1. INTRODUCTION

Metamaterials (MMs) are by now a well-known scheme for engineering the optical response of materials [1,2]. Initial research focused on engineering materials with novel refractive index, $n(\omega )$, and impedance, $Z(\omega )$, with negative index of refraction [2] and electromagnetic cloaking [3] being the most prominent results. But these applications only scratch the surface of MM applications in materials engineering. MMs have potential uses in memory materials [4], thermal detectors [5], waveplates [6], and chemical sensing [7], to name a few examples. Additionally, MM photoexcitation [8], temperature control [9], electrical modulation [10,11], and structural control of intra-unit cell coupling [12–14] combined with microelectromechanical systems actuation [15–17] have opened the door for tunable and broad bandwidth MM modulators and switches [18–20]. The ability to control intra-unit-cell coupling has produced a particularly promising MM device known as a perfect absorber (PA) [21–23]. At terahertz (THz) frequencies, PAs comprise some of the most promising applications, providing new methods for THz sensing and detection, helping to fill the “THz gap” [24]. The use of complex material substrates and optical pumping has made dynamically tunable PAs a reality [11,25].

More recently, the development of high intensity THz sources [26] has jump-started research into “nonlinear” THz MMs, that is, MMs with optical properties dependent on the intensity of incident radiation. At microwave frequencies, nonlinear MMs can be created by introducing (for example) varactors [27] or compressible inclusions [28] into the design. At THz frequencies, such elements are too large to be of use as unit cell inclusions. However, semiconductor inclusions [29] and correlated electron material substrates [30] can be used to produce highly nonlinear MMs. Recent interest has focused on using superconducting thin films (SCs) as the basis of new nonlinear MM designs [31–34]. The complex conductivity, $\sigma (\omega )$, of a SC is sensitive to both temperature and electromagnetic fields, allowing for nonlinear MMs that can be dynamically tuned via temperature, magnetic field, or electric current [35–39]. In particular, superconducting MMs have been used to design gigahertz frequency components that are tunable via an applied RF flux [40–43]. Additionally, the low intrinsic electrical resistance of SCs allows for low loss and compact MM unit cell designs [44].

In this paper, we combine the idea of SC MMs and PAs to create a nonlinear THz wave absorber. As the incident THz field strength increases, the resonant absorption peak of the MM decreases while remaining relatively fixed in frequency. Thus, our superconducting metamaterial acts as a saturable absorber for THz frequencies.

We introduce a superconducting response into the absorber design by etching the split ring resonator (SRR) array out of a commercially obtained ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7(\mathrm{YBCO})$ film of thickness $d_s=100\mathrm{nm}$ [$T_c\approx 75\mathrm{K}$, substrate ${\mathrm{LaAlO}}_3$ (LAO)] using a lithographic wet etching technique [45]. A polyimide spacer layer and gold ground plane are then placed above the SRRs to form a “reflecting” PA [21]. Figure 1(a) shows a schematic of the device with dimensions, while Figs. 1(b) and 1(c) show photos of the mask used for etching and the resulting YBCO SRRs, respectively. Using experimental conductivity measurements of the YBCO film (Fig. 2, discussed more later), we performed CST Microwave Studio simulations (see Fig. 4 later in this paper) and optimized the dimensions of the SRRs such that the resonant absorption is maximized at 0.5 THz for $T=10\mathrm{K}$. The SRR side length is $L=40\mathrm{\mu m}$, the unit cell periodicity is $P=56\mathrm{\mu m}$, the linewidth is $w=5\mathrm{\mu m}$, and the capacitive gap width is $g=3\mathrm{\mu m}$. To create the ground plane and spacer layer, a 3 μm thick polyimide film was spin coated onto a Si substrate. A 10 nm Ti adhesive layer and a 200 nm layer of gold were then deposited on the polyimide subsequently. Finally, the gold-coated polyimide film was pulled from the Si substrate and fixed to the SRR array using the MM “tape” concept [8]. A 200 nm layer of photoresist (polymethyl methacrylate) was used as an adhesive.

 

Fig. 1. (a) Expanded unit cell schematic of a reflecting perfect absorber with relevant dimensions. (b) Mask used to etch YBCO SRRs. (c) Etched YBCO SRRs.

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Fig. 2. (a) Real and (b) imaginary experimental conductivity of the prefabrication YBCO for varying temperature.

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2. PERFECT ABSORPTION THEORY

Unity absorption ($A$) results when a MM’s transmittance ($t$) and reflectance ($r$) are engineered to be simultaneously zero:

In the reflecting PA considered here, the gold ground plane reduces $t$ to negligible values. The normal incidence reflectivity, $r$, from the MM interface is given by [23,46]

Impedance matching is achieved through simultaneous engineering of the MM effective permittivity, $ϵ(\omega )$, and effective permeability, $\mu (\omega )$. THz radiation at normal incidence is polarized so the $E$ field couples to the SRR’s fundamental mode, ${\omega }_o$, via the capacitive gap [Fig. 1(b)]. The resonant $ϵ$ follows a Drude–Lorentz dispersion [47],

Impedance matching to the free space impedance, $z_o$, then ensures that reflection from the MM interface drop to zero:

Reflections from inside the substrate are also possible. But the highly resonant imaginary component of $ϵ,{ϵ}_2$, eliminates these reflections by minimizing the penetration depth [23]:

The temperature dependence of the absorption $A(\omega )$ arises from the complex conductivity, $\sigma (\omega )$, of the YBCO film, which is described by a two-fluid model [49]. In this model, two populations of charge carriers give rise to the complex conductivity. At THz frequencies, the superfluid condensate contributes an $i/\omega$ term to the conductivity dispersion, while normal state carriers contribute a Drude dispersion. The resulting conductivity has the form

Figure 2 plots $\sigma (\omega ,T)$ for an unetched YBCO film at various temperatures, measured using THz time domain spectroscopy (THz-TDS) in transmission. For high temperature, the film conductivity is dominated by a Drude response. As the temperature begins to decrease, Fig. 2(a) shows that the real part of the conductivity, ${\sigma }_1$, increases. This is expected from the Drude model since ${\tau }^{-1}$ decreases for lower temperature. For $T=70\mathrm{K}$ and below, the trend shifts and ${\sigma }_1$ decreases with decreasing $T$ and a $1/\omega$ trend dominates the imaginary component, ${\sigma }_2$, characteristic of a superconducting state. The onset of this $1/\omega$ response allows for the transition temperature to be estimated at $T_c\sim 75\mathrm{K}$. This behavior corresponds well with both the two-fluid model [49] and previously reported data [50].

After fabrication, the YBCO SRRs retain $\sigma (\omega )$. The superconducting fluid affects the SRR resonance through the effective resistance, $R_{\mathrm{eff}}$, and the carrier kinetic inductance, $L_k$, which are functions of $\sigma (\omega )$ [33,51]:

The YBCO SRRs can be treated as lossy oscillator circuits with resonance frequency, ${\omega }_o$, given by the eigenmode equation for an RLC resonator:

Saturable absorption with increasing electric field arises from the suppression of the superconducting state through current-induced pair breaking [52,53]. THz electric fields drive resonant supercurrents in the SRRs. The current pattern in the SRR is, in general, very inhomogeneous. However, in local regions, especially near the corners of the SRR arms, the current density reaches very high values, up to and beyond the critical current density, $j_c$ [54,55]. These localized current spikes quench the superconducting state, raising $R_{\mathrm{eff}}$ [56] and resulting in damping of the resonance with a corresponding decrease of the absorption. In short, the MM absorption depends on the electric field strength of the incident THz radiation as well as the ambient temperature.

3. EXPERIMENTAL RESULTS AND DISCUSSION

High-field THz-TDS in reflection was used to characterize the saturable absorber. THz field strengths up to $E_o=200\mathrm{kV}/\mathrm{cm}$ were generated using the tilted pulse front technique in ${\mathrm{LiNbO}}_3$ [26]. Using two wire grid polarizers to control the electric field strength and polarization, pulses between 20 and 200 kV/cm were directed onto the absorber and polarized such that the $E$ field coupled to the SRR capacitive gap. The absorber was cooled with liquid He, and reflection spectra were measured for different temperatures and electric field strengths. THz transmission through the ground plane is negligible, and the absorption spectra are directly obtained from the measured reflectance spectra:

Due to the ground plane layer, direct illumination of the active layer is not possible. Instead, the incident THz beam must first pass through the LAO substrate, as shown schematically in Fig. 3(a). The LAO substrate acts as a Fabry–Perot etalon, producing multiple pulses in the time domain reflection signal, shown in Figs. 3(a) and 3(b). The first reflected pulse originates from the LAO/air interface and is of no interest here. The second reflected pulse, circled in Fig. 3(b), originates from the MM interface and carries the information about the MM absorption. This second pulse is used to obtain the reflectance spectra. A 500 μm LAO substrate with a 3 μm polyimide spacer layer and gold ground plane is used as a reference. The reference sample consists of a LAO substrate, spacer layer, and gold ground plane, matching the optical dispersion and path length of the MM but without the active YBCO layer. Thus, the absorption values reported subsequently are normalized to the strength of the signal that is transmitted through the substrate, measuring the internal absorbance of the device.

 

Fig. 3. Experimental geometry for THz-TDS of the YBCO absorber. (a) Schematic of MM unit cell with etalon reflections (not to scale). (b) Simulated time domain signals from the absorber showing etalon reflections. The red circles mark the reflection used for this experiment.

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Figure 4(b) shows the experimental absorption curves for a THz field strength of 20 kV/cm at different temperatures. For comparison, Fig. 4(a) shows the simulated absorption spectra. To perform the simulations, the experimental conductivity measurements of the YBCO film from Fig. 2 were imported into CST Studio and used to define the material dispersion of the SRR in the MM unit cell via an interpolative fit in the frequency domain solver. The simulation was solved using a transverse electric and magnetic waveguide port excitation with perfect electric and magnetic conductor boundary conditions that mimic the MM array periodicity. This process was repeated for the different temperature values. In both simulation and experiment, the absorption peak is maximized for $T=10\mathrm{K}$ and decreases with increasing temperature until the peak disappears entirely for $T>Tc\approx 75\mathrm{K}$.

 

Fig. 4. (a) Simulated MM absorption spectrum for varying temperature. (b) Experimental absorption spectra at $E=20\mathrm{kV}/\mathrm{cm}$ and varying temperature. The baseline oscillation is an artifact of an etalon reflection in the measurement.

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There are several likely causes for the discrepancy in peak absorption between simulation and experiment. First, limited spectral resolution as a result of the substrate etalon reflections in the time domain data is the largest contributor. Second, 20 kV/cm fields, though the lowest field strength accessible in this experiment, are still strong enough to excite a nonlinear response in the absorber [37]. Thus it is possible the absorption peak in Fig. 4(b) is already somewhat suppressed by the incident field. In addition, it is possible that as pressure decreases, air bubbles left between the gold polyimide spacer layer and the YBCO layer will expand, causing the two layers to partially detach. This would also result in resonance broadening. Finally, fabrication error in the spacer layer thickness and changes in the material properties of the YBCO during fabrication can also broaden the absorption.

The baseline oscillation in the data is likely an artifact of the experiment. Reflection based measurements present challenges in comparison to more straightforward transmission measurements since the sample is now a reflective optic in the experiment. This is complicated in using high-field pulses from tilted pulse front generation. It is quite possible that this etalon effect results from imperfect alignment between the sample and reference beams, leading to clipping or slight changes of the THz beam in the ZnTe detector crystal. Nonetheless, as shown subsequently, there is reasonable agreement between experiment and simulation in the trends of the absorption as a function of temperature and field strength.

Both simulation and experiment show a redshift in the absorption peak with increasing temperature, caused by the increase in $R_{\mathrm{eff}}$ and corresponding decrease in $L_k$. Here we see the importance of considering the loss in the SRRs. The observed redshift depends on changes to both $L_k$ and $R_{\mathrm{eff}}$. Assuming lossless resonators with $R_{\mathrm{eff}}\approx 0$ for all $T$ and a fundamental mode given by $1/\sqrt{(L+L_k)C}$, one would predict a blueshift with increasing $T$ (lower $L_k$). Instead, in the lossy SRRs, raising temperature causes the second term in Eq. (6), $R_{\mathrm{eff}}^2/{(L+L_k)}^2$, to grow more quickly than the first term, shifting the mode to lower frequencies.

Figure 5 shows the absorption spectra for varying incident field strengths at four temperatures. The absorber is strongly sensitive to incident field strength and functions as a saturable absorber for $10\mathrm{K}\le T\le 70\mathrm{K}$. Though the absorption peak is not completely suppressed at low temperatures, the saturation trend with increasing electric field is most pronounced at $T=10\mathrm{K}$ and $T=40\mathrm{K}$. At $T=10\mathrm{K}$ [Fig. 5(a)], increasing $E$ from $0.2E_o=40\mathrm{kV}/\mathrm{cm}$ to $E_o=200\mathrm{kV}/\mathrm{cm}$ causes the MM absorption peak to decrease from 0.70 to 0.40, corresponding to a modulation of 43% of the peak absorption. Though reduced, saturable absorption is present for higher $T$, up to $T=T_c$. For $T=40\mathrm{K}$ [Fig. 5(b)] and $T=70\mathrm{K}$ [Fig. 5(c)], the saturable absorption effect follows a similar trend, but with less modulation. For $T>T_c=100\mathrm{K}$ [Fig. 5(d)] and higher, the resonance is saturated completely for all field strengths.

 

Fig. 5. Field dependence of the YBCO MM absorption spectrum at (a) 10 K, (b) 40 K, (c) 70 K, and (d) 100 K. $E_o=200\mathrm{kV}/\mathrm{cm}$.

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A notable feature in Fig. 5 is the absence of a redshift in the absorption peak with increasing $E$ field. This is in contrast to the clear redshift in the temperature dependent results in Fig. 4. This has also been seen in previous work [33] on superconducting SRRs. There it was argued that the electric field induced normal carriers screen the superconducting film to incident fields. As a result, $L_k$ is effectively unchanged, eliminating the redshift. This explanation appears to be unlikely since $L_k$ originates from the $i/\omega$ term in Eq. (4) and is characteristic of the superconducting state. A decrease in condensate density, induced by either temperature or electric field, must correspond to a decrease in $L_k$.

It is also possible that $R_{\mathrm{eff}}$ experiences a slower growth and lower peak value as $E$ is increased instead of $T$. This difference could arise from differences in $\tau$ [see Eq. (4)], for carriers induced by supercurrent pair breaking, versus those induced by increasing $T$. Normal state carriers induced via supercurrent sit in a lower temperature environment and have a lower $\tau$. $R_{\mathrm{eff}}$ would thus be lower for the case of electric-field induced normal carriers, resulting in less of a redshift. However, more work is needed before conclusive remarks can be made on this subject.

It is important to note that the nonlinear THz absorption shown in Fig. 5 would be difficult to produce without the strong MM resonance. PA can been implemented at higher frequencies without MMs, using low-permittivity films, for instance [57–59]. But this would be difficult to reproduce at THz frequencies given the lack of natural materials with either plasma frequencies or strong intrinsic optical nonlinearities in the THz range. It is the combination of the strong resonant absorption of the MM with the nonlinear response of the superconductor that produces the saturable absorption band.

To conclude, we combined the ideas of MM PAs and superconducting MMs to create a saturable absorber at terahertz frequencies. The MM consists of an array of SRRs etched from a 100 nm YBCO film. A polyimide spacer layer and gold ground plane are placed above the SRRs to create a reflecting PA. Changing ambient temperature or terahertz field strength altered the complex conductivity of the YBCO SRR array and decreased impedance matching in the absorber, reducing the peak absorption. We used numerical simulations and high-field terahertz time domain spectroscopy to characterize the absorber. For an incident terahertz field strength of $E=20\mathrm{kV}/\mathrm{cm}$, the absorption was optimized near 80% at $T=10\mathrm{K}$ and decreased to 20% at $T=70\mathrm{K}$, the approximate $T_c$ for the YBCO SRRs. For $E=40\mathrm{kV}/\mathrm{cm}$ and $T=10\mathrm{K}$, the peak absorption was 70%. At $E=200\mathrm{kV}/\mathrm{cm}$ and $T=10\mathrm{K}$, the absorption saturates to 40%, for a total modulation of 43%. The saturable absorption effect is present over a broad temperature range and is tunable via changing temperature.