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

Figure 7.6.6a shows the transverse field of the three TE modes of a slab with $V = 3.5$ .

TE $_{m}$ has $m$ zeros across the slab: a half-period more of the standing wave per order.
Outside, the field decays as $e^{- α (∣ y ∣ - d /2)} = e^{- 2 w (∣ y ∣ - d /2) / d}$ , so the decay length is $1/ α = d / (2 w)$ . Near cutoff $w \to 0$ and the tail reaches far out. Far above cutoff $w \approx V$ , the field at the face $cos u$ is small, and the field clings to the slab.
TE $_{2}$ in Figure 7.6.6a works close to its cutoff $V = π$ ( $w = 0.865$ ) and spreads widely into the surrounding medium.

The confinement factor

By Eq. (7.6.14) the power is proportional to $\int ∣ E_{x} ∣^{2} d y$ , since $Z_{T E}$ is the same everywhere. For the even mode Eq. (7.6.9), with $h d /2 = u$ and $α d /2 = w$ ,

\int_{- d /2}^{d /2} cos^{2} h y d y = \frac{d}{2} (1 + \frac{sin 2 u}{2 u}),

2 \int_{d /2}^{\infty} cos^{2} u e^{- 2 α (y - d /2)} d y = \frac{cos ^{2} u}{α} = \frac{d}{2} \frac{cos ^{2} u}{w} .

The fraction of the power carried inside the slab is the ratio of the first integral to the sum:

Fraction of the power inside the slab (even TE modes)

Γ = \frac{1 + \frac{sin 2 u}{2 u}}{1 + \frac{sin 2 u}{2 u} + \frac{cos ^{2} u}{w}} . (7.6.25)

For the odd TE modes $sin 2 u \to - sin 2 u$ and $cos^{2} u \to sin^{2} u$ .

For TE $_{0}$ , $Γ = 33$ , $67$ , $90$ and $96 %$ at $V = 0.5$ , $1$ , $2$ and $3$ (Figure 7.6.6b). A thin slab carries most of its power outside. For TM modes the power density is $τ ∣ H_{x} ∣^{2} / (2 ω ε)$ , which weights the inside by $1/ n_{1}^{2}$ and the outside by $1/ n_{2}^{2}$ , so Eq. (7.6.25) does not apply to them unchanged.

(a) E_x(y) for V=3.5. — (a) $E_{x} (y)$ for $V = 3.5$ .

(b) Power fraction inside the slab. — (a) $E_{x} (y)$ for $V = 3.5$ .

Uses of the tail

The evanescent field is not a nuisance but a tool. It lets two guides placed close together exchange power (directional couplers in integrated optics), it lets a sensor feel the medium just outside the guide (evanescent-wave biosensors), and it is why the cladding of an optical fibre must be many decay lengths thick: whatever touches the tail, a coating or a bend, takes power from the mode.

7.6.7 Field profiles and confinement

The confinement factor