1. Introduction

Cavity quantum electrodynamics investigates strong light-matter interaction in a cavity whose eigenmode is resonant to a fundamental excitation frequency of the matter placed into the cavity. If the coupling strength of light and matter, i.e., the rate of energy exchange between them, is larger than the sum of the dissipation rates of the cavity photons and of the photo-excited matter state, the regime of strong coupling is achieved. It reveals itself by the disappearance of the bare cavity resonance and the bare matter excitation, and their replacement by two polariton modes which represent excitations (quasi-particles) of the coupled system [1]. The difference between the angular frequencies of the two polariton modes is the Rabi splitting $\Omega _R$ which is given by twice the coupling strength. Various types of materials in cavities were studied in the past, and strong coupling was found for both materials with fermionic elementary excitations such as atoms [2], quantum wells [3], and quantum dots [4], and for bosonic excitations such as cyclotron resonances [5,6], plasmon resonances in metamaterials (MMs) [7,8] and molecular vibrations [9,10]. Most of the investigations focused on the interaction between the electric field of the standing electromagnetic waves in the cavity with the electric dipole moments of the material. However, also the strong interaction between the magnetic field and magnetic dipole moments received attention. This involved on one hand naturally magnetic materials as in studies of magnons in microwave resonators [11,12]. In addition, taking advantage of the development of MMs which exhibit a magnetic response upon optical excitation [13–18], strong coupling was demonstrated, for example, in the near-infrared spectral range with Au-MgF$_2$-Au nanowire-pair MMs, which were either arranged with a waveguide to form a one-dimensional photonic crystal [19], or were placed into a gold Fabry-Perot resonator [20].

The observed magnetic response is in both cases the result of the interaction of the metal wires with both the electric and the magnetic radiation field. In the Fabry-Perot arrangement, the magnetic response is strong when the double wires are positioned suitably at electric field nodes of the standing waves in the cavity. There, not only is the magnetic field strongest, but also the electric field could drive currents in opposite directions in the two wires, with the combined effect of creating a loop current configuration which is required for the magnetic moment. The literature also discusses theoretical schemes where only the magnetic field drives such loop currents, e.g. in nanodiscs [21,22]. All scenarios, including those of the experiments addressed above, have in common, that the magnetic moments arise from three-dimensional metallic “magnetic atoms" allowing loop currents around the magnetic field vector.

Here, we investigate purely planar complementary metamaterials (CMMs) based on split-slotring resonators [23] as an alternative material for the study of strong light-matter interaction. A planar CMM is the inverse structure of a parent metamaterial (PMM): the metallic structures of the PMM are replaced by open areas, and the open areas are replaced by metallic ones. According to Babinet’s principle, CMMs exhibit magnetic moments parallel to the metal plane, if their PMMs exhibit an in-plane electric dipole moment [24,25]. Their behavior, however, differs from that of the three-dimensional artificial magnets in several ways, for example with respect to the orientation of the vector of the magnetic dipole moment which has a different sign for reflected (back-scattered) and transmitted (forward-scattered) radiation [24]. On the other hand, there are similarities, such as the fact that pronounced magnetoelectric coupling (often termed as a “cross-polarization" effect) contributes to their optical excitation [23].

In this paper, we explore the behavior of such CMMs as materials which exhibit a magnetic response upon excitation, and the applicability of the concepts of cavity-QED to make predictions of their behavior. The first of such predictions is that a magnetic CMM should interact strongly with the radiation field of the cavity if placed at a node position of the electric field (anti-node position of the magnetic field) of the cavity’s eigenmode. In our earlier paper [8], we demonstrated the validity of the corresponding prediction that the PMM couples best with the cavity field if placed at a node of the magnetic field (position with high electric field strength) of the cavity mode, and that the interaction becomes weaker and finally vanishes as the PMM is placed closer and closer to an electric field node. Unfortunately, we cannot perform the equivalent test with variation of a CMM’s position in the cavity because of the CMM’s high metallic reflectivity (see next point to be discussed) which changes the cavity’s eigenmode drastically when the CMM’s position is changed. We can only investigate the case with the CMM at an electric field node of the cavity’s eigenmode.

Another prediction is that the coupling can be treated as occurring between individual quanta of the light field and the material excitation. In this respect, CMMs offer an interesting feature which arises from their property that they exhibit bandpass characteristics for transmitted electromagnetic radiation, while the corresponding PMMs act as bandstop filters [26]. If placed into a cavity, a CMM – being highly reflective at all frequencies outside of the passband (which itself is cavity-modified and split into polariton branches) – tends to divide the cavity into two sub-cavities, permitting efficient photon exchange between them only at frequencies within the passband(s). We will show that this property leads to interesting coupling phenomena involving two photons (one from each sub-cavity) and one CMM plasmon to form three polariton branches, while insertion of the corresponding PMM in the cavity leads to the formation of only two polariton branches indicative for strong coupling between a single cavity photon with a single plasmon, as we showed in Ref. [8]. It is feasible to build a coupling hierarchy involving more and more quanta if an increasing number of CMM layers are stacked together with cavity sequences, thus establishing implementations of simple model systems of strongly interacting particles.

A final remark: As useful as the concepts of cavity QED are, one has to bear in mind that all our experiments are performed in the realm of classical cavity electrodynamics (cavity ED). There is always a large number of photons in the cavity and a large number of plasmons are excited in the CMMs, with the consequence that the phenomena can be modelled entirely with Maxwell’s equations without recourse to additional quantum-mechanical effects such as quantum-statistical features.

2. Experiments and results

2.1 Coupling of one photon with one plasmon

We first demonstrate strong interaction between just two quanta, a CMM plasmon with a photon in a single-sided cavity. The bare cavity used for that purpose is shown in the inset of Fig. 1(a). The cavity consists of a Bragg reflector and a metallic mirror. The Bragg reflector is composed of two Si slabs, each with a thickness of 100 $\mu$m. A third Si slab (also 100-$\mu$m-thick) serves as a cavity layer (defect layer). It carries a metallic reflector at its backside. All Si slabs are separated from each other by 202-$\mu$m-thick air gaps. The main panel of Fig. 1(a) shows the expected reflectance spectrum calculated with the transfer matrix method (TMM). A single cavity mode at a frequency $\omega _c/2\pi$ of 0.243 THz is identified. Figure 1(b) displays the axial spatial pattern of its electric field (blue curve) and the corresponding magnetic field (red curve). One notices that the electric field has its amplitude maximum at the inner surface of the cavity layer, while the magnetic field peaks on the surface of the Au mirror. The peak amplitude of the electric field is enhanced by a factor of twelve as compared with the amplitude of the incoming radiation. The TMM simulations yield a quality factor (Q-factor) of the cavity of 220. We also note that the cavity (because of its full metallic coverage of the third Si slab) has zero transmittance, which implies that the depth and width of the reflectance dip at the resonance frequency are determined by absorption losses in the cavity. And since the high-resistivity Si is assumed in the model to exhibit no absorption, the losses arise from absorption in the Au reflector into which the electric field of the standing wave penetrates. In the TMM calculations, we used a complex-valued dielectric constant of gold [27] with the value $-5.4\times 10^{4}+ i \cdot 1\times 10^{6}$.

 

Fig. 1. (a) Calculated reflectance spectrum of the cavity which is shown in the inset. (b) Simulated electric and magnetic field pattern in the cavity at the eigenmode frequency. The gray stripes indicate the locations and thicknesses of the Si slabs and the white areas in-between represent the air gaps. The gold mirror is located at the right side (on the outer surface of the rightmost Si slab).

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The experimental cavity and the other cavities discussed below were constructed from commercially available high-resistivity Si wafers (specific resistance: >20 k$\Omega$cm, two-side polished, lateral dimensions of $10\times 10$ mm$^2$), which were not thinned additionally, and from metallic shim rings purchased from MISUMI Europa GmbH [8]. The shim rings had an inner diameter of 6 mm and an outer one of 13 mm. The cavity was constructed by manually stacking the Si layers and shim rings, and clamping them together in a cage holder. Instead of a metallic reflector, the Si cavity slab carried a 200-nm-thick gold CMM metasurface consisting of an array of identical square split-slotring resonators. In various transmission experiments, we varied the structural parameters of the slotring pattern in order to tune its resonance frequency $\omega _m$ sequentially through the eigenmode frequency of the bare cavity. The size parameters of the various CMM patterns are given in the Appendix.

The experimental setup is depicted in Fig. 2. It is based on a TOPTICA TeraScan 1550 radiation source, which generates continuous-wave THz radiation from an InGaAs photoconductive antenna pumped with the beat-note of two telecom lasers (1550 nm) with a tunable difference frequency. The THz wave is detected by a second InGaAs photoconductive antenna with phase-sensitive photocurrent acquisition. For the transmission measurements, the sample was placed in the waist of a THz beam, loosely focused with a paraboloidal mirror with an effective focal length of six inches. With a nominal beam waist diameter of 3 mm and the typical period of the CMM unit cells of 120 $\mu$m, the number of CMM unit cells in the THz beam was about 600. All measurements were performed at room temperature in ambient air.

 

Fig. 2. Experimental setup of the measurements. DFB1 and DFB2 stand for distributed feedback lasers for difference-frequency generation in two photoconductive InGaAs photomixers. InGaAs stands for photomixers employing InGaAs as photoconductive material with an ultrashort conductance lifetime.

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The measured transmission spectra of the CMM-loaded cavity are shown in the color plot of Fig. 3(a) as a function of $\omega _m/2\pi$. The experimental data are compared with the results of simulations with the commercial Maxwell solver CST (Dassault Systèmes SE). All CST simulations described here and below include ohmic losses of the gold metallization using CST’s built-in materials database. The black open circles in Fig. 3(a) are the resonance frequencies of the CMMs on Si as obtained by the simulations (in steps from 0.202 THz to 0.312 THz, for details including a calculated transmission spectrum, see Appendix). As one tunes the resonance of the CMM across the cavity mode (0.243 THz), two polariton branches develop which exhibit an anti-crossing signature typical for strong coupling. With the CST Maxwell solver, we also calculate the theoretically expected transmission spectra. The results are plotted in Fig. 3(b) together with the simulated resonance frequencies of the CMMs on Si substrate (shown by open white circles). The polariton characteristics observed in the experiments are well reproduced including the preponderance of the lower polariton branch over the upper one.

 

Fig. 3. Color plots of the transmission spectra (a) measured with, respectively (b) calculated for the CMM-loaded single-sided cavity as a function of the resonance frequency of the bare CMMs. The calculated resonance frequencies of the bare CMMs are shown as open black circles in (a), respectively open white circles in (b). The dashed lines in both plots represent results obtained with the model of two coupled harmonic oscillators. The inset of (a) shows a segment of a fabricated CMM sample. The inset of (b) in the upper left corner displays both the CMM-loaded cavity, with the CMM metasurface represented by a single CMM unit structure, and the orientation of the electric radiation field indicated by an arrow (the electric field vector is oriented along the symmetry axis of the split slotring). The inset of (b) at the lower right corner shows the measured and calculated transmission spectra at 0.224 THz (black curve: measurement, red curve: simulation).

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The inset at the right-bottom corner of Fig. 3(b) shows transmission spectra at 0.224 THz (i.e., a vertical line scans through the color map of the main panel of the figure) exhibiting the measured polariton branches (black curve) and the simulated ones (red curve) at that frequency. The calculated transmittance values are larger and the lines of the polariton branches narrower than in the experiments. Measurements on bare CMMs showed that these exhibited a two times smaller Q-factor than predicted by CST simulations (see Fig. 6 in the Appendix). This suggests that the reduction of the Q-factor of the CMM-loaded cavities in the experiments in comparison with the simulations arises rather from the metastructures themselves than from the photonic crystal cavity. Upon inspection of the CMM patterns under the microscope, the general quality of the patterns was found to be good, see inset of Fig. 3(a). One discerns upon high magnification, however, a small edge roughness of the gold patterns, which may introduce some performance variations among unit cells. Very possibly, complementary metamaterials (with slot resonators) are more sensitive to such small imperfections than the parent metamaterials (with split-ring resonators). Some of the additional differences are likely due to the different beam divergence in the simulations vs. the experiments; in the former, we assumed a perfect plane wave, while in the latter the sample was placed in the waist of a THz beam which was loosely focused (see Fig. 2). Other experimental factors of relevance are likely to be imperfections of the alignment of the Si slabs during implementation of the cavity.

The light-matter interaction with the anti-crossing of the polariton branches is also well reproduced by a classical model of two coupled harmonic oscillators (more details in the Appendix). The results are shown by the dashed lines in Fig. 3 for both the experiments and the CST simulations. In order to reproduce the experimental polariton frequencies, we had to assume the cavity mode’s frequency $\omega _c/2\pi$ to be slightly down-shifted to 0.225 THz. The calculated TMM-value was 0.243 THz (for a full-metal reflector, not a CMM, on the last Si slab). The CST simulations of the CMM-loaded cavity predicted a value of $\omega _c/2\pi$ of 0.238 THz. For both the experimental and simulation results, good agreement with the anti-crossing features is obtained. The magnitudes of the Rabi splitting $\Omega _R$ – the minimal frequency difference between the two polariton branches – is determined by the mode separation at the frequency $\omega =\omega _c=\omega _m$. One obtains values of $\Omega _R/2\pi$ of 60 GHz in the measurements and of 63 GHz in the simulations. These values amount to 26% of the respective cavity resonance frequency, indicating that ultrastrong coupling ($\Omega _R\,/\,2 \omega _c > 0.1$) is achieved.

Considering the question of the applicability of concepts of cavity QED also in the case of the CMM material with magnetic response, we note that the results presented above confirm the QED prediction of strong light-matter interaction of the magnetic moment with the magnetic field at an electric field node of the cavity’s eigenmode. This is not to dispute that at the polariton frequencies both an electric and a magnetic field are present at the location of the CMM (as confirmed by additional CST simulations), and the interaction of the CMM is likely to occur with both. Another prediction of cavity QED is that the magnitude of the Rabi splitting should be proportional to the strength of the vacuum electromagnetic field in the cavity. This field strength is inversely proportional to the square root of the cavity mode volume [1]. In order to test this prediction by simulations for the CMM-loaded cavity, we increased the thickness of the cavity layer from 100 $\mu$m to 280 $\mu$m, leaving all other cavity elements unchanged. CST calculations show that a higher-order mode resonates at 0.243 THz in this expanded cavity, with a mode volume which is estimated [28] to be 2.3 times larger than that of the fundamental mode of the shorter cavity. The vacuum electromagnetic field is thus reduced by a factor of 1.5. Simulations of light transmission through this cavity reveal a Rabi splitting of 42 GHz, which is indeed about 1.5 times smaller than the 63 GHz obtained for the shorter cavity (for more details, see Appendix, and specifically Fig. 7).

2.2 Coupling of two photons with one plasmon

We now address the coupling of two cavity photons with one CMM plasmon. For this purpose, we employ a dual cavity whose structure is shown in Fig. 4(a). The gray rectangles represent the Si slabs separated from each other by air (white regions). The yellow line represents a continuous (unpatterned) gold layer serving as a mirror and making the two segments of the cavity non-communicating. In the experiments, this gold layer was replaced with the CMM. The composition of the cavity structure from left to right is Si/air/Si/Au/air/Si/air/Si/air/Si with the respective layer thicknesses of 100/202/100/0.2/46/50/96/100/202/100 $\mu$m. TMM calculations yield the reflectance spectra shown in Fig. 4(b) for radiation impinging onto the cavity from either the left or the right side. The spectra reveal eigenmodes at $\omega _{c1}/2\pi =0.241$ THz of the left segment of the structure and at $\omega _{c2}/2\pi =0.299$ THz for the right one. From the line widths of the resonances one derives Q-factors of 33 and 157 for the left and right sub-cavity, respectively.

 

Fig. 4. (a) Structure of the bare dual cavity employed for the demonstration of the coupling of two photons and one plasmon. The gray bars indicate silicon slabs, the yellow line the gold film separating the two parts of the cavity. Also shown are the simulated distributions of the electric and magnetic field on both sides at the respective eigenmode frequencies (0.241 and 0.299 THz). (b) Calculated reflectance spectra of the two sides of the bare dual cavity showing the two resonances at 0.241 and 0.299 THz, respectively. (c) Color plot of the measured transmission spectra of the CMM-loaded dual cavity at different resonance frequencies ($\omega _m/2\pi$) of the CMM. (d) Corresponding simulation results. Open circles in (c) and (d) represent the resonance frequencies of the bare CMMs on Si determined by CST simulations, dashed lines the peaks of the polariton branches obtained with the model of three coupled classical harmonic oscillators.

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Figure 4(a) displays the electric and magnetic field distributions obtained by CST simulations for each of the structures. The calculations were made for the respective resonance frequencies. One observes that the amplitude maxima of the magnetic field in both parts of the dual cavity are located at the position of the Au mirror, where one finds an electric field node, as expected for a metal layer.

Experiments were performed with such a cavity in transmission geometry, with the Au mirror replaced by a CMM whose bare resonance frequency $\omega _m$ is varied through the resonances of the two sub-cavities. The measurements yield the set of transmission spectra shown in Fig. 4(c) as a function of $\omega _m/2\pi$. One discerns three polariton branches – denoted as the upper polariton (UP), middle polariton (MP) and lower polariton (LP) modes – which evolve through two anti-crossings centered at about 0.23 and 0.28 THz (these values denoting the frequencies at which $\omega _m/2\pi$ is in the middle between two adjacent polariton branches). In the spectral regions where all three branches exist simultaneously, one can understand the polaritons as being the result of the simultaneous coupling between a CMM plasmon and two photons, one from each sub-cavity. As shown in Fig. 4(d), CST simulations reproduce the measured frequency dependence of both the resonance frequencies and their relative line strengths well, while the linewidths and the absolute transmission values differ from the experiments, the probable reasons being the same as those stated above in the discussion of Fig. 3.

With a model of three coupled classical harmonic oscillators (outlined in more detail in the Appendix), we derive values of the Rabi splitting for the two anti-crossing features. A representation of the model with three coupled pendula is shown in Fig. 5(a). The center pendulum represents the CMM, while the outer two stand for the sub-cavities, the coupling strengths of neighboring pendula being given by $V_1$ and $V_2$. Note that the model assumes no direct coupling of the two sub-cavities, but only an interaction mediated by the CMM. Fitting of the oscillator-model eigenvalues to the measured and CST-simulated curves of Figs. 4(c) and (d) yield the dashed curves in the two graphs, which reproduce the frequency trends very well except for a deviation at high frequencies for the UP branch in Fig. 4(d). The magnitudes $\Omega _{R1}$ and $\Omega _{R2}$ of the Rabi splitting (the minimal angular frequency spacing between UP and MP, respectively between MP and LP) are determined numerically, yielding the values $\Omega _{R1}/2\pi =63$ GHz and $\Omega _{R2}/2\pi =36$ GHz. The ratios $\Omega _{R1}\,/\,2\omega _{c1}=0.13$ and $\Omega _{R2}\,/\,2\omega _{c2}=0.06$ indicate that the first photon-plasmon interaction is in the regime of ultrastrong coupling ($\Omega _R\,/\,2 \omega _c > 0.1$), while the second one represents a strong-coupling situation ($\Omega _R\,/\,2 \omega _c < 0.1$).

 

Fig. 5. (a) Pendulum representation of the model of three coupled classical oscillators. $\omega _{m}$ represents the resonance frequency of the CMM, $\omega _{c1}$ and $\omega _{c2}$ those of the eigenmodes of the sub-cavities of the bare dual cavity. $V_1$ and $V_2$ are the respective coupling strengths. (b) Structure of the bare dual cavity with (nearly) degenerate sub-cavities. Gray bars indicate silicon slabs, the yellow line the gold film separating the sub-cavities. Red and blue lines: Simulated distributions of the electric and magnetic field on both sides at the respective eigenmode frequencies (0.294 and 0.299 THz). (c) Color plot of the measured transmission spectra of the CMM-loaded dual cavity at different resonance frequencies ($\omega _m/2\pi$) of the CMM. (d) Corresponding simulation results. Open circles in (c) and (d) represent the resonance frequencies of the bare CMMs on Si determined by CST simulations, dashed lines the peaks of the polariton branches obtained with the coupled-harmonic-oscillators model.

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If the sub-cavity modes of the dual cavity are degenerate ($\omega _{c} = \omega _{c1} = \omega _{c2}$), the characteristic equation (Eq. (7)) of the coupled three harmonic oscillators can be simplified to

One of the solutions is $\omega =\omega _c$, i.e., the frequency of the bare cavity mode, independent of the resonance frequency of the CMM. The eigenfrequencies of the other two polariton modes (LP and UP) can be derived from the expression in square brackets, and are identical to those from the characteristic equation (Eq. (3)) of two coupled harmonic oscillators, with $V_1^2+V_2^2=V^2$. Apparently, in the degenerate case, the three coupled harmonic oscillators behave as a system of two coupled oscillators – a CMM plasmon interacting with one cavity photon – plus an independent cavity mode which does not interact with the CMM. One finally can make a statement about the Rabi splitting which is always directly proportional to the coupling strength. For the dual cavity, one hence expects $\Omega _R \propto \sqrt {V_1^2+V_2^2}$, to be compared with the Rabi splitting of the single cavities: $\Omega _{R1} \propto V_1, \; \Omega _{R2}\propto V_2$. If $V_1=V_2$, the Rabi spitting of the degenerate double cavity is found to be increased by a factor of $\sqrt {2}$ compared with the case of a single cavity with the same photon-plasmon coupling strength of $V_1$.

In order to experimentally verify these predictions, we modified the dual cavity to one with (near-)degenerate sub-cavities. For reasons of practical fabrication, instead of two geometrically symmetric structures to the left and right of the CMM layer (which would require to press two fragile 100-$\mu$m-thick Si slabs against each other, with one of them carrying the CMM structure), we realized this with two asymmetric sub-cavities, whereby a higher-order resonance of the right one coincided with the lower-order resonance of the left one. The bare dual-cavity structure, shown in Fig. 5(b), consists of the layer sequence Si/air/Si/Au/air/Si/air/Si/air/Si with layer thicknesses of 100/46/100/0.2/46/50/96/100/202/100 $\mu$m. The frequency of the eigenmode of the left sub-cavity is now 0.294 THz and thus close to the frequency of 0.299 THz of the second eigenmode of the right sub-cavity. Figure 5(b) also displays the calculated electric and magnetic field patterns of the eigenmodes. In the experiments, the Au mirror is replaced by a CMM array.

Figure 5(c) presents the measured transmission spectra of the CMM-loaded cavity for various CMM resonance frequencies which are marked by the open circles. One notices that the MP mode indeed persists at the same frequency when the CMM resonance frequency is varied, and that the UP and LP modes show the typical signature of an anti-crossing behavior between each other. These modes, however, are fairly weak as compared with the dominant MP mode. Figure 5(d) shows the results of CST simulations. The calculated spectra reproduce the frequencies and relative strengths of the polaritons well, albeit again with smaller linewidths as compared to the measured data. Moreover, a slight redshift of the MP mode is visible with increasing $\omega _m$, which results from slight mismatch of the sub-cavities from perfect degeneracy.

The dashed lines in Figs. 5(c) and (d) are again fits with the model of coupled harmonic oscillators. One finds a Rabi splitting between the measured UP and LP modes of 59 GHz (with a rather large uncertainty range). For the simulated spectra, the value is 66 GHz. For the comparison with the Rabi splitting of a corresponding CMM-loaded single cavity, we performed CST simulations for the individual CMM-loaded sub-cavities of the near-degenerate dual cavity of Fig. 5(b) (see Appendix, Sec. 4.5). Values for the Rabi splitting of 44.5 GHz and 52 GHz are obtained. As these values are proportional to the respective plasmon-photon coupling strengths in the cavities, and the near-degenerate dual cavity is formed by these, one expects a Rabi splitting for the dual cavity of $\sqrt {44.5^2+52^2}=68.4$ GHz. This value is indeed close to the Rabi splitting $\Omega _{R}/2\pi$ of 66 GHz found by CST simulations for the near-degenerate dual cavity. The ratio $\Omega _{R}\,/\,2\omega _{c}=0.11$ (using for $\omega _{c}/2\pi$ the average eigenmode frequency of 0.2965 GHz) indicates coupling at the threshold between strong and ultrastrong coupling.

3. Conclusion

We have demonstrated ultrastrong coupling of plasmons of metallic complementary metasurfaces with one and two photons of cavities into which the metasurface is embedded. The fundamental difference in behavior as compared with the parent metamaterials comes on one hand from the different spectral properties of the bare complementary metamaterial (bandpass filtering, instead of bandstop filtering for the parent materials). The other fundamental difference is that, in the picture of cavity quantum electrodynamics, the dominant interaction occurs via the complementary metamaterial’s magnetic dipole moment and the magnetic field of the bare cavity’s eigenmode, while it is via the bare cavity’s electric field and the electric dipole moment of the parent metamaterial, as was shown in Ref. [8]. The findings here extend the available design toolbox for realizing applications based on resonant light-matter interactions, such as modulators and sensors, including scenarios where one can strongly couple electric-/magnetic-dipole material systems with metamaterials via coupled cavities to enhance spectroscopic sensitivity. On the conceptual level, coupling schemes as those discussed in the paper can be generalized to more complex arrangements in order to implement model representations of strongly-coupled many-particle systems.

4. Appendix

4.1. Metamaterials

The metasurfaces employed in this work consisted of arrays of split-slotring resonators which represent the Babinet-complementary structures of split-ring resonators (SRRs). We specify the unit cell in the following as CSRR (complementary SRR). The unit cell and the slotring contained in it had quadratic shapes. A single unit cell is displayed in Fig. 6(a). The metal patterns were defined by optical lithography following the deposition of 5-nm Ti and 200-nm Au by electron-beam evaporation on 100-$\mathrm {\mu }$m-thick Si substrates. The pattern transfer was completed by a lift-off process. The frequencies of the transmission resonances of the CMMs are determined by the choice of the values of four parameters: slot width $w$, slot length $l$, gap width $g$ and period $x$, all shown in Fig. 6(a). The density of the unit cells in the metasurface is determined by the period $x$. For the experiments, ten CMM specimens were fabricated. In order to tune the resonance frequency of the CSRRs while keeping the density of unit cells constant, we only changed the value of the slot length $l$ with the values 50, 54, 58, 62, 68, 72, 78, 82, 86 and 90 $\mu$m, giving respective resonance frequencies of 0.413, 0.371, 0.337, 0.312, 0.276, 0.259, 0.237, 0.224, 0.212 and 0.202 THz, as per simulations with the CST software. The period $x$, the slot width $w$ and the gap width $g$ were kept constant, their values being 120 $\mu$m, 10 $\mu$m, and 10 $\mu$m, respectively. A selection of CMMs with the available CSRR structures were then used in different experiments performed in this report. A representative power transmission curve (simulated and measured) of a bare CMM ($l =78~\mu$m) on a 50-$\mu$m-thick high-resistivity Si substrate is plotted in Fig. 6(b). One has to note that, for this demonstration, the CMM was intentionally patterned on a 50-$\mu$m-thick Si substrate in order to avoid strong spectral modulation by the multiple reflections in the substrate and to show the spectral characteristics of the slotring CMM. The transmission resonance (measured and simulated) lies at 0.229 THz – slightly lower than the simulated resonance (0.237 THz) of a CMM of the same size on an infinitely thick Si substrate. The transmittance curves exhibit the bandpass character typical for CMMs. From the width of the simulated curve, one derives a Q-factor of 9.6, from that of the measured one, a Q-factor of 5.

 

Fig. 6. (a) Schematic of the quadratic CMM unit cell containing a quadratic CSRR. (b) Power transmission spectrum of a CMM ($l =78~\mu$m) on a 50-$\mu$m-thick high-resistivity Si substrate. The red curve represents the simulated spectrum (obtained with the CST software) and the black curve the measured one. The resonance lies at $\Omega _m/2\pi =0.229$ THz for both simulation and measurement.

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Fig. 7. (a) Schematics of the cavities with defect-layer thicknesses of 100 $\mu$m (upper picture) and 280 $\mu$m (lower picture). (b) Distributions of the electric field (blue curve) and the magnetic field (red curve) in the cavity with defect-layer thickness of 280 $\mu$m at the cavity mode of 0.243 THz. (c) Simulation results: Power transmission spectra of the CMM-loaded cavities with defect-layer thicknesses of 100 $\mu$m (red curve) and 280 $\mu$m (blue curve). The same CSRR structure (with $l=78~\mu$m) is used in both cases.

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4.2. Model of two coupled harmonic oscillators

For the modeling of results shown in Figs. 3 and 5, we employed the formalism of frictionless coupling of two harmonic oscillators [8], which is based on differential equations of motion for two generalized coordinates $x_1(t)$ and $x_2(t)$. Based on these equations, we simulated the dependence of the polariton splitting (and avoided-crossing behavior) on the resonance frequency of the CSRR CMM. The equations are

(2)$$\begin{aligned} \ddot{x_1}+\omega_{c}^2x_1+V\dot{x_2}&=0, \\ \ddot{x_2}+\omega_{m}^2x_2-V\dot{x_1}&=0, \end{aligned}$$

where $\omega _{c}$ is the frequency of the cavity mode, $\omega _{m}$ the CSRR resonance frequency, and $V$ the coupling parameter of the oscillators. By requiring these equations of motion to have non-trivial solutions, one arrives at the characteristic equation

(3)$$(\omega^2-\omega_m^2)(\omega^2-\omega_{c}^2)-\omega^2V^2=0.$$

The analytical solutions are then derived as

(4)$$\omega_{{\pm}}^2= \frac{1}{2}\bigg[ (\omega_{c}^2+\omega_{m}^2+V^2) \pm \sqrt{(\omega_{c}^2-\omega_{m}^2)+2(\omega_{c}^2+\omega_{m}^2)V^2+V^4} \bigg].$$

We used this equation to fit the data in Fig. 3(a). The fitted coupling parameter was $V=60$ GHz. The cavity mode frequency for the fitting had to be 0.225 THz instead of 0.243 THz, in order to reproduce the experimental features. The effective resonance shift is attributed to a slight change of the cavity parameters either induced by the finite thickness of the CMM’s metallization or occurring during cavity alignment. For the fit procedures of the results in Fig. 3(b), Fig. 5(c) and Fig. 5(d), we proceeded correspondingly, with the coupling parameters and the fitted cavity modes of 63 GHz and 0.238 THz, 59 GHz and 0.285 THz, 66 GHz and 0.286 THz, respectively.

4.3. Rabi splitting and vacuum electromagnetic field

As discussed in the main text, the Rabi splitting is expected to be proportional to the strength of the vacuum electromagnetic field of the respective cavity mode (more strictly, in the case of CMMs, it is the strength of the magnetic field at the location of the CMM which is relevant). To test this dependence for the coupling between cavity and CMM, we started from the cavity shown in Fig. 1 and designed another one with a thicker defect layer (see both designs in Fig. 7(a)). The thickness of the defect layer was increased from 100 $\mu$m to 280 $\mu$m, because TMM simulations predicted for this longer cavity a second-order mode at 0.243 THz, i.e. at the frequency of the fundamental mode of the shorter cavity. Fig. 7(b) shows the electric and magnetic field distribution of the thick bare cavity with a gold mirror at the end facet (where the CMM is placed in the experiments). The maximum of the magnetic field lies at the position of the Au mirror.

The mode volume can be calculated by the equation [28]

(5)$$\mathrm{Vol}_{c}=\frac{\int \epsilon\,|E|^2\,dV}{\mathrm{max}(\epsilon\, |E|^2)},$$

where $E$ is the local electric vacuum field of the cavity mode and $\epsilon$ the local dielectric function. Using the field distributions from the TMM simulations, we calculated the volumes of the respective modes of the two cavities and found that the volume of the second-order mode of the thick-layer cavity is 2.3 times larger than the volume of the fundamental mode of the thin-layer cavity, and hence the strength of the vacuum electromagnetic field is reduced by a factor of 1.52 [1].

With the CST software, we numerically determined the transmittance spectra of the long cavity when loaded with various CMMs. The blue curve in Fig. 7(c) shows the simulated transmittance spectrum for the case that the resonance frequency of the CMM superposes nearly perfectly with the second-order cavity mode. This is the case for a CSRR with $l=78~\mu$m ($\omega _m/2\pi =0.237$ THz vs. $\omega _c/2\pi =0.243$ THz). The Rabi splitting is found to be 42 GHz. For comparison, the red curve shows the simulated transmittance spectrum of the cavity with the thinner defect layer loaded with the same type of CMM. The Rabi splitting is now 63 GHz. The ratio between the two values of the Rabi splitting (63/42 = 1.5) is very close to what is predicted (1.52) by the reasoning based on the mode volumes.

4.4. Model of three coupled harmonic oscillators

For the simulation of the dual cavities of Fig. 4(a) and Fig. 5(b), we employed the formalism of three coupled harmonic oscillators. With it, we simulated the polariton dispersion curves (and the avoided-crossing behavior) as a function $\omega _m$.

The model is based on differential equations for the three generalized coordinates $x_1(t)$, $x_2(t)$ and $x_3(t)$ of the pendulum model shown in Fig. 5(a). Here, $x_2(t)$ represents the middle pendulum, which stands for the CMM. The equations are

(6)$$\begin{aligned} \ddot{x_1}+\omega_{c1}^2x_1+V_1\dot{x_2}&=0, \\ \ddot{x_2}+\omega_{m}^2x_2-V_1\dot{x_1}-V_2\dot{x_3}&=0, \\ \ddot{x_3}+\omega_{c2}^2x_3+V_2\dot{x_2}&=0, \end{aligned}$$

where $\omega _{c1}$ and $\omega _{c2}$ are the angular frequencies of the cavity modes, $\omega _{m}$ is the angular frequency of the CMM resonance, and $V_1$ and $V_1$ describe the strength of the coupling between the CMM and each of the two cavities, respectively. The characteristic equation can be derived to be

(7)$$\begin{aligned} (\omega^2-\omega_{c1}^2) & (\omega^2-\omega_m^2)(\omega^2-\omega_{c2}^2)\,- \\ & \omega^2V_1^2(\omega^2-\omega_{c2}^2)-\omega^2V_2^2(\omega^2-\omega_{c1}^2)=0 \,. \end{aligned}$$

To solve this cubic polynomial equation, we employed the analytic formula for the roots [29]. With these solutions, we fitted the data shown in Fig. 4(c). The derived coupling parameters were $V_1=63$ GHz and $V_2=36$ GHz. Note that the cavity mode frequencies had to be taken as $\omega _{c1}/2\pi =0.232$ THz and $\omega _{c2}/2\pi =0.28$ THz (instead of 0.241 THz and 0.299 THz) in order to obtain satisfactory fits. The shifts of the effective resonances are again attributed to a slight change of the cavity parameters either induced by the finite thickness of the CMM metallization or imperfect cavity alignment. For the fitting of the data shown in Fig. 4(d), we proceeded correspondingly with the coupling parameters of 65 GHz and 41 GHz, the cavity mode frequencies of 0.235 THz and 0.293 THz.

For a degenerate dual cavity $\omega _{c}=\omega _{c1}=\omega _{c2}$, Eq. (7) is re-written in the form

(8)$$(\omega^2-\omega_c^2)[(\omega^2-\omega_m^2)(\omega^2-\omega_c^2)-\omega^2(V_1^2+V_2^2)]=0.$$

It is clear that $\omega =\omega _c$ is one of the solutions of the equation, and it is independent of the CMM resonance frequency. The other two eigenmode frequencies (those of the UP and LP branches) can be derived from

(9)$$(\omega^2-\omega_m^2)(\omega^2-\omega_c^2)-\omega^2(V_1^2+V_2^2)=0,$$

which is identical to Eq. (3) if one introduces the effective coupling constant V such that $V^2=V_1^2+V_2^2$. One notices that the strength of the coupling with the CMM is larger than in each cavity alone; if $V_1=V_2$, the enhancement amounts to a factor of $\sqrt {2}$. We employed this model to simulate the dispersion curves of Figs. 5(c) and (d), and obtained values of the Rabi splitting of 59 and 66 GHz, respectively.

4.5. Strong coupling in sub-cavities

In the last paragraph of the discussion of Figs. 5(c) and (d) in Sec. 2.2, we compared the Rabi splitting of a CMM-loaded near-degenerate dual cavity with the values of the Rabi splitting of each of the CMM-loaded sub-cavities. Figure 8(a) shows once more the dual cavity of Fig. 5(b), and Figs. 8(b) and (c) display the right and left sub-cavities of the dual cavity, each loaded with the CMM. We numerically simulated the transmittance spectra of these cavities with the CST solver and obtained the curves as shown in Fig. 8(d). The values for the Rabi splitting are 67.5, 52 and 44.5 GHz for the cavities shown in Figs. 8(a), b, and c, respectively. In the case of the dual cavity, this value is the splitting between the LP and UP branches. The relationship of the values fulfills the condition: $67.5^2\approx 52^2+44.5^2$. We note that the splitting (67.5 GHz) between LP and UP branches is very close with the Rabi splitting (66 GHz) derived from a two coupled harmonic oscillator model.

 

Fig. 8. (a) Schematic of the dual cavity, reproduced from Fig. 5(b). (b) Cavity consisting of the right sub-cavity of the dual cavity. The value of 0.299 THz given in the header of the graph is the frequency of the eigenmode, if the CMM is replaced by a full gold mirror. (c) Cavity consisting of the left sub-cavity of the dual cavity. The header of the graph again specifies the frequency of the eigenmode of the gold-mirror-loaded cavity. (d) Simulation results: Power transmission spectra of the CMM-loaded cavities. The full black, red dashed and blue dash-dotted curves correspond to the CMM-loaded cavities as shown in (a), (b), and (c) respectively.

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Funding

Deutsche Forschungsgemeinschaft (770/46-1, RO 770/46-1).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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