{0} n _{1}} = \frac{n _{eff, m}}{n _{1}} . (7.6.31)

For the TE $_{0}$ mode of Problem 7.6.1, $n_{eff} = 2.153$ and $n_{1} = 5$ give $θ_{0} = 74. 3^{\circ}$ , well beyond the critical angle $θ_{c} = 26. 6^{\circ}$ . Every guided mode zig-zags beyond $θ_{c}$ , because $n_{eff} > n_{2}$ is the same statement as $sin θ_{m} > n_{2} / n_{1}$ .

7.6.8 The ray picture: transverse resonance

The phase of total reflection

The round trip