Magnetic Fano resonances by design in symmetry broken THz meta-foils

Magnetic Fano resonances in there-dimensional symmetry broken meta-foils at THz frequencies are theoretically and experimentally studied. Sharp Fano resonances occur due to the interference between different resonances and can be designed by choosing geometric parameters of the meta-foil. At the Fano resonances, the meta-foil supports antisymmetric modes, whereas, at the main resonance, only a symmetric mode exists. The meta-foil is left-handed at the Fano resonances and shows sharp peaks of the real part of the refractive index in transmission with small effective losses opening a way to very sensitive high-speed sensing of dielectric changes in the surrounding media and of mechanical configuration.


Gauss' elimination method leads to
with resonances obtained from Neglecting resistances yields ( 2 ) 2 − 4 2 + 2 = 0 (S10) with solutions ( 2   2 ) 1,2 = 2 ± √2. (S11) The zero resistance currents are For higher meta-foils with ≥ 3, the number of two-capacitor loops exceeds the number of capacitor-resistor loops increasingly. An nSE cell has 2( − 1) two-capacitor loops whereas the number of capacitor-resistor loops stays two, so the ratio of numbers of twocapacitor to capacitor-resistor loops is n-1. These two-capacitor loops will create additional magnetic resonances, which are responsible for the occurrence of Fano resonances. Figure S1. Half unit cell of the 3SE meta-foil.
According to Figure S1, the circuit equations for 3SE are and the matrix form is where 2 = 3 − 1, indicating that there are only two different types of loops. Solving again by Gauss' elimination method, we obtain Resonances occur at 1 2 3 − 1 − 3 = 0 and there are three solutions as expected.
Neglecting resistances as above, we get with the solutions 1 2 = 2 and 2/3 2 = 2 ± √3. The zero-resistance currents are � 10 20 30 � = Figure S2. Normalized currents F(x) (black), G(x) (red), and H(x) (blue) in the three loops of the half 3SE circuit versus normalized angular frequency x. Figure S2 shows the spectrum of such a 3SE meta-foil as calculated from Equation S18, where F(x), G(x) and H(x) are in black, red, and blue, respectively. As expected, three different magnetic resonance peaks are shown, where the resonance peak at x = 1.9 corresponds to the main magnetic resonance peak in Figure  Moreover, the frequency ratios of the three peaks are also fixed, which are independent on the size of the cell. Besides, as shown in Figure S2, the three currents have equal phases at the highest main resonance frequency, whereas only two of the three currents have equal phases at the lower Fano resonance frequencies. This implies that the 3SE meta-foil may only support the symmetric mode at the main resonance frequency, and the modes are both antisymmetric at the Fano resonance frequencies.
Fourthly, the circuit equations for 4SE are and the matrix form is We can see that when we increase n, we do it by intercalating more inner lines in the system of linear equations. We find � 3 = 1223 − 12 − 13 − 23 + 1 = 0 (S21) using the notation = . Resonances occur at ( 23 − 1)( 12 − 1) − 13 = 0.
Finally, the general case nSE leads to Cramer's rule 24 yields the solution with D the determinant of the above matrix and ℎ the algebraic complements to the right-hand side unit vector. The resonances occur at the zeros of D which can be found from the secular equation of D. Note that the determinant comprises the main diagonal, the next upper and lower off-diagonals wherein all elements are equal to one, and zero on all remaining positions. As the secular equation of a half-diagonal matrix is given by the product of the main diagonal elements 24 , the matrix above is transformed into a halfdiagonal obtaining diagonal elements as with definitions 1 = 1 , 2 = 2 , … . , −1 = 2 , = 3 .
From these recursion formulae, any can be computed, and so can their product which is the secular equation providing resonance frequencies and zero-resistance currents for arbitrary n.
From the above equivalent circuit analysis, a symmetry broken nSE meta-foil exhibits one main resonance and n-1 Fano resonances. At the main resonance, the meta-foil supports the symmetric mode while at the Fano resonances, it supports the antisymmetric modes. If n approaches infinity, the main resonance frequency of nSE meta-foils tends to the resonance frequency of the S-strings. Besides, from a mechanical point of view, when n increases, the mechanical strength of the meta-foil decreases, it becomes more flexible.
So, symmetry broken nSE meta-foils with n > 1 are of interest with regard to both supporting multiple Fano resonances and increasing mechanical flexibility.

Potential for sensing and counterfeiting
Since the very first split ring considerations 25 , potential applications of such resonators to sensing were claimed and demonstrated 6,15,26 . Sensing involves two fundamentally different concepts, one detecting a frequency shift in the presence of an agent, the other, the excitation of a resonance by radiation filtered by the agent to be sensed. The nSE meta-foils offer advantages as the Fano resonances are much sharper than the main resonance. The smaller peak width is concomitant with lower loss, and can therefore lead to a better resolution of their shift under changes of the dielectric function of a surrounding medium. In terms of sensitivity, the main magnetic peak shift in the metafoil 13 led already to a value of 5.6 • 10 4 nm/RIU that is quite close to the maximum value of 5.7 • 10 4 nm/RIU as claimed 8,10 . Hence, using the Fano peaks of the meta-foil should further improve the sensitivity beyond these values.
Moreover, mechanical vibrations of strings between interconnecting lines can be used to modulate resonance spectra acousto-optically, e.g., by vibration-induced changes of the gap or the area of the capacitors and inductive loops, respectively. Mechanical vibrations may be excited by external mechanical oscillations or by gas or other fluids flowing past the S-strings. Obviously, modulations of resonance frequency by geometric changes that are very slow compared to THz oscillations can be used for sensing deformations and vibrations. Conversely, as the carrier frequency is larger than 1 THz, changes in the dielectric environment or the meta-foil geometry can be detected up to very high frequencies in the GHz range or down to very short times in the sub-ns range. Finally, the well-defined peak frequencies and their ratios in meta-foils with ≥ 2 may also serve to identify or code specimen by attaching meta-foils and so protect them from counterfeiting.