Transcript Surface Plasmon Theory - Dr. Amir Reza Sadrolhosseini

Surface Plasmon Resonance
Dr. Amir Reza Sadrolhosseini
Dr. A.S.M Noor
The SPR is an optical- electrical phenomenon that due to a charge density oscillation at the interface between a dielectric media
and a metal layer .
Detector
Medium
Sensing Layer
Cladding
Diode
Laser
Core
Detector
Cladding
Metal Layer
1- Refractive index of prism
2-Refractive index and thickness of gold layer
3- Refractive index or thickness of medium
Main question:
What is the relationship between Reflectivity and angle of
incident at the interface?
What is condition of resonance?
How can we determine the thickness and refractive index
of layers?
How can we determine the resonance angle shift?
-Reminder
-Theory
1-Oscillator Model
2-Plasma Osillation and Plasmons
3-Surface Plasmons condition
4- Matrix metodes
-Surface plasmon sensor
-Data Analysis and simulation with matlab
-Condition of sensing layer
Reminder
Maxwell equation:
 D0
 H 0
1 H
 E  
c t
1 D
 H 
c t
D E
Transverse Wave equation
 E
2
i ( q.r t )
E  E0e
 
2
c t
2
2
E
D is displacement field
H is magnetic field
Reminder
     i 
 ( )  2 ( )
 ( ) 

2cn( )
Dielectric Function
Absorption coefficient=2×Extinction coefficient
 ( )
n  n()  ik ()
1
n( ) 
   ( )2  ( )2 

2
  n  k
2
2
Refractive index
k ( ) 
   2nk
1
   ( )2  ( )2 

2
1- Oscillator Model
This model describe the medium and we can obtain absorption
coefficient and refractive index.
We assume that the Crystal consists of charges which can be set in motion an
oscillating electric field of light
.
From Newton´s second law ,we can write equation of motion :
(1)
d 2 x me dx(t )
me 2 
 eE(t )
 dt
d t
For monochromatic field
E  E0 e
it
x  x0 e
it
(2)
x0 
Pc
Pb
e
  i
E0 2 2
me
  1
(3)
polarization density of the conduction electron
ne2
  i
Pc  nex0 (t )  
E 2 2
me    1
(4)
Induce polarization (background polarization )
This background polarization is due to the displacement of bound particles.
TOTAL Polarization is
Susceptibility :
Displacement field:
P  Pb  Pc
P

E
D  E  4P
D  E
Relation between susceptibility and dielectric function
Pb  Pc
  1  4    1  4
E
(5)
substituting
If
Pc in Eq(5) :
   then
4Pb 4ne2   i
  1

 2 2
E
me    1
4Pb
  1
E
4 ne 
4 ne 
   

i
2 2
2 2
me (   1)  me (   1)
2 2
     i 
2
• IF
  1
Low-Frequency
4 ne2
1
4 ne2 2
1
 2 2

 2 2

me    1
me
  1
4 ne2

i
2 2
 me (   1)
4 ne2

i  i 
 me
In This Regime Material Highly Absorptive
Absorption coefficient is:


c
2 
4 ne2
  
 me
 8 ne 

c
me
2
4 ne2 2
4 ne2
   

i
2 2
2 2
me (   1)  me (   1)
• IF
  1
High-Frequency
4 ne2 2
1
4 ne2
1
2 2
 2 2

 2 2
    ( )   4 ne   1
me
  1
me    1
m
 2 2  1
e
4e n
   
me 2
2
    (1 
 p2

2
)
2
4

e
n
2
p 
  me
Plasma frequency
Plasma Frequency is the frequency of collective oscillation of the electron .
 p2
  p  2  1

 p2
  p  2  1

 0
 0
Highly Transparent
Highly Reflecting
Gold
2.183×1015 Hz
Silver
2.18×1015 Hz
300 nm
700 nm
9.993×1014Hz
4.282×1014Hz
 p2
  p  2  1

What is the Plasmon?
n  n0  n1
n1Density at time t
d 2 n1
2


p n1  0
2
dt
Physically, plasma frequency is the frequency of collective
oscillation of electron gas( plasma)
If the electron density n oscillates at plasma frequency,
collective excitations will be occurred.
This oscillation is plasma oscillation and is longitudinal.
The quantum of plasma oscillation is plasmon
or
Quantum of collective electron density oscillation
Eplasmon  p
Plasmons may be exited
for example by inelastic electron scattering
Prism coupling
Grating coupling
Waveguide coupling
2-Plasma Osillations and Plasmons
Mean density of free electron plasma is n0
At time t the electron number density changes slightly from the mean value
n1 is small
nn n
0
Continuity equation:
Velocity at t time:
1

J 
0
t
v  v0  v1  v1
Charge Density:
  en
Current Density:
J  env
dn1
  n0 v1 
0
dt
  E  4
 2 n1
v1
differentiating

n


0
0
2
t
t
d 2 n1
2


p n1  0
2
dt
v1
me
 eE
t
d 2 n1
2


p n1  0
2
dt
The electron density oscillates at the plasma frequency.
The quantum of the plasma oscillation is plasmon.
Plasmons may be exited
for example by inelastic electron scattering
Prism coupling
Grating coupling
Waveguide coupling
3-Surface Plasmons Theory
Surface Plasma waves:
 PRISM  0
Glass
 METAL  0
Metal
medium
Surface Plasmon Wave
The interface between a medium with a positive dielectric constant   0
and a medium with negative dielectric constant   0 such as metals, can
give rise to special propagation electromagnetic waves called surface
plasma waves.
SPR (%90M %10P)
0.9
0.8
0.7
0.6
%R
0.5
0.4
0.3
0.2
0.1
0
30
35
40
45
50
Angle
55
60
65
70
In this case the Energy and
momentum conservation is
satisfied and the wave vector of
light is increased
z
 0
ε0
x
y
z0
  i ( kxt )  z
E  E1e
e 1
  i ( kxt )  z
E  E2 e
e 2


E1  E1 ( A,0, B)


E2  E2 (C,0, D)
z0

 i ( kxt )  z
H  H 1e
e 1


H1  H1 (0, H y ,0)
z0

 i ( kxt )  z
H  H 2e
e 1


H 2  H 2 (0, H y ,0)
Electric Field: z  0
Magnetic Field:
Boundary condition :
E1t  E 2t
 1 E1n   2 E2 n  0

  dE
 d
2
 E 2
  2 ( Ex , E y , Ez )  2 ( Ex , E y , Ez )
c dt
c dt

 k 
2
1
2
1
c
2


 k 
2
2
2
2
2
c
2

2
  E
H y
H y
H y  E x
 
 H 
 (
,0,
)
( E x ,0, E z )  

c t
z
x
c t
z
c t
(k 2 
1
c
1
2
2

)


(
k

2
2
2
c
2
1
2
) 1
Dispersion Relation for surface plasmons
k
2
plasmon
 2  1   2 
 2 

c  1   2 
k
2

nSin  res


1
 1



2
2
   k c
Glass
Metal
medium
air
The surface plasmon and
the photon dispersion curves
do not cross each other
anywhere
Cladding
Diode
Laser
Core
Detector
Cladding
Metal Layer
k glass sin  i 
n12  n22
n p Sin R 
n12  n22

c
In this case the Energy and
momentum conservation is
satisfied and the wave
vector of light is increased
n glass sin  i  k plasmon
n2 
n12  n p sin 2  p
n12 n 2p sin  p
Matrix Methods For Simulation and analysis the SPR Signal
0
n0
a
Ba  n0  0 0 ( E0  Er1 )  n1  0 0 ( Et1  Ei1 )
b
Bb  n1  0 0 ( Ei 2  Er 2 )  n2  0 0 Et1
Ea  ( E0  Er1 ) cos 0  ( Et1  Ei1 ) cos t1
Eb  ( Ei 2  Er 2 ) cos t1  Et 2cost 2
Ei 2  Et1e i
Ei1  Er 2 e i
Bi1  Br 2 e i
Bi 2  Bt1e i

2

B 
c
tn1 cos  t1  N 
n
E
c
1
t
1
t
ei
sin  ) Bb
1
Ba  (1iei sin  )Eb  (ei cos  )Bb
Ea  (e
 i
cos  ) Eb  (i
2
 Ea   m11 m12   Eb 
 B   m
 B 
m
22   b 
 a   21
 0 0
nN
cos t N
n1
n2
 0 0
Matrix of layer

cos 

M

 i 1 sin 
sin  
1 

cos  
i
 Ea 
 B   M1M 2 M 3 .....M N
 a
 Eb 
B 
 b
1
2
3
4
M  M1M 2 M 3.............M N
N
 m11 m12 
M 

m
m
 21
22 
 ( E0  Er1 ) cos 0   m11 m12   Et2 cos t1 





 n0  0 0 ( E0  Er1 )   m21 m22   n2  0 0 Et2 
r
r
Er1
m21  m22 2  m11 0  m12 2 0
m21  m22 2  m11 0  m12 2 0
E0
 n

r
 n
A
 
  i   n cos 
B

 
0 2
 0 0  cos t  i  n12 cos t  0  cos 2 t  0 0
0 2
 0 0  cos t
2
C
2
2
2
1
t2 0
1
 cos 2 t1
D
0
0
R  rr *
Refractive index
1.331+0.001i
1.331+0.005i
1.331+0.01i
1.331+0.05i
1.331+0.1i
1.331+0.2i
SPR
1
ethanol
0.8
R%
0.6
0.4
0.2
0
30
40
50
Angle
60
70
n=.25+3.21i
er =-10.24
1
ei=1.605
Methanol
0.8
Thickness=52.5 nm
R%
0.6
0.4
0.2
0
30
40
50
Angle
60
70
Resonance angle
52.34
52.364
52.405
52.864
53.523
54.628
Reflectance
0.0252
0.0663
0.1221
0.4140
0.5546
0.6587
The effect of refractive index
Data Processing
•
YC 
1-Centroid Method
this method uses a simple algorithm which
finds the geometric center of the portion of the
SPR dip under a certain threshold. Although
the geometric center does not necessarily
coincide with the minimum of the spectrum, as
SPR
sensing usually relies on relative
measurements, the offset of the geometric
center does not affect the final measurements.
The centroid is calculated as follows:
 x j ( I thresh  I j )
j
 ( I thresh  I j )
2
  res
(P


 
 ( P
B
 P( ))d
1
2
B
 P( ))d
1
j
x j represent
the spectral positions of the contributing intensities I j
PB is the baseline,
I thresh denotes the threshold value.
P ( ) the response from a detector array at the angle of incidence onto the metal film θ
2
2
1
1
A0    ( PB  P( ))d  A1   ( PB  P( ))d

A1
A0
A1, A0 are the area above and
under the base line
Base line : PB  P(1 )  P( 2 ),
 0  1  
a Change in light intensity with proportionality factor
New base line:
a
PB  P(1)  P( 2 ),
 0  1  
 2
 2
1
1
   ( PB  P( ))d    ( PB  P( ))d
P( )  aP( )  b
PB  aPB  b
 2
 2
1
1
   ( PB  P( ))d    ( PB  P( ))d 
 2   2
P(1)  P(1 )
1  1
P( 2 )  P( 2 )
 2
 res
2
( P  P( ))d  ( P  P( ))d



 
 
 ( P  P( ))d  ( P  P( ))d
B
1
B
1
2
2
B
1
  res
B
1
SPR (%90M %10P)
0.012
0.9
0.8
0.01
0.7
0.008
0.6
0.5
%R
0.006
0.4
0.004
0.3
0.2
0.002
0.1
0
40
42
44
46
48
50
52
54
56
58
60
0
30
35
40
45
50
Angle
55
60
65
70
SPR FOR BIODIESEL BASE PALM OIL AND METHANOL
0.9
0.8
0.7
%90M %10P
0.6
%80M %20P
%70M %30P
R%
0.5
%60M %40P
0.4
%50M %50P
0.3
%40M %60P
0.2
%30M %70P
%20M %80P
0.1
%10M %90P
0
30
35
40
45
50
Angle
55
60
65
70
PERCENTAGE OF PALM
OIL
%10
%20
%30
%40
%50
%60
%70
%80
%90
Percentage of methanol
%90
%80
%70
%60
%50
%40
%30
%20
%10
Resonance Angle
54.3235
55.0416
55.8516
56.6699
57.4381
59.2083
59.3586
60.3439
61.4043
Refractive Index
1.349
1.359
1.370
1.381
1.391
1.413
1.415
1.427
1.439