Full controlling of Fano resonances in metal-slit superlattice

Abstract

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.

Introduction

Optical Fano resonance has been a focus in nanophotonics field owing to its ultra-narrow linewidth which is available for switching^1,2, lasing^3,4,5,6, biosensing^7,8,9,10, nonlinear optics, slow light application^11,12,13 and so on. Fano resonance with an unusual asymmetric lineshape arises from the interference between a subradiant dark mode and a superradiant bright mode^14,15. When the spectral position of the dark mode coincides with that of the bright mode, there will be an ultra-narrow transmission peak over a broad dip, which is considered as the classic analog of electromagnetically induced transparency (EIT)^16,17,18,19. The system that supports Fano or EIT-like response usually requires complex unit cell geometry with symmetry-breaking structures such as the dipole quadrupole coupled metamaterials or metasurfaces^{17,18,20,21,22}, asymmetric split ring resonators^23,24,25,26, detuned resonator-pair^27,28,29, waveguide-plasmon coupled systems^{30,31,Manipulation of spectral position of the bright and dark modes}

Let us first vary two types of slit distance (d_G and d_L), which influence the coupling strength between the slit cavity modes and thus the spectral positions of the bright and dark modes. To clarify the origin of the sharp resonance and broad resonance, we ignore the cross coupling term between the bright in-phase mode and dark out-of-phase mode in the model expansion method and calculate again the transmission spectra contributed independently by the bright and dark mode as shown in Supplementary Fig. S2. We observe that, there are a broad and a narrow transmission peak contributed independently by the bright and dark mode respectively.

In Fig. 2a,b, the global distance between adjacent supercells d_G is varied from 0.3 to 1.3 (thereafter, all geometric parameters and wavelength are normalized to film thickness h) while the local distance d_L and slit width w are fixed. The in-phase bright mode shifts to longer wavelength linearly with the increasing of d_G, while the position of the out-of-phase dark mode is remained at λ₀ = 2.25 as indicated by Supplementary Fig. S2a,b. It can be explained by the far-field radiative dipolar coupling mechanism with a linear distance dependence^55,56. The in-phase cavity modes in metal slits can be considered as transverse dipole array modes. The increased restoring force makes the blue-shift of the spectral position of bright mode with increasing of coupling strength (decreasing of global slit distance). The out-of-phase dark modes are localized modes with little radiation, therefore the far-field coupling is weak and the spectral position of dark mode is independent of the global distance. The actual position of λ_L (λ_F) fitted by the Fano formula (solid curves in Fig. 2b,d,f)) has a little shift with respect to the position of the bright mode λ_b (dark mode λ_d) (dashed curves in Fig. 2b,d,f)) due to the mutual interaction between the bright and dark modes. The fitted asymmetry factor q is also shown as a function of d_G (purple curves in Fig. 2b), where q continuously changes from negative to zero and finally a positive value when the bright mode changes from the shorter wavelength side of the dark mode to the longer wavelength side. When the bright mode coincides with the dark mode, the asymmetric factor equals to zero exactly and the reflection spectrum exhibits the EIT-like lineshape.

Figure 2

The evolution of Fano spectral profile with respect to different geometric parameters.

In (a,b), d_L = 0.1 is fixed while d_G varying from 0.3 to 1.3; in (c,d), d_G = 0.7 is fixed while d_L varying from 0.1 to 0.4; in (e,f), p = 1.8 is fixed while s varying from 0.1 to 0.55. The slit width w = 0.1 for all cases. (a,c,e) show the phase-maps of transmittance |T₀|² as a function of wavelength λ and geometric parameter (a) d_G, (c) d_L and (e) s respectively. The dashed blue and red curves in (b,d,f) are the resonance wavelengths of the bright and dark modes (the peak wavelengths in Supplementary Fig. S2) respectively; the solid blue and red curves are the Fano fitted parameters λ_L, λ_F, respectively. The solid purple curves are the Fano asymmetry factor q fitted by Equation (2) and the dashed purple curves indicate that the zero position of q represents the position where the dark mode coincides with the bright mode.

Next, let’s keep slit width w and global distance d_G unchanged and inspect the transmission spectra for varying d_L. From the Supplementary Fig. S2c,d, we see that, the spectral position of the bright mode is nearly unchanged with only a little spectral width shrink, but the resonance wavelength of dark mode increases linearly with the local slit distance d_L. As a result, in the Fano lineshape (Fig. 2c) resulting from the interference between the bright and dark modes, the sharp resonance shifts to longer wavelength linearly with the local distance d_L, while the broad resonance peak keeps its spectral position unchanged. Here, the local slit distance determine the near-field coupling strength between adjacent slit cavity modes. The dark mode is strongly influenced by near-field coupling due to the out-of-phase field pattern, while the bright mode is nearly unaffected by the near-field coupling. We note that, the asymmetric factor q linearly depends on d_L (solid purple line in Fig. 2d), which provides us a convenient way to modulate the asymmetry of Fano resonance profile.

In the above, we demonstrated the independent controlling of bright mode and dark mode by d_G and d_L, respectively. Now, we discuss the simultaneous controlling of the bright mode and dark mode. We fix the slit width w and the global period p and vary the local period s from 0.1 to 0.55 as shown in Fig. 2e,f. This is equivalent to decreasing global slit distance (d_G = p-w-2s) from 0 to 0.45 while increasing local slit distance (d_L = s-w) from 1.5 to 0.55 at the same time. Since the spectral position of bright (dark) mode is linearly dependent on d_G (d_L) independently as we just discussed in the above, the simultaneous change of both d_G and d_L will lead to the simultaneous shift of the bright mode and dark mode (Fig. 2e,f) in opposite directions. As a result, we can simultaneously tune the dark and bright modes and obtain nearly arbitrary relative positions between the dark and bright modes by controlling solely the local period s.

Since we have successfully demonstrated the independent and simultaneous controlling of the bright mode and dark mode for single Fano resonance with ‘m = 3’ superlattice, we also attempt the challenging problem of controlling multiple Fano resonances. For this purpose, we consider a superlattice with more slits (m > 3) in one super cell (see Supplementary Notes 2 and Supplementary Figs S3 and S4). The semi-analytical results show that, the positions of multiple dark modes will simultaneously shift to longer wavelength with the increase of the local slit distance d_L and the shift speed of the wider asymmetric resonance is larger than the narrower one. Meanwhile, the single bright mode depends solely on the global distance d_G. The dependence of the spectral position of either the bright or dark mode on geometric parameter is monotonous, which facilitates the implementation for practical applications.

Manipulation of spectral linewidth of the bright and dark modes

Now, let’s turn to study the influence of slit width w on the spectral profile of Fano resonance and discuss the tuning of spectral linewidth of both the bright and dark modes. For a single or periodic metallic subwavelength slit(s), the resonance linewidth increases with slit width due to increased radiation loss perpendicular to the metallic film^50,57. For the case of slit-superlattice in our study, the slit width will influence both the dark and bright modes simultaneously as shown in Fig. 3a,b. Here, we choose appropriate s and p so that the dark mode and bright mode almost overlap and we can clearly see the linewidth of both the bright and dark modes. From the transmission spectra in Fig. 3a and the fitted Fano resonance linewidths γ_F and γ_L in Fig. 3b, we see that the spectral linewidths of both the bright (broad red peak) and dark modes (narrow blue dip) are significantly broadened when the slit width is increased. At the same time, the whole profile of Fano resonance shifts to longer wavelength, which is consistent with the previous works dealing with single and periodic slit systems^50,57. However, in the slit-superlattice system, we have found more interesting phenomena that don’t exist in the common single or periodic slit system. In Fig. 3c,d, we vary slit width w and keep w/p and s/p unchanged. It is found that the narrow resonance dip is broadened as the slit width is increased, but the broad resonance peak sustains its spectral linewidth almost unchanged. It indicates that the duty-cycle w/p determines the spectral linewidth of the bright mode, whereas the slit width w solely determine the linewidth of dark mode. Thus it allows independent tuning of the linewidth of dark mode by varying the grating duty-cycle. Similarly it is also possible to independently tune the linewidth of bright mode by only changing parameter p as shown in Fig. 2a,c. Note that we can also control both the spectral linewidth and position of the Fano resonance in more degrees of freedom by parameter combinations as shown in Supplementary Fig. S5.

Figure 3

Fano profile variation with respect to slit width w.

(a,c) transmittance spectra and (b,d) Fano fitted spectra linewidth γ_F and γ_L normalized to their average values. In (a,b), only slit width w is varied from 0.05 to 0.3 while p = 1.8, s = 0.3 are fixed. In (c,d), slit width w is varied from 0.01 to 0.15 while kee** s/w = 2, p/w = 12.

Influence of dispersion and loss of real metal on the full controlling of Fano resonance

So far, the variation regularity of Fano resonance in metal-slit superlattice has been semi-analytically predicted by the ideal PEC model, in this part, we will validate the applicability of our proposed structure for real metals by the finite element method (FEM) simulation. In the microwave frequency range, the metal can be well described by PEC, thus the semi-analytical prediction is fully applicable for microwave range. In the terahertz (THz) frequency range, we consider the aluminum with finite conductivity σ = 3.72e7 S/m⁵⁸ and numerically calculated the transmission spectra for varying d_G, d_L and s in Fig. 4a–c respectively. We see that the spectral shape and position are similar with the theoretical results (Fig. 3a,c,e), although there are a little discrepancy (Supplementary Fig. S6b) for the resonance frequency and sharpness due to the influence of finite conductivity⁵⁷.

Figure 4

The evolution of Fano spectral profile with respect to different geometric parameters considering the loss and dispersion of metal at different electromagnetic spectral range calculated by FEM simulation.

(a–c) are for Terahertz frequency range with Aluminum film thickness h = 100 μm, conductivity σ = 3.72e7 S/m. (d–f) are for infrared frequency range with silver film thickness h = 1 μm. (h–j) are for visible frequency range with silver film thickness h = 170 nm, w = 45 nm. The complex refractive index of silver in visible and infrared range is taken from the sampled experimental data⁵⁹.

When we move the frequency to optical frequency range (near-infrared and visible range), we consider the metal as silver whose complex refractive indexes are taken from Palik’s experimental data⁵⁹. In the infrared range (Fig. 4d–f), the thickness of silver is scaled to 1 μm and the slit width is w = 0.1h = 100nm. Those geometry parameters are large enough to be fabricated by nowadays nanotechnologies. From Fig. 4d–f, we see that the position of the bright (dark) mode is still separately dependent on d_G (d_L). However, the overall resonance position shifts to longer wavelength. It is because, at infrared range, the gap surface plasmon modes with larger propagating constant (shorter effective wavelength) replace the cavity modes between PEC walls. It requires longer freespace wavelength to match the effective wavelength of the gap surface plasmon modes.

If we further reduce the thickness of silver film to h = 170nm, the operating frequency can be scaled to visible range. Here, the slit width w = 45nm, which is still within the fabrication limit. The transmission spectra in Fig. 4h–j shows that the sharp resonances broaden and the transmission minima become very shallow compared with the semi-analytical results in Fig. 2a,c,e. It is due to significantly enhanced intrinsic losses of metal in visible frequencies. The variation regularity of Fano profile with respect to d_G, d_L and s still survive (Fig. 4h–j). Nevertheless, the tuning range is largely decreased due to due to the strong dispersion of silver in visible frequencies.

Experimental verification of the controlling of Fano resonance in superlattice

To further verify our theory, we make a simple experimental scheme in microwave frequencies to measure the transmission spectra of metal-slit superlattices with different slit distances (d_G and d_L) as shown in Fig. 5. We measured the transmission spectra of five different samples under plane wave excitations in a Microwave anechoic chamber by the vector network analyzer (Fig. 5a). We carefully identify the broad peak representing the bright mode through the overall contour of the measured transmission spectra and the sharp dip representing the dark mode. We note that, there are many fluctuations in the experimental spectra of Fig. 5b–f due to the unavoidable Fabry-Perot (FP) resonances between the sample and antennas. We move the sample in different places and observe that, while a lot of shallow dips shift slightly, there is always a sharp dip sustaining its position unchanged. Then, we can identify the static sharp dip as the spectral position of the dark mode. We see that both the overall contour and the position of the sharp dip of the experimental spectra are consistent with the theoretical curves. In Fig. 5b–d, the three samples are of the same local distance d_L (width of the narrower Aluminum stripe), but different global distance d_G (width of the wider Aluminum stripe). The measured transmission spectra exhibit sharp dips at the same spectral position near 13 GHz, while the broad peak contour shifts to higher frequency (shorter wavelength) when global distance is decreasing. It indicates that the dark mode can be indeed independently controlled by global slit distance with fixed local slit distance. In Fig. 5d–f, the wider Aluminum stripes are of the same width (d_G), while the width of the narrower stripes (d_L) is increasing. As expected, the broad transmission peak contour share the same position near 13.5 GHz, while the sharp transmission dip shifts to lower frequency (longer wavelength). It indicates that, the dark mode could also be separately controlled by d_L with fixed d_G. Thereby, the experimental results further validate the independent controlling of the spectral positions of the bright and dark modes by the global slit distance and the local slit distance, respectively.

Figure 5

Experimental verification of the Full controlling of Fano resonance.

(a) The photograph of the experimental setup. Two horn antennas connected to the vector network analyzer and the sample are placed inside a microwave anechoic chamber. Left panels of (b–f) show the experimental (blue) and theoretical (red) transmissions of metal-slit superlattices constructed by two types of Aluminum plates with different width d_G and d_L respectively. Right panels of (b–f) shows the corresponding photographs of the Aluminum superlattice samples with different d_G and d_L used in the experiment. The thickness and slit width of all the Aluminum superlattices is h = 10 mm, w = 1.3 mm respectively, while global and local distances are (b) d_G = 12 mm, d_L = 1 mm, (c) d_G = 10 mm, d_L = 1 mm, (d) d_G = 6 mm, d_L = 1 mm, (e) d_G = 6 mm, d_L = 2 mm, (f) d_G = 6 mm, d_L = 4 mm, respectively.

Discussion

It is intriguing that we can independently manipulate the sharp resonance and broad resonance with two independent parameters. It provides us with a very easy and flexible way to continuously tune the bright (dark) mode position for a large range at a given dark (bright) mode. Moreover, the linear dependency of λ_L (λ_F) on d_G (d_L) allows a linear modulation and thus further simplifies the tuning procedure. It also allows the continuous tuning of the asymmetric factor of Fano resonance, help us to obtain the desired Fano lineshape of any kind.

The independent and continuous controlling of the sharp and broad resonances have a wide range of practical applications such as nanoscale lasing and biosensing. For example, for the plasmonic nanolaser application, it is desirable to tune the broad resonance to overlap with the absorption frequency of the gain medium in order to enhance the energy conversion efficiency. And it is also essential to make the sharp resonance overlap with the emission frequency of the gain medium for the purpose of high lasing power and low threshold. Our proposal fulfills the criteria to match both the resonances to an arbitrary gain medium and thus providing an easy and effective way to design plasmonic lasing devices. The full controlling of Fano resonance also facilitates the biosensing application by tuning the subradiant resonance to match to the molecule’s vibrational fingerprints⁹.

In summary, by systematically investigating the EM wave response of the subwavelength metal-slit superlattice through the semi-analytic theory, numerical simulation and microwave experiment, we found a convenient and flexible way to independently manipulate the bright and dark modes of Fano resonances. The spectral position of the bright (dark) mode linearly depends on the global (local) slit distance separately owing to the far-field (near-filed) coupling. The spectral linewidth of the bright mode is determined by the duty-cycle w/p, while the linewidth of the dark mode is solely determined by slit width w. The asymmetry of the Fano lineshape can be continuously tuned so that we can easily obtain arbitrary spectral lineshapes such as EIT-like lineshape or Fano lineshape with desired asymmetric factor. The controlling of multiple Fano resonances can also be realized in a similar way with more slits in one supercell. The proposed controlling scheme for Fano resonance is applicable for an extremely wide EM spectrum range from microwave regime to optical frequencies. Our proposal are not restricted in metal-slit structures, but can readily extend to other structures such as metal or dielectric stripe superlattices. Our findings may lead to convenient and flexible applications in switching, sensing, lasing, slow light areas and so on.

Methods

Theory and simulation

The transmission and reflection coefficients of the subwavelength metal-slit superlattice with arbitrary slit number in one supercell are obtained semi-analytically by the model expansion theory. In that model, the cavity mode in slit area is matched to the propagating plane waves outside the grating by the boundary condition which requires the continuity of parallel E-field and H-filed components in each interface between grating and free space. We can first obtain the amplitude coefficients of cavity mode in slit area simply by a matrix inversion and then directly obtain the transmission and reflection coefficients in terms of the amplitude coefficients of cavity mode. The simulation results for real metal are obtained by the finite element method implemented by COMSOL Multiphysics. The Aluminum is represented by a conductive model with finite conductivity σ = 3.72e7 S/m⁵⁸ in Terahertz range. The complex refractive index of silver in optical frequencies are taken from Palik’s experimental data⁵⁹.

Sample fabrication and experiment

Each slit-superlattice sample is fabricated by aligning two types of commercially available Aluminum stripes with standard sizes. After drilling two holes in each side of all the aluminum stripes, we align them together with the help of two long screws for each sample in such a way that the adjacent aluminum stripes are separated by two shims with total thickness 1.3 mm. The total size of each sample is 300 mm×260 mm×10 mm. We fabricated five different samples and put them in a microwave anechoic chamber to measure the scattering parameters (S-parameters) by a vector network analyzer (Agilent E8362B PNA, 10 MHz-20 GHz). Two standard gain Ku-band horn antennas are used as transmitter and receiver. The distance between the two antennas are kept as 600 mm in order to ensure the planewave approximation. The transmission spectra of all the samples are normalized against the background radiation in which scattering parameters are measured without samples between the two horn antennas.