1. Introduction

The discovery of phenomena involving the anomalous transmission of light, such as electromagnetically induced transparency (EIT) [1] and extraordinary optical transmission [2,3], has spurred interest in understanding the interference of light-scattering pathways in matter. The so-called Fano resonance is the universal concept explaining such phenomena that exhibit narrow and asymmetric spectra owing to the constructive or destructive interference between bright and dark modes [4–6]. Since its discovery, numerous attempts have been made to incorporate this concept into the classical coupled-oscillator system to develop new materials for practical applications such as sensing, switching, and lasing devices [7]. Significant progress has been made in realizing and controlling classical Fano systems [8–12]. The broken symmetry of the unit cell has been widely applied in most classical Fano systems using metamaterials and photonic crystals. By utilizing the asymmetry of the unit cell, earlier efforts, such as second harmonic generation [13], photoactive switching [14], flexible active-sensor [15] photoluminescence [16], and cathodoluminescence [17], focused on the coupling strength to maximize the quality factor (Q) in various photonic devices. The ability to control the bright and dark modes independently (not just the magnitude but also the phase) remains a problem as it requires full utilization of Fano resonance. Recently, the Fano asymmetry factor (q), which is the phase-related term of the coupled oscillator, has also attracted attention owing to its potential for phase-sensitive applications that are closely related to the Fano asymmetric spectral line shape [18,19]. The two factors (Q and q) are independent in an ideal Fano system, which can be controlled by adjusting the coupling strength, the relative excitation strength, and the phase between bright (radiative) and dark (non-radiative) modes [4]. However, controlling the bright and dark modes independently is challenging because the structural manipulation of real Fano systems changes both factors in many cases. The absence of the independent controllability of q may impede the development of future applications of high-Q Fano materials that are closely related to phase-sensitive spectral tuning. Such full manipulation of the real system could be realized by adding one more degree of freedom at the unit cell level, as reported in previous studies, despite not being fully realized yet [19–22].

A well-known method for tuning the q parameter involves detuning the frequencies of the bright and dark modes (${\omega _{bright}}$ and ${\omega _{dark}}$) [4,6,23–25]. However, the effectiveness of this tuning approach is limited in most asymmetric Fano metamaterials because both modes share similar structural parameters in asymmetric unit cells. Therefore, ${\omega _{bright}}/{\omega _{dark}}$ is nearly constant. For instance, the frequencies of both the bright and dark modes in asymmetric split-ring resonators are determined from the ring size [26]. Therefore, the Fano asymmetry q and its corresponding spectral line shape remain the same in many cases involving asymmetric Fano metamaterials. Only the Q-factor and the spectral width can be controlled through the coupling strength, which is closely related to the structural asymmetry of the unit cell [27]. In addition, full manipulation requires the variation of q without changing ${\omega _{bright}}/{\omega _{dark}}$ and the coupling constraint. To overcome this limitation, it is necessary to directly modulate the relative excitation phase between both modes rather than indirect manipulation via frequency detuning. Such direct modulation has recently been reported in other types of Fano systems such as cavity–magnon polariton and optomechanical systems [25,28–31]. However, the implementation of this method in asymmetric Fano metamaterials is yet to be realized. In this paper, direct tuning of the dark mode phase in an asymmetric Fano metamaterial is proposed as a method for switching between transparency and absorption using asymmetric subwavelength slits. The structure used in this study supports two propagating modes, the phase velocity of one being faster and the other slower than the speed of light. These modes are, thus, intrinsically radiative (fast/bright) and non-radiative (slow/dark), respectively: ${k_{Bright}}(\omega )< {k_0}(\omega )< {k_{Dark}}(\omega )$, where ${k_0} = \omega /c$, $\omega$, and c represent the wave vector, frequency, and speed of light in free space, respectively. This feature ensures that both modes are completely independent from each other, which means that the dark mode phase can be independently controlled by introducing an additional degree of freedom in the unit cell. This selective control capability provides a universal scheme for full (independent) control of the four major parameters of Fano resonance, thereby paving the way for a novel design of classical Fano systems. The conventional bright and dark modes localized in each unit cell, such as the asymmetric split-ring or dielectric resonators, cannot be independently controlled because ${k_{Bright}}(\omega )= {k_{Dark}}(\omega )= constant$ (zero if light is incoming from the normal direction). The phase of both modes always remains the same owing to the simultaneous excitation condition under the long-wavelength limit, that is, the subwavelength nature of the unit cells of metamaterials.

2. Results and discussion

2.1 Independent control of Q and Fano asymmetry q by decomposing the structural asymmetry of the unit cell in metallic subwavelength slits

Subwavelength slits are used to form asymmetric unit cells in this Fano metamaterial, as shown in Fig. 1(a). The p-polarized electromagnetic wave (${H_x} = {H_y} = 0$) was incident on the structure for the excitation of Fano resonance. We utilized the two independent optical modes of the subwavelength slits: waveguide mode in each slit [32] and near-field coupling of the modes at the apertures [33]. These properties produce a bright Fabry–Perot (FP) mode and a dark surface mode (SM). Because of its extremely high anisotropic dielectric behavior [34,35], this structure forms nearly zero group velocity on the surface-like dark mode. This exotic behavior resulted in high-Q ($> 100$) Fano resonance with a high photonic density of states [17]. The mechanisms for exciting FP and SM modes originate from effective dipole scattering at the apertures of the slits [36] and diffractive coupling from the structural asymmetry, respectively [14]. These could be explained by taking into account the band structure $\omega (k )$, as shown in Fig. 1(b). The resonant frequency of the FP mode (${\omega _{FP}}$) was approximately $\pi c/h$ ($\lambda \approx 2h$) [35]. The spectral width of the FP mode is broad (${\omega _{FP}}/\Delta {\omega _{FP}} \approx 2$); therefore, it acted as the radiative (bright) continuum [37]. The dark modes frequencies (${\omega _m}$) were determined at the band-folding position (${k_x} = 2\pi m/D$, where m represents an integer) of the surface mode with correction of the perturbation of the band-gap formation [17,38,39], which is similar to that of surface plasmons on gratings [39]. Because five slits exist in a unit cell, there are four possible phase configurations of near-field coupling ($m ={-} 1, + 1, - 2, + 2$) in increasing order of the resonant frequency over the first harmonic of the FP background [17,35,40]. Their coupling strength and relative excitation phase were manipulated by decomposing the structural asymmetry into two parameters: ${\Delta _\rho }$ and ${\Delta _\phi }$, as shown in Fig. 1(a). The radial asymmetry ${\Delta _\rho }$ mainly affects Q [27,41], whereas the angular asymmetry ${\Delta _\phi }$ manipulates the relative excitation phase between both modes. The change in the Fano spectra for these manipulations is depicted in Fig. 1(c) and 1(d). The oblique incident ($\theta = {30^ \circ }$) condition was considered in this study owing to its otherwise mirror symmetry in the y–z plane, thereby limiting the degree of freedom for q control. This study focused on the $m ={-} 1$ condition (${\omega _{ - 1}} = {\omega _{SM}}$); other conditions exhibit similar behavior. A high Q can be achieved by lowering the coupling strength, as shown in Fig. 1(c). The phenomenon involving the reduction of the asymmetry size (${\Delta _\rho }$ in this structure) has been investigated in previous studies [17,26,27,41,42]. The lower size of the asymmetry causes lower radiative damping owing to the dipole scattering of the dark modes [17,26,43]. Currently, this is a well-known phenomenon observed in various asymmetric metamaterials [41] used in various high-Q Fano applications [13–17]. The asymmetry of the Fano line shape (q) cannot be controlled using the size of the asymmetry. However, only the spectral broadness ($\gamma \propto {Q^{ - 1}}$) was affected, as demonstrated in most previous studies, such as those that focused on two-dimensional metamaterials with asymmetric split-rings [27] and nano [13] resonators. The inset graph in Fig. 1(c) shows the independency of Fano q from the size of asymmetry ${\Delta _\rho }$ in this structure. This structure possesses an additional property: q can be manipulated from $- \infty$ to $+ \infty$ while the radiative Q remains nearly constant, as shown in the inset figure of Fig. 1(d). The two inset graphs shown in Fig. 1(c) and (d) were obtained by fitting the Fano formula with several correction factors (see Supplement 1 for supporting content). Meanwhile, the resonant frequencies of the bright (${\omega _{FP}}$) and dark (${\omega _{SM}}$) modes remained nearly unchanged and ${\omega _{FP}} > {\omega _{SM}}$ was always true during the process of Q and phase tuning in this structure. Thus, q in this structure was not determined using the conventional approach for detuning the frequencies of the bright and dark modes for EIT applications: ${q_{EIT}} = 0$ [4]. This suggests that the angular asymmetry directly controls the dark mode phase.

 

Fig. 1. Independently controlled quality (Q) and Fano asymmetry (q) factors, which correspond to the coupling status and the relative phase between the FP (bright) and surface(dark) modes. (a) Geometrical configuration and parameters used in this study ($a = 0.5$ mm, $d = 2a$, $D = 10a$, $h = 7.4a$). Each unit cell comprises five subwavelength slits and two structural asymmetries: ${\Delta _\rho }$ and ${\Delta _\phi }$. Only two unit cells are shown in this figure. (b) Schematic views of the band structure and the corresponding three-level diagram explaining the excitation and coupling mechanisms of both the FP (bright) and surface (dark) modes in this Fano system. $\varphi$ is the relative excitation phase. (c) Simulated transmission spectra with varying radial asymmetry ${\Delta _\rho }$ and constant angular asymmetry (${\Delta _\phi } ={+} \pi /2$). Each curve is shifted by +1 with respect to the previous one (inset graph). Q decreases and q remains nearly constant (approximately $q = 0.324 \pm 0.05$) as ${\Delta _\rho }$ increases owing to the increase in the coupling of the radiative continuum. (d) Simulated transmission spectra with varying angular asymmetry ${\Delta _\phi }$ and ${\Delta _\rho }/\lambda = 0.05$ (inset graph). Q remains nearly constant (approximately $Q = 641 \pm 15$), whereas q varies from $0$ to ${\pm} \infty$. The gray solid line represents the fitted values: ${q_{fit}}({{\Delta _\phi }} )= 0.3 \times \cot ({\pi /4 - {\Delta _\phi }} )$. Further details about the two inset graphs in (c) and (d) are presented in Supplement 1 for supporting content.

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2.2 Direct tuning of the dark mode phase by varying angular asymmetry

This structure exhibited highly transparent and absorptive properties for ${\Delta _\phi } ={+} \pi /4$ and $- \pi /4$, respectively, as shown in Fig. 2(a). Both cases exhibited no reflection, whereas other ${\Delta _\phi }$ values showed fringe patterns owing to the interference between incoming and reflected waves. Meanwhile, the spatial phase of the dark mode changes by $\pi /2$ while ${\Delta _\phi }$ varies from $- \pi /4$ to $+ \pi /4$. It is interesting to note that under perfect conduction conditions (i.e., ignoring the dissipative loss), the magnitudes of the transmission spectra are similar for ${\Delta _\phi } ={+} \pi /4$ and $- \pi /4$, resulting in a perfect transmission with a Lorentzian shape (see Supplement 2 for supporting content). Despite the equal magnitude, a $\pi$-phase difference was observed for the transmitted light, as shown in Fig. S3. The transparent Lorentzian peak with ${\Delta _\phi } ={-} \pi /4$ transformed into an absorption dip for a real metal (the black solid line in Fig. 1(d)). However, the case with ${\Delta _\phi } ={+} \pi /4$ maintained nearly perfect transmission. The negligible reflection suggests that a significant amount of incident light energy was captured in the dark mode and dissipated during resonant scattering for ${\Delta _\phi } ={-} \pi /4$. These two Lorentzian transmission and absorption spectra have a strong relationship with the relative excitation strength and phase between the bright and dark modes (${\varepsilon _{FP}}$ and ${\varepsilon _{SM}}{e^{i\varphi }}$ in the three-level scheme of the Fano system shown in Fig. 1(b)) [4,44]. The other spectra with ${\Delta _\phi } \ne \pm \pi /4$ show the intermediate state between $q = 0$ and ${\pm} \infty$, clearly manifesting the asymmetric Fano line shape. Fig. 2(b) and (c) show the magnitude and phase of the transmission spectra around ${\Delta _\phi } = 0$, respectively, exhibiting an abrupt $\pi$-phase shift with a negligible change in magnitude at the resonant frequency. This phase transition at ${\Delta _\phi } = 0$ enables switching between radiative (${\Delta _\phi } > 0$) and non-radiative (${\Delta _\phi } < 0$) damping with constant Q, resulting in nearly perfect transmission (${\Delta _\phi } ={+} \pi /4$) and absorption (${\Delta _\phi } ={-} \pi /4$), respectively. This observation is similar to that reported in a recent study on loss engineering via phase transition between the radiative and absorptive states of metasurfaces [45–46]. Additionally, the strong scattering of high-Q supercavity modes with $q = \infty$ was similarly observed in a subwavelength dielectric resonator [47].

 

Fig. 2. Control of phase transition between transparent and absorptive states using the dark mode phase via manipulation of the angular asymmetry ${\Delta _\phi }$. (a) Snapshots of the field distribution of ${H_z}$ with ${\Delta _\rho }/\lambda = 1/40$ at the resonant frequency, showing $\pi /2$ phase shifts of the dark mode with varying from ${\Delta _\phi } ={-} \pi /4$ to $+ \pi /4$. Conversion between the transparent and absorptive states is achieved. (b) and (c) show the calculated and measured transmission and its phase spectra, respectively around ${\Delta _\phi } = 0$ with ${\Delta _\rho }/\lambda = 0.3$. Each curve is shifted by 1 in (b) and $2\pi$ in (c) with respect to the previous one. An abrupt phase shift occurs at ${\Delta _\phi } = 0$, while the magnitude remains nearly unchanged. Nearly zero phase shift is achieved at the resonant frequency when ${\Delta _\phi } > 0$, indicating that the incoming light directly captured in the slow mode escapes immediately with no time delay. $\pi$ shift is achieved when ${\Delta _\phi } < 0$, indicating that the incoming light is coupled to the broad FP mode and indirectly captured in the dark mode. Because the dark mode is not coupled to the radiative continuum ($q = 0$), the light energy is trapped long enough to dissipate. Further analysis in the time-domain is presented in Supplement 3 for supporting content.${\Delta _\phi }$

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To elucidate this phenomenon, light scattering processes were analyzed in the time domain (see Supplement 2 and 3 for supporting content). Fig. S3 and Fig. S4 show the time-domain signal of the electric field of the transmitted light for different values of ${\Delta _\phi }$ obtained using the fast Fourier transform algorithm. The first chirped peak at approximately 1 ns corresponds to the FP (bright) scattering of incoming light from the symmetry part of the unit cell. The exponentially decaying (${e^{ - {\omega _{SM}}t/Q}}$) signal at approximately 2 ns originated from the radiation damping of the surface (dark) mode from the asymmetry of the unit cell [17]. The gradual increase in the bright mode and decrease in the dark mode signal strength are clearly visible. The degradation of the dark mode signal as ${\Delta _\phi }$ increases represents transition from the absorptive state to the transparent state. Fig. S3(a) and (b) show the transmission and reflection spectra, respectively under perfectly conducting conditions. Fig. S3(c) and (d) show the time-domain signals of transmitted and reflected electric fields, respectively. The gray lines represent FP background signals uncoupled with the dark modes (${\Delta _\rho } = 0$). For the coupled case (${\Delta _\rho } \ne 0$), the off-resonant portion of the incident chirped pulse is swiftly transmitted through or reflected back from the metamaterial as the FP mode (${t_{FP}}$ position in Fig. S3(c) and (d)). For ${\Delta _\phi } ={-} \pi /4$, the resonant portion appears to be captured and released for a long lifetime $\tau = Q/{\omega _{SM}}$. This temporal observation suggests that the incoming light did not directly excite the dark mode, otherwise the trapped electromagnetic energy would have escaped to free space via coupling. Instead, the resonant portion of the incident pulse drove the FP background first, and the dark mode was sequentially excited by coupling to the FP mode. The magnified views of the time-domain signal at ${t_{SM}}$ are shown at the bottom of Fig. S3, further verifying the $\pi$-phase shift difference arising from the $\pi /2$-phase shift of the dark mode. This suggests that the bright and dark modes were out-of-phase and coupled in a destructive manner, seemingly delaying the radiative damping signal of the dark mode (${t_{FP}} \to {t_{SM}}$). In contrast, the resonant portion was directly coupled to the dark mode for ${\Delta _\phi } ={+} \pi /4$. This direct excitation of the dark mode was achieved through the diffraction of incoming light from the asymmetry, that is, phase matching between diffractive light and the dark mode at ${k_x} = 2\pi /D$. In addition, the time-domain signal (the black solid lines in Fig. S3(c)) indicates that the bright and dark modes were in-phase coupled so that the captured light energy was released quickly in a constructive manner. Therefore, the resonant portion of the light pulse passed through without time delay together with the off-resonant portion, thereby restoring the complete shape of the incoming chirped pulse. For the reflected light, there was no phase difference between ${\Delta _\phi } ={\pm} \pi /4$ and the nearly zero reflection at the resonant frequency. The time-domain analysis under perfectly conducting conditions clearly demonstrates the two different light scattering pathways for ${\Delta _\phi } ={+} \pi /4$ and $- \pi /4$. The observation and analysis of the phase shift are in good agreement with theoretical interpretations in both the frequency-domain and time-domain [4,44].

2.3 Efficient switching between transparency and absorption in the high-Q regime

Fig. 3(b) shows the independency of Q from the phase modulation. Q was measured as the full width at half maximum (${\sigma _{\max }}/2$) of the absorption peak (black solid lines in Fig. 3(a)). The maximum amplitude of fluctuation of Q at varying ${\Delta _\phi }$ was approximately 0.033, indicating that the Q values remained nearly constant with phase modulation. At the low asymmetry regime (${\Delta _\rho } \ll \lambda$), Q increased in inverse proportion to ${({{\Delta _\rho }/\lambda } )^2}$. This quadratic relation is interpreted as dipole scattering of the dark mode from the asymmetry, as noted in a previous study on similar structures [17]. A recent study also showed that this relation is universal for asymmetric metamaterials [41]. For ${\Delta _\phi } ={+} \pi /4$, nearly perfect transmission is achieved with any Q value owing to the highly radiative behavior of the dark mode, as discussed in the previous section [45]. For ${\Delta _\phi } ={-} \pi /4$, strong absorption occurs as a result of the long-time oscillation of the dark mode. The absorption peak (${\sigma _{\max }}$) gradually increases as Q increases, and nearly perfect absorption is achieved for $Q \approx 800$. The absorption decreases again with further increase in Q. Therefore, the highest figure of merit for the absorbing performance is $Q \approx 800$ for ${\Delta _\phi } ={-} \pi /4$, as shown in Fig. 3(c).

 

Fig. 3. Fully manipulated switching between total transmission and absorption. Mapping and analysis of Q and the absorption peak height ${\sigma _{\max }}$ in ${\Delta _\rho }$–${\Delta _\phi }$. (a) Transmission T (grey solid lines), reflection R (grey dashed lines), absorption $\sigma$ (black solid lines) spectra with different ${\Delta _\rho }$ and ${\Delta _\phi }$. Each curve is shifted by +1 with respect to the previous one. Zero reflection is clearly shown at the resonant frequency with ${\Delta _\phi } ={\pm} \pi /4$. (b) Mapping of Q as a function of ${\Delta _\rho }$ and ${\Delta _\phi }$. Q was measured as the full width at half maximum (${\sigma _{\max }}/2$) of the absorption peak. Q is nearly independent of ${\Delta _\phi }$ (approximately 3.3% fluctuation). (c) Mapping of the absorption peak ${\sigma _{\max }}$. ${\Delta _\phi } ={+} \pi /4$ represents nearly perfect transmission with low absorption and ${\Delta _\phi } ={-} \pi /4$ indicates nearly perfect absorption with $Q \approx 800$.

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Mapping analysis of the tunability shown in Fig. 3 was performed at $0.01 < {\Delta _\rho }/\lambda < 0.05$ ($Q > 200$). Under the given geometric parameters (Fig. 1(a)) in this study, ${\Delta _\phi }$ cannot reach up to ${\pm} \pi /4$ if ${\Delta _\rho }/\lambda > 0.0707$ owing to the geometric limit, as shown in Fig. S5; that is, the asymmetry of the gratings touches the sidewalls of the symmetric ones. In the experiment, the transmission spectra were measured from ${\Delta _\rho }/\lambda = 0.075$ ($Q \approx 80$) to ${\Delta _\rho }/\lambda = 0.3$ ($Q \approx 10$), as shown in Fig. S6, mapping with ${\Delta _\rho }$ and ${\Delta _\phi }$. Drastic degradation of the absorption with decreasing ${\Delta _\rho }/\lambda$ should start to occur around ${\Delta _\rho }/\lambda = 0.025$ ($Q \approx 800$) according to the calculated results (Fig. 3). However, in experiments, it started to occur around ${\Delta _\rho }/\lambda = 0.075$ ($Q \approx 80$). This is because the structural disorder (${\Delta _d}$), which is inevitable during the fabrication of the experimental sample, was not considered in the calculation. ${\Delta _d}$ is not negligible anymore when compared with ${\Delta _\rho }/\lambda = 0.075$. However, it was fortunate that the switching between transmission and absorption with sufficient efficiency (≈ 70%) was observable with ${\Delta _\rho }/\lambda = 0.075$. For ${\Delta _\rho }/\lambda > 0.15$ ($Q < 40$), the degradation of the absorption dip is not clearly shown, whereas this is gradually achieved for ${\Delta _\rho }/\lambda < 0.15$ ($Q > 40$).We distinguished the high-Q and low-Q regimes based on this observation. The Q values at ${\Delta _\rho }/\lambda = 0.05$ are approximately 200, as shown in Fig. 3(b), indicating that the tendency of inverse quadratic behavior is further confirmed at the low Q regime (Fig. 4(b)). However, this relationship disappears when the evanescent decay length of the dark mode is smaller than the size of the asymmetry, as reported in our previous study [17]. For ${\Delta _\rho }/\lambda > 0.075$ ($Q \approx 80$), the disappearance of the absorption dip in Fig. 4(a) is clearly shown as the angular asymmetry moves from ${\Delta _\phi } ={-} \pi /4$ to ${\Delta _\phi } ={+} \pi /4$. The switching efficiency shown in Fig.$Q \propto {({{\Delta _\rho }/\lambda } )^{ - 2}}$ 4(b) is calculated as $[{T({{\Delta _\phi } ={+} \pi /4} )- T({{\Delta _\phi } ={+} \pi /4} )} ]\times 100$ in the measured spectra shown in Fig. S6. The drastic increase in the minimum position when sweeping from ${\Delta _\phi } ={-} \pi /4$ to ${\Delta _\phi } ={+} \pi /4$ with ${\Delta _\rho }/\lambda = 0.075$ clearly demonstrates the switching between nearly perfect transmission and absorption. However, the switching efficiency gradually decreases as the value of Q decreases, as shown in Fig. 4(b). This is because the dark mode has an insufficient lifetime to lose its electromagnetic energy in a non-radiative manner. However, the abrupt phase transition of the dark mode at ${\Delta _\phi } = 0$ still exists at the low-Q regime, as shown in Fig. 2(c), although the magnitude of the transmission remains nearly unchanged. This implies that the low-Q operation allows for the development of dynamic phase shifter applications.

 

Fig. 4. Experimental verification of the Q-dependence of the efficiency of switching between transparent and absorptive states. (a) Measured transmission spectra with varying and ${\Delta _\rho }/\lambda = 0.075$ ($Q \approx 80$). The degradation of the absorption dip as it sweeps from ${\Delta _\phi } ={-} \pi /4$ to ${\Delta _\phi } ={+} \pi /4$ clearly demonstrates the switching between the transparency and absorption states at the high-Q regime (${\Delta _\rho }/\lambda < 0.15$). (b) Q and switching efficiency as a function of ${\Delta _\rho }$. An increase in efficiency at the high-Q regime is shown.

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3. Conclusions

In conclusion, we demonstrated a method for achieving full control of high-Q Fano resonances: quality factor (Q) and Fano asymmetry (q). Two independent modes of the subwavelength slits in this Fano metamaterial allow for the separation of the geometric origins of the broad FP (bright) and high-Q dark modes, both of which propagate over the entire structure. In addition, the conventional size-dependent asymmetry ${\Delta _\rho }$ and angular asymmetry ${\Delta _\phi }$ are introduced as independent control parameters. During this process, a phase shift of both the transmitted wave and dark mode is observed. It is interesting to note that this phase transition is closely related to the switching between the transparent and absorptive states of the dark mode [45]. This results in tunability between the nearly perfect transmission and absorption of Fano metamaterials at the high-Q regime (${\Delta _\rho }/\lambda < 0.15$). Unlike the conventional EIT, the resonant frequencies of the bright and dark modes are not necessarily tuned for the realization of transparency, which is enabled through direct tuning of the dark mode phase. At the low-Q regime (${\Delta _\rho }/\lambda > 0.15$), this switching phenomenon is degraded as a result of the insufficient lifetime of the dark mode for dissipative loss. However, the abrupt phase shift of the dark mode at ${\Delta _\phi } = 0$ is still valid, thereby allowing for the development of dynamic phase shifter applications [19]. Although it has been demonstrated at millimeter wavelength in this study, the concept of the dark mode’s phase tuning can be extended to higher frequencies, such as the terahertz (THz) frequency and infrared. This is because of the universal nature of Fano resonance in various fields despite the different mechanisms for the formation of bright and dark modes. For instance, tuning the relative phase between order-induced and disorder-induced scattered light results in reversed Bragg rise in photonic crystals [48,49]. A demonstration of this concept will be further explored in the future at the THz regime where noble metals still act as good conductors. Because of the difficulty in the fabrication process of this structure, it is desirable to actualize this concept in two-dimensional metasurfaces such as the mushroom structure, which provides the surface-like dark mode in flat metasurfaces [50,51].

4. Experimental section

4.1 Sample fabrication

Fano metamaterials were fabricated by stacking copper slits with spacers to create slit openings. The copper slits and spacers were prepared using the wire electrical discharge machining method. All slits and spacers were carefully polished using abrasive papers and diamond compounds to decrease the unwanted radiative and absorptive loss resulting from surface roughness and disorder, that is, the size mismatch of the geometric parameters shown in Fig. 1(a). The symmetry (four slits in a unit cell) and asymmetry (a single slit in a unit cell) parts were individually prepared to manipulate the radial (${\Delta _\rho }$) and angular (${\Delta _\phi }$) asymmetries. Twenty-five unit-cells were prepared. The measured fabrication error of the distance between adjacent slits ($d \approx 1000$µm) in the completely assembled structure was approximately $8$µm.

4.2 Transmission experiment

A vector network analyzer (Anritsu VectorStar MS4644A) and two standard gain horns (MTG SGH-28) were employed for the millimeter-wave (Ka-band) transmission experiment in which the Fano resonances in metallic metamaterials were measured. A sample holding and rotating fixture was used to vary the angles of the incident waves. An additional fixture for manipulating the asymmetry part was employed through xyz-microstages.

4.3 Numerical methods

The magnitude and phase of the transmission spectra were obtained from the real and imaginary value of the two-port scattering parameters measured in the experiment (${|{{S_{21}}} |^2}$ and $\arctan \{{Im({{S_{21}}} )/Re({{S_{21}}} )} \}$ respectively). As a post-process, the time-domain signals of the measured spectra were obtained using the fast Fourier Transform algorithm (MATLAB code). These experimental results were compared with the numerical ones obtained from a commercial software (Frequency-domain solver in CST Microwave Studio).