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

Recap

Guiding: a wave in a layer of index $n_{1}$ is trapped by total internal reflection at faces with $n_{2} < n_{1}$ beyond $θ_{c} = arcsin (n_{2} / n_{1})$ . Along the guide it has $τ = k_{0} n_{1} sin θ$ and $n_{2} < n_{eff} = τ / k_{0} < n_{1}$ , Eq. (7.6.3). Outside, the field is evanescent with $α = τ^{2} - k_{0}^{2} n_{2}^{2}$ , and no power leaves the guide.
Symmetric slab: TE ( $E_{x}$ , $H_{y}$ , $H_{z}$ ) and TM ( $H_{x}$ , $E_{y}$ , $E_{z}$ ) families, $cos h y$ or $sin h y$ inside, $e^{- α (∣ y ∣ - d /2)}$ outside, $h^{2} + α^{2} = k_{0}^{2} (n_{1}^{2} - n_{2}^{2})$ , Eq. (7.6.8). Eigenvalue equations $α = h tan (h d /2 - mπ /2)$ (TE $_{m}$ ) and the same times $n_{2}^{2} / n_{1}^{2}$ (TM $_{m}$ ), Eq. (7.6.17).
Normalised form: $u = h d /2$ , $w = α d /2$ , $u^{2} + w^{2} = V^{2}$ , $V = (π d / λ_{0}) NA$ , solved graphically or by bisection. Cutoff $V = mπ /2$ , $f_{c, m} = m c / (2 d NA)$ , TE $_{0}$ and TM $_{0}$ without cutoff, $M = 1 + ⌊ 2 V / π ⌋$ TE modes, single pair for $d < λ_{0} / (2 NA)$ . Below cutoff a mode radiates.
Confinement: decay length $d / (2 w)$ , power fraction $Γ$ , Eq. (7.6.25), $33$ to $96 %$ for TE $_{0}$ between $V = 0.5$ and $3$ .
Ray picture: $∣ Γ_{⊥} ∣ = 1$ , $Γ_{⊥} = e^{j 2 ϕ}$ with $tan ϕ = α / h$ , and the transverse resonance $- 2 h d + 4 ϕ = - 2 mπ$ gives the same equations.
Coated conductor: the modes of a slab of thickness $2 t$ with $E_{tan} = 0$ at the metal, TM $_{0}$ (no cutoff), TE $_{1}$ from $c / (4 t NA)$ , TM $_{2}$ , …, and the surface impedance $j α / (ω ε_{0})$ (TM, inductive).
Fibre: $NA = sin θ_{a} = n_{1}^{2} - n_{2}^{2} = n_{1} 2Δ$ , $V = k_{0} a NA$ , LP modes, Eq. (7.6.37), single-mode for $V < 2.405$ , about $V^{2} /2$ modes for $V ≫ 1$ . Modal spread $L n_{1} (n_{1} - n_{2}) / (c n_{2}) \approx L n_{1} Δ/ c$ (step), $L n_{1} Δ^{2} / (8 c)$ (optimum graded), chromatic spread $∣ D ∣ L Δ λ$ .

7.6.13 Recap