Transcript Simulation Tools for Passive Waveguides

Department of Electrical and Computer Engineering
hotonics
research
aboratory
Nano-Photonics (1)
W. P. Huang
Department of Electrical and Computer Engineering
McMaster University
Hamilton, Canada
1
Agenda
 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
– Scattering of Light by Metal Particles
– Surface plasma polariton resonators

Light-matter interaction in nano-crystals
– Optical properties of nano-crystals
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
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 sufficient 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, at the
lowest energy level, and within shortest temporal scale.
5
Methods for Construction of Nano-Structures
6
Nature and Properties of Light
7
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)
8
Maxwell’s Equations
Faraday Law of Induction:
  Er, t   

Br, t 
t
Generalized Amplere’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
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
2
9
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.
10
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
11
About Maxwell’s Equations
Boltzmann:
Was it a God who wrote these lines…
Pierce:
To anyone who is motivated by anything beyond the most narrowly practical,
it is worthwhile to understand Maxwell’s equations simply for the good of his
soul…
Crease:
When we discover that two or more heterogeneous empirical laws of nature
can be unified under one principle that comprises them both, the discovery
does give rise to a noticeable pleasure...even an admiration that does not
cease when we have become fairly familiar with its object. This delight is
more than having our expectations fulfilled or surprised, more than about the
domination and control of nature, more than a biological product. The
pleasure is a feature of the exercise of the human intellect.
12
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.
13
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
The sun is about 1.5x1011 m from the earth. How long does it take light to get here?
1.5 1011 m
t
 500 s  8.8 min
3 108 m/s
Maxwell’s Predictions:
1. Electromagnetic fields may exist in the form of transverse waves
2. Light is electromagnetic in nature
14
Historical Accuracy for speed of light
Date
1600
1676
1729
1849
1879
1950
1958
1972
1974
1976
1983
Experimentor Country
Galileo
Roemer
Bradley
Fizeau
Michelson
Michelson
Essen
Froome
Evenson et al.
Blaney et. al
Woods et al.
Italy
France
England
France
United States
United States
England
England
United States
England
England
International
Experimental
Method
Speed
(10^8m/s)
Lanterns and shutters "Fast"
Moons of Jupiter
2.14
Aberration of Light 3.08
Cog Wheel
3.14
Rotating mirror
2.9991
Rotating mirror
2.99798
Microwave cavity
2.997925
Interferometer
2.997925
Laser Method
2.99792457
Laser Method
2.99792459
Laser Method
2.99792459
2.99792458
Relative
Uncertainty (m/s) Error
?
?
?
?
75000.0
22000.0
1000.0
100.0
1.1
0.6
0.2
0.0
28%
2.70%
4.70%
400 in 10^6
18 in 10^6
0.1 in 10^6
0.1 in 10^6
2 in 10^9
3 in 10^9
3 in 10^9
Exact
Maxwell: The velocity of transverse undulations in our hypothetical medium, calculated
from the electromagnetic experiments of MM Kohlrausch and Weber, agrees so exactly
with the velocity of light calculated from the optical experiments of M Fizeau, that we can
scarcely avoid the inference that light consists in the transverse undulations of the same
medium which is the source of the electric and magnetic phenomena.
15
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 vs Frequency
λf  c
16
Relationship between E and H Fields
  Er, t    μ
  Hr, t   ε

H r, t 
t


E x z, t    μ H y z, t 
z
t

Er, t 
t


H y z, t    ε E x z, t 
z
t
x
 z
 z
H y z, t   H   t    H   t  
 c
 c
Ex
Ex  Hy  z
z
E and H are perpendicular to each other and
also to the direction of wave propagation. The
three orientations obey the right hand rule.
1
E    H 
c
1
H    E
c
E

Z 
H

Zo 
0
 377
0
y
Hy
E and H are proportional to each other
and the ratio is the characteristic wave
impedance of the media
17
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  cos 2   t   
  c 
Relationship between Stored Energies and Power Flow
Sz  v p  we  wo 
Power Flow = Velocity  Total Stored EM Energy
18
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
19
Momentum and Pressure
  z 
E x z, t   E  cos ω t  
  c 
H y z, t  
Maxwell’s Stress Tensor
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 
 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

0

0

 ε o E 2x  μ o H 2y 

  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
20
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.
21
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
22
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  f 
c
λ
p
 f

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.
23
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 / ħf
24
Wave-Particle Duality of EM Fields
Wave
E  f 
p
c
λ
Particle
 f

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
25
The Electromagnetic Spectrum
Shortest wavelengths
(Most energetic photons)
More particle-like
E =hf = hc/
h = 6.6x10-34 [J*sec]
(Planck’s constant)
More wave-like
Longest wavelengths
(Least energetic photons)
26
Notes on EM Spectrum

Radio Waves
 Used in radio and television communication systems

Microwaves
 Wavelengths from about 1 mm to 30 cm
 Well suited for radar systems
 Microwave ovens are an application

Infrared waves
 Incorrectly called “heat waves”
 Produced by hot objects and molecules
 Readily absorbed by most materials

Visible light
 Part of the spectrum detected by the human eye
 Most sensitive at about 560 nm (yellow-green)

Gamma rays
 Emitted by radioactive nuclei
 Highly penetrating and cause serious damage when absorbed by living tissue
27
Human Visual Response of Colors
Red + Blue = Magenta
Red + Green = Yellow
Blue + Green = Cyan
The way electronic display
(e.g., computer screen,
TV, etc.) makes colors.
Magenta + Green = White
Cyan + Red = White
Yellow + Blue = White
28
Gamut of Display Technologies
Conventional
Technologies
Laser Technologies
CIE 1976 UCS Chromaticity Diagram. Source: Joe Kane Productions, California
29
Laws of Electrons and Photons
Laws of Electron
Laws of Photon
– Primarily particle like
– Primarily wave like
– Localization and
– Globalization and
concentration
– Properties easy to change
and control
– Rich in information
processing power
– Lack in information
transmission
distribution
– Properties hard to change
and control
– Rich in information
transmission power
– Lack in information
processing
It is much more difficult to confine
and control light than current!
30
Light Propagation in Free Space
31
Classical Optical Waveguides
Key Idea: To Use Objective Lens to Re-Focus
the Diffracted Light so as to maintain guidance
Device Feature Size >> Optical Wavelength
32
Focus of Parallel Light by Lens: Geometric Optics
A lens followed by propagation by one focal length:
x out   1 f  1 0 x in   0 f  x in   0
θ    0 1  1 f 1 0    1 f 1 0     x
  
   in
 out   
f
Parallel rays
Focused rays
f


f
For all rays
xout = 0!
Assume all input rays have qin = 0
At the focal plane, all rays
converge to the z axis (xout = 0)
independent of input position.
Parallel rays at a different angle
focus at a different xout.
33
The limitation of Geometric Optics
The focused EM waves or beams will diffract as they propagate in free space and therefore can only focus
with a minimum width (i.e., the diffraction limit) which is NOT zero as predicted by the ray theory under
the geometrical optics approximation.
The expression for a beam electric field is given by:
E ( x, y, z ) 
exp  ikz  i ( z )
w( z )
 x2  y 2  x2  y 2 
exp  2
i

w
(
z
)

R
(
z
)


w(z): the spot size vs. distance from the waist
R(z) : the beam radius of curvature, and
Ψ(z) : a phase shift.
This expression is the solution to the wave equation when we require that the beam be well
localized at some point (i.e., its waist).
34
Focusing a Gaussian beam
winput
f
wfocus
f
A lens will focus a collimated Gaussian beam to a new spot size:
wfocus   f / winput
which is limited by wavelength, focus length and the spot size of the input beam
35
Gaussian beam spot size, radius, and phase
The expressions for the spot size,
radius of curvature, and phase shift:
w( z )  w0
1   z / zR 
Rz  z  z / z
2
R
where zR is the Rayleigh Range
(the distance over which the
beam remains about the same
diameter), and given by:
2
zR   w0 / 
2
 ( z )  arctan( z / z R )
36
Light Diffraction and Spatial resolution
– If S1, S2 are too close together the Airy patterns will overlap and become
4 0
2 -2
0 -4
-2 -6
-4 -8
S2

37
-6-10
S1
0.5
6 2
8 4
10 6
g
8
10
g
indistinguishable
0.5

1.0

Suppose two point sources or objects are far away (e.g. two stars)
Imaged with some optical system
Two Airy patterns
1.0

Maximum Resolution due to Diffraction Limit

Rayleigh criterion
– Resolving two equal intensity airy spots
– Separation ~ 0.61 l0 / NA (circular aperture)
– Practical limitations ~ 0.5um @ visible
38
Light Propagation in Waveguides
39
Conventional Dielectric Optical Waveguides
John Tyndall
Jean-Daniel Colladon
40
Glass Optical Fibers
Kao, K.C. and Hockham, G.A., "Dielectric-fibre Surface Waveguides
for Optical Frequencies", Proc. I.E.E. Vol. 113, No. 7, July 1966, pp.
1151-1158.
41
Integrated Optics
S. E. Miller, “Integrated optics: an introduction,” Bell Syst.
Tech. J., 48, 2059-2069, 1969.
E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell
Syst. Tech. J., 48, 2103-2132, 1969.
P. K. Tien, Light waves in thin films and integrated optics,
Appl. Opt. 10, 1971.
``Research in integrated optics has two goals: One is to apply thin-film technology to the formation
of optical devices and circuits. The other is the integration of a large number of optical devices on a
small substrate, so forming an optical circuit reminiscent of the integrated circuit in
microelectronics‘’, P. K. Tien, 1978.
``Integrated Optics has a long history. Yet, practical applications of integrated optics are still only
few. Optical components in current use are large compared with a wavelength. This puts a
fundamental limit on the density of the integrated components. By using structures with a large
index contrast one may, at best, reduce the structure size to the order of one wavelength. In this
limit, the structures resemble microwave components that are of the order of a single wavelength
in size.’’ H.A.Haus,2002
42
Waveguide Models
Assumptions
– CW steady state
– Longitudinally invariant structures
– Arbitrary transverse geometries and index
profiles
Concepts
– Propagation constants (modal indices)
– Field patterns (modal fields)
Forward Propagating Modes
z
y
Backward Propagating Modes
General Solutions for Waveguides
A general solution to Maxwell’s equations under waveguide assumption
ErT , z   E rT  e

 jβ  z
HrT , z   H rT  e

Modal Field
Patterns
 jβ  z
rT  x xˆ  y y
The transverse position vector
Modal
Propagation Constants
(Eigen Values)
(Eigen Functions)
The transverse fields
E  rT   E t  E z
The longitudinal fields
H  rT   H t  H z
+: Forward Traveling Wave
_: Backward Traveling Wave
1D Waveguide Structures



0
x y
TEM Fields
Hx 
TEM Fields
Maxwell’s Equations are
reduced to two identical
sub-sets as
1 
Ey
jω   z
Hz  0
1 
Ey  
Hx
j z
Ex  
1 
Hy
j z
Ez  0
Ex  E y  Et
H x  H y  Ht
Hy 
Et  
dH t
j dz
1
Ht 
Et  H t
1 
Ex
j z
1 dEt
j dz
45
Governing Equations and Modal Solutions
Et  
1 dH t
j dz
d 2 Et
2


 Et  0
2
dz
Et  z   E e j z  E e j z
Forward Propagating Wave
Ht 
1 dEt
j dz
d 2 Ht
2


 H t  0
2
dz
H t  z   H  e j  z  H  e j  z
Backward Propagating Wave
Propagation Constant
   
TEM Modes
46
2D Waveguide Structures
z
x
z
x
n
n2
y
Parallel-Plate
y
n1
n2
Dielectric Slab
The y-dimension variation is much slower than that of x so that only onedimensional structure needs to be considered over the waveguide cross
section

0
TM Fields
TE Fields
y
1 
Hx 
Ey
j z
1 
Hz  
Ey
j x
1
Ey 
j



H

H
z
x
 x
z 
Maxwell’s Equations are
reduced to two decoupled
equations such that
Ex  
1 
Hy
j z
Ez 
1 
Hy
j x
Hy  
1 
 
E

Ex
z
j  x
z 
47
Guided Waves in Parallel Metal Plate Structures
x
+a
–a
R+
A
A'
B'
R–
z
B
The plane wave is reflected at the upper and lower interfaces and propagates
along a zig-zeg path (e.g., A'ABB‘).
Phase-shifts due to
transverse round-trip
Conditions for Guidance
1.
Total reflection at the upper and lower interfaces
2.
Round-trip transverse phase-shift leads to
constructive interference
The transverse
resonance condition
R  R  1
    k x (4a)  2m
R R exp   jkx 4a   1
Phase-shifts due
to reflection
m=0,1,2,…
48
TE Modes on Parallel Plate Waveguide
R  R  1
R  R  1
kx 
    π
 m  1 
2a
k x2   2  n 2 k 2
2  k x (4a)  2m
  m  1  
2 2
  n k 

2
a


2
m=0,1,2,…
The transverse wave vector
kx and the propagation
constant β becomes discrete!
For a given k, only certain discrete transverse wave number
and therefore propagation constants are allowed!
β
nk
For a given k2a, there
β0
β1
are only limited number
of real β that exist
k
π/2a
π/a
49
Field Patterns of the TE Modes
in Parallel Plate Waveguide
z
x
–a
+a
  m  1 
Ey  Eo sin 
 2a

x  exp   j m z 

Standing wave
along x
Traveling wave
along z
  m  1  
Hz   j
cos 
x  exp   j  m z 

2a


Eo k x
Only specific standing-wave field patterns are permitted due to the transverse
resonance condition
Due to the constructive interference between the forward and the back
propagating waves along x, the guided waves are "quantized " into
modes with discrete propagation constants and specific field patterns
50
Different modes correspond to plane waves at different angles
q3
q2
q1
 π 
β1  n 2 k 2   
 2a 
m=0
 2π 
β2  n k   
 2a 
m=1
2
Ey(x)
2
–a
+a
Ey(x)
–a
+a
2
2
Ey(x)
m=2
 3π 
β3  n k   
 a 
2
2
2
–a
+a
m+1=number of peaks in the field patterns
The mode index m increases
Decrease in propagation constants
Increase in field oscillations 51
Mode Cut-Off Conditions
Guided Modes
Above Cut-Off:
λ  λ Cm 
If
m 
f  fC
2
  
 fC
 m  nk 1    m    nk 1  

f


 C 

then
The highest order guided modes is therefore
Below Cut-Off:
If
  C m
f  fC m
real
 2nka 
M max  
 1



 m 1 
n k 
 0
2
a


2
Cut-off Condition:
The cutoff
m 1
kC 

wave number
2na
 m  2
2
The cutoff
wavelength
C 
4na
m 1
2
C
The cutoff
frequency
fC 
m 1 c
4a n
Evanescent Modes
2
  
then  m   j m   jnk   m    1   jnk
 C 
2
 fC m  

  1 imaginary
 f 
52
Field Profiles of Evanescent Modes
 4 
3     n2 k 2
 2a 
Ey(x)
2
m=3
–a
+a
z
x
Ey(x)
m=4
 5 
 4     n2 k 2
 2a 
2
–a
+a
z
x
m+1=number of nulls in the field patterns
The mode index m increases
Increase in field oscillations
Increase in decay constants
53
Mode spectrum for parallel plate waveguide
α
– nk –β0 – β1– β2
– α3
– α4
Backward Guide Modes
– α5
Backward Evanescent Modes
–α6
α6
α5
α4
α3
Forward Evanescent Modes
Forward Guide Modes
β2 β1 β0 nk
β
Guided Modes + Evanescent Modes = Complete Orthogonal Modal Set
54
How Small A Metallic Waveguide Can Be?
Fundamental Guided Mode
If
λ
a
4n
If
λ
a
4n
then
  
1  nk 1  

 4na 
  
 0   jnk 
1

 4na 
2
then
2
real
imaginary
The total width of the waveguide must be greater than half the
wavelength in the media filling the waveguide to support the lowestorder guided mode. For waveguide of width smaller than λ/4n, no
propagating mode exists.
55
Guided Waves in Dielectric Slab Structures
Bound Rays→Guided Modes
Conditions for Guidance
Refracting Rays→Leaky Modes
Conditions for Leakage
n core  n cladding
n core > n cladding
and
q  qc
Limited guided modes with
discrete mode indices and
specific standing-wave
field patterns are allowed.
Mode Equation
R 2exp   jk x 4d   1
Transverse Resonance
Conditions
or
q  qc
For anti-guides or guides below
cut-off, leaky modes with complex
propagation constants exist (as
an approximate representation of
the radiation modes).
56
TE Modes on Slab Waveguide
R  R  exp  j 2TE 
R  R  1

n22
2
 sin q1   2
n1
1 
4 tan 
cos q1 



      2 TE

n 22
2
 sin θ1   2
n1

 2 tan 1 
cosθ1 






  n1k cos q1  (4a)  2m



m=0,1,2,…
kx
q1 n k
1
β
Only certain discrete angles are allowed!
The dispersion Relations
kx  n1 k cos q1 
k x2   2  n12 k 2
k x2   x2   n12  n22  k 2
 x2   2  n22 k 2
 x  n12 k 2 sin 2 q1   n22 k 2

tan  k x a    x
Even Modes
kx
1   x 
4 tan    4k x a  2m
kx
k
tan
k
a


 x 
Odd Modes
 x
x
57







Solutions for Propagation Constants
 x a   kx a  tan  kx a 
Even Modes
n
2
1


 x a    kx a  cot  kx a 
k x2   x2  n12  n22 k 2
αxa
Odd Modes

 n 22 ka
kxa
The transverse wave vector kx
and the propagation constant β
becomes discrete!
For a given kd, there are only
limited number of real β that
exist
58
Mode Indices for Slab Waveguide
n2
n
q
1
q
n2
1 Ray Cycle
3.60
90º
88º
87º
86º
85º
Effective Refractive Index
TEo, TMo
TE 1, TM1
3.59
TE 2, TM2
TE 3, TM3
3.58
84º
83º
TE4, TM4
3.57
82º
3.56
81º
TE 5, TM5
3.55
80.4º
0
1
2
d/
3
4
5
59
Mode Fields for Slab Waveguides
The different guided modes correspond to the plane waves at different angles
q3
q2
q1
TE0
TE
0
TE1
d
TE
1
TE2
TE 2
TE3
TE
3
m+1=number of peaks in the field patterns
The mode index
m increases
Decrease in propagation constants
Increase in field oscillations
60
Cut-off Condition For Slab Waveguides
αx  0


 x a   kx a  tan  kx a 
k x2   x2  n12  n22 k 2
tan  kx a   0
k x2  n12  n22 k C2

k x a  m
VC  m
Below Cut-Off:
qC
q
cot  kx a   0
m 1
kxa 

2

Define
V  ka

n12  n22
q>qC
 x a    kx a  cot  kx a 

VC 
m 1

2
The total internal reflection
condition is no longer satisfied
 n2 
qC  cos  
 n1 
1
Radiation Modes
61
Waveguide Parameters
Single Mode Condition
2d
1

λ
2 2n
For symmetric
step-index slab
Field Confinement Factor



 Eˆ  Hˆ  ẑda
ˆ H
ˆ   ẑda

E

GuidingRegion


n1  n 2
n
n
n1  n 2
2
Mode Attenuation Factor
   

Entire Region
Total Loss=Absorption + Leakage
62
Single Mode Condition
For single-mode dielectric waveguide, the full width of waveguide
depends on index contrast and average index of the waveguide.
63
Bending Radius Vs Index Contrast
Slab Dielectric Waveguide
TE Mode
TM Mode
The minimum bending radius is defined for bending loss of 0.5dB/cm and
waveguide width adjusted to meet single-mode condition at λ=1.550μm
64
Field Confinement Vs Index Contrast
Symmetric Slab Dielectric Waveguide
TE Mode
TM Mode
The calculations are performed at the minimum bending
radius for the single-mode slab waveguide at λ=1.550μm
65
Polarization Dependence Vs Index Contrast
66
Confinement Factors



ˆ H
ˆ  ẑda


E

ˆ H
ˆ   ẑda

E

GuidingRegion

Entire Region
Symmetrical Step-Index
Slab Waveguide
e
TE
 2k x cos 2  k x w 2  
 1 


k
w

sin
k
w
 x  
x
x

1

 2k x sin  k x w 2  
 1 


k
w

sin
k
w




x
x
x
1
2
o
TE
What if the slab is
not symmetric?
e
TM
  n  2k cos  k w 2  
x

 1   1  x
  n2   x k x w  sin  k x w  
1
o
TM
  n  2k sin  k w 2  
x

 1   1  x
  n2   x k x w  sin  k x w  
1
2

2

2
2
Confinement Factors for Guided Modes in
Symmetric Step-Index Slab
Note that if d/λ→0, thenΓ →0 for the fundamental mode. Therefore, confinement
of light by a waveguide of sub-wavelength dimension is difficult
S. -L. Chuang, Phys. Optoeletron. Dev., John Wiley, pp. 253-257, 1995
Advanced Waveguides
69
Advanced Waveguides



Bragg Waveguides (or Photonic Crystal Waveguides) χ
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
70
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
71
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
g c  x  d 2 
d 2
o
e g l x
x
ws
ws
ws
ws  1 g l
ws  1 g l
72
Slot Waveguides
V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, Guiding and
confining light in void nanostructure, Opt. Lett., vol.29, 1209-1211, 2004
73
Confinement Factor vs. Slot Layer Thickness and Index
0.5
tl=80nm
tl=60nm
tl=40nm
tl=20nm
tl=10nm
tl=5nm
Total Confinement in Nano-layer
Single Slot
tl increases
0.4
0.3
0.2
0.1
0.0
1.0
1.5
2.0
2.5
3.0
3.5
Refractive Index of the Slot Layer
74
Bending Loss and Confinement Factor
For Different Slot Indices
I
SiO2
n
w1
w
II
III
Silicon
w1
IV
V
n
n
SiO2 Silicon
SiO2
y
z
x
10
10
10
0
0.4
Air
SiO2
-5
-10
-15
0
20
40
60
Radius (um)
80
100
Confinement
Loss (dB/um)
10
0.35
Air
SiO2
0.3
0.25
0.2
0
20
40
60
80
100
Radius (um)
75
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
76
Surface Plasma Polariton
77
Why metals for optical waveguides?
Higher Frequencies at smaller dimensions
78
Dielectric Constants of Bulk Metal
-20
Model Drachev
Model Vial
Data J&C
Data Palik
-40
Drude Model
Real Part of Permittivity

 p2
      
   ig 
: plasma frequency
p
g : damping coefficient
 Gold:
g  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
79
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
80
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
81
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.
82
Guided Waves by Interface between Media
ε o ε̂ 2 , μ o
x
z
ε o ε̂1 , 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  1 x  jNkz 
1 dHˆ y
ˆ
Hy,
continuous
ˆ dx
12   N 2  ˆ1  k 2
1  0
Region 2

Decay
along –x
Evanescent Wave
Ey  Eo 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
85
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!
86
Negative Dielectric Functions
Conductors
The Drude model for simple metals
        p2  2
    0
if
 p2  4 ne2 me the plasma frequency
  p

Insulators
The Lorentz model for insulators
     
 0     02  0    0       


    0 if
02   2
0 the natural frequency
 
2
0
 0     02



87
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
88
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   mo
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
89
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
90
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)
91
Surface Plasmon Polariton Dispersion Relation

Radiative modes
'm > 0)
Volume Plasma
real β
imaginary α
p
c
sp
p
 do
Quasi-bound modes
d < 'm < 0)
imaginary β
imaginary α
1   do
 mo  1
Surface Plasma
1/ 2
 ˆmˆd 
 k

ˆ
ˆ
 m  d 
Bound modes
('m < d)
Re β
real β
real α
92
Critical Points for SPP Dispersion Relation
Frequency
ω0
ω  ω sp
ω  ωp
ω  
Metal Ɛ
SPP
Mode Index
ε̂ m ω  
N 1
ε̂ m ω sp    ε̂ d
ε̂ m ω p   0
ε̂ m    1
N
N
Decay
in Metal
α m  ωP c
ω2ε d
αd  2
ωP c
αm  
αd  
ε do ε mo  1
ε mo  12
α m   jk
ε mo  1  ε do
ε do  ε mo  1
N
ωsp 
ε do ε mo
ε mo  ε do
Decay
in Dielectric
ε 2mo
α m   jk
ε mo  ε do
2
ε do
α d   jk
ε mo  1  ε do
ε do2
α d   jk
ε mo  ε do
ωP
ε mo  ε do
93
SPP Mode Dispersion Relation (Ag/SO2)
β<ω/c
β=ω/c
β>ω/c
ωs 
ω0
1  ε mo
ne 2
ωp 
moεo
β
A.M.Gadou, et.al., Epypt.J.Sol.,Vol.23,No.1,pp.13-26,2000
94
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
95
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
96
Case A: Poor Conductor
If
ˆR  ˆI
ˆR  ˆd
12
 ˆRˆd 
R  k 

ˆ
ˆ



 R d
1 ˆ
 I   k I2
2 ˆR
 ˆd ˆR 


ˆ
ˆ



 R d
32
1 ˆI  R3

2 k 2ˆR2
 R  k ˆd
Case B: Good Conductor
If
ˆR  ˆI
ˆR  0
or

ˆI  ˆI2  ˆd2
 R  k ˆd ˆI
2 ˆd2  ˆI2



12



 R  k ˆd

ˆd3ˆI

I  k 
2
2
2
2
ˆ
ˆ
ˆ
ˆ
ˆ
2








d
I
I
I
d



12





97
SPP Dispersion Relations
F. Yang, et.al., Physical Review B, Vol.44, No.11, pp.5855-5872,1991
98
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.
99
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, et.al., Nature, Vol. 424, pp. 824-840, 2003
100
Propagation Distance of SPP at Different Wavelengths
100 m
High loss
in region
of small SP
101
Transverse Skin Depths of SPP
Confinement in Dielectric
Reduces for Longer
Wavelength
Confinement in Metal Does not
Change for Longer Wavelength
102
Intensity Profiles of SPPs
x (nm)
103
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
 

104
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
ˆ
 2


ˆ 
ˆ   1
1
M.P.Neshad,et.al.,Opt.Express,Vol.12,No.17,pp.4072-4079,2004
ˆ2
2
2
ˆ2
105
Various SPP Propagation Regimes
ε̂ 2
Gain Required for Lossless SPP Propagation
ε̂1 3 2
γk
ε̂ 
2
2 2
ε̂2   ε̂1
M.P.Neshad,et.al.,Opt.Express,Vol.12,No.17,pp.4072-4079,2004
106
Longitudinal and Transverse Characteristics
Lossless Propagation
δ SP
β
α 2 
Loss of Transverse Confinement?
M.P.Neshad,et.al.,Opt.Express,Vol.12,No.17,pp.4072-4079,2004
107
Summary




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 TE 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
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