EM WaveguidingOverview

• 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 WaveguidingElectromagnetic wave reflection by perfect conductor

E┴ can be finite just outside

conducting surface

E|| vanishes just outside and

inside conducting surface

qi qr

EI

ER

z

y

EI||

EI┴

ER┴

ER||

z

y

EI┴ ER┴

- - - - - - -

D┴2 = eoeE ┴2

D┴1 = eo E ┴1

D┴1 = D┴2

z

y

EoI + EoR = 0

E||1 = E||2

EI|| ER||EI|| ER|| EI|| ER||

EoT = 0

EM WaveguidingElectromagnetic 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)

B┴ = 0 at surface since B1 = 0

Two parallel plates, TE mode

b

E1E2k1

k2

y

xzb

q

EM WaveguidingE = E1 + E2

= ex Eo eiwt (ei(-ky sinq + kz cosq) - ei(ky sinq + kz cosq))

= ex Eo eiwt e-ikz cosq 2i sin( ky sinq )

Boundary condition E||1 = E||2 = 0

means that E = E|| vanishes at y = 0, y = b

E||(y=0,b) if ky sinq = np n = 1, 2, 3, ..

Fields in vacuum

E1 = ex Eo ei(wt - k1.r)

k1 = -ey k sinq + ez k cosq

k1.r = - ky sinq + kz cosq E2 = -ex Eo ei(wt - k2.r)

k2 = +ey k sinq + ez k cosq

k2.r = + ky sinq + kz cosq

EM WaveguidingAllowed field between guides is

E = ex Eo eiwt e-ikz cosq 2i sin( ky sinq )

= ex Eo eiwt e-ikz cosq 2i sin(npy/b)

Since

The wavenumber for the guided field is

kg = k cosq n = 1, 2, 3, ..

Profile of the first transverse electric mode (TE1)

Fields

E1 = ex Eo ei(wt - k1.r)

k1 = -ey k sinq + ez k cosq

k1.r = - ky sinq + kz cosq E2 = -ex Eo ei(wt - k2.r)

k2 = +ey k sinq + ez k cosq

k2.r = + ky sinq + kz cosq

Ex

y

sin(npy/b)

EM Waveguiding

Magnetic component of the guided field from Faraday’s Law

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

B = i x E / w = 2 Eo / w (0, ikg sin(npy/b), √( - kg) cos(npy/b) ) ei(wt - kgz)

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 = w / kg = c k / kg

kg = Hence vp = c

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

vp vg = c2

EM Waveguiding

Frequency Dispersion and Cutoff

cutoff when → 1

w = ck = 2 pn n = =

ncutoff =

kg==

b b

q q’

0 1 2 3 4 5 6

1

2

3

4

5

6

kg

wc

n  = 3

1 propagating mode

2 modes

n  = 1 

n  = 2

vacuum propagation

EM Waveguiding

Summary of TEn modes

E = 2 Eo (i sin(npy/b), 0 ,0) ei(wt - kgz) kg =

B = 2 Eo / w (0, ikg sin(npy/b), √( - kg) cos(npy/b) ) ei(wt - kgz)

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

Group velocity vg = ∂w / ∂kg = c kg / k

ncutoff,n = = x

y

x

y viewed along kg

EM WaveguidingElectric 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)

z

y

z

y

EM WaveguidingRectangular 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

Infinite, rectangular conduit

0b

y

xz

a

EM WaveguidingTEmn 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

Let Ex = C f(x) sin(npy/b) ei(wt - kgz)

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

Hence ∂Ex/∂x = -∂Ey/∂y

f(x) = -np / b cos(mpx/a)

satisfies this condition

By integration

Ex = -C np / b cos(mpx/a) sin(npy/b) ei(wt - kgz)

Ey = C mp / a sin(mpx/a) cos(npy/b) ei(wt - kgz)

Ez = 0

EM WaveguidingDispersion Relation

Substitute into wave equation (2 - 1/c 2 ∂ 2/∂t2 )E = 0

2Ex,y = Ex,y

∂ 2/∂t2 Ex,y = - w2 Ex,y

- w2 / c 2 = 0

kg =

Magnetic components of the guided field from Faraday’s Law

Bx = -C mp / a / w sin(mpx/a) cos(npy/b) ei(wt - kgz)

By = -C np / b / w cos(mpx/a) sin(npy/b) ei(wt - kgz)

Bz = i C√) / w cos(mpx/a) cos(npy/b) ei(wt - kgz)

EM WaveguidingCutoff Frequency

kg =

ncutoff =

EM WaveguidingElectric 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

x

y

x

y

EM Waveguiding

Comparison of fields in TE and TM modes

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

EM WaveguidingThe TE01 mode

Most commonly used since a single frequency ncutoff,02 > n > ncutoff,01 can be

selected so that only one mode propagates.

Example 3 cm radar waves in a 1cm x 2 cm guide

ncutoff,01= c = 7.5 x 109 Hz

ncutoff,01= c = 7.50 x 109 Hz

ncutoff,10= c = 1.50 x 1010 Hz

ncutoff,11= c = 1.68 x 1010 Hz