nano-photonics and plasmonics in comsol...

Nano-Photonics and Plasmonics in COMSOL Multiphysics

Speaker: Dr. Thierry Luthy (COMSOL GmbH, Zurich)Credits: Dr. Yaroslav Urzhumov (COMSOL Inc, Los Angeles)

ETH Zürich08.07.2009

complexity

Outline

COMSOL product overview: company, product and RF module

DEMO: An illustrated surface plasmon example

Dealing with periodicity, dispersion and infinity

Customizing equations

Equation-based modeling

Introduction COMSOL

Basic concepts

Product structure

The RF module

The Multiphysics perspective

Chemical Reactions Acoustics

Electrodynamics Heat

Fluid DynamicsMechanics

Beneficial for both single- and multi-field analysis:

Reality, Flexibility, Synergy, Openness

User defined PDE

Multiphysics

COMSOL - the Multiphysics people

Spin-off of KTH Stockholm (1986)Science & Engineering Softwaretoday 16 branch officesworldwide net of distributors12’500 licenses and 50’000 usersannual growth in CH ~36%

COMSOL Multiphysics Product Structure

Plasmonics model challenges and their COMSOL solutions

Large field discontinuities, currents and charges on curved boundaries

Automatically and accurately handled by Vector Element FEM; both FD and TD

Accurate spectra, effective medium parameters Parametric sweeps, or solve once for one wavelength

Temporal dispersion model FEFD method needs only εc(ω)

Light scattering Scattered field formulation

Infinitely extended domains and objects Scattering/Matched b.c., PMLs, Impedance b.c.

Radiation and scattering cross-sections Far Field integral; S-parameters

Launching specific wave forms Port b.c., Boundary Mode Analysis

Resolving plasmonic skin depth Boundary layer mesh; Impedance b.c.

Nonlinear effects FETD formulations

You name it!.. We have probably seen it…

Overview of Analysis Types

Three levels of difficulty:i. Fully predefined in COMSOLii. Minimum changes to the predefined equationsiii. Equation-based modeling: maximum flexibility

Photonics

Frequency-Domain Time-Domain

Linear Nonlinear

Non-dispersive

Trivial dispersion

Driven Eigenfrequency Eigenmode

Total-field

Scattered-field

Quadratic Eigenvalue

Nonlinear Eigenvalue

Perpendicular

Periodic

Drude-LorentzParaxial

Levels of working with COMSOL• Ready-to-use interface for standard problems

– Fully-predefined equations and BND conditions– Powerful drawing and meshing interface– Solver defaults– Help, Report Generator etc.

• Customizing COMSOL-defined equations– Slight modifications of existing equations– e.g. magneto-electric (chiral) media– e.g. Bloch-Floquet eigenmode analysis of dispersive

periodic structures

• Fully equation-based modeling – Full flexibility– Time domain models of dispersive media

DEMO – Surface Plasmons

Draw and Mesh

Perfectly matched layers (PMLs)

Modification of expressions (incident angle)

Parametric solver

Resolution of skin-depth vs. Impedance BND conditions

4 μm

Air

Metal

Surface Plasmons Demo

H field perpendicular to the „wall“Wavelength 600 nm


4 μm

Air

Perfectly Matched Layer (PML)

Metal

4 μm

Air


Metal


Air


Impedance Boundary Condition


The 5 Steps of Modeling


Periodic boundary conditions

Periodic meshes

Dispersive media in the frequency domain

Scattered field analysis

Unlimited Mesh functionality

Periodic boundary conditions

Goal of simulation: find eigenmodes of a honeycomb lattice photonic crystal, and view them in a large domain

This lattice is a common motif in carbon-based crystals (graphite, graphene) and organic polymers (C6 rings).

Honeycomb lattices have been used in design of photovoltaic cells, photonic crystal fibers and negative-index super-lenses

Periodic boundary conditionsIrreducible unit cell: solution space Larger domain, multiple periods

?

Example: honeycomb lattice crystal

Visualizing the Bloch wave

Periodicity tools even more powerful in 3D

20

Reflection/transmission spectra of a periodic structure

Goal: calculate normalized reflectance, transmittance and absorbance of a perforated nano-film (photonic crystal slab)

Set-up tricks: double-periodic boundary conditions,user-defined port boundaries, S-parameters.

Applications:Optical characterization of nanostructuresExtracting effective medium parameters of metamaterials

incident

reflected

transmitted

21

Above: a generic geometry (hole array)Draw any unit cell for your own metamaterial design!

Air

Dielectric film

Air-filled hole

Periodic mesh generation: Node identity!

1. Select boundaries 2 and 5 (the first equivalent pair).2. Click Copy Mesh button (red double triangle).3. Go to the Mesh Mode to see that the boundary mesh

has been translated.

Dielectric film may have dispersive permittivity

Just enter the relation!

Three COMSOL ways of entering material data:

Analytic expression (Global, Scalar, Subdomain, etc.)Interpolation function – provide ASCII file with a lookup tableReference to external m-function (MATLAB interface)

)()(

2

γννν

ενεi

pb −−=

S-parameters and metamaterial characterizationPort boundary: easy way to launch a specific wave form

Provides complex-valued S-parameter matrix

S11=r=reflectance (1 1) S21=t=transmittance (1 2)

Effective medium approximation: use Fresnel-Airy formulas for a finite-thickness slab

Metamaterial analysis: invert those formulas to extract effective medium parameters from S-parameters

{S11,S21} -> {Zeff, neff} -> {εeff, μeff}

Reference: Smith D. R., Schultz S., Markos P. and Soukoulis C. M. 2002, Phys. Rev. B 65 195104

transmitted

absorbed

reflected

Scattered-Field Formulation

Illustration: cloak of invisibility on human head

Basic idea:

Instead of solving

L [u] = 0,

solve

L [uin + usc] = 0

L [usc] = - L [uin]

Customized Scattered-Field formulationsProblem: A single particle (or group of particles) on infinitely extended substrateSet-up issue: PML can only be perfectly matched to one medium (either air/solvent or the substrate)Avoiding artificial reflections on the boundary between two PMLs may be difficultSolution: modify “incident” field expressionsIf the “incident” field is an exact solution without the particle, then the “scattered” field is small at some distance away from the particle.For infinite metallic domains don’t use PMLs but scattering BND condition.

Particle

Air-matched PML Glass-matched PML

Air Glass

Plot of the “scattered” field

Far Field feature:Used for calculating radiation pattern and differential scattering cross-section

near field radiation pattern

phi component of the electric field far field

Far Field – Antenna - Scattering

Unlimited 3D Meshing Functionality

Free combination of mesh types

Customizing COMSOL equations

Modifying constitutive relations:

Magneto-electric (chiral) media

Modifying built-in equations:

Time-domain modeling of lossless plasma with dispersive permittivity

Modeling chiral (magneto-electric) media

The most general dispersion relation for a linear medium includes 4 electromagnetic response tensors:

For an isotropic medium consisting on non-centrosymmetric unit cells (crystals or metamaterials):

Chirality parameter χ controls polarization rotation

EHB

HED

BE

DHrtrtr

rtrtr

ζμ

ζε

+=

+=

EiHB

HiEDrrr

rrr

χμ

χε

+=

−=

Geometry

Geometry consists of 5 adjacent rectangular blocks, each 1x1 “meter” in cross-section (could be 1 micron as well – only the ratio wavelength/size matters)

Physical domain: 3m long

PML 1 and 2: thickness 0.2m, centered at x1=-1.6 and x2=1.6

Chiral slab: thickness L=1m, centered at the origin (x=y=z=0)

PML

Chiral medium

Air

Chiral

Modifying built-in constitutive relationsin chiral medium

Ec

iHB

Hc

iED

rrr

rrr

χμ

χε

+=

−=

Results: polarization rotation

Click Solve

Open Postprocessing Plot Parameters, enable Slice and Arrow plots

Slice tab: type expression

atan(abs(Ey)/abs(Ez))/pi

This is polarization rotation angle in fractions of pi radian

Arrow tab: choose “Electric field”from “Predefined quantities”

Electric field polarization is clearly rotated by 45 degrees (or 0.25*pi radian)

Negative refraction of circular polarized wave

For circularly polarized waves, effective indices are n±=1±χSufficiently large chiralityparameter rotates properly handed waves so much as to fully compensate (and win over) natural rotation of the circular polarizationBackward waves => Negative refraction!Reference: J.B. Pendry, “A chiralroute to negative refraction”, Science 306, 1353 (2004).

clockwise

counterclockwise

clockwise

k

To excite this wave, use surface currentJs=[0 –i 1]

Time-domain modeling of lossless plasma with dispersive permittivity

Finite Element Time Domain (FETD) analysis in COMSOL is implemented in terms of vector potential A using the V=0 gauge:

It satisfies equation

The most general isotropic dielectric function that can be modeled without additional degrees of freedom:

The final equation after factorizing ε0,μ0 this becomes in SI units:

01 =×∇×∇+∂−∂+∂∂ − APAA tttt

rrrrμσε

ABAE t

rrrr×∇=−∂= ,

)()( 2

2

02 ωω

ωσεε

ωωωε picba −−=++= ∞

012000 =×∇×∇++∂+∂∂ −

∞ AAkAA pttt

rrrrμσμεεμ

Plasmonic term

Implementation

The major part of the equation is predefined in COMSOL standard GUI. You only need to enter the plasmonic term

012000 =×∇×∇++∂+∂∂ −

∞ AAkAA pttt

rrrrμσμεεμ

Plasmonic term

Results: plasma echo in linear electron density gradient

COMSOL Equation-based modeling

Non-linear Eigenvalue problems

classical Eigenvalue Problem (EP)

Quadratic Eigenvalue Problem (QEP)

Generalized Eigenvalue Differential Equation (GEDE)

Bloch-Floquet-Eigenmode

Surface charge integral equations (SCIE)

Examples of non-linear eigenvalueproblems

The resonance in PEC waveguides is a classical eigenvalue problem (EP). If the walls are not PEC but lossy (using Impedance BC.) the waveguide becomes dispersive, the EP nonlinear

Dispersive photonic band structuresEigenvalue problem becomes quadratic (QEP) regardless of the complexity of temporal dispersion, ε(ω)

Surface Plasmon Resonances (e.g. of Nano-holes) as Electrostatic EigenvaluesGeneralized Eigenvalue Differential Equation (GEDE)

zz EE 22 )( ωωε=∇−

nnn ϕλϕθ 2)( ∇=∇∇rr

Traditionally, nonlinear eigenvalue problems are hard to solve.– Iterative approach to nonlinear eigenvalue problems requires a good

initial guess; convergence is not guaranteed.– One can only obtain a single eigenmode at a time, from a given initial

guess.

QEP [1] and GEDEs [3,4] are easily implemented in COMSOL's weak mode.

[1] Credit: Dr. Marcelo Davanco, Univ. of Michigan, 2007, Published in: Davanco, Urzhumov, Shvets, Opt. Express 15, p.9681 (2007).

[2] Bergman D., PRB 19, 2359 (1979); Bergman D., Stroud D., Solid StatePhys. 46, 147 (1992);Stockman M., Faleev S., Bergman D., PRL 87, 167401 (2001).

[3] Shvets, Urzhumov, PRL 93, p. 243902 (2004).

COMSOL approach of treating nonlinear EP

COMSOL access to the weak formPDE equations are easily converted to the “weak form”

– multiply with the test function (u_test)

– integration by parts (Gauss-Stokes theorems)

Example: Laplace operator

Weak term:

Example GEDE

0)()()()(0 22 =∇⋅∇−=∇→=∇ ∫∫ dVuudVuuu testtest

rrrr

ux*ux_test + uy*uy_test + uz*uz_test

nnn ϕλϕθ 2)( ∇=∇∇rr

COMSOL Implementationnnn uu 22 ∇=∇ λ

r

Enter weak terms just as you write them on paper!

weak = ux*test(ux)+uy*test(uy)+uz*test(uz)dweak= -(uxt*test(ux)+uyt*test(uy)+uzt*test(uz))

Note: -ut is the same as lambda*u

Example Surface Plasmon Resonance (GEDE)Sample surface plasmon resonances of a plasmonic tetramer1st and 20th eigenvalue

Emerging field: plasmonic metafluidsManoharan et al.: colloidal solutions with clusters of various symmetric forms [Science, 2003]Some clusters are useful as building blocks for photonic crystalsOthers may be useful even in solutionResonances of plasmonic clusters modify electromagnetic properties of liquidsManipulate electric permittivity, magnetic permeability, chirality of liquids

Electric dipoleresonance

Magnetic dipoleresonance

fccblock

Plasmonic crystal superlens (doable with QEP)

Magnetic field behind plane wave illuminated double-slit:

D = λ/5, separation 2D

Blue w/wp = 0.6, X = -0.2l

Red w/wp = 0.6, X = 0.8 λ no damping

Black same as red, but with damping Dotted w/wp = 0.606 (outside of the left-handed band)

Nanostructuredsuper-lens*

Hot spots at the super-lens Electric field profiles

Shvets, Urzhumov, PRL 93, 243902 (2004); Davanco, Urzhumov, Shvets, Opt. Express 15, p.9681 (2007).

Surface charge integral equations (SCIE)Surface integral eigenvalue equation for surface charge [3]:

[3] Mayergoyz I.D., Fredkin D.R., Zhang Z., Phys. Rev. B 72, 155412 (2005)

Quadrupole plasmonresonance of a nanoring

∫ = )(')'()',( sudSsussK λ

Fredholm integral = Boundary Integration VariableUsage of this variable (sigmaint) in the weak mode

Input as -u_time

complexity

Outline

COMSOL product overview: company, product and RF module

DEMO: An illustrated surface plasmon example


Customizing equations

Equation-based modeling

Concluding remarksCOMSOL covers the majority of standard simulation tasks in Plasmonicsand Nano-Photonics

Frequency-domain, time-domain, modal analyses

Unprecedented flexibility combined with hi-end numerical analysis tools

Users can invent new types of analysis; creativity is welcomed

Every new version brings more powerful features! E.g. In Release 3.5:New time-dependent solvers (generalized-alpha, segregated)Optimization and sensitivity analysisParametric sweeps wrapped around eigenmode or time-dependent analysis

How to get started?Tell us about your plans, requirements, models.

Order your free trial version: www.comsol.com/[email protected]

Get: - support from an international team- free introductory kits to various applications- free conference proceedings- free minicourses

- Intensive training coursesAC/DC, Chem. Engineering, Mechanics, RF

Join the Leaders in Multiphysics Simulation

• Hands-on Minicourses• Keynote Talks and User Presentations• Exhibition and Poster Session• Demo Stations• Tutorial Presentations• Awards Banquet

www.comsol.eu/conference2009

nano-photonics and plasmonics in comsol...

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