SUPPLEMENTAL MATERIAL
Experimental Demonstration of the Optical Lattice Resonance in Arrays of Si Nanoresonators
Stanislav Tsoi,1 Francisco J. Bezares,2 Alexander Giles,1 James P. Long,1 Orest J. Glembocki,1 Joshua D.
Caldwell,1 and Jeffrey Owrutsky1
1
US Naval Research Laboratory, Washington, DC 20375, USA
2
ICFO, Barcelona, 08860, Spain
S1. Additional experimental results
Figure S1 shows reflectivity spectra of the arrays of nanopillars with a 150 nm diameter recorded with a
40X objective. Figure S2 shows reflectivity spectra of the arrays of nanopillars with 200 nm and 250 nm
diameters recorded with a 10X objective.
D=150 nm
40X, NA=0.6
Reflectance (arb. u.)
P=700nm
P=650nm
P=600nm
P=550nm
P=500nm
P=450nm
P=400nm
 (nm)
400
600
800
1000
Figure S1 (color online). Reflectivity spectra of arrays of nanopillars with a 150 nm diameter and various
periods (shifted vertically for clarity) recorded with a 40X objective.
(A) D=200nm
Reflectance (arb. u.)
1
10X, NA=0.2
(B) D=250nm
P=800nm
P=750nm
P=700nm
P=650nm
P=750nm
P=700nm
P=650nm
P=600nm
P=550nm
P=600nm
P=550nm
P=500nm
P=500nm
P=450nm
random
random
4 3 2
400
800
1200
 (nm)
4
400
3 2
800
1200
 (nm)
Figure S2 (color online). (A) Reflectivity spectra of arrays of nanopillars with a 200 nm diameter and
various periods (shifted vertically for clarity) recorded with a 10X objective. Vertical dashed lines and
corresponding numbers designate localized Mie resonances seen in the random array. The gray arrow
emphasizes the lattice resonance dispersing with the period. (B) Same for arrays with a nanopillar
diameter of 250 nm.
S2. Reflectivity fits
To determine spectral positions and widths of the dielectric resonances, the reflectivity spectra of the
nanopillar arrays are fitted with Gaussian dips according to
−(𝜆−𝜆𝑖 )
2𝜎2
𝑖
2
𝑅(𝜆) = 𝑅0 − ∑𝑁
,
(S1)
𝑖=1 𝐴𝑖 ∙ 𝑒
where R is the reflectance,  the wavelength, R0 the reflectance offset, N the total number of the
resonances; and Ai, i, and 2i the amplitude, wavelength, and width of the ith resonance, respectively.
Because of the large number of the parameters and potential unaccounted contributions to the
reflectivity, the fits are not based on minimization of the deviation from the experimental data. Instead,
the parameters R0, Ai, i, and i are adjusted manually until the fits yield a close approximation to the
experimental curves. Figures S3-S5 exhibit the fits to the individual spectra and Tables S1-S3 list the
parameters determined.
Table S1. Parameters of the reflectivity fits for the arrays with the nanopillar diameter of 150 nm.
Random
R0=35 %
A1=14.0 % 1=689 nm 1=77 nm
A2=2.0 % 2=538 nm 2=33 nm
P=400 nm
R0=39 %
A1=21.5 % 1=675 nm 1=65 nm
A2=2.2 % 2=505 nm 2=20 nm
P=450 nm
R0=37 %
A1=20.0 % 1=685 nm 1=62 nm
A2=5.0 % 2=511 nm 2=15 nm
P=500 nm
R0=36 %
A1=17.5 % 1=700 nm 1=65 nm
A2=9.5 % 2=532 nm 2=19 nm
P=550 nm
R0=38 %
A1=20.0 % 1=712 nm 1=71 nm
A2=10.0 % 2=568 nm 2=30 nm
P=600 nm
R0=39 %
A1=19.0 % 1=730 nm 1=73 nm
A2=9.0 % 2=610 nm 2=37 nm
A3=2.0 % 3=550 nm 3=40 nm
P=650 nm
R0=39.5 % A1=17.0 % 1=755 nm 1=70 nm
A2=11.5 % 2=644 nm 2=42 nm
A3=3.0 % 3=550 nm 3=35 nm
P=700 nm
R0=40 %
A1=13.5 % 1=820 nm 1=67 nm
A2=16.0 % 2=680 nm 2=60 nm
A3=2.5 % 3=545 nm 3=35 nm
Random
D=150nm
P=400nm
40
30
Exp
Fit
20
400
 (nm)
600
800
Reflectance (%)
Reflectance (%)
40
30
Exp
Fit
20
1000
400
600
P=450nm
1000
Reflectance (%)
Reflectance (%)
40
30
30
Exp
Fit
20
400
Exp
Fit
20
 (nm)
600
800
1000
400
 (nm)
600
800
1000
P=600nm
P=550nm
40
40
30
Exp
Fit
20
400
 (nm)
600
800
Reflectance (%)
Reflectance (%)
800
P=500nm
40
30
Exp
Fit
20
1000
400
P=650nm
 (nm)
600
800
1000
P=700nm
40
30
Reflectance (%)
40
Reflectance (%)
 (nm)
30
Exp
Fit
20
400
600
Exp
Fit
20
 (nm)
800
1000
400
600
 (nm)
800
1000
Figure S3 (color online). Reflectivity spectra from arrays of nanopillars with a diameter of 150 nm fitted
with Gaussian dips. The individual dips are offset vertically from the net fit and experimental data for
clarity.
D=200nm
Random
P=450nm
40
30
Reflectance (%)
Reflectance (%)
40
30
Exp
Fit
20
400
 (nm)
600
800
Exp
Fit
20
1000 1200
400
600
800
1000 1200
P=550nm
P=500nm
Reflectance (%)
Reflectance (%)
40
40
30
30
Exp
Fit
20
400
600
800
Exp
Fit
20
400
1000 1200
600
800
1000 1200
P=650nm
P=600nm
Reflectance (%)
Reflectance (%)
40
40
30
30
Exp
Fit
20
400
 (nm)
600
800
Exp
Fit
20
400
1000 1200
 (nm)
600
P=700nm
800
1000 1200
P=750nm
Reflectance (%)
Reflectance (%)
40
40
30
30
Exp
Fit
20
400
600
Exp
Fit
20
 (nm)
800
1000 1200
400
600
 (nm)
800
1000 1200
Figure S4 (color online). Reflectivity spectra from arrays of nanopillars with a diameter of 200 nm fitted
with Gaussian dips. The individual dips are offset vertically from the net fit and experimental data for
clarity.
D=250nm
Random
P=500nm
Reflectance (%)
30
30
20
10
Reflectance (%)
40
40
Exp
Fit
400
600
800
20
 (nm)
1000 1200
10
Exp
Fit
400
P=550nm
Reflectance (%)
Reflectance (%)
30
20
Exp
Fit
400
600
800
20
10
1000 1200
Exp
Fit
400
1000 1200
30
Reflectance (%)
Reflectance (%)
800
40
30
20
20
Exp
Fit
400
600
800
1000 1200
10
Exp
Fit
400
P=750nm
600
800
1000 1200
P=800nm
40
40
30
Reflectance (%)
Reflectance (%)
600
P=700nm
P=650nm
40
30
20
10
1000 1200
P=600nm
30
10
800
40
40
10
600
20
Exp
Fit
400
600
 (nm)
800
1000 1200
10
Exp
Fit
400
600
 (nm)
800
1000 1200
Figure S5 (color online). Reflectivity spectra from arrays of nanopillars with a diameter of 250 nm fitted
with Gaussian dips. The individual dips are offset vertically from the net fit and experimental data for
clarity.
Table S2. Parameters of the reflectivity fits for the arrays with the nanopillar diameter of 200 nm.
Random
R0=31.5 % A1=10 %
1=910 nm 1=130 nm
A2=5.5 % 2=655 nm 2=50 nm
A3=2.0 % 3=535 nm 3=20 nm
A4=2.0 % 4=495 nm 4=15 nm
P=450 nm
R0=32 %
A1=11 %
1=880 nm 1=160 nm
A2=3.5 % 2=640 nm 2=40 nm
A3=4.0 % 3=505 nm 3=35 nm
A4=9.0 % 4=475 nm 4=15 nm
P=500 nm
R0=32 %
A1=12.0 % 1=900 nm 1=150 nm
A2=5.5 % 2=633 nm 2=40 nm
A3=10.0 % 3=522 nm 3=26 nm
A4=8.5 % 4=486 nm 4=22 nm
P=550 nm
R0=34 %
A1=16.0 % 1=920 nm 1=140 nm
A2=11.5 % 2=630 nm 2=35 nm
A3=11.5 % 3=550 nm 3=25 nm
A4=10.5 % 4=495 nm 4=25 nm
P=600 nm
R0=31 %
A1=16.0 % 1=940 nm 1=120 nm
A2=14.0 % 2=647 nm 2=30 nm
A3=6.0 % 3=565 nm 3=22 nm
A4=4.0 % 4=503 nm 4=23 nm
P=650 nm
R0=32 %
A1=16.0 % 1=950 nm 1=110 nm
A2=14.0 % 2=680 nm 2=40 nm
A3=2.0 % 3=565 nm 3=25 nm
A4=2.0 % 4=503 nm 4=23 nm
P=700 nm
R0=34%
A1=18.0 % 1=970 nm 1=110 nm
A2=14.0 % 2=710 nm 2=55 nm
A3=2.0 % 3=557 nm 3=24 nm
A4=3.0 % 4=497 nm 4=20 nm
P=750 nm
R0=35 %
A1=18.0 % 1=980 nm 1=105 nm
A2=13.0 % 2=740 nm 2=65 nm
A3=2.0 % 3=557 nm 3=24 nm
A4=2.5 % 4=500 nm 4=20 nm
Table S3. Parameters of the reflectivity fits for the arrays with the nanopillar diameter of 250 nm.
Random
R0=31 % A1=7.0 % 1=995 nm
1=120 nm
A2=8.0 % 2=750 nm
2=85 nm
A3=6.5 % 3=630 nm
3=35 nm
A4=9.5 % 4=562 nm
4=35 nm
P=500 nm
R0=30 % A1=8.5 % 1=962 nm
1=120 nm
A2=4.5 % 2=750 nm
2=69 nm
A3=8.0 % 3=560 nm
3=35 nm
A4=12.0 % 4=517 nm
4=22 nm
P=550 nm
R0=30 % A1=9.0 % 1=980 nm
1=130 nm
A2=6.0 % 2=740 nm
2=70 nm
A3=15.0 % 3=578 nm
3=22 nm
A4=14.0 % 4=537 nm
4=22 nm
P=600 nm
R0=29 % A1=9.0 % 1=1000 nm 1=150 nm
A2=6.0 % 2=740 nm
2=70 nm
A3=15.0 % 3=617 nm
3=27 nm
A4=11 %
4=565 nm
4=25 nm
P=650 nm
R0=31 % A1=11.0 % 1=1010 nm 1=150 nm
A2=11.0 % 2=730 nm
2=65 nm
A3=12.0 % 3=641 nm
3=30 nm
A4=9.0 % 4=580 nm
4=30 nm
P=700 nm
R0=32 % A1=12.0 % 1=1030 nm 1=180 nm
A2=13.0 % 2=750 nm
2=50 nm
A3=10.5 % 3=660 nm
3=30 nm
A4=8.0 % 4=590 nm
4=33 nm
P=750 nm
R0=34%
A1=17.0 % 1=1080 nm 1=200 nm
A2=13.0 % 2=778 nm
2=50 nm
A3=8.0 % 3=673 nm
3=40 nm
A4=7.0 % 4=590 nm
4=45 nm
P=800 nm
R0=37 % A1=17.0 % 1=1080 nm 1=220 nm
A2=12.0 % 2=815 nm
2=45 nm
A3=8.0 % 3=690 nm
3=65 nm
A4=7.0 % 4=590 nm
4=50 nm
S3. Numerical simulations
Simulations are performed with CST Studio1 using the frequency solver and Lumerical2 using finite
difference time domain which employs a time domain solver. We calculate the reflectivity spectra using
both programs and generate electric field maps using CST Studio. In our studies, both CST Studio and
Lumerical generate identical reflectivity spectra. The optical constants for Si used in the simulations are
obtained from Palik’s Handbook of Optical Constants of Solids.3 The incident plane wave is normal to the
array surface (z-axis in Fig. S6), while periodic boundary conditions are applied in the plane of the array
(xy-plane), corresponding to an infinitely large two-dimensional array. The electric field amplitude of the
incident radiation is 5.6107 V/m.
Figure S6(A) shows a simulated reflectivity map for Si nanopillar arrays on a Si substrate with the
nanopillar diameter of 150 nm as a function of the array period and incident wavelength. The map
clearly exhibits the avoided crossing between the diffracted wave and the second localized Mie
resonance, in close agreement with the experimental results (Fig. 1C). The range of the array periods
over which the avoided crossing takes place appears narrower in the simulations than the experiment,
possibly due to the significantly spectrally narrower diffracted wave realized in the idealized simulations.
Figure S6(B) demonstrates reflectivity spectra for the periods of 480 and 530 nm, the former featuring
well separated diffracted wave and second localized Mie resonance, while the latter the collective lattice
mode.
(A)
36%
36
600
P=480nm
P=530nm
Reflectance (%)
 (nm)
580
30
560
24
540
18
520
500
480
(B)
1
12
450
12%
2
3
500
550
600
 (nm)
480 500 520 540 560 580
P (nm)
(C)
1 (=480nm)
2 (=520nm)
9.75107 V/m
3 (=537nm)
0
9.75107 V/m
z
x
y
Figure S6 (color online). (A) Simulated reflectivity map for Si nanopillar arrays on a Si substrate with the
nanopillar diameter of 150 nm, as a function of the array period and incident wavelength. The dashed
line emphasizes the avoided crossing trajectory of the lattice mode. (B) Simulated reflectivity spectra for
arrays with the period of 480 nm and 530 nm. The vertical dashed lines indicate the diffraction condition
=P. Features marked ‘1’, ‘2’ and ‘3’ correspond to the diffracted wave, second localized Mie resonance,
and the lattice resonance, respectively. (C) Near field distribution of the x-component of the electric
field in the yz-plane for the modes identified in B.
The simulations are used to yield near field distributions of the electric field, shown for the three above
modes in Fig. S6(C). A regular interference pattern commensurate with the period is seen for the
diffracted wave in panel ‘1’ suggesting that scattered waves arrive at each pillar in phase, consistent
with the diffraction. In contrast, the distribution for the second localized Mie mode in panel ‘2’ exhibits
almost unperturbed incident plane wave. Because the wavelength of the Mie mode is noticeably longer
than the array period (Fig. S6B), the scattered waves arrive at most positions in the near field with
unmatched phases effectively cancelling each other. This leaves each pillar to respond mostly to the
incident wave, in accordance with the localized nature of the Mie resonance. Finally, the hybrid lattice
mode seen in panel ‘3’ bears close resemblance to both the diffracted wave and localized Mie
resonance. The field of the lattice mode inside the pillars is very similar to that of the localized Mie
resonance, however because its wavelength is close to the period, a definitive interference pattern also
develops around the pillars similar to the diffracted wave, but with the highest field concentrated
between the pillars. This interference pattern suggests that each pillar responds not only to the incident
wave, but also to the scattered waves, underscoring interaction among the pillars and thus the collective
character of the lattice resonance.
1
2
3
Computer Simulation Technology AG, (www.cst.com).
Lumerical Solutions Inc., (www.lumerical.com).
Edward D. Palik, Handbook of optical constants of solids. (Academic Press, Orlando, 1985),
pp.xviii.