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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

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Stefan Maier – Bath Complex Systems 2005 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). Sensing in hot spots Highly integrated optical chips Optical nanolithography High density data storage Novel microscopy techniques Enhancement of light/matter interactions

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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… … 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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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

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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.

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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 )

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Stefan Maier – Bath Complex Systems 2005 Lower dimensional waves: Surface Plasmon Polaritons 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

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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)

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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

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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

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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

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Stefan Maier – Bath Complex Systems 2005 The Purcell effect and the effective mode volume 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. Normalize the (classical) electric field E: Consider dipole aligned with field in highest intensity spot of cavity field:

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Stefan Maier – Bath Complex Systems 2005 The effective mode volume concept Quantification of the spatial energy density of an electromagnetic mode Example: 2D – analogy applied to HE 11 mode of silica fibre taper:

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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 )

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Stefan Maier – Bath Complex Systems 2005 50 nm100 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

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Stefan Maier – Bath Complex Systems 2005 Effective mode length of the Au/air/Au system 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

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Stefan Maier – Bath Complex Systems 2005 A simple threedimensional resonator Approximate fundamental cavity mode 3D FDTD validates analytical approximations, taking into account field penetration into end mirrors and radiative losses. Maier and Painter, PRB (submitted)

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Stefan Maier – Bath Complex Systems 2005 Cavity model of SERS Raman Scattering Excited molecule in hot site with field E loc Incoming beamStokes shifted beam Incoming beam power: Raman enhancement: 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: A c is the effective radiation cross-section of the resonant cavity mode, bound by the diffraction limit

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Stefan Maier – Bath Complex Systems 2005 Cavity model of SERS (cont.) Steady state mode amplitude: Dielectric cavityMetallic 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 0 =980 nm for diffraction-limited radiation cross-section: R ~ 1600

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Stefan Maier – Bath Complex Systems 2005 Hot Sites at particle junctions 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.

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Stefan Maier – Bath Complex Systems 2005 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 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 For our particle crevice, this yields an enhancement of 1.5 x 10 12 !

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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)

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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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