Nano-Photonics and Plasmonics in COMSOL Multiphysics

Nano-Photonics and Plasmonics in COMSOL Multiphysics

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

ETH Zürich 08.07.2009

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

complexity

Introduction COMSOL Basic concepts Product structure The RF module

The Multiphysics perspective Mechanics

Fluid Dynamics

Electrodynamics

Heat

Multiphysics Chemical Reactions

Acoustics

User defined PDE Beneficial for both single- and multi-field analysis: Reality, Flexibility, Synergy, Openness

COMSOL - the Multiphysics people

Spin-off of KTH Stockholm (1986) Science & Engineering Software today 16 branch offices worldwide net of distributors 12’500 licenses and 50’000 users annual 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 Photonics

Frequency-Domain

Driven

Eigenfrequency

Total-field

Time-Domain

Eigenmode

Quadratic Eigenvalue

Scattered-field

Nonlinear Eigenvalue

Perpendicular Periodic

Linear

Nonlinear

Non-dispersive Trivial dispersion Drude-Lorentz

Paraxial Three levels of difficulty: i. ii. iii.

Fully predefined in COMSOL Minimum changes to the predefined equations Equation-based modeling: maximum flexibility

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

Surface Plasmons Demo

Air

4 μm

H field perpendicular to the „wall“ Wavelength 600 nm

Metal

Surface Plasmons Demo Perfectly Matched Layer (PML) Air

4 μm

Metal


4 μm

Metal


Impedance Boundary Condition

The 5 Steps of Modeling

Dealing with periodicity, dispersion and infinity 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 conditions Irreducible unit cell: solution space

Larger domain, multiple periods

?

Example: honeycomb lattice crystal

Visualizing the Bloch wave

Æ Periodicity tools even more powerful in 3D

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.

transmitted

incident reflected

Applications: Optical characterization of nanostructures Extracting effective medium parameters of metamaterials 20

Air

Air-filled hole

Dielectric film

Above: a generic geometry (hole array) Draw any unit cell for your own metamaterial design! 21

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 Æ Just enter the relation!

ν p2 ε (ν ) = ε b − ν (ν − iγ ) Three COMSOL ways of entering material data:

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

S-parameters and metamaterial characterization

Port boundary: easy way to launch a specific wave form

Provides complex-valued S-parameter matrix S11=r=reflectance (1 Æ 1) S21=t=transmittance (1 Æ 2)

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

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

transmitted

absorbed reflected

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

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

Scattered-Field Formulation Basic idea: Instead of solving L [u] = 0, solve L [uin + usc] = 0 Ù L [usc] = - L [uin]

Illustration: cloak of invisibility on human head

Customized Scattered-Field formulations

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

Far Field – Antenna - Scattering

phi component of the electric field

near field radiation pattern

far field

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

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: r tr t r

D = εE + ζ DHH r tr t r B = μH +ζ BEE

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

D = εE − iχH r r r B = μH + iχE

Chirality parameter χ controls polarization rotation

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

Air

Chiral Chiral medium

Modifying built-in constitutive relations in chiral medium r r χ r D = εE − i H c r r χ r B = μH + i E c

Results: polarization rotation Click Solve Open Postprocessing Æ 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±χ Sufficiently large chirality parameter rotates properly handed waves so much as to fully compensate (and win over) natural rotation of the circular polarization Backward waves => Negative refraction! Reference: J.B. Pendry, “A chiral route to negative refraction”, Science 306, 1353 (2004).

k

clockwise counterclockwise clockwise To excite this wave, use surface current Js=[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:

r r r r E = −∂rt A, B =r ∇ × rA

r ∂tε ∂t A +σ∂t A − ∂t P + ∇× μ ∇× A = 0 −1

It satisfies equation

The most general isotropic dielectric function that can be modeled without additional degrees of freedom: ε (ω ) = a +

b

ω

+

c

ω

2

= ε 0 (ε ∞

ω − − ω ω iσ

2 p 2

)

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

r r 2r r −1 μ0∂tε 0ε ∞∂t A + μ0σ∂t A + k p A + ∇ × μ ∇ × A = 0 Plasmonic term

Implementation

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

r r 2r r −1 μ0∂tε 0ε ∞∂t A + μ0σ∂t A + k p A + ∇ × μ ∇ × A = 0 Plasmonic term

Results: plasma echo in linear electron density gradient

COMSOL Equation-based modeling Non-linear Eigenvalue problems Æ classical Eigenvalue Problem (EP) Æ Quadratic Eigenvalue Problem (QEP)

Æ Generalized Eigenvalue Differential Equation (GEDE) Æ Bloch-Floquet-Eigenmode Surface charge integral equations (SCIE)

Examples of non-linear eigenvalue problems

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 structures Eigenvalue problem becomes quadratic (QEP) regardless of the complexity of temporal dispersion, ε(ω)

Surface Plasmon Resonances (e.g. of Nanoholes) as Electrostatic Eigenvalues Æ Generalized Eigenvalue Differential Equation (GEDE)

− ∇ 2 E z = ε (ω )ω 2 E z

r r ∇(θ ∇ϕ n ) = λn ∇2ϕ n

COMSOL approach of treating nonlinear EP

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 State Phys. 46, 147 (1992); Stockman M., Faleev S., Bergman D., PRL 87, 167401 (2001). [3] Shvets, Urzhumov, PRL 93, p. 243902 (2004).

COMSOL access to the weak form

PDE equations are easily converted to the “weak form” – multiply with the test function (u_test) – integration by parts (GaussStokes theorems)

r r r r ∇ u = 0 → ∫ (∇ u )utest dV = ( −) ∫ (∇u ) ⋅ (∇utest )dV = 0

2Example: Laplace2 operator

Weak term:

ux*ux_test + uy*uy_test + uz*uz_test

Example GEDE

r r ∇(θ ∇ϕ n ) = λn ∇2ϕ n

COMSOL Implementation 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

Enter weak terms just as you write them on paper!

r2 ∇ un = λn ∇2un

Example Surface Plasmon Resonance (GEDE)

Sample surface plasmon resonances of a plasmonic tetramer 1st and 20th eigenvalue

Emerging field: plasmonic metafluids

Manoharan et al.: colloidal solutions with clusters of various symmetric forms [Science, 2003] Some clusters are useful as building blocks for photonic crystals Others may be useful even in solution Resonances of plasmonic clusters modify electromagnetic properties of liquids Manipulate electric permittivity, magnetic permeability, chirality of liquids Electric dipole resonance

fcc block

Magnetic dipole resonance

Plasmonic crystal superlens (doable with QEP) Nanostructured super-lens*

Hot spots at the super-lens

Electric field profiles

Blue Æ w/wp = 0.6, X = -0.2l Magnetic field behind plane wave illuminated double-slit: D = λ/5, separation 2D

Red Æ w/wp = 0.6, X = 0.8 λ no damping Black Æ same as red, but with damping Dotted Æ w/wp = 0.606 (outside of the left-handed band)

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]:

∫ K ( s, s ' )u ( s ' ) dS ' = λu ( s )

Input as -u_time

Quadrupole plasmon resonance of a nanoring

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

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

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

complexity

Concluding remarks

COMSOL covers the majority of standard simulation tasks in Plasmonics and 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: Æ New time-dependent solvers (generalized-alpha, segregated) Æ Optimization and sensitivity analysis Æ Parametric sweeps wrapped around eigenmode or time-dependent analysis

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

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