On the menu today - ETH Z · On the menu today Decay rate engineering •The electric dipole •Green function •Fields of electric dipole ... • The power dissipated by a dipole

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: The Purcell effect• Example: classical analogue of Drexhage experiment• Example: optical antenna

Optical antennas• Dipolar scattering theory• Radiation damping

www.photonics.ethz.ch 1

Power radiated in inhomogeneous environment


Local density of optical states

• The power dissipated by a dipole depends on its 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.

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



Local density of optical states

current

Power dissipated in an electrical circuit:

resistance

LDOS!

Radiation resistance!

Via the local density of states (LDOS) the power radiated by a dipole depends on• location of source within its environment• frequency of source• orientation of source

The LDOS can be interpreted as a radiation resistance



In analogy with

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

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

Think about an application (beyond academic interest) for fluorescence lifetime measurements!

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)

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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Decay rate engineering

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

Drexhage’s experiment (late 1960s)

Emitter sees its own mirror image.Gs is given as the field generated by the mirror dipole.


Mirror

Eu3+

d

HW3

Drexhage, J. Lumin. 1,2; 693 (1970)

A classical analogy for Drexhage’s experiment


Langguth et al., PRL 116, 224301 (2016)

Fourier-trafoDecay rate g




Fourier-trafoDecay rate g




Fourier-trafoDecay rate

Make clear to yourself how the radiated power for a dipole with constant current/amplitude is related to the decay rate!

Density of states in a realistic resonator

21

How many modes in frequency band [w, w+Dw] and resonator volume V?

In free space (large resonator):

Small resonator

Large resonator

• Losses broaden delta-spike into Lorentzian

• Area under Lorentzian is unity

• The lower the loss, the higher the density of states on mode resonance

• Density of states on resonance exceeds that of free space

The Purcell effect

Free space: Density of states via Green function.Alternatively, count states in large box (see EM course).

In cavity: Lorentzian with essentially one mode per Δωand cavity volume V

ω

ρ(ω)

Δω


Literally: “1 mode per volume V and frequency band “

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


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



Back to our optical antennas

• We saw that metal nanoparticles provide a near-field enhancement

• Do metal nanoparticles provide an LDOS enhancement?


without Au particle : with Au particle:

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

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 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)

Nanoparticles: resonators at optical frequencies


100 nm Au particle

“damping”

• Metal nano-particles show resonances in the visible

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


+++

- - --

+

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

• 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.)

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

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On the menu today - ETH Z · On the menu today Decay rate engineering •The electric dipole •Green function •Fields of electric dipole ... • The power dissipated by a dipole