UEE 3201 Electromagnetics II
8
Fall, 2014
Plane Electromagnetic Waves
8.2
Plane Waves in Lossless Media
1. Homogeneous wave equation in free space
∇2 E − 12 ∂E2 = 0
c ∂t
For a time harmonic (sinusoidal) field,
∇2 E + k02 E = 0, (E = phasor)
√
where k0 = ω µ0 ϵ0 = ω/c = free space wave number.
In Cartesian corrdinate,
2
2
2
( ∂ 2 + ∂ 2 + ∂ 2 + k02 )Ex,y,z = 0
∂x
∂y
∂z
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 E0+ and E0− are complex constant to be determined from boundary or initial conditions.
4. The real-time solution represented by the first phasor term is
= wave traveling in the +z direction
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UEE 3201 Electromagnetics II
Fall, 2014
wave velocity (phase velocity up )
=
5. Real-time solution represented by the second phasor term is
Ex− (z, t) = ℜ[E0− ej(ωt+k0 z) ]
= E0− cos(ωt + k0 z)
= 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 E0− = 0.
6. H field can be found from E and Maxwell’s equations:
⇒
For +z wave,
Similarly for −z wave,
Hy− (z) = −
1 −
E (z)
η0 x
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UEE 3201 Electromagnetics II
7. η0 =
Fall, 2014
Ex+ (z)
= intrinsic impedance of the free space
Hy+ (z)
If η0 = real → Ex+ (z) and Hy+ (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 = ax Ex propagates in +z direction,
ϵr = 4, µr = 1, σ = 0, f = 100MHz
(a) instantaneous expression of E
(b) instantaneous expression of H
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UEE 3201 Electromagnetics II
Fall, 2014
(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 lossless medium, the solution to the wave equation ∇2 E + k 2 E = 0 has the
general form (from separation of variables, let E = E0 X(x)Y (y)Z(z))
E(x, y, z) = E0 e−jkx x e−jky y e−jkz z
= E0 e−j(kx x+ky y+kz z)
= E0 e−jk·R
where R = ax x + ay y + az z = position vector
k = ax kx + ay ky + az kz = wave vector
|k|2 = kx2 + ky2 + kz2 = k 2 = ω 2 µϵ
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UEE 3201 Electromagnetics II
Fall, 2014
Wavefront of the wave E(x, y, z) = set of points with same phase
⇒ k = an k = 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) = η1 (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)
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UEE 3201 Electromagnetics II
Fall, 2014
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
)2 (
)2
(
E2 (0, t)
E1 (0, t)
+
= cos2 ωt + sin2 ωt = 1
E10
E20
→ ellipse in the counterclockwise sense
→ elliptically polarized if E10 ̸= E20
circularly polarized if E10 = E20
→ right-hand or positive circularly polarized wave
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UEE 3201 Electromagnetics II
Fall, 2014
15. If E2 (z) leads E1 (z) by 90◦ (π/2),
E(z) = ax E10 e−jkz + ay jE20 e−jkz
E(0, t) = ax E10 cos ωt − ay jE20 sin ωt
• elliptically or circularly polarized wave
E2 (0, t)
= −ωt for E20 = E10
E1 (0, t)
→ left-hand or negative circularly polarized wave
• α = tan−1
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
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UEE 3201 Electromagnetics II
8.3
Fall, 2014
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 ejkc z 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 distance 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
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Fall, 2014
For a good but imperfect insulator, ϵ′′ ≪ ϵ′ , or σ/ωϵ ≪ 1,
√
√
⇒ γ = α + jβ = jω µϵc = jω µϵ′ (1 − jϵ′′ /ϵ′ )−1/2
≈
′′ √ µ
⇒ attenuation constant α ≃ ωϵ
∝ω
2
ϵ′
[
( ′′ )2 ]
√ ′
1
phase constant β ≃ ω µϵ 1 + 8 ϵ ′
ϵ
2. intrinsic impedance
√ (
)−1/2
ϵ′′
µ
ηc =
1−j ′
ϵ′
ϵ
≈
⇒ Ex and Hy are not in time phase
3. phase veolcity (
( ′′ )2 )
ω
1
1
up = ≈ √ ′ 1 − 8 ϵ ′
β
ϵ
µϵ
8.3.2
Good Conductor
1. propagation constant
)
√ (
σ 1/2
γ = α + jβ = jω µϵ 1 + jωϵ
≈
2. intrinsic impedance
3. phase velocity
√
2ω
ω
up = ≈ µσ
β
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UEE 3201 Electromagnetics II
Fall, 2014
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)
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UEE 3201 Electromagnetics II
Fall, 2014
At f = 5 MHz,
σ/ωϵ = 200 ≫ 1 ⇒ good conductor,
attenuation constant α = phase constant β =
√
πf µσ = 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 incorrect 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
Ex (z)
,
ηc
[
]
Ex (z) jωt
Hy (z, t) = ℜ
e
ηc
Hy (z) =
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
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UEE 3201 Electromagnetics II
Fall, 2014
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
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Fall, 2014
(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
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UEE 3201 Electromagnetics II
8.4
Fall, 2014
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 finite 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 amplitude: ω = ω0 ± ∆ω, β = β0 ± ∆β.
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UEE 3201 Electromagnetics II
Fall, 2014
(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
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UEE 3201 Electromagnetics II
8.5
Fall, 2014
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
∫
∫
I
∂
(we + wm )dv +
Pσ dv
− P · ds =
∂t V
V
where we = 21 ϵE 2 = electric energy density,
wm = 21 µH 2 = magnetic energy density,
Pσ = σE 2 = 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)
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UEE 3201 Electromagnetics II
Fall, 2014
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] = 12 ℜ[A × B∗ + A × B],
⇒
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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
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UEE 3201 Electromagnetics II
8.6
Fall, 2014
Normal Incidence at a Plane Conducting Boundary
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
cos β1 z
⇒ H1 (z) = Hi (z) + Hr (z) = ay 2 Eηi0
1
2. Average power
(a) incident wave:
reflected wave:
total average power flow in medium 1
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UEE 3201 Electromagnetics II
Fall, 2014
, θη1 = − π
2 ⇒ (Pav )1 = 0
(b) Or, η1 =
3. Instantaneous expression of fields
standing wave:
propagation wave:
4. For
 a standing wave,
 zero of E (z, t) :
1
 max of H (z, t) :
1

 max of E (z, t) :
1
 zero of H (z, t) :
1
(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
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UEE 3201 Electromagnetics II
Fall, 2014
Ex 8-9:
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UEE 3201 Electromagnetics II
8.7
Fall, 2014
Oblique Incidence at a Plane Conducting Boundary
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) = η11 ani × Ei (x, z)
= Eηi0
(−ax cos θi + az sin θi )e−jβ1 (x sin θi +z cos θi )
1
2. Reflected wave (θr = angle of reflection)
⇒ Er (x, z) =
= ay Er0 e−jβ1 (x sin θr −z cos θr )
Er0 and θr can be found from boundary conditions:
⇒
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Fall, 2014
⇒
⇒
Hr (x, z) = η11 anr × Er (x, z)
= Eη1i0 (−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)
[
= −2 Eηi0
ax cos θi cos(β1 z cos θi )e−jβ1 x sin θi
1
]
+ az j sin θi sin(β1 z cos θi )e−jβ1 x 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 :
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UEE 3201 Electromagnetics II
Fall, 2014
→ A conducting plane can be inserted at z = − mλ1 without
2 cos θi
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)
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UEE 3201 Electromagnetics II
Fall, 2014
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) = −ay Eηr0
e−jβ1 (x sin θr −z cos θr )
1
B.C.: at z = 0,
3. Total wave
(a) In the normal direction (z), E1x and H1y have standing wave patterns 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 = − mλ1 to form a
2 cos θi
parallel plate waveguide for the TM wave
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UEE 3201 Electromagnetics II
8.8
Fall, 2014
Normal Incidence at a Plane Dielectric Boundary
1. Incident wave
2. Reflected wave
3. Transmitted wave

 E (z) = a E e−jβ2 z , z > 0
t
x t0
 H (z) = a Et0 e−jβ2 z
t
y η
2
4. Boundary conditions (tangential components)
5. Reflection and transmissin coefficients

η2 − η1
r0

Γ = reflection coefficient = E


Ei0 = η2 + η1




 τ = transmission coefficient = Et0 = 2η2
η2 + η1
Ei0
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Fall, 2014
- 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) = ax Ei0 e−jβ1z (1 + Γej2β1 z )
For lossless media, η1 , η2 , Γ, τ = real,
(a) Γ > 0 (η2 > η1 )
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(b) Γ < 0 (η2 < η1 )
Standing Wave Ratio (SWR)
8. Magnetic field
H1 (z) = Hi (z) + Hr (z)
= ay Eη1i0 (e−jβ1 z − Γejβ1 z )
= ay Eηi0
e−jβ1 z (1 − Γej2β1 z )
1
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 = 21 ℜ{E × H∗ })
(a) medium 1
(Pav )1 = 12 ℜ{[ax Ei0 e−jβ1 z (1+Γej2β1 z )]×[ay Eη1i0 e−jβ1 z (1−Γej2β1 z )]∗ }
(b) medium 2
(Pav )2 =
η
⇒ 1 − Γ2 = η12 τ 2 (can also be verified from Eqs. 8-140 and 8-141)
⇒ incident power - reflected power = transmitted power
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UEE 3201 Electromagnetics II
8.9
Fall, 2014
Normal Incidence at Multiple Dielectric Interfaces
1. Dielectric coating can be found in eyeglass, camera lens, optical communication 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 regions can be expressed as


 H = a 1 (E e−jβ1 z − E ejβ1 z )
1
yη
i0
r0
1


 H = a 1 (E + e−jβ2 z − E − ejβ2 z )
2
yη
2
2
2


 H = a 1 E + e−jβ3 z
3
yη
3 3
- The four unknowns Er0 , E2+ , E2− , and E3+ can be solved from the
boundary conditions on the two interfaces:

 E (0) = E (0), H (0) = H (0)
1t
2t
1t
2t
 E (d) = E (d), H (d) = H (d)
2t
3t
2t
3t
- The algebraic procedure is straightforward. But it lacks physical insight and becomes tedious when the number of interfaces
increases.
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UEE 3201 Electromagnetics II
8.9.1
Fall, 2014
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)
η2 cos β1 ℓ + jη1 sin β1 ℓ
= η1
H1y (−ℓ)
η1 cos β1 ℓ + jη2 sin β1 ℓ
(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 β2 d + jη2 sin β2 d
η2 cos β2 d + jη3 sin β2 d
(η3 is transformed to Z2 (0))
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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
η −η
- In comparison to Γ = η2 + η1
2
1
- 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 β2 d + jη2 sin β2 d) = η1 (η2 cos β2 d + jη3 sin β2 d)
⇒
(a) η1 = η3 , η2 =
√
η1 η3 = η1
⇒trivial solution
(b) η1 = η3 , η2 ̸=
√
η1 η3
(c) η1 ̸= η3
- Arbitray η2 may not be available
- For optical applications, complex multiple coating is necessary for
wideband or other types of filter
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UEE 3201 Electromagnetics II
8.10
Fall, 2014
Oblique Incidence at a Plane Dielectric Boundary
1. Snell’s law (see textbook)

 θ =θ
(reflection)
r
i
 n sin θ = n sin θ
(refraction)
1
i
2
t
- n1 , n2 = index of refraction and
√
n1 = up2 = β1 = µ1 ϵ1
n2
up1
µϵ
β2
√ 2 2 √
= ϵϵ12 = ϵϵ1r
2r
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
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UEE 3201 Electromagnetics II
Fall, 2014
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
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UEE 3201 Electromagnetics II
Fall, 2014
2. Reflected wave

 E (x, z) = a E e−jβ1 (x sin θr −z cos θr )
r
y r0
E
 H (x, z) =
−jβ1 (x sin θr −z cos θr )
r0
r
η1 (ax cos θr + az sin θr )e
3. Transmitted wave

 E (x, z) = a E e−jβ2 (x sin θt +z cos θt )
t
y t0
E
 H (x, z) =
−jβ2 (x sin θt +z cos θt )
t0
t
η2 (−ax cos θt + az sin θt )e
4. Boundary conditions: tangential components of E, Ey , and H, Hx , are
continuous across the boundary

 E e−jβ1 x sin θi + E e−jβ1 x sin θr = E e−jβ2 x sin θt
i0
r0
t0
⇒
 1 (−E cos θ e−jβ1 x sin θi + E cos θ e−jβ1 x sin θr ) = − Et0 cos θ e−jβ2 x sin θt
i0
i
t
r0
r
η1
η2
”phase matching” condition
⇒


 Er0 = Γ⊥ = η2 cos θi − η1 cos θt = η2 / cos θt − η1 / cos θi
Ei0
η2 cos θi + η1 cos θt
η2 / cos θt + η1 / cos θi
⇒
2η
/
cos
θ
E
2
t

′
t0

Ei0 = τ⊥ = η2 / cos θt + η1 / cos θi (Fresnel s eq.)
- For normal incidence, θi = 0 = θr = θt
- 1 + Γ⊥ = τ⊥
- Brewster angle (θB⊥ )
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UEE 3201 Electromagnetics II
Fall, 2014
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 = θi
 sin θt = β1 = n1
n2
sin θi
β2

 Γ∥ = Er0 = η2 cos θt − η1 cos θi
Ei0
η2 cos θt + η1 cos θi
2η2 cos θi
 τ = Et0 =
∥
Ei0
η2 cos θt + η1 cos θi
2. Brewster’s angle
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UEE 3201 Electromagnetics II
Fall, 2014
- 1 + Γ∥ = τ∥ cos θt
cos 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 ∥ polarization
Ex 8-15: Reflection from water
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UEE 3201 Electromagnetics II
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Chiu, ECE Dept., NCTU
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UEE 3201 Electromagnetics II
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UEE 3201 Electromagnetics II
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