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Radiation sources

• 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

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A dipole – the elementary source of radiation

An oscillating dipole is a point-like time-harmonic current source.

Harmonic time dependence:


The Green function

Field at r generated by dipole at r0

In a homogeneous medium:


Fields in inhomogeneous environment

Field at r generated by dipole at r0

In an inhomogeneous environment:


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Radiation sources


• Green function







The energy radiated by a dipole equals the work done by the dipole’s own field on the dipole itself!


Power radiated by a dipoleThought experiment:Displace positive and negative charge with respect to each other and let go.

+ -


Power dissipated in volume V (c.f. Poynting’s theorem): Cycle averaged (monochromatic case):



Power radiated by a dipoleThought experiment:Displace positive and negative charge with respect to each other and let go.

+ -

Power dissipated in volume V (c.f. Poynting’s theorem): Cycle averaged (monochromatic case):

The environment determines the radiated power via the imaginary part of the GF at the origin.

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 position of the dipole).

• 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!


Why is it called density of states?

Expression “density of states” becomes clear when considering modes of resonator.

Determine DOS for free space by counting states in cubic resonator and letting resonator size become much larger than the wavelength.

Density of states in the lossless resonator

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

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V: resonator volumeN: number of modes

Density of states in the lossless resonator


19


In free space (large resonator):

Density of states in a realistic resonator

20




Via the local density of states (LDOS)• Radiated power depends on location of source within its

environment• Radiated power depends on frequency of source• Radiated power depends on orientation of source

The LDOS can be interpreted as a radiation resistance



In analogy with

Normalize emitted power to power emitted in free space:

Depending on the sign (phase) of the scattered field returning to the dipole, it enhances or suppresses power dissipation.


Power enhancement by photonic system

Warning: The term LDOS (enhancement) is used sloppily to refer to

and more

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Radiation sources


• Green function







Radiating source up to GHz:

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Radiation sources

Radiating source up to GHz:

Radiating sources at 1000 THz (visible):

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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

Probability to find the system in the excited state decays exponentially with rate γ.

How can we measure the population of the excited state?


Decay rate of a quantum emitter

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 times

detectormolecule

time


Fluorescence lifetime measurements



detectormolecule t1

time





detectormolecule t1 t2 t3



time

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




time

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




time

Now we know how to measure the fluorescence lifetime (spontaneous emission rate) of a quantum emitter.

Can we also calculate it?

Fermi’s Golden Rule:

Initial state (excited atom, no photon):

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


Calculation of decay rate g

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

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38

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)

First observation of the local (!) character of the DOS!


Mirror

Eu3+

d

HW3

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


How did he come up with that?

Density of states in a realistic resonator

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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.

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

ω

ρ(ω)

Δω


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.


1. Because it is fun!2. Some people like bright sources. Increase photon production rate of

emitter by LDOS enhancement (see homework problem).

3. Some people like efficient sources. Increase quantum efficiency of emitter by LDOS enhancement (see homework problem).

4. Some people (physicists) like to investigate the excited states of quantum emitters. Increase lifetime of excited state by LDOS reduction (see quantum optics lecture).


Why engineer the decay rate?

Micro-cavities in the 21st century


Vahala, Nature 424, 839



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HW3


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



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