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We experimentally demonstrate planar plasmonic THz waveguides using metal films that are periodically perforated with complementary split ring resonators (CSRRs). The waveguide transmission spectra exhibit numerous transmission resonances. While the geometry is commonly used in developing negative index materials, the excitation geometry used here does not allow for conventional metamaterial response. Instead, we show that all of the observed resonances can be determined from the geometrical properties of the CSRR apertures. Surprisingly, the Bragg condition does not appear to limit the frequency extent of the observed resonances. The results suggest that metamaterial-inspired geometries may be useful for developing THz guided-wave devices.
©2011 Optical Society of America
1. Introduction
Although the conductivity of conventional metals is finite, metals are often approximated as perfect electrical conductors (PECs) in the far-infrared and beyond. Under this approximation, planar metal films are not capable of supporting surface plasmon-polaritons (SPPs). However, there have been a number of theoretical studies demonstrating that patterning the surface of a PEC can alter the dielectric properties of the effective medium, which, in turn, can be manipulated by simply altering the geometrical parameters of the pattern [1–6]. This concept has potential utility in a broad range of applications, including the development of guided-wave devices. An early embodiment of this concept is a plasmonic slab waveguide demonstrated by Williams et al. [7]. In their demonstration, they used a two-dimensional periodic array of square blind holes (i.e. holes that do not perforate the metal film). More recently, we have shown that a one-dimensional array of rectangular apertures that do perforate the metal film can be used to create a variety of guided-wave devices, including a straight waveguide, a y-splitter, and a 3 dB coupler [8,9]. Subsequently, there have been several theoretical papers suggesting the use rectangular and wedge-shape protrusions [10–12]. In each of these cases, relatively simple shapes have been used. This raises the obvious question of whether more complex structuring would allow for greater control of the propagation parameters or offer any new or unexpected capabilities.
As a starting point for considering more complex artificially structured materials, we look at the significant amount of work that has been done in the area of metamaterials [13–15]. While the topic is very broad at this point, the basic geometries that have been most extensively studied are variants of split ring resonators. Composite materials based on these structures have been shown to exhibit negative values of µ and ε over narrow frequency ranges. Recently, Navarro-Cia et al. reported numerical simulations suggesting that ultrathin complementary split ring resonators (CSRRs) can give rise to an SPP-like response that supports the guiding of broadband THz radiation [16]. At THz frequencies, the fabrication of such ultrathin structures (typically ≤ λ/100) would require a dielectric substrate for support. However, as we noted above, the inclusion of dielectrics typically increases the propagation losses. In related work, Reinhard et al. have examined the excitation of surface waves on thin films covered with a two-dimensional array of in-plane SRRs using free-space transmission measurements [17].
In this submission, we experimentally demonstrate low-loss guided-wave capability using one-dimensional array of CSRRs periodically perforated in free-standing metal films. We measure the propagation properties of the devices using THz time-domain spectroscopy. In contrast to our earlier studies on waveguides based on rectangular apertures, devices based on CSRRs exhibit much richer spectral characteristics over a broader frequency range than expected. Using a simple effective cavity resonance model and finite-difference time-domain (FDTD) simulations, we explain the origin of the various modes. A surprising feature of the measurements is the fact that the Bragg condition does not appear to limit the frequency extent of the observed resonances. We also measure the propagation properties for a linear waveguide. The observed modes appear to be somewhat less well confined than observed using simple rectangular apertures [8,9]. Nevertheless, the use of metamaterial inspired geometries appears to be a promising approach for developing new THz guided-wave devices.
2. Experimental details
We fabricated a number of straight plasmonic THz waveguides by periodically perforating 300 µm thick stainless steel foils with CSRRs using conventional laser micromachining (i.e. the apertures go completely through the metal sheet). A schematic diagram of a section of the waveguide is shown in Fig. 1 , along with a photograph of an individual structure. The typical CSRR dimensions were s = 500 µm, b = 300 µm, a = 50 µm, g = 100 µm and h = 300 µm. The center-to-center spacing, d, between individual CSRRs was 400 µm. Normally incident broadband THz radiation was coupled to a broadband THz SPP using a chemically etched semi-circular groove at the one end of the waveguide. The groove had a radius of 1 cm and rectangular cross-section (500 µm wide by 100 µm deep) and was chemically etched at the input side of each waveguide, with the origin of the circle lying at the center of the first CSRR. This groove coupled and subsequently focused freely propagating THz pulses to SPP waves propagating along the waveguide [7,8,18].
Fig. 1 (a) Schematic diagram of a terahertz waveguide based on CSRR apertures. A semi-circular groove is used to couple a broadband THz waveform to the patterned structure. For clarity, a small section of the waveguide structure is expanded in the figure panel. The CSRR design parameters are s = 500 µm, b = 300 µm, a = 50 µm, g = 100 µm and h = 300 µm. The center-to-center spacing, d, between individual CSRRs is 400 µm. (b) a photograph of an individual CSRR fabricated in a 300 µm thick stainless steel foil.
We used a modified THz time-domain spectroscopy setup to characterize the waveguiding properties of the periodically spaced apertures, which has been described in detail elsewhere [4]. For completeness, we provide a brief description here. Broadband THz radiation was generated using a 1 mm thick <110> ZnTe crystal, which was then collected and collimated using an off-axis paraboloidal mirror. The collimated THz radiation was normally incident on the semi-circular groove fabricated on the waveguide samples and was subsequently coupled to SPP waves propagating towards the input end of the CSRR array. Using a second <110> ZnTe crystal, we measured the out-of-plane component of the surface electric field via electro-optic sampling. It should be noted that the vector components of the surface electric field could be measured at any point above the metal surface by moving the optical probe beam and the ZnTe detector simultaneously [19].
In the numerical FDTD simulations of the propagation properties, the metal was modeled as a perfect electrical conductor, which is a reasonable approximation for real metals in the THz regime, and the surrounding dielectric medium was air. We used a spatial grid size of 10 µm, which is sufficient to ensure convergence of the numerical calculations, and perfectly matched layer absorbing boundary conditions for all boundaries. For the input electric field, we used the derivative of a Gaussian pulse that had the same bandwidth (and similar pulse shape) as what was available in the experimental work. All simulated results were obtained using measurements taken at specific spatial points, typically centered on the waveguide (except for y-axis measurements) in order to match the experimental measurements.
3. Experimental results, simulations and discussion
In Fig. 2(a) , we show both the experimentally measured and simulated waveguide transmission spectra for a 7 cm long linear waveguide measured 2 mm after the last CSRR aperture. There are a number of characteristics that are immediately apparent. In contrast to predictions for waveguides based on ultrathin CSRRs, the transmission spectra associated with CSRRs fabricated in thick metal films do not appear to support broadband propagation of THz radiation. This is particularly evident from the numerical simulations. It is also clear that the experimentally measured transmission resonances are considerably broader than those observed in simulations. As we pointed out above, in the numerical simulations, the metal was modeled as a PEC with idealized geometrical properties. Since metals are characterized by a finite conductivity, there is an associated loss at THz frequencies, which leads to linewidth broadening. Furthermore, imperfections in the individual apertures, as well as variations between apertures, are also expected to broaden the resonances.
Fig. 2 (a) The experimentally measured and numerically simulated waveguide transmission spectra. Both spectra are normalized relative to the corresponding input spectra. (b) the simulated AR frequencies of the lowest order plasmonic mode (filled circles). The solid black trace corresponds to a fit to Eq. (1) with m = 1 and n = p = 0.
In both the experimental data and the simulation results shown in Fig. 2(a), it is clear that there are a number of transmission resonances, characterized by both resonance and anti-resonance (AR) frequencies (i.e. the frequencies corresponding to the sharp dips on the high frequency side of each resonance). In order to analyze the data, we examine the anti-resonance frequencies, as opposed to the resonance peaks. We have previously shown that AR frequencies remain fixed even when the other geometrical parameters (a and d, in the present case) are varied [8,9], demonstrating the fundamental nature of the AR frequencies. From the data, there are four clear transmission resonances in common between the two spectra, with associated AR frequencies occurring at 0.12 THz, 0.35 THz, 0.58 THz, and 0.76 THz. The simulation results show additional amplitude dips, discussed below, that are not apparent in the experimental data.
As we noted above, there have been significant studies on a variety of metamaterial designs. In the case of CSRRs, when the incident electric field is perpendicular to the plane of the structure, no metamaterial response is expected [20,21]. Therefore, alternative explanations are necessary to explain the origin of the transmission resonances. In previous work, we have shown that the AR frequencies are directly associated with the geometrical parameters of the structures. In order to examine whether or not such an explanation is valid here, we begin by considering the lowest order resonance at 0.12 THz. In the case of rectangular apertures, the lowest order AR frequency was given by the cutoff frequency of the aperture, νc = c/2s, where s is the aperture length and c is the speed of light in vacuum [22]. In the case of CSRRs, the lowest order resonance corresponds to the cutoff frequency νc = c/2L, where L = 2s + 2b-4a-g is the “unwrapped” aperture length. For the structure described in Fig. 2, L = 1300 μm, corresponding to a lowest order mode frequency of 0.12 THz.
While the CSRR aperture shape is obviously more complex than a simple rectangular aperture, we find that the all of the AR frequencies can be determined by considering the entire unwrapped CSRR length, L, rather than any subsections of the structure. The AR frequencies correspond to effective cavity resonance frequencies, since the top and bottom of each aperture is open, and are approximately given by
where, m, n, and p are integers (m = 1,2,3… and n, p = 0,1,2…). Using this approach, the lowest order resonance discussed above can be indexed as m = 1, n = 0, and p = 0. Therefore, ν100 = c/2L = νc = 0.12 THz. With the theory of Pendry and associates in mind [9,10], since ν100 = νc for the “unwrapped” aperture, we refer to it as the plasmonic mode [9]. Based on FDTD simulations, where we vary the number of CSRR apertures in the waveguide, we find that the lowest order mode diminishes in amplitude with increasing number of apertures at a rate that is faster than the other resonances. This may occur because the associated wavelength (0.12 THz corresponds to a wavelength of ~2.5 mm) is significantly larger than the CSRR structures and may result in greater loss due to interaction with the unstructured metal region adjacent to the waveguide. In what follows, we will index the modes as (m,n,p).To further demonstrate that the lowest order resonance is directly related to the “unwrapped” CSRR aperture length, we performed numerical simulations in which the value s, and therefore L, varied, while all other parameters were kept fixed. In Fig. 2(b), we plot the value of the lowest order AR frequency as a function of the aperture length, L. A fit to the equation νc = c/2L shows excellent agreement.
In contrast to transmission spectrum for a rectangular aperture-based waveguide [8,9], the present CSRR-based waveguide exhibits richer spectral characteristics. In addition to the ARs at 0.35 THz, 0.58 THz, and 0.76 THz, the simulated spectrum shows two additional ARs at 0.69 THz and 0.81 THz. All of these frequencies lie above νc and correspond to dielectric slab modes, where energy can flow into the apertures and bounce back and forth between the interfaces like Fabry-Perot resonances. Using Eq. (1), these AR frequencies can be indexed as (3,0,0) at 0.35 THz, (5,0,0) at 0.58 THz, (6,0,0) at 0.69 THz, (7,0,0) at 0.81 THz, and (5,0,1) at 0.76 THz. Since the aperture array is symmetrically excited by the semi-circular groove (i.e. the groove is symmetrically etched about the waveguide), we would normally expect that only symmetric modes where m is an odd integer, such as the (1,0,0), (3,0,0), (5,0,0), (7,0,0) and (5,0,1) modes, could be excited. However, clearly, the (6,0,0) mode is also observed in the numerical simulations. We discuss this in greater detail below. Furthermore, while the dip associated with the (5,0,1) mode is evident, dips associated with the (1,0,1) mode at 0.51 THz and the (3,0,1) mode at 0.61 THz are not. The dip associated with the (3,0,1) resonance appears very weakly in the simulated spectrum, while the (3,0,1) AR frequency is very close to the (5,0,0) AR frequency, making it difficult to differentiate between the two.
The mode designations above arise from fitting the observed AR frequencies to Eq. (1) and are in excellent agreement. Nevertheless, more evidence is necessary to validate these assignments. To do so, we used FDTD simulations with a sinusoidal input corresponding to each individual AR frequency and computed the steady state electric field distributions for each mode. In Fig. 3 , we show the simulated total electric field in the xy plane for the six different modes shown in Fig. 2(a). These distributions clearly show the value of the index, m. Specifically, we expect to observe m-1 nodes in the electric field distribution. Thus, for m = 1, no nodes should be evident in the xy electric field distribution.
Fig. 3 Steady state total electric field distribution in the xy plane using a sinusoidal input frequency of (a) 0.12 THz corresponding to the (1,0,0) mode (b) 0.35 THz corresponding to the (3,0,0) mode (c) 0.58 THz corresponding to the (5,0,0) mode and (d) 0.69 THz corresponding to the (6,0,0) mode and (e) 0.76 THz corresponding to the (5,0,1) mode and (f) 0.81 THz corresponding to the (7,0,0) mode.
There are several important points to note about these field distributions. First, using the aforementioned FDTD simulations, we have verified that n = 0 for all of the modes and that p = 1 only for the (5,0,1) mode (not shown). Second, as we noted above, the symmetric nature of the coupling mechanism leads to the suppression of modes when m is even. Nevertheless, in the simulated spectrum shown in Fig. 2(a), the AR frequency associated with the TM600 mode is evident. While modes with even values of m are not expected due to symmetry considerations theoretically, they may arise in numerical simulations, albeit with very low associated magnitudes. For example, the magnitude of the (6,0,0) mode, shown in Fig. 3(d), is approximately 30 dB lower in magnitude than the other modes shown in Fig. 3. In addition, the AR frequencies associated with the (2,0,0) mode at 0.23 THz and (4,0,0) mode at 0.46 THz occur in spectral regions where the transmission amplitude is already low and, therefore, are not expected to be visible. Field distributions associated with these two modes exhibit even weaker field strengths than were computed for the (6,0,0) mode. Finally, the xy plane field distributions for the (5,0,0) and (5,0,1) modes do not look the same. However, since the electric field distributions along each of the three axes are not wholly independent, there is no reason to believe that the (5,0,0) and the (5,0,1) modes will look the same.
In our previous work on rectangular aperture-based waveguides, no resonances existed beyond the Bragg frequency, νB = c/2d [8,9], in either experimental measurements or numerical simulations, since any mode with a transverse wave number beyond the first Brillouin zone would exhibit high radiative loss. In the case of the CSRR waveguides, we noted above that the entire unwrapped aperture length was used to assign each of the modes. Therefore, it is appropriate to use the center-to-center periodicity between individual CSRRs of d = 400 µm to compute the corresponding Bragg frequency, νB = 0.375 THz. Clearly, higher order modes appear above the Bragg frequency. This raises an important question. What is the role of periodicity on the waveguide transmission frequency range? In order to answer this question, we performed FDTD simulations in which all of the waveguide parameters were kept fixed, except the periodicity, d. In Fig. 4 , we show three representative waveguide transmission spectra with d varying between 500 µm and 900 µm. There appears to be no connection between the aperture spacing, which determines the Bragg condition, and the number of observed resonances. Simulation results show that modes are present beyond 1 THz. Thus, the high frequency limit appears to be determined by issues such as coupling efficiency, radiation loss, etc. It should be noted that the spectral amplitudes vary with aperture spacing, which is consistent with earlier observations using rectangular aperture based waveguides [8,9].
Fig. 4 Simulated waveguide transmission spectra as a function of CSRR periodicity. The red trace corresponds to a periodicity of 500µm. The blue trace corresponds to a periodicity of 700µm, and the black trace corresponds to a periodicity of 900µm.
In order to further examine this phenomenon, we simulated the transmission properties of complementary closed ring resonators (i.e. CSRR apertures with g = 0 and all other parameters the same as in Fig. 1) and periodically spaced rectangular apertures with s = 500 µm, a = 50 µm and d = 400 µm. The complementary closed ring resonator waveguide and the CSRR waveguide both exhibited well-defined transmission resonances well beyond the Bragg condition, while the rectangular waveguide did not. Apparently, the more complex electric field properties within each ring resonator, caused by the center metal region in each aperture, allows for higher frequency operation. Further work is necessary to fully elucidate this effect.
Finally, we examined the waveguiding characteristics of a linear device based on periodically spaced CSRRs, as summarized in Fig. 5 . Since the lowest order (1,0,0) (plasmonic) mode is extremely weak, we measured the propagation properties of the (3,0,0) mode, which was the most prominent resonance. Due to the orientation of the crystal and the polarization of the THz and optical probe beam, we were sensitive only to the out-of-plane (Ez) component of the propagating surface field, rather than the Ex and Ey field components, which dominate within the apertures. In Fig. 5(a), we show the magnitude of the Ez field component measured along the length (x-axis) of the waveguide. From these measurements, we find that the loss is ~0.0166 mm−1, corresponding to a 1/e propagation length of 60 mm. This value is smaller, but comparable, to the loss observed in other planar waveguide geometries [8,9]. However as previously mentioned, the structure of a split ring resonator allows for a broader range of higher order modes that can be supported with low loss.
Fig. 5 The measured electric field amplitude |Ez| (a) along the x-axis at different positions along the length of the waveguide. (b) along the y-axis approximately 4 cm from the waveguide input. (c) along the z-axis at different heights above the sample surface.
In Fig. 5(b), we show the magnitude of the Ez field component measured along the y-axis of the waveguide, 4 cm from the waveguide input. The lateral field distribution at the cross-section exhibits a Gaussian mode profile with a full-width at half maximum (FWHM) mode size of ~9.8 mm. When rectangular apertures were used, we observed a much tighter Gaussian mode profile. The change in mode profile may arise from the more complex aperture geometry, including the fact that there is metal at the center of the CSRR aperture. In Fig. 5(c), we show the magnitude of the Ez field component as a function of distance above the waveguide surface (along the z-axis). The field decays exponentially from the metal dielectric interface with a 1/e decay length of ~4 mm. While the field confinement along the y and z axes is greater than observed using rectangular apertures, it is important to note that these measurements do not correspond to the lowest order mode. Measurements on the TM100 mode are expected to show a greater level of field confinement. With that in mind, we have previously found that the 1/e decay length for an unstructured planar metal film was ~4-5 mm or ~4-5 λ at ~0.3 THz (corresponding to λ = 1 mm) [19].
4. Conclusion
In conclusion, we have demonstrated that a one-dimensional array of complementary split ring resonators can act as effective terahertz waveguides. In contrast to what has been predicted for CSRRs in ultrathin metal films [16], we find that CSRR-based planar waveguides in free-standing thick metal films support a number of well-defined narrowband modes. The observed mode frequencies extend well beyond what is expected based upon the Bragg condition. Each of the observed modes can be ascribed to an effective cavity resonance of the unwrapped CSRR aperture. We have demonstrated guided-wave propagation in a linear waveguide device that exhibits low loss. Based on these results, we believe that other metamaterial inspired geometries offer promising approaches for developing new THz guided-wave devices.
Acknowledgements
We gratefully acknowledge support of this work through National Science Foundation (NSF) grants ECCS-0824025 and DMR-0415228.
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