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Nano-Photonics (1)
1
Outline
 Introduction:
 Nature of light:
• Light as electromagnetic waves
• Light as a quantum photons
 Confinement and guidance of light at nano-scale
• Conventional waveguides
• Advanced waveguides
• Surface plasma polariton waveguides


Confinement and resonance of light at nano-scale
Light-matter interaction at nano-scale
2
Introduction to Nano-Photonics
3
What is Nano-Photonics?

Science and technologies that investigate and utilize
phenomena of light confinement, guidance, resonance,
and interaction with nano-structures

Topics:
– Near Field Optics
– Surface Plasma Polariton (SPP) Waveguides and
Resonators
– Dipole Energy Transfer
– Photon-Matter Interaction in Nano-scale
4
Why Nano-Photonics?


Optical wavelength (100nm -- 1000nm) is much larger than
typical dimension of nano-structures and hence interaction
between light and nanostructures is usually very weak, less
understood, and not sufficiently utilized
We need to understand how we may enhance, engineering and
utilize the interaction between light and nano-structures to develop
new optical functionalities on the smallest spatial dimension and
within shortest temporal scale.
5
Nature and Properties of Light
6
Waves Theory
Nature of Light
Huygens (1690)
Light is a wave
traveling through
ether (invisible
medium)
Newton (1704):
Light is a
stream of
corpuscles
Planck (1900):
Radiation theory
included
“quantization” of
energy
Young(1801):
Interference
experiments
supporting the
wave theory
Maxwell (1865):
Light is
electromagnetic
wave.
Particle Theory
Quantum Theory of
Light
Bohr
1913
Bose
(1924)
de Broglie
(1924)
Einstein (1905):
Photoelectric effect
light is a stream of
particles (photons)
7
Maxwell’s Equations
Faraday Law of Induction:
  Er, t   

Br, t 
t
Generalized Ampere’s
Law :

  Hr, t    Dr, t   J r, t 
t
Gauss’s Law for magnetic field :
  Br, t   0
Gauss’s Law for electric field :
  Dr, t   ρr, t 
Er, t  : volts/m
Br, t  : webers/m 2
Dr, t  : coulombs/m 2
Hr, t  : amperes/m
J r, t  : amperes/m 2
ρr, t  : coulombs/m
Conservation Law for electric charge and current:
  J r, t  

ρr, t   0
t
8
3
James Clerk Maxwell
(1831 - 1879)
A Scottish mathematical physicist who is
widely regarded as the nineteenth century
scientist who had the greatest influence on
twentieth century physics. Maxwell
demonstrated that electrical and magnetic
forces are two complementary aspects of
electromagnetism. He showed that
electromagnetic fields travel through space,
in the form of waves, at a constant velocity
of 3.0 × 108 m/s. He also proposed that light
was a form of electromagnetic radiation.
Heinrich Hertz
(1857 - 1894)
A German physicist who was the first to
broadcast and receive radio waves.
Between 1885 and 1889, he produced
electromagnetic waves in the laboratory
and measured their wavelength and
velocity. He showed that the nature of
their reflection and refraction was the
same as those of light, confirming that
light waves are electromagnetic radiation
obeying the Maxwell equations.
9
Maxwell’s Equations and EM Theory

The study of Maxwell’s equations, devised in 1863, to represent
the relationships between electric and magnetic fields in the
presence of electric charges and currents, whether steady or
rapidly fluctuating, in a vacuum or in matter.

The equations represent one of the most elegant and concise
ways to describe the fundamentals of electricity and magnetism.
They pull together earlier results known from the work of Gauss,
Faraday, Ampère, Biot-Savart and others, in a consistent and
unified manner.

Remarkably, Maxwell’s equations are perfectly consistent with
the transformations of special relativity as well as the quantum
field theory of electrodynamics
10
Maxwell’s Equations
in Linear, Isotropic and Stationary Media
Constitutive Relations
Dr, t   εr Er, t 
Br, t   μr Hr, t 
Jr, t   σr Er, t 
Curl Equations
  Er, t    μ

Hr, t 
t

  Hr, t   ε Er, t   σ Er, t 
t
The fields can be expressed in terms of E and H and described by the curl equations
Divergence Equations
  ε Er, t   ρr, t 

  σ Er, t   ρr, t   0
t
  μ Hr, t   0
The electric charge can be obtained as an left-over after E and H are solved.
11
Wave Equation in Free Space


 2E
    E    E   E     B   μ o   H   μ o ε o 2
t
t
t
2
1  2E
 E 2 2  0
c t
2
c
Similarly,
1
μ oεo
1  2H
 H 2
0
2
c t
2
 299,792,458 m/s
Maxwell’s Predictions:
1. Electromagnetic fields may exist in the form of transverse waves
2. Light is electromagnetic in nature
12
A Special Solution: Sinusoidal Traveling Plane Wave
E x z, t   E  cosωt kz
H y z, t   H  cosωt  kz 
General solution
for forward wave
Dispersion
Relation
Phase Velocity
vp 
k  ω μ oεo
Group Velocity
ω
c
k
vg 
Characteristic
Impedance
Zo 
μo
E

 377
H
εo
Field Expressions
  z 
E x z, t   E  cos ω t  
  c 
H y z, t  
E
  z 
cos ω t  
Zo
  c 
ω
c
k
Wavelength v. s. Frequency
λf  c
13
Power Flow and Stored EM Energies
  z 
Ex  z, t   E cos   t   
  c 
Stored Electric Energy Density
2
  z 
1
1
2
we  Ex  z, t    E cos2   t  
2
2
  c 
H y  z, t  
  z 
E
cos   t   
Zo
  c 
Stored Magnetic Energy Density
2
  z 
1
1
2
wm   H y  z, t    H  cos 2   t   
2
2
  c 
Power Flow Density
  z 
S z  Ex H y  E H  cos2   t   
  c 
Relationship between Stored Energies and Power Flow
Sz  v p  we  wo 
Power Flow = Velocity  Total Stored EM Energy
14
Time Average Power Flow and Stored Energies
Stored Electric Energy Density
2
  z 
1
1
2
we  Ex  z, t    E cos2   t  
2
2
  c 
Stored Magnetic Energy Density
2
  z 
1
1
2
wm   H y  z, t    H  cos2   t  
2
2
  c 
T
2
1
1
2
wm    H y  z, t  dt   H 
2
4
0
T
2
1
1
2
we   Ex  z, t  dt   E
2
4
0
Power Flow Density
  z 
S z  Ex H y  E H  cos   t   
  c 
2
T
1
S z   Ex H y dt  E H 
2
0
Relationship between Stored Energies and Power Flow
Sz  vp w
w  we  wo
Power Flow = Velocity  Total Stored EM Energy
15
Particle Nature of Light

Quantum theory describes light (or
more generally EM fields) as a particle
called a photon

According to quantum theory, a
photon has an energy and momentum
given by
E  hf 
hc
λ
p
h hf

λ
c
h= 6.6x10-34 [Js] Planck’s constant

The energy and momentum of the light are proportional to the frequency
(inversely proportional to the wavelength) ! The higher the frequency (lower
wavelength) the higher the energy and momentum of the photon.
One photon of visible light contains about 10^-19 Joules!

Quantum theory describes experiments to astonishing precision, whereas the
classical wave description cannot.
16
Illustration of Particle Characteristics of Light
Photographs taken in dimmer light look grainier.
Very, very dim
Bright
Very dim
Very bright
Dim
Very, very bright
When detecting very weak light, we find that
it is made up of particles.
 is the "photon flux," i.e., number of
photons/sec in a beam of power P
 = P / hf
17
Wave-Particle Duality of EM Fields
Wave
E  hf 
p
hc
λ
Particle
h hf

λ
c
Characterized by:
Characterized by:
 Amplitude
(A)
 Frequency (f)
 Wavelength ()
 Amplitude (A)
 Energy
(E)
 Momentum (p)
Energy of wave: A2
Probability of particle:  A2
18
Momentum and Pressure
  z 
E x z, t   E  cos ω t  
  c 
H y z, t  
Momentum Density
pz 
Sz 1
1
z 
2 

E
H

E
H
cos
ω
t



x y
 

c2 c2
c2
c

 
pz 
ε o 2 2   z 
E  cos ω t  
c
  c 
T
ε
1
p z  2  E x H y dt  o E 2
c 0
2c
pz 
w
c
E
  z 
cos ω t  
Zo
  c 
Maxwell’s Stress Tensor
 ε o E 2x  μ o H 2y
 1
T 
0
2
0

μ0
Ex
 Zo 
Hy
εo
0
 ε o E 2x  μ o H 2y
0



 ε o E 2x  μ o H 2y 
0
0

  z 
T   ε o E 2 cos 2 ω t   zˆ zˆ
  c 
T

1
T   ε o zˆ zˆ  E 2x dt   ε o E 2 zˆ zˆ
2
0

T   w zˆ
z
19
Application of Radiation Pressure:
Optical Tweezers

A low power, continuous wave
laser that is focused through a
high N.A. objective can trap
particles of diameter 10 m.
 Can move the trapped particle
by moving the laser or stage,
hence the laser acts as a
“tweezer” by picking up and
moving an individual particle.
20
Working Principle of Optical Tweezers

Two regimes of operation
– Rayleigh regime (diameter of
particle << )
– Mie regime (diameter of particle >>
)

Two main forces
– Scattering force caused by
reflection of incident beam
– Gradient force caused by the
deflection (transmission gradient
force) of incident beam

Gradient force dominates
scattering force (for
“transparent” particles)
21
Light Propagation in Free Space
22
Wave Equation

From Maxwell’s equations to the wave equation
source less, harmonic

B
t

 H  J 
D
t
 E  
homogeneous medium
 E  j B  j H
 E   2 E
 H   j D   j E
   H   2 H
(  E )   2 E   2 E
coupling term
independent term
Wave in Free Space

Simplified wave equation
sourceless, homogenous
 D    0    E    E   E   E  0   E  0
simplified wave equation
 2 E   2 E  0
from
 B  0   H  0
we have 2 H   2 H  0

Obviously, to manipulate the polarization through coupling among different field
components, the introduction of medium inhomogeneity and/or structure is
necessary

Free space solution of the wave equation – plane harmonic wave
 j ( kr t )
E0 e
 j ( kr t )
H 0e
where
 2
| k |   2  or
| k |  
Wave in Free Space


Dispersion in free space
1/ 
|k |
1/ 

Plane harmonic wave characteristics
 E  0   [ E0e j ( k r t ) ]  e j ( k r t )  E0  ( jk  E0 )e j ( k r t )  0  k  E0  0
for the same reason k  H 0  0
we have k0  E0   /  H0


also from   E  j H   H   jE
k0  H0    /  E0
where
k0  k / | k |
Wave in Free Space

Therefore, the plane harmonic wave in free space has its electric field,
magnetic field, and propagation direction all orthogonal – it is a condition
forced by the electric and magnetic divergence free requirement


| E0 |  /  | H 0 |
E [V/m]
k
free space impedance
H [A/m]

Example of plane wave polarization splitting – sun light scattering by small
particles
Wave in Free Space

Significance of the plane harmonic wave – eigen solution of the wave equation
in free space, i.e., for whatever excitation, after waiting for infinitely long, the
solution at infinitely far distance from the source in free space can only be the
plane harmonic wave

Therefore, regardless of the source distribution, the electromagnetic wave in
free space will become the plane harmonic wave after infinitely long at infinity,
the evolution process is the spatial diffraction that eventually smears out any
initial nonuniformity
Plane Harmonic Wave at Boundary
z
ki
i r
[ E0i e j ( ki r t )  E0 r e j ( kr r t ) ]t  [ E0t e j ( kt r t ) ]t
kr
z=0
t
kt
must hold for any point (x, y) at plane z=0, hence
ki  r |z 0  kr  r |z 0  kt  r |z 0
or
kix  krx  ktx , kiy  k ry  kty the three wave vectors have identical projection
on the boundary plane
Feature 1：The incident wave vector can always be selected in a plane (y=0)
perpendicular to the boundary plane (z=0). As such, the reflected and
refracted wave vectors must be in the same plane (y=0), since kiy=0 requires
kry= kty=0 as well. This plane (y=0) is called the incident plane.
Feature 2: ki sin i  kr sin r  kt sin t or  11 sin i   11 sin r    2 2 sin t
i.e., i  r , 11 sin i   2 2 sin t Snell’s law
Plane Harmonic Wave at Boundary
z
E-field parallel to the boundary (s-wave):
i r
ki
E0i  E0 r  E0t
kr
z=0
t
E0i  E0 r  E0t
kt
E
or
H
Hence
E0 r

E0i
H 0i cos i  H 0 r cos  r  H 0t cos t
1
cos i 
1
1
cos i 
1
1


E0i cos i  1 E0 r cos r  2 E0t cos t
1
1
2
2
cos t
2
2
cos t
2
1
cos i
1
E0t

E0i
1

cos i  2 cos t
1
2
2
Plane Harmonic Wave at Boundary
H-field parallel to the boundary (p-wave):
z
i r
ki
z=0
t
H
Hence
H 0i  H 0 r  H 0t
kr
kt
E0i cos i  E0 r cos  r  E0t cos t
H 0i  H 0 r  H 0t
or
1

2
H 0i cos i  1 H 0 r cos  r 
H cos t
1
1
 2 0t
E
H 0r

H 0i
1
cos i 
1
1
cos i 
1
2
1
cos t
cos i 
2

E
 0r  1
2
E0i
1
cos t
cos i 
2
1
1
cos i

H 0t
1

H 0i
1
2
cos i 
cos t
1
2
2

E0t

E0i
2
cos t
2
2
cos t
2
2
cos i
2
1
2
cos i 
cos t
1
2
2
Dielectric-Dielectric Boundary
For non-magnetic materials 1  2  0
E0 r

E0i
s-wave
E0t

E0i
E0 r

E0i
1 cos i   2 cos t
k  ktz
 iz
kiz  ktz
1 cos i   2 cos t
2 1 cos  i
1 cos i   2 cos  t

2kiz
kiz  ktz
 2 cos i  1 cos t
k /   ktz /  2
 iz 1
kiz / 1  ktz /  2
 2 cos  i  1 cos  t
p-wave
E0t

E0i
2 1 cos i
 2 cos i  1 cos t

2kiz / 1
1
 2 kiz / 1  ktz /  2
Dielectric-Dielectric Boundary
1
E0 r / E0i
s
B
0
p
1
E0 r / E0i
1   2
1   2
p
C
90

0
90
s
-1
-1
B

Total Internal Reflection

Under the internal
sin t  1 sin i  sin i
reflection scheme
2
t  i for i  C  sin 1  2
1
2
total reflection happens. Since cos t  1  sin t  1 
1 2

sin i  j | 1 sin 2 i  1|
2
2
is purely imaginary, the refracted wave vector becomes:
ktz    2 0 cos t  j  2 0 |
ktx    2 0 sin t    2 0
1
sin 2  i  1|
2
1
sin  i   10 sin  i  kix
2
i.e., the refracted wave is propagating along the boundary, decaying in the direction
perpendicular to the boundary.
Therefore, the refracted wave under TIR is reduced to a surface wave propagating
along the boundary only, formed by the projection of the incident and reflected
wave vectors on the boundary plane.
Total Internal Reflection
Once i  C we find
s-wave
p-wave
E0 r kiz  j | ktz |
|k |

 e2 js , s  tan 1 tz  tan 1
E0i kiz  j | ktz |
kiz
2 |
E0 r kiz / 1  j | ktz | / 2
 |k |
2 j

 e p ,  p  tan 1 1 tz  tan 1
E0i kiz / 1  j | ktz | / 2
 2 kiz
1
sin 2 i  1|
2
1 cos i
1 |
1 2
sin i  1|
2
 2 cos i
Light Propagation in Waveguides
35
Concept of Waveguide

The wave has to localized in certain directions

How to localize the wave? – Convert the traveling wave into the standing
wave

Introduce the transverse resonance
kx2
| k2 |   2 

k x1
x
| k2 |    k
2
2
2
x2
| k1 |  1

s-wave reflection at the boundary:
z
| k1 |2   2  k x21
2
1
R
k x1  k x 2
k x1  k x 2
standing wave is formed underneath
the boundary
Concept of Waveguide
2
d
E0
1
x
E0 R 2 e jk x1 (2 d )
z
The resonance condition for standing
wave in transverse direction (x):
E0  E0 R 2e jkx1 (2 d )  R 2e jk x1 (2 d )  1
A necessary condition is: | R | 1
How to make it possible?
TIR – dielectric waveguide
Conductor reflection – metallic waveguide
Photonic crystal – Bragg waveguide
Plasma reflection – plasmonic polariton waveguide
Dielectric Waveguide
If k x 2 becomes purely imaginary, or:  2 | k2 |2   2 2   kx 2  | k2 |2  2  j  2   2 2 
k  k x 2 k x1  j | k x 2 |
|k |
R  x1

 e2 j ,   tan 1 x 2  tan 1
k x1  k x 2 k x1  j | k x 2 |
k x1
 2   2 2 
 21   2
The resonance condition becomes: e4 j e2 jk d  1  2k x1d  4  2m
x1
d  1    2 tan
2
2
1
 2   2 2 
 2   2 2 
d
m
2
2
 m  tan(  1   
)
2
2
 21   2
 21   2
Even mode tan  1   
2
2
 2   2 2 
 1  
2
2
Odd mode
Obviously, we have:   2      1
we find:
 r 2  n2  neff  n1   r1
1
tan  21   2
or:
Dispersion relation for the
dielectric slab waveguide

 2   2 2 
 21   2
With definition     0  neff
neff - waveguide effective index
Dielectric Waveguide
Dispersion relation
1/  2  c / n2
1/ neff  0   c / neff
1/ 1  c / n1

E-field E0 ( x )e j ( z t ) (y-component)
Symmetric (even mode):

Ae  k x 2 ( x  d / 2 ) , x  d / 2

E0 ( x)   B cos( k x1 x),  d / 2  x  d / 2
k x 2 ( x d / 2)

Ce
, x  d / 2

Anti-symmetric (odd mode):

Ae  k x 2 ( x  d / 2 ) , x  d / 2

E0 ( x )   B sin( k x1 x ),  d / 2  x  d / 2
 Ce k x 2 ( x  d / 2 ) , x   d / 2

A, B, C – given by the tangential boundary condition
H-field is given by the Faraday’s law, with x and z
components only – that’s the TE wave
Dielectric Waveguide
Similarly, the guided TM wave solution can be derived from the reflection of
the p-wave at the boundary
The E-field of the TM wave has abrupt change
at the boundary!
Hence, the effective index of the TM wave is smaller than that of the TE wave.
Application examples:
Single mode waveguide – higher order mode cut-off
2D waveguide – no analytical solution
Slot waveguide – utilizing the abrupt change of the E-field normal to the
boundary, for TM wave guidance only
Advanced Waveguides
41
Advanced Waveguides



Bragg Waveguides (or Photonic Crystal Waveguides)
Slot Waveguides 
Surface Plasma Polariton (SPP) Waveguides 
Features of the Advanced Optical Waveguides:
1) Confining and guiding light in low-index media
2) Confinement and guidance of light in sub-wavelength scale
42
Coupled Slab Waveguide
TE Mode
TE Mode
n
1
Normalized Intensity
nc=1.44, nh=1.99
Normalized Intensity
1
0.8
0.6
0.4
0.2
0
-2
-1
0
X ( m)
1
0.8
0.6
0.4
0.2
0
-2
2
-1
TM Mode
hy
0
X ( m)
1
2
1
2
TM Mode
W
1
Normalized Intensity
X
Normalized Intensity
Y
1
0.8
0.6
0.4
0.2
0
-2
-1
0
X ( m)
1
W=1 μm
2
0.8
0.6
0.4
0.2
0
-2
-1
0
X ( m)
W=50nm
43
Guiding Mechanism for Slot Waveguides
 
 
nh2 E xh 0   nl2 E xl 0
E xc
e
 
 
Exl 0 nh2
 2 1
h 
Ex 0
nl
E xl
E xh
 c  x  d 2 
d 2
o
e  l x
x
ws
ws
ws
ws  1  l
ws  1  l
44
Bending Characteristics of 2D Slot WG
10
4
10
3
10
2
10
1
10
0
0.46
0.44
10
-1
10
-2
Horizontal Slots
Vertical Slots
0.42
0.40
0.38
0.36
2
4
6
Bending Radius (m)
8
y (m)
5
x (m)
Ex component
y (m)
10
Ey component
Total Confinement in Slot-layers
Bending Loss (dB/cm)
5 times better
10
x (m)
Bending radius: R=3m
45
Surface Plasma Polariton
46
Why metals for optical waveguides?
Higher Frequencies at smaller dimensions
47
Dielectric Constants of Bulk Metal
-20
Model Drachev
Model Vial
Data J&C
Data Palik
-40
Drude Model
Real Part of Permittivity

 p2
      
   i 
: plasma frequency
p
 : damping coefficient
 Gold:
  2  18.36  1012
 Aluminium:
p
-120
900
1000
1100
1200
1300
1400
Wavelength/nm
1500
1600
1700
16
14
 100nm
Imaginary Part of Permittivity
h p  14eV
-100
-160
800
   9.0685 p  2  2.1556 10
2 c
-80
-140
15
p 
-60
12
10
8
6
4
Model Drachev
Model Vial
Data J&C
Data Palik
2
Vial et al. Phys. Rev. B 71, 085416 (2005)
0
800
900
1000
1100
1200
1300
1400
Wavelength/nm
1500
1600
48
1700
Propagation in Bulk Metal

E  x, y, z   E exp i t   z    exp  i z 
Plane wave along z-axis:
H  x, y, z   H exp i t   z    exp  i z 
   r  i i , i  0
The propagation length in optical wavelengths
1
2i
Propagation in bulk gold
26
24
Power Propagation Distance (nm)

B 
850nm
22
EM Wave Can Not Propagate in
Bulk Metal at Optical Wavelengths
20
18
1310nm
16
14
1550nm
12
10
800
900
1000
1100
1200
1300
1400
Wavelength (nm)
1500
1600
1700
49
Surface Electromagnetic Waves
Zenneck Modes
Radio frequency surface electromagnetic waves
that occur at the surface of absorbent medium
Brewster Modes
Damping brings ‘Brewster case’ rays into two
exponentially decaying away from the interface waves
Fano Modes
The only surface normal modes that exist at the surface
in absence of damping
50
Concepts of Surface Polariton
A polariton is an electromagnetic wave that is linearly
coupled to an electric or magnetic dipole active
elementary excitation in a condensed medium.
A surface polariton is a polariton whose associated
electromagnetic field is localized at the surface of the
medium.
51
Guided Waves by Interface between Media
ε o ε̂ 2 , μ o
x
z
ε o ε̂ 1 , μ o
TM Fields
TE Fields
Hx 
1 
Ey
jω μo  z
Ex  
1 
Ey
jω  o  x
Ez 
Hz  
Ey 
1  


H

H
z
x
jω    x
 z 
Hy  
1 
Hy
jω   z
1 
Hy
jω   x
1
jω  o
 

E

  x z  z E x 
TE Guided Mode Solution
d 2 Ey
dx
2
 ˆ  x   N 2  k 2 E y  0
Region 1
Evanescent Wave
Ey  Eo exp  1 x  jNkz 
12   N 2  ˆ1  k 2
N: Mode Effective Index
Region 2
Interface
ˆ
dE
y
Eˆ y ,
continuous
dx
1   2
Evanescent Wave
Ey  Eo exp  2 x  jNkz 
 22   N 2  ˆ2  k 2
N 2  ˆ1   N 2  ˆ2
Impossible!
No TE Modes Can Be Supported by the Interface
TM Guided-Mode Solution
d  1 dHˆ y 
ˆ

  ˆ  x   N 2  k 2 Hˆ y  0 N: Mode Effective Index
dx  ˆ dx 
Region 1
Interface
Evanescent Wave
H y  Ho exp  1x  jNkz 
1 dHˆ y
ˆ
Hy,
continuous
ˆ dx
12   N 2  ˆ1  k 2
1  0
Region 2

Decay
along –x
Evanescent Wave
H y  Ho exp 2 x  jNkz 
1  2

ˆ
1
ˆ2
N 2  ˆ1


ˆ1
 22   N 2  ˆ2  k 2
N 2  ˆ2
ˆ2
2  0
ˆ1 ˆ2
N 
ˆ1  ˆ2
Decay
along +x
2
ˆ12
 
k2
ˆ1  ˆ2
2
1
 
2
2
ˆ22
ˆ1  ˆ2
k2
54
Conditions for Surface Plasma Polariton
(SPP)
Surface Wave with Field Decay
Away from the Interface
1
ˆ
 1 0
2
ˆ2
ˆ1
0
ˆ
2
Decay along x
ˆ1 ˆ2
N 
0
ˆ1  ˆ2
Guided Wave Propagating
along the interface
or
2
ˆ1  ˆ2
Propagate along z
ε̂ 1
 ε̂ 2
ˆ12
 
k2  0
ˆ1  ˆ2
2
1
ˆ1  ˆ2  0
ε̂
0
 
2
2
ˆ22
ˆ1  ˆ2
k2  0
If ε2>0, then ε1<0, so the condition for lossless guidance with transverse confinement
is that the dielectric function of the metal must be sufficiently negative!
55
Negative Dielectric Functions
Conductors
The Drude model for simple metals
        p2  2
    0
if
 p2  4 ne2 me the plasma frequency
  p

Insulators/Semiconductors
The Lorentz model for insulators/semiconductors
     
 0     02  0    0       


    0 if
02   2
0 the natural frequency
 
2
0
 0     02



56
Guided Modes
Effective Mode Index
ε̂1ε̂ 2
N
R
ε̂1  ε̂ 2
ε̂ 22
α2  k 
ε̂ 1  ε̂ 2
ε̂ 12
α1  k 
ε̂ 1  ε̂ 2
Mode Field Distribution
Hy
exp  α1 x  x  0
H y  H o exp  jNkz
exp  α 2 x  x  0
ε2
z
x
ε1
x
Hy
ε1
ε2
z
z
57
Guided Mode on Dielectric-Metal Interface:
Ideal Metal
ε̂ d  ε do  constant
Dielectric: d
P2
ˆm     mo  2

x
z
If the frequency is smaller than the plasma frequency,
the dielectric constant of metal becomes negative
Metal: m = m' + m"
ˆm    0
Further, if
 
ˆmˆd
  Nk  k
k
ˆm  ˆd
P
 mo   do
m  k 


mo   
mo
     do
2
P
2
P
2

2

2
0
 do  mo  P2  2 

mo
      do
2
P
2
0
 do2
d  k 
0
2
2
 mo  P     do
58
Drude Models and Experimental Data for Ag
Drude model:
 p2
 p2
 m  1  2 ,  m   3


50
"
Modified Drude model:
0

-50
-100
Measured data:
'
"
Drude model:
'
"
Modified Drude model:
'
 p2
 m   mo  2

 p2
 m   3

'
-150
200
400
600
800
1000
1200
Wavelength (nm)
1400
1600
1800
Contribution of
bound electrons
Ag:
 mo  3.4
59
Existence of the Bound SPP modes: m < − d
Drude model
50
ε m  1 
"
-d

-100
Measured data:
'
"
'
Drude model:
'
"
bound SP mode: m < -d
Modified Drude model:
'
400
600
800
1000
1200
, ε m  
1400
1600
 p2
 m   mo  2

 p2
 m  
 3
Contribution of
bound electrons
-150
200
ω
2
ω 2p
τω 3
Modified Drude model
0
-50
ω 2p
1800
Ag: ε mo  3.4
Wavelength (nm)
60
SPP Mode Fields

 exp   d x 
H y  H o exp   jNkz  

exp   m x 
x0
x0
x
d 
d
,m 
1
m
D  d  m
1
 ˆ exp   d x  x  0
 d
Ex  o NH o exp   jNkz  
 1 exp   x  x  0
d

ˆ

 d
 d
x0
 ˆ exp   d x 
1
 d
Ez 
H o exp   jNkz  
j o
  m exp   x  x  0
m

ˆ

m

x
1
Ex
x
61
Propagation Losses of SP Modes
ˆmˆd
 k
ˆm  ˆd
   R  j I
Let
12


ˆ

d

R  k 
2
 ˆ  ˆ  ˆ 2 
d
I 
 R



ˆd
I  k 
 ˆ  ˆ
d
 R

ˆm  ˆR  jˆI
 ˆ 2  ˆ 2ˆ 2  ˆ 4
d I
e
 e
2

12



12



2
 ˆI2 

ˆd ˆI

12
 2 ˆ 2  ˆ 2ˆ 2  ˆ 4 
e
d I
e 


ˆe2  ˆR2  ˆI2  ˆd ˆR
Surface Plasmon
Propagation Length
 SP 
1
 ˆI1
2 I
62
SPP Dispersion Relations
F. Yang, etc., Physical Review B, Vol.44, No.11, pp.5855-5872,1991
63
Transverse and Longitudinal SPP Scales
Transverse Length
Longitudinal Length
Au/air interface
Au/air interface
1.8
260
1.6
240
Propagation Length (um)
Transverse Length (um)
1.4
1.2
1
0.8
0.6
0.4
0.2
220
200
180
160
140
0.8
1
1.2
1.4
Wavelength (um)
1.6
1.8
2
120
0.8
1
1.2
1.4
Wavelength (um)
1.6
1.8
2
The transverse confinement length is in sub-wavelength,
whereas the longitudinal propagation length is more than 100
times wavelength.
64
Surface Plasmon Polariton Length Scales
Transverse
Length Scale
Longitudinal
Length Scale
Ideal Scenario:
SP Waveguide
Design Criteria
δ m , δ d  λ  δ SP
Small transverse δt
Large longitudinal δl
Barnes, etc., Nature, Vol. 424, pp. 824-840, 2003
65
Average Power Flow Density of SPPs


1
1
S   E  H    Ex xˆ  Ez zˆ   H y yˆ
2
2
Ex 

Hy

Ez 
1 
Hy
j x
2
1
1  
1  j


S   E  H     H y zˆ    H y H y  xˆ
2
2   
2  
x 

S 

1
1
2
  zˆ  j xˆ 
 E  H   H o exp 2    z  

2
2
 



66
Gain-Assisted Lossless Propagation
Gain Medium: 1
 k
ˆ1  jˆ1ˆ2  jˆ2 
ˆ1  ˆ2   j ˆ1 ˆ2 
n  k
ˆn  jˆn 

ˆ1  ˆ2   j ˆ1 ˆ2 
x
z
2
Metal:ε2
12

n  k
k
ˆ1  ˆ2   ˆ1 ˆ2
2
2
2
2
 



ˆ
ˆ
ˆ
ˆ




2
2
2
2 
1
2
2 1
 ˆ2    jˆ2 ˆ1  
 ˆ1  
ˆ ˆ2  
ˆ1
ˆ2


 
 
ˆ1  ˆ2
ˆ1  ˆ2 
2
  ˆ1  ˆ2 
2

ˆ1  ˆ2 
ˆ
ˆ


  n  j n  1  j ˆ ˆ 
2  1   2  

Gain Required
Lossless
Transmission
2


ˆ2 ˆ1
2
2
ˆ2 ˆ1  
 ˆ1   0
ˆ2


ˆ 
ˆ   1
1
M.P.Neshad,et.al.,Opt.Express,Vol.12,No.17,pp.4072-4079,2004
ˆ2
2
2
ˆ2
67
Various SPP Propagation Regimes
ε̂ 2
Gain Required for Lossless SPP Propagation
32

ε̂1 
γk
ε̂ 
2
2 2
ε̂2   ε̂1
M.P.Neshad,et.al.,Opt.Express,Vol.12,No.17,pp.4072-4079,2004
68
Summary on Light Confinement




Conventional metallic and dielectric waveguide structures can
only confine and guide light at transverse dimension greater
than quarter of the wavelength due to the wave nature and can
achieve low-loss propagation for distance of many wavelength
in length
Slot waveguides can confine and guide light at sub-wavelength
nano dimension and through distance of many wavelength in
length, but limited to only the TM mode
SPP waveguide can confine light within sub-wavelength nano
dimension near the metal-dielectric interface by plasmonic
resonance, but limited in propagation distance due to
attenuation of the metal
Long-reach sub-wavelength optical waveguides are important
for miniaturized, high-density photonic ICs, but yet to be
developed for practical applications
69