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Wideband dielectric metamaterial reflectors: Mie scattering or
leaky Bloch mode resonance?: supplementary material YEONG HWAN KO
AND ROBERT MAGNUSSON* Department of Electrical Engineering,
University of Texas at Arlington, Box 19016, Arlington, Texas
76019, USA
*Corresponding author: [email protected]
Published 9 March 2018
This document provides supplementary information to “Wideband
dielectric metamaterial reflectors: Mie scattering or leaky Bloch
mode resonance?,” https://doi.org/10.1364/OPTICA.5.000289. Here, we
explain further details on simulation and modeling. Section 1
presents the effective medium conversion of a 2D resonant lattice
to 1D slab waveguide. In conversion to 1D from 2D, Section 2 shows
the modal processes with two equivalent 1D models. Section 3
illustrates the discussion of Mie scattering in isolated Si rods.
Finite-difference time-domain (FDTD) simulations are employed to
quantify the total scattering cross section (TSCS) spectra of an
isolated Si rod. Utilizing an enclosed launch excitation type, only
scattered fields outside the launch boundary are numerically
encountered. Section 4 explains the specific boundary conditions in
the FDTD simulations to make the four visualizations of the
electrodynamics of light propagation in periodic and finite Si
gratings.
1. Effective Medium Conversion
Figure S1. Effective-medium conversion of a 2D resonant lattice
to 1D slab representations. Using the zero-order effective index,
the 2D constituent grating is decomposed into two quasi-equivalent
1D grating components, where TM and TE modes have their
electric-field vectors perpendicular and parallel to the grooves of
the 1D grating. Then, the two equivalent 1D-slab waveguide
structures are modeled using the second-order effective index.
https://doi.org/10.1364/OPTICA.5.000289
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Tolatlatesby
Th1Dsehan∥n
whin∥aneffde
Figfunm
o enable clear uttices, we decottices. As depictstablished by
zey N
N
when, two equivaD TM and TE grecond-order EMTave second-or∥ N ,
thaa, and n∥( ) ( )
( ) ( )
2
2
n ,
n a,b
a b
⊥
=
=
( )
( )
1C a,b3
C a,b
f Dwhere
⊥
=
=
=
( )
( )
0
0
n
n⊥here n∥ , dices as func∥ , and nd TM polarizatffective indices
ecoupled two 1
gure S2 Modal pronction of Dh are dodal processes in
understanding ompose the 2D ted in Fig. S1, thero-order effecti(
) (( ) (
0 2H
0 2H
N Fn 1
N Fn
F πD /where
−⊥
= +
= +=alent 1D-slab strrating componenT [1,2]. Thereforder
effectiveat are given a, .
( ) ( )( )
( ) ( )( )
20
20
n ,
n a,b
a b
⊥
+
+
( ) (
( )
22 2
22 2
1 π f 1 f31 π f 1 f3
D / Λ
−
−
( )(
2
2
a, b fa
fa 1 f−
= +
= + −and n , ctions of two , are constantions as seen in in the
zero-o1D grating stru
ocesses in 2D guiddisplayed for (a) tn the 2D resonant
of the physics golattice into quahe correspondinive-medium
the))
1/22L
1/22L
1 F n
1 F n
4Λ
an−−
−
−
ructures are connt (see definitionre, the TM and Te indices nby
the follow(
(
2
2
Λ ,
Λ C a, bλ
C baλ
⊥
+
+
)((
22 2
2
2 2
a b
1 1 na b
and
⊥
−
−
( ))
1/ 22
1/22
1 f b
f b
an−−
−
are the secondifferent indnt coefficient mEq. (3). The n∥order
EMT apuctures. To calc
ded-mode resonathe original 2D ret grating are expla
overning resonaasi-equivalent 1ng 1D gratings aeory (EMT)
givd
(S
nverted from ean in Fig. S1) usinTE slab structurN∥ , anwing
functions )
)
1/2
1/2
b
anb d
(S
) ) ( )( )6 20 0n⊥ (S
nd (S
nd-order effectidices , . Thmultipliers for T and n are
thpproximation fculate the mod
ant lattices and coectangular gratinainable by more t
ant 1D are en 1)
ach ng res nd of 2)
3)
4)
ive he TE the for dal
curves formula
( )
where
γi
tan
tan k
κ
λ
Here thslab wan neffectivwith grthe diffunctionTM mo2. EquIn the
csquare the reflthe speapprox(2nd EMquasi-eeach
otcalculatpreviouseen thfrom a the TMDegradin the 2be impl
orrelations with prng, (b) 1D TM poltransparent proc
using the estimas3 for the slab s( )
( )
( ) () ( )(
2
4
2
2 2
γ
e
ii
slabi
c
i sla
i
d
nk d
n
n
n
κ λκ
λ
κ λ
β λ
=
=
=
= −he estimated efwaveguide by , . The prve propagation c ( )iβ
λ =rating period Λ affracted waves. n of the slab thodes.
uivalent 1D Mconversion to 1Dpillars having lectance
spectraectra in Fig. S2(aximation. UtiliziMT) [2], we
reduequivalent 1D gther and they hted by 2nd EMTus paper.4 Fromhat
all main refleblend of TM andM modes contrded reflection ap2D
grating. Optimlemented to avo
rocesses in effectilarized reflector, cesses occurring i
mated effective structure in air (( ) ( )
( ) (( )
( )( ))
) ( )
2
2 2
4 2 4
12 2
1/22 2
δγ δ
γ
γ
,δ
i
i i i
b c i c
c i slab
ab i
C i
n nn
k
k
λ λ κκ λ λ
λκ λ
β λ
λ
+ −
+−
−
ffective refractiv n∥ropagation consconstant ( )( slabk n sin θ
−and angle of incFinally, the slhickness d are c
Models D, we approximthe same diama of Fig. 2(b) maa) (square
pillarng second-ordeuce the dimensiogratings. The 1Dhave TM and TT.
The detailedm the R0 maps iectance featuresd TE resonant leribute
the broappears where thmization of the 2oid the degraded
ive 1D decomposand (c) 1D TE poin the 1D resonan
index, we use(nc=1) as follows( ))
( ) ( )( ) ( )
( )(
2
/2
2 2
( )
δ
γ
,
δ
i
i i
i i
i C
TE
n
κ λλ
λ κ λλ λ
β λ
= −ve index is appn∥ , andstant is then d)/ Λiλ− idence θ. The
ilab waveguidecharacterized fo
mate the cylindrimeter and heightain text (cylindrirs) shows
that ter effective-medon of the 2D graD gratings are oTE effective
refrd method is exn Figs. S2(b) ans of the 2D grateaky modes. It
isad band of highe TM and TE m2D resonant refd reflection zone
sitions. The R0 coolarized reflectornt gratings.
ed the classic s
)1/22
( )
C
T
k
M (S5)
lied for each efined as an (S6) integers label modes as a r the
TE and
ical pillars by t. Comparing ical pillars) to this is a good
dium theory ating into two orthogonal to ractive index xplained in
a nd S2(c), it is ting originate s notable that gh reflection.
modes coexist flector should e.
olor maps as a r. The complex
2
-
Figdis3.Figpeof sinpeAssigSi at co(m
Figusm
gure S3 Mie scatstribution in (a) a. Mie Scatterigure S3 shows
teriodic Si rods atf the main text. ngle Si rod of eriodically
arrans displayed in Fignificantly differrod even at the t λ = 1.426
µm ounterpropagatimain text).
gure S4 Arrangemsing periodic bouodeled with PML
ttering and guidea single Si rod anding In Isolatethe comparison
t wavelengths (iNotably, the eleFig. S3(a) doesnged to enter theig.
S3(b), the locrent from the MMie electric-dipis close to the fng
lateral Bloch
ment of simulationundary conditionLs at both sides of
ed-mode resonand in a (b) periodicd Rods of field distributi)
1.426 µm as mectric dipole fields not maintain e guided-mode
ralized field and Mie resonance prpole wavelength.field
distributionh modes as pro
n boundaries for rens at both sides f the simulation b
nce in relation to c Si grating at samtions in single anmarked
in Fig. 4(d signature in thas the rods aresonance regimphase
profiles arofile of the sing. In fact, the profn associated
wiovided in Fig. 3(
ecording the dynaof the simulationboundary.
the 1D Si gratinme wavelength (i)nd (b) the are me. are gle file
ith (c)
4. NumWe visreflectoconditio(b) a suthe lateat the t(PMLs)finite
gmodelein Fig. S
amic resonant fien boundary. Fin
ng in Fig. 3(c) of t) 1.426 µm. merical Visuasualize the dynor
by recording ons as shown inublayer is modeeral sides of the top
and bottom). To eliminategrating reflectored by the PMLs aS4.
lds. The reflectornite grating reflec
the main text. Malizations namic field distFDTD simulation Fig.
S4. The reeled using periosimulation bounm boundaries
wicounterpropagrs without (c) aat the sides of si
without (a) and ctors without (c)
Magnitude and p
tributions in thons with differeeflector withoutodic boundary
cndary. We remoith perfectly magation and stanand with (d) a
imulation bound
with (b) a sublay) and with (d) a
phase of field
he Si-grating ent boundary (a) and with conditions at ove
scattering atched layers nding waves, sublayer are dary as
noted
yer is modeled a sublayer are
3
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References 1. R.Bräuer and O. Bryngdahl, “Design of
antireflection gratings with
approximate and rigorous methods,” Appl. Opt. 33, 7875-7882
(1994) 2. D. L.Brundrett, E. N. Glytsis, and T. K. Gaylord,
“Homogeneous layer
models for high-spatial-frequency dielectric surface-relief
gratings: Conical diffraction and antireflection designs,” Appl.
Opt. 33, 2695–2706 (1994).
3. S. S. Wang and R. Magnusson, “Theory and applications of
guided-mode resonance filters,” Appl. Opt. 32, 2606-2613
(1993).
4. Y. H. Ko, M. Shokooh-Saremi, and R. Magnusson, “Modal
processes in two-dimensional resonant reflector and their
correlation with spectra of one-dimensional equivalents,” IEEE
Photon. J. 7, 4900210 (2015).
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