Effective mode volume in plasmonic ... - University of Bath

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Nov 16, 2013 (3 years and 8 months ago)

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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

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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