advertisement

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 1 Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology(KAIST), Daejeon 305-701, Republic of Korea 2 The Blackett Laboratory, Department of Physics, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom 3 Advanced Photonics Research Institute, GIST, Gwangju, 500-712, Republic of Korea 4 Department of Physics, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 305-701, Republic of Korea 1 I. Equivalent RLC model for CDZM A. Numerical simulation of fields at resonances B. Derivation of resonance frequencies of CDZM C. Derivation of effective chirality parameters of CDZM D. Definition of three regimes of coupling II. Intra-molecular coupling in CDZM A. Single layer double Z metasurface B. Surface current of cut wires, double cross-wires and CDZM C. Dependence of effective parameters on geometrical parameters III. Comparison between numerical simulations and equivalent RLC model IV. CDZM at THz frequencies A. Fabrication process B. Optical characterization V. Electric field profiles for CDZM VI. Gap width dependent circular dichroism η A. Two possible loss channels in CDZM B. High loss dielectric substrate – FR4 2 I. Equivalent RLC model for CDZM A. Numerical simulation of fields at resonances To identify the capacitive and inductive elements in CDZM, we performed the finitedifference 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) direction for RCP (LCP) waves. Therefore, we can describe both 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 by the 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. 3 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 the CDZM. For example, we can assign a capacitive element, ′ denoted by πΆπ , at the side strips of metallic layers since electric fields between the side strips of top and bottom metallic layers are strong at the resonance frequencies π1 , π2 . 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 πΆπ . Similarly, we can assign a capacitive element πΆπ at the both ends of the central arms as shown ′ in Fig. S1a. Please note that the two capacitors πΆπ and πΆπ are activated at both ′ resonance frequencies, but the relative magnitude and sign of electric fields of πΆπ and πΆπ vary for different resonances. For instance, at the frequency π1 (π2 ), the πΈπ§ ′ ′ field at πΆπ is stronger (weaker) than the one at πΆπ and the πΈπ§ fields at πΆπ and πΆπ have the same (opposite) signs. The different sign becomes a characteristic of the two resonances in the Lagrangian description of the CDZM as described below in Section I.B and I.C In addition, from the πΈπ¦ field plot in Fig. S1b, we can identify the gap capacitance πΆπ that induces electric fields between side strips over unit cell boundaries. πΆπ is ′ normally weaker than πΆπ and πΆπ but can be extremely larger and dominant when the gap width is very 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. Figure S2 clearly shows that there is an inductive element πΏπ composed of the central strips of top and bottom metallic layers. 4 Figure S2. Magnetic fields at resonance frequencies. The inductive elements πΏπ at the central strips are dominant at both resonance frequencies. B. Derivation of resonance frequencies of CDZM The resonance frequencies π1 and π2 of 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 of the 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 of the CDZM. 5 Figure S3. Schematics of an equivalent RLC circuit for CDZM (a) Equivalent RLC circuit for intra-molecular couplings. C′m is the capacitance of the side strips and Lm and Cm 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 couplings between adjacent metamolecules are indicated by the additional capacitance πΆπ . (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 circuit ′ using the three capacitive elements πΆπ , πΆπ , πΆπ and one inductive element πΏπ as shown in Fig. S3. The equivalent RLC circuit can be regarded as 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 πΏπ and one of πΆπ , πΆπ , πΆπ . To take into account the 6 effect of these couplings, we adopt the Lagrangian formulation for chiral metamaterials [2]. Then, the total Lagrangian becomes 2 2 Γ = πΏπ (πΜ1 ± πΜ3 + πΜ1π ) + πΏπ (πΜ2 ± πΜ4 + πΜ2π ) 1 1 π π 1 2 2 − πΆ (π12 + π22 ) − πΆ ′ (π32 + π42 ) − πΆ (π1π + π2π ) π (S1) where π1, π3 , π1π (π2 , π4 , π2π ) are charges accumulated at the capacitance πΆπ , ′ πΆπ and πΆπ , respectively (Fig. S3c). Here, the ± sign corresponds to the lowest two resonance frequencies of the equivalent RLC circuit. Subsequently, by putting this into the Euler-Lagrange equation π πΓ πΓ π π ( ) − ππ = 0, ππ‘ ππΜ π = 1, 2, 3, 4, 1π, 2π, (S2) we have 1 πΏπ (πΜ1 ± πΜ3 + πΜ1π ) + πΆ π1 = 0, π 1 πΏπ (πΜ1 ± πΜ3 + πΜ1π ) + πΆ ′ π3 = 0, (S3a) (S3b) π 1 πΏπ (πΜ1 ± πΜ3 + πΜ1π ) + πΆ π1π = 0. π (S3c) Here, we omit three equations for π2 , π4 , π2π , since they have identical forms with (S3) and they are not coupled to these equations. We assume a solution of the form π1 = π1 π −πππ‘ , π3 = π3 π −πππ‘ , π1π = π1π π −πππ‘ , . Then, this leads to the form 7 (S4) (−π2 + πΏ 1 π πΆπ ) π1 β π2 π3 − π2 π1π = 0, −π2 π1 + (βπ2 + πΏ 1 ′ π πΆπ ) π3 − π2 π1π = 0, −π2 π1 β π2 π3 + (−π2 + πΏ 1 π πΆπ ) π1π = 0. (S5a) (S5b) (S5c) This can be written in a matrix equation as −π2 + π12 ( −π2 −π2 where π12 = πΏ 1 π πΆπ , π32 = πΏ β π2 βπ2 + π32 βπ2 1 ′ π πΆπ −π2 π1 0 2 π −π 3 ) ( ) = (0 ) 2 π1π 0 −π2 + π1π 2 , and π1π =πΏ 1 π πΆπ (S6) . The above equation can have solutions only when the determinant of the matrix becomes 0. −π2 + π12 | −π2 −π2 β π2 βπ2 + π32 βπ2 −π2 −π2 | = 0 2 −π2 + π1π (S7) From this condition, we have two positive resonance frequencies 1 1 π± = 2π π± = 2π 1 1 1 1 1 √π2 ±π2 +π2 1 3 = 2π 1 ′ +πΆ ) √πΏπ (πΆπ ±πΆπ π (S8) 1π where + and – signs corresponds to the first and second resonance frequencies π1 , π2 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 (π₯, π¦) and (π₯ ′ , π¦ ′ ) as shown in Fig. S3c. In the primed coordinate system, the incident waves are expressed as 8 π¬(π§, π‘) = πΈπ₯ ′ (π§, π‘)πΜ′ + πΈπ¦ ′ (π§, π‘)πΜ′ = (πΈπ₯ πΜ′ + πΈπ¦ ′ πΜ′ )eπ(ππ§−ππ‘) (S9a) π―(π§, π‘) = π»π₯ ′ (π§, π‘)πΜ′ + π»π¦ ′ (π§, π‘)πΜ′ = (π»π₯ ′ πΜ′ + π»π¦ ′ πΜ′ )eπ(ππ§−ππ‘) (S9b) The field components in the original coordinate systems can be expressed 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 1 πΏπ (πΜ1 ± πΜ3 − πΜ1π ) + πΆ π1 = −π½πΏπ π»Μπ¦ ′ (π‘) (S11a) 1 πΏπ (πΜ1 ± πΜ3 − πΜ1π ) + πΆ ′ π3 = −π½πΏπ π»Μπ¦ ′ (π‘), (S11b) 1 πΏπ (πΜ1 ± πΜ3 − πΜ1π ) + πΆ π1π = −π½πΏπ π»Μπ¦ ′ (π‘) . (S11c) π π π where π½ = π0 π/πΏπ and π 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 the electric field terms and the damping constants (for example, π /πΏπ = πΎ 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 −π2 + π12 ( −π2 −π2 β π2 βπ2 + π32 βπ2 −πππ½π»π¦ ′ −π2 π1 2 −π ) ( π3 ) = (−πππ½π»π¦ ′ ), 2 2 π1π −πππ½π»π¦ ′ −π + π1π Then, the solutions are given as 9 (S12) 1 1 1 1 ( β − 2 ) π12 π 2 π32 π1π π1 2 π2 π12 π32 π1π ( π3 ) = β π1π 1 2 2 π1 π3 − ( ± 1 π12 π32 1 2 π12 π1π 1 1 1 1 2 (π 2 + 2 − 2 ) π3 π1 π1π 1 2 π12 π1π ± 1 2 π32 π1π −πππ½1 π»π¦′ 1 2 2 π3 π1π 1 1 1 1 2 (π 2 − 2 β 2 ) π1π π1 π3 ) ( −πππ½3 π»π¦′ ) −πππ½1π π»π¦′ (S13) where the determinant of the 3×3 matrix in (S12) β is 2 β= π2 π12 π32 π1π [ 1 1 1 1 − ( ± + 2 2 2 )] π2 π1 π3 π1π 2 = π2 π12 π32 π1π ( 1 1 − 2) . π 2 π± By expanding the matrix multiplication, we have 2 π32 π1π π1 −πππ½π»π¦′ 2 ( π3 ) = (π12 π1π ). β π1π π12 π32 (S14) The electric polarization components due to the gap capacitance can be expressed using the charge π1π and the effective length between the gap charges ππ = π π ππ¦ ′ = π1π ππ ( ) = −ππ½ππ ( ) π π π 1 1 2 π 2 π1π ( 2 − 2) π π± π = −ππ½ππ ( π ) π2 2 ππ± 2 2 1π ( π± −π ) π»π¦ ′ , 1 √2 π as π»π¦ ′ (S15) where N 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 of π as follows: π π = − π2± π± π 2 2 1π π± −π π π½ππ π0 ( π ) = − 10 πΊπ π± π 2 −π2 π± (S17) where the resonant strengths are given as π π πΊπ = − π2± π½ππ π0 ( π ) (S18) 1π If we use the definition of the π± and π1π , we obtain πΊπ ∝ πΏπ πΆπ ′ +πΆ ) √πΏπ (πΆπ ±πΆπ π . (S19a) ′ When πΆπ β« πΆπ , πΆπ , we have πΊπ ∝ √πΆπ . (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, πΆ = πΆπππ‘ππ + πΆπππ‘ππ 2π‘ 2π€ 0.22 β π0 ππ {1.15 ( π ) + 2.80 ( π ) } ππ + πΆπππ‘ππ = πΆπ‘ + πΆπ€ + πΆπππ‘ππ , (S20) where π0 is the vacuum permittivity, ππ is the relative permittivity of the substrate material, π‘ is the thickness of the strip, ππ is the effective length of the side strip and π€ is the width of the side strip. Thus, the resonant frequency π1 = 2π(πΏπΆ)−1/2 can be written as, 2π‘ 2π€ 0.22 π1 = 2π [πΏ × {α × (1.15 ( π ) + 2.80 ( π ) where α = π0 ππ ππ = 0.0449 pF. 11 −1/2 ) + πΆπππ‘ππ }] , (S21) 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 ( πΆπππ‘ππ ≥ πΆπ€ ), weak inter-molecular coupled regime (πΆπππ‘ππ ≤ πΆπ€ and πΆπ‘ ≤ πΆπ€ ), and strong inter-molecular coupled regime (πΆπ‘ ≥ πΆπ€ ). 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 frequency is scaling with a rate of π−0.11. In the strong inter-molecular coupled regime (red shaded area), the resonance frequency is scaling with a rate of π−0.5. Here, the fitting parameter πΏ = 2.19 μH and πΆπππ‘ππ = 0.11 pF. Figure S4. Simulated resonance frequency ππ (scatters) and parallel plate capacitor approximation (lines). 12 II. Intra-molecular coupling in CDZM A. Single layer double Z metasurface Figure S5 shows the calculated transmission amplitude and chirality π and ellipticity π for single layer double Z metasurface with different gap width π. 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 widths g = 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 and g = 1.0 mm. 13 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 ′ πΆπ . 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. 14 Figure S6. Surface current density for various structures Calculated 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 κ and ellipticity π are numerically estimated for samples having different geometrical parameters l and d. First, the dependency of π and π on the size l is plotted in Figure S7a. For this simulation, gap width π 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 π π1 increases gradually, π π=0 does not change significantly as unit cell size 15 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 and π with 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, π 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 π comes from the reduced thickness of lossy dielectric resulting in lower loss for both LCP and RCP waves. 16 Figure S7. Geometrical parameter dependent optical parameters Effective parameters chirality π and ellipticity π as a function of (a) size of CDZM π and (b) thickness of substrate π . Here, the gap width π is set to 0.1 mm. 17 III. 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 strengths are plotted as a function of the gap width π. 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 π−1/2 dependence of the resonance strength coefficient can be clearly seen in Figure S8a,b. Figure S8. Comparison of simulation results with the equivalent RLC model. Simulated (circles) and fitting (lines) results of (a) chirality parameter π , (b) resonant frequencies π1 and π2. (c) Fitting result of resonance strength coefficients πΊπ 1 and πΊπ 2 with the gap width g ranging from 10 to 100 μm using adjusted analytical Eqs. S19a and S19b. 18 IV. CDZM at THz frequencies A. 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). B. Optical characterization The THz CDZM were characterized by terahertz time domain spectroscopy (THzTDS). 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 π|| with φ = 0° and π⊥ with φ = 90° (see inset of Fig. S9b). The simulated and measured transmission amplitudes for π|| and π⊥ are shown in Figure S9b and are in good agreement with each other. Figure S9c shows the ellipticity η for the gap width π = 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 π. It is clearly shown that the gap width plays a crucial role not only at microwave frequencies but also in the THz regime. 19 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 π+ and π− 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 η in the case of g = 1.5 μm. The shaded region indicates the pure optical activity region, η = 0. (d) The chirality κ at the η = 0 as a function of g. 20 V. Electric field profiles for CDZM Figure S10 shows the calculated electric field distributions in the unit cells of CDZM at the frequencies of (a) π1 , (b) πη=0 and (c) π2 with gap width π = 1.0 mm and π = 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) π1, (b) πη=0 and (c) π2 with gap width (left) π = 1.0 mm and (right) π = 0.1 mm. 21 VI. Gap width dependent circular dichroism η A. 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. Figure S11. 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. B. 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 tan πΏ = 0.028. The chiral metamaterial with different gap width ranging from 0.12 to 5 mm has geometric parameters l = 5.0 mm, w = 0.8 mm and r = 0.65 mm. In Figure S12a, it is clearly shown that the chirality parameter π at η = 0 also represent gap-dependent behaviour. In Figure S12b, the ellipticity η is plotted with the gap width π 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). 22 Figure S12. Optical parameters for the CDZM with high-loss substrate FR4. (a) The simulated (black circled line) and measured (red circle) chirality κ at the frequencies η = 0 as a function of the gap width g. (b) Simulated ellipticity η as a function of the gap width g. The black dashed line represents the frequency where the η = 0. (c) The simulated (black circled line) and measured (red circle) η at the resonance frequency f1 with the different gap width g. References 1. R. Zhao, et al., Optics Express 18, 14553, 2010 2. H. Liu, et al., Nature Photonics 3, 157, 2009 3. S. Lee, et al., Nature Materials 11, 936, 2012 23