Post on 16-Jul-2020
Nanoscale antennas
Said R. K. Rodriguez24/04/2018
The problem with nanoscale optics
ฮ๐๐(๐๐๐๐โ๐๐๐๐)
~1-10 nm
~400-800 nm
How to interface light emitters & receiverswith plane waves?
Antenna
Tx/Rx
An antenna is a device that converts free-space radiation into localized energy, and vice versa
Radiation
Novotny & van Hulst, Nat.Photon. 5, 83 (2011).
What is an antenna?
Antenna
Tx/Rx
An antenna is a device that converts free-space radiation into localized energy, and vice versa
Radiation
What is an antenna?
At radio frequencies, E = 0 inside the metal โ perfect metal
Novotny & van Hulst, Nat.Photon. 5, 83 (2011).
Antenna
Tx/Rx
An antenna is a device that converts free-space radiation into localized energy, and vice versa
Radiation
What is an antenna?
At optical frequencies, E โ 0 inside the metal.
Consequence: radio-freq. antenna designs cannot be directly scaled
Novotny & van Hulst, Nat.Photon. 5, 83 (2011).
200 400 600 800 1000 1200 1400 1600 1800-150
-100
-50
0
50
Measured data: ฮต' ฮต"
Drude model: ฮต' ฮต"
Modified Drude model: ฮต'
ฮต"
ฮต
Wavelength (nm)
ฮต'
Dielectric constant for Ag
Finite ฮตโ leads to field penetration
Plasmons in the bulk oscillate at ฯp determined by the free electron density and effective mass
Plasmons confined to surfaces that can interact with light to form propagating โsurface plasmon polaritons (SPP)โ
Localized surface plasmons in nanoparticles
+ + +
- - -
+ - +
k 0
2
ฮตฯ
mNedrude
p =
From plasmons to plasmonics
2/1
"'
+
=+=dm
dmxxx c
ikkkฮตฮต
ฮตฮตฯ
optical resonance frequency depends on shape & size; k is irrelevant
transmissionreflection
Colors of gold nanoparticles Stained glass @ Notre Dame de Paris
Lycargus cup, 4thC AD 1260
Observables
Extinction cross section [m2]
Power removed from beamIncident intensity
Extinction = scattering + absorption
removed from the beam re-radiated into all angles lost as heat in the scatterer
Linear response to applied field
Small object kd <<1 - incident field is approximately constant
Volume polarization (weak index so E=Ein)
Total dipole moment
Larger particles & ฮต : larger dipole moments
Electrostatic sphere
Consider a sphere in a static field E0 ฮตmฮต
z
rฮธ
a( )( )ar
ar>=โฮฆ<=โฮฆ
00
2
1
Laplace equation:
( ) ( ) zEarrr
arr
m 0221
21 โ=ฮฆ=โฮฆโ
=โฮฆโ
=ฮฆ=ฮฆโโ
lim,, ฮตฮต
Boundary conditions set by 0)()( =ฮฆโโ โโ=โ โ=โ โ ฮตฮต ED
Solution
1 0 0 0
32 0 0 02 2
0
3cos cos cos2 2
cos coscos cos2 4
m m
m m
m
m m
E r E r E r
pE r a E E rr r
ฮต ฮต ฮตฮธ ฮธ ฮธฮต ฮต ฮต ฮต
ฮต ฮต ฮธ ฮธฮธ ฮธฮต ฮต ฯฮต ฮต
โฮฆ = โ + = โ + +
โฮฆ = โ + = โ + +
E0 ฮตm
ฮตz
rฮธ
a[ see J. D. Jackson, Classical Electrodynamics, Ch. 4]
30 0 with 4
2m
SI SI mm
p E a ฮต ฮตฮฑ ฮฑ ฯฮต ฮตฮต ฮต
โ= = +
rr
Inside sphere: homogeneous fieldOutside sphere: background field plus field of a dipole with
In the ball:
Outside:
Metal sphere
Drude model for a metal: Lorentzian `plasmon resonanceโ
โข Resonance at ฮต(ฯ0) = -2 ฮตmโข Response scales with the volume Vโข ฮฑ exceeds V by factor 5 to 10โข Shape shifts condition ฮต = -2 ฮตmโข ฮณ still needs to include radiation damping
๐๐ = 4๐๐๐๐0๐ผ๐ผ๐ธ๐ธ0 ๐ผ๐ผ = ๐๐3 ๐๐ โ ๐๐๐๐
๐๐ + 2๐๐๐๐ 0
๐ผ๐ผ = ๐๐3 ๐๐02
๐๐02 โ ๐๐2 + ๐๐๐๐๐๐ 0
๐๐ = 1 โ๐๐๐๐
2
๐๐(๐๐ + ๐๐๐๐๐๐)means
Revisiting polarizabilityClassical model of harmonically bound electron describes atom, and scatterer alike, as an oscillating dipole
20 0
2 20
3 ( ) ( )i t i tSI
Vt e ei
ฯ ฯฮต ฯ ฮฑ ฯฯ ฯ ฯฮณ
= =โ โ
p E E
Lorentzian resonance
Extinction: how much power is taken from the beam ?
Cycle average work done by E on p
in ImdpW Edt
ฮฑโ โ โ
Revisiting polarizabilityExtinction: how much power is taken from the beam (in SI units) ?
0 0
1 1Re[ ] Re[ ] Re[ ] Re[ ]T Ti t
i t i t i td eW e dt e i e dtT dt T
ฯฯ ฯ ฯฯฮฑ= โ = โ โซ โซ
pE E E
* * *
0
1 ( ) ( )4
Ti t i t i t i tW e e i e i e dt
Tฯ ฯ ฯ ฯฯฮฑ ฯฮฑโ โ= + โ โโซ E E E E
* 2 2
0
1 ( | | | | ) oscill.terms ( 2 )4
T
W i i dtT
ฯฮฑ ฯฮฑ ฯ= โ + + ยฑโซ E E
2Im | |2
W ฯ ฮฑ= E
Revisiting polarizabilityClassical model of harmonically bound electron describes atom, and scatterer alike as an oscillating dipole
20 0
2 20
3 ( ) ( )i t i tSI
Vt e ei
ฯ ฯฮต ฯ ฮฑ ฯฯ ฯ ฯฮณ
= =โ โ
p E E
Lorentzian resonance
Scattering: how much power does p radiate ?
22
0
2
0 0
22dipole
2
0 0
2 ||4
sinsin||sin W ฮฑฯฮต
ฮธฮธฯฮธฯฯ ฯฯ ฯ
โ=โโ โ โซ โซโซ โซโซ rprdErddA
sphere
nS
Equate extinction to scattering (energy conservation)
Scattering
Rayleigh / Larmor
Extinction
Work done to drive p
โฅ
Optical theorem
1. Very small particles scatter like r6/ฮป4 (Rayleigh)2. For very small particles absorption wins ~ r3/ฮป3. Big |ฮฑ|2 implies large Im ฮฑ
4๐๐๐๐ Im ๐ผ๐ผ [๐๐2] 8๐๐3
๐๐4 ๐ผ๐ผ 2 [๐๐2]
ScatteringExtinction
โฅ
Optical theorem
Since
Upper bound on the strongest possible dipole scatterer
Rayleigh / LarmorWork done to drive p
Equate extinction to scattering (energy conservation)
4๐๐๐๐ Im ๐ผ๐ผ [๐๐2] 8๐๐3
๐๐4 ๐ผ๐ผ 2 [๐๐2]
๐ผ๐ผ โค32
๐๐2๐๐
3
Im ๐ผ๐ผ < ๐ผ๐ผ
Extinction โ Interference effect๐๐๐๐๐๐๐๐ = โฌ๐ท๐ท Sext๏ฟฝ๏ฟฝerdA = โฌ๐ท๐ท
12
Re EiรHsโ +Es รHiโ ๏ฟฝ๏ฟฝerdA
๐๐๐๐๐๐๐๐ =๐๐๐๐๐๐๐๐
๐ด๐ด๐๐๐๐๐๐๐๐Out of resonanceOn resonance
2๐๐๐๐๐๐
= 0.3
Extinction โ Interference effect๐๐๐๐๐๐๐๐ = โฌ๐ท๐ท Sext๏ฟฝ๏ฟฝerdA = โฌ๐ท๐ท
12
Re EiรHsโ +Es รHiโ ๏ฟฝ๏ฟฝerdA
๐๐๐๐๐๐๐๐ =๐๐๐๐๐๐๐๐
๐ด๐ด๐๐๐๐๐๐๐๐
ฯext = Apart
r=20 nm Ag particle, in n=1.5 (glass)
Extinction โ Interference effect๐๐๐๐๐๐๐๐ = โฌ๐ท๐ท Sext๏ฟฝ๏ฟฝerdA = โฌ๐ท๐ท
12
Re EiรHsโ +Es รHiโ ๏ฟฝ๏ฟฝerdA
๐๐๐๐๐๐๐๐ =๐๐๐๐๐๐๐๐
๐ด๐ด๐๐๐๐๐๐๐๐
2๐๐๐๐๐๐
= 0.3
Question: what does the above expression tells us aboutthe detector needed to measure the full extinction?
Summary
โข Antennas convert free-space radiation into localized energy & viceversa
โข At optical frequencies, E-field penetrates into the metal. This leads to surface plasmon resonances
โข Extinction:โ Work done by E on pโ โ Im(ฮฑ)โ Interference of incident & scattered field
โข Subwavelength particles can absorb and scatter much more light than is geometrically incident on them. In general, Qext >1 on resonance and Qext <1 off resonance
10 min. break
Approaches to controlling lightResonant nanoparticles Photonic crystals
Surface Plasmon Polaritons
Dipole radiation
๐ฌ๐ฌ(๐๐) = ๐๐0๐๐2๐บ๐บ(๐๐) ๏ฟฝ ๐๐1 dipole (vector):
1 dipole (scalar): ๐๐1 ๐๐ =๐๐๐๐๐๐๐๐
๐๐๏ฟฝ ๐๐
Dipole arrays
๐ฌ๐ฌ(๐๐) = ๐๐0๐๐2๐บ๐บ(๐๐) ๏ฟฝ ๐๐1 dipole (vector):
1 dipole (scalar): ๐๐1 ๐๐ =๐๐๐๐๐๐๐๐
๐๐๏ฟฝ ๐๐
Dipole array (scalar): ๐๐๐ก๐ก๐ก๐ก๐ก๐ก ๐๐ = ๐๐1(๐๐) ๏ฟฝ ๐ด๐ด๐ด๐ด
Depends on positions & complex amp. of scatterersFourier transform of geometry (more ahead)
a โ ฮป
Far-field of 2 dipoles
๐๐๐ก๐ก = ๐๐1 + ๐๐2 = ๐๐๐๐โ๐๐(๐๐๐๐1โ๐ฝ๐ฝ/2)
๐๐1cos ๐๐1 +
๐๐โ๐๐(๐๐๐๐2+๐ฝ๐ฝ/2)
๐๐2cos ๐๐2
โr1
r2
ฮธ1
ฮธ2
ฮฒ = phase difference between dipoles
๐๐๐ก๐ก ๐๐ = ๐๐๐๐๐๐๐๐๐๐
๐๐๏ฟฝ ๐ด๐ด๐ด๐ด
โr1
r2
ฮธ1
ฮธ2 ๐ด๐ด๐ด๐ด = cos12
(๐๐๐๐๐๐๐ก๐ก๐๐ ๐๐ + ๐ฝ๐ฝ)
ฮฒ = phase difference between dipoles
Exercise:
Far-field of 2 dipoles
๐๐๐ก๐ก = ๐๐1 + ๐๐2 = ๐๐๐๐โ๐๐(๐๐๐๐1โ๐ฝ๐ฝ/2)
๐๐1cos ๐๐1 +
๐๐โ๐๐(๐๐๐๐2+๐ฝ๐ฝ/2)
๐๐2cos ๐๐2
๐๐๐ก๐ก ๐๐ = ๐๐๐๐๐๐๐๐๐๐
๐๐๏ฟฝ ๐ด๐ด๐ด๐ด
โr1
r2
ฮธ1
ฮธ2 ๐ด๐ด๐ด๐ด = cos12
(๐๐๐๐๐๐๐ก๐ก๐๐ ๐๐ + ๐ฝ๐ฝ)
ฮฒ = phase difference between dipoles
Exercise:
Note: ฮฒ=ฯ & ฮธ=ฯ/2 โ AF = 0 โ kd
Far-field of 2 dipoles
๐๐๐ก๐ก = ๐๐1 + ๐๐2 = ๐๐๐๐โ๐๐(๐๐๐๐1โ๐ฝ๐ฝ/2)
๐๐1cos ๐๐1 +
๐๐โ๐๐(๐๐๐๐2+๐ฝ๐ฝ/2)
๐๐2cos ๐๐2
Dimer in static approximationDimer in a static approximation
Linear problem
โข Symmetric, but not real matrixโข 1/polarizability on the diagonalโข Interaction on the off-diagonal - this will shift resonances
Hybrid modes
Hybridization (exercise)
Arrays of coupled dipoles
Arrays of coupled dipoles
d= 100 nmax = ay = 450 nmn = 1.5
ax n
1 dipole
array of dipoles
Light cone & diffraction
Light emission from plasmonic arrayNA of objective
kx
ky
S.R.K. Rodriguez et al., Phys. Rev. X 1, 021019 (2011).
LSPR
Extinction of Au nanorod arrays
Bright โ even / Dark - odd
Diffraction / Bloch theorem determines mode dispersionMode symmetry + illumination determines what you excite
S.R.K. Rodriguez et al., Physica B 407, 4081 (2012).
Coupled dipole calculations
Measurements
Shaper resonances by adding nanoparticles
Collective resonances
Uses of resonant nanostructures
Enhanced local fieldsOn resonance, ~ 104 enhanced intensity
Au spheres 5,8,20 nm, gaps of 1-3 nm
100 102 104
|E|2/|Ein|2
Single molecule Fluorescence Enhancement
100 nm
๐ด๐ด๐ธ๐ธ =๐๐(๐๐0, ๐๐๐๐๐๐)๐๐0(๐๐0, ๐๐๐๐๐๐)
๐ท๐ท(๐๐0, ๐๐๐๐๐๐)๐ท๐ท0(๐๐0, ๐๐๐๐๐๐)
๐ธ๐ธ (๐๐0, ๐๐๐๐๐๐๐๐) 2
๐ธ๐ธ0(๐๐0, ๐๐๐๐๐๐๐๐) 2
A. Kinkhabwala et al., Nat. Phot. 3, 654 (2009)
Yagi-Uda nanoantenna
A. F. Koenderink, Nano Lett. 9, 4228 (2009)
Directional emission from localized sources
100 nm
A. Curto et al., Science 329, 930 (2010)
100 nm
Directional emission from extended sources
emitting layer
G. Lozano et al., Light Sci. Appl. e66 (2013)
100 nm
Directional emission from extended sources
emitting layer emitting layer
SensingSingle protein binding/unbinding Refractive index sensing
n
P. Offermans et al., ACS Nano 5, 5151 (2011)
Biological imaging
Novotny & van Hulst, Nat.Photon. 5, 83 (2011).
Diffraction unlimited resolution
ฮป= 633 nm
Nonlinear effectsGenerating new frequenciesEnhanced mode mapping
P. Ghenuche et. al. Phys. Rev. Lett. 10, 116805 (2008) H. Harutyunyan et. al. Phys. Rev. Lett. 108, 217403 (2012)
Summary
โข Small particles of size < ฮป/10 scatter like dipolesโข Arrays of dipoles can be described by effective polarizabilityโข Nanoantennas can be used to enhance:
โ local fieldsโ absorption & spontaneous emissionโ Sensingโ Biological imagingโ Nonlinearities