Finite difference time domain modeling of subwavelength-structured
anti-reflective coatings
Han, K., Han, H. Y., Stack Jr, J., & Chang, C. H. (2014). Finite Difference Time
Domain Modeling of Sub-Wavelength Structured Anti-Reflective Coatings. Applied
Computational Electromagnetics Society Journal, 29(1), 1-8.
The Applied Computational Electromagnetics Society
Accepted Manuscript
http://cdss.library.oregonstate.edu/sa-termsofuse
Finite difference time domain modeling of subwavelengthstructured anti-reflective coatings
Katherine Han1,2, Hai-Yue Han3, James Stack Jr.4, and Chih-hung Chang1,2
1
2
School of Chemical, Biological and Environmental Engineering
Oregon State University, Corvallis, OR 97331, USA
vanwormk@onid.orst.edu, chih-hung.chang@oregonstate.edu
Oregon Process Innovation Center, Microproduct Breakthrough Institute
Oregon State University, Corvallis, OR 97330, USA
3
Inspired Light, Llc., Corvallis, OR, 97330, USA
haiyuehan@gmail.com
4
Remcom, Inc., State College, PA, 16801, USA
jim.stack@remcom.com
Abstract - The finite difference time domain
(FDTD) method was used to model anti-reflective
properties of a variety of sub-wavelength
structures for 300 to 1300 nm input light. Light
hitting non-tapered nanostructures exhibited
interference patterns similar to thin film antireflective coatings (ARCs), increasing the antireflective effect at several wavelengths. The
lowest reflectance was observed with conical and
pyramidal nano-structures with bases of 100 or
200 nm and heights of 400 or 800 nm.
Index Terms - anti-reflective, FDTD, modeling,
nanostructure, sub-wavelength.
I. INTRODUCTION
Improvements in solar module efficiency
have been a popular topic of research these past
few decades, usually focusing on either decreasing
costs or increasing efficiency. One method to
increase efficiency is by reducing reflections off
solar module lamination materials. Reflections
occur when light travels between media with two
different refractive indices (RI). The amount of
light reflected at a normal incidence can be
described by
n n 
R 1 2 
 n1  n2 
2
(1)
where n1 is the index of refraction of the low RI
material (usually air) and n2 is the index for the
high RI material. One technique to reduce
reflection is to gradually increase the index of
refraction from that of air to that of the surface so
that there are little to no abrupt changes in RI that
would induce reflections. This can be done by
creating a sub-wavelength structured antireflective coating (ARC) at the reflective interface.
Sub-wavelength sized structures (SWS) are
perceived by light as a material with an
intermediate index of refraction between that of air
and that of the surface.
Advances in computing resources have
made it possible to quickly and accurately model
the anti-reflective properties of 3-dimensional subwavelength structures. The FDTD method
performs numerical time-stepping of Maxwell’s
curl equations to describe the behavior of electric
and magnetic fields. The FDTD method has been
used to model ARCs in 2 and 3 dimensions of
pillars[1], pyramids[2-5], slots[6], cones[3, 4, 7,
8], conical cylinders[9], triangles[10], spheres[11],
hemispherres[5], nano
oholes[12], thin
t
films[13
3],
nanoporou
us films[14
4], and nanowires[15
5].
Experimeental
photo
onic
nanosttructures
are
a
reviewed by Chattopad
dhyay et al[16
6].
To
T date FDTD
D has been ussed primarily to
compare modeling ARC
A
results for a small
number of
o SWS. This paper will model
m
a varieety
of shapes and sizes in three
t
dimensiions to quantiify
the reflecctivity of AR
R sub-waveleength structurres
(ARSWSss), compare this
t calculated
d reflectivity to
results frrom transfer matrix metthod modelin
ng,
describe trends
t
in the effects of shaape and size of
the ARSW
WSs, and pro
ovide a direcction for futu
ure
experimen
ntal research.
II. METHODS
M
Remcom’s
R
XF
F 7.3.1 was used to mod
del
the behaavior of lig
ght reflection
ns from an
ntireflective coatings (AR
RCs). All sim
mulations weere
performed
d on an NVID
DIA M1060 Tesla
T
GPU. The
T
geometry was drawn using the XF
F CAD GUI to
produce an area of air
a on the +Z
+ end of the
t
simulation
n, a SWS in the
t middle, and
a a solid bu
ulk
material on
o the –Z end
d (see Fig. 1). In all cases the
t
SWS and
d solid material were assiigned the sam
me
RI, n=1.5
5, except the thin film AR
RC, which was
w
1.22. To simulate sollar input, a polarized
p
plaane
wave wiith an auto
omatic waveeform with a
Gaussian distribution of
o frequencies between 30
001300 nm was input at the eighth ceell from the +Z
+
boundary toward the –Z
– direction (see Fig. 1 parts
B and D). Boundary conditions
c
in the +Z and –Z
–
directionss were perfeectly matched
d layer (PM
ML)
with 7 cells
c
of padd
ding and in the X and Y
directionss were perio
odic. Electricc field senso
ors
were placced to the +Z
+ side of th
he plane waave
generation
n plane (refleected light sensor; see Fiig.
1A) and just
j
inside the bulk material (transmittted
light sensor; see Fig. 1C). Cell sizess were 5 nm for
f
n
sttructures and 2.5 nm for the
t
short or non-tapered
cone and pyramid sweeeps. The bullk material was
w
2,000 nm x 2,000 nm x 1,000 nm, resulting
r
in 400
x 400 poiints output fo
or each senso
or at each tim
me
point. Hollow spheres were drawn in a hexagon
nal
closest paacked structu
ure with one modeling un
nit
consisting
g of 4 sphere diameters on
n one axis and
d4
rows of closest
c
packed
d spheres on the other ax
xis.
All hollow
w spheres were
w
constructed with 5 nm
n
shell thick
kness and thee grid sizing was 1/200th of
the overalll modeling dimension
d
in question for X
and Y axees, keeping th
he mesh size between 0.866
nm andd 3 nm. Thesse small meshh sizes are reqquired
to ressolve the shhell thicknesss features oof the
hollow
w spheres withh the FDTD m
method.
Hollow sphheres were m
modeled with ss- and
p-polar
arized plane waves and the results were
averagged to report unpolarized results. Thiss is a
requireement due too their non-sqquare packingg. For
many semi-analyticcal methods pp-polarized liight is
compliicated to sim
mulate[17], bbut FDTD haandles
either ppolarization nnaturally.
Fig. 11. Schematicc of modeliing domain with
reflectted light planne sensor (A)), light sourcce (B)
and traansmitted lighht plane sensoor (C), the shaape of
the inpput waveform
m (D, red linee) and the inteensity
of solaar input (D, bllue line).
Electric fiield data was post-proccessed
using MATLAB R
R2012a. A discrete fast Foourier
transfoorm (DFT) w
was performeed for each space
point for each of Ex, Ey, and Ez in botth the
transm
mitted and reeflected plannes. The absolute
value ssquared of thee FFT result aat each point in the
plane oof the sensor was multipliied by the inddex of
refracttion and the vvalues for Exx, Ey, and Ezz were
added together, ressulting in thee light intenssity at
each ffrequency. Inntensities foor each frequuency
were aaveraged oveer the area oof the sensorr. The
reflected and transmiitted light in
ntensities weere
then addeed and compaared to the in
nput wave lig
ght
intensitiess to determinee the error off the simulatio
on;
only simu
ulations with
h <2% errorr are reporteed.
Reported reflection vaalues are an average of the
t
t solar inp
put
percent reeflection weiighted over the
spectrum as shown in Eqn.
E 2.
%
∑%
∑
∗
.
.
(2)
Transfer
T
matriix method sim
mulations weere
performed
d on the Lux
xpop softwaree (luxpop.com
m).
The Loreentz-lorentz approximation
a
n[18] was ussed
to determ
mine an effective index off refraction for
f
the pyraamids and nanorods. For pyramiids
approximations were calculated for
f each 1 nm
n
slice.
III. RESULTS
R
In
n this experriment all sub-waveleng
s
gth
structures, as well as thin film
m ARCs wiith
intermediate RI vaalues, resultted in low
wer
reflectancce than a sim
mple flat su
urface of glaass
(4.0%). Overall,
O
ARSW
WSs that hav
ve full or nearrly
full coverrage at the su
ubstrate and taper
t
to a poiint
with high
her aspect raatios (2 to 8)) exhibited the
t
lowest reeflection. Stru
uctures with
h widths larg
ger
than 200 nm req
quired prohiibitively long
calculation times due to the multiple reflectio
ons
induceed by thee higher frequency input
waveleengths.
To investiggate the anti-rreflective effeects of
well-orriented nanorods on a suurface, a seriies of
cylindeers of the same materiaal as the subbstrate
(n=1.55) were simullated (see Figg. 2). All cyliinders
in thiss sweep werre 375 nm inn length andd they
rangedd in fill factorrs of 20, 50, aand 79%. Forr each
densityy, the distaance betweeen cylinders was
modeleed to be 50, 100, or 200 nnm. The fill factor
of thhe cylinderss was the most important
determ
mining factoor in reduucing reflecctions;
cylindeers covering about 20% off the area refl
flected
2.3 to 3.9% of the llight, cylinderrs covering 500% of
the areea reflected 2.0 to 2.1%,, and the 79%
% fill
factor cylinders haad reflectancees of 2.8 to 3.1%.
The rreflectance spectrum oof the 3755 nm
cylindrrical ARSWS
Ss was charaacteristic of thhat of
thin fillms with inteermediate inddexes of refraaction;
both A
ARC types exhibit interfeerence patternns, as
seen inn Fig. 2, andd the lowest rreflection wass seen
in the 50% fill facttor cylinders and the thinn film,
both oof which had an effective iindex of refraaction
of 1.222.
FDTD-preddicted reflecttion characterristics
for nannorods matchhed FDTD results for thin films
will thhe same effecctive refractivve index as w
well as
TMM results for eeffective meddium-approxim
mated
nanoroods (see Figurre 2, red and light blue linnes for
FDTD
D results and ccircles for TM
MM results).
Fig. 2. Reeflectance speectra of nanorods of 375 nm
m length. Vollumetric fill fa
factors are inddicated by collor.
best
The
T
sub-waveelength strucctures with the
t
antti-reflective properties were taperred
structuures with higgh aspect ratios. Pyramidss with
lengthss of 800 oor 400 nm had the lowest
reflectancce (<0.1%; see
s Fig. 3). Cones, whiich
were mod
deled here in a grid array with
w their basses
of each touching
t
its four closest neighbors had
h
reflectancces of <0.25%
% for the sam
me lengths (ssee
Fig. 4). For
F both shap
pes, the conffigurations wiith
the lowesst reflectance (<0.25%) were
w
100 to 200
nm acrosss at the base and
a 400 to 80
00 nm long. The
T
pyramids with the worst
w
AR pro
operties had a
length of 100 nm (1.7--1.9%). Conees of length 100
nm reflectted between 1.6
1 and 1.75%
% of light. Th
hus
the heigh
ht of the taapered structu
ure was mo
ore
importantt to anti-reflective propeerties than was
w
base widtth or shape. The
T main diffe
ference betweeen
the refflectivity behaavior of the cones and pyraamids
is that cones had a slightly higheer reflection ddue to
the steep change inn effective m
medium wherre the
cone bbase meets thee substrate.
Comparisonn of FDT
TD with T
TMM
simulaation resultss for pyrramidal AR
RSWS
structuures indicatess a consistenttly lower preddicted
reflecttance from TM
MM for all hheights below
w 800
nm. F
For the 800 nm structurees the TMM
M and
FDTD
D simulations matched cloosely. Upon close
examinnation of Figg. 3 it appeaars that the T
TMM
resultss are not shifteed down, butt are shifted tooward
longerr wavelengthss.
Fig. 3. Reeflectance of four-sided
f
pyramid-shaped
d ARSWSs. L
Line color inddicates pyram
mid height.
c
ARSWSs.
A
Lin
ne color indiccates cone heiight.
Fig. 4. Reeflectance of cone-shaped
Several structtures with asspect ratios of
approximately one were investtigated. Theese
structuures include densely paccked hemisphheres,
cubes (checkerboarrd pattern), aand cylinderss (see
Fig. 5) as well as sparsely packed hemispherees,
cubes, an
nd cylinders (see
(
Fig. 6). All except the
t
hemispherres had an aspect ratio of one, while
hemispherres have, by definition, an
n aspect ratio of
one half. The sparseely packed structures
s
weere
arranged in a grid wiith the spacee between eaach
structure equal to the width
w
of thatt structure (Fiig.
6, right siide), while th
he densely paacked structurres
were arran
nged in a grid
d pattern with
h each structu
ure
touching its four nearest neighborrs (see Fig. 5,
right sidee). Overall, these smalleer aspect rattio
structuures did not hhave as low off reflectance aas did
the talller cones annd pyramids, with the llowest
reflecttance cominng from densely packed
hemisppheres of 2200 nm at 0.9% reflecttance.
Overalll, the checkkerboard connfiguration (ddotted
lines inn Fig. 5) preesented the loowest reflectiion at
betweeen 1.1 and 1.66%. Larger structures exhhibited
less rreflectance ffor both spparse and dense
structuures and thee cylinders and cubes both
exhibitted mild interrference patteerns at sizes larger
than 500 nm.
Fig. 5: Reeflectance speectra ARSWS
Ss of aspect raatio one. Coloors indicate feeature size.
RSWSs. Coloors indicate feeature sizes.
Fig. 6: Reeflectance speectra of sparseely packed AR
Hexaganol
H
clo
osest packed hollow spherres
were mod
deled for theeir anti-reflecctive propertiies
(see Fiig. 7). One, ttwo, or fourr layers of sppheres
were m
modeled for sspheres of 500, 100, and 1550 nm
diameters, all of which
h had shell th
hicknesses off 5
nm. In general
g
the smaller
s
spherres, 50 nm in
diameter, exhibited th
he lowest reeflectances; the
t
structures of two layeers of hollow
w spheres of 50
nm in diameter had 0.7%
0
reflectaance. The neext
best ARC
C in this grou
up was one laayer of 100 nm
n
hollow
w spheres at 1.4% reflectaance. These vvalues
were ffollowed by thhe other 50 nnm hollow sphheres,
then thhe rest of thee 100 nm holllow spheres,, with
the 1550 nm holloow spheres m
making the worst
ARCs of this subseet at between 3.1% (1 layerr) and
3.8% ((4 layers).
Fig. 7: Reeflectance speectra of hollo
ow spheres laayered in a heexaganol clossest packed sstructure withh shell
thicknessees of 5 nm. Diameter
D
is ind
dicated by collor and numbber of layers iss indicated byy line type.
IV. CO
ONCLUSION
N
Optical
O
simu
ulations of a variety of
subwaveleength structu
ures were peerformed usin
ng
the finitee-difference time-domain
n method and
confirmed
d using the trransfer matriix method. The
T
goal of th
his study waas to providee a quantitatiive
compariso
on of a variety of ARSW
WSs that can aid
a
in further design of AR
RCs.
Nanostructures
N
s with non-ttapered shap
pes
reflected more electromagnetic rad
diation than did
d
his is due to
o difference in
tapered structures. Th
reflective behavior between gradient
g
ind
dex
materials and the inteerference pro
operties of th
hin
film and thin
t
film-like ARCs. Thin film-like ARC
Cs
consist off a layer of material
m
that has the same fill
f
factor an
nd RI at eveery level off cross section
normal to
o the travel of the lightt. These film
ms,
consisting
g of both air and
a nanostruccture materiaals,
have an intermediatee “effective”” n, which is
between that
t
of the airr and the bulk
k solid. As lig
ght
hits the to
op of these laayers it encou
unters an abru
upt
change in
n n and a portion of light reeflects, dictatted
by eqn. (1). The samee event occurs at the botto
om
of the lay
yer as the ligh
ht enters the bulk substraate.
When the film thicckness is an odd multiplee of a
quarterr wavelengthh, the reflectted light from
m the
two innterfaces cauuse interfereence, resultinng in
reduceed reflectionn. This periiodic interfeerence
effect results in wh
what appears tto be waveleengthdependdent oscillatioons in the refflectance spectrum
for theese materials.
Tapered naanostructures resulted in more
broadbband anti-refllective properrties than didd nontaperedd nanostructuures. This occurs becausse the
effectivve n of the ARSWS layyer is continuuously
increassing from thaat of air to thhat of the subbstrate
materi al over the llength of the structures, sso the
incidennt light does not detect aan abrupt inteerface
that w
would induce reflections. Of the strucctures
studiedd, the minim
mum reflecttion was thhrough
conicaal and pyramiddal structuress with bases oof 200
nm annd heights off 800 or 400 nm. These rresults
are in accordance with the thheory that opptimal
textureed surface ARCs havee structures with
diametters smaller th
than but heighhts that are att least
a signiificant fractioon of the waveelengths.
Hollow naanospheres w
were modeled for
their aanti-reflectivee properties as a real world
example of a nanostructure that can be easily
created in the laboratory. Hollow nanospheres
were modeled in a hexaganol closest packed
structure with a variety of number of layers. These
results indicate a recommendation for two layers
of smaller (50 nm) nanospheres for best AR
properties. These simulation results indicate that,
like the non-tapered structures investigated in this
paper, hollow nanospheres exhibit interferencelike patterns in their reflectance spectra. This can
likely be explained by an effective media theory,
where the hollow nanospheres and the non-tapered
nanostructures interact with light in a similar
manner as a thin film of intermediate index of
refraction.
The transfer matrix method was used to
confirm the FDTD methods reported here. Very
good correlation was found between the two
methods for non-tapered nanorod structures.
However, some discrepancy was found for the
tapered pyramidal structures. This can be
explained in part by the methods used to discretize
the two simulations. While the TMM simulation
was smoothly discretized every 1 nm of height of
the pyramids, the FDTD method was meshed at
2.5 nm. This would affect the description of the tip
of the structure, making the effective height of the
structure shorter, which would target a shorter
wavelength. This effect is seen in the apparent
shift of the TMM data toward a longer
wavelength. Discretizing the FDTD simulations
further becomes very computationally intensive.
Models in this study covered feature
widths between 50 and 200 nm, which are
appropriate sub-wavelength widths for 300 to
1300 nm light. Attempts were made to investigate
the anti-reflective properties of features larger than
200 nm, but multiple reflections in the structures
not only made modeling times prohibitively long,
but also resulted in regions of the spectrum that
erroneously reported much higher than 100%
transmission. For these simulations, error averaged
over the spectrum was between 2 and 7%, while
for all reported simulations the error was well
below 2% and usually below 0.4%.
Future work will investigate the effects of
angle of incidence and index of refraction on the
anti-reflective properties of ARSWSs. The index
of refraction of real materials is wavelengthdependent. A thorough study of ARSWSs within a
range of indexes of refraction that is reasonable for
the materials used to make those structures could
provide researchers with more realistic modeling
capabilities.
ACKNOWLEDGEMENT
The authors like to acknowledge the
support from NSF STTR Phase II, Oregon Process
Innovation Center/Microproduct Breakthrough
Institute, ONAMI, Oregon BEST, and Remcom,
Inc. The authors would like to thank Yujuan He
for her contributions to the design of sizing and
configurations of the hollow spheres simulations.
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Linear Modeling of Active or Passive Optical Lamellar Nanostructures," presented at the 27th Annual Review of Progress in Applied Computational Electromagnetics, Williamsburg, Virginia, U.S.A., 2011. D. Aspnes, "Local‐field effects and effective‐medium theory: A microscopic perspective," American Journal of Physics, vol. 50, p. 704, 1982. Ka
atherine Han is a Ph.D.
stu
udent in th
he School of
Ch
hemical,
Biiological,
and
a
En
nvironmental Engineering.
E
She
S
is a co-founder of
o the OSU So
olar
Veehicle Team an
nd has served as
graduate studentt president in her
h
departm
ment. Kat is the lab managerr for the Oreg
gon
Process Innovation Ceenter.
Drr. Chih-hung (Alex) Chang
is a Professor in
n the School of
o
Ch
hemical, Bio
ological, and
Engineering
En
nvironmental
g.
Hiis group has studied
s
solution
baased thin fillm deposition
prrocesses, ink jet printing
g,
microreacction technolog
gy, and X-ray absorption fin
ne
structure. He has more than
t
80 refereeed publicationss,
8 issued patents, and 4 pending paatents in thesse
areas. Ch
hih-hung is a SHARP Lab
bs of Americca
scholar, an
a Intel Facultty Fellow, and
d a recipient of
o
AVS Graaduate Researrch award, Naational Sciencce
Foundatio
on’s CAREER
R award, and
d awardees of
o
W.M. Keeck Foundatio
on. He is thee founder and
Director of Oregon Prrocess Innovattion Center an
a
Oregon BEST
B
signatu
ure research lab.
l
He is th
he
founder of
o a start-up ven
nture, CSD Naano Inc.
Dr.. Hai-Yue Ha
an received hiis
Ph.D. in electricall engineering at
a
Oreegon State Uniiversity. He is a
co-founder of thee Oregon Statte
Uniiversity Solar Vehicle Team
m
and
d has built threee solar powered
carss at OSU. He currently
c
work
ks
as the leead balance of systems engin
neer at Inspire d
Light.
Jam
mes F. Stack, Jr. received a
B.S
S. degree in electricaal
eng
gineering in 20
001 and an M.E
E.
deg
gree in systemss engineering in
201
10 from the Pennsylvaniia
Staate University. He has worked
forr Remcom, Innc. since 20000 where he cuurrently
maanages the appplications enggineering depaartment.
Hiis research innterests includde high perforrmance
coomputing (SSE
E/AVX vectorizzation, GPGPU
U, MPI
annd threading) annd using optim
mization techniques to
achhieve novel RF
F structures.