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Plasmon polariton
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Plasmonics and WaveguidesEngineering plasmon-polaritons in metallic nanostructures
Luca Dal Negro ECE & Photonics Center, Boston University, USA
dalnegro@bu.eduhttp://www.bu.edu/nano/
Plasmonics: engineering charge density oscillations bound at metal-dielectric interfaces
Design Parameters
200nm
Size
Walsh et al., Nano Lett., 13 (2), 786 (2013)
Hanke et al., Nano Lett., 12 (4) 2037 (2012)
Shape
Yan et al., Opt. Mat. Exp., 1, 8, 1548 (2011) http://www2.mpip-mainz.mpg.de/groups/bonn/research/menges_bernhard/cleanroom
Suh, Nano Lett., 12, 1 269 (2012)
Near-Field Coupling
Hanke et al., Nano Lett., 12 (4) 2037 (2012)Toroghi et al, Appl. Phys. Lett. 100, 183105 (2012)
Y. Chu, et. al. Applied Physics Letters 93, 181108 (2008).
Photonic / Diffractive CouplingMaterial
Nanoplasmonics
• Electron density oscillations coupled to EM waves • Strong field intensity enhancement at metal surfaces • Optical fields nanoscale localized (sub-)• Largely tunable optical properties
Key Features:
• Enormous polarizability
• Resonant behavior
• Classical EM description
Why metals:
Engineering polarization bound charges in nanomaterials (=Polaritonics) enables the control of strongly enhanced radiative and non-radiative fields
Example: Plasmon Resonances in Metallic Nanostructures
Localized Surface Plasmons=
Collective electron oscillations
• Field enhancement • Resonant response
Example: Surface modes of small particles
3
2
d
NP md
NP m
f a
f
Collective, in-phase motion of electrons
Quadrupole plasmon resonance
For larger particles
Half of the electron clouds moves parallel to the applied field and half moves anti-parallel
3
3 / 2
q
NP mq
NP m
f a
g
Enhanced light-matter coupling
•Optical sensing / spectroscopy•Nanoscale nonlinear optics •Field concentrators•Nanoscale light sources•Solar energy harvesting•Imaging / optical detection•Singular optics Hao, E.; Schatz, et al., J. Chem. Phys, 120, 357 (2004)
Molecular Emission
Chen et al., Sci. Rep., 3, 1505 (2013)
Solar
Atwater & Polman, Nat. Mat.. 9, 205 (2010)
Sensors
Semrock, www.semrock.com
COBRA DANE phased-array radar34,000 antennas, Siberia's Kamchatka Peninsula
Radiation engineering with plasmonic coupled arrays
Near Field Coupling in Clusters
Near-field of one particle induces additional polarization in close particles
Diffractive Coupling between Clusters
-Propagating scattered light from one particle can be scattered by another
d
d
Photonic-plasmonic coupling regimes in arrays
Transverse field 1/rRadiative
2
ˆ ˆ( ) ( )4
jkrk eE r r p r
r
• Propagating;
• Diffraction-limited;
• Transverse;
Radiative vs Non-radiative fields: always think of Hertzian dipoles first
Longitudinal fields 1/r2, 1/r3
ˆ ˆ1 1 3 ( )( )
4
jkrr r p p eE r jk
r r r
• Non-propagative (reactive);
• Sub-wavelength localized;
• Large nanoscale intensity;
Principles of Nano-Optics, L. Novotny, .B. Hecht, Cambridge (2006)
Quasi-static near-field coupling
Heisenberg principle for photons
x xp x k x
1
x
xk
Using evanescent (non-propagative) fields the spatial bandwidth of photons can be dramatically enhanced.
Copyright 2008, Boston University
Theory of Optical Constants
• Why polished aluminum cooking pan is shiny but opaque transmitted light ?• Why glass windows are transparent but weakly reflecting ?• Why all materials behave like metals at high frequencies ?• Why radio-waves do not escape from the atmosphere while satellite waves do ?
Models are needed to explain the (linear) opticalresponse of materials
Optics of conducting media
extJt
DH
0
Bt
BE
extD
extJDjH
For monochromatic waves:
Assuming:EJ
EED
)1(00
EjH eff
jjeff )1(00
Where we defined the: Effective electric permittivity:
jt /
r
,, are generally complex!
We can write:
tjerEtrE )(Re),(
Physical meaning
Where the complex permittivity is defined as:
0 (1 ) i
“Bound” charge current density
Free charge current density
Im( ) Im( ) Re
Both conductivity and susceptibility contribute to the imaginary part of the permittivity:
If the imaginary part of the material coefficients µ or ε is nonzero, the amplitudeof a plane wave will decrease as it propagates through the medium absorption
Complex phenomenological coefficients of a medium are equivalent to a phase difference between P and E (or H and B) and are manifested by absorption
0
2exp expc
zE E i nkz t
Complex refractive indexUsing the ansatz: 0 ( )E E f k r t we can obtain the dispersion relation:
2 2
2( )
c k
complex dielectric function
( ) ( )n n i Where we can define:
Complex refractive index
0
( )n
0
4( ) ( )
Snell’s law refractive index (real dispersion)
Intensity extinction(energy dissipation)
Re( ) ( )ph
g
cv
k n
vk
)(~)(
n
cc
k
Absorption coefficient
General relations
2 ( )n ( )
( ) ' ''
n n i
i
2 2'
'' 2
n
n
2 2
2 2
' '' '
2
' '' '
2
n
Or, equivalently: ( , ) ( ', '')n
are connected and describe the intrinsic optical properties of matter
1
)( in
ck
Both the real and imaginary parts of contribute to the attenuation of the optical field!
)(
Metal Skin Depth)(
inc
k
22
21
1
2
1
2
)(0 )(),( tkzierEtrE
zk
/)/(0 )(),( ztczni eerEtrE
2
c
21 iin
Skin depth – determines the attenuation length of the field in metals!
/20
zeII
For the intensity attenuation:
Useful only when the distances associated with spatial changes of the field are large compared to the mean free path l of the conduction electrons
0)( k
Skin depth data
Element Na Al Cu Ag Au Hg
δ(2eV) 38 13 30 24 31 255
δ(3eV) 42 13 30 29 37 141
δ(4eV) 48 13 29 82 27 115
lBulk 34 16 42 52 42 11*
All the lengths are expressed in nanometers and are measured at RT
*measured at 77K
Copyright 2008, Boston University
The Drude-Sommerfeld (DS) model of free electrons – lossless metals
• Physical picture: Treat the metal as a gas of electrons
2
2
de ma m
dt
rF E Re expt i t E E
e2
2
dm e
dt
rE
( )t e tp r
22
2
dm e
dt
pE
Re expt i t p p 2
2
1e
m
p E
22
2 20 0
11 1 1 p
r
N Ne
m
p
E
ENpP 0
r
p m
Nep
0
2
Plasma frequency:Drude dielectric function – negligible damping
Dipole moment:
Copyright 2008, Boston University
• Only conduction e’s contribute to: r
2
21 p
r
• Last Page
• becomes 1 at high r
Metal becomes transparent
Aluminum
r
n’ n’’
Photon energy (eV)
0
-2
Die
lect
ric f
unct
ions
-4
-6
-8
0 5 10 15
-10
0
-2
-4
-6
-8
-10
ħp
The dielectric function is negative at optical frequencies!
Optical response of simple Drude metals
• becomes 0 at =pr
For most metals, the plasma frequency is in the ultraviolet range (5-15eV)
Copyright 2008, Boston University
Drude Model
Real and imaginary part of the dielectric constant for gold according to the Drude-Sommerfeld free electron model. The blue solid line is the real part, the red, dashed line is the imaginary part. Note the different scales for real and imaginary part.
114
115
10075.1
108.13
s
sp
Copyright 2008, Boston University
General Drude model (with losses)
Optical response of a collection of free electrons
Lorentz model with “clipped” springs: 00 ( 0)K
2
2
2
2 2
2
2 2
1
' 1
''
p
p
p
i
Drude-Sommerfeld model = Lorentz model without springscollisionless gas of free electrons moving against A fixed background of positive ions
20/p Ne m
Free electron plasma frequency:
Density of free electrons
Effective electron mass
Collision timeLow T: impurity, imperfections
RT: electron-phonon scattering
1
l
vF
Fermi velocity
Electron mean free path
Copyright 2008, Boston University
At the plasma frequency electrons undergo longitudinal oscillations. The quantum quasi-particle associated with these oscillations is called a plasmon
:p Natural frequency of oscillation of the free electron “sea”
Due to their longitudinal nature, volume plasmons do not couple to transverse electromagnetic waves, and can only be excited by particle impact.
Longitudinal electron waves can be excited at the plasma frequency!
ED )(0 at 0)( pp
0)()(0 0 EikD p
0 0 if k // E, the wave is longitudinal!
Longitudinal (bulk) waves in metals
Copyright 2008, Boston University
Physical interpretation of plasma frequency
++
+
+
+
++
++
+
+
+
++
+
+- -
--
--
--
-
--
--
-
-
Equilibrium, neutrality
-
Non equilibrium, applied field E
++
+
+
+
++
++
+
+
+
++
+
+- -
--
--
--
-
--
--
-
--
- +
E
Homogeneous electron density
Non-homogeneous electron density
N
N N
0
e NE
Induced electric field:
u eE
t m
Continuity equation:
Nu
t N
Equation of motion:2
22
0p
N N
t N N
Plasma oscillationsCollective oscillations of the electron gas
Copyright 2008, Boston University
Plasma frequencyN (cm-3) ωp λp Type of
plasma
1024 5.7 x 1016 33 nm metals
1022 5.7 x 1015 330 nm metals
1020 5.7 x 1014 3.3 µm Doped semiconductors
1018 5.7 x 1013 33 µm Doped semiconductors
1017 1.8 x 1013 105 µm Doped semiconductors
1016 5.7 x 1012 330 µm Doped semiconductors
106 5.7 x 107 33 m Ionosphere
105 1.8 x 107 105 m Ionosphere
p
Opaque Transparent
Copyright 2008, Boston University
Aluminum
2
2
2
3
' 1
''
p
p
( )
Identical to the high frequency limit of the Lorentz model!
Plasma frequencies of metals are in the VIS and UV
3 eV 20 eVp
p
Opaque Transparent
Luca Dal Negro, ECE Department, Boston University
The resonant medium: Lorentz model
NpP
exp
Dipole moment
Polarization density
Lorentz model of materials
K
,m e“There is not a single granule of light which is not the fruit of an oscillating charge” A. Lorentz
Copyright 2008, Boston University
The Lorentz model
Lorentz model of materials
K
,m e
i tc
i tloc c
x x e
E E e
Ansatz:
2 20
( / ) cc
e m Ex
i
20 /
/
K m
b m
Complex representation of the real time-harmonic quantities
2
2 loc
d x dxm b Kx eE
dt dt
Driving force
General solution: transient + oscillatory
Most general classical response model
The proportionality function between the field and the electron displacement is complex!
Copyright 2008, Boston University
ConsequencesIf 0 the proportionality factor between F and x is complex
( / )ix Ae eE m where:
1/ 222 2 2 20
12 2
0
1
tan
A
A
0 In-phase response
0
0 180° out of phase response
0
Max(A) occurs at
FWHM
Max(A)
0
)(for 0
/1
Copyright 2008, Boston University
Collection of oscillators
Induced dipole moment of an oscillator is: p exThe polarization of a medium containing N oscillators per unit volume is: P Np Nex
2
02 20
pP Ei
Plasma frequency: 2
2
0p
Ne
m
Particular example of the constitutive
relation: 0P E
2
2 20
1 1 p
i
0L
Copyright 2008, Boston University
0
0 for 0
1 for 0L
n
n
n
Copyright 2008, Boston University
Dielectric function of Lorentz oscillators
2
2 20
1 1 p
i
' ''i
2 2 20
22 2 2 20
2
22 2 2 20
' 1
''
p
p
1
L
0P E
Longitudinal Waves in Matter
0
1
P E
Longitudinal waves:
0
1( ) 0L E P
0
0)(
01
0
B
EkH
D
Pure polarization waves (not EM!)
ED )(0 at 0)( pp
0)()(0 0 EikD p
0 0 if k // E, the wave is longitudinal!
Klingshirn, Semiconductor opticsSpringer Verlag (2004)
Polarization mechanisms in dielectrics
Debye relaxation: alignment of permanent dipoles along E against thermal buffeting
Lattice vibrationsvibrational oscillators
Electronic transitionselectronic oscillators
ECE Department, Boston University
Stop band (Reststrahlbande)
0
0 for 0
1 for 0L
n
n
n
No wave propagation possible exponentially decaying wave amplitude
In this range matter is optically thinner than vacuum (n<<1) !!
cr
1 2n n
vacuum
2
1
sin cr
n
n
For:
2 0 0crn
Total internal reflectionat normal incidence!!
1n
medium
Copyright 2008, Boston University
Important remarks
For metals:
00
T
LP
Surface polaritons in metals (called surface-plasmon polaritons) can be excited in the wide range:
P 0
Metals are ideal materials for broadband engineering of surface-polariton waves
Copyright 2008, Boston University
Free and bound electrons in metals: Ag
Silver cannot be explained within the simple Drude model, why?
Bound charge effect : 4eV plasma frequency for Ag, 2 eV for Cu
2 2
2 2 21 pe pj
je j ji i
Free electrons Bound electrons
Bound (d electrons) and free charges are competing:• shift (decrease) of plasma frequency• extra reflectance peaks
To describe noble metals (Au,Ag, Cu, etc) in the visible, more general Lorentz-Drude models are needed!
Scattering losses/heat Radiation damping
ECE Department, Boston University
Mixed excitations in a solid: polaritons
0 ( ) 1P E The electric field in matter is always accompanied by polarization waves
Waves traveling in solids are always a mixture of an electromagnetic wave and a mechanical polarization wave (as long as
( ) deviates from 1)
Light in matter: mixture of photons and quanta of the polarization field. The mixed state is quantized in polaritons
2 2
2( )
c k
Classical Polaritons equationFrom:
)( c
k
Material responseOptical response
( ) 1
ECE Department, Boston University
Dispersion of waves
The fact that the eigenfrequency ω0 of some excitations of a solid depends on k is called “spatial dispersion”.
The term “dispersion relation” means the relation E(k) or ω(k) for all wave-like excitations. It can be a simple horizontal-line, a linear or parabolic relation, or something
more complicated. Every excitation which has a wave-like character has a dispersion relation.The detailed shape of ω(k) depends on the physical nature of the oscillators and the coupling mechanism.
The term “spatial dispersion” means that the eigenfrequency ω0 of one of the elementary excitations in a solid depends on k
The term “spatial dispersion” means that the eigenfrequency ω0 of one of the elementary excitations in a solid depends on k
Bulk plasmon: dispersion relation (Drude metal)
Dispersion relation for
bulk plasmons
2
22 2
,,r
tt
c
E r
E rt
• The wave equation is given by:
• For a propagating wave solution: , Re , expt i i t E r E r k r
2 2 2r c k
2
21 p
r
• Dielectric constant:
22 2 2 2 2
21 p
p c k
k
L p
No allowed propagating modes into the metal (imaginary k)
ck
2 2 2p c k
• Dispersion relation:
Note1: Solutions lie above light line
Note2: Metals: ħp 10 eV; Semiconductors ħp < 0.5 eV (depending on dopant conc.)
n2
Copyright 2008, Boston University
Volume (bulk) Plasmons: Drude dispersion
Bulk Polaritons Dispersion
2 2
2 2 20
b
c k f
i
Using the general Lorentz model (or Kramers-Heisenberg) dielectric function we can write the polariton equation as:
Implicit representation of ( )k for polaritons.
Re k
Photon-like
0
LGap
LPB
UPB
Phonon/Exciton-like
Simple example: uncoupled oscillators at vanishing damping
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