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
Surface Plasmons Demo
4 μm
Air
Perfectly Matched Layer (PML)
Metal
4 μm
Air
Perfectly Matched Layer (PML)
Metal
Surface Plasmons Demo
Air
Perfectly Matched Layer (PML)
Impedance Boundary Condition
Surface Plasmons Demo
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 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
Dealing with periodicity, dispersion and infinity
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]
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