Plasmonics: elementary excitation of a plasma (gas of free charges) nano-scale optics done with plasmons at metal interfaces

Transcription

1 Plasmonics Plasmon: Plasmonics: elementary excitation of a plasma (gas of free charges) nano-scale optics done with plasmons at metal interfaces Femius Koenderink Center for Nanophotonics AMOLF, Amsterdam

2 Wrap up Surface charge density wave excited by light - Extremely dispersive, low group velocity wave - Kramers Kronig limits on material response means loss intrinsic to dispersion

3 Wrap up Plasmon waveguides Nano-scale integrated optics Very dense chips for signals Plasmon antennas Strong scattering Shrink light to single molecules

4 Figures of merit strongly scattering α V Cross section vz geometric area, λ 2 Polarizability vz Physical volume, λ 3

5 Observables Extinction cross section [m 2 ] Power removed from beam Incident intensity Extinction = scattering + absorption removed from the beam Re-radiated into all angles Lost as heat in the scatterer

6 Lycurgus cup 500 A.D. Scattered Transmitted

7 Mie theory dielectric n=2.7 sphere Multipole resonances Rayleigh/dipole (r/λ) 4

8 Mie theory dielectric n=2.7 sphere Rayleigh/dipole p 4 πε α E with α a ε ε 3 m = 0 easy 0 easy = ε + 2εm Weak scattering limit, unless (r/λ) 4 ε =-2ε m Metal particle

9 Physical quantity - polarizability Small object kd <<1 - incident field is approximately constant Volume polarization (weak index so E=Ein) Total dipole moment Larger particles & ε means - modified dipole moment - also higher order multipoles

10 Solution r p r E r E a r E r E r E r E m m m m m m m πε θ θ θ ε ε ε ε θ θ ε ε ε θ ε ε ε ε θ cos cos cos cos cos cos cos + = + + = Φ + = + + = Φ E 0 ε m ε z r θ a Easy to verify with 4 2 m SI SI m m p E a ε ε α α πε ε ε ε = = + Inside sphere: homogeneous field Outside sphere: background field plus field of a dipole with In the ball: Outside:

11 Plot of solution

12 Measured data and model for Ag ε Measured data: ε' ε" Drude model: ε' ε" Modified Drude model: ε' ε" Wavelength (nm) ε' Drude model: ω ε' = 1 ω ε' ε = 2 p 2 ω ω, 2 p 2, ε" = ω ω ε" = 2 p 2 ω ω γ Modified Drude model: 2 p 2 Contribution of bound electrons Ag: ε = 3. 4 γ

13 Metal sphere p 4 πε α E with α a ε ε 3 m = 0 easy 0 easy = ε + 2εm Drude model for a metal: Lorentzian `plasmon resonance 2 2 ω p 3 ω 0 ε = 1 means αeasy = a 2 2 ω( ω + iγ ) ω0 ω + iωγ Resonance where ε(ω 0 ) = -2 ε m Response scales with the volume V α exceeds V by factor 5 to 10 Shape shift condition ε = -2 ε m

14 General properties Extinction crosssect. (µm 2 ) nm Ag particle in glass E 2 / E in Wavelength (nm) Cross section ~ 10x πr 2 Strong dipolar near field Q ~5 means 95% of loss is radiation into free space σ as large as σ 3 λ 2 Plasmon particle is a solid state `strongest point-scatterer 2π

15 Revisiting polarizability Extinction: how much power is taken from the beam (in SI units)? T iωt 1 iωt dpe 1 iωt iωt W = Re[ Ee ] Re[ ] dt = Re[ Ee ] Re[ iωα Ee ] dt T dt T 0 0 T 1 iωt * iωt iωt * * iωt W = ( e + e )( iωα e iωα e ) dt 4T E E E E 0 Only cross terms survive cycle average T 1 * 2 2 W = ( iωα + iωα ) + oscill.terms ( ± 2 ω) dt 4T E E 0 W = ω Im α E 2 2 T

16 Radiation pattern of a dipole

17 Revisiting polarizability Classical model of harmonically bound electron Describes atom, and scatterer alike 2 p 03V 0 i t i t () t = ε ω ω ω e α ( ) 2 2 SI ω e ω0 ω iωγ E = E Lorentzian resonance Scattering: how much power does p radiate? W 2π π 2π π 2 2 S nda dϕ r sinθ Edipole = dϕ sphere r 2 psinθ sinθ 4πε r 0 2 α 2

18 Optical theorem Equate extinction to scattering (energy conservation) ω c Extinction 2 4 π Im αeasy [ m ] Work done to drive p 8π ω 3 c Scattering α Easy [ m ] Rayleigh / Larmor 1. Very small particles scatter like r 6 /λ 4 (Rayleigh) 2. For very small particles absorption wins ~ r 3 /λ 3. Big α 2 implies large Im α

19 p 4 πε α E with α a ε ε 3 m = 0 easy 0 easy = ε + 2εm Paradox: dielectric constant only knows about absorption How could α possibly satisfy the optical theorem? The quantity k=ω/c does not even appear!

20 Optical theorem Receipe to fix quasi-static (Rayleigh) dipoles P p α α α 3 ik 2 2 Radiated power P W 2 Intrinsic P damping α 2 a 1 a= = k 2 W Imα 3 dp Ein dt 3 = It is easy to check that this secures Im α = α [ k Im ] α Int [ k ] 3 α 3 Radiative damping α ( ω) = Vω ω ω iωγ 0 Work done to drive p 3 Γ= γ / 3 kvω ω Imα

21 Example: simple spheres Calculated exact cross section of Au spheres r=10-50 nm Dashed line: σ = 4π k Im α easy Surprises -Peak bounded by

22 Silver particles on resonance, vs size α x 2/3 k Radius (nm) Quality factor ω res /γ 1/(kr) Radius (nm) Albedo ~(kr) Radius (nm) Universal features for any particle shape (if dipole dominates) - Small particles have small cross sections - Albedo - fraction of scattering to extinction, increases with size - Scattering is a loss that reduces the linewidth or quality factor

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24 Example dimer in static approximation Dimer in a static approximation Linear problem Symmetric, but not real matrix 1/polarizability on the diagonal Interaction on the off-diagonal - this will shift resonances

25 Hybridization picture Approximating this in near field electrostatics Means:

26 Hybridization picture New eigen-frequencies Hybridization (as in: orbitals) Normal modes (as in: pendulums)

27 Hybridization (exercise)

28 Exact calculation

29 Green function of free space The field at r of a dipole at the origin depends linearly on p Dyadic form Green dyadic: 3 x 3 tensor that quantifies field in response to source at r oriented along p

30 General form in dipole approximation Linear problem - response can be found by simple inversion of M Symmetric, but not real matrix 1/polarizability on the diagonal (3x3 blocks) Dipole Interaction on the off-diagonal

31 Example dimer in scalar approximation Short hand Note how the inversion of M is equivalent to an infinite series Matrix inversion = infinite number of multiple scattering orders

32 Plasmon ruler Idea: Alivisatos group

33 AC current 1 meter

34 Yagi-Uda Quantum emitter λ = 650 nm 100 nm silver nanospheres 1 micrometer Snap shot of electric field - Far-field: photon with 90% probability in a narrow beam - Broadband (> 300 nm bandwidth in the visible) - 90% chance that the photon is not lost to heat de Waele Nano Lett 2007 / Koenderink Nano Lett 2009 / Curto Science 2010, Coenen Nano Lett. 2011

35 Yagi Uda antenna Curto et al., Science (2010).

36 Phased array Suppose I have N dipoles, radiating into the far field as Taylor expand for large Viewing distance r 1 r 2

37 Far field If all dipoles are parallel (not meaning in phase) Overall spherical wave Array geometry Structure factor Single scatterer Form factor

38 Example Suppose I can set up a phase advance from dipole to dipole in a 1 D chain that reads:

39 Yagi-Uda Optical domain: a chain of plasmon particles driven by a single molecule

40 Funneling light into a single beam Sample: perforated Au film - hexagons of 440 nm pitch Sources: dilute fluorophores Atto 640 dye diffusing in H 2 O We pump the central hole only in a confocal microscope L. Langguth, D. Punj, J. Wenger, A. F. Koenderink ACS Nano 7, 8840 (2013)

41 Emission can be redirected Single hole One shell Two shells Three shells k y Fourier image k x (up to NA=1.2) Emission strongly redirected in a narrow beam Single aperture: 10x brightness enhancement (full NA), pump E 2 Array: 40x enhancement in forward direction L. Langguth, D. Punj, J. Wenger, A. F. Koenderink ACS Nano 7, 8840 (2013)

42 Emission can be redirected Single molecule phased array antenna k y Fourier image Optimal for normal outcoupling Emission strongly redirected in a narrow beam - Use a waveguide mode in this case the gold/glass SPP Single aperture: 10x brightness enhancement (full NA), pump E - Favorite periodicity a=λ 2 Array: 40x enhancement SPP in forward direction - Plasmon also enhances pump light Quantitative per molecule improvement (FCS - <N>=3.5) L. Langguth, D. Punj, J. Wenger, A. F. Koenderink ACS Nano 7, 8840 (2013)

43 Plasmon antenna uses Diffractive arrays - LEDs & solar cells Lozano, Light Sci Appl (2013) Torma & Barnes, Rep Prog Phys (2015) Single photon sources Aouani Nano Lett (2011), Langguth ACS Nano (2013) Scanning probe microscopy Nano-photodetectors Nano-thermal control Nano-photochemistry HAMR Magnetic recording

44 Metamaterials & magnetic scatterers Femius Koenderink Center for Nanophotonics FOM Institute AMOLF Amsterdam

45 Optical response of natural materials Damped solutions Propagating waves Free electron plasma Drude model Localized polarizabilities Lorentz oscillator model

46 Thought experiment Damped solutions Propagating waves Propagating waves Damped solutions

47 How ε,µ come about Conventional material `Meta material Artificial atoms Magnetic polarizability

48 What is special about ε<0, µ<0 Veselago (1968, Russian only) / Pendry (2000) Conventional choice: If ε<0, µ<0, one should choose: Propagating waves with `Negative index of refraction

49 Snell s law with negative index Povray raytrace of Snells law S 1 S 2 Does negative index mean negative refraction of rays?

50 Refraction movies Positive refraction n=1 n=2 Negative refraction n=1 n=-1 W.J. Schaich, Indiania

51 Energy flow and k If both ε = µ = -1, plane wave: (1) k, E, B means (2) Poynting vector sets energy flow S Energy flow to the left S E B H k Phase fronts to the right

52 Refraction k n1 n2 n 1 ω/c n 2 ω/c Generic solution steps: 1) Plane waves in each medium to be matched at boundary 2) Use k conservation to find allowed refracted wave vectors 3) Use causality to keep only outgoing waves 4) Match field continuity at boundary to find r and t

53 Snell s law n=1 n=-1 k in k k Energy flux Exactly what does negative refraction mean?? (1) k is conserved (2) Energy flows away from the interface (3) Phase advances towards the interface (4) Snell s law for rays holds with negative refractive index

54 Negative index lens A flat n=-1 Negative index slab focuses light NIM slab The image is upright The lens position is irrelevant Object-image spacing is 2d

55 Negative refraction Negative refraction makes a perfect lens Arbitrary ε, µ makes an invisibility cloak