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5 nm
S. Lin et al, Nature, vol. 394, p. 251-3, (1998)
J.R. Krenn et al., Europhys.Lett. 60, 663-669 (2002)
T.Thio et al., Optics Letters 26, 1972-1974 (2001).
K.S. Min et al. PhD Thesis
J. D. Joannopoulos, et al, Nature, vol.386, p.143-9 (1997)
K.V. Vahala et al, Phys. Rev. Lett, 85, p.74 (2000)
MotivationMajor breakthroughs are often materials related
• Stone Age, Iron Age, Si Age,….
Is it possible to engineer new materials with useful optical properties• Yes !
• Wonderful things happen when structural dimensions are ≈ λlight
What are the smallest possible devices with optical functionality ?
• People realized the utility of naturally occurring materials
• Scientists are now able to engineer new functional nanostructured materials
This course talk about what these “things” are…and why they happen
• Scientists have gone from big lenses, to optical fibers, to photonic crystals, to…
• Does the diffraction set a fundamental limit ?
• Possible solution: metal optics/plasmonics
Designing New Functional DevicesWe need to be able to solve the following problem:
ε3
εh
ε1
ε4 ε5
ε6
?
Light Interaction with MatterMaxwell’s Equations
fρ∇ ⋅ =D
0∇⋅ =B
∂∇× = −
∂BEt
∂∇× = +
∂DH Jt
Divergence equations Curl equations
D = Electric flux density
E = Electric field vector
ρ = charge density
B = Magnetic flux density
J = current density
H = Magnetic field vector
Constitutive Relations
( )0ε ε= + =D E P E EElectric polarization vector…… Material dependent!!
Total electric flux density = Flux from external E-field + flux due to material polarization
ε = Material dependent dielectric constant
ε0 = Dielectric constant of vacuum = 8.85 ⋅ 10-12 C2N-1m-2 [F/m]
( )0 0μ μ= +B H M H
Magnetic flux density Magnetic field vector Magnetic polarization vector
μ0 = permeability of free space = 4πx10-7 H/m
Note: For now, we will focus on materials for which 0=M 0μ=B H
Constitutive relations relate flux density to polarization of a medium
Electric When P is proportional to E
Magnetic
E
-+
+++
---
Divergence Equations
Coulomb
How did people come up with: ?
• Charges of same sign repel each other (+ and + or – and -)
• Charges of opposite sign attract each other (+ and -)
+
• He explained this using the concept of an electric field : F = qE
Every charge has some field lines associated with it
-
• He found: Larger charges give rise to stronger forces between charges• Coulomb explained this with a stronger field (more field lines)
ρ∇⋅ =D
Divergence Equations
E
dS
Gauss’s Law (Gauss 1777-1855)
A A V
d E d dvε ρ⋅ = ⋅ =∫ ∫ ∫D S S
E-field related to enclosed charge
Gauss’s Theorem (very general)
A V
d dv⋅ = ∇ ⋅∫ ∫F S F
Combining the 2 Gauss’s
A V V
d dv dvρ⋅ = ∇ ⋅ =∫ ∫ ∫D S D ρ∇⋅ =D
The other divergence eq. 0∇⋅ =B is derived in a similar way from 0A
d⋅ =∫B S
++ +
+
+
H
Curl Equations
How did people come up with:∂
∇× = +∂DH Jt
?
J
H
D
D increasing whencharging the capacitor
JC
A
Ampere (1775-1836)
C A
d dt
∂⎛ ⎞⋅ = + ⋅⎜ ⎟∂⎝ ⎠∫ ∫iDH l J S Magnetic field induced by: Changes in el. flux
Electrical currents
Ampere:C A
d dt
∂⎛ ⎞⋅ = + ⋅⎜ ⎟∂⎝ ⎠∫ ∫iDH l J S
Stokestheorem:
Curl Equations
( )C A
d d⋅ = ∇× ⋅∫ ∫i F l F S
( )C A A
d d dt
∂⎛ ⎞⋅ = ∇× ⋅ = + ⋅⎜ ⎟∂⎝ ⎠∫ ∫ ∫iDH l H S J S
∂∇× = +
∂DH Jt
∂∇× = −
∂BEt
Other curl eq.
Derived in a similar way fromC A
d dt
∂⋅ = − ⋅
∂∫ ∫iBE l S
B
E
CA( )
C A A
d d dt
∂⋅ = ∇× ⋅ = − ⋅
∂∫ ∫ ∫iBE l E S S
Stokes
Summary Maxwell’s Equations
ρ∇ ⋅ =D
0∇⋅ =B
∂∇× = −
∂BEt
∂∇× = +
∂DH Jt
Divergence equations Curl equations
Flux lines start and end on charges or poles
Changes in fluxes give rise to fields
Currents give rise to H-fields
Note: No constants such as μ0 ε0, μ ε, c, χ,……. appear when Eqs are written this way.
The Wave Equation
Curl equations: Changing E-field results in changing H-field results in changing E- field….
Plausibility argument for existence of EM waves
The real thing
EH
E HE
…….
Goal: Derive a wave equation:
( ) ( ) ( ){ }0, Re expt i tω=U r U r
for E and H
Solution: Waves propagating witha (phase) velocity v
( ) ( )22
2 2
,1,r t
r tv
∂∇ =
∂U
Ut
Position Time
Starting point: The curl equations
0 tμ∂ ∂
∇× = − = −∂ ∂B HEt
∂∇× = +
∂DH Jt
The Wave Equation for the E-field
( ) ( )22
2 2
,1,r t
r tv
∂∇ =
∂E
Et
Goal:
Curl Eqs: (Materials with M = 0 only) a)
b)
Step 1: Try and obtain partial differential equation that just depends on E
( )0 0t t
μ μ∂ ∇×∂⎛ ⎞∇×∇× = ∇× − = −⎜ ⎟∂ ∂⎝ ⎠
HHE
Step 2: Substitute b) into a)2 2 2
0 0 0 0 0 02 2 2t t t t tμ μ μ ε μ μ∂ ∂ ∂ ∂ ∂
∇×∇× = − − = − − −∂ ∂ ∂ ∂ ∂
D J E P JE
Apply curl on both side of a)
0ε= +D E P Cool!....looks like a wave equation already
2 2
0 0 0 02 2t t tμ ε μ μ∂ ∂ ∂
∇×∇× = − − −∂ ∂ ∂
E P JE
The Wave Equation for the E-field2
22 2
1v
∂∇ =
∂EEt
Compare:
With:
!( ) 2∇×∇× = ∇ ∇⋅ −∇E E EUse vector identity:
Verify that ∇⋅E = 0 when 1) ρf = 0
2) ε(r) does not vary significantly within a λ distance
2 22
0 0 0 02 2t t tμ ε μ μ∂ ∂ ∂
∇ = + +∂ ∂ ∂
E P JEResult:
1) Find P(E)
2) Find J(E)…something like Ohm’s law: J(E) = σE… we will look at this later..for now assume: J(E) = 0
In order to solve this we need:
2 22
0 0 0 02 2t t tμ ε μ μ∂ ∂ ∂
∇ = + +∂ ∂ ∂
E P JE
P linearly proportional to E: 0
Dielectric Media
Linear, Homogeneous, and Isotropic Media
ε χ=P E
χ is a scalar constant called the “electric susceptibility”
( )2 2 2
20 0 0 0 0 02 2 21
t t tμ ε μ ε χ μ ε χ∂ ∂ ∂
∇ = + = +∂ ∂ ∂
E E EE
Define relative dielectric constant as: 1rε χ= +
22
0 0 2r tμ ε ε ∂
∇ =∂
EE
(2) 2 (3) 30 0 0 ......ε χ ε χ ε χ= + + +P E E E
Note 1 : In anisotropic media P and E are not necessarily parallel: 0i ij jj
P Eε χ=∑Note2 : In non-linear media:
All the materials properties
Results from P
Properties of EM Waves in Bulk MaterialsWe have derived a wave equation for EM waves!
22
0 0 2r tμ ε ε ∂
∇ =∂
EE
Now what ?
Let’s look at some of their properties
Euh….
22
2 2
1v
∂∇ =
∂EEt
Speed of the EM wave:
Compare
22 0
0 0
1 1
r r
cvμ ε ε ε
= =
Speed of an EM Wave in Matter
22
0 0 2r tμ ε ε ∂
∇ =∂
EE and
Where c02 = 1/(ε0 μ0) = 1/((8.85x10-12 C2/m3kg) (4π x 10-7 m kg/C2)) = ( 3.0 x 108 m/s)2
Refractive index is defined by:
Optical refractive index
1rcnv
ε χ= = = +
Note: Including polarization results in same wave equation with a different εr c becomes v
2.0
3.0
1.0
3.4
0.1 1.0 10 λ (μm)
Ref
ract
ive
inde
x: n
Refractive Index Various Materials
( ) ( ) ( ){ }, Re , expz t z ikr i tω ω= − +E E
( ) ( )222
2 2
,,
tntc
∂∇ =
∂E r
E rt
Derived from wave equation
Dispersion RelationDispersion relation: ω = ω(k)
Substitute:
22 2
2
c kn
ω =
Result:
1gr
d c c cvdk k nω ω
ε χ≡ = = = =
+
ω
k
gv
Phase velocity:
Group velocity:
phvkω
≡
22 2
2
nkcω=
Check this!
Electromagnetic Waves
( ) ( )222
2 2
,,
tntc
∂∇ =
∂E r
E rt
Solution to:
Monochromatic waves: ( ) ( ) ( ){ }, Re , expt i i tω ω= − ⋅ +E r E k k rCheck theseare solutions!
TEM wave
Symmetry Maxwell’s Equations result in E ⊥ H ⊥ propagation direction
Optical intensity
( ) ( ) ( ){ }, Re , expt i i tω ω= − ⋅ +H r H k k r
Time average of Poynting vector: ( ) ( ) ( ), , ,t t t= ×S r E r H r
Light Propagation Dispersive MediaRelation between P and E is dynamic
( ) ( ) ( )0, ' ' , 't dt x t t tε+∞
−∞
= −∫P r E r
P results from response to E over some characteristic time τ :
The relation : assumes an instantaneous response( ) ( )0, ,t tε χ=P r E r
In real life:
Function x(t) is a scalar function lasting a characteristic time τ :
t’
E(t’)
x(t-t’)
t’ = tt’ = t - τ
x(t-t’) = 0 for t’ > t (causality)
EM waves in Dispersive Media
( ) ( ) ( ){ }, Re , expt i i tω ω= − ⋅ +E r E k k r
( ) ( ) ( )0, ' ' , 't dt x t t tε+∞
−∞
= −∫P r E r
Relation between P and E is dynamic
EM wave:
( ) ( ) ( ){ }, Re , expt i i tω ω= − ⋅ +P r P k k r
This follows by equation of the coefficients of exp(iωt) ..check this!
Relation between complex amplitudes
( ) ( ) ( )0, ,ω ε χ ω ω=P k E k
( ) ( )0 1ε ω ε χ ω⎡ ⎤= +⎣ ⎦It also follows that:
(Slow response of matter ω-dependent behavior)
Absorption and Dispersion of EM Waves
EM wave: ( ) ( ) ( ){ }, Re , expz t z ikz i tω ω= − +E E
22 2
2
c kn
ω =Dispersion relation k ncω
= ±
' ''n n in= +Absorbing materials can be described by a complex n:
( )' '' ' ''2
k n in n i n ic c cω ω ω αβ⎛ ⎞ ⎛ ⎞= ± + = ± + ≡ ± −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠It follows that:
Investigate + sign: ( ) ( ), Re , exp2
z t z i z z i tαω β ω⎧ ⎫⎛ ⎞= − − +⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭
E E
Traveling wave Decay
0' 'n k ncωβ = =Note: n’ act as a regular refractive index
02 '' 2 ''n k ncωα = − = − α is the absorption coefficient
Transparent materials can be described by a purely real refractive index n
Absorption and Dispersion of EM Waves
Complex n results from a complex χ: ' ''iχ χ χ= +
n is derived quantity from χ (next lecture we determine χ for different materials)
' '' 1 1 ' ''n n in iχ χ χ= + = + = + +
1n χ= +
0
' 1 ' ''2
n n i ikα χ χ= − = + +
02 ''k nα = −
Weakly absorbing media
When χ’<<1 and χ’’ << 1: ( )11 ' '' 1 ' ''2
i iχ χ χ χ+ + ≈ + +
Refractive index:1' 1 '2
n χ= +
Absorption coefficient: 0 02 '' ''k n kα χ= − = −
SummaryMaxwell’s Equations
fρ∇ ⋅ =D 0∇⋅ =B∂
∇× = −∂BEt
∂∇× = +
∂DH Jt
Curl Equations lead to2 2
20 0 02 2t t
μ ε μ∂ ∂∇ = +
∂ ∂E PE
0ε χ=P E
Wave Equation with v = c/n
Linear, Homogeneous, and Isotropic Media
(under certain conditions)
( ) ( )222
2 2
,,
tntc
∂∇ =
∂E r
E rt
In real life: Relation between P and E is dynamic
( ) ( ) ( )0, ' ' , 't dt x t t tε+∞
−∞
= −∫P r E r ( ) ( ) ( )0, ,ω ε χ ω ω=P k E k
This will have major consequences !!!
Next 2 LecturesReal and imaginary part of χ are linked
● Kramers-Kronig
Derivation of χ for a range of materials
● Insulators (Lattice absorption, Urbach tail, color centers…)
● Semiconductors (Energy bands, excitons …)
● Metals (Plasmons, plasmon-polaritons, …)
● Origin frequency dependence of χ in real materials
Useful Equations and Valuable Relationsρ∇⋅ =D
( )0 0 0 0 1 o rε ε ε χ ε χ ε ε= + = + = + =D E P E E E
( )0 0 0 0 0 01m m rμ μ μ μ χ μ χ μ μ= + = + = + =B H M H H H H
0∇⋅ =B
Divergence EquationsMaxwell’s Equations Curl Equations∂
∇× = −∂BEt
∂∇× = +
∂DH Jt
Handy Math Rules
( ) 2∇×∇× = ∇ ∇⋅ −∇E E EVector identities:
ε ε ε∇ ⋅ = ∇ ⋅ + ⋅∇E E E
Stokes
A V
d dv⋅ = ∇ ⋅∫ ∫
Constitutive relations:
F S F
( )C A
d d⋅ = ∇× ⋅∫ ∫i F l F S
Gauss theorem
Gauss’s LawA A V
d E d dvε ρ⋅ = ⋅ =∫ ∫ ∫D S S
C A
d dt
∂⎛ ⎞⋅ = + ⋅⎜ ⎟∂⎝ ⎠∫ ∫iDH l J SMaxwell (also)
( ) ( ) ( )0, ' ' , 't dt x t t tε+∞
−∞
= −∫P r E rDynamic relation between P and E: ( ) ( ) ( )0, ,ω ε χ ω ω=P k E kand
( ) ( ), Re , exp2
z t z i z z i tαω β ω⎧ ⎫⎛ ⎞= − − +⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭
E EDispersive and absorbing materials:
0' 'n k ncωβ = = 02 '' 2 ''n k n
cωα = − = −where ,absorption coefficient