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
EI
EI┴
EI┴
ER┴
EI||
θi θr
D┴1 = D┴2
ER┴
D┴1 = εo E ┴1
ER
ER||
-
-
-
-
y
-
-
D┴2 = εoεE ┴2
y
z
E||1 = E||2
z
E┴ can be finite just outside
conducting surface
E|| vanishes just outside and
inside conducting surface
EI|| ER|| EI|| ER||
EI|| ER|| EoI + EoR = 0
y
EoT = 0
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
k1 E1
Two parallel plates, TE mode
b
E2
x
θ
b
z
y
EM Waveguiding
E = E1 + E2
Fields in vacuum
= ex Eo eiωt (ei(-ky sinθ + kz cosθ) - ei(ky sinθ + kz cosθ))
= ex Eo eiωt e-ikz cosθ 2i sin( ky sinθ )
Boundary condition E||1 = E||2 = 0
means that E = E|| vanishes at y = 0, y = b
E||(y=0,b) if ky sinθ = nπ
k=
nπ
b sinθ
sinθ =
nπ
kb
n = 1, 2, 3, ..
E1 = ex Eo ei(ωt - k1.r)
k1 = -ey k sinθ + ez k cosθ
k1.r = - ky sinθ + kz cosθ
E2 = -ex Eo ei(ωt - k2.r)
k2 = +ey k sinθ + ez k cosθ
k2.r = + ky sinθ + kz cosθ
EM Waveguiding
Allowed field between guides is
E = ex Eo eiωt e-ikz cosθ 2i sin( ky sinθ )
= ex Eo eiωt e-ikz cosθ 2i sin(nπy/b)
Since
nπ
sinθ =
kb
n2π2 1/2
cosθ = 1 − 2 2
kb
The wavenumber for the guided field is
kg = k cosθ =
k2−
n2π2 1/2
n = 1, 2, 3, ..
b2
Fields
E1 = ex Eo ei(ωt - k1.r)
k1 = -ey k sinθ + ez k cosθ
k1.r = - ky sinθ + kz cosθ
E2 = -ex Eo ei(ωt - k2.r)
k2 = +ey k sinθ + ez k cosθ
Ex
k2.r = + ky sinθ + kz cosθ
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
B = i∇ x E /ω = 2 Eo / ω (0, ikg sin(nπy/b), √(k2 - kg2) cos(nπy/b) ) ei(ωt - 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 = ω / kg = c k / kg
kg =
k2 −
n2π2 1/2
n2π2 −1/2
Hence vp = c 1 − 2 2
b2
kb
The group velocity for the guided wave is vg = ∂ω / ∂kg= c ∂k / ∂kg = c kg / k
vp vg = c2
EM Waveguiding
Frequency Dispersion and Cutoff
b
b
θ
θ’
6
k=
nπ
b sinθ
sinθ =
nπ
kb
cutoff when sinθ → 1
ck
cn
ω = ck = 2πν ν =
=
2π 2b sinθ
c
ω
π
νcutoff =
or cutoff =
(n = 1)
c
b
2b
n2π2 1/2
ω2 n2π2 1/2
2
k g= k − 2
= c 2 − b2
b
5
ω
c
4
n=33
2 modes
n=22
n = 11
0
1 propagating mode
vacuum propagation
1
2
3
4
5
kg
6
EM Waveguiding
Summary of TEn modes
E = 2 Eo (i sin(nπy/b), 0 ,0)
kg =
ei(ωt - kgz)
k2 −
n2π2 1/2
b2
B = 2 Eo / ω (0, ikg sin(nπy/b), √(k2 - kg2) cos(nπy/b) ) ei(ωt - kgz)
Phase velocity vp = ω / kg = c k / kg
E
B
Group velocity vg = ∂ω / ∂kg = c kg / k
x
x
νcutoff,n =
ckcutoff,n
c
=
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
z
y
Magnetic components of TEn guided fields viewed along x (plan view)
z
y
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
b
z
y
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
Let Ex = C f(x) sin(nπy/b) ei(ωt - kgz)
In free space ∇.E = 0 and Ez = 0 for a TEmn mode and ∂Ez/∂z = 0
Hence ∂Ex/∂x = -∂Ey/∂y
f(x) = -nπ / b cos(mπx/a)
satisfies this condition
By integration
Ex = -C nπ / b cos(mπx/a) sin(nπy/b) ei(ωt - kgz)
Ey = C mπ / a sin(mπx/a) cos(nπy/b) ei(ωt - kgz)
Ez = 0
EM Waveguiding
Dispersion Relation
Substitute into wave equation (∇2 - 1/c 2 ∂ 2/∂t2 )E = 0
mπ 2
nπ 2
∇2Ex,y = −
−
− kg2 Ex,y
a
b
∂ 2/∂t2 Ex,y = - ω2 Ex,y
mπ 2
nπ 2
−
−
− kg2 - ω2 / c 2 = 0
a
b
2
kg =
k2 −
m2π2 n2π2
− 2
a2
b
Magnetic components of the guided field from Faraday’s Law
Bx = -C mπ / a kg / ω sin(mπx/a) cos(nπy/b) ei(ωt - kgz)
By = -C nπ / b kg / ω cos(mπx/a) sin(nπy/b) ei(ωt - kgz)
Bz = i C (k2 −kg2) / ω cos(mπx/a) cos(nπy/b) ei(ωt - kgz)
EM Waveguiding
Cutoff Frequency
2
kg =
k2 −
m2π2 n2π2
− 2
a2
b
ckcutoff c m2π2 n2π2 1/2
m2 n2 1/2
νcutoff =
=
+ 2
=c
+
b
4a2 4b2
2π
2π a2
EM Waveguiding
Electric components of TEmn guided fields viewed along kg
m=0n=1
m=1n=1
m=2n=2
m=3n=1
x
y
Magnetic components of TEmn guided fields viewed along kg
x
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
1/2
12
02
νcutoff,01= c
+
4a2 4x4.10−4
1/2
12
02
νcutoff,01= c
+
4x1.10−4 4x4.10−4
= 7.5 x 109 Hz
= 7.50 x 109 Hz
1/2
02
12
νcutoff,10= c
+
4x1.10−4 4x4.10−4
= 1.50 x 1010 Hz
1/2
12
12
νcutoff,11= c
+
4x1.10−4 4x4.10−4
= 1.68 x 1010 Hz