8 Plane Electromagnetic Waves - National Chiao Tung …ocw.nctu.edu.tw/course/emii031/CH08.pdfUEE...

UEE 3201 Electromagnetics II Fall, 2014

8 Plane Electromagnetic Waves

8.2 Plane Waves in Lossless Media

1. Homogeneous wave equation in free space

∇2E− 1c2

∂E∂t2

= 0

For a time harmonic (sinusoidal) field,

∇2E+ k20E = 0, (E = phasor)

where k0 = ω√µ0ϵ0 = ω/c = free space wave number.

In Cartesian corrdinate,

( ∂2

∂x2 + ∂2

∂y2+ ∂2

∂z2+ k2

0)Ex,y,z = 0

2. wavefront: set of points/positions in space with same phase

plane wave: wave with planar wavefront perpendicular to the direction

of propagation

spherical wave: wave with spherical wavefront

uniform wave: wave with constant E and H field magnitude across the

wavefront

3. Now consider a uniform plane wave with constant Ex on the wavefront

xy plane,

→ General solution in phasor form is

where E+0 and E−

0 are complex constant to be determined from bound-

ary or initial conditions.

4. The real-time solution represented by the first phasor term is

= wave traveling in the +z direction

c⃝Yi Chiu, ECE Dept., NCTU 8-1


wave velocity (phase velocity up)

=

5. Real-time solution represented by the second phasor term is

E−x (z, t) = ℜ[E−

0 ej(ωt+k0z)]

= E−0 cos(ωt+ k0z)

= wave traveling in −z direction with phase velocity up = −c

- Sign of up (or k0) represents direction of wave propagation.

- If there is only one wave in the +z direction, then E−0 = 0.

6. H field can be found from E and Maxwell’s equations:

⇒

For +z wave,

Similarly for −z wave,

H−y (z) = − 1

η0E−

x (z)



7. η0 =E+

x (z)H+

y (z)= intrinsic impedance of the free space

If η0 = real → E+x (z) and H+

y (z) are in phase

8. Real time solution of the H field

(directions of wave propagation (az), electric field (ax) and magnetic

field (ay) are mutually perpendicular.)

Ex 8-1: A uniform plane wave E = axEx propagates in +z direction,

ϵr = 4, µr = 1, σ = 0, f = 100MHz

(a) instantaneous expression of E

(b) instantaneous expression of H



(c) positions for postitive peaks Ex at t = 10−8 sec

- E and H are perpendicular to each other and both are transverse

(perpendicular) to propagation direction

→ transverse electromagnetic wave (TEM wave)

9. Doppler effect (see the textbook)

10. For a uniform plane wave propagating in an arbitrary direction in a loss-

less medium, the solution to the wave equation ∇2E+k2E = 0 has the

general form (from separation of variables, let E = E0X(x)Y (y)Z(z))

E(x, y, z) = E0e−jkxxe−jkyye−jkzz

= E0e−j(kxx+kyy+kzz)

= E0e−jk·R

where R = axx+ ayy + azz = position vector

k = axkx + ayky + azkz = wave vector

|k|2 = k2x + k2

y + k2z = k2 = ω2µϵ



Wavefront of the wave E(x, y, z) = set of points with same phase

⇒ k = ank = wave vector

an = propagation direction

k = |k| = ω√µϵ = 2π/λ = wave number

11. For a uniform plane wave in a charge free region,

⇒ E0 is transverse to the propagation direction

12. H(R) =

where η = ωµ/k =√µ/ϵ (Ω) = intrinsic impedance of the medium,

→ H(R) = 1η (an × E0) e

−jk·R

→ H ⊥ an and E, an is in the direction of E×H (right hand rule)

→ TEM wave in the an (or k) direction

Ex 8-2: Find E(R) in terms of H(R)

(自己看)

⇒ E(R) = −ηan ×H(R)



13. Polarization of plane waves (偏極化, 偏振)

Polarization is the time varying behavior of the orientation of the E

field at a given point in space. For example, in Ex 8-1,

The ”tip” of E field oscillates only along the x axis

→ linear polarization (linaerly polarized)

14. Now consider another wave in the +z direction,

= superposition of two orthogonal linearly polarized wave with

y polarization lagging x polarization by π/2.

The instantaneous E field is

At a given position z = 0,

Locus of ”tip” of E(0, t) is(E1(0, t)E10

)2

+

(E2(0, t)E20

)2

= cos2 ωt+ sin2 ωt = 1

→ ellipse in the counterclockwise sense

→ elliptically polarized if E10 = E20

circularly polarized if E10 = E20

→ right-hand or positive circularly polarized wave



15. If E2(z) leads E1(z) by 90 (π/2),

E(z) = axE10e−jkz + ayjE20e

−jkz

E(0, t) = axE10 cosωt− ayjE20 sinωt

• elliptically or circularly polarized wave

• α = tan−1 E2(0, t)E1(0, t)

= −ωt for E20 = E10

→ left-hand or negative circularly polarized wave

16. If E1(z) and E2(z) are in time phase,

→ linearly polarized

17. In general,

→ elliptically polarized

Ex 8-3: Linear and circular waves

Consider a linearly polarized wave propagating in +z direction,

= superposition of a right-hand and a left-hand circular waves

Note: - AM wave: linearly polarized with E field perpendicular to ground

- TV wave: linearly polarized in the horizontal direction

- FM wave: circularly polarized

→ orientation of receiving antenna should be adjusted accordingly



8.3 Plane Waves in Lossy Media

1. wave equation in lossy media

- kc = ω√µϵc = complex wave number (assume µ = real)

- plane wave ejkz in lossless medium becomes ejkcz in lossy medium

We can define a propagation constant γ

* For lossless media, σ = 0, ϵ′′ = 0 ⇒ α = 0, β = k = ω√µϵ

The Helmholtz wave equation becomes

For a linearly polarized uniform plane wave in +z direction,

e−αz: attenuation factor

α: attenuation constant (Np/m, 1/m)

(wave amplitude decresed by a factor of e−1 after traveling a dis-

tance of 1/α meters)

e−jβz: phase factor

β: phase constant (rad/m)

(similar to the wave number in lossless media)

8.3.1 Low-Loss Dielectrics

1. propagation constant



For a good but imperfect insulator, ϵ′′ ≪ ϵ′, or σ/ωϵ ≪ 1,

⇒ γ = α + jβ = jω√µϵc = jω

√µϵ′(1− jϵ′′/ϵ′)−1/2

≈

⇒ attenuation constant α ≃ ωϵ′′

2

√µϵ′

∝ ω

phase constant β ≃ ω√µϵ′

[1 + 1

8

(ϵ′′

ϵ′

)2]

2. intrinsic impedance

ηc =

√µ

ϵ′

(1− j

ϵ′′

ϵ′

)−1/2

≈

⇒ Ex and Hy are not in time phase

3. phase veolcity

up =ωβ

≈ 1√µϵ′

(1− 1

8

(ϵ′′

ϵ′

)2)

8.3.2 Good Conductor

1. propagation constant

γ = α + jβ = jω√µϵ

(1 + σ

jωϵ

)1/2

≈

2. intrinsic impedance

3. phase velocity

up =ωβ

≈√

2ωµσ



4. example: copper

⇒ after a distance of δ = 1/α = 0.038 mm, the wave amplitude will be

attenuated by a factor of e−1 = 0.368.

(At f = 10 GHz, δ = 0.66µm ⇒ high frequency EM wave is attenuated

very rapidly in a good conductor.)

5. skin depth or depth of penetration

δ = distance over which the wave amplitude is attenuated by a factor

of 1/e

=

=

=

→ E,J are distributed near the surface of a conductor at high

frequency.

(See Table 8-1)

Ex 8-4: Seawater (Summary)



At f = 5 MHz,

σ/ωϵ = 200 ≫ 1 ⇒ good conductor,

attenuation constant α = phase constant β =√πfµσ = 8.89 (Np/m,

rad/m),

wavelength in water λ = 2π/β = 0.707 m,

skin depth δ = 0.112 m

(a) At 5 MHz, the wavelength in air is 60 m. It is very difficult to

communicate with a submarine due to the small penetration depth

in seawater and long wavelength in air (difficult to build efficient

tranmission antenna).

(b) After the real-time electric field Ex(z, t) is calculated, it is in-

correct to calculate the real-time magnetic field Hy(z, t) from

Hy(z, t) = Ex(z, t)/ηc. Since the complex impedance is defined

in the phasor form (frequency domain), the correct formula are

Hy(z) =Ex(z)

ηc,

Hy(z, t) = ℜ[Ex(z)

ηcejωt

]8.3.3 Ionized Gas (plasma)

Plasma - assembly of equal numbers of positive ions and negative electrons

and possibly other neutral species

- electrons are much lighter than ions

- motion of ions can be neglected and electrons can be viewed as a

free electron gas (collision is neglected)

- neutral species, if any, are not affected by electric field



1. From Newton’s force law, force on an electron in a time harmonic E

field is

In phasor form,

Displacement x from the background positive ions gives rise to a dipole

moment

If there are N electrons per unit volume, the polarization vector is

Equivalent permittivity of a plasma is

- propagation constant in a plasma

- intrinsic impedance



(a) f < fp , γ = pure real

⇒ pure attenuation of wave in plasma

ηp = E/H = pure imaginary

⇒ plasma is a reactive load

⇒ no power transmission (electrons can move fast enough to

screen the EM fields)

(fp ≈ 9√N ≈ 0.9− 9 MHz for ionosphere)

(b) f > fp , γ = pure imaginary

⇒ no attenuation in plasma

Ex 8-5: Communication with space ship



8.4 Group Velocity

1. Phase velocity is defined as up = ω/β. In a lossless medium, β =

ω√µϵ is a linear function of ω, so the phase velocity is a constant. In

cases where the phase constant β is not a linear function of ω (e.g.

lossy dielectric, transmissioin line, or waveguide), the phase velocity is

not a constant. Therefore, different frequency components in a wave

propagate with different phase velocity. This phenomenum is called

dispersion and the medium or structure is dispersive.

2. In information transmission, informatin signals are usually composed

of a high frequency carrier surrounded by a signal sideband with fi-

nite bandwidth. Therefore, the information waveform with a group of

frequencies will be affected by dispersion.

3. Now consider a signal with two frequency components with equal am-

plitude: ω = ω0 ±∆ω, β = β0 ±∆β.



(a) phase velocity of carrier, up

(b) phase velocity of envelop, ug

ug = group velocity of the information signal

4. ω − β diagram of a plasma

Ex 8-6: Narrow band signal in lossy medium



8.5 Flow of Electromagnetic Power and the Poynting

Vector

1. From the two curl equaitons,

it can be shown that for a simple, time invariant medium,

For a volume V bounded by a surface S,

⇒ P , E×H = Poynting vector

= power flow per unit area

= power density vector

2. Poynting theorem

−∮

P · ds = ∂

∂t

∫V

(we + wm)dv +

∫V

Pσdv

where we =12 ϵE2 = electric energy density,

wm = 12 µH2 = magnetic energy density,

Pσ = σE2 = Ohmic power density.

note: (1) P is perpendicular to E and H (P is parallel to k for a uniform

plane wave in a simple medium)

(2) for lossless media, Pσ = 0, and energy is stored in the electric and

magnetic fields

(3) in a static case, ∂/∂t = 0, all power flow into a closed region is

disscipated by the ’equivalent’ conduction current (Ohmic loss)



Ex 8-7: Verify the Poynting vector for a straight wire

Consider a section of wire with length ℓ,

8.5.1 Instantaneous and average power density

1. phasor form of field and power density

⇒ P can not be defined as E×H in the phasor form.

2. Average power density

Note that ℜ[A]×ℜ[B] = 12ℜ[A×B∗ +A×B],

⇒



Consider a time-averaged power density Pav,

3. For a x−polarized TEM wave in the +z direction, the Poynting vector

expressed in terms of intrinsic impedance of the medium is



8.6 Normal Incidence at a Plane Conducting Bound-

ary

1. Incident wave in medium 1

Reflected wave in medium 1

Total E field in medium 1

Boundary conditions at z = 0: tangential component of E is continuous

Magnetic field

⇒ H1(z) = Hi(z) +Hr(z) = ay2Ei0η1 cos β1z

2. Average power

(a) incident wave:

reflected wave:

total average power flow in medium 1



(b) Or, η1 = , θη1 = −π2 ⇒ (Pav)1 = 0

3. Instantaneous expression of fields

standing wave:

propagation wave:

4. For a standing wave, zero of E1(z, t) :

max of H1(z, t) : max of E1(z, t) :

zero of H1(z, t) :

(a) E1 = 0 on the surface

(b) H1 = max on the surface

(c) temporal phase difference between E and H = 90

(d) spatial separation of E and H field patterns = λ/4



Ex 8-9:



8.7 Oblique Incidence at a Plane Conducting Bound-

ary

1. plane of incidence = plane that contains ki and an

2. With respect to the plane of incidence, any polarization of Ei can be

decomposed into two components: Ei⊥ and Ei∥

8.7.1 Perpendicular polarizatioin (Ei⊥, TE wave)

1. Incident wave

Hi(x, z) = 1η1 ani × Ei(x, z)

= Ei0η1 (−ax cos θi + az sin θi)e

−jβ1(x sin θi+z cos θi)

2. Reflected wave (θr = angle of reflection)

⇒ Er(x, z) = = ayEr0e−jβ1(x sin θr−z cos θr)

Er0 and θr can be found from boundary conditions:

⇒



⇒

⇒

Hr(x, z) = 1η1 anr × Er(x, z)

= Ei0η1 (−ax cos θi − az sin θi)e

−jβ1(x sin θi+z cos θi)

3. Total field

H1(x, z) = Hi(x, z) +Hr(x, z)

= −2Ei0η1

[ax cos θi cos(β1z cos θi)e

−jβ1x sin θi

+ azj sin θi sin(β1z cos θi)e−jβ1x sin θi

]Compared to a plane wave

(a) In the normal direction (z), E and H form standing wave patterns

described by β1z = β1 cos θi. No average power is transmitted.

(b) In the transverse direction (x), E and H form propagation wave

described by β1x = β1 sin θi, and

(c) The plane wave in the x−direction is non-uniform (interference

pattern).

(d) Zeros of E1:



→ A conducting plane can be inserted at z = − mλ12 cos θi

without

changing the field distribution

→ TE wave in a parallel waveguide

Ex 8-10: (a) Current on the surface of the conductor

→ There is a discontinuity of H across the interface

→ From B.C., the discontinuity is caused by a surface current Js

Instantaneous expression of surface current Js

(b) Poynting vector in medium 1

– E1y and H1x are in time quadrature → no net power in z

direction

– (Pav)1 is a function of z because total wave in medium 1 is a

non-uniform plane wave

8.7.2 Parallel polarization (Ei∥, TM wave)



1. Incident wave

2. Reflected wave

Er(x, z) = Er0(ax cos θr + az sin θr)e−jβ1(x sin θr−z cos θr)

Hr(x, z) = −ayEr0η1 e−jβ1(x sin θr−z cos θr)

B.C.: at z = 0,

3. Total wave

(a) In the normal direction (z), E1x and H1y have standing wave pat-

terns described by β1z = β1 cos θi.

(b) In the transverse direction (x), E1z and H1y are non-uniform

propagating wave described by β1x = β1 sin θi, u1x = ω/β1x =

u1/ sin θi, λ1x = 2π/β1x = λ1/ sin θi.

(c) A conducting plane can be inseted at z = − mλ12 cos θi

to form a

parallel plate waveguide for the TM wave



8.8 Normal Incidence at a Plane Dielectric Boundary

1. Incident wave

2. Reflected wave

3. Transmitted wave Et(z) = axEt0e−jβ2z, z > 0

Ht(z) = ayEt0η2 e−jβ2z

4. Boundary conditions (tangential components)

5. Reflection and transmissin coefficientsΓ = reflection coefficient = Er0

Ei0=

η2 − η1η2 + η1

τ = transmission coefficient = Et0Ei0

=2η2

η2 + η1



- if η1, η2 = real, → Γ > 0 or < 0 (in- or 180 out-of-phase)

τ > 0 (in-phase)

- if η1, η2 = complex (dissipative media)

→ Γ, τ = complex,

→ phase shift in reflected and transmitted waves

- 1 + Γ = τ

- if medium 2 is a perfect conductor,

6. In general, total field in medium 1 is

7. Pattern of the standing wave

E1(z) = axEi0e−jβ1z(1 + Γej2β1z)

For lossless media, η1, η2,Γ, τ = real,

(a) Γ > 0 (η2 > η1)



(b) Γ < 0 (η2 < η1)

Standing Wave Ratio (SWR)

8. Magnetic field

H1(z) = Hi(z) +Hr(z)

= ayEi0η1 (e−jβ1z − Γejβ1z)

= ayEi0η1 e−jβ1z(1− Γej2β1z)

In a lossless medium, |H1(z)| is max (min) where |E1(z)| is min (max)

9. Transmitted wave in medium 2

Ex 8-11: Power density in lossless media (Γ = real,Pav =12ℜE×H∗)

(a) medium 1

(Pav)1 =12ℜ[axEi0e

−jβ1z(1+Γej2β1z)]×[ayEi0η1 e−jβ1z(1−Γej2β1z)]∗

(b) medium 2

(Pav)2 =

⇒ 1−Γ2 =η1η2 τ

2 (can also be verified from Eqs. 8-140 and 8-141)

⇒ incident power - reflected power = transmitted power



8.9 Normal Incidence at Multiple Dielectric Interfaces

1. Dielectric coating can be found in eyeglass, camera lens, optical com-

munication systems, and laser systems to reduce or enhance reflection.

2. Consider the following figure. Multiple reflection occurs at z = 0 and

z = d. These waves can be summarized as forward and backward

waves.

Assume an x-polarized incident wave, the total waves in the three re-

gions can be expressed as H1 = ay1η1 (Ei0e

−jβ1z − Er0ejβ1z)

H2 = ay1η2 (E

+2 e

−jβ2z − E−2 e

jβ2z)

H3 = ay1η3E

+3 e

−jβ3z

- The four unknowns Er0, E+2 , E

−2 , and E+

3 can be solved from the

boundary conditions on the two interfaces: E1t(0) = E2t(0), H1t(0) = H2t(0)

E2t(d) = E3t(d), H2t(d) = H3t(d)

- The algebraic procedure is straightforward. But it lacks physi-

cal insight and becomes tedious when the number of interfaces

increases.



8.9.1 Wave Impedance of the Total Field

1. Wave impedance of total wave is defined as the ratio of total electric

field intensity to the toal magnetic field intensity at a particular position

z.

- For for an unbound medium, Z(z) = ±η for a ± z wave for all z.

- For two media separated by a plane boundary,

The wave impedance of total field in medium 1 at a distance z

from the boundary is

At z = −ℓ to the left of the boundary,

Z1(−ℓ) =E1x(−ℓ)

H1y(−ℓ)= η1

η2 cos β1ℓ+ jη1 sin β1ℓ

η1 cos β1ℓ+ jη2 sin β1ℓ

(note: Γ = (η2 − η1)/(η2 + η1))

- Total wave impedance is a function of both medium properties

(η and β) and distance to the boundary (z and ℓ)

8.9.2 Impedance Transformation with Multiple Dielectrics

1. At z = 0+ in medium 2 and looking into +z direction, the situation is

same as that of Eqs. 8-169 ∼ 8-171. Therefore η2, η1, β1 and ℓ can be

replaced by η3, η2, β2 and d.

⇒ Z2(0+) = η2

η3 cos β2d+ jη2 sin β2d

η2 cos β2d+ jη3 sin β2d

(η3 is transformed to Z2(0))



2. For the incident wave in medium 1, it sees an equivalent infinite medium

with instrinsic impedance Z2(0) at the boundary z = 0. Therefore the

effective reflection coefficient at z = 0 is

- In comparison to Γ =η2 − η1η2 + η1

- In medium 1, Γ0 and Er0 can be calculated by the transformed

impedance Z2(0).

- Fields in medium 2 and medium 3 can be calculated from B.C.

Ex 8-12: Find η2 and d for anti-reflection coating

⇒ Γ0 = 0, Z2(0) = η1

⇒ η2(η3 cos β2d+ jη2 sin β2d) = η1(η2 cos β2d+ jη3 sin β2d)

⇒

(a) η1 = η3, η2 =√η1η3 = η1

⇒trivial solution

(b) η1 = η3, η2 =√η1η3

(c) η1 = η3

- Arbitray η2 may not be available

- For optical applications, complex multiple coating is necessary for

wideband or other types of filter



8.10 Oblique Incidence at a Plane Dielectric Boundary

1. Snell’s law (see textbook) θr = θi (reflection)

n1 sin θi = n2 sin θt (refraction)

- n1, n2 = index of refraction and

n1n2

=up2up1

=β1

β2=

√µ1ϵ1µ2ϵ2

=√

ϵ1ϵ2 =

√ϵ1rϵ2r for non-magnetic media

- Both Snell’s laws are independent of polarization.

8.10.1 Total reflection

1. If ϵ1 > ϵ2 (n1 > n2), there is a critial incident angle θc such that

θt = π/2,

2. When θi > θc, there is no real solution for θt

→ no transmitted wave in medium 2 from a geometric point of view

3. From a physical point of view, when θi > θc,

In this case, the transmitted wave in medium 2 is



Ex 8-14: dielectric waveguide (e.g. optical fibers)

When θ1 > θc, power can be guided along the waveguide by total

internal reflection. The condition for waveguiding is

8.10.2 Perpendicular Polarization (TE wave)

1. Incident wave



2. Reflected wave Er(x, z) = ayEr0e−jβ1(x sin θr−z cos θr)

Hr(x, z) = Er0η1 (ax cos θr + az sin θr)e

−jβ1(x sin θr−z cos θr)

3. Transmitted wave Et(x, z) = ayEt0e−jβ2(x sin θt+z cos θt)

Ht(x, z) = Et0η2 (−ax cos θt + az sin θt)e

−jβ2(x sin θt+z cos θt)

4. Boundary conditions: tangential components of E, Ey, and H, Hx, are

continuous across the boundary

⇒

Ei0e−jβ1x sin θi + Er0e

−jβ1x sin θr = Et0e−jβ2x sin θt

1η1 (−Ei0 cos θie

−jβ1x sin θi + Er0 cos θre−jβ1x sin θr) = −Et0

η2 cos θte−jβ2x sin θt

”phase matching” condition

⇒

⇒

Er0Ei0

= Γ⊥ =η2 cos θi − η1 cos θtη2 cos θi + η1 cos θt

=η2/ cos θt − η1/ cos θiη2/ cos θt + η1/ cos θi

Et0Ei0

= τ⊥ =2η2/ cos θt

η2/ cos θt + η1/ cos θi(Fresnel′s eq.)

- For normal incidence, θi = 0 = θr = θt

- 1 + Γ⊥ = τ⊥

- Brewster angle (θB⊥)



According to Snell’s law,

(For non-magnetic materials, µ1 = µ2 = µ0, sin2 θB⊥ → ∞, θB⊥

does not exist)

8.10.3 Parallel Polarization (TM wave)

1. Similarly, θr = θisin θtsin θi

=β1

β2= n1

n2

Γ∥ =Er0Ei0

=η2 cos θt − η1 cos θiη2 cos θt + η1 cos θi

τ∥ =Et0Ei0

=2η2 cos θi

η2 cos θt + η1 cos θi

2. Brewster’s angle



- 1 + Γ∥ = τ∥cos θtcos θi

- Γ⊥ = Γ∥, τ⊥ = τ∥ unless θi = θr = θt = 0 (normal incidence)

- |Γ⊥|2 > |Γ∥|2 except at θi = 0,

→ unpolarized incident wave upon reflection

→ more reflected power in the ⊥ polarization than in the ∥ polar-

ization

Ex 8-15: Reflection from water


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