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even
odd
semi�inf .
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0.1
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0.3
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even
odd
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a) b)
0
12
m � 4
edgereflection
0.5 1 2 5 10 20k0R
0.2
0.4
0.6
0.8
1.0�rm�
m � 0
12
4
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1 2 5 10 20 50k0R
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�40
�20
0
20
40
Φmr
a) b)
n � 1
n � 5
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6 10 20 40 80 160d�nm0
200
400
600
800
1000
1200
Rn�nm
300 400 500 600 700Ν�THz0.5
1.0
1.5
2.0|Emax
z |2 /a.u.
R200 500
Ρ�nm�0.5
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R200 500
Ρ�nm�0.5
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a)
c)b)
1st 2nd
3rd
4th
5th
pred. resonances4
6nm
160nm
4 10 16xn�J 1�0
600
1200Rn�nm
J 1�kSPP' Ρ�R1
R4
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�0.5
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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
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
(√εdk2
0 − k2zρ
)eikzzdkz
Hϕ and∫
Ez · Hϕdz at ρ = R
Im ≡∫ ∞
−∞
H1m
(√εdk2
0 − k2zR
)
DH1m
(√εdk2
0 − k2zR
)
·√
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