Metallic and Graphene-based

Nano-Antennas

Dmitry N. Chigrin

Institute of High-Frequency and Communication TechnologyFaculty of Electrical, Information and Media EngineeringUniversity of Wuppertal, Germany

My group:● Dr. Christian Kremers● Fereidoon Ahmedi● Ignacio Llatser (DAAD, Universitat Politecnica de Catalunya)● Alexander Malashtan (DAAD, Belarusian Academy of Science)● Dr. Sergei V. Zhukovsky (University of Toronto, Canada)

Collaborators:● Prof. A. Cabellos-Aparicio, Prof. E. Alarcón

(Universitat Politecnica de Catalunya, Spain)● Prof. H. Giessen (Universität Stuttgart, Deutschland)● Prof. A. V. Lavrinenko (NanoDTU, DTU, Denmark)

Projects:DFG KR-1726/4-1, DFG KR-1726/5-1,DFG CH-407/2-1, DFG CH-407/5-1

Acknowledgments

(after Barnes)

E

Ex

Ez

Surface Plasmon Polaritons

● Surface wave (surface plasmon polaritons-SPP) could be excited at the dielectric-metal interface

Surface Plasmon Polaritons

Gold:

● Dispersion relation of SPP

Particle Plasmon Polariton Resonance

+++---

Metallic particles

● Resonant response at optical frequency at nanometer length-scale(20-200 nm)

Examples

[After Novotny (2011)]

[After Zheludev (2009)]

[After Giessen (2009)]

Optical Nano-Antennas Chiral Surfaces

Metamaterials

[After Oulton (2011)]

Nanolaser

Particle Plasmon Polariton Resonance

Start with Green-function solution of the wave equation

Effective Dipole Moment

Resulting effective dipole moment of each rod:

copper

Optical Frequency Range

Scattering cross section of the single gold nanoblocks

Lengths are left to right from 60 nm to 120 nm with 10 nm step. Width and height of all nanoblocks are 20 nm.

Graphene Plasmonics

[after Jablan]

Support TM surface plasmon polariton modes at THz frequencies

What is about THz frequency range?● Metal antennas are several hundreds micrometer long

… is not acceptable for nano-applications.● Graphenas - Graphene-based plasmonic nano-antennas

(Collaboration with Llatser, Cabellos-Aparicio and Alarcón)

SPP Resonances

● SPP dispersion relation:

● Fabry-Perot resonance condition:

● Complex frequencies to model SPP cavity modes as uncoupled damped harmonic oscillators

SPP Resonances

● Multiple, resonator length dependent resonances

Cross Section

First resonance is well described using single dipole!

Numerical Methods

● Thin-layer approximation: slab with a thickness Δ and normalized conductivity

● Impedance surface model: 2D surface with imposed BC

Fundamental Resonance

Resonance frequency decreases when increasing the antenna length… but how does it compare to metallic antennas?

in THz band graphenas are just a few micrometers

Graphennas vc Metallic Antennas

green line: metallic antennasblue line: graphennasdots: simulation results

Graphennas: (i) lower frequencies, but also(ii) better scaling!

Influence of Substrate

● Resonant shifts towards lower frequencies

● Scattering efficiency reduces

● … but could be compensated using substrate resonances

Tunability

● Very sensitive to bias (dopping)!

● … but temperature stable!

Plasmonic Dimer

Field Enhancement Mode Hybridization

What happens if two resonant particlesare placed close to each other?

Hydrogen Sensing

(Collaboration with Giessen)

● Gold bowtie antenna is placed next to a palladium nanodisk● Presence of hydrogen changes both:

● palladium refractive index● and lattice constant (leads to the

volume change of the dot)

● … due to strong field enhancement scattering resonance is very sensitive both to index and geometry changes

Hydrogen Sensing

● … due to strong field enhancement scattering resonance is very sensitive both to index and geometry changes

Emission Modification

Two-level system in the center of bowtie antenna gap dipole moment is parallel to the antenna axis

Emission Modification

To enter the strong coupling regime one need stronger fields

Candidate: bowtie antenna

plane wave scatteringproblem calculatedwith HFSS

Emission Modification

Radiation dynamics of two-level system (“atom”) in nano-structures with a loss channel is treated using the Welsch quantization approach

Within the electric-dipole and rotating wave approximations the minimal-coupling Hamiltonian is given by

here are bosonic elementary excitation operators, atom position, transition frequency and dipole moments.

(Dung et al., PRA 62 (2000) 053804)

Upper state occupation probability amplitude is given via integro-differential equation:

with all parameters of the inhomogeneous environment relevant for atomic evolution included in classical Green's function via:

Emission Modification

z

y

x

xz

xx

x y

r ' r

Ez

Ex

E y

Dyadic Green's function is a solution of wave equation for elementary dipole excitations:

dyadic product

Matrix representation

Emission Modification

Definition of the i’th column of the dyadic Green’s function:

(i) in free space

(ii) with scatterer

(ii)-(i)

Dyadic Green's function can be calculated using 1D integral equation formalism

Solution is given by equation:

electric field of point dipole in free space

Emission Modification

In this case integro-differential equation for upper state occupation probability amplitude can be solved analytically [Dung (2000)]

resulting in

;

Strong coupling condition:

In the weak coupling regime (Markovian approximation) we have well known single exponential decay

Emission Modification

Green's function: direct numerical calculations (HFSS)

Resonances are well separated

First resonance can be approximated with Lorentzian

Fit

HFSS

Emission Modification

Emission decay for transition frequency near the first resonance:

Non-Markovian dynamics for realistic dipole moments!

d=10 debye

d=15 debye

d=30 debye

Full solutionMarkovian

1

2

3

1

2

3

Nano-Laser

● Gold bowtie antenna in dye-doped polymer layer● Active medium is modeled as 4-level system● Coupled to 3D FDTD solver

Glass (n=1.5)

Active layer (n=1.5)

Optical pump

30nm

15nm

90nm

4nm

Nano-Laser

pumpinggain

Cold spectrum (solid line) and first two modes

● Gain is tuned to the first mode● Absorption and pumping are tuned to the second mode

Nano-Laser

● Structure lases at the first resonance frequency● Lasing intensity grows linearly

Nano-Laser

● After some time a population inversion is build up … followed by the build up of the laser field

Mode Hybridization

Dimer Polarizabilities

For elongated particles one can approximate induced dipole moment as

Effective dimer polarizability of planar meta-atom can be expressed as:

with couplings constant:

Gold Dimer

Effective electric quadrupol moment:

Effective electric dipole moment:

Effective magnetic dipole moment:

Total cross section: Gold Dimer

How accurate is the coupled dipole model?

Pretty good for moderate dipole couplings!

a1=80 nm; a2=75 nmb= 20 nm; φ=45°

R=100 nm

R=60 nm

Differential cross section: Gold Dimer

How accurate is the coupled dipole model?

Pretty good for moderate dipole couplings!

a1=80 nm; a2=75 nmb= 20 nm; φ=45°

R=100 nm

Planar chiral materials

3D enantiomers

Planar chiral materials

● Transmission is different for LH/RH polarized EM waves

Effective material parameters

“Microscopic field” – field between “meta”-atoms is described by Maxwell's equations for homogeneous space

“Macroscopic field” – field averaging has to be introduced

Averaging is in general different near the surface (transition layer)and in the bulk

Effective material parameters

Microscopic fields,charges and currents:

Macroscopic fields

Effective material parameters

Average current – multipole expansion

Vortex free (electric) current Vortex like (magnetic) current

Material equations:

Averaged: electric dipole electric quadrupole magnetic dipole

Effective material parameters

Material equations:Maxwell's equations:

...but higher order multipoles are not origin independent

Effective material parameters CANNOTdependent on the definition of the multipoles!

Raab's material parameters

Maxwell's equations: ● Fields H and D are notuniquely defined!!!● So there are some sets of transformations keeping Maxwell's equations invariant● Raab proposed the following one:

Raab, R. E.; De Lange, O. L., “Multipole Theory In Electromagnetism”, (Oxford University Press, 2005)

… allows to formulate origin independent material equations in terms of multipoles!

Raab's material equations

Average multipole moments: multipole expansion

Material equations in terms of multipoles:

Post form: Lindell-Sihvola form:

Effective material equations

● Self-consistent introduction of the macroscopic fileds leads in general to bi-anisotropic effective medium

Now we need averaged electric and magnetic polarizability of the meta-atoms!

● Intrinsic contribution: depends on the form and materials of the meta-atom● Extrinsic contribution: depends on the surrounding, namely, background medium, substrate, neighbors

Influence of neighbors: Planar chiral MTM

Transmission/reflection of left- (right-) circular polarized EM waves on array of copper chiral split-rings on FR-4 printed circuit board substrate.

Method: finite-difference time-domain (FDTD) scheme with auxiliary differential equations (ADE) for media polarizability.

Influence of neighbors: Planar chiral MTM

● Transmission/reflection is different for LH/RH polarized EM waves● For smaller period (stronger inter-cell coupling) peak is broaden and shifted towards higher frequencies

Planar chiral MTM

● To understand the physics behind the planar chirality we need a simple model.● Extrinsic contribution to asymmetric transmission (background medium, substrate, neighbors) does not change qualitative picture of the effect

We focus on the intrinsic effects

What is the simplest shape of a planar chiral meta-atom?

Planar chiral materials

Transmission asymmetry (HFSS)

copper

Effective parameters: Planar chiral medium

...leads to following effective material parameters:

Both dielectric and magneto-electric tensorsare origin independent!

Effective parameters: Planar chiral medium

a1=13 mm; a2=10 mmb= 0.8 mm; R=10mmφ=45°

Effective parameters: Planar chiral medium

… so effective material parameters are given by:

We have got anisotropic, absorbing, non-magnetic medium!

Such a medium does not possess chirality, but possesses elliptical dichroism, which could lead to

asymmetric transmission and optical activity!

Elliptical dichroism: Copper PCM

H=C1h

1 + C

2h

2

● In every direction two eigenwaves can propagate with different(i) refraction index, (ii) attenuation and (iii) polarization ● Incident wave will be decomposed into these two wavesleading to the polarization rotation

Transmission Asymmetry

Numerical (top) and analytical (bottom) dependence of transmission asymmetry

Experiment: THz chiral filter

(Collaboration with Lavrinenko)

2μm-thick nickel membrane

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H

0°- pol.H

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90°- pol.

THz chiral filter

Frequency [THz]

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smis

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Conversion difference of several percent on a single film!

Linear polarization basis Circular polarization basis

Graphene-based planar chirality

Planar chirality (dichroism) at a single atomic layer!

induced current

Transmission asymmetry

Thank you for attention!