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 surfaceE|| 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

yx

zb 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 = 0means 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 iskg = 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 WaveguidingMagnetic 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 WaveguidingFrequency Dispersion and Cutoff

cutoff when → 1 w = ck = 2pn 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 WaveguidingSummary 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

yx

z

a

EM WaveguidingTEmn modes in rectangular waveguidesTEn modes for two infinite plates are also solutions for the rectangular guideE 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 RelationSubstitute 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 WaveguidingComparison 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 beselected so that only one mode propagates.

Example 3 cm radar waves in a 1cm x 2 cm guidencutoff,01= c = 7.5 x 109 Hzncutoff,01= c = 7.50 x 109 Hzncutoff,10= c = 1.50 x 1010 Hzncutoff,11= c = 1.68 x 1010 Hz