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On the menu today...The Purcell factor is the maximum rate enhancement provided by a cavity given...
Transcript of 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
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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?
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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
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Fluorescence lifetime measurements
time
decay rate lifetime
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)
Power radiated in inhomogeneous environment
In an inhomogeneous environment:
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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.
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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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Decay rate engineering
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
Micro-cavities in the 21st century
How to squeeze more light out of a source:
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Vahala, Nature 424, 839
HW3
Micro-cavities in the 21st century
How to squeeze more light out of a source:
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Vahala, Nature 424, 839
HW3
This is relevant for engineering light sources of all sorts• Lighting• Lasing• Information transmission• …
Micro-cavities in the 21st century
A cavity is a tool to increase light-matter interaction.
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Vahala, Nature 424, 839
Micro-cavities in the 21st century
A cavity is a tool to increase light-matter interaction.
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Vahala, Nature 424, 839
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?
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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 16
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
www.photonics.ethz.ch 17
Nanoparticles: resonators at optical frequencies
www.photonics.ethz.ch 18
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
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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)
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
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source
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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
www.photonics.ethz.ch 22
Molecule (λem~600nm)
Kühn et al., PRL 97, 017402 (2006)Au particle(80nm diam.)
Optical antennas – an intuitive approach
www.photonics.ethz.ch 23
source
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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
The electrodynamic polarizability
• 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
The electrodynamic polarizability
• 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
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Ohmic damping
Radiation damping
Metallic particle (Drude model for e)
HW4
Nanoparticles: resonators at optical frequencies
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nanoparticle
100 nm Au particle
“damping”
• Metal nano-particles show resonances in the visible
• 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
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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
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Buchler et al., PRL 95, 063003 (2005)
Scatterer (metal nanoparticle)
(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.
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Buchler et al., PRL 95, 063003 (2005)
Scatterer (metal nanoparticle)
(weak) mirror
Optical antennas
• 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 31
Molecule (λem~600nm)
Kühn et al., PRL 97, 017402 (2006)Au particle(80nm diam.)
Optical antennas – revisited
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source
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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
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Calculate rate enhancement via power enhancement
A=const.
Source@ r0
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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
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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
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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
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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
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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
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Optical antennas
Fermi’s Golden Rule
Local Density of Optical States
Cavities
• Modulate LDOS on a λ scale
• Rely on interference of propagating waves
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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)
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Feed element
Passive directors
Passive reflectors
p=aE
Yagi-Uda antenna (1926)
Yagi-Uda antenna
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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
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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
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Antennas – resonators with engineered radiation loss
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In the µwave-regime:
Total internal reflection Metallic reflection
From resonators to antennas
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Kü
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RL
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Curto et al., Science 329, 930 (2010)
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Near-field antennas• Sub-l-sized resonators• Naturally high radiation loss
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Cavity-based antennas:• l-sized resonators• Deliberately introduced
radiation loss
Is this an antenna?
From resonators to antennas
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Vah
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RL
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Mu
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L 2
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Curto et al., Science 329, 930 (2010)
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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
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Curto et al., Science 329, 930 (2010)
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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
Antennas – radio vs. optical
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Curto et al., Science 329, 930 (2010)
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
Antennas – radio vs. optical
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Curto et al., Science 329, 930 (2010)
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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.
Radio-antennas Optical antennas
Antenna theory: Maxwell Maxwell
Resonance mechanisms: Geometric resonances Geometric resonancesMaterial resonances
Sources: Classical current source Quantum emitter
Summary – light matter interaction
Quantum emitters are probes for their electromagnetic environment.
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Curto et al., Science 329, 930 (2010)
Kühn et al., PRL 97, 017402 (2006)
Vah
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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.
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Curto et al., Science 329, 930 (2010)
Kühn et al., PRL 97, 017402 (2006)
Vah
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Quantum emission can be tailored via the emitter’s electromagnetic environment.
10 eV1 eV100 meV
kT Ry
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