# Lecture07 by rite2methun

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```									WAVES IN MEDIA

Significance to Communications
Air and space Ionosphere (plasma) Satellite Cloud, Rain Refraction, moist or dense air Troposcatter Blue sky, red sunset Reflection Optical Fibers θ Polarization-based optoelectronic devices Linear Circular polarization fiber Optoelectronics on chips

L7-1

WAVES IN MEDIA

Constitutive Relations + + + + +
E

Vacuum: + + + + + -

D = εo E

-

ρf = free charge density

∇iD = ρf

P

Dielectric Materials:

D = εE = εo E + P ∇ iεo E = ρf + ρp ∇iP = −ρp polarization charge density P = “Polarization Vector”

ρp e + + e
B B

e +

Magnetic Materials:

∇iB = 0 B = µo H in vacuum B = µH = µo H + M

e

(

)

M = “Magnetization Vector”
L7-2

TYPES OF MEDIA

Properties are a function of:
Field direction Position Time: ≠ f(t) ≠ f(history) Frequency E or H Temperature Pressure Designation:
Anisotropic D = εE , B = µH

Inhomogeneous
Stationary
Amnesic
Dispersive
Non-linear
Temperature dependent
Compressive

L7-3

ANISOTROPIC DIELECTRICS

D = Dx = Dy = Dz = εE
ε xx E x + ε xy E y + ε xz Ez
ε yx E x + ε yy E y + ε yz Ez ε zx E x + ε zy E y + ε zz Ez 0  0 εz  

y Dy = εyEy E

D

ε x  Let ε =  0 0 

0 εy 0

EY

0

x EX Dx = εxEx

x,y,z are “principal axes”
Note: D // E iff E // x, y, or z for ε x ≠ ε y ≠ ε z ˆ ˆ ˆ

Real ε, µ ⇒ Lossless medium
L7-4

HOW TO MAKE ANISOTROPIC MATERIALS
Consider: ε>> εo (capacitors) Q = CV Area A (m2) V + d -Q ε ( A 2) C≅ d εeff ≅ ε 2 ε A C≅ o (d 2 ) εeff ≅ 2εo ε A C = eff d εeff = ε “uniaxial medium” d/2 Q ε εo A εo ε

atom, molecule

εx = εy = εo “ordinary” εz = εe “extraordinary” z y

εo < εe εo > εe

x

L7-5

WAVE BEHAVIOR IN UNIAXIAL MEDIUM

ˆ Assume wave in + z direction, σ = 0 Derive wave equation:
∇ × E = − jωB ∇ × H = jωD ∇iD = ρf = 0 ∇iB = 0 D = εE

εe  ε=0 0 

 ε 0 , µ = µ 0 ε 
2

0

0

Therefore ∇ × ∇ × E = ∇ ∇iE − ∇ E = − jωµ∇ × H = ω µεE
Does ∇iE = 0 here?
2 2

(

)

(

)

2

Yes, (let’s skip proof) can test final solution

Therefore ∇ E + ω µεE = 0 ⇒ 3 equations (x,y,z components)

 ∂2 ∂2 ∂2  2 ˆ ˆ ˆ  2 + 2 + 2  [ xE x + yE y + zE z ] + ω µεE = 0  ∂x ∂y ∂z 
Assume = 0 (UPW in z direction)
This leads to 2 decoupled equations for x and y polarization

L7-6

BIREFRINGENT MEDIA

Decoupled wave equations:
 2  
∂ + ω2µεe  E x = 0 , k e = ω µεe ,  ∂z2 #\$%   & (k e )2 (x-pol equation)  2  ∂ + ω2µε  E = 0 , k o = ω
µε  y  ∂z2 "   & (k o )2 (y-pol equation)

Where E x ∝ e− jk z = e

e

− j ω ve z

(

)

 e e v = 1 µε ⇒ v o = 1 µε 

Thus the x- and y-polarized waves propagate independently at different velocities If ve < vo then ve → “slow-axis velocity”

L7-7

BIREFRINGENT MEDIA

Example:
1 d LHC π/2 ∆φ π RHC 3π/2 0 Linear pol. x z y Demo; Polaroids 1) 2)	 3)	 ⊕ ⊕
S F

ˆ ˆ E1 = Eo ( x + y ) 45° linear pol. input

2

z

− jk d

e

− jk d

o

X

− jφe + ye− jφo ˆ ˆ E2 = Eo xe #''\$''' ' % What pol.?

Say, “slow axis” Z 45° d

∆φ & φe − φo = (k e − k o )d x “Quarter wave plate” z y Output: d ∋ ∆φ = π/2 ⇒ ⇒ 0

Y “fast axis”

⇒ ⇒ ⇒

F

4) 5)

S

MICA

Blue Red 1 µm

6) GEARS

λ
L7-8

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