Circular Optical Nanoantennas - An Analytical Theoryrobertfilter.net/files/Poster_Bad_Honnef.pdf1 P....

1
even odd semiinf . 1.0 10.0 5.0 2.0 3.0 1.5 7.0 k 0 d 0.0 0.1 0.2 0.3 0.4 r 1 even odd semiinf . 10.0 5.0 2.0 20.0 3.0 1.5 15.0 7.0 k 0 d 90 80 70 60 50 40 Φ 1 r a) b) 0 1 2 m 4 edgereflection 0.5 1 2 5 10 20 k 0 R 0.2 0.4 0.6 0.8 1.0 r m m 0 1 2 4 edgereflection 1 2 5 10 20 50 k 0 R 60 40 20 0 20 40 Φ m r a) b) n 1 n 5 6 10 20 40 80 160 dnm 0 200 400 600 800 1000 1200 R n nm 300 400 500 600 700 ΝTHz 0.5 1.0 1.5 2.0 |E max z | 2 /a.u. R 200 500 Ρnm 0.5 0.5 1.0 E z E z max R 200 500 Ρnm 0.5 0.5 1.0 E z E z max a) c) b) 1 st 2 nd 3 rd 4 th 5 th pred. resonances D EȞ 4 6nm 160nm 4 10 16 x n J 1 0 600 1200 R n nm J 1 k SPP ' Ρ R 1 R 4 200 600 1000 1400 Ρnm 0.5 0.0 0.5 1.0 E z E z,max a) b) c) A B and HANKEL plasmons propagate outwards accross the resonator and get reflected composition of the stack fixes dispersion relation and mode profile • neglecting reflection into other modes; field representation (inner, outer): ansatz: continuity of to derive the reflection coefficient with the abbreviations Institute of Condensed Matter Theory and Solid State Optics, Abbe Center of Photonics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany Robert Filter, Jing Qi (戚婧), Carsten Rockstuhl, and Falk Lederer Circular Optical Nanoantennas - An Analytical Theory r m = 2π d k SPP σH 1 m (k SPP R) - DH 1 m (k SPP R) I m -2π d k SPP σH 2 m (k SPP R)+ DH 2 m (k SPP R) I m E m,- z (ρ, z ) = A m (k SPP ρ) · a (z ) with A m (k SPP ρ) = H 1 m (k SPP ρ)+ r m · H 2 m (k SPP ρ) 2 · k SPP R n,m + φ r m =2 · x n (J m ) E m,- z (ρ, z )= J m (k SPP ρ) · a (z ) E m,+ z (r)= -∞ c m (k z ) H 1 m d k 2 0 - k 2 z ρ e ik z z dk z H ϕ and E z · H ϕ dz at ρ = R I m -∞ H 1 m d k 2 0 - k 2 z R DH 1 m d k 2 0 - k 2 z R · d k 2 0 - k 2 z · B - (k z ) · B + (k z ) dk z Subject and Motivation DH 1/2 m (x) x H 1/2 m (x) -∞ ( z ) a (z ) 2 dz , B ± (k ) -∞ ( z ) a (z ) e ±ikz dz antennas for visible light now feasible 1,2 metals no perfect conductors in this spectral domain goal: understand nanoantenna characteristics on analyti- cal grounds - self consistent predictions of supported plasmonic eigenmodes requires three ingredients profile of plasmonic field, dispersion relation and reflection coefficient to calculate resonances using a FABRY-PEROT model 3 circular nanoantennas axial symmetry properties tunable via stack composition Theory of HANKEL Reflection Field Profile and Scaling robertfilter.de A Simple Resonator Model HANKEL functions diverge at origin - cannot be eigenmodes • stationary solutions in given symmetry: BESSEL functions - field inside: apparant length change due to phase of reflection •FABRY-PEROT resonance condition: A plane wave excites a dipolar plasmonic BESSEL-resonance of an 80 nm thick metallic disc. The reflection phase leads to an apparent length change. resonator model implies form of the field and scaling of resonant radii • educated guesses; have to be verified; simplest structure: a metallic disc • numerical simulations: plane wave excitation at ν = 625 Thz a) & c) field profile • 80 nm thick disc, excitation of even mode • the electric field shows a qualitative agree- ment to a BESSEL type plasmonic field b) scaling • resonant radii for thicknesses from 6 nm to 160 nm are linearly related to the roots of J 1 resonant radii: theory (full lines) vs. full wave simulations (dots) resonance condition with calculated phases of reflection predict resonant disc radii for different thicknesses d agreement for all observed orders n 1 P. MÜHLSCHLEGEL et al. Resonant optical antennas, Science 308, 1607 (2005) 2 J. DORFMÜLLER et al. Near-field Dynamics of Optical Yagi-Uda Nanoantennas, Nano Lett. 11, 2819 (2011) 3 T. H. TAMINIAU et al. Optical Nanorod Antennas Modeled as Cavities for Dipolar Emitters, Nano Lett. 11, 1020 (2011) 4 R. GORDON Vectorial method for calculating the Fresnel reflection of surface plasmon polaritons, Phys. Rev. B 74, 153417 (2006) References Conclusions Spectral Predictions Limiting Cases resonances of a silver disc, nm a) spectrum with predicted resonances b) & c) actual field distributions for 4 th and 5 th resonance • SPP highly damped for 5 th order - deviation from BESSEL form, other effects important • also spectral very good agreement to numerics within limitations of theory A: verification of known results 4 in infinite disc limit for several orders B: even and odd modes converge to same result for increasing thickness theory for radially propagating HANKEL-type SPPs in piecewise homogeneous circular nanoantennas properties explained by FABRY-PEROT model using phase of reflection in agreement to simulations antenna properties tunable via stack composition How to use this results to understand the characteristics of circular nanoantennas? a(z ) R = 900

Transcript of Circular Optical Nanoantennas - An Analytical Theoryrobertfilter.net/files/Poster_Bad_Honnef.pdf1 P....

Page 1: Circular Optical Nanoantennas - An Analytical Theoryrobertfilter.net/files/Poster_Bad_Honnef.pdf1 P. MÜHLSCHLEGEL et al. Resonant optical antennas, Science 308, 1607 (2005) 2 J. DORFMÜLLER

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• HANKEL plasmons propagate outwards accross the resonator and get reflected• composition of the stack fixes dispersion relation and mode profile • neglecting reflection into other modes; field representation (inner, outer):

• ansatz: continuity of to derive the reflection coefficient

with the abbreviations

Institute of Condensed Matter Theory and Solid State Optics,Abbe Center of Photonics, Friedrich-Schiller-Universität Jena,

Max-Wien-Platz 1, 07743 Jena, Germany

Robert Filter, Jing Qi (戚婧), Carsten Rockstuhl, and Falk Lederer

Circular Optical Nanoantennas - An Analytical Theory

rm =2πεd kSPP σ H1

m(kSPP R) − DH1m(kSPP R) Im

−2πεd kSPP σ H2m(kSPP R) + DH2

m(kSPP R) Im

Em,−z (ρ, z) = Am (kSPP ρ) · a (z) with

Am (kSPP ρ) = H1m (kSPP ρ) + rm · H2

m (kSPP ρ)

2 · k′SPP Rn,m + φr

m = 2 · xn (Jm)

Em,−z (ρ, z) = Jm (kSPP ρ) · a (z)

Em,+z (r) =

∫ ∞

−∞cm (kz)H1

m

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0 − k2zρ

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DH1m

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·√

εdk20 − k2

z · B− (kz) · B+ (kz) dkz

Subject and Motivation

DH1/2m (x) ≡ ∂xH1/2

m (x) , σ ≡∫ ∞

−∞ε (z) a (z)2 dz ,

B± (k) ≡∫ ∞

−∞ε (z) a (z) e±ikzdz

• antennas for visible light now feasible 1,2

• metals no perfect conductors in this spectral domain

• goal: understand nanoantenna characteristics on analyti- cal grounds - self consistent predictions of supported plasmonic eigenmodes requires three ingredients• profile of plasmonic field, dispersion relation and• reflection coefficient

to calculate resonances using a FABRY-PEROT model 3

• circular nanoantennas• axial symmetry• properties tunable via

stack composition

Theory of HANKEL Reflection

Field Profile and Scaling

robertfilter.de

A Simple Resonator Model• HANKEL functions diverge at origin - cannot be eigenmodes• stationary solutions in given symmetry: BESSEL functions - field inside:

• apparant length change due to phase of reflection• FABRY-PEROT resonance condition:

A plane wave excites a dipolar plasmonic BESSEL-resonance of an 80 nmthick metallic disc. The reflection phase leads to an apparent length change.

• resonator model implies form of the field and scaling of resonant radii• educated guesses; have to be verified; simplest structure: a metallic disc• numerical simulations: plane wave excitation at ν = 625 Thz

• a) & c) field profile • 80 nm thick disc, excitation of even mode• the electric field shows a qualitative agree-

ment to a BESSEL type plasmonic field• b) scaling• resonant radii for thicknesses from 6 nm to

160 nm are linearly related to the roots of J1

• resonant radii: theory (full lines) vs. full wave simulations (dots)• resonance condition with calculated

phases of reflection predict resonant disc radii for different thicknesses d• agreement for all observed orders n

1 P. MÜHLSCHLEGEL et al. Resonant optical antennas, Science 308, 1607 (2005)2 J. DORFMÜLLER et al. Near-field Dynamics of Optical Yagi-Uda Nanoantennas, Nano Lett. 11, 2819 (2011)3 T. H. TAMINIAU et al. Optical Nanorod Antennas Modeled as Cavities for Dipolar Emitters, Nano Lett. 11, 1020 (2011)4 R. GORDON Vectorial method for calculating the Fresnel reflection of surface plasmon polaritons, Phys. Rev. B 74, 153417 (2006)

References

ConclusionsSpectral Predictions Limiting Cases

• resonances of a silver disc, nm • a) spectrum with predicted resonances • b) & c) actual field distributions for 4th and 5th resonance

• SPP highly damped for 5th order - deviation from BESSEL form, other effects important• also spectral very good agreement to numerics within limitations of theory

• A: verification of known results 4 in infinite disc limit for several orders• B: even and odd modes converge to same result for increasing thickness

• theory for radially propagating HANKEL-type SPPs in piecewise homogeneous circular nanoantennas• properties explained by FABRY-PEROT model using phase of reflection in agreement to simulations• antenna properties tunable via stack composition

How to use this results to understand the characteristics of circular nanoantennas?

a(z)

R = 900