Supplementary Information for “Optical Activity Enhanced by Strong Inter-molecular Coupling in Planar Chiral Metamaterials”
Teun-Teun Kim1, Sang Soon Oh2, Hyun-Sung Park1,Rongkuo Zhao2,Seong-Han Kim3,Wonjune Choi4, Bumki Min1 and Ortwin Hess2
1Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology(KAIST), Daejeon 305-701, Republic of Korea
2The Blackett Laboratory, Department of Physics, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom
3Advanced Photonics Research Institute, GIST, Gwangju, 500-712, Republic of Korea
4Department of Physics, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 305-701, Republic of Korea
I.Equivalent RLC model for CDZM
- Numerical simulation of fields at resonances
- Derivation of resonance frequencies of CDZM
- Derivation of effective chirality parameters of CDZM
- Definition of three regimes of coupling
II.Intra-molecular coupling inCDZM
- Single layer double Z metasurface
- Surface current of cut wires, double cross-wires and CDZM
- Dependence of effective parameters on geometrical parameters
III.Comparison between numerical simulations and equivalent RLC model
IV.CDZM at THz frequencies
- Fabrication process
- Optical characterization
V.Electric field profiles for CDZM
VI.Gap width dependent circular dichroism η
- Two possible loss channels in CDZM
- High loss dielectric substrate – FR4
- Equivalent RLC model for CDZM
- Numerical simulation of fields at resonances
To identify the capacitive and inductive elements in CDZM, we performed the finite-difference time-domain simulations and plotted the electric and magnetic fields at resonant frequencies. The field distributions for RCP (top in Fig. S1a) and LCP (bottom in Fig. S1a) waves at resonances are very similar to each other except the different handedness (rotation direction) of electric or magnetic dipoles around the axis of propagation, that is,clockwise (anti-clockwise) directionfor RCP (LCP) waves. Therefore, we candescribeboth the RCP and LCP excitations using an RLC circuit with the same capacitive and inductive elements.
Figure S1. Snap shots of electric fields at resonant frequencies. (a) z component of electric field in the middle of dielectric substrate. Outlines of the top metallic structure are drawn with the solid lines. (b) Cross sectional view of electric field at the plane denoted bythe horizontal dashed line in (a).Metallic structures are indicated by the six horizontal lines and the direction of the incident waves are indicated by the arrows.
As shown in Fig. S1, electric fields are highly enhanced in several specific locations of the CDZM. This local field enhancement allows us to identify capacitive and inductive elements of theCDZM. For example, we can assign a capacitive element, denoted by , at the side strips of metallic layers since electric fields between the side strips of top and bottom metallic layers arestrong at the resonance frequencies, .From the cross-sectional view of electric fields in Fig. S1b, we confirm that the enhancement of the electric fields are due to the capacitive element. Similarly, we can assigna capacitive element at the both ends of the central arms as shown in Fig. S1a. Please note that the two capacitors and are activated at both resonance frequencies, but the relative magnitude and sign of electric fields of and vary for different resonances.For instance, at the frequency (), the field at is stronger (weaker) than the one at and the fields at and have the same (opposite) signs. The different sign becomesacharacteristic of the two resonances in the Lagrangian description of the CDZMas described below in Section I.B and I.C
In addition, from the field plot in Fig. S1b, we can identify the gap capacitance that induces electric fields between side strips over unit cell boundaries.is normally weaker than and but can be extremely larger and dominant when the gap width isvery small.This will be discussed in Section I.D in more detail.
In a similar way, we can also identify inductive elements from calculated magnetic field plots. FigureS2 clearly shows that there is an inductive elementcomposed of the central strips of top and bottom metallic layers.
Figure S2. Magnetic fields at resonancefrequencies. The inductive elements at the central strips are dominant at both resonance frequencies.
- Derivation of resonance frequencies of CDZM
The resonance frequencies andof a CDZM can be obtained using an equivalent RLC model; however, it is challenging to consider all inductive and capacitive elements and their connections in the CDZM and solve the resulting coupled equations. Therefore, it is reasonable to simplify the coupled equations by considering only the dominant elements among various inductors and capacitors ofthe CDZM, as is the case for an Ω-particle model [1]. Here, we will use the capacitive and inductive elements identified in Section I.A to derive analytical expressions for resonance frequencies ofthe CDZM.
Figure S3. Schematics of anequivalent RLC circuit for CDZM(a) Equivalent RLC circuitfor intra-molecular couplings. is the capacitance of the side stripsandand are the inductance and capacitance of the central strips respectively.The Solid and dashed lines correspond to the top and bottom metallic layers, respectively. (b) The inter-molecular couplingsbetween adjacent metamolecules are indicated by the additional capacitance .(c) Electric charges induced by an electromagnetic excitation at the top metallic layer.
By analysing the connections in a CDZM, we can draw an equivalent RLC circuitusing the three capacitive elements, , and one inductive element as shown in Fig. S3. The equivalentRLC circuit can be regardedas a coupled resonator system composed of six resonators with two inductive elements and two capacitive elements. The six resonators can be classified as one of three types of resonators composed of an inductance and one of , , .To take into account the effect of thesecouplings, we adopt the Lagrangian formulation for chiral metamaterials [2]. Then, the total Lagrangian becomes
(S1)
where , , (, ,) are charges accumulated at the capacitance , and , respectively (Fig. S3c). Here, the sign corresponds to the lowest two resonance frequenciesof the equivalent RLC circuit.
Subsequently, by putting this into the Euler-Lagrange equation
, (S2)
we have
, (S3a)
,(S3b)
.(S3c)
Here, we omit three equations for , ,, since they have identical forms with (S3) and they are not coupled to these equations.
We assume a solution of the form
. (S4)
Then, this leads to the form
(S5a)
(S5b)
. (S5c)
This can be written in a matrix equation as
=(S6)
where , , and .
The above equation can have solutions only when the determinant of the matrix becomes 0.
=0(S7)
From this condition, we have two positive resonance frequencies
(S8)
where + and – signs corresponds to the first and second resonance frequencies , in the main text,respectively.
C. Derivation of effective chirality parameters of CDZM
To derive the effective chirality parameters of the CDZM, we calculate the induced polarization and magnetization upon electromagnetic wave excitation. For the sake of convenience, we will use two coordinate systems () and () as shown in Fig. S3c. In the primed coordinate system, the incident waves are expressed as
(S9a)
(S9b)
The field components in the original coordinate systems can beexpressed to the ones in the primed coordinate system as follows:
The equations for the motion of electric charges with this field excitation can be written as
(S11a)
,(S11b)
. (S11c)
whereand is the cross-sectional area between top and bottom metal layers.For simplicity, we do not take into account electric field excitations and the dissipative damping with the electric resistances of the RLC circuits since we are interested in derivation of the effective chirality parameter from electric polarization induced by magnetic field excitation. Please note that theelectric field terms and the damping constants (for example, with a resistance R)can be added to these equations for complete derivation of all effective parameters including the effective electric permittivity and magnetic permeability.
This linear equation can be written in a matrix form
(S12)
Then, the solutions are given as
(S13)
where the determinant of the 3×3 matrix in (S12) is
By expanding the matrix multiplication, we have
(S14)
The electric polarization components due to the gap capacitance can be expressed using the charge and the effective length between the gap charges as
(S15)
whereN is the number of the resonators in the system and V is the total volume of the system.
Finally, we have the expressions for chirality parameter in terms ofas follows:
(S17)
where the resonant strengths are given as
(S18)
If we use the definition of the and , we obtain
.(S19a)
When , we have
.(S19b)
D. Definition of three regimes of coupling
As stated in the main manuscript, the total capacitance of a single Z element is composed of three capacitive contributions that are scale differently with the gap width. For the sake of clarity, the formula for the total capacitance is rewritten here as,
(S20)
where is the vacuum permittivity,is the relative permittivity of the substrate material, is the thickness of the strip,isthe effective length of the side strip and isthe width of the side strip.
Thus, the resonant frequency can be written as,
, (S21)
where0.0449pF.
Accordingly, depending on which of the three terms is dominating over the resonance frequency, three regimes of coupling can be defined for the ranges of gap width: Uncoupled regime (), weak inter-molecular coupled regime ( and), and strong inter-molecular coupled regime (). As shown in Figure S4, in the uncoupled regime (grey shaded area), the resonance frequencies show negligible shift because the internal capacitance does not depend on the gap width. In the weak intermolecular coupled regime (blue shaded area), the second term in the curly bracket of Eq. (S26) is dominating and the resonance frequencyis scaling with a rate of . In the strong inter-molecular coupled regime (red shaded area), the resonance frequency is scaling with a rate of . Here, the fitting parameter and .
Figure S4. Simulated resonance frequency (scatters) and parallel plate capacitor approximation (lines).
- Intra-molecular coupling in CDZM
A. Single layer double Z metasurface
Figure S5shows the calculated transmission amplitude and chirality and ellipticity for single layer double Z metasurface with different gap width . The chirality of a single layer chiral metasurface is one order of magnitude smaller than CDZM and one single resonance is observed in the frequency range of interest. Therefore, it is clearly shown that strong chirality comes from the double-layering that induces parallel (antiparallel) current flows along the two (top, bottom) central strips of CDZM.
Figure S5. Optical parameters for single layer double Z metasurface(a) Calculated transmission spectra of RCP (solid line) and LCP (dashed line) waves with different gap widthsg = 0.1 mm (red) and g = 1.0 mm (blue). Effective parameters for (b) chirality κ and (c) ellipticity η for different gap widths g = 0.1 mm andg = 1.0 mm.
B. Surface current of cut wires, double cross-wires and CDZM
Figure S6 shows the transmission amplitude and the surface current of (a) cut wires, (b) double cross-wires and (c, d) CDZM with different side metallic strips. Here, gap width g is fixed at 1.0 mm.This clearly shows how the magnetic resonance and electric resonance evolve as we change the geometry from cut-wire pairs to CDZM. As can be seen in Fig. S6, the surface currents for MR and ER are antiparallel and parallel for cut-wire pairs and double-crosses. However, the surface currents in the central strips of CDZM cannot be classified clearly as antiparallel and parallel for MR and ER due to the additional coupling between the top and bottom at the capacitance . This also confirms the fact that CDZM is chiral, literally meaning that it breaks the mirror symmetry and the oscillating modes along each central arm are not decoupled to each other under linear polarization excitation.
Figure S6. Surface current density for various structuresCalculated transmission amplitude (left) and surface current density (right) for (a) cut wires, (b) double cross-wires and (c,d) CDZM with different side metallic strips.
C. Dependence of effective parameters on geometrical parameters
In order to verify an intra-molecular coupling in the CDZM, chirality κand ellipticity are numerically estimated for samples having different geometrical parameters l and d. First, the dependency of and on the size l is plotted in Figure S7a. For this simulation, gap width is fixed at 0.1 mm. In this plot, it is shown that the resonances are significantly red-shifted as unit cell size increases. It is noteworthy that while increases gradually, does not change significantly as unit cell size increases. Another important parameter is the thickness of the substrate d (i.e. the inter-planar spacing). In Figure S7b, the dependency of effective parameters κ andwith a variation in the thickness of substrate d is plotted. It is shown that κ increases gradually as d becomes smaller. As briefly discussed in the main manuscript, this dependence clearly show that the intra-molecular coupling depend on the geometric parameters of one unit cell. Moreover, decreases as d becomes smaller. This seems supportive that the ellipticity becomes smaller when the inter-planar coupling becomes strong. However, in fact the decrease of comes from the reduced thickness of lossy dielectric resulting in lower loss for both LCP and RCP waves.
Figure S7. Geometrical parameter dependent optical parametersEffective parameters chirality and ellipticity as a function of (a) size of CDZM and (b) thickness of substrate . Here, the gap width is set to 0.1 mm.
- Comparison between numerical simulations and equivalent RLC model
To test the validity of the equivalent RLC model, we compared numerical simulations with the data fitted by analytical expressions (Eqs. (S17), (S21) and (S19)). In Figure S8, frequency-dependent chirality parameters, resonant frequencies, and resonant strengthsare plotted as a function of the gap width . The fitted data are in excellent agreement with the simulated ones, which indicates that the physics can be described well by the equivalent RLC model for all three coupling regimes. Moreover, the red shift of resonance frequencies and the dependence of theresonance strength coefficient can be clearly seen in FigureS8a,b.
Figure S8. Comparison of simulation results with the equivalent RLC model. Simulated (circles) and fitting (lines) results of (a) chirality parameter , (b) resonant frequencies and . (c) Fitting result of resonance strength coefficients andwith the gap width g ranging from 10 to 100 μm using adjusted analytical Eqs. S19a and S19b.
- CDZM at THz frequencies
- Fabrication process
Fabrication of a terahertz CDZM started with a bare silicon substrate as a sacrificial wafer, and a polyimide solution (PI-2610, HD MicroSystems) was used for the spacer material. The polyimide solution was then spin-coated on the bare silicon wafer and pre-baked at 180°C in a convection oven for 30 min. After the curing, a negative photoresist (AZnLOF2035, AZ Electronic Materials) was spin-coated and patterned with conventional photolithography. Metallic patterns were defined by the evaporation and lift-off process. After spin-coating the spacer layer (2 μm, PI-2610), the same process was repeated for the second layer, and the CDZM were finally peeled from the silicon substrate (Fig. S9a).
- Optical characterization
The THz CDZM were characterized by terahertz time domain spectroscopy (THz-TDS). Two free-standing extraordinary optical transmission (EOT) polarizers [3], one in front of and the other after the sample, were used to measure the transmission with φ = 0° and with φ = 90° (see inset of Fig. S9b). The simulated and measured transmission amplitudes for and are shown in Figure S9band are in good agreement with each other. Figure S9cshows the ellipticity η for the gap width = 1.5 μm. The shaded region represents the regime of pure optical activity, i.e. η ~ 0. In Figure S9d, the chirality κ at η = 0 are plotted as a function of gap width . It is clearly shown that the gap width plays a crucial role not only at microwave frequencies but also in the THz regime.
Figure S9. CDZM at THz frequencies(a) Optical micrograph of fabricated CDZM with a unit cell of l = 40 um, d = 2 um, w = 5 um, and different gap width g. (b) Comparison between simulation and measurement of transmission amplitude for and in the case of g = 1.5 μm. Two EOT polarizers [3] are employed to measure the parallel and perpendicular polarization states (inset), respectively. (c) The ellipticity ηin the case of g = 1.5 μm. The shaded region indicates the pure optical activity region, η = 0. (d) The chirality κat the η = 0 as a function of g.
- Electric field profiles for CDZM
Figure S10 shows the calculated electric field distributions in the unit cells of CDZM at the frequencies of (a) , (b) and (c) with gap width = 1.0 mm and = 0.1. As described in the main manuscript, the electric field is more strongly concentrated not only at the resonance frequencies but also non-resonance frequencies as gap width decreased.
Figure S10. Field localization and enhancement in the small gap. Simulated electric field for CDZM at (a) , (b) and (c) with gap width (left) = 1.0 mm and (right) = 0.1 mm.
- Gap width dependent circular dichroism η
- Two possible loss channels in CDZM
Fig. S11 shows comparison of simulated ellipticity spectra between lossy (black line) and lossless (red line) Teflon substrates, and PEC (black line) and lossy metallic (copper, blue line) elements with g = 0.1 mm. It is shown that dielectric losses in the substrate are the main sources of ellipticity in the CDZM.
FigureS11. Two possible loss channels of CDZM. Loss of dielectric substrate is the main reason for the large ellipticity in the CDZM due to negligible loss in the metallic element at microwave frequencies.
- High loss dielectric substrate – FR4
Figure S12 shows the optical parameters for the CDZM patterned on a FR-4 substrate with a thickness d = 0.4 mm. The dielectric constant of the FR-4 substrate is Re(εr) = 4.0 with a dielectric loss tangent of = 0.028. Thechiral metamaterial with different gap width ranging from 0.12 to 5 mmhas geometric parameters l = 5.0 mm, w = 0.8 mm andr = 0.65 mm. In Figure S12a, it is clearly shown that the chirality parameter at η = 0 also represent gap-dependent behaviour. In Figure S12b, the ellipticity η is plotted with the gap width ranging from 0.1 mm to 5 mm. As expected, due to the high loss of the FR-4 material, the ellipticity η of the transmitted wave is significantly larger than the chiral metasurfaces with Teflon substrate. As the gap becomes smaller, the ellipticity η decreases at the resonance frequency f1 as well (Fig. S12c).