Nanophotonics

5 nm

Overview of the CoursePart I: Introduction to light interaction with matter

Part II: Photonic Crystals

Part III: Metal optics (evanescent fields)

Derivation Wave Equation in matter from Maxwell’s equationsDielectric properties of insulators, semiconductors and metals (bulk)Light interaction with nanostructures and microstructures (compare d with λ)

Electromagnetic effects in periodic media (Bloch Theorems)Media with periodicity in 1, 2, and 3-dimensionsApplications: Omni-directional reflection, sharp waveguide bends,

Light localization, Superprism effects, Photonic crystal fibers

Light interaction with 0, 1, and 2 dimensional metallic nanostructures Guiding of light below the diffraction limit Near-field optical microscopyTransmission through subwavelength aperturesMaterials with a negative refractive indexPerfect lens

Overview in Images

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

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

• Yes !

• Wonderful things happen when structural dimensions are ≈ λ light

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

Light Interaction with Matter

Maxwell’s Equations

fρ∇ ⋅ =D

0∇ ⋅ =B

∂∇× = −∂B

Et

∂∇× = +∂D

H 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 E

Electric 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

Magnetic

When P is proportional to EE

-

++++

---

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:∂∇× = +∂D

H Jt

?

J

H

D

J

D increasing whencharging the capacitor

C

A

Ampere (1775-1836)

C A

d dt

∂ ⋅ = + ⋅ ∂ ∫ ∫iD

H l J S Magnetic field induced by: Changes in el. flux

Electrical currents

Ampere:C A

d dt

∂ ⋅ = + ⋅ ∂ ∫ ∫iD

H l J S

Curl Equations

Stokestheorem: ( )

C A

d d⋅ = ∇× ⋅∫ ∫i F l F S

( )C A A

d d dt

∂ ⋅ = ∇× ⋅ = + ⋅ ∂ ∫ ∫ ∫iD

H l H S J S

∂∇× = +∂D

H Jt

∂∇× = −∂B

Et

Other curl eq.

Derived in a similar way fromC A

d dt

∂⋅ = − ⋅∂∫ ∫iB

E l S

B

E

CA

( )C A A

d d dt

∂⋅ = ∇× ⋅ = − ⋅∂∫ ∫ ∫iB

E l E S S

Stokes

Summary Maxwell’s Equations

ρ∇ ⋅ =D

0∇ ⋅ =B

∂∇× = −∂B

Et

∂∇× = +∂D

H Jt

Divergence equations Curl equations

Flux lines start and end on charges or poles

Changes in fluxes give rise to fields

Note: No constants such as µ0 ε0, µ ε, c, χ,……. appear when Eqs are written this way.

Currents give rise to H-fields

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 tr t

v

∂∇ =

∂U

Ut

Position Time

Starting point: The curl equations

0 tµ∂ ∂∇× = − = −

∂ ∂B H

Et

∂∇× = +∂D

H Jt

The Wave Equation for the E-field

( ) ( )22

2 2

,1,

r tr t

v

∂∇ =

∂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 J

E

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 J

E

The Wave Equation for the E-field

22

2 2

1

v

∂∇ =∂

EE

tCompare:

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 J

EResult:

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 J

E

Dielectric Media

P linearly proportional to E:

Linear, Homogeneous, and Isotropic Media

0ε χ=P E

χ is a scalar constant called the “electric susceptibility”

( )2 2 2

20 0 0 0 0 02 2 2

1t t t

µ ε µ ε χ µ ε χ∂ ∂ ∂∇ = + = +∂ ∂ ∂

E E EE

Define relative dielectric constant as: 1rε χ= +

22

0 0 2r tµ ε ε ∂∇ =

∂E

E

(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 Materials

We have derived a wave equation for EM waves!

22

0 0 2r tµ ε ε ∂∇ =

∂E

E

Now what ?

Let’s look at some of their properties

Euh….

22

2 2

1

v

∂∇ =∂

EE

t

Speed of the EM wave:

Compare

22 0

0 0

1 1

r r

cv

µ ε ε ε= =

Speed of an EM Wave in Matter

22

0 0 2r tµ ε ε ∂∇ =

∂E

E and

Where c02 = 1/(ε0 µ0) = 1/((8.85x10-12 C2/m3kg) (4πx 10-7 m kg/C2)) = ( 3.0 x 108 m/s)2

Refractive index is defined by:

Optical refractive index

1r

cn

vε 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

,,

tnt

c

∂∇ =

∂E r

E rt

Derived from wave equation

Dispersion Relation

Dispersion relation: ω = ω(k)

Substitute:

22 2

2

ck

nω =

Result:

1g

r

d c c cv

dk k n

ω ωε χ

≡ = = = =+

ω

k

gv

Phase velocity:

Group velocity:

phvk

ω≡

22 2

2

nk

cω=

Check this!

Electromagnetic Waves

( ) ( )222

2 2

,,

tnt

c

∂∇ =

∂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 Media

Relation 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

ck

nω =Dispersion relation k n

c

ω= ±

' ''n n in= +Absorbing materials can be described by a complex n:

( )' '' ' ''2

k n in n i n ic c c

ω 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 nc

ωβ = =Note: n’ act as a regular refractive index

02 '' 2 ''n k nc

ωα = − = − 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 ik

α χ χ= − = + +02 ''k nα = −

Weakly absorbing media

When χ’<<1 and χ’’ << 1: ( )11 ' '' 1 ' ''

2i iχ χ χ χ+ + ≈ + +

Refractive index:1

' 1 '2

n χ= +

Absorption coefficient: 0 02 '' ''k n kα χ= − = −

Summary

Maxwell’s Equations

fρ∇ ⋅ =D 0∇ ⋅ =B∂∇× = −∂B

Et

∂∇× = +∂D

H 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

,,

tnt

c

∂∇ =

∂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

∂∇× = −∂B

Et

∂∇× = +∂D

H Jt

Handy Math Rules

( ) 2∇× ∇× = ∇ ∇ ⋅ − ∇E E EVector identities:

ε ε ε∇ ⋅ = ∇ ⋅ + ⋅ ∇E E E

Stokes

Constitutive relations:

A V

d dv⋅ = ∇ ⋅∫ ∫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

∂ ⋅ = + ⋅ ∂ ∫ ∫iD

H 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 nc

ωβ = = 02 '' 2 ''n k nc

ωα = − = −where ,absorption coefficient