Transcript Effective mode volume in plasmonic

Effective Mode Volume in Plasmonic Nanoresonators
Towards a common description of dielectric and
metallic cavities
[email protected]
Funding provided by EPSRC
Stefan Maier – Bath Complex Systems 2005
Stefan Maier
Photonics and Photonic Materials Group
Department of Physics, University of Bath
Different approaches to nanophotonics
Nanophotonics 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).
High-index
Dielectrics
Plasmonics
Sensing in “hot spots”
Quantum
Confinement
Optical nanolithography
Molecular
Photonics
Novel microscopy techniques
Enhancement of light/matter interactions
Stefan Maier – Bath Complex Systems 2005
High density data storage
Highly integrated optical chips
Diffraction and the Rayleigh limit
Diffraction of 3D waves (3 real phase constants) limits
the resolving power of optical instruments…
qAiry  1.22
f
D
and cavities
 k k 
2
2
x
 dx , d y 
2
y
2
c
2
 core  0
  core   , k x , k y 

c
 core 
2
0
ncore
0
2ncore
This limit can be broken with lower-dimensional waves with 1 or 2 imaginary
phase constants.
Junichi Takahara et al, Optics Letters 22, 475 (1997)
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… and also the size of optical modes in dielectric waveguides
Size mismatch between electronics and photonics
SOI Waveguide
CMOS
transistor:
Medium-sized
molecule
Rectangular
Dielectric
Waveguide
Dimension
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Photonic integrated system with
subwavelength scale components
Light localization in biophotonics
Levene et al, Science 299, 682 (2003)
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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.
Nanophotonics and quantum optics
Some important processes depending on Q and
Veff include:
–
Spontaneous emission control (Purcell
factor ~ Q/Veff)
–
Strong matter-photon coupling in
cavity QED ~ Q/(Veff)1/2
–
Non-linear thresholds (Raman laser ~
Vnl,eff/Q2)
–
Biomolecular sensing (abs. or phase
spectroscopy ~ Q/Veff)
Where and how do plasmonic and
other novel light-confining
structures fit into this picture?
Stefan Maier – Bath Complex Systems 2005
Microcavity influences light-matter interaction
Function of spectral (Q) and spatial (Veff)
energy density within the cavity
Lower dimensional waves: Surface Plasmon Polaritons
Dispersion relation of surface plasmons propagating at Ag/air interface:
kx 
=c kx
=337 nm; 1= -1
15
-1
 (10 s )
6

1 2
c 1   2
Large lateral wave vectors imply
short wavelengths and
high localization to the interface
4
2

0
0
20
40
60
80
100
k z  ( ) 2  2  k x2
c
-1
kx (m )
Propagation lengths up to 100 m in the visible/near-IR
Au 1.11 m
Si
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8
Two-dimensional optics with surface plasmons
Au
glass
Bozhevolnyi, PRL 86 (14), 3008 (2001)
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Ditlbacher et al, APL 81 (10), 1762 (2002)
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.
Stefan Maier – Bath Complex Systems 2005
Thin Ag film in glass
Passive devices: Engineering localization and loss
Below the diffraction limit
Well above the diffraction limit
Krenn et al, Europhysics Letters 60 (5), 663 (2002)
Typical attenuation lengths
span from the sub-micron to the
millimetre regime
50 nm
Maier et al, Nature Materials 2, 229 (2003)
Emerging geometry:
metal/insulator/metal gap and wedge
waveguides
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Berini et al, JAP 98, 043109 (2005)
Passive devices for light transmission and localization
Barnes et al, Nature 424, 824 (2003)
Martin-Moreno et al, PRL 86, 1114 (2001)
Hot-spot sensing
Xu et al, PRE 62, 4318 (2000)
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Apertures
The Purcell effect and the effective mode volume
Spontaneous emission rate of 2-level system interacting with a cavity in
perturbative (weak coupling) limit:
e  c 

 
Normalize the (classical) electric field E:
c 2
2
8 2 nd 2
0 
3 0 3
 0   r Er 2 dV 
0
0
2 
2
2
2 0   r Er  dV
Consider dipole aligned with field in highest intensity spot of cavity field:
2
2
4d 2 2 Emax
4Qd 2 2 Emax


 2 c
 20
2
 3 0Q3 Emax
3 

 2  2 Q 
2
0
2 0 n
4  n 
3
 r E 
2
max
2
  r Er  dV

3  Q
3 Q
 2 
 2
0 4  n  Veff 4 Veff
3
Enhancement driven by quality factor Q alone is limited to spectral width
of the transition; thus, a small mode volume becomes important.
Stefan Maier – Bath Complex Systems 2005
c
2
2
  2 d  Er 
2

2   0  
The effective mode volume concept
Quantification of the spatial energy density of an electromagnetic mode
 0 Emax Veff    0 E dV
2
2
Stefan Maier – Bath Complex Systems 2005
Example: 2D – analogy applied to HE11 mode of silica fibre taper:
Comparisons with established dielectric optics
Some important processes depending on Q and
Veff include:
–
Spontaneous emission control (Purcell
factor ~ Q/Veff)
–
Strong matter-photon coupling in
cavity QED ~ Q/(Veff)1/2
–
Non-linear thresholds (Raman laser ~
Vnl,eff/Q2)
–
Biomolecular sensing (abs. or phase
spectroscopy ~ Q/Veff)
Where and how do
plasmonic structures fit into
this picture?
Stefan Maier – Bath Complex Systems 2005
Microcavity influences light-matter interaction
Function of spectral (Q) and spatial (Veff)
energy density within the cavity
A simple metallic heterostructure revisited
1 µm/single
interface 100 nm
50 nm
As a simple and well-studied model
system, look at the odd vector parity
mode of a planar Au-air-Au
heterostructure…
 = 850 nm
Re 
 = 600 nm
 = 850 nm
10x Im 
 = 1.5 m
 = 10 m
 = 100 m
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(e.g. Prade et al, PRB 44, 13556 (1991)
Effective mode length of the Au/air/Au system
 0 Emax Leff    0 E dz
2
 = 600 nm
2
 = 850 nm
 = 1.5 m
 = 10 m
Superlinear decrease in Leff 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
Stefan Maier – Bath Complex Systems 2005
 = 100 m
A simple threedimensional resonator
Qabs 
0
2vgroup Im 

 

 Lx 
; Ly  0 
 0 , a 
2

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
Approximate fundamental cavity mode
Cavity model of SERS
0
Incoming beam power:
s 
2
2
Stokes shifted beam
  0  
Raman enhancement:
R
Eloc
Ei
Excited molecule in “hot site”
with field Eloc
4
4
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:

u t    u t   s
2
   rad   abs
Energy decay rate
  i
Coupling constant
Estimate contribution of excitation channel to total radiative decay rate for two-sided cavity:
i 
 rad Ac
2 Ai
Ac is the effective radiation cross-section
of the resonant cavity mode, bound by the
diffraction limit Ad  Ac  Ai
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Incoming beam
Ei 0  Ai
2
Raman Scattering
Cavity model of SERS (cont.)
Steady state mode amplitude: u 
2 rad
Ac
Ai
 rad   abs
1
 rad

 rad Ac Ei
  rad   abs 
Metallic cavity
 Q
 abs   rad  u 
1
 abs
Q
Assuming a metallic cavity, express Raman enhancement in terms of quality factor
and effective mode volume:
u   0 E loc Veff
R
Eloc
Ei
2
2
 rad Ac
Q2
 2 2
4 c  0 0 Veff
Estimate for simple Au plate resonator with 50 nm gap and 0=980 nm for diffraction-limited
radiation cross-section: R ~ 1600
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Dielectric cavity
 rad   abs  u 
s
“Hot Sites” at particle junctions
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 1010
Cavity model yields same order of magnitude for
Raman enhancement in geometries thus far
treated using direct numerical calculation of Eloc.
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L y  Lx 
Xu et al, PRE 62, 4318 (2000)
Total enhancement of Stokes emission
Total observable enhancement of Stokes emission =
field enhancement of incoming radiation x enhanced radiative decay rate
3 Q2
Q
 2

4 Veff Qrad
This yields a total observable Raman cross-section enhancement of
R 
For our particle crevice, this yields an enhancement of 1.5 x 1012!
Stefan Maier – Bath Complex Systems 2005
The observable emission enhancement  at peak Stokes emission frequency
can be expressed as the product of Purcell factor and an extraction efficiency:
Some theoretical challenges…
Fine submeshing for FDTD algorithm
to model metallic nanostructures in
extended dielectric environments
Solving the inverse problem: How to create a specific near-field pattern
using metallic nanostructures while minimizing loss (field inside the metal)
New effects in very thin films or very small particles where the dielectric
approach breaks down?
Interested mathematicians are invited to join in the game!!
Stefan Maier – Bath Complex Systems 2005
Circular resonator structures
Summary
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
Stefan Maier – Bath Complex Systems 2005
The field of plasmonics offers unique
opportunities for the creation of a nanoscale
photonic infrastructure that could allow largescale optical integration on a chip.