Plane EM waves - San Jose State UniversityElectromagnetic Waves EE142 Dr. Ray Kwok •reference:...

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Plane Electromagnetic Waves EE142 Dr. Ray Kwok •reference: Fundamentals of Engineering Electromagnetics, David K. Cheng (Addison-Wesley) Electromagnetics for Engineers, Fawwaz T. Ulaby (Prentice Hall)

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  • PlaneElectromagnetic Waves

    EE142Dr. Ray Kwok

    •reference:

    Fundamentals of Engineering Electromagnetics, David K. Cheng (Addison-Wesley)

    Electromagnetics for Engineers, Fawwaz T. Ulaby (Prentice Hall)

  • Plane EM Wave - Dr. Ray Kwok

    Source-free Maxwell Equations

    t

    EH

    0Ht

    HE

    0E

    ∂ε=×∇

    =⋅∇∂

    ∂µ−=×∇

    =⋅∇

    rr

    r

    rr

    r

    t

    EJH

    0Ht

    HE

    E

    f

    f

    ∂ε+=×∇

    =⋅∇∂

    ∂µ−=×∇

    ρ=⋅∇ε

    rrr

    r

    rr

    r

    in source-free medium

    (homogeneous,

    linear, isotropic)

    ρ = 0, J = 0

  • Plane EM Wave - Dr. Ray Kwok

    Wave Equation

    ( ) ( )

    ( ) ( )

    2

    22

    22

    2

    2

    t

    EE

    EEEE

    t

    EH

    tt

    HE

    t

    EH

    t

    HE

    ∂µε=∇

    −∇=∇−⋅∇∇=×∇×∇

    ∂µε−=×∇

    ∂µ−=

    ∂µ−×∇=×∇×∇

    ∂ε=×∇

    ∂µ−=×∇

    rr

    rrrr

    rr

    rr

    rr

    rr

    plane wave equation with v2 = 1/µε

  • Plane EM Wave - Dr. Ray Kwok

    Traveling Wave

    2

    2

    22

    2

    2

    2

    2

    2

    2

    t

    f

    v

    1

    x

    f

    )u("fvt

    u)u("vf

    t

    f

    )u('vft

    u)u('f

    t

    f

    )u("fx

    u)u("f

    x

    f

    )u('fx

    u)u('f

    x

    f

    )u(f)vtx(f

    ∂=

    =∂

    ∂±=

    ±=∂

    ∂=

    =∂

    ∂=

    =∂

    ∂=

    ≡± reverse / forward traveling wave

    wave equation

    ( )

    ( )

    ( )

    ( )rktfkxtf)vtx(f

    kxtk

    1

    ft2kxk

    1

    vt2

    x2

    k

    1vtx

    rr⋅−ω

    ±ω=±

    ±ω±=

    π±=

    λ

    π±

    λ

    π=±

    note:

    (3D)

  • Plane EM Wave - Dr. Ray Kwok

    Plane Wave

    µε==

    ω

    µεω=

    µεω−=−

    ω−=∂

    −=∇

    =

    ∂µε=∇

    ⋅−ω

    1v

    k

    k

    k

    Et

    E

    EkE

    eE)t,r(E

    t

    EE

    22

    2

    2

    2

    22

    )rkt(j

    o

    2

    22

    rr

    rr

    rrr

    rr

    rr

    )rkt(j

    o

    2

    22

    eH)t,r(H

    t

    HH

    rrrrr

    rr

    ⋅−ω=

    ∂µε=∇

    Similarly for B or H field

  • Plane EM Wave - Dr. Ray Kwok

    Source-free EM wave)kzt(j

    oeEx̂)t,r(E−ω=

    rr

    { })kzt(jo

    )kzt(j

    o

    )kzt(j

    o

    ekE

    ŷ)t,r(H

    ejkEŷE

    HjE

    00eE

    zyx

    ẑŷx̂

    E

    −ω

    −ω

    −ω

    ωµ=

    −=×∇

    ωµ−=×∇

    ∂=×∇

    rr

    r

    rr

    r

    V/m in vacuum with no current source.

    What is H ? { }

    )kzt(jo

    )kzt(jo

    )kzt(jo

    )kzt(jo

    )kzt(j

    o

    ev

    Eŷ)t,r(B

    ev

    EŷHB

    eE

    ŷ)t,r(H

    1

    v

    1

    f

    1k

    ekE

    ŷ)t,r(H

    ejkEŷE

    −ω

    −ω

    −ω

    −ω

    −ω

    =

    µ

    µ=µ=

    η=

    η≡

    µ

    ε=

    µ

    µε=

    µ=

    λµ=

    ωµ

    ωµ=

    −=×∇

    rr

    rr

    rr

    rr

    r

    ηηηη = waveImpedance

  • Plane EM Wave - Dr. Ray Kwok

    EM wave properties

    )kzt(jo

    )kzt(jo

    )kzt(j

    o

    ev

    Eŷ)t,r(B

    eE

    ŷ)t,r(H

    eEx̂)t,r(E

    −ω

    −ω

    −ω

    =

    η=

    =

    rr

    rr

    rr

    ĤÊk̂

    kHE

    ×=

    ⊥⊥rrr

    (1) E & H are in phase.

    ε

    µ=η=

    µε===

    o

    o

    rro

    o

    H

    E

    c

    n

    cv

    B

    E

    (2)

    (3)

    Index of refraction

  • Plane EM Wave - Dr. Ray Kwok

    Electromagnetic wave

  • Plane EM Wave - Dr. Ray Kwok

    Earlier exercise

    )zt900cos(01.0x̂)t,r(H β+=rr

    A/m in vacuum with no current source.

    What is f, λ, direction of propagation, E ?

    ω = 2πf = 900, f = 143 Hz

    λ = c/f = (3 x 108)/(143) = 2094 km

    propagate to the –z direction

    EjHrr

    ωε=×∇

    E(r) = ay 3.77 cos[900t + 3(10-6)z] V/m

    ĤÊk̂

    kHE

    ×=

    ⊥⊥rrr

    (1) E & H are in phase.

    η=

    =

    o

    o

    o

    o

    H

    E

    vB

    E

    (2)

    (3)

    ηo = 377 Ω

  • Plane EM Wave - Dr. Ray Kwok

    Example)1.0zt10sin()02.0ŷ01.0x̂()t,r(H 9 −β+−=

    rrA/m. What is E ?

    β = ω/c = 109/(3 x 108) = 3.33 rad/m

    (1) E & H are in phase.

    ĤÊk̂

    kHE

    ×=

    ⊥⊥rrr

    (2)

    η=

    =

    o

    o

    o

    o

    H

    E

    vB

    E(3)

    ηo = 377 Ω

    )1.0z33.3t10sin(E)t,r(E9

    o −+=rrr

    V/m

    ŷ447.0x̂894.05

    ŷx̂2Ê +=

    +=x

    y

    H

    E

    Ho2 = 0.012 + 0.022 → Ho = 0.022 = Eo/ηo = Eo/377

    Eo = 8.3 V/m

    )1.0z33.3t10sin()7.3ŷ4.7x̂()t,r(E

    )1.0z33.3t10sin()447.0ŷ894.0x̂(3.8)t,r(E

    9

    9

    −++=

    −++=rr

    rr

    V/m

  • Plane EM Wave - Dr. Ray Kwok

    Exercise)yz3t10cos(20x̂)t,r(E 8 −+=

    rrV/m in non-magnetic material.

    What is H ?

    v = ω/k = 108/√10 = 3.16 x 107 m/sv = c/n = c/√εr → εr = (c/v)

    2 = 90

    η = √(µ/ε) = ηo/√εr = 377/√90 = 39.7

    (1) E & H are in phase.

    ĤÊk̂

    kHE

    ×=

    ⊥⊥rrr

    (2)

    )yz3t10cos(H)t,r(H 8o −+=rrr

    A/m

    Ho = Eo/η = 20/39.7 =0.5

    )yz3t10cos()158.0ẑ475.0ŷ()t,r(H

    )yz3t10cos()316.0ẑ949.0ŷ(5.0)t,r(H

    8

    8

    −++−=

    −+−−=rr

    rr

    A/m

    ẑ316.0ŷ949.010

    ẑŷ3Ĥ −−=

    −−=

    y

    z

    k

    E

    H

    η=

    =

    o

    o

    o

    o

    H

    E

    vB

    E(3)

    ηo = 377 Ω

    ŷẑ3k +−=r

  • Plane EM Wave - Dr. Ray Kwok

    EM wave properties - generalized

    ĤÊk̂

    kHE

    ×=

    ⊥⊥rrr

    (1) E & H are in phase.

    ε

    µ=η=

    µε===

    o

    o

    rro

    o

    H

    E

    c

    n

    cv

    B

    E

    (2)

    (3)

    Index of refraction

    Ek̂1

    Hrr

    ×η

    =

    k̂HE ×η=rr

  • Plane EM Wave - Dr. Ray Kwok

    Earlier exercise (free space))1.0zt10sin()02.0ŷ01.0x̂()t,r(H 9 −β+−=

    rrA/m. What is E ?

    β = ω/c = 109/(3 x 108) = 3.33 rad/m

    )1.0z33.3t10sin()54.7x̂77.3ŷ(E

    )ẑ()1.0ztsin()02.0ŷ01.0x̂(377E

    k̂HE

    9 −++=

    −×−β+ω−=

    ×η=

    r

    r

    rr

    V/m.

  • Plane EM Wave - Dr. Ray Kwok

    Earlier exercise)yz3t10cos(20x̂)t,r(E 8 −+=

    rrV/m in non-magnetic material.

    What is H ?

    A/m

    ŷẑ3k +−=r

    ( )[ ]

    ( ) ( )yz3t10cos159.0ẑ478.0ŷH

    yz3t10cos20x̂10

    ŷẑ3

    7.39

    1H

    Ek̂1

    H

    8

    8

    −+−−=

    −+×

    +−=

    ×η

    =

    r

    r

    rr

    v = ω/k = 108/√10 = 3.16 x 107 m/sv = c/n = c/√εr → εr = (c/v)

    2 = 90

    η = √(µ/ε) = ηo/√εr = 377/√90 = 39.7

  • Plane EM Wave - Dr. Ray Kwok

    Poynting Theorem

    ( )

    ( )

    ( )

    ( )HEt

    HHHE

    HEEHHE

    t

    EEHEJE

    t

    EJH

    dVJEdt

    dW

    dtJEVdtvEVdW

    dtvEqdW

    dtvBvEqdFdW

    f

    rrr

    rrr

    rrrrrr

    rrrrrr

    rrr

    rr

    rrrr

    rr

    rrrrlrr

    ×⋅∇−

    ∂µ−⋅=×∇⋅

    ×∇⋅−×∇⋅=×⋅∇

    ∂⋅ε−×∇⋅=⋅

    ∂ε+=×∇

    ⋅=

    ⋅δ=⋅ρδ=

    ⋅δ=

    ⋅×+δ=⋅=

    Work done on a charge δδδδq by EM force:

    ( )

    ( )

    ( )

    ( )

    adSt

    U

    dt

    dW

    adHEdVt

    E

    t

    H

    2

    1

    dt

    dW

    dVHEdVt

    E

    t

    H

    2

    1

    dt

    dW

    HEt

    E

    2t

    H

    2JE

    t

    EEHE

    t

    HHJE

    EM

    2

    o

    2

    22

    22

    rr

    rrrrr

    rrrr

    rrrr

    rr

    rrrr

    rrrr

    ∫∫

    ∫∫

    ⋅−∂

    ∂−≡

    ⋅×−

    ∂ε+

    ∂µ−=

    ×⋅∇−

    ∂ε+

    ∂µ−=

    ×⋅∇−∂

    ∂ε−

    ∂µ−=⋅

    ∂⋅ε−×⋅∇−

    ∂⋅µ−=⋅

    (math)

    Poynting Vector

    Power flow out of the enclosed surface

    EM power stored in the volume

    The decrease of the EM energy stored is to do work on

    a charge or due to the net energy escape through the

    surface.

    the work-energy theorem

    for electrodynamics

  • Plane EM Wave - Dr. Ray Kwok

    Poynting Vector

    HESrrr

    ×≡

    Magnitude gives power intensity = power/area

    Direction gives direction of propagation

    2

    o

    2

    o

    ave

    ave

    H22

    ES

    *]HE[e2

    1S

    η=

    η=

    ×ℜ≡rrr

    This is similar to P = ½ (V2/Z) = ½ (I2Z)

    Time average

  • Plane EM Wave - Dr. Ray Kwok

    ExampleIf solar illumination is characterized by a power density of 1 kW/m2 at Earth’s surface,

    find (a) the total power radiated by the sun,

    (b) the total power intercepted by Earth, and

    (c) the electric field of the power density incident upon Earth’s surface, assuming that

    all the solar illumination is at a single frequency. The radius of Earth’s orbit around the

    sun, Rs, is approximately 1.5 x 108 km, and Earth’s mean radius RE is 6380 km.

    (a) Average power radiated Psun = Save(4πRs2) = (103)(4π)(1.5 x 1011)2 = 2.8 x 1026 W

    (b) Average power intercepted PE = Save(πRE2) = (103)(π)(6.38 x 106)2 = 1.28 x 1017 W

    (c) Electric field intensity …

    870)10)(377(2S2E

    2

    ES

    3

    aveoo

    2

    o

    ave

    ==η=

    η=

    V/m

  • Plane EM Wave - Dr. Ray Kwok

    Linear Polarization

    )ztcos(Ex̂)t,r(E 1o1 β−ω=rr

    x

    y Defined by the direction of E-field. For example:

    Horizontal polarization

    )ztcos(Eŷ)t,r(E 2o2 β−ω=rr

    Vertical polarization

    x

    y

    21TOTAL EEErrr

    +=ETOTAL is still linearly polarized

    If Eo1 = Eo2, then ETOTAL is at 45 degrees.

    A linear combination of these….

    E1

    E2

    Any EM wave can be expressed as a linear combination ofhorizontal & vertical polarized field.

  • Plane EM Wave - Dr. Ray Kwok

    Circular Polarization

    )ztcos(Ex̂)t,r(E 1o1 β−ω=rr

    21T EEErrr

    +=

    x

    y

    If E1 & E2 are 90 degrees out-of-phase, & Eo1 = Eo2:

    Horizontal polarization

    )ztsin(Eŷ)t,r(E 2o2 β−ω=rr

    Vertical polarization

    ET(t=0)

    ET(t>0)RHCP

    Right-Hand Circularly Polarized

    )zt(j

    o

    )2/zt(j

    o

    )zt(j

    o

    oo

    eE)ŷjx̂()t,r(E

    eEŷeEx̂)t,r(E

    )ztsin(Eŷ)ztcos(Ex̂)t,r(E

    β−ω

    π−β−ωβ−ω

    −=

    +=

    β−ω+β−ω=

    rr

    rr

    rr

  • Plane EM Wave - Dr. Ray Kwok

  • Plane EM Wave - Dr. Ray Kwok

    LHCP

    x

    y

    ET(t=0)

    ET(t>0)

    )zt(j

    o

    )2/zt(j

    o

    )zt(j

    o

    oo

    eE)ŷjx̂()t,r(E

    eEŷeEx̂)t,r(E

    )ztsin(Eŷ)ztcos(Ex̂)t,r(E

    β−ω

    π+β−ωβ−ω

    +=

    +=

    β−ω−β−ω=

    rr

    rr

    rr

    LHCP

  • Plane EM Wave - Dr. Ray Kwok

    Circular polarization in nature

    Rose Chafer (a beetle)& others…

    Reflection from its surface is LHCP.,due to the helical molecular structureof their shells.

  • Plane EM Wave - Dr. Ray Kwok

    Mantis Shrimp (Stomatopod Crustacean)

    Able to sense LHCP & RHCP,

    over a wide spectrum of color,

    better than man-made instrument.

    Most complex eye structure in

    animal kingdom.

    Fastest animal (striking) & produces

    over 200 lbs of striking force !!!

    Nature Photonics 3, 641-644 (2009)Roberts, Chiou, Marshall & Cronin,

    Sheila Patek at TED.com

    a popular dish known as

    "pissing shrimp" (攋尿蝦)

  • Plane EM Wave - Dr. Ray Kwok

    Circular polarization in starlight

    Most starlight are slightly circularly polarized.

    The interstellar medium (low density dust) can produce circularly polarized

    (CP) light from unpolarized light by sequential scattering from elongated

    interstellar grains (with complex index of refraction) aligned in different

    directions. One possibility is twisted grain alignment along the line of sight

    due to variation in the galactic magnetic field; another is the line of sight

    passes through multiple clouds.

  • Plane EM Wave - Dr. Ray Kwok

    Polarizer

    θunpolarized

    Io

    polarized

    I1 = ½ Io

    polarized

    I2 = I1cos2θ = ½ Iocos

    complex n

    Linearly polarized

    intensity

  • Plane EM Wave - Dr. Ray Kwok

    Elliptical Polarization

    )zt(j

    2o1o

    2o1o

    e)EŷjEx̂()t,r(E

    )ztsin(Eŷ)ztcos(Ex̂)t,r(E

    β−ω=

    β−ω±β−ω=

    mrr

    rrIf Eo1 ≠ Eo2

    Or, if the phase difference is not 90 degrees

    RHEP

    LHEP

    )zt(j

    o

    j

    )zt(j

    o

    )zt(j

    o

    eE)eŷx̂()t,r(E

    eEŷeEx̂)t,r(E

    β−ωδ

    δ+β−ωβ−ω

    +=

    +=rr

    rr

    RHEP (δ < 0)

    LHEP (δ > 0)

    Special cases:

    if δ = −π/2, → RHCPif δ = π/2, → LHCP

    x

    y

    ET(t=0)

    LHEP

    ET(t>0)

    E02

    E01

  • Plane EM Wave - Dr. Ray Kwok

    Polarization Generalization)zt(j

    y

    )zt(j

    x eaŷeax̂)t,r(Eδ+β−ωβ−ω +=

    rr

    0

    0

    cos)2tan(2tan

    sin)2sin(2sin

    a

    atan

    R

    1tan

    a

    aR

    o

    o

    x

    y

    o

    γ

    δψ=γ

    δψ=χ

    ≡ψ

    ±≡χ

    ≡η

    ξaxial ratio ≥ 1= 1 circular

    = ∞ linear

    ellipticity angle

    χ > 0 when sinδ > 0 → LHχ < 0 when sinδ < 0 → RH

    auxiliary angle

    0 ≤ ψο ≤ π/2

    −π/4 ≤ χ ≤ π/4

    rotation angle −π/2 ≤ γ ≤ π/2

    if cosδ > 0

    if cosδ < 0

  • Plane EM Wave - Dr. Ray Kwok

    Polarization States

  • Plane EM Wave - Dr. Ray Kwok

    e.g. what’s the polarization?

    ( )( ) )6/kzt(j105j

    )6/kzt(j12/7j

    )4/3kzt(j)6/kzt(j

    )2/4/kzt(j)6/kzt(j

    )2/4/kzt(j)6/kzt(j

    ee4ŷ3x̂)t,z(E

    ee4ŷ3x̂)t,z(E

    e4ŷe3x̂)t,z(E

    e4ŷe3x̂)t,z(E

    e4ŷe3x̂)t,z(E

    o π+−ω

    π+−ωπ

    π+−ωπ+−ω

    π+π−π+−ωπ+−ω

    π−π+−ωπ+−ω

    +=

    +=

    +=

    +=

    −=

    r

    r

    r

    r

    r)4/kztsin(4ŷ)6/kztcos(3x̂)t,z(E π+−ω−π+−ω=

    r

    o

    oo

    o

    o

    oo

    o

    o

    o

    x

    y

    o

    69

    )105cos()106tan(cos)2tan(2tan

    34

    )105sin()106sin(sin)2sin(2sin

    53

    3

    4

    a

    atan

    −=γ

    =δψ=γ

    =δψ=χ

    =≡ψ00256.0)105cos(cos

    LH00966.0)105sin(sin

    105

    o

    o

    o

    ==δ

    x

    y

    ay=4

    ax=3

    χχχχ

    γ γ γ γ = −−−− 69o

    ψψψψo

    LHEP

  • Plane EM Wave - Dr. Ray Kwok

    Linear combination

    )ztcos(Ex̂)t,r(E o β−ω=rr

    Any LP EM wave can be written as linear combination of RHCP + LHCP

    ( ) ( )[ ]

    2

    EEE

    eEEŷjEEx̂)t,r(E

    )t,r(E)t,r(E)t,r(E

    eE)ŷjx̂()t,r(E

    eE)ŷjx̂()t,r(E

    o12

    )zt(j

    1221T

    LHCPRHCPT

    )zt(j

    2LHCP

    )zt(j

    1RHCP

    ==

    −++=

    +=

    +=

    −=

    β−ω

    β−ω

    β−ω

    rr

    rrrrrr

    rr

    rr

    e.g.

    More generally, any EM wave can be written as linear combination of RHEP + LHEP

  • Plane EM Wave - Dr. Ray Kwok

    ExerciseThe amplitudes of an elliptically polarized plane wave traveling in a lossless,

    nonmagnetic medium with εr = 4 and Hxo= 8 mA/m, and Hyo= 6 mA/m. Determine the average power flowing through an aperture in the y-z plane if its area is 20 m2.

    η = 60π = 188.5 ΩSave = 9.43 mW/m

    2 along the x-axis

    Pave = 0.19 W