Stefan Maier
–
Bath Complex Systems 2005
Towards a common description of dielectric and
metallic cavities
Stefan Maier
Photonics and Photonic Materials Group
Department of Physics, University of Bath
Effective Mode Volume in Plasmonic Nanoresonators
S.Maier@bath.ac.uk
Funding provided by EPSRC
Stefan Maier
–
Bath Complex Systems 2005
Different approaches to nanophotonics
Nano
photonics is concerned with the localization, guiding and
manipulation of electromagnetic fields on the nanoscale,
i.e. over dimensions comparable or smaller than the wavelength
of the electromagnetic mode(s).
Sensing in “hot spots”
Highly integrated optical chips
Optical nanolithography
High density data storage
Novel microscopy techniques
P
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P
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t
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n
i
c
s
Q
u
a
n
t
u
m
C
o
n
f
i
n
e
m
e
n
t
H
i
g
h

i
n
d
e
x
D
i
e
l
e
c
t
r
i
c
s
Enhancement of light/matter interactions
Stefan Maier
–
Bath Complex Systems 2005
Diffraction and the Rayleigh limit
Diffraction of 3D waves (
3 real phase constants
) limits
the resolving power of optical instruments…
2
0
2 2 2
2
0
0
2
,,
,
2
core
x y core x y core core
x y
core
k k k k n
c c
d d
n
D
f
q
22
.
1
Airy
…
and also the size of optical modes in dielectric waveguides
and cavities
Junichi Takahara et al, Optics Letters
22
, 475 (1997)
This limit can be broken with
lower

dimensional waves with 1 or 2 imaginary
phase constants
.
Stefan Maier
–
Bath Complex Systems 2005
Rectangular
Dielectric
Waveguide
Dimension
SOI Waveguide
CMOS
transistor:
Photonic integrated system with
subwavelength scale components
Medium

sized
molecule
Size mismatch between electronics and photonics
Stefan Maier
–
Bath Complex Systems 2005
Light localization in biophotonics
Levene et al, Science 299, 682 (2003)
Breaking the diffraction limit is a prerequisite for understanding cell
biology on a molecular level, since molecular interactions
(e.g. pathways of enzyme kinetics) are concentration

dependent.
Stefan Maier
–
Bath Complex Systems 2005
Where
and
how
do plasmonic and
other novel light

confining
structures fit into this picture?
Nanophotonics and quantum optics
Microcavity influences light

matter interaction
Function of
spectral
(Q) and
spatial
(V
eff
)
energy density within the cavity
Some important processes depending on Q and
V
eff
include:
–
Spontaneous emission control (Purcell
factor ~ Q/V
eff
)
–
Strong matter

photon coupling in
cavity QED ~ Q/(V
eff
)
1/2
–
Non

linear thresholds (Raman laser ~
V
nl,eff
/Q
2
)
–
Biomolecular sensing (abs. or phase
spectroscopy ~ Q/V
eff
)
Stefan Maier
–
Bath Complex Systems 2005
Lower dimensional waves: Surface Plasmon Polaritons
2
2
2
)
(
x
z
k
c
k
0
20
40
60
80
100
0
2
4
6
8
=c k
x
=337 nm;
1
= 1
(10
15
s
1
)
k
x
(
m
1
)
2
1
2
1
c
k
x
Dispersion relation of surface plasmons propagating at Ag/air interface:
Large lateral wave vectors imply
short wavelengths
and
high localization
to the interface
1.11
m
Si
Au
Propagation lengths up to 100
m in the visible/near

IR
Stefan Maier
–
Bath Complex Systems 2005
Two

dimensional optics with surface plasmons
Ditlbacher et al,
APL
81
(10), 1762 (2002)
glass
Au
Bozhevolnyi,
PRL
86
(14), 3008 (2001)
Stefan Maier
–
Bath Complex Systems 2005
Coupled modes in thin films
–
go far (x)or be tight
Jennifer Dionne, Caltech
In thin metal films embedded in homogeneous host, plasmons
can couple between the top and bottom interfaces…
the mode of odd

vector parity looses confinement as the metal
thickness approaches zero, and can guide up to cm

distances
In general, there exists a trade

off between confinement and loss.
Thin Ag film in glass
Stefan Maier
–
Bath Complex Systems 2005
Passive devices
: Engineering localization and loss
Krenn et al,
Europhysics Letters
60
(5), 663 (2002)
Below the diffraction limit
50 nm
Maier et al,
Nature Materials
2
, 229 (2003)
Well above the diffraction limit
Berini et al, JAP 98, 043109 (2005)
Emerging geometry:
metal/insulator/metal gap and wedge
waveguides
Typical attenuation lengths
span from the sub

micron to the
millimetre regime
Stefan Maier
–
Bath Complex Systems 2005
Passive devices for light transmission and localization
Barnes et al,
Nature
424
, 824 (2003)
Martin

Moreno et al, PRL 86, 1114 (2001)
Apertures
Xu et al,
PRE
62
, 4318 (2000)
Hot

spot
sensing
Stefan Maier
–
Bath Complex Systems 2005
The Purcell effect and the effective mode volume
eff
2
eff
3
2
0
4
3
4
3
V
Q
V
Q
n
dV
dV
2
0
0
2
0
2
0
2
2
r
E
r
r
E
r
3
0
2
2
0
3
8
nd
Spontaneous emission rate of 2

level system interacting with a cavity in
perturbative (weak coupling) limit:
Enhancement driven by quality factor
Q
alone is limited to spectral width
of the transition; thus, a small mode volume becomes important.
c
e
2
2
2
0
2
2
2
2
c
c
r
E
d
0
2
2
max
2
2
2
2
max
2
2
4
4
E
Qd
E
d
c
dV
E
n
Q
n
E
Q
2
max
2
3
2
2
0
2
2
max
3
0
0
4
3
2
3
r
E
r
r
Normalize the (classical) electric field
E
:
Consider dipole aligned with field in highest intensity spot of cavity field:
Stefan Maier
–
Bath Complex Systems 2005
The effective mode volume concept
dV
V
eff
2
0
2
max
0
E
E
Quantification of the spatial energy density of an electromagnetic mode
Example: 2D
–
analogy applied to HE
11
mode of silica fibre taper:
Stefan Maier
–
Bath Complex Systems 2005
Where
and
how
do
plasmonic structures fit into
this picture?
Comparisons with established dielectric optics
Microcavity influences light

matter interaction
Function of
spectral
(Q) and
spatial
(V
eff
)
energy density within the cavity
Some important processes depending on Q and
V
eff
include:
–
Spontaneous emission control (Purcell
factor ~ Q/V
eff
)
–
Strong matter

photon coupling in
cavity QED ~ Q/(V
eff
)
1/2
–
Non

linear thresholds (Raman laser ~
V
nl,eff
/Q
2
)
–
Biomolecular sensing (abs. or phase
spectroscopy ~ Q/V
eff
)
Stefan Maier
–
Bath Complex Systems 2005
50 nm
100 nm
1
µm/single
interface
A simple metallic heterostructure revisited
As a simple and well

studied model
system, look at the odd vector parity
mode of a planar Au

air

Au
heterostructure…
(e.g. Prade et al,
PRB
44
, 13556 (1991)
= 600 nm
= 850 nm
= 1.5
m
= 10
m
= 100
m
= 850 nm
Re
10x Im
Stefan Maier
–
Bath Complex Systems 2005
Effective mode length of the Au/air/Au system
dz
L
eff
2
0
2
max
0
E
E
Superlinear decrease
in
L
eff
for small gaps and frequencies close
to the surface plasmon resonance frequency as more and more
energy enters metal and
gets increasingly localized to the interfaces
= 600 nm
= 850 nm
= 1.5
m
= 100
m
= 10
m
Stefan Maier
–
Bath Complex Systems 2005
A simple threedimensional resonator
2
;
,
0
0
y
x
L
a
L
Approximate fundamental cavity mode
Im
2
group
0
abs
v
Q
3D FDTD validates analytical approximations, taking
into account field penetration into end mirrors and
radiative losses.
Maier and Painter,
PRB
(submitted)
Stefan Maier
–
Bath Complex Systems 2005
Cavity model of SERS
2
2
0
2
i
i
A
s
E
0
0
Raman Scattering
Excited molecule in “hot site”
with field
E
loc
Incoming beam
Stokes shifted beam
Incoming beam power:
Raman enhancement:
4
4
i
loc
R
E
E
Consider this problem as the coupling of an input channel (incoming beam) to a cavity.
Expression for on

resonance mode amplitude
u
inside the cavity:
s
t
u
t
u
2
abs
rad
Energy decay rate
i
Coupling constant
Estimate contribution of excitation channel to total radiative decay rate for two

sided cavity:
i
c
i
A
A
2
rad
A
c
is the
effective radiation cross

section
of the resonant cavity mode, bound by the
diffraction limit
i
c
d
A
A
A
Stefan Maier
–
Bath Complex Systems 2005
Cavity model of SERS (cont.)
abs
rad
rad
abs
rad
rad
2
i
c
A
A
A
s
u
i
c
E
Q
u
rad
abs
rad
1
Q
u
abs
rad
abs
1
Steady state mode amplitude:
Dielectric cavity
Metallic cavity
Assuming a metallic cavity, express Raman enhancement in terms of quality factor
and effective mode volume:
eff
loc
0
E
V
u
eff
2
0
0
2
2
rad
2
2
loc
4
V
Q
c
A
E
E
R
c
i
Estimate for simple Au plate resonator with 50 nm gap and
0
=980 nm for diffraction

limited
radiation cross

section:
R
~ 1600
Stefan Maier
–
Bath Complex Systems 2005
“Hot Sites” at particle junctions
x
y
L
L
Xu et al,
PRE
62
, 4318 (2000)
Application to a crevice between two Ag nanoparticles:
Crevice can be approximately modelled as
capacitor

like cavity with reduced lateral width
For 1 nm gap and
0
=400 nm, this yields
R
~ 2.7 x 10
10
Cavity model yields same order of magnitude for
Raman enhancement in geometries thus far
treated using direct numerical calculation of
E
loc
.
Stefan Maier
–
Bath Complex Systems 2005
Total enhancement of Stokes emission
rad
eff
2
2
4
3
Q
Q
V
Q
Total observable enhancement of Stokes emission =
field enhancement of incoming radiation x enhanced radiative decay rate
The observable emission enhancement
at
peak
Stokes emission frequency
can be expressed as the product of Purcell factor and an extraction efficiency:
This yields a total observable Raman cross

section enhancement of
R
For our particle crevice, this yields an enhancement of 1.5 x 10
12
!
Stefan Maier
–
Bath Complex Systems 2005
Some theoretical challenges…
Circular resonator structures
Interested mathematicians are invited to join in the game!!
Fine
submeshing
for FDTD algorithm
to model metallic nanostructures in
extended dielectric environments
New effects
in very thin films or very small particles where the dielectric
approach breaks down?
Solving the inverse problem
: How to create a specific near

field pattern
using metallic nanostructures while minimizing loss (field inside the metal)
Stefan Maier
–
Bath Complex Systems 2005
Summary
The field of plasmonics offers unique
opportunities for the creation of a nanoscale
photonic infrastructure that could allow large

scale optical integration on a chip.
The effective mode volume concept translated to plasmonics allows quick
estimates of the “performance” of a given metallic nanocavity structure,
thus guiding efforts for designing cavities for specific sensing purposes.
Acknowledgement: Oskar Painter, Caltech
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