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 ε c (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 Time-Domain
Driven Eigenfrequency Eigenmode
Total-field
Scattered-field
Paraxial
Quadratic Eigenvalue
Nonlinear Eigenvalue
Perpendicular
Periodic
Three levels of difficulty:
i. Fully predefined in COMSOL
ii. Minimum changes to the predefined equations
iii. Equation-based modeling: maximum flexibility
Linear Nonlinear
Non-dispersive
Trivial dispersion
Drude-Lorentz

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
4 μm
H field perpendicular to the „wall“
Wavelength 600 nm
Metal
Air

Surface Plasmons Demo
4 μm
Perfectly Matched Layer (PML)
Metal
Air

Surface Plasmons Demo
4 μm
Perfectly Matched Layer (PML)
Metal
Air

Surface Plasmons Demo
Impedance Boundary Condition
Perfectly Matched Layer (PML)
Air

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
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.
incident
reflected
transmitted
Applications:
Optical characterization
of nanostructures
Extracting effective
medium parameters of
metamaterials
20

Air
Above: a generic geometry (hole array)
Draw any unit cell for your own metamaterial design!
Dielectric film
Air-filled hole
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
Just enter the relation!
2
ν p
ε ( ν ) =
εb
−
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
transmitted
reflected
absorbed
Port 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 finitethickness
slab
Metamaterial analysis: invert those
formulas to extract effective medium
parameters from S-parameters
{S11,S21} -> {Z eff , n eff } -> {ε eff , μ eff }
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
solve
L [u] = 0,
L [u in + u sc] = 0
L [u sc] = - L [u in]
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
perfectly matched to one medium
(either air/solvent or the substrate)
Air Glass
Avoiding artificial reflections on the
boundary between two PMLs may be
difficult
Particle
Solution: modify “incident” field
expressions
If the “incident” field is an exact
solution without the particle, then the
Air-matched PML Glass-matched PML
“scattered” field is small at some
distance away from the particle.
Plot of the “scattered” field
For infinite metallic domains don’t use
PMLs but scattering BND condition.

Far Field – Antenna - Scattering
near field radiation pattern
phi component of the electric field 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 r t r
t
D=
εE+
ζ
r t r t
B = μH
+ ζ
For an isotropic medium consisting on non-centrosymmetric unit cells
(crystals or metamaterials):
Chirality parameter χ controls polarization rotation
DH
BE
r r r
D=
εE
−iχH
r r r
B = μH
+ iχE
H
r
E

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 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
clockwise
counterclockwise
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:
It satisfies equation
r r r r
E = −∂r
t A,
B = r ∇×
r
A r
−1
∂ ε ∂ + σ∂
A−∂
P+
∇×
μ ∇×
A=
0
t
tA
t t
The most general isotropic dielectric function that can be modeled
without additional degrees of freedom:
2
b c
iσ
ω p
ε ( ω ) = a + + = ε 0 ( ε
2
∞ − − 2
ω ω
ω ω
The final equation after factorizing ε 0 ,μ 0 this becomes in SI units:
μ ∂ ε ε ∂
0
t
0
∞
r
r
2
tA
+ μ0σ∂
tA
+ kp
r
r
−1
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
μ ∂ ε ε ∂
0
t
0
∞
r
r
2
tA
+ μ0σ∂
tA
+ kp
r
r
−1
A+
∇×
μ ∇×
A = 0
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 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
Generalized Eigenvalue Differential Equation
(GEDE)
− ∇
2
E
z
2
= ε ( ω)
ω E
∇( θ ∇ϕ
) = λ ∇
r r
n
n
2
ϕ
n
z

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 (Gauss-
Stokes theorems)
r
2Example:
Laplace 2operator
u = 0 → ( ∇ u)
utestdV
= ( −)
( ∇u)
⋅(
∇utest
∇ ∫ ∫
Weak term:
Example GEDE
n
r
∇( θ ∇ϕ
) = λ ∇
r r
n
r
r
) dV
ux*ux_test + uy*uy_test + uz*uz_test
2
ϕ
n
=
0

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!
r
2
∇ u = λ ∇
n
n
2
u
n

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
Magnetic dipole
resonance
fcc
block

Plasmonic crystal superlens (doable with QEP)
Nanostructured
super-lens*
Magnetic field behind plane
wave illuminated double-slit:
D = λ/5, separation 2D
Hot spots at the
super-lens
Shvets, Urzhumov, PRL 93, 243902 (2004);
Davanco, Urzhumov, Shvets, Opt. Express 15, p.9681 (2007).
Electric field profiles
Blue w/w p = 0.6, X = -0.2l
Red w/w p = 0.6, X = 0.8 λ no damping
Black same as red, but with damping
Dotted w/w p = 0.606 (outside of the
left-handed band)

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
Fredholm integral = Boundary Integration Variable
Usage of this variable (sigmaint) in the weak mode
Quadrupole plasmon
resonance of a
nanoring
[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:
New time-dependent solvers (generalized-alpha, segregated)
Optimization and sensitivity analysis
Parametric 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/
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