Nanoscale antennas. Said R. K. Rodriguez 24/04/2018

Transcription

1 Nanoscale antennas Said R. K. Rodriguez 24/04/2018

2 The problem with nanoscale optics How to interface light emitters & receivers with plane waves? Ε ii(kkkk ωωωω) ~1-10 nm ~ nm

3 What is an antenna? An antenna is a device that converts free-space radiation into localized energy, and vice versa Tx/Rx Antenna Radiation Novotny & van Hulst, Nat.Photon. 5, 83 (2011).

4 What is an antenna? An antenna is a device that converts free-space radiation into localized energy, and vice versa Tx/Rx Antenna Radiation At radio frequencies, E = 0 inside the metal perfect metal Novotny & van Hulst, Nat.Photon. 5, 83 (2011).

5 What is an antenna? An antenna is a device that converts free-space radiation into localized energy, and vice versa Tx/Rx Antenna Radiation At optical frequencies, E 0 inside the metal. Consequence: radio-freq. antenna designs cannot be directly scaled Novotny & van Hulst, Nat.Photon. 5, 83 (2011).

6 Dielectric constant for Ag 50 0 ε" ε Measured data: ε' ε" Drude model: ε' ε" Modified Drude model: ε' Wavelength (nm) Finite ε leads to field penetration ε'

7 From plasmons to plasmonics Plasmons in the bulk oscillate at ω p determined by the free electron density and effective mass 2 drude Ne ω p = mε k Plasmons confined to surfaces that can interact with light to form propagating surface plasmon polaritons (SPP) k x = k' x + ik" x ω ε mε d = c ε m + ε d 1/ 2 Localized surface plasmons in nanoparticles optical resonance frequency depends on shape & size; k is irrelevant

8 Colors of gold nanoparticles reflection Lycargus cup, 4thC AD transmission Stained Notre Dame de Paris 1260

9 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

10 Linear response to applied field Small object kd <<1 - incident field is approximately constant Volume polarization (weak index so E=E in ) Total dipole moment Larger particles & ε : larger dipole moments

11 Electrostatic sphere Consider a sphere in a static field E 0 ε m Laplace equation: ε Φ1 = 0 ( r < a) a Φ = 0 r > a 2 ( ) θ r z Boundary conditions set by = ( ) = ( Φ) = 0 D ε E ε Φ Φ 1 2 ( r = a), ε = ε ( r = a), lim Φ = E z 1 = Φ 2 m 2 0 r r Φ r

12 Solution [ see J. D. Jackson, Classical Electrodynamics, Ch. 4] In the ball: Outside: ε ε m 3ε m Φ 1 = Er 0 cosθ + Er 0 cosθ = Er 0 cosθ ε + 2εm ε + 2εm Φ = Ercosθ + a ε ε E cosθ = Ercosθ + p cosθ 3 m ε + 2ε m r 4πε 0ε mr Inside sphere: homogeneous field Outside sphere: background field plus field of a dipole with r r p α E with α 4πε ε a ε ε 3 m = SI 0 SI = 0 m ε + 2εm E 0 ε m ε a θ r z

13 Metal sphere pp = 4ππεε 0 ααee 0 αα = aa 3 εε εε mm εε + 2εε mm 0 Drude model for a metal: Lorentzian `plasmon resonance εε = 1 ωω pp 2 ωω(ωω + iiiiii) means αα = aa 3 ωω 0 2 ωω 0 2 ωω 2 + iiiiii 0 Resonance at ε(ω 0 ) = -2 ε m Response scales with the volume V α exceeds V by factor 5 to 10 Shape shifts condition ε = -2 ε m γ still needs to include radiation damping

14 Revisiting polarizability Classical model of harmonically bound electron describes atom, and scatterer alike, as an oscillating dipole p 3V i t () t = ε ω ω e αsi ( ω) e ω ω ωγ E = E i Extinction: how much power is taken from the beam? iωt Lorentzian resonance Cycle average work done by E on p W dp Ein dt Imα

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[ e ] Re[ ] dt Re[ e ] Re[ iωα e ] dt T E = dt T E E 0 0 T 1 iωt * iωt iωt * * iωt W = ( e + e )( iωα e iωα e ) dt 4T E E E E 0 T 1 * 2 2 W = ( iωα + iωα ) + oscill.terms ( ± 2 ω) dt 4T E E 0 W = ω Im α E 2 T 2

16 Revisiting polarizability Classical model of harmonically bound electron describes atom, and scatterer alike as an oscillating dipole p 3V i t () t = ε ω ω e αsi ( ω) e ω ω ωγ E = E i iωt 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

17 Optical theorem Equate extinction to scattering (energy conservation) Extinction 4ππkk Im αα [mm 2 ] Work done to drive p Scattering 8ππ 3 kk4 αα 2 [mm 2 ] 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

18 Optical theorem Equate extinction to scattering (energy conservation) Extinction 4ππkk Im αα [mm 2 ] Scattering 8ππ 3 kk4 αα 2 [mm 2 ] Work done to drive p Rayleigh / Larmor Since Im αα < αα αα 3 2 λλ 2ππ 3 Upper bound on the strongest possible dipole scatterer

19 Extinction Interference effect 1 PP eeeeee = DD Sext e r da = Re E DD 2 i H s +Es H i er da On resonance QQ eeeeee = σσ eeeeee AA pppppppp Out of resonance 2ππππ λλ = 0.3

20 Extinction Interference effect 1 PP eeeeee = DD Sext e r da = Re E DD 2 i H s +Es H i er da QQ eeeeee = σσ eeeeee AA pppppppp r=20 nm Ag particle, in n=1.5 (glass) σ ext = A part

21 Extinction Interference effect 1 PP eeeeee = DD Sext e r da = Re E DD 2 i H s +Es H i er da σσ eeeeee Question: what doesqqthe eeeeee = above expression tells us about AA the detector needed to measure pppppppp the full extinction? 2ππππ λλ = 0.3

22 Summary Antennas convert free-space radiation into localized energy & viceversa At optical frequencies, E-field penetrates into the metal. This leads to surface plasmon resonances Extinction: Work done by E on p Im(α) Interference of incident & scattered field Subwavelength particles can absorb and scatter much more light than is geometrically incident on them. In general, Q ext >1 on resonance and Q ext <1 off resonance

23 10 min. break

24 Approaches to controlling light Resonant nanoparticles Photonic crystals Surface Plasmon Polaritons

25 Dipole radiation 1 dipole (vector): EE(rr) = μμ 0 ωω 2 GG(rr) pp 1 dipole (scalar): ψψ 1 rr = eeiiiiii rr pp

26 Dipole arrays 1 dipole (vector): EE(rr) = μμ 0 ωω 2 GG(rr) pp a λ 1 dipole (scalar): ψψ 1 rr = eeiiiiii rr pp Dipole array (scalar): ψψ tttttt rr = ψψ 1 (rr) AAAA Depends on positions & complex amp. of scatterers Fourier transform of geometry (more ahead)

27 Far-field of 2 dipoles ψψ tt = ψψ 1 + ψψ 2 = pp ee ii(kkkk 1 ββ/2) rr 1 cos θθ 1 + ee ii(kkkk2+ββ/2) rr 2 cos θθ 2 θ 1 r 1 β = phase difference between dipoles θ 2 r 2

28 Far-field of 2 dipoles ψψ tt = ψψ 1 + ψψ 2 = pp ee ii(kkkk 1 ββ/2) rr 1 cos θθ 1 + ee ii(kkkk2+ββ/2) rr 2 cos θθ 2 θ 1 r 1 β = phase difference between dipoles ψψ tt rr = pp eeiiiiii AAAA Exercise: rr θ 2 1 r 2 AAAA = cos (kkkkkkkkkk θθ + ββ) 2

29 Far-field of 2 dipoles ψψ tt = ψψ 1 + ψψ 2 = pp ee ii(kkkk 1 ββ/2) rr 1 cos θθ 1 + ee ii(kkkk2+ββ/2) rr 2 cos θθ 2 θ 1 r 1 β = phase difference between dipoles ψψ tt rr = pp eeiiiiii AAAA Exercise: rr θ 2 1 r 2 AAAA = cos (kkkkkkkkkk θθ + ββ) 2 Note: β=π & θ=π/2 AF = 0 kd

30 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

31 Hybrid modes Hybridization (exercise)

32 Arrays of coupled dipoles

33 Arrays of coupled dipoles 1 dipole a x n array of dipoles d= 100 nm a x = a y = 450 nm n = 1.5

34 Light cone & diffraction

35 Light emission from plasmonic array NA of objective k y k x

36 Extinction of Au nanorod arrays LSPR S.R.K. Rodriguez et al., Phys. Rev. X 1, (2011).

37 Bright even / Dark - odd Diffraction / Bloch theorem determines mode dispersion Mode symmetry + illumination determines what you excite

38 Collective resonances Measurements Shaper resonances by adding nanoparticles Coupled dipole calculations S.R.K. Rodriguez et al., Physica B 407, 4081 (2012).

39 Uses of resonant nanostructures

40 Enhanced local fields On resonance, ~ 10 4 enhanced intensity Au spheres 5,8,20 nm, gaps of 1-3 nm E 2 / E in 2

41 Single molecule Fluorescence Enhancement 100 nm A. Kinkhabwala et al., Nat. Phot. 3, 654 (2009) FFFF = ηη(rr 0, ωω eeee ) DD(rr 0, ωω eeee ) ηη 0 (rr 0, ωω eeee ) DD 0 (rr 0, ωω eeee ) EE (rr 0, ωω aaaaaa ) 2 EE 0 (rr 0, ωω aaaaaa ) 2

42 Yagi-Uda nanoantenna A. F. Koenderink, Nano Lett. 9, 4228 (2009)

43 Directional emission from localized sources 100 nm A. Curto et al., Science 329, 930 (2010)

44 Directional emission from extended sources emitting layer 100 nm G. Lozano et al., Light Sci. Appl. e66 (2013)

45 Directional emission from extended sources emitting layer emitting layer 100 nm

46 Sensing Single protein binding/unbinding Refractive index sensing n P. Offermans et al., ACS Nano 5, 5151 (2011)

47 Biological imaging Diffraction unlimited resolution λ= 633 nm Novotny & van Hulst, Nat.Photon. 5, 83 (2011).

48 Nonlinear effects Enhanced mode mapping Generating new frequencies P. Ghenuche et. al. Phys. Rev. Lett. 10, (2008) H. Harutyunyan et. al. Phys. Rev. Lett. 108, (2012)

49 Summary Small particles of size < λ/10 scatter like dipoles Arrays of dipoles can be described by effective polarizability Nanoantennas can be used to enhance: local fields absorption & spontaneous emission Sensing Biological imaging Nonlinearities