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Transcript of Lesson 16 Plane Waves in Homogeneous Media sdyang/Courses/EM/Lesson16_Std.pdf · PDF file...

  • Electromagnetics P16-1

    Edited by: Shang-Da Yang

    Lesson 16 Plane Waves in Homogeneous Media

    Introduction

    By eq’s (15.5-7), E v

    and H v

    in a charge-free ( 0=ρ ), nonconducting ( 0=J v

    ), simple

    (linear, homogeneous, isotropic) medium are governed by the vector wave equations:

    01 2 2

    2 2 =

    ∂ ∂

    −∇ t E

    u E

    p

    v v

    , 01 2 2

    2 2 =

    ∂ ∂

    −∇ t H

    u H

    p

    v v

    , where µε 1

    =pu .

    The general solutions are waves propagating with phase velocity pu .

    A plane wave is a particular solution to the vector wave equation, where every point on an

    infinite plane perpendicular to the direction of propagation has the same electric field E v

    and

    magnetic field H v

    (in terms of magnitude and direction, see Fig. 16-1).

    Fig. 16-1. A general plane wave propagates in +z-direction illustrated at some time instant. The

    fields E v

    and H v

    are constant throughout the plane 1zz = , but may differ from those on another plane 2zz = .

    Time-harmonic (sinusoidal) plane waves are of particular importance because:

    1) The mathematical treatments are greatly simplified.

    2) Superposition of time-harmonic plane waves remains a solution to the (linear) vector

    wave equation. They can be used as a basis to describe general electromagnetic waves.

  • Electromagnetics P16-2

    Edited by: Shang-Da Yang

    E.g. Superposition of time-harmonic plane waves of different frequencies and common

    direction of propagation can describe plane wave packets. Superposition of time-

    harmonic plane waves of the same frequency and different directions of propagation can

    describe Gaussian beam.

    3) It is a good approximation for far-field EM waves (e.g. sunlight received on the earth),

    and waves whose cross-sections have a linear dimension much larger than the wavelength

    (e.g. typical laser beam).

    16.1 Plane Waves in Vacuum

    Most simplified time-harmonic (sinusoidal) plane waves

    The vector phasor E v

    of a time-harmonic E-field is satisfied with:

    022 =+∇ EkE vv

    (15.13)

    where puk ω= [eq. (15.15)] denotes the wavenumber. For simplicity, consider a

    time-harmonic plane wave propagating in the z-direction and the electric field is polarized in

    the x-direction, ⇒

    )( zEaE xx vv = . (16.1)

    Note that eq. (16.1) does represent a plane wave, for any point on an infinite plane 0zz =

    (perpendicular to the direction of propagation za v ) must have the same electric field

    )( 0zEa xx v . In this particular case, eq. (15.13) reduces to a scalar ordinary differential equation:

    022 2

    =+ x x Ek

    dz Ed

    (16.2)

    In the absence of physical boundary, the general solution to eq. (16.2) consists of two

    counter- propagating waves:

    jkzjkz x eEeEzE

    +−−+ += 00)( , (16.3)

  • Electromagnetics P16-3

    Edited by: Shang-Da Yang

    where +++ = φjeEE 00 and

    −−− = φjeEE 00 are complex amplitudes corresponding to

    time-harmonic waves propagating in the +z and −z directions, respectively.

    1) As in transmission lines [eq’s (4.7), (4.8)], phasor jkzx eEzE −++ = 0)( describes a

    time-harmonic wave with velocity kup ω= because:

    { } ( )++++ +−== φωω kztEezEtzE tjxx cos)(Re),( 0

    is a function of variable ( )k zt

    ω τ −= . By eq. (15.15),

    µεµεω ω 1

    ==pu , which is the

    same as eq. (15.7).

    2) As in transmission lines, reflected wave jkzx eEzE +−− = 0)( can emerge if the medium is

    nonhomogeneous along the z-direction (i.e. ε or µ changes with z ).

    Characteristics of plane waves

    The electrical field of a plane wave propagating in an arbitrary direction ka v is of the form

    (Fig. 16-2):

    ( ) rkjzyxrkj eEaaaeEeE vvvv vvvvv ⋅−⋅− ++== 00 coscoscos γβα (16.4)

    (1) The phase evolution term kz in eq. (16.3) is generalized by rk v v ⋅ , where

    kakakakak kzzyyxx vvvvv =++=

    is the wavevector, and zayaxar zyx vvvv ++= is the position vector of the observation point. (2)

    The direction of electric field (polarization direction) xa v in eq. (16.1) is generalized by a

    unit vector:

  • Electromagnetics P16-4

    Edited by: Shang-Da Yang

    γβα coscoscos zyx aaae vvvv ++=

    where { }γβα ,, are the included angles between ev and { }zyx aaa vvv ,, , respectively. For any

    point rv on an infinite plane perpendicular to the direction of propagation ka v (intensity &

    phase front, Fig. 16-2), constantcos ==⋅ θkrrk v v

    , ⇒ rkjeEeE vvvv ⋅−= 0 is a constant vector.

    Therefore, eq. (16.4) does represent a plane wave.

    Fig. 16-2. A plane wave propagates in an arbitrary direction on the xz-plane.

    The three vectors E v

    , H v

    , k v

    of a time-harmonic plane wave are mutually orthogonal in a

    charge-free simple medium.

    Proof:

    (1) In a charge-free ( 0=ρ ) simple medium, eq. (7.8) reduces to:

    0=⋅∇ E v

    .

    (2) By the vector identity ( ) ( ) ( )ψψψ ∇⋅+⋅∇=⋅∇ AAA vvv (where ψ , Av are arbitrary scalar and vector fields, respectively), eq. (16.4) reduces to:

    ( ) ( ) ( ) ( )[ ]rkjrkjrkjrkj eEeeEeeeEeEeE vvvvvvvv vvvvv ⋅−⋅−⋅−⋅− ∇⋅=∇⋅+⋅∇=⋅∇=⋅∇ 0000 . (3) By the formula of gradient in Cartesian coordinate system,

    ( ) ( ) rkjrkjzzyyxxzkykxkjzyxrkj ekjekakakajezayaxae zyx

    vvvvvv vvvvvvv ⋅−⋅−++−⋅− −=++−= 

      

     ∂ ∂

    + ∂ ∂

    + ∂ ∂

    =∇ )( .

  • Electromagnetics P16-5

    Edited by: Shang-Da Yang

    (4) ( )[ ] ( ) 000 =⋅−=−⋅=⋅∇ ⋅−⋅− krkjrkj aeejkEekjEeE vvvvv v vvv

    , ⇒ kae vv ⊥ , i.e.,

    kE vv

    ⊥ (16.5)

    If E v

    has been known, one can derive H v

    by HjE vv

    ωµ−=×∇ [eq. (15.9)]:

    (5) By the formula of curl in Cartesian coordinate system,

    ( ) γβα

    ωµωµ coscoscos

    11

    000 rkjrkjrkj

    zyx

    eEeEeE zyx

    aaa

    j E

    j H

    vvvvvv

    vvv

    vv

    ⋅−⋅−⋅−

    ∂∂∂∂∂∂ −

    =×∇ −

    =

    γβα ωµ

    coscoscos

    1

    000 rkjrkjrkj zyx

    zyx

    eEeEeE jkjkjk aaa

    j vvvvvv

    vvv

    ⋅−⋅−⋅−

    −−− −

    =

    ( ) ( )k EaEkEkj

    j k

    ωµωµωµ

    vvvvvv × =

    × =×−

    − =

    1 , ⇒

    η EaH k vvv ×

    = , (16.6)

    where ε µ

    µεω ωµωµη ===

    k is the intrinsic impedance of the medium:

    ε µη = (Ω) (16.7)

    For vacuum, ( )Ω== 377000 εµη . Eq. (16.6) implies that kH vv

    ⊥ and EH vv

    ⊥ .

    Combined with eq. (16.5), ⇒ E v

    , H v

    , k v

    of a time-harmonic plane wave are mutually

    orthogonal in a charge-free simple medium. The fact that both E v

    and H v

    are perpendicular

    to the direction of propagation ( ka v ) makes the plane wave a transverse electromagnetic

    (TEM) wave.

    Example 16-1: (1) If jkzxxx eEazEaE −++ == 0)(

    vvv , ⇒ kak z vv = . By eq. (16.6), ⇒

    ηη )()( zEazEaaH xyxxz

    ++

    = ×

    = v vvv

    , where η

    )()( zEzH xy +

    + = . (2) If jkzxxx eEazEaE +−− == 0)(

    vvv , ⇒

  • Electromagnetics P16-6

    Edited by: Shang-Da Yang

    kak z vv −= ,

    ηη − =

    ×− =

    −− )()( zEazEaaH xyxxz v

    vvv , where

    η− =

    − − )()( zEzH xy . (3) If

    jkz yyy eEazEaE

    −++ == 0)( vvv , ⇒ kak z

    vv = , ηη −

    = ×

    = ++ )()( zE

    a zEaa

    H yx yyz v

    vv v

    , where

    η− =

    + + )()(

    zE zH yx .

    1) Intrinsic impedance η [eq. (16.7)] of plane wave is analogous to the characteristic

    impedance 0Z [eq. (2.8)] in transmission lines:

    0Z VIEaH k

    + + =↔

    × =

    η

    vvv implies { vE↔

    v , iH ↔

    v }.

    C LZ =↔= 0ε

    µη implies { L↔µ , C↔ε }.

    2) Not all TEM waves are plane waves. E.g. Coaxial transmission lines support TEM waves

    ( raE vv // , φaH

    vv // , zak vv // ) with −r dependent (non-uniform) fields.

    What is the state of polarization of time-harmonic plane waves?

    By eq. (16.4), the electric field phasor of a time-harmonic plane wave propagating in the

    +z-direction is:

    ( ) jkzyxrkj eEaaeEeE −⋅− +== 00 coscos βα vvv v vv

    .

    At 0zz = , the time-dependent electric field vector:

    { } ( )0000 cos)(Re),( kztEeezEtzE tj −== ωω v vv

    will traverse a line of length 02E along e v in one period ωπ2=T (Fig. 16-3a). In

    general, the electric field of a time-harm