Full controlling of Fano resonances in metal-slit superlattice

Controlling of the lineshape of Fano resonance attracts much attention recently due to its wide capabilities for lasing, biosensing, slow-light applications and so on. However, the controllable Fano resonance always requires stringent alignment of complex symmetry-breaking structures and thus the manipulation could only be performed with limited degrees of freedom and narrow tuning range. Furthermore, there is no report so far on independent controlling of both the bright and dark modes in a single structure. Here, we semi-analytically show that the spectral position and linewidth of both the bright and dark modes can be tuned independently and/or simultaneously in a simple and symmetric metal-slit superlattice, and thus allowing for a free and continuous controlling of the lineshape of both the single and multiple Fano resonances. The independent controlling scheme is applicable for an extremely large electromagnetic spectrum range from optical to microwave frequencies, which is demonstrated by the numerical simulations with real metal and a microwave experiment. Our findings may provide convenient and flexible strategies for future tunable electromagnetic devices.

. The asymmetric Fano spectral lineshape and EIT-like lineshape for superlattice consisting of real metal with finite conductivity or finite permittivity in different electromagnetic spectral range. In (a, b) the metal is considered as Aluminum with conductivity σ=3.72e7 S/m. The thickness of metal is (a) h=10 mm and (b) h=100 μm respectively. In (c, d) the metal is considered as silver with Drude-Lorentz model. The thickness of metal film is (c) h=1 μm and (d) h=170 nm respectively. The dashed line indicates the position of the sharp Fano resonance predicted by model expansion theory which consider the metal as PEC. In microwave range, the spectral shape is identical to the results by the expansion theory with a negligible difference of the spectral position of the dark mode. In Terahertz range, the lineshape is still similar with the theoretic model. The finite conductivity makes a little shift of the dark mode and a reduction of the peak value of resonance. In infrared range, the overall spectral position shifts to longer wavelength and the resonance shapes become very shallow. In visible range, the dark modes are largely broadened due to strong loss of metal in optical frequencies.

Supplementary Note 1. Model expansion theory of metal-slit superlattice
Let us consider the general case as shown in and TM polarization (Magnetic fields is always parallel to the slit direction) illuminates the structure from the superstrate. Owing to the diffraction effect of grating, the electromagnetic (EM) field in superstrate and substrate can be expanded as the plane waves of all diffraction orders. As a result, the transverse EM field components in superstrate and substrate have the forms as, where, ( 0 is the wavevector of incident plane wave, and is the wavevector of the n-th diffraction order in superstrate and substrate respectively. Tn and Rn are the complex transmission and reflection coefficients for the n-th diffraction order respectively. ε0 is the permittivity of vacuum; ω is the angular frequency and H0 is the amplitude of incident wave.
In the slit area, there are generally multiple waveguide modes in each metallic slit.
However, for the deep subwavelength slit, the higher order modes are in cut-off range, and hence only the fundamental mode needs to be considered. For the m slits in each supercell, the complex amplitude coefficients for each slit may be different due to their mutual coupling. We assume ak + (k=1,2, …, m) and akare the forward and backward amplitude coefficient of the k-th slit in each supercell respectively, then the transverse EM field components in metallic slit area are, Multiplying Eq. (4a), (5a) by   ' k M x and then integrating in terms of x in region -p/2<x<-p/2 respectively, it yields,  Fig. S3 (b). Moreover, the shift speed of the wider asymmetric resonance is larger than the narrow one. When s is varied from 0.06 to 0.4 while other parameters are fixed, the two sharp resonances shift to short wavelength and at the same time the broad resonance shifts to longer wavelength. The multiple sharp resonances can be tuned to overlap with the broad resonance to form a multiple EIT-like lineshape, which is an interesting phenomenon studied widely in recent years.
Supplementary Figure S4 shows the transmission spectra and field patterns for multiple numbers of slit in one supercell and we have found that the positions of the multiple sharp resonances can be tuned to be at any places relative to the broad resonance by the local period s. And the dependence of the spectral position on parameter s is monotonous, which facilitates the implementation for practical applications.

Supplementary Note 3. Manipulation of Fano resonance by certain parameter combinations of the superlattice
Besides the simple tuning of a single parameter of the superlattice, the tuning of several parameters together can may provide more degrees of freedom and larger tuning range different ways of the manipulation of the Fano resonance as we have already seen in Fig. 3c in the maintext. In Supplementary Fig. S5, we simultaneously vary two of the three typical parameters w, s, p, but keeps its ratio unchanged to study the combined effect. From Supplementary Fig. S5a-5b, we see that it is possible to separately tune the position of bright mode while keeping the dark mode nearly unchanged by varying both p and w increase with fixed radio p/w. The spectral linewidth position of the bright mode keeps unchanged when both p and w increase, and the larger the p/w, the smaller the linewidth of the bright mode. It indicates that the spectral width of bright mode is indeed determined by the ratio p/w. When both s and w are increasing (s/w is fixed), the spectral position of bright mode (red peaks in Fig. 4c-4d)) is unchanged whereas the dark mode (blue dips in Supplementary Fig.   S5c-5d)) shifts to longer wavelength. It is different from Fig. 2c that, here the dark mode can cross the whole lineshape of the bright mode, and thus allowing for an even larger tuning range for the dark mode. Moreover, the linewidth of bright mode broadens due to the increase of the overall duty-cycle. Comparing Supplementary Fig.   5c and 5d, we found that different ratio s/w makes different overall spectral linewidth, providing a convenient way to separately manipulate the spectral linewidth and relative spectral position. The relative position between dark mode and bright mode is mainly determined by the ratio p/s as indicated in Supplementary Fig. S5e-5f. When p/s=6 ( Supplementary Fig. S5f), the dark mode overlaps with the bright mode for any p (or s) values. Similarly, the dark mode will be above ( Supplementary Fig. S5e) or below the dark mode when p/s is smaller or larger than 6. It provides us a clear guideline to obtain an EIT-like lineshape or a Fano lineshape with particular asymmetry factor.