On the menu today...The Purcell factor is the maximum rate enhancement provided by a cavity given...

On the menu today

Decay rate engineering

• The electric dipole

• Green function

• Fields of electric dipole

• Power dissipated by an oscillating dipole

• The local density of optical states (LDOS)

• Decay rate of quantum emitters

• Decay rate engineering

• Example: Drexhage experiment

• Example: classical analogue of Drexhage experiment

• Example: optical antenna

Optical antennas

• Dipolar scattering theory

• Radiation damping

www.photonics.ethz.ch 1

Radiating source up to GHz:

Radiating sources at 1000 THz (visible):

Wikimedia; Emory.edu

Quantum dotsDye moleculesAtoms


Light sources

Radiating sources at 1000 THz :

Quantum dotsDye moleculesAtoms

Optical emitters have discrete level scheme (in the visible)Let’s focus on the two lowest levelsHow long will the system remain in its excited state?


Quantum emitters – optical sources

The probability to detect a photon at time t is proportional to p(t)!1. Prepare system in excited state with

light pulse at t=02. Record time delay t13. Repeat experiment many times4. Histogram arrival time delays

detectormolecule t1 t2 t3


Fluorescence lifetime measurements

time

decay rate lifetime

Sum over final states is sum over photon states (k) at transition frequency ω.


Calculation of decay rate g

Fermi’s Golden Rule:

Initial state (excited atom, no photon):

Final state (de-excited atom, 1 photon in state k at frequency w):

Interaction Hamiltonian:

Atomic part: transition dipole moment (quantum)

Field part: Local density of states (classical)

Power radiated in inhomogeneous environment

In an inhomogeneous environment:


Local density of optical states

• The power dissipated by a dipole depends on it’s environment and is proportional to the local density of optical states (LDOS).

• The LDOS is (besides prefactors) the imaginary part of the Green’s function evaluated at the origin, i.e., the location of the source.

• Controlling the boundary conditions (and thereby the LDOS) allows us to control the power radiated by a dipole!

Transition dipole moment is NOT classical dipole moment, but

Classical electromagnetism CANNOT make a statement about the absolute decay rate of a quantum emitter.BUT: Classical electromagnetism CAN predict the decay rate enhancement provided by a photonic system as compared to a reference system.


Rate enhancement – quantum vs. classical

EmitterTransition dipole moment:Wave function engineering by synthesizing molecules, and quantum dots

Chemistry, material science

EnvironmentLDOS: Electromagnetic mode engineering by shaping boundary conditions for Maxwell’s equations

Physics, electrical engineering

www.photonics.ethz.chantennaking.com, Wikimedia, emory.edu

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The Purcell effect

The Purcell factor is the maximum rate enhancement provided by a cavity given that the source is1. Located at the field maximum of the mode2. Spectrally matched exactly to the mode3. Oriented along the field direction of the mode

Caution: Purcell factor is only defined for a cavity. The concept of the LDOS is much more general and holds for any photonic system.


Micro-cavities in the 21st century


Vahala, Nature 424, 839


How to squeeze more light out of a source:


www.photonics.ethz.ch


HW3


How to squeeze more light out of a source:


www.photonics.ethz.ch


HW3

This is relevant for engineering light sources of all sorts• Lighting• Lasing• Information transmission• …


A cavity is a tool to increase light-matter interaction.




A cavity is a tool to increase light-matter interaction.



These tools have in common that they shape the LDOS on a length scale comparable to the wavelength (interference of propagating waves).

What about tools to control the LDOS on the sub-wavelength scale?

Optical antennas

• Do metal nanoparticles provide an LDOS enhancement?


Optical antennas for LDOS engineering

• Metallic nanoparticles can act as “antennas” and boost decay rate of quantum emitters in their close proximity

• Effect confined to length scale of order λ/10


Molecule (λem~600nm)

Kühn et al., PRL 97, 017402 (2006)Au particle(80nm diam.)

On the menu today


• The electric dipole

• Green function

• Fields of electric dipole

• Power dissipated by an oscillating dipole

• The local density of optical states (LDOS)

• Decay rate of quantum emitters

• Decay rate engineering

• Example: Drexhage experiment

• Example: classical analogue of Drexhage experiment

• Example: optical antennas for decay rate engineering

Optical antennas

• Dipolar scattering theory

• Radiation damping


Nanoparticles: resonators at optical frequencies


100 nm Au particle

“damping”

• Metal nano-particles show resonances in the visible

Nan

op

arti

cle.

com

Nanoparticle(smaller than wavelength)



100 nm Au particle

“damping”


Lycurgus Cup (glass with metal nano-particles):Green when front lit Red when back lit

How does that work? W

ikip

edia

.org

Nanoparticle(smaller than wavelength)

The electrostatic polarizability

• Static polarizability: induced dipole moment due to static E-field

• For a sphere in vacuum:

• For a Drude metal, we find a Lorentzian polarizability


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+

HW4

Ohmic damping rate

plasma frequency

Optical antennas – an intuitive approach

• assume an oscillating dipole close to a polarizable particle

• Assume that particle is small enough to be described as dipole

• Assume distance d<<λ, near field of source polarizes particle

• If polarizability α is large, antenna dipole largely exceeds source dipole

• Radiated power dominated by antenna dipole moment


source

-+++-

-

antenna

Optical antenna is a dipole moment booster!

α

d

LDOS enhancement:

Optical antennas for LDOS engineering






Optical antennas – an intuitive approach


source

-+++-

-

antenna

According to this, the only limit to the LDOS enhancement provided by an optical antenna is the material damping rate (Ohmic loss rate) g.

Are there other damping mechanisms?

α

d

LDOS enhancement:

HW4

The electrodynamic polarizability

• Static polarizability: induced dipole moment due to static E-field

• Dynamic case: additional field generated by induced dipole moment

• Define effective electrodynamic polarizability “dressed” with Green function

• ReG0 diverges at origin! Fact that we describe the scatterer as a mathematical point backfires. Choose to fit experimentally found resonance frequency.

• ReGs shifts resonance frequency depending on environment.

• ImG represents radiation damping term: essential for energy conservation


+++

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+

HW4


• This is a recipe to amend any electrostatic polarizability α0 with a radiation damping term to ensure energy conservation

• Electrodynamic polarizability depends on position within photonic system

• Radiation correction small for weak scatterers (small α0)

• Radiation correction significant for strong scatterers (large α0)

• Limit of maximally possible scattering strength


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+

HW4


• Compare static and dynamic α

• Static α0 may be huge, dynamic αeff

is always bounded by inverse LDOS

• Radiation damping is a loss channel and dampens resonance

• Radiation damping is given by the LDOS at the scatterer’s position


Ohmic damping

Radiation damping

Metallic particle (Drude model for e)

HW4



nanoparticle

100 nm Au particle

“damping”


• Resonance frequency given by

• plasma frequency of Drude metal (first order)

• Size of particle (second order)

• Width of resonance given by

• Ohmic damping of Drude metal

• Radiation damping (LDOS)

Drexhage’s experiment with a scatterer

• Metal nanoparticle on a scanning probe close to a reflecting substrate

• Measure width of scatterer’s resonance as a function of distance to substrate


Buchler et al., PRL 95, 063003 (2005)

Scatterer (metal nanoparticle)

(weak) mirror

Drexhage’s experiment with a scatterer

• Spectral width of scattering cross section (i.e. damping) can be tuned by changing scatterer-mirror distance

• LDOS determines damping rate of scatterer


Buchler et al., PRL 95, 063003 (2005)


(weak) mirror

Optical antennas for spontaneous emission control

We can use material (plasmonic) resonances to build resonant optical antennas of sub-wavelength size.

The Q-factor of strongly polarizable antennas is dominated by radiation loss.


Buchler et al., PRL 95, 063003 (2005)


(weak) mirror

Optical antennas






Optical antennas – revisited


source

-+++-

-

antenna

Optical antenna is a dipole moment booster!

α

d

Remember our (sloppy) derivation?

Optical antennas – a cleaner derivation

• Field at source is primary field + field generated by induced antenna dipole

• Assume source is close to particle (near-field terms dominate)

• Close to source ReG along dipole axis diverges as 1/d³, ImG is constant


Calculate rate enhancement via power enhancement

A=const.

Source@ r0

-+++-

-

Antenna@ rant

α

d

Optical antennas – a cleaner derivation

• Rate enhancement goes with the imaginary part of polarizability

• Rate enhancement goes with inverse source-antenna distance d-6


Calculated rate enhancement (equals power enhancement):

Wait a minute! Didn’t we say earlier that the enhancement for a strong antenna should go as |α|²?True. But for a strong scatterer

Source@ r0

-+++-

-

Antenna@ rant

α

d

HW4

Optical antennas …

• Modulate LDOS on sub-λ length scale

• Can boost decay rates of quantum emitters

• Can direct the emission of quantum emitters

• Rely on resonances in the polarizability of their constituents


The polarizability of strong dipolar scatterers …

• has to take radiation effects into account

• depends on position within photonic system

The local density of optical states (LDOS) …

• Is (essentially) the imaginary part of the Green function

• Governs light-matter interaction, e.g.

• Determines the decay rate (enhancement) of quantum emitters

• Determines the linewidth of dipolar scatterers

• Determines the power dissipated by a classical constant-current source


Photonic structures to control LDOS

• Modulate LDOS on a sub-λ scale

• Rely on resonances of conduction electrons of metal nanoparticles

• Rely on evanescent fields


Optical antennas

Fermi’s Golden Rule

Local Density of Optical States

Cavities

• Modulate LDOS on a λ scale

• Rely on interference of propagating waves

Vah

ala,

Nat

ure

42

4,

83

9

Kü

hn

et a

l., P

RL

97

, 01

74

02

(2

00

6)

From radio to optical antennas

• Single active element

• Field of active element polarizes passive elements

• Passive elements generate fields and polarize each other (self-consistent solution)

38

Feed element

Passive directors

Passive reflectors

p=aE

Yagi-Uda antenna (1926)

Yagi-Uda antenna

39

Taminiau et al., 2008

Optical antennas for directional photon emission

• Optical antennas allow control of directionality of light emission for quantum emitters

• Antenna is photonic environment that offers a large density of states with specific k-vector


Driven elementPassive scatterers

emission

Scale down size, scale up frequency

106

Yagi and Uda (1920s) Curto et al., Science 329, 930 (2010)

emission

Quantum emitter

Metal nanoparticles

Rad

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Antennas – resonators with engineered radiation loss


In the µwave-regime:

Total internal reflection Metallic reflection

From resonators to antennas


Kü

hn

et a

l., P

RL

97

, 01

74

02

(2

00

6)

Curto et al., Science 329, 930 (2010)

Rad

io-e

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ron

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Near-field antennas• Sub-l-sized resonators• Naturally high radiation loss

Vah

ala,

Nat

ure

42

4,

83

9

Mu

nsc

het

al.,

PR

L 2

01

3

Cavity-based antennas:• l-sized resonators• Deliberately introduced

radiation loss

Is this an antenna?

From resonators to antennas


Vah

ala,

Nat

ure

42

4,

83

9

Kü

hn

et a

l., P

RL

97

, 01

74

02

(2

00

6)

Mu

nsc

het

al.,

PR

L 2

01

3


Rad

io-e

lect

ron

ics.

com

Is this an antenna?

Antennas are devices which mediate between far-field (=propagating) radiation and localized fields.Antennas boost light-matter interaction.Discuss antennas using LDOS.Do not discuss antennas in terms of Purcell factors (mode volume V not well defined).

Near-field antennas• Sub-l-sized resonators• Naturally high radiation loss

Cavity-based antennas:• l-sized resonators• Deliberately introduced

radiation loss

Antennas – radio vs. optical



Rad

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Radio-antennas Optical antennas

Antenna theory: Maxwell Maxwell

Resonance mechanisms: Geometric resonances Geometric resonancesMaterial resonances

Sources: Classical current source Quantum emitter




Rad

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Rad

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• Nano-optics with optical antennas relies on classical antenna theory due to the scale invariance of Maxwell’s equations.

• Difference 1: Frequency dependence of the material constants.At radio frequencies we have practically perfect metals.At optical frequencies metals are imperfect and show material resonances.

• Difference 2: Emitters in the optical regime show quantum behavior.





Summary – light matter interaction

Quantum emitters are probes for their electromagnetic environment.



Kühn et al., PRL 97, 017402 (2006)

Vah

ala,

Nat

ure

42

4,

83

9

Quantum emission can be tailored via the emitter’s electromagnetic environment.

Radiation carries information about• The emitter• The emitter’s environment• The emitter-environment interaction

Summary – light matter interaction

Quantum emitters are probes for their electromagnetic environment.



Kühn et al., PRL 97, 017402 (2006)

Vah

ala,

Nat

ure

42

4,

83

9

Quantum emission can be tailored via the emitter’s electromagnetic environment.

10 eV1 eV100 meV

kT Ry

Rad

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