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NUS/ECE
EE2011
Plane Wave Propagation in Lossless Media
See animation “Plane Wave Viewer”
1 Plane Waves in Lossless Media
In a source free lossless medium, J = ρ = σ = 0.
Maxwell’s equations:
∂H
∇ × E = -μ
∂t
∂E
∇×H = ε
∂t
ε∇ ⋅ E = 0
J =current density
ρ =charge density
σ =conductivity
μ∇ ⋅ H = 0
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Plane Wave Propagation in Lossless Media
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Take the curl of the first equation and make use of the
second and the third equations, we have: Note :
2
∂
∂
∇ 2 E = μ ∇ × H = με 2 E
∂t
∂t
∇ × ∇ × E = ∇(∇ ⋅ E ) − ∇ 2 E
This is called the wave equation:
2
∂
∇ 2E − με 2 E = 0
∂t
A similar equation for H can be obtained:
∂
∇ H − με 2 H = 0
∂t
2
2
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Plane Wave Propagation in Lossless Media
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In free space, the wave equation for E is:
∂
∇ E − μ 0ε 0 2 E = 0
∂t
2
2
where
1
μ 0ε 0 = 2
c
c being the speed of light in free space (~ 3 × 108 (m/s)).
Hence the speed of light can be derived from Maxwell’s
equation.
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To simplify subsequent analyses, we consider a special
case in which the field (and the source) variation with
time takes the form of a sinusoidal function:
sin(ωt + φ ) or cos(ωt + φ )
Using complex notation, the E field, for example, can be
written as:
•
⎧
⎫
E( x, y , z , t ) = Re⎨E( x, y , z )e jωt ⎬
⎩
⎭
•
where E( x, y, z ) is called the phasor form of E(x,y,z,t) and
is in general a complex number depending on the spatial
coordinates only. Note that the phasor form also includes
the initial phase information and is a complex number.
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The benefits of using the phasor form are that:
⎧ ∂n •
∂n
j ωt ⎫
E( x, y, z , t ) = Re ⎨ n E( x, y, z )e ⎬
n
∂t
⎩ ∂t
⎭
•
n
⎧
jωt ⎫
= Re ⎨( jω ) E( x, y, z )e ⎬
⎩
⎭
•
⎧
⎫
jωt
∫ "∫ E(x, y, z, t )dt " dt = Re⎨⎩∫ "∫ E(x, y, z )e dt " dt ⎬⎭
⎧ 1 •
jωt ⎫
= Re⎨
E( x, y, z )e ⎬
n
⎩ ( jω )
⎭
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Therefore differentiation or integration with respect to
time can be replaced by multiplication or division of the
phasor form with the factor jω. All other field functions
and source functions can be expressed in the phasor form.
As all time-harmonic functions involve the common
factor ejωt in their phasor form expressions, we can
eliminate this factor when dealing with the Maxwell’s
equation. The wave equation can now be put in phasor
form as (dropping the dot on the top, same as below):
2
•
•
∂
2
∇ 2 E − μ0ε 0 2 E = 0 ⇒ ∇ 2 E− μ0ε 0 ( jω ) E = 0
∂t
(dropping the dot sign) ⇒ ∇ 2 E + μ0ε 0ω 2 E = 0
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Plane Wave Propagation in Lossless Media
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In phasor form, Maxwell’s equations can be written as:
∇ × E = - jωB
∇ × H = jωD
∇⋅D = ρ
∇⋅B = 0
Using the phasor form expression, the wave equation for
E field is also called the Helmholtz’s equation, which is:
∇ 2 E + μ0ε 0ω 2 E = ∇ 2 E + k 2 E = 0
where k = ω μ0ε 0
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k is called the wavenumber or the propagation
constant.
2πf 2π
k = k 0 = ω μ 0ε 0 =
=
λ0
c
where λ0 is the free space wavelength.
In an arbitrary medium with ε =ε0εr and μ =μ0μ r,
2πf
k = ω μ 0ε 0 μ r ε r =
c
We call,
2π
λ=
=
k
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μ rε r =
2π
λ0
μrε r
λ0
= wavelength in the medium
μrε r
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Plane Wave Propagation in Lossless Media
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In Cartesian coordinates, the Helmholtz’s equation can
be written as three scalar equations in terms of the
respective x, y, and z components of the E field. For
example, the scalar equation for the Ex component is:
⎛ ∂2
∂2
∂2
2⎞
⎜⎜ 2 + 2 + 2 + k ⎟⎟ Ex = 0
∂y
∂z
⎠
⎝ ∂x
Consider a special case of the Ex in which there is no
variation of Ex in the x and y directions, i.e.,
∂2
∂2
Ex = 2 Ex = 0
2
∂y
∂x
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This is called the plane wave condition and Ex(z) now
varies with z only. The wave equation for Ex becomes:
d 2 Ex (z ) 2
+ k Ex (z ) = 0
2
dz
Note that a plane wave is not physically realizable
because it extends to an infinite extent in the x and y
directions. However, when considered over a small plane
area, its propagation characteristic is very close to a
spherical wave, which is a real and common form of
electromagnetic wave propagating.
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Plane Wave Propagation in Lossless Media
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Solutions to the plane wave equation take one form of the
following functions, depending on the boundary
conditions:
1. E x ( z ) = E0+ e − jkz
2. E x ( z ) = E0− e + jkz
3. E x (z ) = E0+ e − jkz + E0− e + jkz
E0+ and E0- are constants to be determined by boundary
conditions.
+ − jkz
0
− + jkz
0
are plane waves propagating along the
E e and E e
+z direction and –z direction.
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1.1 Solutions to plane wave equation
In time domain,
(1) E x ( z ) = E0+ e − jkz → E x ( z , t ) = Re{E0+ e − jkz e jωt }
= E0+ cos(ωt − kz )
Assume E0+ = ω = k = 1, then E x (z , t ) = cos(t − z ).
We can plot this solution for several seconds to
see its motion in space. We focus on one period
of the sine function only while keeping in mind
that this period repeats itself continuously in
both left and right directions.
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At t = 0, E x (z ,0 ) = cos(− z )
-2π -3π/2
-π
Ex
1
0
-π/2
π/2
π
3π/2
2π
π
3π/2
2π
π
3π/2
2π
z
-1
At t = 1s, E x ( z ,1) = cos(1 − z )
-2π -3π/2
-π
Ex
1
0
-π/2
π/2
z
-1
At t = 2 s, E x (z ,2 ) = cos(2 − z )
-2π -3π/2
-π
Ex
1
0
-π/2
π/2
z
-1
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( 2)
− + jkz
− + jkz jωt
(
)
(
)
{
E x z = E0 e → E x z , t = Re E0 e e }
= E0− cos(ωt + kz )
Assume E0− = ω = k = 1, then E x ( z , t ) = cos(t + z ).
The solution is plotted on the next page for the
first several seconds.
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At t = 0, E x (z ,0 ) = cos( z )
-2π -3π/2
Ex
1
-π
0
-π/2
π/2
π
3π/2
2π
π
3π/2
2π
π
3π/2
2π
z
-1
At t = 1s, E x (z ,1) = cos(1 + z )
-2π -3π/2
Ex
1
-π
0
-π/2
π/2
z
-1
At t = 2 s, E x ( z ,2 ) = cos(2 + z )
-2π -3π/2
-π
Ex
1
0
-π/2
π/2
z
-1
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E0+ e − jkz is a wave propagating in the + z direction.
E0− e + jkz is a wave propagating in the − z direction.
For the wave moving in +z direction, in a time of 1 second,
the wave moves in 1 unit of distance (for example, meter).
Then the speed of propagation is (1 m/1 s = 1ms-1). A
similar result can be obtained for the wave moving in –z
direction.
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1.2 Propagation speed of a general plane wave
If ω ≠ 1, k ≠ 1, and E+0 ≠ 1, then for the wave moving
to the right:
E x ( z , t ) = E0+ cos(ωt − kz ).
+
(
)
E
z
,
0
=
E
(1) At t = 0, x
0 cos(− kz ).
Consider a zero point (z coordinate = z0) of the
wave (for example the first one to the right of the
origin), then
E x (z0 ,0 ) = 0 ⇒ cos(− kz0 ) = 0
⇒ − kz0 = −
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π
2
⇒ z0 =
π
2k
Plane Wave Propagation in Lossless Media
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(2) At t = 1s, E x ( z ,0 ) = E0+ cos(ω − kz ).
Consider the same zero point of the wave as in (1) but
now its z coordinate has move from z0 to z1, then
E x (z1 ,1) = 0 ⇒ cos(ω − kz1 ) = 0
⇒ ω − kz1 = −
π
2
⇒ z1 =
Distance change in 1s = z1 − z0 =
Thus propagation speed =
ω
ω
k
+
π
2k
/ 1s =
ω
ω
k
−
+
π
π
2k
2k
ms
=
ω
k
−1
k
k
A same result can be obtained for the wave moving to
the left.
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Plane Wave Propagation in Lossless Media
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1.3 Solution for the magnetic field
Once the electric field is known, the accompanying
magnetic field H can be found from the Maxwell’s
equation
∇ × E = - jωμH
For example, if the solution for E is,
E( z ) = xˆ E x = xˆ E0+ e − jkz ,
then the solution for H is:
E0+ ∂e − jkz
k + − jkz
H (z ) = yˆ
= yˆ
E0 e
= yˆ H y
ωμ
− jωμ ∂z
Note that H is ⊥ to E and they are shown on next page.
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Plane Wave Propagation in Lossless Media
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H and E propagate in free space
See animation “Plane Wave E and H Vector Motions ”
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Plane Wave Propagation in Lossless Media
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The ratio of Ex to Hy is called the intrinsic impedance of
the medium, η.
ωμ
E x ( z ) ωμ
η=
=
=
=
H y (z ) k
ω με
μ
(Ω)
ε
Note that η is independent of z. In free space,
μ0
η0 =
= 120π ≈ 377 Ω
ε0
and Ex and Hy are in phase (as η0 is a real number).
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The phase velocity (propagation speed of a constantphase point) of the wave up is given by:
1
ω
ω
up = =
=
(m/s)
k ω με
με
See animation “Plane Wave in 3D”
The plane wave is also called the TEM wave (TEM =
Transverse ElectroMagnetic) in which Ez = Hz = 0 where
z is the direction of propagation.
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Plane Wave Propagation in Lossless Media
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1.4 Expressions for a general plane wave
The general expressions of a plane wave are:
E = E0 e − jk ⋅r
H = H 0e
− jk ⋅r
E0 and H0 are vectors in arbitrary directions. k is the
vector propagation constant whose magnitude is k
and whose direction is the direction of propagation of
the wave. r is the observation position vector.
k = k x xˆ + k y yˆ + k z zˆ , k = k x2 + k y2 + k z2
r = xxˆ + yyˆ + zzˆ
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Right-hand rule:
k
(index finger) (thumb)
r E
E
k
H
H
(middle finger)
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Plane Wave Propagation in Lossless Media
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Using Maxwell’s equations
∇ × E = - jωμH
∇ × H = jωεE
it can be shown that we have the following relations
for the field vectors and the propagation direction.
E⊥H⊥k
1ˆ
H = k × E,
E = ηH × kˆ
η
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Example 1
A uniform plane wave with E = xˆ E x propagates in the +zdirection in a lossless medium with εr = 4 and μr = 1.
Assume that Ex is sinusoidal with a frequency of 100 MHz
and that it has a positive maximum value of 10-4 V/m at t = 0
and z = 1/8 m.
(a) Calculate the wavelength λ and the phase velocity up,
and find expressions for the instantaneous electric and
magnetic field intensities.
(b) Determine the positions where Ex is a positive
maximum at the time instant t = 10-8s.
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eˆ = unit vector of E
Solutions
ˆ ⋅r = kz,
, kkk̂
(a) k̂
k
up =
ω
k
=
1
c
=
= 1.5 × 108 (m/s)
4
με
(phasor form)
k̂
(instantaneous form)
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Plane Wave Propagation in Lossless Media
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The cosine function has a positive maximum when its argument
equals zero (ignoring the 2nπ ambiguity). Thus at t = 0 and z = 1/8,
k̂
k̂
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Plane Wave Propagation in Lossless Media
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kˆ = zˆ
(m)
See animation “Plane Wave Simulator”
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Plane Wave Propagation in Lossless Media
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2 Polarization of Plane Waves
The polarization of a plane wave is the figure the tip of
the instantaneous electric-field vector E traces out with
time at a fixed observation point. There are three types of
polarizations: the linear, circular, and elliptical
polarizations.
Ey
Ey
Ey
Ex
Eectric-field vector
Linearly polarized
Ex
Ex
Eectric-field vector
Eectric-field vector
Circularly polarized
Elliptically polarized
See animation “Polarization of a Plane Wave - 2D View”
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See animation “Polarization of a
Plane Wave - 3D View”
Polarization of a Plane Wave
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(a) Linear polarization
A plane wave is linearly polarized at a fixed
observation point if the tip of the electric-field vector at
that point moves along the same straight line at every
instant of time.
(b) Circular Polarization
A plane wave is circularly polarized at a a fixed
observation point if the tip of the electric-field vector
at that point traces out a circle as a function of time.
Circular polarization can be either right-handed or lefthanded corresponding to the electric-field vector
rotating clockwise (right-handed) or anti-clockwise
(left-handed).
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Plane Wave Propagation in Lossless Media
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(c) Elliptical Polarization
A plane wave is elliptically polarized at a a fixed
observation point if the tip of the electric-field vector
at that point traces out an ellipse as a function of time.
Elliptically polarization can be either right-handed or
left-handed corresponding to the electric-field vector
rotating clockwise (right-handed) or anti-clockwise
(left-handed).
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Plane Wave Propagation in Lossless Media
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For example, consider a plane wave:
E x = E x 0 e − jkz
E = xˆ E x + yˆ E y
= xˆ E x 0 e
− jkz
Ex0 and Ey0 are
both real numbers
− yˆ jE y 0 e
− jkz
E y = − jE y 0 e − jkz
Note that the phase difference between Ex and Ey is 90º.
The instantaneous expression for E is:
{
E( z , t ) = Re xˆ E x 0 e jωt − jkz − yˆ jE y 0 e jωt − jkz
}
= xˆ E x 0 cos(ωt − kz ) + yˆ E y 0 sin (ωt − kz )
Let:
X = Ex =Ex 0 cos (ωt − kz ) , Y = E y = E y 0 sin (ωt − kz )
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Plane Wave Propagation in Lossless Media
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Case 1: Exo = 0 or E yo = 0, then
X = 0 or Y = 0
Both are straight lines. Hence the wave is linearly
polarized.
Case 2: Exo = E yo = C , then
X 2 + Y 2 = C 2 ⎡⎣cos 2 (ωt − kz ) + sin 2 (ωt − kz ) ⎤⎦ = C 2
X and Y describe a circle. Hence the wave is
circularly polarized.
Case 3: Exo ≠ E yo , then
X2 Y2
2
2
t
kz
+
=
cos
ω
−
+
sin
(
)
(ωt − kz ) = 1
2
2
Ex 0 E y 0
X and Y describe an ellipse. Hence the wave is
elliptically polarized.
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Plane Wave Propagation in Lossless Media
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Example 2
Two circularly polarized plane waves watched at z = 0 are
given by:
E1 (t ) = xˆ 5 cos(ωt + 53.1°) + yˆ 5 sin (ωt + 53.1°)
E 2 (t ) = xˆ 5 cos(ωt − 53.1°) − yˆ 5 sin (ωt − 53.1°)
Show that they combine together to form a linearly
polarized wave.
Solutions:
E = E1 + E2 = xˆ [5 cos(ωt + 53.1°) + 5 cos(ωt − 53.1°)]
+ yˆ [5 sin (ωt + 53.1°) − 5 sin (ωt − 53.1°)]
= xˆ 10 cos(ωt )cos(53.1°) + yˆ 10 cos(ωt )sin (53.1°)
= xˆ 6 cos(ωt ) + yˆ 8 cos(ωt )
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Plane Wave Propagation in Lossless Media
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Now,
E x = 6 cos(ωt ), E y = 8 cos(ωt )
Let
X = Ex , Y = E y
Y 8 cos(ωt ) 4
=
=
Then
X 6 cos(ωt ) 3
4
4
Y = X ⇒ equation of a straight line with slope =
3
3
Hence the locus of the combined electric field falls on a
straight line and the polarization of the combined wave is
thus linear.
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Generic Polarization Description Method
In general, the polarization state of an EM wave is
characterized by two parameters.
E x = E x 0 e − jkz , E y = e jδ E y 0 e − jkz
1. Ratio of E y 0 to E x 0
⎛ Ey0 ⎞
⎟⎟,
⇒ γ = tan ⎜⎜
⎝ Ex0 ⎠
−1
0 ≤ γ ≤ 90°
2. Phase difference between E x and E y ,
i.e., δ ,
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- 180° ≤ δ ≤ 180°
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Plane Wave Propagation in Lossless Media
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For example:
γ = 0 or 90º and for any value of δ
⇒ linearly polarized
γ = 45° and δ = 90°
⇒
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(right - hand) circularly polarized
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Plane Wave Propagation in Lossless Media