Lecture 4

Polarization of EM Waves

Field Polarization

definition: the locus traced by the extremity of a harmonic time-varying field vector at a given observation point

• plane of polarization is orthogonal to direction of propagation

• there are three types of polarization: linear, circular, and elliptic

EE Ex x x

y y y

z z z

ω

linear circular elliptic ElecEng 4FJ4 Nikolova LECTURE 04: POLARIZATION 2

Field Polarization: Time Domain Analysis

field of any polarization can be represented in terms of 2 orthogonal linearly polarized components:

( ) cos( ) and ( ) cos( )x x y yE t m t E t m tω ω δ= = +

0( )He tω

0( )He tω π+

linearly (horizontally) polarized field

H

V

( ) cos( ) cos( )ˆ ˆx yt m t m tω ω δ= + +E x y

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Field Polarization: Time Domain Analysis (2)

linearly (vertically) polarized field

0( )Ve tω

0( )Ve tω π+

V

H

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Field Polarization: Time Domain Analysis (3) case 1: linear polarization of arbitrary direction

nδ π=

example: direction at 45 deg. (δ = 0, mH = mV)

0tω

0tω π+

no constraints on magnitudes

H

V

( ) cos( ) cos( ), 0,1,2,ˆ ˆx yt m t m t n nω ω π= + ± =E x y

arctan V

H

mm

φ

=

φ

• •

Field Polarization: Time Domain Analysis (4)

/ 2Lδ π= ±case 2: circular polarization

example: clockwise circularly polarized vector, δL = π/2

magnitudes must fulfill mH = mV = m

1tω

2 1 2t t πω ω= +

(0) 1(0) 0

H

V

EE

==

(90 ) 0(90 ) 1

H

V

EE

== −

H

V

( ) cos( ) cos( / 2), 1,3,ˆ ˆt m t m t n nω ω π= + ± =E x y

• •

ˆ k ⊗u

Field Polarization: Time Domain Analysis (5) case 3: elliptic polarization

elliptic polarization is any polarization different from linear or circular

( 0)

( 0)

10

H t

V t

EE

ω

ω

=

=

==

2 , / 2H V Lm m δ π= = −example: counter-clockwise elliptically polarized vector

( 90 )

( 90 )

00.5

H t

V t

EE

ω

ω

=

=

==

H

V

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( ) 2 cos( ) cos( / 2)ˆ ˆt m t m tω ω π= + −E x y

ˆ k ⊗u

Field Polarization and Wave Propagation

linear horizontal linear vertical

linear 45 deg. circular

LH circularly polarized wave ElecEng 4FJ4 Nikolova LECTURE 04: POLARIZATION 8

Field Polarization in Terms of CW-CP and CCW-CP Terms

any field can be represented by 2 circularly polarized components — one clockwise and one counter-clockwise

example: linearly polarized wave decomposed into two CP components

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Field Polarization – Phasor Analysis

• work with decomposition into 2 orthogonal linear components

( ) ( ) ( )ˆ ˆ ˆ ˆx y x yt E t E t E E= + ⇒ = +E x y E x y

( )( )

( ) cosˆ ˆ

( ) cosx xx x j

x yjy y y y

E mE t m t kzm m e

E t m t kz E m eδ

δ

ωω δ

== −⇒ ⇒ = +

= − + =E x y

• drop the propagation factor, exp(−jkz), because it is common to both components and we can always choose z = 0

• IMPORTANT: the direction of propagation (+z or −z) determines whether the wave is left-hand (LH or CCW) or right-hand (RH or CW) polarized in the case of circular or elliptical polarizations

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Field Polarization – Phasor Analysis – Linear Polarization , 0,1,2,n nδ π= =

( ) cos( ) cos( )ˆ ˆ

ˆ ˆ

x y

x y

t m t m t n

m m

ω ω π= + ±

= ±

E x y

E x y

arctan y

x

mm

φ = ±

φ

20

nδ πφ=

⇒ >

x

y

E

xE

yE

(a)

(2 1)0nδ π

φ= +

⇒ <

(b)

x

y

φxEyEE

sign+ sign−

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Field Polarization – Phasor Analysis – Circular Polarization ( )and / 2 , 0,1,2,x ym m m n nδ π π= = = ± + =

( ) cos( ) cos[ ( / 2 )] ( )ˆ ˆ ˆ ˆt m t m t n m jω ω π π= + ± + ⇒ = ±E x y E x y

RH (CW) for ˆ ˆLH (CCW) for ˆ ˆ

= −=

u zu z

LH (CCW) for ˆ ˆRH (CW) for ˆ ˆ

= −=u z

u z

x

y

0tω =

4t πω =

2t πω =

34

t πω =

tω π=

54

t πω =

32

t πω =74

t πω =

ˆ ˆ( )/ 2

m jδ π= +=

E x y

zx

y

0tω =

4t πω =

2t πω =

34

t πω =

tω π=

54

t πω = 32

t πω =

74

t πω =

ˆ ˆ( )/ 2L

m jδ π= −= −

E x y

z

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Field Polarization – Phasor Analysis – Circular Polarization (2)

• sense of rotation is determined when looking along the direction of propagation (be aware: in optics, the agreement is opposite)

(CCW) LH-CP wave is( ( ˆ ˆ) ) jkzz em j −= +x yE

(CW) RH-CP wave is( )ˆ( ) ˆ jkzz em j ++=E x y

• if wave propagates along +z

• if wave propagates along −z x

y

0tω =

4t πω =

2t πω =

34

t πω =

tω π=

54

t πω =

32

t πω =74

t πω =

ˆ ˆ2

( )/

m jδ π= +=

E x y

z

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Circular Polarization: Sense of Rotation

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• snapshot of E-field along direction of propagation for RH-CP wave

[Hayt & Buck, Engineering Electromagnetics, 8th ed., p. 399]

Field Polarization – Phasor Analysis – Elliptic Polarization

• phase difference δ can be any

• magnitude ratio mx/my can be any

• the term elliptic polarization is used to indicate polarizations other than linear or circular

( ) cos( ) cos( )ˆ ˆ

ˆ ˆ

x y

jx y

t m t m t

m m e δ

ω ω δ= + +

= +

E x y

E x y

• elliptic polarization has sense of rotation – the sign of δ determines the sense of rotation

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Field Polarization in Terms of Circularly Polarized Components

Any field can be written in terms of two CP terms – one RHCP, the other LHCP (wave propagating along +z)

( ) ( )ˆ ˆ ˆ ˆR LE j E j= − + +E x y x y

Bearing in mind that this field can also be represented in terms of two LP terms

ˆ ˆx yE E= +E x y

find the equations allowing for the calculation of ER and EL if Ex and Ey are known.

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Summary: Field Polarization

• harmonic plane waves are characterized by their polarization

• polarization is defined by the locus traced by the tip of the E-field vector as time flows

• polarization can be linear, circular, or elliptic (do not identify with the other two!)

• circular and elliptic polarizations can be right-hand (clockwise) or left-hand (counter-clockwise)

• in microwave and antenna engng: the sense of rotation is determined by looking along the direction of propagation (thumb points along direction of propagation)

• every plane-wave field can be decomposed into: (1) two mutually orthogonal linearly polarized terms, or (2) two circularly polarized terms of opposite sense of rotation

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