Plane Wave Propagation in Lossless Media

Hon Tat Hui Plane Wave Propagation in Lossless Media

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Plane Wave Propagation in Lossless Media

1 Plane Waves in Lossless MediaIn a source free lossless medium, .0=== σρJ

00=⋅∇=⋅∇

∂∂

=×∇

∂∂

=×∇

HE

EH

H-E

με

ε

μ

t

t

Maxwell’s equations:

See animation “Plane Wave Viewer”

J =current densityρ =charge densityσ =conductivity


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Take the curl of the first equation and make use of the second and the third equations, we have:

EHE 2

22

tt ∂∂

=×∇∂∂

=∇ μεμ( ) EEE 2

:Note∇−⋅∇∇=×∇×∇

This is called the wave equation:

02

22 =

∂∂

−∇ EEt

με

A similar equation for H can be obtained:

02

22 =

∂∂

−∇ HHt

με


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In free space, the wave equation for E is:

02

2

002 =

∂∂

−∇ EEt

εμ

where

2001c

=εμ

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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where 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.

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:

)(cosor )(sin φωφω ++ ttUsing complex notation, the E field, for example, can be written as:

( ) ( )⎭⎬⎫

⎩⎨⎧=

•tjezyxtzyx ω,,Re,,, EE

( )zyx ,,•

E


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The benefits of using the phasor form are that:

( ) ( )

( ) ( )⎭⎬⎫

⎩⎨⎧=

⎭⎬⎫

⎩⎨⎧∂∂

=∂∂

•

•

tjn

tjn

n

n

n

ezyxj

ezyxt

tzyxt

ω

ω

ω ,,Re

,,Re,,,

E

EE

( ) ( )

( )( )

⎭⎬⎫

⎩⎨⎧

=

⎭⎬⎫

⎩⎨⎧=

•

•

∫∫∫∫tj

n

tj

ezyxj

dtdtezyxdtdttzyx

ω

ω

ω,,1Re

,,Re,,,

E

EE


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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 phasorform as (dropping the dot on the top, same as below):

( )2

22 20 0 0 02

2 20 0

0 0

(dropping the dot sign) 0

jt

μ ε μ ε ω

μ ε ω

• •∂∇ − = ⇒ ∇ − =

∂⇒ ∇ + =

E E E E

E E


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In phasor form, Maxwell’s equations can be written as:

0=⋅∇=⋅∇=×∇=×∇

BD

DHB-E

ρjωjω

00

22200

2

where

0

εμω

ωεμ

=

=+∇=+∇

k

k EEEE

Using the phasor form expression, the wave equation for E field is also called the Helmholtz’s equation, which is:


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k is called the wavenumber or the propagation constant.

0000

22λππεμω ====

cfkk

where λ0 is the free space wavelength.

In an arbitrary medium with ε =ε0εr and μ =μ0μ r,

rrrrrr cfk εμ

λπεμπεμεμω0

0022

===

We call,

medium in the wavelength2 0 ===rrk εμ

λπλ


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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:

022

2

2

2

2

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

+∂∂

+∂∂

xEkzyx

Consider a special case of the Ex in which there is no variation of Ex in the x and y directions, i.e.,

02

2

2

2

=∂∂

=∂∂

xx Ey

Ex


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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:

( ) ( ) 022

2

=+ zEkdz

zEdx

x

Note that a plane wave is not physically realizable because it extends to an infinite extent in the x and ydirections. 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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are plane waves propagating along the +z direction and –z direction.

Solutions to the plane wave equation take one form of the following functions, depending on the boundary conditions:

( ) jkzx eEzE −+= 0 .1( ) jkz

x eEzE +−= 0 .2( ) jkzjkz

x eEeEzE +−−+ += 00 .3

E0+ and E0

- are constants to be determined by boundary conditions.

jkzjkz eEeE +−−+00 and


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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.

1.1 Solutions to plane wave equation

( ) ( ) ( )kztE

eeEtzEeEzE tjjkzx

jkzx

−=

=→=+

−+−+

ω

ω

cos

Re, )1(

0

00

In time domain,

( ) ( ).cos , then ,1 Assume 0 zttzEkE x −====+ ω


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( ) ( )zzEt x −== cos 0, ,0At Ex

z0 π/2 π 3π/2 2π-π/2-π-3π/2-2π

1

-1

( ) ( )zzEst x −== 1cos 1, ,1At Ex

z0 π/2 π 3π/2 2π-π/2-π-3π/2-2π

1

-1

( ) ( )zzEst x −== 2cos 2, ,2At Ex

z0 π/2 π 3π/2 2π-π/2-π-3π/2-2π

1

-1


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The solution is plotted on the next page for the first several seconds.

( ) ( ) ( )kztE

eeEtzEeEzE tjjkzx

jkzx

+=

=→=−

+−+−

ω

ω

cos

Re, )2(

0

00

( ) ( ).cos , then ,1 Assume 0 zttzEkE x +====− ω


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( ) ( )zzEt x cos 0, ,0At == Ex

z0 π/2 π 3π/2 2π-π/2-π-3π/2-2π

1

-1

( ) ( )zzEst x +== 1cos 1, ,1At Ex

z0 π/2 π 3π/2 2π-π/2-π-3π/2-2π

1

-1

( ) ( )zzEst x +== 2cos 2, ,2At Ex

z0 π/2 π 3π/2 2π-π/2-π-3π/2-2π

1

-1


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direction. in the gpropagatin wave a is

direction. in the gpropagatin wave a is

0

0

zeE

zeEjkz

jkz

−

++−

−+

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 –zdirection.


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If ω ≠ 1, k ≠ 1, and E+0 ≠ 1, then for the wave moving

to the right: ( ) ( ).cos, 0 kztEtzEx −= + ω

(1) At t = 0, ( ) ( ).cos0, 0 kzEzEx −= +

Consider a zero point (z coordinate = z0) of the wave (for example the first one to the right of the origin), then ( ) ( )

kzkz

kzzEx

2

2

0cos 00,

00

00

ππ=⇒−=−⇒

=−⇒=

1.2 Propagation speed of a general plane wave


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( ) ( )

kkzkz

kzzEx

2

2

0cos 01,

11

11

πωπω

ω

+=⇒−=−⇒

=−⇒=

kkkkzzs ωππω

=−+=−=22

1in change Distance 01

1ms 1/speedn propagatio Thus −==k

sk

ωω

(2) At t = 1s, ( ) ( ).cos0, 0 kzEzEx −= + ωConsider the same zero point of the wave as in (1) but now its z coordinate has move from z0 to z1, then

A same result can be obtained for the wave moving to the left.


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Once the electric field is known, the accompanying magnetic field H can be found from the Maxwell’s equation

H-E μjω=×∇

For example, if the solution for E is, ( ) ,ˆˆ 0

jkzx eEEz −+== xxE

then the solution for H is:

( ) yjkz

jkz

HeEkz

ejEz yyyH ˆˆˆ 0

0 ==∂

∂−

= −+−+

ωμωμ

Note that H is ⊥ to E and they are shown on next page.

1.3 Solution for the magnetic field


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H and E propagate in free spaceSee animation “Plane Wave E and H Vector Motions ”


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The ratio of Ex to Hy is called the intrinsic impedance of the medium, η.

( )( ) )( Ω

kzHzE

y

x

εμ

μεωωμωμη ====

Note that η is independent of z. In free space,

Ω 3771200

00 ≈== π

εμη

and Ex and Hy are in phase (as η0 is a real number).


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The phase velocity (propagation speed of a constant-phase point) of the wave up is given by:

(m/s) 1μεμεω

ωω===

kup

The plane wave is also called the TEM wave (TEM = Transverse ElectroMagnetic) in which Ez = Hz = 0 where z is the direction of propagation.

See animation “Plane Wave in 3D”


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The general expressions of a plane wave are:

rk

rk

HH

EE⋅−

⋅−

=

=j

j

e

e

0

0

1.4 Expressions for a general plane wave

E0 and H0 are vectors in arbitrary directions. k is the vector propagation constant whose magnitude is kand whose direction is the direction of propagation of the wave. r is the observation position vector.

zyxr

zyxk

ˆˆˆ

,ˆˆˆ 222

zyx

kkkkkkk zyxzyx

++=

++=++=


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rk

E

HE

H

k(index finger) (thumb)

(middle finger)

Right-hand rule:


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kHEEkH

kHE

ˆ ,ˆ1

×=×=

⊥⊥

ηη

Using Maxwell’s equations

EHH-E

εμ

jωjω

=×∇=×∇

it can be shown that we have the following relations for the field vectors and the propagation direction.


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A uniform plane wave with propagates in the +z-direction 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.

xExE ˆ=Example 1

(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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Solutions Ee ofvector unit ˆ =

k k(a)

k

(m/s) 101.54

1 8×====c

kup με

ω

(phasor form)

(instantaneous form)

,ˆ , kzk =⋅rk

k


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k

k

The cosine function has a positive maximum when its argument equals zero (ignoring the 2nπ ambiguity). Thus at t = 0 and z = 1/8,


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zk ˆˆ =

(m)

See animation “Plane Wave Simulator”


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2 Polarization of Plane Waves

Ex

Ey

Eectric-field vector

Ex

Ey


Ex

Ey


Linearly polarized Circularly polarized Elliptically polarized

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 ellipticalpolarizations.

See animation “Polarization of a Plane Wave - 2D View”


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31Polarization of a Plane Wave

See animation “Polarization of a Plane Wave - 3D View”


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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.

(a) Linear polarization

(b) Circular PolarizationA 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 left-handed corresponding to the electric-field vector rotating clockwise (right-handed) or anti-clockwise (left-handed).


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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).

(c) Elliptical Polarization


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The instantaneous expression for E is:

( ) ( ) ( )kztEkztE

ejEeEtz

yx

jkztjy

jkztjx

−+−=

−= −−

ωω

ωω

sinˆcosˆ

ˆˆRe,

00

00

yx

yxE

For example, consider a plane wave:

jkzy

jkzx

yx

ejEeE

EE−− −=

+=

00 ˆˆ

ˆˆ

yx

yxEjkz

yy

jkzxx

ejEE

eEE−

−

−=

=

0

0

( ) ( )0 0= cos , sinx x y yX E E t kz Y E E t kzω ω= − = = −

Note that the phase difference between Ex and Ey is 90º.

Let:

Ex0 and Ey0 are both real numbers


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Case 1: 0 or 0, thenxo yoE E= =0 or 0X Y= =

Both are straight lines. Hence the wave is linearlypolarized.

Case 2: , thenxo yoE E C= =( ) ( )2 2 2 2 2 2cos sinX Y C t kz t kz Cω ω⎡ ⎤+ = − + − =⎣ ⎦

X and Y describe a circle. Hence the wave iscircularly polarized.

Case 3: , thenxo yoE E≠

( ) ( )2 2

2 22 20 0

cos sin 1x y

X Y t kz t kzE E

ω ω+ = − + − =

X and Y describe an ellipse. Hence the wave iselliptically polarized.


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Example 2Two circularly polarized plane waves watched at z = 0 are given by: ( ) ( ) ( )

( ) ( ) ( )°−−°−=°++°+=

1.53sin5ˆ1.53cos5ˆ1.53sin5ˆ1.53cos5ˆ

2

1

tttttt

ωωωω

yxEyxE

Show that they combine together to form a linearly polarized wave.Solutions:

( ) ( )[ ]( ) ( )[ ]( ) ( ) ( ) ( )( ) ( )tt

tttttt

ωωωω

ωωωω

cos8ˆcos6ˆ 1.53sincos10ˆ1.53coscos10ˆ

1.53sin51.53sin5ˆ 1.53cos51.53cos5ˆ21

yxyx

yxEEE

+=°+°=

°−−°++°−+°+=+=


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Now,

( ) ( )

( )( )

34slopewith linestraight a ofequation

34

34

cos6cos8Then

Y ,Let cos8 ,cos6

=⇒=

==

==

==

XY

tt

XY

EEXtEtE

yx

yx

ωω

ωω

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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In general, the polarization state of an EM wave is characterized by two parameters.

jkzy

jy

jkzxx eEeEeEE −− == 00 , δ

°≤≤⎟⎟⎠

⎞⎜⎜⎝

⎛=⇒ − 900 ,tan

to of Ratio .1

0

01

00

γγx

y

xy

EE

EE

°≤≤° 180018- , i.e., ,and between difference Phase .2

δδyx EE

Generic Polarization Description Method


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γ = 0 or 90º and for any value of δ⇒ linearly polarized

polarized circularly hand)-(right 90 and 54

⇒°=°= δγ

For example:

Plane Wave Propagation in Lossless Media

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