on the menu today - eth z · on the menu today decay rate engineering •the electric dipole...
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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
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Power radiated in inhomogeneous environment
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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!
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Power radiated in inhomogeneous environment
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Local density of optical states
current
Power dissipated in an electrical circuit:
resistance
LDOS!
Radiation resistance!
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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
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Power radiated in inhomogeneous environment
In analogy with
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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?
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Quantum emitters
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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
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Fluorescence lifetime measurements
time
decay rate lifetime
Think about an application (beyond academic interest) for fluorescence lifetime measurements!
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Sum over final states is sum over photon states (k) at transition frequency ω.
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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)
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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
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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.
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Rate enhancement – quantum vs. classical
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Drexhage’s experiment (late 1960s)
Emitter sees its own mirror image.Gs is given as the field generated by the mirror dipole.
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Mirror
Eu3+
d
HW3
Drexhage, J. Lumin. 1,2; 693 (1970)
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A classical analogy for Drexhage’s experiment
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Langguth et al., PRL 116, 224301 (2016)
Fourier-trafoDecay rate g
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A classical analogy for Drexhage’s experiment
www.photonics.ethz.ch 17
Langguth et al., PRL 116, 224301 (2016)
Fourier-trafoDecay rate g
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A classical analogy for Drexhage’s experiment
www.photonics.ethz.ch 18
Langguth et al., PRL 116, 224301 (2016)
Fourier-trafoDecay rate
Make clear to yourself how the radiated power for a dipole with constant current/amplitude is related to the decay rate!
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Density of states in a realistic resonator
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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
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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
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Literally: “1 mode per volume V and frequency band “
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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.
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Micro-cavities in the 21st century
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Vahala, Nature 424, 839
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Micro-cavities in the 21st century
How to squeeze more light out of a source:
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www.photonics.ethz.ch
Vahala, Nature 424, 839
HW3
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Micro-cavities in the 21st century
A cavity is a tool to increase light-matter interaction.
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Vahala, Nature 424, 839
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Back to our optical antennas
• We saw that metal nanoparticles provide a near-field enhancement
• Do metal nanoparticles provide an LDOS enhancement?
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without Au particle : with Au particle:
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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
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Molecule (λem~600nm)
Kühn et al., PRL 97, 017402 (2006)Au particle(80nm diam.)
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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
www.photonics.ethz.ch 33
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Nanoparticles: resonators at optical frequencies
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100 nm Au particle
“damping”
• Metal nano-particles show resonances in the visible
Nan
op
arti
cle.
com
Nanoparticle(smaller than wavelength)
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Nanoparticles: resonators at optical frequencies
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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)
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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
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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
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source
-+++-
-
antenna
Optical antenna is a dipole moment booster!
α
d
LDOS enhancement:
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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
www.photonics.ethz.ch 38
Molecule (λem~600nm)
Kühn et al., PRL 97, 017402 (2006)Au particle(80nm diam.)
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Optical antennas – an intuitive approach
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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