chapter 5 electromagnetic opticsfaculty.washington.edu/lylin/ee485w04/ch5.pdf · 1 chapter 5...
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EE 485, Winter 2004, Lih Y. Lin
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Chapter 5 Electromagnetic Optics
- Introduce “vector” nature of light Electrical field: ),( trEEEE Magnetic field: ),( trHHHH
5.1 Electromagnetic Theory of Light
),( trEEEE and ),( trHHHH must satisfy Maxwell’s equations. In free space
t∂
∂ε=×∇ EEEEHHHH 0 (5.1-1)
t∂
∂µ−=×∇ HHHHEEEE 0 (5.1-2)
0=⋅∇ EEEE (5.1-3) 0=⋅∇ HHHH (5.1-4)
units MKSin )MeterFarad( 10
361ty permittivi Electric : 9
0−×
π=ε
units MKSin )MeterHenry( 104ty permeabili Magnetic : 7
0−×π=µ
The wave equation All components of ),( trEEEE and ),( trHHHH (Ex , Ey , Ez , Hx , Hy , Hz ) satisfy the wave equation:
012
2
20
2 =∂∂−∇
tu
cu (5.1-5)
00
01µε
=c
In free space, Maxwell equations and wave equations are linear, therefore principle of superposition applies. Maxwell’s equations in a medium Assume no free electric charges or currents. Need two more vector fields: ),( trDDDD : Electric flux density, or electric displacement ),( trBBBB : Magnetic flux density
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t∂
∂=×∇ DDDDHHHH (5.1-7)
t∂
∂=×∇ BBBBEEEE (5.1-8)
0=⋅∇ DDDD (5.1-9) 0=⋅∇ BBBB (5.1-10)
PPPPEEEEDDDD +ε= 0 (5.1-11) MMMMHHHHBBBB 00 µ+µ= (5.1-12)
PPPP : Polarization density (macroscopic sum of the electric dipole moments that the electric field induces)
MMMM : Magnetization density (macroscopic sum of the magnetic dipole moments that the magnetic field induces)
In a non-magnetic medium (assumption of this course), MMMM = 0 → HHHHBBBB 0µ= (5.1-13)
Boundary conditions Assume no free charges or surface currents. In a homogeneous medium, E E E E , HHHH , DDDD , BBBB are continuous. At the boundary between two dielectric media, the tangential components of EEEE and HHHH, and the normal components of DDDD and BBBB are continuous. Intensity and power
HHHHEEEESSSS ×= : Poynting vector (direction and magnitude of power flux)
S=I : Power flow across a unit area normal to SSSS 5.2 Dielectric Media - Linear: ),( trPPPP = constant ),( trEEEE× - Non-dispersive: Response is instantaneous. )( 0tPPPP does not depend on
)( 0tt <EEEE (An idealization. Any physical system has a finite response time). - Homogeneous: )(EEEEPPPP does not depend on r . - Isotropic: )(EEEEPPPP is independent of the direction of EEEE . A. Linear, Nondispersive, Homogeneous, and Isotropic Media
EEEEPPPP χε= 0 at any position and time (5.2-1) χ : Electric susceptibility (scalar constant) EEEEEEEEDDDD ε≡χ+ε= )1(0 (5.2-2,3)
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ε: Electric permittivity of the medium
χ+=εε≡ε 10
r : Dielectric constant
t∂∂ε=×∇ EEEE
HHHH (5.2-4)
t∂
∂µ−=×∇ HHHHEEEE 0 (5.2-5)
0=⋅∇ EEEE (5.2-6) 0=⋅∇ HHHH (5.2-7)
Wave equation:
012
2
22 =
∂∂−∇
tu
cu (5.2-8)
ncc 0
0
1 =εµ
= (5.2-9)
χ+=ε= 1rn : Refractive index (5.2-10) B. Nonlinear, Dispersive, Inhomogeneous, or Anisotropic Media Inhomogeneous media )( ,)( ,)(0 rnnrr =ε=χε= EEEEDDDDEEEEPPPP If the medium is locally homogeneous, that is, )(rε varies sufficiently slowly,
0)(
12
2
22 =
∂∂−∇
trcEEEE
EEEE
(5.2-12)
Anisotropic media PPPP and EEEE are not necessarily parallel. If the medium is linear, non-dispersive, and homogeneous, ∑ χε=
jjiji EP 0 i, j = 1, 2, 3 denote the x, y, and z components
ijχ : Susceptibility tensor (3 × 3 matrix) ∑ ε=
jjiji ED
ijε : Electric permittivity tensor
Dispersive media
')'()'()( 0 dttttxtt
EEEEPPPP ∫∞−
−ε= (5.2-17)
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)(νχ is the Fourier transform of )(tx → )( ,)( ,)( ω=ωε=εωχ=χ nn Nonlinear media If the medium is homogeneous, isotropic, and nondispersive, )(EEEEPPPP Ψ= for all position and time, Ψ is a nonlinear function. Wave equations:
2
2
02
2
20
2 )(1ttc ∂Ψ∂µ=
∂∂−∇ EEEEEEEE
EEEE (5.2-20)
→ Principle of superposition no longer applicable 5.3 Monochromatic Electromagnetic Waves
Define complex amplitude:
)exp()(Re),(
)exp()(Re),(tjt
tjtω=
ω=rHr
rErHHHH
EEEE (5.3-1)
Likewise, for BDP , , .
Maxwell’s equations DH ω=×∇ j (5.3-2)
BE ω−=×∇ j (5.3-3) 0=⋅∇ D (5.3-4)
0=⋅∇ B (5.3-5) PED +ε= 0 (5.3-6)
HB 0µ= (5.3-7)
Optical intensity and power
*21 HES ×≡ : Complex Poynting vector (5.3-9)
Optical intensity SRe=S (5.3-8) Linear, nondispersive, homogeneous, and isotropic media
EH ωε=×∇ j (5.3-11) HE 0ωµ−=×∇ j (5.3-12) 0=⋅∇ E (5.3-13)
0=⋅∇ H (5.3-14) Components of E and H must satisfy the Helmholtz equation:
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022 =+∇ UkU (5.3-15) 0nkk = Inhomogeneous media Eqs. (5.3-11) ~ (5.3-14) remain applicable, )(rε→ε . For locally homogeneous, )(rε varies slowly with respect to wavelength
→ 0
0)()( ,)(
εε== rrr nknk
Dispersive media EP )(0 νχε= (5.3-16) ED )(νε= (5.3-18) [ ])(1)( 0 νχ+ε=νε (5.3-19) → The only difference between non-dispersive medium and dispersive medium is that ε and χ are frequency-dependent. The Helmholtz equation applicable with
0
0)()( ,)(
ενε=νν= nknk (5.3-20)
5.4 Elementary Electromagnetic Waves Consider monochromatic waves. Assume the medium is linear, homogeneous, and isotropic. The transverse electromagnetic (TEM) plane wave )exp()( 0 rkErE ⋅−= j (5.4-1) )exp()( 0 rkHrH ⋅−= j (5.4-2) 0kk n= Substituting into Maxwell’s equations, we obtain:
000
00
HEkEHk
ωµ=×ωε−=×
→ E, H, and k are mutually orthogonal.
nn
0
0
0
0
0 1 η=η≡εµ=
HE
: Impedance of the medium (5.4-5,6)
Ω=π=εµ=η 377 120
0
00 : Impedance of free space (5.4-7)
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Magnitude of Poynting vector IE
=η
=2
20S : Intensity (5.4-8)
5.5 Absorption and Dispersion A. Absorption
Light-absorbing dielectric materials have complex susceptibility, "' χ+χ=χ j (5.5-1) and comlex permittivity, )1(0 χ+ε=ε The Helmholtz equation is still valid, but 0"'1 kjk χ+χ+= (5.5-2) is now complex-valued.
α−β≡21jk (5.5-3)
)exp()21exp()exp( zjzjkz β−α−=−
→ Intensity attenuated by exp(-αz) after propagating a distance z. α: Absorption coefficient (attenuation coefficient, or extinction
coefficient) β: Propagation constant = 0nk (5.5-4)
The wave travels with a phase velocity ncc 0= .
"'12 0
χ+χ+=α−k
jn (5.5-5)
→ Relating the refractive index and the absorption coefficient to the real and imaginary parts of the susceptibility.
Weekly absorbing media 1" ,1' <<χ<<χ
→ '211 χ+≈n (5.5-6)
"0χ−≈α k (5.5-7) B. Dispersion
)(/)( ,)( ,)( 0 ν=ν=ν=νχ=χ ncccnn
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Pulse broadening (example: chromatic dispersion in optical fibers):
Measures of dispersion Examples:
1) For glass optical components used with white light,
CF
D
nnn−−≡ 1number V
nF, nD, nC: Refractive indices at blue (486.1 nm), yellow (589.2 nm), and red (656.3 nm)
2) 0λ=λ
λddn
Prism: λ
θ=λθ
ddn
dnd
dd dd
C. The Resonant Medium
EPPP
0020
202
2
χεω=ω+σ+dt
ddt
d (5.5-12)
↔ Classical harmonic oscillator for bound charges in the medium
m
xdtdx
dtxd F=ω+σ+ 2
02
2
(5.5-13)
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m: Mass of the bound charge
mκ=ω0 : Resonant angular frequency
κ: Elastic constant σ: Damping coefficient EF e= : Force
Polarization density Nex=P , N: Number of charges per volume
→ 200
2
0 ωε=χ
mNe
Substituting )exp(Re)( tjEt ω=E and )exp(Re)( tjPt ω=P into Eq. (5.5-12), we obtain:
EEj
P )(0220
200
0 νχε=
σω+ω−ωωχε= (5.5-14)
ν∆ν+ν−ν
νχ=νχj22
0
20
0)( (5.5-15)
πσ=ν∆2
( )( ) ( )2222
0
220
20
0)('ν∆ν+ν−ν
ν−ννχ=νχ (5.5-16)
( ) ( )22220
20
0)("ν∆ν+ν−ν
ν∆ννχ−=νχ (5.5-17)
0)(" litysusceptibifrequency -Low :)(' 00
≈νχχ≈νχ→ν<<ν
space free like acts medium the,0)(" ,0)('0 ≈νχ≈νχ→ν>>ν
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00
00 )("- ,0)(' χ
ν∆ν=νχ=νχ→ν=ν
Near resonance:
2/)(
2/)(0
00 ν∆+ν−ν
νχ=νχj
(5.5-18)
( ) ( )220
00 2/
14
)("ν∆+ν−ν
ν∆νχ−=νχ (5.5-19)
)("2)(' 0 νχν∆ν−ν=νχ (5.5-20)
ν∆ : FWHM of )(" νχ
5.6 Pulse Propagation in Dispersive Media Assume the medium is linear, homogeneous, and isotropic,
0/)(2)( ),( ),( cnnn νπν=νβν=να=α Pulsed plane wave in z-direction: [ ])(exp),(),( 00 ztjtztzU β−ω= A (5.6-1) )( 00 νβ=β
),( tzA : Complex envelope of the pulse, slow varying in comparison with 0ν
Knowing ),0( tA → Need to determine ),( tzA
Linear-system description Suppose )2exp(),0(),0( ftjfAt π=A )(),0(),( ffAfzA H= (5.6-2)
( )
β−β−α−= νν+ν+ zjzf ff 00021exp)(H (5.6-3)
Determine ),( tzA from ),0( tA : Fourier transform
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dtftjtfA )2exp(),0(),0( π−= ∫∞
∞−
A
)(),0(),( ffAfzA H= Inverse Fourier transform
dfftjfzAtz )2exp(),(),( π= ∫∞
∞−
A
Or by convolution:
')'()',0(),( dttthttz −= ∫∞
∞−
AA (5.6-5)
dfftjfth )2exp()()( π= ∫∞
∞−
H
The slowly varying envelope approximation
),( tzA slowly varying in comparison with central frequency 0ν → ),( fzA a narrow function of f with 0ν<<ν∆ Assume within ∆ν centered about ν0, α=να constant ~)(
πνν=νβ0
2)()(c
n varies slightly and gradually with ν.
)exp()2exp()( 2
0 zfDjfjf d νπ−τπ−=HH (5.6-7) )2/exp(0 zα−≡H
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gd z υ=τ / , gυ : Group velocity
ωβ=
νβ
π=
υ dd
dd
g 211 (5.6-8)
υν=
ωβπ=
νβ
π=ν
gdd
dd
ddD 12
21
2
2
2
2
: Dispersion coeff. (5.6-9)
Dispersion coefficient )()( ντ=τ→νυ=υ ddgg
δν=δν
υν=δν
ντ=δτ ν zDz
dd
dd
g
d
Normal dispersion: 0>νD . Anomalous dispersion: 0<νD . If the pulse has a spectral width νσ (Hz), the spread of the temporal width zD νντ σ=σ (5.6-10)
:νD second/m·Hz → Measure of the pulse time broadening per unit spectral width per unit distance
Determine the shape of the transmitted pulse:
')'()',0(),( dttthttz −= ∫∞
∞−
AA
τ−π=νν zD
tjzDj
th d2
0)(exp1)( H (5.6-11)
Wavelength dependence of group velocity and dispersion coefficient
0
)()( 00
0
λ=λλλλ−λ=
=υ
ddnnN
Nc
g
(5.6-19)
0
2
2
20
30 )(
λ=λν λ
λλ=dnd
cD (
Hzmsec⋅
) (5.6-20)
In terms of wavelength: ν=λ νλ dDdD
0
2
2
0
0 )(
λ=λλ λ
λλ−=dnd
cD (
nmmsec⋅
) (5.6-21)
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