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

The TM modes follow by the dual steps. Take $H_{x} = H_{0} cos h y$ inside and $H_{0} cos (h d /2) e^{- α (∣ y ∣ - d /2)}$ outside. The second set Eq. (7.6.6) gives the electric field

E_{y} = - \frac{τ}{ω ε} H_{x}, E_{z} = \frac{j}{ω ε} \frac{d H _{x}}{d y}, (7.6.15)

with $ε = ε_{0} n_{1}^{2}$ inside and $ε_{0} n_{2}^{2}$ outside. The tangential fields $H_{x}$ and $E_{z}$ must be continuous at the faces. $H_{x}$ is continuous by construction, and continuity of $E_{z}$ now means continuity of $(1/ ε) d H_{x} / d y$ , not of the slope itself:

\frac{1}{n _{1}^{2}} (- h H_{0} sin \frac{h d}{2}) = \frac{1}{n _{2}^{2}} (- α H_{0} cos \frac{h d}{2}) .

The odd TM mode ( $H_{x} = H_{0} sin h y$ inside) is treated in the same way.

Eigenvalue equations of the TM modes

even TM: odd TM: α = \frac{n _{2}^{2}}{n _{1}^{2}} h tan \frac{h d}{2}, α = - \frac{n _{2}^{2}}{n _{1}^{2}} h cot \frac{h d}{2} . (7.6.16)

In plain words

The TM equations are the TE equations with the right-hand side reduced by the factor $n_{2}^{2} / n_{1}^{2}$ , which comes from the permittivity in $E_{z} = (j / ω ε) d H_{x} / d y$ . When the two indices are close, as in an optical fibre, TE and TM modes are almost identical.

$E_{y}$ is normal to the faces, so it is $D_{y} = ε E_{y}$ that is continuous: $E_{y}$ jumps up by the factor $n_{1}^{2} / n_{2}^{2}$ on leaving the slab. The electric field lies in the plane of incidence, and TM modes are made of parallel (P) polarised plane waves, reflected with $Γ_{∥}$ .

Names of the modes: two conventions

Here, as in Orfanidis (Electromagnetic Waves and Antennas, Section 9.11) and in the optics literature, a mode is called even or odd after its transverse field, $E_{x}$ for TE and $H_{x}$ for TM, and TE $_{m}$ or TM $_{m}$ has $m$ zeros of that field inside the slab. Cheng (Field and Wave Electromagnetics, Section 10-6) labels the modes after the longitudinal field $H_{z}$ or $E_{z}$ , which has the opposite parity. His "odd" TE and TM modes are our even ones, TE $_{0}$ and TM $_{0}$ included, and his equations (10-257), (10-263), (10-278) and (10-281) are Eqs. (7.6.12) and (7.6.16) with the names exchanged.

Both families can be written as one formula, numbered by the number of zeros $m$ . Since $tan (x - π /2) = - cot x$ and $tan (x - π) = tan x$ , even $m$ gives the tangent form and odd $m$ the cotangent form:

All slab modes ($m=0,1,2,\dots$)

TE_{m} : TM_{m} : α = h tan (\frac{h d}{2} - \frac{mπ}{2}), α = \frac{n _{2}^{2}}{n _{1}^{2}} h tan (\frac{h d}{2} - \frac{mπ}{2}) . (7.6.17)

Common mistake · Using the TE equation for a TM mode

Forgetting the factor $n_{2}^{2} / n_{1}^{2}$ . For a slab of $ε_{r} = 5$ in air the factor is $0.2$ , and the TE $_{0}$ and TM $_{0}$ effective indices differ visibly ( $2.153$ against $2.119$ in Problem 7.6.1). The factor is there because the boundary condition on $E_{z}$ involves the permittivity, while the one on $H_{z}$ for TE modes does not involve $μ$ , which is $μ_{0}$ on both sides.

7.6.4 TM modes of the slab