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Prof. David R. Jackson

Notes 17

Polarization of Plane Waves

ECE 3317

Spring 2013

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Polarization

The polarization of a plane wave refers to the direction of the electric field vector in the time domain.

x

y

z

( , )E z t

We assume that the wave is traveling in the positive z direction.

,S z t

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Consider a plane wave with both x and y components

ˆ ˆE( ) ( E E ) jkzx yz x y e

( = E E )y x In general, phase of phase of

Assume:

Ex

Ey

x

y

E

E

x

jy

a

be

real number

Polarization (cont.)

Phasor domain:

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Time Domain:

Re cos

Re cos

j tx

j j ty

E ae a t

E be e b t

Depending on b/a and b, three different cases arise:

x

y

E (t)

At z = 0:

Polarization (cont.)

Linear polarization Circular polarization Elliptical polarization

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Power Density: *1S E H

2

*

E E1ˆ ˆ ˆ ˆS E E

2y x

x yx y x y

H

H

xy

yx

E

E

221ˆS E E

2 x yz

Assume lossless medium ( is real):

21ˆS E

2z

or

Polarization (cont.)

From Faraday’s law:

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At z = 0:

ˆ ˆ cosE x a y b t

cos

cosx

y

E a t

E b t

a

b

x

y x

y

E

b = 0 or

(shown for b = 0)

Linear Polarization

This is simply a “tilted” plane wave.

cos

cos

x

y

E a t

E b t

+ sign: = 0- sign: =

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At z = 0:

2 2 2 2 2 2 2 2cos sinx yE E E a t a t a

cos

cos( / 2) sinx

y

E a t

E a t a t

b = a AND b = p /2

Circular Polarization

cos

cos

x

y

E a t

E b t

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a

x

y

( )E t

f

1 1tan tan ( tan )y

x

Et

E

t

Circular Polarization (cont.)

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t

a

x

y

( )E t

f

b = + p / 2

b = - p / 2

RHCP

LHCP

IEEE convention

Circular Polarization (cont.)

d

dt

cos

cos

x

y

E a t

E b t

b = p /2

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There is opposite rotation in space and time.

Circular Polarization (cont.)

Examine how the field varies in both space and time:

Rotation in space vs. rotation in time

ˆ ˆE( ) j jkzz x a y be e

ˆ ˆ( , ) cos cosE z t x a t kz y b t kz

Phasor domain

Time domain

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RHCP

Notice that the rotation in space matches the left hand!

z

Circular Polarization (cont.)

A snapshot of the electric field vector, showing the vector at different points.

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Circular Polarization (cont.)

Animation of LHCP wave

http://en.wikipedia.org/wiki/Circular_polarization

(Use pptx version in full-screen mode to see motion.)

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Circular Polarization (cont.)

Circular polarization is often used in wireless communications to avoid problems with signal loss due to polarization mismatch.

Misalignment of transmit and receive antennas

Reflections off of building

Propagation through the ionosphere

Receive antenna

The receive antenna will always receive a signal, no matter how it is rotated about the z axis.

z

However, for the same incident power density, an optimum linearly-polarized wave will give the maximum output signal from this linearly-polarized antenna (3 dB higher than from an incident CP wave).

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Circular Polarization (cont.)

Two ways in which circular polarization can be obtained:

This antenna will radiate a RHCP signal in the positive z direction, and LHCP in the negative z direction.

Method 1) Use two identical antenna rotated by 90o, and fed 90o out of phase.

Antenna 1

V 1x oV 1 90y j

Antenna 2

x

y

+

-

+

-

/ 2

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Circular Polarization (cont.)

Realization with a 90o delay line:

2 1 / 2l l

01ANT

inZ Z

Antenna 1

Antenna 2

x

y

l1

l2

Z01

Z01

Feed line

Power splitter

Signal

02 01 / 2Z Z

02Z

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Circular Polarization (cont.)

An array of CP antennas

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Circular Polarization (cont.)

The two antennas are actually two different modes of a microstrip or dielectric resonator antenna.

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Circular Polarization (cont.)

Method 2) Use an antenna that inherently radiates circular polarization.

Helical antenna for WLAN communication at 2.4 GHz

http://en.wikipedia.org/wiki/Helical_antenna

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Circular Polarization (cont.)

Helical antennas on a GPS satellite

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Circular Polarization (cont.) Other Helical antennas

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Circular Polarization (cont.)

An antenna that radiates circular polarization will also receive circular polarization of the same handedness (and be blind to the opposite handedness).

It does not matter how the receive antenna is rotated about the z axis.

Antenna 1

Antenna 2

x

y

l1

l2

Z0

Z0

Output signal

Power combinerRHCP wave

02 01 / 2Z Zz

RHCP Antenna

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Circular Polarization (cont.)

Summary of possible scenarios

1) Transmit antenna is LP, receive antenna is LP

Simple, works good if both antennas are aligned. The received signal is less if there is a misalignment.

2) Transmit antenna is CP, receive antenna is LP

Signal can be received no matter what the alignment is. The received signal is 3 dB less then for two aligned LP antennas.

3) Transmit antenna is CP, receive antenna is CP

Signal can be received no matter what the alignment is. There is never a loss of signal, no matter what the alignment is. The system is now more complicated.

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Includes all other cases that are not linear or circular

x

y

( )E t

f

Elliptic Polarization

The tip of the electric field vector stays on an ellipse.

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0

0

LHEP

RHEP

x

LHEP

RHEP

y

f

Elliptic Polarization (cont.)

Rotation property:

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cos( )

cos cos sin sin

yE b t

b t b t

so2

cos sin 1x xy

E EE b b

a a

22 2cos siny x x

b bE E b E

a a

2 22 2 2 2 2cos 2 cos siny x x y x

b b bE E E E b E

a a a

cosxE a t

Elliptic Polarization (cont.)

or

Squaring both sides, we have

Here we give a proof that the tip of the electric field vector must stay on an ellipse.

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This is in the form of a quadratic expression:

2

2 2 2 2 2 2cos sin 2 cos sinx y x y

b bE E E E b

a a

22 2 2 22 cos sinx x y y

b bE E E E b

a a

2 2x x y yA E B E E CE D

Elliptic Polarization (cont.)

2 22 2 2 2 2cos 2 cos siny x x y x

b b bE E E E b E

a a a

or

Collecting terms, we have

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2

2 22

22

4

4 cos 4

4 cos 1

B AC

b b

a a

b

a

224 sin 0

b

a

Hence, this is an ellipse.

so

Elliptic Polarization (cont.)

(determines the type of curve)

Discriminant:

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cos

cos( )

cos cos sin sin

x

y

E a t

E b t

b t b t

x

y

( )E t

f

Elliptic Polarization (cont.)

Here we give a proof of the rotation property.

0

0

LHEP

RHEP

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Take the derivative:

tan cos tan siny

x

E bt

E a

2 2sec sec sind b

tdt a

Hence

2 2sin cos secd b

tdt a

( ) 0 0

( ) 0 0

da

dtd

bdt

LHEP

RHEP (proof complete)

Elliptic Polarization (cont.)

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Rotation Rule

Here we give a simple graphical method for determining the type of polarization

(left-handed or right handed).

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First, we review the concept of leading and lagging sinusoidal waves.

B leads A

Reb

B

A

Im

B lags A

Reb B

A

Im

Two phasors: A and B

0

0

Note: We can always assume that the phasor A is on the real axis (zero degrees phase) without loss of generality.

Rotation Rule (cont.)

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( ) 0a LHEP

Rule:

The electric field vector rotates from the leading axis to the lagging axis

Ey leads Ex

Phasor domainRe

b

Ey

Ex

Im

Now consider the case of a plane wave

Rotation Rule (cont.)

( )E t

x

y

Time domain

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( ) 0b RHEP

Rule:

The electric field vector rotates from the leading axis to the lagging axis

Rotation Rule (cont.)

Ex

Ey

Im

Reb

Ey lags Ex

Phasor domain

x

y

( )E t

Time domain

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The electric field vector rotates from the leading axis to the lagging axis.

Rotation Rule (cont.)

The rule works in both cases, so we can call it a general rule:

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ˆˆE (1 ) (2 ) jkyz j x j e

y

Ez

z

xEx

What is this wave’s polarization?

Rotation Rule (cont.)

Example

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Example (cont.)

Therefore, in time the wave rotates from the z axis to the x axis.

Ez leads Ex Ez

Ex

Im

Re

Rotation Rule (cont.)

ˆˆE (1 ) (2 ) jkyz j x j e

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LHEP or LHCP

Note: and 2x zE E (so this is not LHCP)

Example (cont.)

Rotation Rule (cont.)

ˆˆE (1 ) (2 ) jkyz j x j e

z

y

Ez

xEx

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LHEP

Rotation Rule (cont.)

Example (cont.)

ˆˆE (1 ) (2 ) jkyz j x j e

z

y

Ez

xEx

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x

y

( )E t

A

B

C

D

1AB

ARCD

major axis

minor axis

Axial Ratio (AR) and Tilt Angle ()

Note: In dB we have 1020logdBAR AR

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o o45 45

LHEP

RHEP

sin 2 sin 2 sin

cot

0 :

0 :

AR

Axial Ratio and Handedness

tan 2 tan 2 cos

Tilt Angle

Note: The tilt angle is ambiguous by the addition of 90o.

1

o

tan

0 90

b

a

Axial Ratio (AR) and Tilt Angle () Formulas

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tan 2 tan 2 cos Tilt Angle:

Note: The title angle is zero or 90o if:

/ 2

x

y

( )E t

Note on Title Angle

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Example

z

y

Ez

xEx

LHEP

Relabel the coordinate system:

ˆˆE (1 ) (2 ) jkyz j x j e

x x

z y

y z

Find the axial ratio and tilt angle.

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Example (cont.)

LHEP

ˆ ˆE (2 ) (1 ) jkzx j y j e

y

z

Ey

xEx

1ˆ ˆE (1)

2jkzj

x y ej

Renormalize:

1.249ˆ ˆE (1) 0.6324 j jkzx y e e

or

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Example (cont.)

LHEP

oE E 71.565y x

1 1tan tan 0.632 0.564b

a

Hence

1.249ˆ ˆE (1) 0.6324 j jkzx y e e

o

1

0.6324

1.249[rad] 71.565

a

b

y

z

Ey

xEx

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Example (cont.)

tan 2 tan 2 cos

o o45 45

LHEP

RHEP

sin 2 sin 2 sin

cot

0 :

0 :

AR

o16.845

o29.499

LHEP

Results

1.768AR

o71.565

0.564

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Example (cont.)

o16.845

1.768AR

x

y

( )E t

o

o o

16.845

16.845 90

Note: We are not sure which choice is correct:

We can make a quick time-domain sketch to be sure.

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Example (cont.)

x

y

( )E t

cos

cos

x

y

E a t

E b t

o

1

0.6324

1.249[rad] 71.565

a

b