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Towards a common description of dielectric and metallic cavities

Stefan Maier

Photonics and Photonic Materials GroupDepartment of Physics, University of Bath

Effective Mode Volume in Plasmonic Nanoresonators

S.Maier@bath.ac.uk

Funding provided by EPSRC

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Different approaches to nanophotonics

Nanophotonics is concerned with the localization, guiding andmanipulation of electromagnetic fields on the nanoscale,i.e. over dimensions comparable or smaller than the wavelengthof the electromagnetic mode(s).

Sensing in “hot spots”

Highly integrated optical chips

Optical nanolithography

High density data storage

Novel microscopy techniques

Plasmonics

MolecularPhotonics

QuantumConfinement

High-indexDielectrics

Enhancement of light/matter interactions

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Diffraction and the Rayleigh limit

Diffraction of 3D waves (3 real phase constants) limits the resolving power of optical instruments…

2 02 2 2

20

0

2, ,

,2

core

x y core x y core core

x ycore

k k k k nc c

d dn

D

fq

22.1Airy

… 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.

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RectangularDielectric

WaveguideDimension

SOI Waveguide

CMOS transistor:

Photonic integrated system with subwavelength scale components

Medium-sized molecule

Size mismatch between electronics and photonics

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Light localization in biophotonics

Levene et al, Science 299, 682 (2003)

Breaking the diffraction limit is a prerequisite for understanding cellbiology on a molecular level, since molecular interactions (e.g. pathways of enzyme kinetics) are concentration-dependent.

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Where and how do plasmonic and other novel light-confining structures fit into this picture?

Nanophotonics and quantum optics

Microcavity influences light-matter interactionFunction of spectral (Q) and spatial (Veff) energy density within the cavity

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)

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Lower dimensional waves: Surface Plasmon Polaritons

22

2)( xz kc

k 0 20 40 60 80 100

0

2

4

6

8=c k

x

=337 nm; 1= -1

(1

015 s

-1)

kx (m-1)

21

21

c

kx

Dispersion relation of surface plasmons propagating at Ag/air interface:

Large lateral wave vectors implyshort wavelengths andhigh localization to the interface

1.11 m

Si

Au

Propagation lengths up to 100 m in the visible/near-IR

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Two-dimensional optics with surface plasmons

Ditlbacher et al, APL 81 (10), 1762 (2002)

glass

Au

Bozhevolnyi, PRL 86 (14), 3008 (2001)

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Coupled modes in thin films – go far (x)or be tight

Jennifer Dionne, Caltech

In thin metal films embedded in homogeneous host, plasmonscan couple between the top and bottom interfaces…the mode of odd-vector parity looses confinement as the metalthickness 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

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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 wedgewaveguides

Typical attenuation lengthsspan from the sub-micron to themillimetre regime

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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

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The Purcell effect and the effective mode volume

eff2

eff

3

20 4

3

4

3

V

Q

V

Q

n

dVdV

20

02020

22 rErrEr

30

22

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 widthof the transition; thus, a small mode volume becomes important.

ce

2

22

0

2

22

2c

c

rEd

02

2max

22

2

2max

22 44

EQdEd

c

dV

E

nQ

n

EQ2

max23

22

02

2max

30

0 4

3

2

3

rEr

r

Normalize the (classical) electric field E:

Consider dipole aligned with field in highest intensity spot of cavity field:

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The effective mode volume concept

dVVeff 2

0

2

max0 EE

Quantification of the spatial energy density of an electromagnetic mode

Example: 2D – analogy applied to HE11 mode of silica fibre taper:

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Where and how do plasmonic structures fit into this picture?

Comparisons with established dielectric optics

Microcavity influences light-matter interactionFunction of spectral (Q) and spatial (Veff) energy density within the cavity

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)

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50 nm100 nm1 µm/single interface

A simple metallic heterostructure revisited

As a simple and well-studied modelsystem, look at the odd vector paritymode of a planar Au-air-Auheterostructure…(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

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Effective mode length of the Au/air/Au system

dzLeff 2

0

2

max0 EE

Superlinear decrease in Leff for small gaps and frequencies closeto the surface plasmon resonance frequency as more and moreenergy enters metal and gets increasingly localized to the interfaces

= 600 nm= 850 nm= 1.5 m

= 100 m

= 10 m

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A simple threedimensional resonator

2;

,0

0

yx La

LApproximate fundamental cavity mode

Im2 group

0abs vQ

3D FDTD validates analytical approximations, takinginto account field penetration into end mirrors and radiative losses.

Maier and Painter, PRB (submitted)

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Cavity model of SERS

2

2

02 ii As

E

0 0

Raman Scattering

Excited molecule in “hot site”with field Eloc

Incoming beam Stokes shifted beam

Incoming beam power:

Raman enhancement:4

4

i

locRE

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:

stutu 2

absrad Energy decay rate

i Coupling constant

Estimate contribution of excitation channel to total radiative decay rate for two-sided cavity:

i

ci A

A

2rad

Ac is the effective radiation cross-sectionof the resonant cavity mode, bound by thediffraction limit icd AAA

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Cavity model of SERS (cont.)

absrad

rad

absrad

rad2

icAA

Asu i

c E

Qu rad

absrad

1

Qu

absradabs

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:

effloc0 E Vu eff

2

0022

rad2

2

loc

4 V

Q

c

A

E

ER c

i

Estimate for simple Au plate resonator with 50 nm gap and 0=980 nm for diffraction-limitedradiation cross-section: R ~ 1600

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“Hot Sites” at particle junctions

xy LL

Xu et al, PRE 62, 4318 (2000)

Application to a crevice between two Ag nanoparticles:

Crevice can be approximately modelled ascapacitor-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 fartreated using direct numerical calculation of Eloc.

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Total enhancement of Stokes emission

radeff

2

24

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 frequencycan 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 1012!

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Some theoretical challenges…

Circular resonator structures

Interested mathematicians are invited to join in the game!!

Fine submeshing for FDTD algorithmto model metallic nanostructures inextended dielectric environments

New effects in very thin films or very small particles where the dielectricapproach breaks down?

Solving the inverse problem: How to create a specific near-field patternusing metallic nanostructures while minimizing loss (field inside the metal)

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Summary

The field of plasmonics offers unique opportunities for the creation of a nanoscalephotonic infrastructure that could allow large-scale optical integration on a chip.

The effective mode volume concept translated to plasmonics allows quickestimates of the “performance” of a given metallic nanocavity structure,thus guiding efforts for designing cavities for specific sensing purposes.

Acknowledgement: Oskar Painter, Caltech