Towards a common description of dielectric and metallic cavities

1. Effective Mode Volume in Plasmonic Nanoresonators Towards a common description of dielectric and metallic cavities Stefan Maier Photonics and Photonic Materials GroupDepartment of Physics, University of Bath S.Maier@bath.ac.uk Funding provided by EPSRC

2. 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). Highly integrated optical chips High density data storage Optical nanolithography Sensing in “hot spots” Novel microscopy techniques Enhancement of light/matter interactions

3. Diffraction and the Rayleigh limit Diffraction of 3D waves (3 real phase constants) limits the resolving power of optical instruments… … and also the size of optical modes in dielectric waveguides and cavities 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)

4. SOI Waveguide Rectangular Dielectric Waveguide Dimension Photonic integrated system with subwavelength scale components CMOS transistor: Medium-sized molecule Size mismatch between electronics and photonics

5. Light localization in biophotonics 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. Levene et al, Science 299, 682 (2003)

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

7. Lower dimensional waves: Surface Plasmon Polaritons Dispersion relation of surface plasmons propagating at Ag/air interface: Large lateral wave vectors implyshort wavelengths andhigh localization to the interface 1.11 mm Au Propagation lengths up to 100 mm in the visible/near-IR Si

8. Au glass Two-dimensional optics with surface plasmons Ditlbacher et al, APL 81 (10), 1762 (2002) Bozhevolnyi, PRL86 (14), 3008 (2001)

9. Coupled modes in thin films – go far (x)or be tight Thin Ag film in glass 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.

10. 50 nm Passive devices: Engineering localization and loss Below the diffraction limit Well above the diffraction limit Berini et al, JAP 98, 043109 (2005) Krenn et al, Europhysics Letters 60 (5), 663 (2002) Typical attenuation lengthsspan from the sub-micron to themillimetre regime Emerging geometry:metal/insulator/metal gap and wedgewaveguides Maier et al, Nature Materials 2, 229 (2003)

11. Passive devices for light transmission and localization Apertures Hot-spot sensing Barnes et al, Nature 424, 824 (2003) Xu et al, PRE 62, 4318 (2000) Martin-Moreno et al, PRL 86, 1114 (2001)

12. The Purcell effect and the effective mode volume Spontaneous emission rate of 2-level system interacting with a cavity in perturbative (weak coupling) limit: Normalize the (classical) electric field E: Consider dipole aligned with field in highest intensity spot of cavity field: Enhancement driven by quality factor Q alone is limited to spectral widthof the transition; thus, a small mode volume becomes important.

13. The effective mode volume concept Quantification of the spatial energy density of an electromagnetic mode Example: 2D – analogy applied to HE11 mode of silica fibre taper:

14. 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) Where and how do plasmonic structures fit into this picture?

15. A simple metallic heterostructure revisited 1 µm/single interface 100 nm 50 nm As a simple and well-studied modelsystem, look at the odd vector paritymode of a planar Au-air-Auheterostructure… (e.g. Prade et al, PRB44, 13556 (1991) Re b l = 850 nm l = 600 nm l = 850 nm 10x Im b l = 1.5 mm l = 10 mm l = 100 mm

16. Effective mode length of the Au/air/Au system l = 850 nm l = 600 nm l = 1.5 mm l = 10 mm l = 100 mm 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

17. A simple threedimensional resonator Approximate fundamental cavity mode 3D FDTD validates analytical approximations, takinginto account field penetration into end mirrors and radiative losses. Maier and Painter, PRB (submitted)

18. Cavity model of SERS Raman Scattering Incoming beam power: Incoming beam Stokes shifted beam Raman enhancement: Excited molecule in “hot site”with field Eloc 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: Energy decay rate Coupling constant Estimate contribution of excitation channel to total radiative decay rate for two-sided cavity: Ac is the effective radiation cross-sectionof the resonant cavity mode, bound by thediffraction limit

19. Cavity model of SERS (cont.) Steady state mode amplitude: Dielectric cavity Metallic cavity Assuming a metallic cavity, express Raman enhancement in terms of quality factor and effective mode volume: Estimate for simple Au plate resonator with 50 nm gap and l0=980 nm for diffraction-limitedradiation cross-section: R ~ 1600

20. “Hot Sites” at particle junctions 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 l0=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. Xu et al, PRE 62, 4318 (2000)

21. Total enhancement of Stokes emission Total observable enhancement of Stokes emission =field enhancement of incoming radiation x enhanced radiative decay rate The observable emission enhancement G 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 For our particle crevice, this yields an enhancement of 1.5 x 1012!

22. Some theoretical challenges… Circular resonator structures Fine submeshing for FDTD algorithmto model metallic nanostructures inextended dielectric environments Solving the inverse problem: How to create a specific near-field patternusing metallic nanostructures while minimizing loss (field inside the metal) New effects in very thin films or very small particles where the dielectricapproach breaks down? Interested mathematicians are invited to join in the game!!

23. 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