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: magnusson@uta.edu

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

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

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).

4