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```Surface Plasmons
Surface plasmons: outline
1. Time-line of major discoveries
2. Surface plasmons - surface mode of
electromagnetic waves on a metal
surface
3. Spectroscopy of SPs in nanostructures:
(a) Nanoparticles
(b) Gratings, nanostructures
4. Applications: sensors, nanophotonics,
surface enhanced Raman spectroscopy
(SERS)
Surface plasmons, A. Kolomenski, S. Peng, 9/24/2012
Time line
19931991
1974
1968
SPs allow to localize and guide
EM waves!!!
Nanoplasmonics, extraordinary
transmission, etc.
First biosensor on SPs
Surface Enhaced Raman Spectroscopy
Excitation of SPs with a
prism: Raether, Kretschmann
1941
Fano: role of surface
waves, surface plasmons
1907
Rayleigh’s explanation (anglediffraction orders)
1902
Wood anomalies: reflection
on gratings (two types)
Surface plasmons, A. Kolomenski, S. Peng, 9/24/2012
Maxwell’s equations (SI units) in a material,
differential form
 f density of introduced
charges in the medium
J f density of currents introduced in the
medium
Surface plasmons, A. Kolomenski, S. Peng, 9/24/2012
Wave equation
0



2

2

  (  B )  (  B )   B   B
2


 


1  
1 2 B
  ( E )  (  E )  (  2
B)   2
t
t
t
c t
c t 2
Double vector product rule is used
a x b x c = (ac) b - (ab) c

2

1  B
 B 2
0
2
c t

2

1  E
2 E  2
0
2
c t
Surface plasmons, A. Kolomenski, S. Peng, 9/24/2012
Plane waves
Thus, we seek the
solutions of the form:


 


 
B  B 0 Exp[i ( k  r  t )]
E  E 0 Exp[i( k  r  t )]


From Maxwell’s equations
one can see that

 
 B  i k  B


E
B

k

is parallel to E

 E  i k  E
is parallel to
B
Surface plasmons, A. Kolomenski, S. Peng, 9/24/2012
Simple system of a metal bordering a
dielectric with incident plane wave
Dielectric, refractive index n 
 2 is dielectric permittivity
2
Incident light
Reflected light
Transmitted light
Metal (gold) 1   r  i im
Surface plasmons, A. Kolomenski, S. Peng,
9/24/2012
Waves at the interface
k1 x  k 1 z 
2
In medium 1, z<0,
2
2
c1
z
y
2
x
Assume that incident light is p-polarized, which means that the E-vector is
parallel to the incidence plane

E1  ( E1x ,0, E1z ) Exp[i (k1x x  k1z z  t )]
Then the vector of the magnetic field is perpendicular to the incidence plane and

has the form
E
E

B1
1z
 (0, B1 y ,0) Exp[i ( k1x x  k1z z  t )]
In medium 2, z>0, k 2 x  k 2 z 
2
2
2
c2
2

E 2  ( E 2 x ,0, E 2 z ) Exp[i ( k 2 x x  k 2 z z  t )]

B2
 (0, B2 y ,0) Exp[i (k2 x x  k2 z z  t )]
Surface plasmons, A. Kolomenski, S. Peng,
9/24/2012
1
E1x
x
Boundary conditions
z
dl
y
B1 y / 1  B2 y / 2 , i.e. H1 y  H 2 y
x
Stokes's theorem
 ( Bt /  )dl   B1 / 1dl1   B2 / 2dl2  l1( B1 y / 1  B2 y / 2 )   (  B t /  )ds   ds 
l1
l2
E1x  E2 x
s  0
Stokes's theorem
 Edl   E1dl1   E2dl2  l1( E1x  E2 x )     Eds    dsi
l1
l2
Bi
0
t
s  0
1E1z   2 E2 z
Gauss’s theorem
  Eds   1E1ds   E2ds  S1(1E1z   2 E2 z )     ( E )dv   0
S
S1
S2
E
0
t
V
Surface plasmons, A. Kolomenski, S. Peng,
9/24/2012
V 0
Relations in an E-M wave

xˆ


 A 
x
Ax
the curl operator
yˆ

y
Ay
zˆ

z
Az


Exp(ikr )  Exp(ik x x  ik y y  ik z z )  ik x Exp(ikr )
x
x
 i[k  E ]  i B
 i[k  B /  ]  i E
Ex  
1

[k  B /  ] x  
1

( k y Bz  k z B y ) 
Surface plasmons, A. Kolomenski, S. Peng,
9/24/2012
kz By

Derivation of the dispersion equation
Assume no external currents or free charges, 1  2  0 magnetic permeability.
One boundary condition is
H1y  H 2 y
From the other condition
E1x  E2 x =>  H1 y   H 2 y
1
2
k1z
k2 z
H1y  H 2 y  0
k1z
k2z
H1y 
H2y  0
1
2
Therefore we have a system of 2 homogeneous equations and
a nontrivial solution is possible only if the determinant of this
system is equal to 0.
1
1
k1z k 2 z
k
k
D0  1z  2 z 

0
1
2
1
Surface plasmons, A. Kolomenski, S. Peng,
9/24/2012
2
Surface plasmon dispersion equation
k1z
1

k2 z
2
We square both sides
 (
2
1
2
c2
2  k )   (
2
2
2
2
c2
1  k 2 )
We introduce k  k x , wavenumber of the surface plasmon along the
propagation direction, then we obtain
2
( 22  12 )k 2  ( 221  12 2 ) 2 
c
2
 1 2

2
k  2
c 1   2
Surface plasmons, A. Kolomenski, S. Peng, 9/24/2012
Dispersion equation and properties of
surface plasmons
We would like to have a solution which is localized to the surface, i.e. it
decays with distance from on both sides from the interface.
This is possible, if
z 
Exp[ik1z z ]  0
z 
Exp[ik 2 z z ]  0
k1z 
k2z 
2
c
2
2
c
2
 2  k 2  iq1 ,
q1  0
 1  k 2  iq 2 ,
q2  0
Indeed, then we have waves localized near the interface
z 
Exp[ik1z z]  Exp[i(iq1z )]  Exp[q1z ] 
0
z 
Exp[ik2 z z]  Exp[i(iq2 z )]  Exp[q2 z ] 0
Surface plasmons, A. Kolomenski, S. Peng, 9/24/2012
Dispersion equation analysis
k1z
1

k2 z
2

This is only possible, if
q1 q2

and q1, q2  0
1  2
1  0 or  2  0 1  0 and  2  0 (dielectric)
If we look again at the dispersion equation
2
 1 2

2
k  2
c 1   2
,k must be real (propagating wave!), then with 1  0 or  2  0
, we see that the condition for surface waves to exist is
1   2  0, i.e.  1   2
Surface plasmons, A. Kolomenski, S. Peng, 9/24/2012
Relation of Plasmonics to SOME other fields
Electronics
Opto-electronics
Optics
SERS
High harmonics generator
coherent control
imaging
Nanotechnology
nanostructures
nanophotonics
nanoantennas
Plasmonics
Biotechnology
Metamaterials
molecular interactions
nano-sensors
proteomics
The Growth of the Field of Surface Plasmons
illustrated by the
number of scientific
articles published
annually containing
the phrase “surface
plasmon” in either
the title or abstract
PIETER G. KIK and MARK L. BRONGERSMA SURFACE PLASMON NANOPHOTONICS, (2007)
Surface plasmons (or surface plasmon
polaritons), Part 2: outline
1. Why SP named so?
2. Excitation of SPs: with a prism or a
grating
3. Spectroscopy of SPs in nanostructures:
(a) Nanoparticles
(b) Gratings, nanostructures
4. Applications: sensors, nanophotonics,
surface enhanced Raman spectroscopy
(SERS)
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Dielectric constant of a metal, Drude model
d 2x


eE
,
E
~
E
exp
(
i

t
)
0
dt 2
eE0
then x ~ x0exp(it )  x0 
2
me
me
D   0 E  P   r 0 E
N
eE0
i 1
me
P    exi   Nex0
Consequently,
Ne
2
 2p
2

Ne2 E0
me
2
2
Ne
r  1

1

,
where


p
2
2
 0me
 0me

Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
plasmon frequency
Remarks to Drude’s formula
Bound electrons should be taken into account, then 1->  b ,
which takes into account the contribution of bound electrons.
Also the mass of electron should be replaced with
* .
the effective mass of electron in the metal, me
1   b 
 2p
, where  p 
Ne2

 0me*
Plasmons correspond to   0 , these are eigen (free)
oscillations of the electronic plasma.
2
Influence of attenuation
d 2x
dx
m 2  mg
 eE0 e it
dt
dt
For g << p:
 2p
 2p
1 '  1  2 , 1 "  3 g


Surface Plasmons, Part 2,
A. Kolomenski, 9/26/2012
Electrons oscillating in the SP field
dielectric
Interface
     
metal
There is a longitudinal component in the electric field of SP, because E-M field is
coupled to oscillations of the electronic density (plasmonic oscillations).
This is why to excite SPs one needs a p-polarization of the incident light.
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Graphing dispersion equation of SPs
1/2
 1   2 
  ck 



 1 2 
1   b 

2
, where  p 
Ne2
 0me*
 ck
,
Light line: 
 2p
For excitation of SPs we need
to slow down light!
(
,k
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Resonance excitation with a prism

ck /(  2 sin  )
ck
p
SP
0
ck /  2
k
ksp
Conditions for the Surface Plasmon Resonance (SPR): phase matching!!!
Energy conservation
 Light   SP
Momentum conservation
2n

sin   k sp ( )
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Surface plasmon excitation:
Coupling of light to SPs with a prism
0: critical angle
incident
laser
beam
reflected
beam
Optical arrangement used to
excite the surface-plasmon wave
based on the KretschmannRaether configuration where a
thin metal film is sandwiched
between the prism and the
sample.
prism (n0)
metal film (n1)
sample (n2)
SPW
evanescent
wave
E. Kretschmann, Z. Phys. 241, 313-324 (1971).
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
SPR curves for different wavelengths
Gold film (d=47nm) contacting water
=1230 nm
REFLECTION COEFFICIENT
1.0
=633 nm
0.8
0.6
0.4
=490 nm
0.2
0.0
50
60
70
80
INCIDENCE ANGLE (deg)
90
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Surface Plasmon Part 3
Graphing dispersion equation of SPs

 p2

1/2
 b  2  2
 1   2 

  ck 
 ck 


 p2 
 1 2 
  b  2 2

 

1/2








1/2
 

 p2 
  b 
2 
2

 
  

,k  
2
c 


p


 b  2  2 





1   b 
 p2

2
where  p 
,
Ne2
 0me*


 p2
p
p


k   , when  b  2   2   0; then m 
.For  b   2  1 we have m 
1/2


2

( b   2 )


2
  m
,
Light line:   ck /
For excitation of SPs we need
to slow down light!
(
,k
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Approximation of small losses
R  1
4(i1  i 2 ) rad
2
[ k  ( k p  k p )]  (i1  i 2  rad )
2
k p  ( 2n p /  0 ) the metal film is infinitely thick
n p  [ 1r  2r / ( 1r   2r )]1/ 2
k p describes the correction due to finite thickness
i1,2 internal losses in the film and in the medium
A. Kolomenski et al., Applied Optics, Vol. 48, 5683-5691 (2009)
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
The influence of the thickness of the gold film
on the properties of SPs
Air
Gold
Lsp  (  0 k 0 cos res  ) -1
Glass
1.0
Attenuation length (m)
0.9
0.8
Reflectivity
180
(a)
0.7
0.6
0.5
10 nm
20 nm
50 nm
80 nm
120 nm
0.4
0.3
(b)
=633 nm
=805 nm
160
140
120
100
80
60
40
20
0
0.2
40
42
44
46
Incidence angle (deg)
48
50
20
40
60
80
100
120
140
Film thickness (nm)
(a) SP resonance curves at 633 nm for different film thicknesses.
(b) The dependence of the attenuation length on the film thickness for 633 nm and
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Examples: changes in the flow cell, biomolecular binding reactions
Example: binding of monoclonal antibody to
0.45
C=0%
C=0.82%
0.40
0.64 deg
0.35
0.30
70.50 70.75 71.00 71.25 71.50
SPR angle (pixels)
0.50
550
A
500
B
450
400
B
NHS/EDC
HRP
350
300 B
250
0
B
10
20
30
40
Time (min)
INCIDENCE ANGLE (deg)
A. A. Kolomenskii, P. D. Gershon, and H. A. Schuessler, Applied Optics 36, 6539-6547 (1997).
Applied this sensing technique to myofibers and tubulin molecule.
50
60
Sensitivity and detection limit
(relationships between different quantities)
angular resolution -4deg=2 RU
changes of the refractive index n-6
average thickness of the protein layer d=0.03 Å
surface concentration d=3 pg/mm2
with mprotein=24 Da surface concentration of molecules ns=1010 cm-2
A. A. Kolomenskii, P. D. Gershon, and H. A. Schuessler, Applied Optics 36, 6539-6547 (1997).
Attenuation lengths of SPs for gold and silver films in
contact with air, calculated for a broad spectral range
Au
1000
100
exact,  from [1]
10
approximate,  from [1]
exact,  from [2]
exact,  from [3]
Attenuation length (m)
Attenuation length (m)
1000
Ag
100
exact,  from [1]
10
1
600 800 1000 1200 1400 1600 1800 2000 2200 2400
Wavelength (nm)
approximate,  from [1]
exact,  from [2]
exact,  from [3]
1
600 800 1000 1200 1400 1600 1800 2000 2200 2400
Wavelength (nm)
1. American Institute of Physics Handbook, D. E. Gray, ed. (McGraw-Hill, 1972), p.
105.
2. U. Schröder, Surf. Sci. 102, 118-130 (1981).
3. Handbook of Optical Constants of Solids, E.D. Palik, ed. (Academic1985).
A. Kolomenski et al., Applied Optics, Vol. 48, 5683-5691 (2009)
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Summary of surface plasmons
 2  dielectricPropagating wave with k 2   2
~ exp(| k 2z | z )
Z
E
 1  metal
~ exp(| k1z | z)
Condition of existence:
x
1 2
c 2 1   2
 2p
Approximat ion of free electrons: 1   2 ,
b 
ne 2 
plasmon frequency; 1  0   < p
p 
 0me
Re( 1 )   2
SPs:
•Spatially localized to the surface E-M wave
•Oscillations of the electronic density.
•Have E -longitudinal component
•Are excited with p-polarized light and the local
field can significantly exceed the field in the
exciting beam.
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Dependence of the near field intensity enhancement factor on the back side
of the gold film
vs. the angle for two wavelengths 633 nm and 805
nm (dashed lines – smooth surface, solid lines – surface with roughness)
110
633nm
633nm with1,eff.
100
90
805nm
805nm with1,eff.
|t012()|
2
80
70
60
50
40
30
20
10
0
42
43
44
45
46
Angle  (deg.)
A. Kolomenski et al., Applied Optics, Vol. 48, 5683-5691 (2009)
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
SP resonance: coupling with a grating
(conservation of momentum)
ki
ki
θ
θ
grating
kSP
ki sin(θ)
kg
kSP = ki sin(θ) + kg
+1 order coupling
kSP
kg
ki sin(θ)
kSP = ki sin(θ) - kg
-1 order coupling
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Conditions for the resonance excitation of SPs with a
grating
,
Light line
  ( c / n )k
SP dispersion
curve
required
momentum
Light line, suited
for resonance excitation
0  frequency of the source
The crossing of the SP curve and the light
line means resonance excitation for
desired frequency 0
,k
SPs are slower than light, and therefore for the same frequency their momentum is larger.
To enable the resonance excitation additional momentum must be provided.
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Conditions for the resonance
excitation of SPs
Conditions for the resonance excitation of SPs:
a photon is converted into a surface plasmon.
kx
General laws must be observed:
kz
(1) Energy conservation,
k SP  k x
h light  (h / 2 )(2 )  hlight  hSP  light  SP
(2) Momentum conservation,
k z is changing
k x is not changing
hk x,light  hkSP  k x,light  kSP
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Schematic of experiment on spectroscopy of SP
modes in nanostructures :transmission
measurements in the far field
This setup maps intensity distribution
over angle and wavelength and thus reveals
SP modes that affect transmission.
Charge
Coupled
Device
(CCD)
λ
θ
Laser
beam
Grating
Sample
(nanostructure)
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Study of the Interaction of 7 fs Rainbow Laser Pulses with Gold
Nanostructure Grating: Coupling to Surface Plasmons
Transmission dependence
AFM image of the nanostructure:
5°
Angle of
Incidence
Intensity
0°
-5°
650
Wavelength (nm)
800
The valley area (x-structure) the laser light is
efficiently converted into SPs, about 80% .
A. Kolomenskii et al., Optics Express, 19, 6587-6598 (2011).
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Mie theory and dipole approximation
t=0
Ionic cluster
t=T/2
Electric field
Electronic plasma
oscillations
Light
Electronic cluster
For small nanoparticles (R<<, or roughly 2R< /10): dipole
approximation
i ()

ext ()  9 3m/ 2V
c
[ r ()  2 d )]2  i 2 ()
where V is the particle volume,  frequency light, εm and  ( )   r ( )  i i ( )  0
are the dielectric functions of the surrounding medium and the particle
material.
When  2 () is small or varies slowly, the resonance takes place
at
p
 2p
max 
=>
 r ( )  (2 d )  0,  r  1 
2
1  2 d
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Extinction spectra of Ag n-particles
in solution
Extinction (a.u.)
1.2
1.0
0.8
0.6
0.4
Ag 27 nm particles
Ag 48 nm particles
0.2
0.0
350
375
400
425
450
475
Wavelength, nm
The oscillations of a n-particle, induced by a pump pulse, modulate (displace) the plasmon
absorption band. For efficient detection the probe wavelength was selected at the steeper
portion of the slope of this band.
S. N. Jerebtsov et al. Phys. Rev. B Vol. 76, 184301 (2007).
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Bowtie nano-antenna and measured
intensity enhancement
Intensity enhancement vs wavelength
Fabricated by Electron Beam Lithography
(EBL) bowtie antennas. Indium tin oxide
substrate. Gap was varied, thickness 20 nm.
3D finite difference time domain (FDTD)
simulations
Kino et al. In: Surface Plasmon nanophotonics, p.125 (2007).
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Experimental setup for study of “hot spots” for SERS
Raman signals from individual Ag n-particles
Futamata et al. Vibrational Spectroscopy 35, 121-129 (2004).
Raman microscope with sensitive CCD cameras for imaging the sample in scattering
and using Raman signal. Notch filters were used to suppress the excitation light. Low
concentration of n-particles needed to separate individual particles.
Raman spectroscopy
Photon scattering on molecules
Elastic or
Rayleigh scattering
Inelastic or
Raman scattering
Stocks
h
h( - )
Anti-Stocks
h( + )
h
Raman scattering increases when h produces electronic transition
Surface Plasmons, Part 2, A. Kolomenski, 9/26/2012
Surface Enhanced Raman Spectroscopy (SERS) of DNA bases
Futamata et al. Vibrational Spectroscopy 35, 121-129 (2004).
Spectra of
individual
n-particles
Characteristic stretching modes in heterocycles suited
for DNA sequencing :
adenine 718 and 893 cm-1;guanine 641cm-1;
cytosine 791 cm-1; thymine 616, 743 and 807 cm-1.
Time evolution (whole scale 1 s) demonstrates
Raman peaks and blinking effect, known
for single molecule detection.
Stongest enhancement ~1010 from pairs
of particles with axis parallel to polarization
Energy level diagram for sum-frequency generation
(SFG), difference-frequency generation (DFG), and
• The lower inset is
photograph of Raman
sidebands by time
delayed linear chirped
pulses
• Positive chirp(below)
• Negative chirp opposite
Beam crossing setup, BS, beam splitter
Raman sideband generation with optical vortices
Line Profiles of Spectral Lines
Full width
at half
maximum
Natural Linewidth
Solution:
Note: Damping of molecular oscillators is
very small
From the Fourier Transform:
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