EM Waveguiding

Overview

• Waveguide may refer to any structure that conveys electromagnetic waves between its endpoints • Most common meaning is a hollow metal pipe used to carry radio waves • May be used to transport radiation of a single frequency

• Transverse Electric (TE) modes have E ┴ kg (propagation wavevector)

• Transverse Magnetic (TM) modes have B ┴ kg

• Transverse Electric-Magnetic modes (TEM) have E, B ┴ kg • A cutoff frequency exists, below which no radiation propagates

EM Waveguiding Electromagnetic wave reflection by perfect conductor

D┴1 = D┴2 EI┴ ER┴ EI EI┴ D 1 = ε E 1 ER ┴ o ┴ E ER┴ I|| ------θ θ E i r R|| y D┴2 = εoεE ┴2 y

E 1 = E 2 z || ||

E can be finite just outside EoI + EoR = 0 ┴ EI|| ER|| EI|| ER|| EI|| ER|| conducting surface y E|| vanishes just outside and EoT = 0 inside conducting surface

z EM Waveguiding

Electromagnetic wave propagation between conducting plates

Boundary conditions B┴1 = B┴2 E||1 = E||2 (1,2 inside, outside here)

E|| must vanish just outside conducting surface since E = 0 inside

E┴ may be finite just outside since induced surface charges allow E = 0 inside (TM modes only)

k2 B┴ = 0 at surface since B1 = 0 E k E 2 Two parallel plates, TE mode 1 1 x b z θ b y EM Waveguiding

E = E1 + E2 Fields in vacuum

= e E eiωt (ei(-ky sinθ + kz cosθ) - ei(ky sinθ + kz cosθ)) x o i(ωt - k1.r) E1 = ex Eo e

iωt -ikz cosθ θ = ex Eo e e 2i sin( ky sin ) k1 = -ey k sinθ + ez k cosθ

k .r = - ky sinθ + kz cosθ Boundary condition E||1 = E||2 = 0 1 means that E = E vanishes at y = 0, y = b || i(ωt - k2.r) E2 = -ex Eo e

θ π E||(y=0,b) if ky sin = n n = 1, 2, 3, .. k2 = +ey k sinθ + ez k cosθ

nπ nπ k .r = + ky sinθ + kz cosθ k = sinθ = 2 b sinθ kb EM Waveguiding Allowed field between guides is Fields iωt -ikz cosθ E = ex Eo e e 2i sin( ky sinθ ) iωt -ikz cosθ = E ei(ωt - k1.r) = ex Eo e e 2i sin(nπy/b) E1 ex o

nπ n2π2 1/2 Since sinθ = cosθ = 1 − = - k sinθ + k cosθ kb k2b2 k1 ey ez

k1.r = - ky sinθ + kz cosθ The wavenumber for the guided field is n2π2 1/2 E = -e E ei(ωt - k2.r) k = k cosθ = k2− n = 1, 2, 3, .. 2 x o g b2

k2 = +ey k sinθ + ez k cosθ Ex θ θ k2.r = + ky sin + kz cos sin(nπy/b)

y Profile of the first transverse electric mode (TE1) EM Waveguiding

Magnetic component of the guided field from Faraday’s Law

∇ x E = -∂B/∂t = -iω B for time-harmonic fields

2 2 i(ωt - kgz) B = i∇ x E /ω = 2 Eo / ω (0, ikg sin(nπy/b), √(k - kg ) cos(nπy/b) ) e

The BC B┴1 = B┴2 = 0 is satisfied since By = 0 on the conducting plates. The E and B components of the field are perpendicular since Bx = 0.

The phase velocity for the guided wave is vp = ω / kg = c k / kg

n2π2 1/2 n2π2 −1/2 k = k2 − Hence v = c 1 − g b2 p k2b2

The group velocity for the guided wave is vg = ∂ω / ∂kg= c ∂k / ∂kg = c kg / k

2 vp vg = c EM Waveguiding

Frequency Dispersion and Cutoff

b b

θ θ’

6 ω

nπ nπ c k = sinθ = cutoff when sinθ → 1 5 b sinθ kb ck cn 4 ω = ck = 2πν ν = = 2π 2b sinθ n = 3 3 c ωcutoff π 2 modes νcutoff = or = (n = 1) 2b c b n = 2 2 1 propagating mode 1 n2π2 1/2 ω2 n2π2 1/2 n = 1 k k2 − − vacuum propagation k g= b2 = c2 b2 g 0 1 2 3 4 5 6 EM Waveguiding

Summary of TEn modes

n2π2 1/2 = 2 E (i sin(nπy/b), 0 ,0) ei(ωt - kgz) k k2 − E o g = b2

2 2 i(ωt - kgz) B = 2 Eo / ω (0, ikg sin(nπy/b), √(k - kg ) cos(nπy/b) ) e

Phase velocity vp = ω / kg = c k / kg E B

Group velocity vg = ∂ω / ∂kg = c kg / k x x ck c ν = cutoff,n = n cutoff,n 2π 2b

y y n = 1 mode viewed along kg EM Waveguiding

Electric components of TEn guided fields viewed along x (plan view)

n = 1 n = 2 n = 3 n = 4

Magnetic components of TEn guided fields viewed along x (plan view)

EM Waveguiding

Rectangular waveguides

Boundary conditions B┴1 = B┴2 E||1 = E||2

E|| must vanish just outside conducting surface since E = 0 inside

E┴ may be finite just outside since induced surface charges allow E = 0 inside

B┴ = 0 at surface

a Infinite, rectangular conduit x z y b EM Waveguiding

TEmn modes in rectangular waveguides

TEn modes for two infinite plates are also solutions for the rectangular guide E field vanishes on xz plane plates as before, but not on the yz plane plates Charges are induced on the yz plates such that E = 0 inside the conductors

i(ωt - kgz) Let Ex = C f(x) sin(nπy/b) e

In free space ∇.E = 0 and Ez = 0 for a TEmn mode and ∂Ez/∂z = 0

Hence ∂E /∂x = -∂E /∂y x y By integration

E = -C nπ / b cos(mπx/a) sin(nπy/b) ei(ωt - kgz) f(x) = -nπ / b cos(mπx/a) x E = C mπ / a sin(mπx/a) cos(nπy/b) ei(ωt - kgz) satisfies this condition y Ez = 0 EM Waveguiding Dispersion Relation Substitute into wave equation (∇2 - 1/c 2 ∂ 2/∂t2 )E = 0 mπ 2 nπ 2 ∇2E = k 2 E x,y a b g x,y 2 2 ω2 ∂ /∂t Ex,y− = - E−x,y − mπ 2 nπ 2 k 2 - ω2 / c 2 = 0 a b g

− − − m2π2 n2π2 k 2 = k2 − − g a2 b2

Magnetic components of the guided field from Faraday’s Law

i(ωt - kgz) Bx = -C mπ / a kg / ω sin(mπx/a) cos(nπy/b) e i(ωt - kgz) By = -C nπ / b kg / ω cos(mπx/a) sin(nπy/b) e

2 2 i(ωt - kgz) Bz = i C (k −kg ) / ω cos(mπx/a) cos(nπy/b) e

EM Waveguiding

Cutoff Frequency

m2π2 n2π2 k 2 = k2 − − g a2 b2 ck c m2π2 n2π2 1/2 m2 n2 1/2 ν = cutoff = + = c + cutoff 2π 2π a2 b2 4a2 4b2

EM Waveguiding

Electric components of TEmn guided fields viewed along kg

m = 0 n = 1 m = 1 n = 1 m = 2 n = 2 m = 3 n = 1

Magnetic components of TEmn guided fields viewed along kg

y EM Waveguiding

Comparison of fields in TE and TM modes

www.opamp-electronics.com/tutorials/waveguides_2_14_08.htm EM Waveguiding

The TE01 mode

Most commonly used since a single frequency νcutoff,02 > ν > νcutoff,01 can be selected so that only one mode propagates.

Example 3 cm radar waves in a 1cm x 2 cm guide 02 12 1/2 ν = c + = 7.5 x 109 Hz cutoff,01 4a2 4x4.10−4 02 12 1/2 ν = c + = 7.50 x 109 Hz cutoff,01 4x1.10−4 4x4.10−4 12 02 1/2 ν = c + = 1.50 x 1010 Hz cutoff,10 4x1.10−4 4x4.10−4 12 12 1/2 ν = c + = 1.68 x 1010 Hz cutoff,11 4x1.10−4 4x4.10−4