ELEG 648Plane waves

Mark Mirotznik, Ph.D.Associate Professor

The University of DelawareEmail: mirotzni@ece.udel.edu

SUMMARY

mBD

Jt

DHM

t

BE

mBD

JDjHMBjE

~~~~

~~~~~~

0)~~

(ˆ~)~~

(ˆ

~)

~~(ˆ0)

~~(ˆ

1212

1212

BBnDDn

JHHnEEn

s

s

0)(ˆ)(ˆ

)(ˆ0)(ˆ

1212

1212

BBnDDn

JHHnEEn

s

s

EJ

HB

ED

c~~

~~

~~

EJ

HB

ED

c

2

)1( jjXRZ sss

0,0][

]2

1[,]

2

1[

)(

2

22

vdv ii

vevm

ss

dvEPdvJEP

dvEWdvHW

dsHEP

0~

2

1,0]

~~2

1[

]~

4

1[,]

~4

1[

)~~

(

2*

22

*

vdv ii

vevm

ss

dvEPdvJEP

dvEWdvHW

dsHEP

Frequency DomainTime Domain

Wave Equation

0

B

E

JEt

EH

t

HE

Ht

E

t

HE

)(][

t

J

t

E

t

EE

JEt

E

tE

2

2

AAA

2)( Vector Identity

t

J

t

E

t

EEE

2

22)(

t

J

t

E

t

EE

2

221

Time Dependent Homogenous Wave Equation (E-Field)

1

2

22

t

J

t

E

t

EE

Wave EquationSource-Free Time Dependent Homogenous Wave Equation (E-Field)

1

2

22

t

J

t

E

t

EE

0,0 J

Source Free

02

22

t

E

t

EE

Wave EquationSource-Free Time Dependent Homogenous Wave Equation (E-Field)

1

2

22

t

J

t

E

t

EE

0,0 J

Source Free

02

22

t

E

t

EE

Source-Free Lossless Time Dependent Homogenous Wave Equation (E-Field)

0Lossless

02

22

t

EE

Wave Equation: Time Harmonic

1

2

22

t

J

t

E

t

EE

0,0 J

Source Free

02

22

t

E

t

EE

0Lossless

02

22

t

EE

Time Domain Frequency Domain

~1~~~~ 22 JEjEE

0,0 J

Source Free

0~~~ 22 EjEE

0Lossless

0~~ 22 EE

“Helmholtz Equation”

General Solution Case: Time HarmonicRectangular Coordinates

0~~ 22 EE 0

~~ 22 EE Wave Number

0

0

0

22

2

2

2

2

2

22

2

2

2

2

2

22

2

2

2

2

2

zzzz

yyyy

xxxx

Ez

E

y

E

x

E

Ez

E

y

E

x

E

Ez

E

y

E

x

E

Separation of Variable Solutions

022

2

2

2

2

2

xxxx E

z

E

y

E

x

E

Assume Solution of the form: )()()(),,( zhygxfzyxEx

Separation of Variable Solutions

022

2

2

2

2

2

xxxx E

z

E

y

E

x

E

Assume Solution of the form: )()()(),,( zhygxfzyxEx

0)()()()()()()()()()()()( 2 zhygxfzhygxfzhygxfzhygxf

0)()()(

)]()()()()()()()()()()()([ 2

zhygxf

zhygxfzhygxfzhygxfzhygxf

Separation of Variable Solutions

022

2

2

2

2

2

xxxx E

z

E

y

E

x

E

Assume Solution of the form: )()()(),,( zhygxfzyxEx

0)(

)(

)(

)(

)(

)( 2

zh

zh

yg

yg

xf

xf

function of x

function of y

function of z

constant

0)()()(

)]()()()()()()()()()()()([ 2

zhygxf

zhygxfzhygxfzhygxfzhygxf

Separation of Variable Solutions

2222

2

2

2

)(

)(

)(

)(

)(

)(

zyx

z

y

x

zh

zh

yg

yg

xf

xf

2222

2

2

2

0)()(

0)()(

0)()(

zyx

z

y

x

zhzh

ygyg

xfxf

0)(

)(

)(

)(

)(

)( 2

zh

zh

yg

yg

xf

xf

function of x

function of y

function of z

constant

Separation of Variable Solutions

Solutions:

zzjzzj

yyjyyj

xxjxxj

eBeAzh

eBeAyg

eBeAxf

33

22

11

)(

)(

)(

zjzjyjyjxjxjx

zzyyxx eBeAeBeAeBeAzyxE 332211),,(

)()()(),,( zhygxfzyxEx

22222

2zyxc

2222

2

2

2

0)()(

0)()(

0)()(

zyx

z

y

x

zhzh

ygyg

xfxf

Separation of Variable Solutions

2222

2

2

2

0)()(

0)()(

0)()(

zyx

z

y

x

zhzh

ygyg

xfxf

Solutions:

zzjzzj

yyjyyj

xxjxxj

eBeAzh

eBeAyg

eBeAxf

33

22

11

)(

)(

)(

purely real purely

imaginary complex

xxjxxj eBeA 11

Traveling andstanding waves Evanescent waves

xx eBeA 11

Exponentially modulatedtraveling wave

xxjxxxjx eeBeeA 11

xxjxxxjx eeBeeA 11

oror

)sin()cos( 11 xDxC xx

Wave Propagation and PolarizationTEM: Transverse Electromagnetic Waves

“A mode is a particular field configuration. For a given electromagnetic boundary value problem, manyfield configurations that satisfy the wave equation, Maxwell’s equations, and boundary conditions usuallyexits. A TEM mode is one whole field intensities, both E and H, at every point in space are contained ina local plane, referred to as equiphase plane, that is independent of time”

E

H

E

H E

H

Plane Waves“If the space orientation of the planes for a TEM mode are the same (equiphase planes are parallel) then the fieldsform a plane wave.

E

H

E

H

E

H

E

H

“If in addition to having planar equiphases the field has equiamplitude (the amplitude of the field is the sameover each plane) planar surfaces then it is called a uniform plane wave.”

Uniform Plane Waves

Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation

0

0

0

22

2

2

2

2

2

22

2

2

2

2

2

22

2

2

2

2

2

zzzz

yyyy

xxxx

Ez

E

y

E

x

E

Ez

E

y

E

x

E

Ez

E

y

E

x

E

Let’s begin by assuming the solution is only a function of z and has only the x component of electric field.

zjozj

oxxx eEeEazEazE ˆ)(ˆ)(~

Let’s also look at the term that represents a traveling wave moving in the +z direction

zjoxxx eEazEazE ˆ)(ˆ)(~

Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation

For uniform plane wave assume the solution is only afunction of z and has only the x component of electricfield.

zjoxxx eEazEazE ˆ)(ˆ)(~

zjoy

zjoy

zjoy

zjoy

zjoy

zjo

zyx

zyx

zyx

eEaeEaeEaH

eEaeEzj

a

eEzyx

aaa

jH

EEEzyx

aaa

jH

1ˆˆˆ

~

ˆ)(1

ˆ

00

ˆˆˆ1~

ˆˆˆ1~

11

Lets find H

Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation

zjoy

zjox

eEaH

eEaE

1

ˆ~

ˆ~

Several observations:

(1) E and H are orthogonal to each other and to the direction of energy propagation

(2) E and H are in phase with each other

(3) H is smaller in amplitude than E by the term

y

xw H

EZ (for a uniform plane wave)

Wave Impedance

Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation

How fast does the wave move?

Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation

How much power does the wave carry?

Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation

How fast does the power flow?

Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation

Relationship between phase and group velocity

Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation

Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation

Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation

Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation

Uniform Plane Waves in Unbounded Lossless MediumPrincipal Axis Propagation

Uniform Plane Waves in Unbounded Lossy MediumPrincipal Axis Propagation

Uniform Plane Waves in Unbounded Lossy MediumPrincipal Axis Propagation

Uniform Plane Waves in Unbounded Lossy MediumPrincipal Axis Propagation

Uniform Plane Waves in Unbounded Lossy MediumPrincipal Axis Propagation

Uniform Plane Waves in Unbounded Lossy MediumPrincipal Axis Propagation

Uniform Plane Waves in Unbounded Lossy MediumPrincipal Axis Propagation

Frequency, GHz

Ski

n D

epth

, cm

Example: Skin Depth in Sea Water

Frequency, GHz

Ski

n D

epth

, mic

rons

Example: Skin Depth in Copper

Frequency, GHz

Ski

n D

epth

, met

ers

Example: Skin Depth in Teflon

Polarization

Polarization

Polarization

Polarization

Polarization

Polarization

Uniform Plane Waves: Propagation in Any Arbitrary Direction

E

H

y

x

z

zzyyxxo

zyx

zzyyxx

rjo

EaEaEaE

zayaxar

aaa

eErE

ˆˆˆ

ˆˆˆ

ˆˆˆ

)(~

222zyx

222

22

)sin(zyx

yx

22)cos(

yx

y

Uniform Plane Waves: Propagation in Any Arbitrary Direction

0

)(~

o

rjo

E

eErE

E

H

y

x

z

Since E and are at right anglesfrom each other.

EaEa

EaE

EeE

Eejj

Eej

eEj

Ej

H

rjo

orj

orj

rjo

ˆ1

ˆ

ˆ1

11

11

1~1~

where 222

ˆˆˆˆ

zyx

zzyyxx aaaa

and

Uniform Plane Waves: Propagation in Any Arbitrary Direction

aaa

aaH

aaE

aErtE

trHEarH

aErtEtrEeErE

HE

Ho

Eo

Hoo

o

Eoorj

o

00

00

)cos(),(1

)(~

)cos(),()(~

Observation 1. E, H and vectors are pointing in orthogonal directions.

Summary and Observations:Frequency Domain Time Domain

Observation 2. E and H are in phase with each other, however, H’s magnitude is smaller by the amount of the wave impedance