L04 Polarization

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    Lecture 4

    Polarization of EM Waves

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    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

    E E Ex x x

    y y y

    z z z

    linear circular elliptic

    Nikolova 2012 2LECTURE 04: POLARIZATION

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    Field Polarization: Time Domain Analysis

    field of any polarization can be represented in terms of 2 orthogonallinearlypolarized components:

    ( ) cos( ) and ( ) cos( )x x y yE t m t E t m t

    0( )He t

    0( )He t

    linearly (horizontally) polarized field V

    ( ) cos( ) cos( ) x yt m t m t

    E x y

    Nikolova 2012 3

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    Field Polarization: Time Domain Analysis (2)

    linearly (vertically) polarized field

    0( )Ve t

    0( )Ve t

    V

    Nikolova 2012 4LECTURE 04: POLARIZATION

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    5

    Field Polarization: Time Domain Analysis (3)

    case 1: linear polarization of arbitrary directionn

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

    0t

    0t

    no constraints on magnitudes

    V

    ( ) cos( ) cos( ), 1,2, x y

    t m t m t n n E x y

    arctan V

    H

    m

    m

    Nikolova 2012 LECTURE 04: POLARIZATION

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    6

    Field Polarization: Time Domain Analysis (4)

    / 2L

    case 2: circular polarization

    example: clockwise circularly polarized vector, L= /2

    magnitudes must fulfill mH= mV= m

    1t

    2 12

    t t

    (0) 1

    (0) 0

    H

    V

    E

    E

    (90 ) 0

    (90 ) 1

    H

    V

    E

    E

    V

    ( ) cos( ) cos( / 2), 1,2, t m t m t n E x y

    Nikolova 2012 LECTURE 04: POLARIZATION

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    Field Polarization: Time Domain Analysis (5)

    case 3: elliptic polarizationelliptic polarization is any polarization different from linear

    or circular

    (0) 1(0) 0

    H

    V

    EE

    2 , / 2H V Lm m example: counter-clockwise elliptically polarized vector

    (90 ) 0

    (90 ) 0.5H

    V

    E

    E

    V

    Nikolova 2012 7LECTURE 04: POLARIZATION

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    Field Polarization and Wave Propagation

    linear horizontal linear vertical

    linear 45 deg. circular

    LH or RH?

    Nikolova 2012 8LECTURE 04: POLARIZATION

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

    any field can be represented by 2 circularlypolarized components

    one clockwise and one counter-clockwise

    example: linearly polarized wave decomposed into two CP components

    Nikolova 2012 9LECTURE 04: POLARIZATION

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    Field Polarization Phasor Analysis

    work with decomposition into 2 orthogonal linear components

    ( ) ( ) ( ) x y x yt E t E t E E E x y E x y

    ( ) cos

    ( ) cos

    x xx x jx yj

    y y y y

    E mE t m t kzm m e

    E t m t kz E m e

    E x y

    drop the propagation factor, exp(jkz), because it is common to

    both components and we can always choosez = 0

    Nikolova 2012 10LECTURE 04: POLARIZATION

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    Field Polarization Phasor Analysis Linear Polarization

    , 0,1,2,n n

    ( ) cos( ) cos( )

    x y

    x y

    t m t m t n

    m m

    E x y

    E x y

    arctan yx

    mm

    2

    0

    n

    x

    y

    E

    xE

    yE

    (2 1)

    0

    n

    x

    y

    xE

    yEE

    Nikolova 2012 11LECTURE 04: POLARIZATION

    sign sign

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    Field Polarization Phasor Analysis Circular Polarization

    and / 2 , 0,1,2,x ym m m n n

    ( ) cos( ) cos[ ( / 2 )] ( ) t m t m t n m j E x y E x y

    RH (CW) for LH (CCW) for

    u zu z

    LH (CCW) for RH (CW) for

    u z

    u z

    x

    y

    0t

    4

    t

    2t

    3

    4t

    t

    5

    4t

    3

    2t

    7

    4t

    ( )

    / 2

    m j

    E x y

    zx

    y

    0t

    4t

    2t

    3

    4t

    t

    5

    4t

    3

    2t

    7

    4

    t

    ( )

    / 2L

    m j

    E x y

    z

    Nikolova 2012 12LECTURE 04: POLARIZATION

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    Field Polarization Phasor Analysis Circular Polarization (2)

    sense of rotation is determined when looking along the direction ofpropagation (caution: in optics, the agreement is opposite)

    (CCW) LH-CP wave is

    ( ( ) ) jkzz em j x yE

    (CW) RH-CP wave is

    ( )( ) jkzz em j E x y

    if wave propagates along +z

    if wave propagates along z

    Nikolova 2012 13LECTURE 04: POLARIZATION

    x

    y

    0t

    4t

    2t

    3

    4t

    t

    5

    4t

    3

    2t

    7

    4t

    2

    ( )

    /

    m j

    E x y

    z

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    Field Polarization Phasor Analysis Elliptic Polarization

    phase difference 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( )

    x y

    jx y

    t m t m t

    m m e

    E x y

    E x y

    elliptic polarization has sense of rotation the sign of determinesthe sense of rotation

    Nikolova 2012 14LECTURE 04: POLARIZATION

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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, theother LHCP (wave propagating along +z)

    ( ) ( ) R LE j E j E x y x y

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

    x yE E E x y

    find the equations allowing for the calculation ofER andEL ifEx and

    Ey are known.

    Nikolova 2012 15LECTURE 04: POLARIZATION

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    Field Polarization

    What is the angular speed of rotation of the field vectors in the caseof a circularly polarized wave of radian frequency ?

    ( ) cos( ) cos( / 2), 1,2, m mt e t e t n E x y

    Nikolova 2012 16LECTURE 04: POLARIZATION

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    Summary: Field Polarization

    Nikolova 2012 17LECTURE 04: POLARIZATION

    harmonic plane waves are characterized by their polarizationpolarization 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 withthe 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 alongthe 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