# <Chap. 8> Plane Electromagnetic waves Section 8.5 ~ 8.8 Electromagnetics

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## Electromagnetics

### (1st of week 4)

Textbook: Field and Wave Electromagnetics, 2E, Addison-Wesley

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st

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### Electromagnetic power

(Proof of identity; HW)

• EM waves carry EM power with them (Energy is transported by EM waves through space to distant receiving points)

• Derivation

### ∂t− E ⋅ J

By assuming a simple medium whose ε, μ, and σ are not functions of t,

2

2

2

2

2

2

Product rule

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e

V

### (J)

where V is the potential at a point where the volume charge density is ρ and V’ is the volume of the region where ρ exists.

e

V

V

### (∇ ⋅ D)V dv

• We in terms of field quantities E and D without explicitly knowing ρ

V

V

S

V

### D⋅ E dv

Refer Section 3-11 for derivation

Vanishes if we choose V’ to be a very large sphere of a radius R → ∞ since

2

2

e

### ∫

V

• For a linear medium where D = εE

e

2

V

V

2

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m

V

### (J)

where V’ is the volume of the loop or the linear medium where J exists

• Wm in terms of field quantities B and H without explicitly knowing J

Refer to Section 6-12 for derivation

m

V

V

V

Thus,

V

V

S

### (A× H)⋅d ′s

Vanishes if we choose V’ to be a very large sphere of a radius R → ∞ since

2

2

m

V

### (J)

• For a linear medium where H = B/μ

m

2

V

V

2

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### Energy transfer rate vs. E, H (cont’d)

• Poynting’s theorem

2

2

2

V

S

V

2

2

V

2

### dv

Energy stored in

electric & magnetic fields

Ohmic power dissipated in the volume V

due to conduction current (σE)

S

### ( E× H ) ⋅ds

Power leaving the volume V enclosed by S

=

Decreasing electric and magnetic power + Dissipating ohmic power

2

### )

Poynting’s vector: "Power density” vector associated with an electromagnetic field (* Perpendicular to electric and magnetic fields)

:

S

### =

Power leaving the enclosed volume Poynting’s theorem

(Not limited to plane waves)

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S

V

e

m

V σ

### dv

John Henry Poynting (1852~1914)

e

2

*

m

2

*

σ

2

2

*

*

### σ

where = Electric energy density

= Magnetic energy density

= Ohmic power density

: Power flowing into a closed surface

= Increasing stored electric + magnetic + ohmic power

### Special cases

• Poynting vector in lossless medium (σ=0)

S

V

e

m

### )dv

• Poynting vector in static case

S

V σ

### dv

Img src: Wikipedia

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### Instantaneous expression for Poynting vector

• Instantaneous expressions for time-harmonic E and H-fields

x

x

x

0

γ z

x

0

### e

(α+ jβ)z

• Instantaneous expression for P or power density vector

jωt

x

0

αz

j(ωtβz)

x

0

αz

y

y

y

x

y

0

α

z

− j

βz+θη

where

### η = η e

θη is the intrinsic impedance of the “lossless” medium

jωt

y

0

αz

− j

ωtβzθη

y

0

αz

η

jωt

jωt

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### Instantaneous expression for Poynting vector (Cont’d)

• Instantaneous expression for P or power density vector

jωt

jωt

x

0

αz

y

0

α

z

η

z

02

−2α

z

η

z

02

−2α

z

η

η

av

z

0T

z

0T

0

2

−2αz

η

η

z

02

−2αz

η

av

z

02

−2αz

η

2

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### General formula for time-average Poynting vector

• Vector identity

*

*

### ) ,

With given complex vectors A and B such that

*

*

*

*

*

*

*

jωt

jωt

### ⎤⎦.

• Time-average Poynting vector

jωt

jωt

By replacing

jωt

jωt

jωt

jωt

jωt

*

− jωt

j 2ωt

*

av

0T

0T

j 2ωt

*

*

av

*

2

### )

Average power density vector of electromagnetic wave

propagating “in an arbitrary direction”

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### In practical cases

• Waves propagate in “bounded regions” where several media with different constitutive parameters (ε, μ, and σ) are present

### Normal incidence on a perfect conductor

x

y z

Er

Hr

anr

Ei

Hi ani

Reflected wave

Incident wave

Perfect conductor

Medium 1 1 = 0)

z = 0

Medium 2 2 = ∞)

• EM wave traveling in a lossless medium impinges on another medium with a different η (intrinsic impedance)

→ Reflection occurs

• Incident wave

i

x

i 0

− jβ1z

i

y

i 0

1

− jβ1

z

i

i

i

### ( ) z

* Poynting vector direction: az

• Reflected wave

r

x

r 0

+ jβ1z

r

y

r 0

1

+ jβ1

z

### ⎩⎪

• Wave in medium 1

1

i

r

1

i

r

### ⎩⎪

• Wave in medium 2

2

2

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### Normal incidence on a perfect conductor

Boundary condition: “Continuity of tangential component of E-field interface”

1

x

i 0

− jβ1z

r 0

+ jβ1z

1

x

i 0

r 0

2

r 0

### = −E

i 0

At z = 0,

• Associated magnetic field

r

1

nr

r

1

z

r

y

1

r 0

+ jβ1

z

y

i 0

1

+ jβ1

z

1

i

r

y

i 0

1

1

1

y

i 0

1

1

1

x

i 0

− jβ1z

+ jβ1z

x

i 0

1

1

x

i 0

1

av

1

1

*

z

i 02

1

1

### ⎠⎟ = 0

• Electromagnetic power

av

### ( ) z= 0

No average power associated with total EM wave in medium 1

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1

1

jωt

x

i 0

1

jωt

x

i 0

1

x

i 0

1

1

1

jωt

y

i 0

1

1

jωt

y

i 0

1

1

1

when

1

1

1

when

1

1

### 2

x

y

z

Occurs at “Fixed Positions”

### Standing wave

• Superposition of two wave traveling in opposite directions

• Standing wave is NOT a traveling wave

E1 and H1 are in time quadrature (90º phase difference)

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## Electromagnetics

### (2nd of week 4)

Textbook: Field and Wave Electromagnetics, 2E, Addison-Wesley

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nd

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### Chap. 8 | Polarization

Plane of incidence

- Containing the propagation vector of the incident wave - Normal to the surface

Transverse Electric (TE) wave - E-field ⊥ Plane of incidence - H-field // Plane of incidence

- a.k.a. s-polarized wave, “s”: senkrecht (German; vertical)

Transverse Magnetic (TM) wave - E-field // Plane of incidence

- H-field ⊥ Plane of incidence

- a.k.a. p-polarized wave, “p”: parallel

### Plane of incidence and TE & TM waves

TEM wave propagation (In transmission lines)

### Chapter. 9

TE & TM wave propagation (In waveguides)

### TM waveTE wave

(Image source: Clker)

(Image source: Specac)

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### Chap. 8 | Polarization

Img src: Lucid Vision Lab

“Polarization” direction of EM wave = Orientation of E-field

- e.g.) TEM wave propagating in z-direction with E- and H-fields oscillating in x- and y-directions = Polarized in “x-direction”

• EM waves (including light) characterized by their polarization

- Most light sources (e.g. sunlight, LED, halogen lamp and so on) = Unpolarized - Reflected light from surfaces = Partially polarized

- Laser = Fully polarized

Light-matter interaction (e.g. transmission, reflection, absorption and many) dependent on polarization

### Usage of polarization detection

Non-uniformity inspection Improving contrast Reduced reflection

Scratch inspection Object detection

“Oblique Incidence at a Plane Dielectric Boundary”

(Chap. 8-10, next class)

θB

θB: Brewster’s angle

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### Chap. 8 | Polarization (example)

https://blog.pugsgear.com/What-Exactly-Do-Aﬀordable-Polarized-Sunglasses-Do LG 디스플레이 블로그

Polarized sunglasses reduced glare

### Most representative example: Polarizer

Optical filter that allows EM wave with “only a specific polarization” to pass through

Blocking EM wave of other polarizations (abs or ref)

(Image source: Lucid Vision Lab)

Linear polarizer

Metal Wire Grid

3D Display

Binocular disparity → Depth perception

Thin & parallel metallic wires

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### x

Diﬀuser &

Light guide plane

Polarizer (90º)

Liquid Crystal TFT

Polarizer (180º)

Color Filter

### Operating Principle

“Unpolarized” white light generated by LED backlight unit (BLU)

90º Polarizer allows only “90º linearly polarized” light

TFT applies E-field to liquid crystals such that they rotate in desirable orientation

Rotated liquid crystals change the angle of polarization of light (θp)

Image source (FlatPanelshd.com)

If θp = 90º → Completely blocked (min brightness)

If θp = 180º → Completely transmitted (max brightness) If 90º < θp < 180º → Partially transmitted (mid brightness)

Liquid Crystals

n0

ne

Ordinary refractive index

Extraordinary refractive

index

### Birefringent material

(i.e. anisotropic material)

### Birefringence

: Optical property of the material having a refractive index depending on polarization /

propagation direction of the light

p

e

o

ep

### < u

op

where up is propagation of light in the medium n is the refractive index of the medium

c is the speed of light

(Sec. 8-10 in next class!)

### Chap. 8 | Polarization (example)

Img src: flatpanelshd.com, Wikipedia

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### Organic Light Emitting Diodes (OLED) display

• AR coating = Linear polarizer + quarter-wave plate (a.k.a phase retarder by λ/4)

Circularly polarized wave = Sum of TWO linearly polarized waves in both space and time quadrature

Phase change (180º) upon reflection by metal

“Anti-reflecting (AR) coating”

(for better outdoor visibility)

### Composition of AR coating

No AR coating With AR coating

Img src: flatpanelshd.com, LGD blog, eeNews Europe, Wikipedia

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### Perpendicular polarization = Transverse Electric (TE) wave

x

y z

Er

Hr

anr

Ei

ani

Perfect conductor

Medium 1 1 = 0)

z = 0

Medium 2 2 = ∞) Hi

θr

θi

Plane of incidence

• Ei-field ⊥ Plane of incidence (xz plane)

• Propagation directions of incident and reflective E-field

ni

x

i

z

i

nr

x

r

z

### cos θ

r

• Incident and reflective E-field

i

y

i 0

− jβ1ani⋅R

y

i 0

− jβ1

x sinθi+zcosθi

r

y

r 0

− jβ1anr⋅R

y

r 0

− jβ1

x sinθr−zcosθr

### )

• Boundary condition (@ z = 0): total E-field in medium1 = total E-field in medium 2

1

i

r

2

y

i 0

− jβ1x sinθi

r 0

− jβ1x sinθr

r 0

i 0

r

### = θ

i

• To satisfy above condition,

Snell’s law of reflection Phase is shifted by 180º

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### Total electric and magnetic fields

x

y z

Er

Hr

anr

Ei

ani

Perfect conductor

Medium 1 1 = 0)

z = 0

Medium 2 2 = ∞) Hi

θr

θi

Plane of incidence

• Total Electric Field E1

1

i

r

y

i 0

− jβ1

x sinθi+zcosθi

r 0

− jβ1

x sinθr−zcosθr

y

i 0

− jβ1z cosθi

jβ1z cosθi

− jβ1x sinθi

y

i 0

1

i

− jβ1x sinθi

y

1y

1

1

nr

1

1

x

i

z

i

y

1y

x

y

i

1y

1

z

y

i

1y

1

i 0

1

x

i

1

i

z

i

1

i

− jβ1x sinθi

1

x

1x

z

1z

### ( ) x, z⎤⎦

• Total Magnetic Field H1

r 0

i 0

r

i

nr

x

i

z

### cos θ

i

“Perpendicular” to Plane of Incidence

“Transverse Electric (TE)” wave

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### Induced surface current on the conducting wall (Example 8-10)

x

y z

Er

Hr

anr

Ei

ani

Perfect conductor

Medium 1 1 = 0)

z = 0

Medium 2 2 = ∞) Hi

θr

θi

Plane of incidence

• Boundary condition for H-field at the interface (z=0)

n2

1

2

s

s

z

1

2

z

1

1

i 0

1

x

i

1

i

z

i

1

i

### ⎡⎣ ⎤⎦e

− jβ1x sinθi

• Since H1 is given by

1

x

i 0

1

i

− jβ1x sinθi

s

z

x

i 0

1

i

− jβ1x sinθi

y

i 0

1

i

### e

− jβ1x sinθi

• Thus,

• Instantaneous expression

s

s

jωt

y

i 0

1

i

i

### ⎠⎟

• This induced Js results in the reflected wave in medium 1

• Reflected wave cancels the incident wave in the wall

• Microscopically, incident wave absorbed by free electrons

• Oscillating electrons (Js) re-radiate EM wave

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### Parallel polarization = Transverse Magnetic (TM) wave

x

y z

Er

Hr

anr

Ei

ani

Perfect conductor

Medium 1 1 = 0)

z = 0

Medium 2 2 = ∞) Hi

θr

θi

Plane of incidence

Ei-field // Plane of incidence (xz plane)

• Total Electric Field E1

1

i

r

x

1x

z

1z

1

1

nr

1

y

1y

### ( ) x, z

• Total Magnetic Field H1

“Perpendicular” to

Plane of Incidence “Transverse Magnetic (™)” wave (Refer to Sec. 8-7.2 for derivation)

(Refer to Sec. 8-7.2 for derivation)

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### Two different dielectric media

x

y z

Er

Hr

anr

Ei

ani

Medium 1 1, μ1)

z = 0

Medium 2 2, μ2) Hi

• When EM wave is incident on the interface between two media with different intrinsic impedance, part of incident power is reflected and part is transmitted

i

x

i 0

− jβ1z

i

z

1

i

y

i 0

1

− jβ1z

Et

ant

Ht

• Incident wave

r

x

r 0

+ jβ1z

r

z

1

r

y

r 0

1

+ jβ1

z

• Reflected wave

t

x

t 0

− jβ2z

t

z

2

t

y

t 0

2

− jβ1

z

### ⎩⎪

• Transmitted wave

• Boundary conditions at z = 0

→ Tangential E and H should be continuous

i

r

t

i 0

r 0

t 0

i

r

t

2

i 0

r 0

### ) =Eη

t 0

2

Both are lossless (σ1, σ2 = 0)

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