Supplemental material for “Optical Möbius Symmetry in
Metamaterials”
Chih-Wei Chang, Ming Liu, Sunghyun Nam, Shuang Zhang, Yongmin Liu, Guy Bartal
and Xiang Zhang
Detailed sample dimensions
We have designed two kind of meta-molecular trimers.
Trimer Ĉ3 highlights a
meta-molecule with Möbius C3 symmetry whereas Trimer C3 (not shown in the paper)
demonstrates a uniform negative coupling configurations with ordinary C3 symmetry
(discussed later). The detailed sample dimensions are shown in Fig. S1. Trimer C3 has a
symmetric design consisting of three split-ring resonators with 0º, 60º, and 120º-orientated
gaps, respectively. On the other hand, Trimer Ĉ3 has one 60º-orientated gap and two
120º-orientated gaps.
Fig. S1 Detailed dimensions of Trimer C3 (left) and Trimer Ĉ3 (right).
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For the hexamers, they can be decomposed into three kinds of dimers with different
coupling constants. Figure S2 shows detailed dimensions of the constituent dimers of our
designs.
Fig. S2. Detailed dimensions of three kinds of constituent dimers for the hexamers.
Sample fabrication
A double-side polished sapphire substrate was spin coated by 50nm thick of
Poly(methyl methacrylate) (PMMA) 495, 50nm of PMMA 950 and then deposited by
5nm of chromium, subsequently. Different designs of the structures were then defined by
standard electron-beam lithography followed by lift-off processes. For the trimers, arrays
of size 20µm × 20µm with a lattice constant of 1µm were fabricated. For the hexamers,
arrays of size 30µm × 30µm with a lattice constant of 2µm were fabricated. After
chromium etch (Cr-7) and cold development (0°C), a 45nm thick gold film deposition
was carried out at the same time for all samples so that uncertainties from thickness can
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be eliminated. Various metamaterial dimers consisting of two split-ring resonators were
also made for control experiments.
Optical measurements and characterization
A Fourier-transform infrared spectrometer (FTIR) (Bruker IFS125, tungsten lamp)
equipped with a microscope (Bruker Hyperion 2000) was employed for transmission
measurements. For all measurements, the incident light was normal to the sample surface.
An infrared linear polarizer was put in front of the light source and polarization
dependence of transmission was measured. The measured spectra were normalized with
respect to a bare sapphire substrate. Finite-difference time-domain simulations using a
commercial software package CST Microwave Studio were used for analyzing
experimental results. The permittivity of gold film was described by the Drude model
with plasma frequency 1.37×1016 s-1 and the damping rate 1.2×1014 s-1. The refraction
index of sapphire is 1.7.
The signs of coupling of the dimers
Due to the strong current-current exchange interactions in connected split-ring
resonators, the magnetic interactions are much stronger than the electric dipolar
interactions (1). Thus the symmetric and antisymmetric modes of the dimers are defined
by the orientations of magnetic fields, whose information can be obtained by numerical
simulations. The definition is justifiable when we show the results of Trimer C3, which
agrees with the expectations from strong magnetic interactions.
Depending on the
structural designs of the meta-molecules, parallel (symmetric mode) or anti-parallel
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(antisymmetric mode) magnetic dipolar oscillations of a dimer can occur at either low or
high frequency, which determines the signs of coupling between two split-ring resonators,
or meta-atoms. The detailed discussions for the signs of coupling of Dimer 12, Dimer 23,
and Dimer 13 are in the following:
Dimer 12
As shown in Fig. S3, the frequency of the symmetric mode (96 THz) is lower than
that of the antisymmetric mode (107 THz). Thus Dimer 12 exhibits a positive coupling,
i.e. 12 > 0.
Fig. S3. Simulative magnetic field distribution of the two eigenmodes of Dimer 12.
Dimer 23
As shown in Fig. S4, the frequency of the symmetric mode (106 THz) is higher than
that of the antisymmetric mode (94 THz). Thus Dimer 23 exhibits a negative coupling,
i.e. 23 < 0. Notably, Dimer 23 belongs to a point group C2v and the symmetric mode has
an irreducible representation B2. According to the selection rule, it cannot be excited by
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the electric field of a normal-incident light. Thus the symmetric mode is not observed in
the FTIR measurement using a normal-incident light.
Fig. S4. Simulative magnetic field distribution of the two eigenmodes of Dimer 23.
Dimer 13
As shown in Fig. S5, the frequency of the symmetric mode (106 THz) is higher than
that of the antisymmetric mode (88 THz), which indicates that Dimer 23 exhibits a
negative coupling, i.e. 13 < 0.
Fig. S5. Simulative magnetic field distribution of the two eigenmodes of Dimer 13.
Relations between the trimers and the dimers
We use an important relation between a trimer and its constituent dimers to unravel
the sign of 122313. In general, the eigenfrequency () of a three-body system (i.e. a
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trimer) with nearest-neighbor interactions can be determined by solving the following
determinant (1-3):
1 2
12
12
1 
13
 23
2
13
 23  0
1 
(1)
2
where ij’s are the coupling constants between two units. Here we have normalized the
resonance frequency of the decoupled, individual elements.
Notably, when the
magnitude of the coupling between all bodies is identical (i.e. 12 =  23 = 13 = ), the
resonant frequencies are of the form

 1   , 1   , 1  2 for 12 2313  0


 1   , 1   , 1  2 for 12 2313  0
(2)
Importantly, the system always possesses two-fold degeneracy regardless of the sign of
each individual ij’s. Moreover, whether the degenerate mode of the trimer occurs at the
low or the high frequency depends on the sign of the product of 122313. On the other
hand, when two split-ring resonators are coupled (with a coupling constant ) to form a
dimer, the resonance will split into low ( 1   ) and high ( 1   ) frequencies. Notably,
for a trimer with 122313 < 0, its degenerate resonance will occur at 1   , which is
identical to the low resonance frequency of the corresponding dimer. The relation of the
resonance frequencies between a single split-ring resonator (i.e. a meta-atom), a dimer,
and a trimer with 122313 < 0 (Trimer C3) and can be seen in Fig. S6.
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Fig. S6. The relation of resonance frequencies of a meta-atom, Dimer 13, and Trimer C3. When
122313 < 0, the degenerate resonance frequency of Trimer C3 coincides with the symmetric
mode of Dimer 13.
On the other hand, for a trimer with 122313 > 0 (Trimer Ĉ3), the degenerate
frequency will be identical to the high resonance frequency (
1   ) of the
corresponding dimer, as shown in Fig S7. In contrast, the non-degenerate resonance
( 1  2 ) in both cases cannot be matched to any resonance in the dimers. Therefore, by
comparing the resonance frequencies of a trimer with those of the corresponding dimers,
we can identify the degenerate frequency, and the sign of the product of 122313. We
emphasize that we have experimentally and numerically verified the relation in Fig. S6
(shown below) and Fig. S7 (shown in the paper).
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Fig. S7. The relation of resonance frequencies of a meta-atom, Dimer 12, Dimer 23, Dimer 13,
and Trimer Ĉ3. When 122313 > 0, the degenerate resonance frequency of Trimer Ĉ3 coincides
with the symmetric mode of Dimer 12 and the antisymmetric modes of Dimer 23 and Dimer 13,
respectively.
Experimental and simulation results for Trimer C3
The experimental and simulative transmission spectra of Trimer C3, Dimer 13 and
Dimer 13΄ (which is Dimer 13 rotated by 120º) are shown in Fig. S8. The C3 symmetry
of Trimer C3 inhibits a direct excitation of the non-degenerate mode by a normally
incident plane wave. Hence, the observed magnetic resonance at 94 THz must be a
degenerate mode. Furthermore, the resonance at 94 THz does not depend on the incident
polarization and it significantly overlaps with the resonance of Dimer 13 and Dimer 13΄,
which are subsets of Trimer C3. The result suggests that this resonance can be always
decomposed into two independent modes of the same frequency.
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Thus Trimer C3
exhibits a two-fold degeneracy. The result also confirms Fig. S6 that 122313 = 133< 0
for Trimer C3.
Fig. S8. Experimental (upper panel) and simulative (lower panel) transmission spectra of Trimer
C3, Dimer 13 and Dimer 13΄. The number in the parenthesis denotes the polarization angle. The
shaded area highlights the overlapping resonances. Physically, Dimer 13΄ is obtained by rotating
Dimer 13 by 120º. Thus their transmission spectra are identical once the corresponding incident
polarization is also rotated by 120º.
The fact that the degenerate resonance occurs at the low frequency instead of high
frequency demonstrates that magnetic interactions are much stronger than the electric
dipolar interactions.
Correspondingly, it also justifies our choices of using the
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orientations of magnetic fields in each split-ring resonators to be the definition of
symmetric or antisymmetric modes.
Simulation results for Trimer Ĉ3
The simulation results for Trimer Ĉ3 and its corresponding dimers are shown in Fig.
S9. The simulation results agree with experimental measurements shown in the paper.
The resonance at 106 THz of Trimer Ĉ3 overlaps with the antisymmetric mode of Dimer
12 and the symmetric mode of Dimer 13, which confirms that it is a doubly-degenerate
mode.
Or equivalently, the resonance at 106 THz of Trimer Ĉ3 can be always
decomposed into two independent magnetic resonances. Thus the result also confirms
Fig. S7 that 122313 > 0 for Trimer Ĉ3.
Fig. S9. Simulative transmission spectra of Trimer Ĉ3, Dimer 12, Dimer 23, and Dimer 13. The
number in the parenthesis denotes the polarization angle. The shaded area highlights the region
where Trimer B and the dimers exhibit overlapping resonances.
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Topological properties of the hexamers
For meta-molecules consisting of six split-ring resonators, various configurations are
possible. In general, since the nearest neighbor couplings dominate the interactions
between split-rings resonators, the eigenfrequency () of the hexamers can be understood
by solving the following determinant:
1 2
12
0
0
0
16
12
1 
 23
0
0
0
0
 23
1 2
 34
0
0
0
0
 34
1 
 45
0
0
0
0
 45
1 2
 56
16
0
0
0
 56
1 2
2
2
0
(3)
where ij’s are the coupling constants. Here we have normalized the resonance frequency
of individual split-ring resonator. Remarkably, when the magnitudes of ij’s are equal,
the eigenfrequencies of Eq. (3) are a periodic function of the number of positive ij’s (or
equivalently, the number of negative ij’s). In fact, when there are even numbers of
positive ij’s, Eq. (3) gives one set of eigenfrequencies. When there are odd numbers of
positive ij’s, Eq. (3) yields another set. More interestingly, once the number of positive
ij’s is fixed, the eigenfrequencies are independent of the location of positive ij’s in the
hexamers. Importantly, the distribution of eigenfrequencies with respect to the number of
positive ij’s is similar to energy levels of idealized Hückel and Möbius annulenes (4).
Thus the number of positive ij’s in the hexamers represents an equivalent “twist” in a
Möbius strip. Accordingly, it highlights the topological properties of the hexamers.
Simulation results of the hexamers
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The simulative resonance frequencies of the (symmetric, antisymmetric) modes for
the dimers shown in Fig. S2 are respectively: Dimer A (77 THz, 98 THz), Dimer B (97
THz, 90 THz), Dimer C (102 THz, 81 THz). Thus we know that Dimer A exhibits a
positive ; whereas Dimer B and Dimer C exhibit negative ’s. The significant overlaps
of resonance frequencies between different dimers indicate that the magnitudes of ij’s
are almost equal. Furthermore, due to the loss of the system, the magnetic resonance is
broadened by ~10 THz and the criteria for degeneracy are less constrained than the ideal
case.
The simulative results for Hexamer 2A and Hexamer 2B with y-polarized incident
light is shown in Fig. S7.
Like experimental results, the simulation displays a
pronounced overlap of two magnetic resonances from two structurally-different but
topologically-identical hexamers, which clearly demonstrates a topological effect.
Fig. S7. Transmission spectra of Hexamer 2A and Hexamer 2B for y-polarized incident light.
The shaded area highlights the overlapping magnetic resonances.
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References
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2.
3.
4.
H. Liu et al., Phys. Rev. Lett. 97, 243902 (2006).
N. Liu, S. Kaiser, H. Giessen, Adv. Mater. 20, 1 (2008).
T. Li et al., Opt. Express 17, 11486 (2009).
R. Herges, Chem. Rev. 106, 4820 (2006).
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