Nanophotonics

Class 4

Density of states

Outline

Spontaneous emission: an exited atom/molecule/.. decays to the ground state and emits a photon

• Emission rates are set by Fermi’s Golden Rule

• Fermi’s Golden Rule & the number of available photon states (LDOS)

• Experiments demonstrating emission rate control via LDOS

• Conclusion

Fermi’s Golden Rule

• Consider an atom, molecule or quantum dot with eigenstates .

• Suppose the system is perturbed, e.g. by incident light. Perturbing term in hamiltonian:

V μ E

The coupling can take the atom in initial state i to another state f

Fermi’s Golden Rule: rate of decay of the initial state i

light

Dipole operator

2

2all finalstates

2( )f i f i

f

V E E

Understanding Fermi’s Golden Rule

2

2all finalstates

2( )f i f i

f

V E E

Energy conservationMatrix elements:Transition strengthSelection rules

Spontaneous emission of a two-level atom:

Initial state: excited atom + 0 photons.

Final state: ground state atom + 1 photon in some photon state

Question: how many states are there for the photon ??? (constraint: photon energy = atomic energy level difference)

How many photon states are there in a box of vacuum ?

( , ) sin( ) with ( , , )i tE x t Ae l m nL

k r k

States in an LxLxL box:

l,m,n positive integers

Number of states with |k|between k and k+dk:3

24( ) 2

8

LN k dk k dk

l,m,n > 0

fill one octantfudge 2 for polarization

As a function of frequency ck

2 23 3

2 2 2 3( )

dkN d L d L d

c d c

Picture fromhttp://britneyspears.ac

k

dk

Density of states in vacuum2 2

3 32 2 2 3

( )dk

N d L d L dc d c

Example: ~50000 photon states per m3 of vacuum per 1 Hz @ =500 nm

0 2 4 60

50000

100000

150000

Den

sity

of

phot

on s

tate

s pe

r un

it vo

lum

e (s

-1)

Frequency (1015 s-1)

nm

~ 50000 states

Controlling the DOS

Photonic band gap material

Example: fcc close-packed air spheres in n=3.5Lattice spacing 400 nm

0 2 40

200000

400000

600000

800000

Low group velocity modes:high DOS

1st Bragg condition:fewer modes

Den

sity

of

phot

on s

tate

s pe

r un

it vo

lum

e (s

-1)

Frequency (1015 s-1)

vacuum

Enhanced DOS:as in high index homogeneous material

Ban

d g

ap:

no

mo

des

Photonic band gap: no states = no spontaneous emission

Enhanced DOS: faster spontaneous emission according to Fermi G. Rule

Local DOSAn emitter doesn’t just count modes (as in DOS)It also feels local mode strength |E|2.It can only emit into a mode if the mode is not zero at the emitter

all modes

( ) ( )m

m

N DOS: just count states

2

all modes

( , , ) | ( ) | ( )mmN r d d E r

Local DOS

A B

Atom at position A can not emit into cavity mode.

Atom at position B can emit into cavity mode.

LDOS: emission in front of a mirror

Drexhage (1966): fluorescence lifetime of Europium ions depends on source position relative to a silver mirror (=612 nm)

Silver mirror

Spacer thickness d

Europium ions

Example II: dielectric nano-sphere

ar

n0 .4

0 .8

5001000

1500 2000

0

0.5

1

1

1.5

r/a

RR bulk

Eu ions in 100 nm – 1 m polystyrene spheres [1]Er ions in 340 nm SiO2 spheres [2]

[1] Schniepp & Sandoghdar, Phys. Rev. Lett 89 (2002)[2] de Dood, Slooff, Polman, Moroz & van Blaaderen, Phys. Rev. A 64 (2001)

LDOSnormalized toLDOS in SiO2

Dielectric nanosphere

1 80 0 2 00 0

1 .0

1 .5

2 .0

100 1000

0.4

0.6

0.8

1.0

1.2

S phere d iam eter [nm ]

Nor

mal

ize

d ra

diat

ive

deca

y ra

te

b) AFM

Confocal

AFM to check individual particle diametersConfocal microscopy to collect luminescence

n=1.52

n=1.33

n=1

Index matching of spherewith fluid droplets:

Emitter stays the sameLifetime change disappears

[1] Schniepp & Sandoghdar, Phys. Rev. Lett 89 (2002)

LDOS & measuring nonradiative decay

A real emitter often also decays nonradiatively (no photons but heat)

total non radiative radiative

Measured in experiment Unknown losslocal chemistry at source

Fermi’s Golden RuleLDOS

Measurement technique: vary the nanophotonic configuration vary LDOS and not the chemistry

ExampleEmitter in sphere: index match sphere to vary

Assignment: you can find by varying LDOS

radiative

non radiative

Conclusions• Spontaneous emission rates are controlled by nanophotonic structures

• Fermi’s Golden Rule: transition rate depends on availability of final states

• Spontaneous emission: final states for photon ?

• Density of states (DOS): number of photon states depending on frequency

• Local density of states (LDOS): number of photon states available locally for spontaneous emission

Applications

• Enhance the efficiency of light sources

• Characterize non-radiative mechanisms