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In optical frequency, surface plasmons of metal provide us a prominent way to build compact photonic devices or circuits with non-diffraction limit. It is attributed by their extraordinary electromagnetic confining effect. But in the counterpart of lower frequencies, plasmonics behavior of metal is screened by eddy current induced in a certain skin depth. To amend this, spoof plasmons engineered by artificial structures have been introduced to mimic surface plasmons in these frequencies. But it is less useful for practical application due to their weak field confinement as manifested by large field decaying length in the upper dielectric space. Recently, a new type of engineered plasmons, domino plasmon was theoretically proposed to produce unusual field confinement and waveguiding capabilities that make them very attractive for ultra-compact device applications [Opt. Exp. 18, 754-764 (2010)]. In this work, we implemented these ideas and built three waveguiding devices based on domino plasmons. Their strong capabilities to produce versatile and ultra-compact devices with multiple electromagnetic functions have been experimentally verified in microwaves. And that can be extended to THz regime to pave the way for a new class of integrated wave circuits.
©2011 Optical Society of America
1. Introduction
Surface plasmonic polaritons (SPPs) have the ability to generate strongly localized electromagnetic field and tightly confined wave beam with cross section far smaller than wavelength in optics. Their versatile nonlinear photon-matter interactions have inspired extensive investigation and been deeply explored for broad technological applications, for example acting as critical media to enhance performance of many optical functionalities such as Raman scattering or solar light harvesting [1–3]. As a promising approach, the field concentration capacity of SPPs’ in principle also can be applied to build compact optical circuits/devices with non-diffraction limit [4–6]. Plasmonic behavior originating from collective oscillation of surface electrons, however, occurs only within a limit frequency range usually from infrared to visible light. In the lower frequency counterpart, dielectric response of a metal is generally dominated by its conductivity by contributing a huge imaginary permittivity. At these frequencies, wave beam of confined cross section or enhanced localization is often equally desired or demanded for many practical applications [7, 8]. To this end, spoof plasmons with SPP-like mode dispersions can be constructed by periodically texturing a metal surface. These engineered plasmonics have similar wave guiding characteristics as SPP’s and can be manipulated in proper manners to mimic SPP-like electromagnetic behaviors such as strong mode localization and field concentration. Consequently, spoof plasmons have potential to enable us to achieve new electromagnetic devices in non-SPP frequencies that are not conceivable before using natural materials. One of potential applications is to build planar electromagnetic devices/circuits preferably with subwavelength transverse dimension. This is particularly important for THz wave where highly compact THz devices or integrated THz circuits are critically demanded in current rapid development of THz technologies [9–11]. Waveguide of tight transverse mode may also find important applications in non-diffraction limited THz sensing, imaging or in THz time domain spectroscope [12–15].
Approaches of corrugating surface structures in either grating or hole array geometry with subwavelength periods are mostly considered to induce spoof surface plasmonics [16–19]. They are effective in demonstrating physical concepts but less useful for practical application due to their weak field confinement as manifested by large field decaying length in the upper dielectric space [20]. To solve this problem, introduction of a cavity mode with strong field localization is very helpful to spatially redistribute the mode pattern and concentrate electromagnetic energy, for example by coupling an internal Fabry-Perot like mode. This can be easily realized in grating structures by coupling pure surface traveling mode with a standing slit mode [21]. But in the past, gratings of deep subwavelength slits are usually discussed in two dimensions with the focus mainly on light scattering features such as enhanced transmission [22, 23]. Recently Martin-Cano el al proposed that rectangular metallic bars arrayed in domino line on a conducting ground plane had an unusual ability to excite bounding wave with tightly confined cross section, and named domino plasmons (DPs) [24]. This kind of localized traveling mode has a prominent character that its mode dispersion is insensitive to the lateral dimension of constituent element in a wide size range from infinity to a small fraction of wavelength [25, 26]. Accordingly, a planar single mode THz waveguide or THz beam adaptor/concentrator by tapered waveguide was proposed [24]. Compared to fiber waveguide, domino waveguide is planar, monolithic and can be easily integrated with current circuit design technologies. Propagation loss of DP waveguide is relatively small compared to SPP or dielectric optic waveguide [24]. In this case losses mainly arise from ohmic conducting and bend radiation, while in general bend loss can be engineered to be very small by controlling the bend curvature. Application of DP in THz or millimeter band is very versatile and can be utilized to design key functional components/devices. In this paper we demonstrated these ideas and reported our experimental results for three domino plasmon devices implemented in microwaves which generally can be extended to THz domain by geometrical miniaturization. We show that domino plasmons are very versatile and effective to configure wave devices with multiple wave functions that may find very important applications in electromagnetic (EM) circuit or radiation controlling in THz region.
2. Mode analysis
Figure 1 shows a schematic structure of one portion of domino waveguide consisting of metal bars and a conducting ground together with the coordinates used. Structural parameters l, h a, and d describe the transverse length, height, spacing and period of metallic elements, respectively. Aluminum with conductivity σ = 3.8 × 107 s/m is used for metal bar and ground plane as well. Electromagnetically it is plausible to assume aluminum as perfect electric conductor up to THz. Similar results in THz will be anticipated for the discussions made in this paper. The red sine curve vividly shows travelling of DP surface wave along the top of metal bars in x direction.
Fig. 1 Schematic of domino waveguide. It is composed of aluminum metal bars and a conducting ground plane with structural parameters described in the text. The red sine curve vividly shows traveling of domino plasmonic wave along the top surface of domino waveguide.
Figure 2 plots the band diagram of single domino array at different transverse lengths l with other parameters h = 9.2 mm and a = 0.5d = 6 mm, numerically calculated up to 25 GHz using eigenmode solver of AnsoftTM HFSS. In calculation we use a unit volume of 6 (//x) × 20 (//y) × 40 (//z) mm3 with the metallic element in the center. The shaded region under light line cone denotes the non-radiative f-k space. Here we plot more bands than those reported in the literature [24] to give complete mode pictures, although the actual interesting ones are the lowest bands. These bands and their corresponding mode patterns can be understood with the help from previous studies on one-dimensional metallic reflection gratings [27] which are different only in transverse physical length. It has been concluded that both bounding wave mode on grating surface and standing wave mode inside slit cavities will be excited in these deep metallic gratings and they will electromagnetically couple together to form hybrid mode patterns around metallic elements [20,28]. This is electromagnetically similar with spoof plasmonics induced in periodically hole-perforated metal surface [29]. In our domino chain case, there basically exist two kinds of hybrid electromagnetic modes arising from the coupling of longitudinal surface traveling wave with y (vertical) and z (transverse) direction Fabry-Perot cavity modes, respectively. The vertical hybrid (VH) mode is sensitive to the cavity depth h and the transverse hybrid (TH) mode sensitive to its lateral length l. Hence the TH mode bands move toward higher frequencies as l reduces while the VH modes show no obvious change. The typical E_field patterns for these modes at an example case of l = 9 mm and kxd = π were plotted in Figs. 2(b), 2(c) and 2(d), respectively corresponding to the second and the first VH modes and the first TH mode. Their strong field confining effect as well as their individual features was manifested. For wide domino waveguide, e.g., l > λ (λλis the local wavelength in the surrounding dielectric space), the asymptotic plasma frequency of different bands could be estimated by f = (2m-1)c/4nh for the VH mode and f = mc/2nl for the TH mode with m an integer, n local permittivity inside cavity and c light velocity in vacuum. These formulas loss their accuracy and become invalid until l is reduced to be far smaller than local wavelength (e.g. l < 12 mm in this case). However, the increased capacity responses at the lateral ends of cavity need to be considered from the point of view of equivalent circuit [30].
Fig. 2 Band dispersions of domino plasmon modes. The bands in figure (a) were calculated for one domino waveguide up to 25 GHz at different lateral width l. They were divided into three bunches of modes: the first and second order VH modes and the first order TH mode. The corresponding mode patterns illustrated by E fields in the xy plane at the slit center were plotted in the right figures (b), (c) and (d) at an example case of l = 9 mm and kxd = π. Here we didn’t plot the modes inside the light cone, for they are radiative and out of our interest.
3. Results and Discussions
Based on the above discussion, the lowest band that could be exhibited as a VH or a TH mode depends on the relative values of h and l, i.e., whether it is a wide-thin or narrow-thick domino waveguide. From the viewpoint of application, the VH mode is more favorable due to its stronger field confinement to support compact device design. In experiment described below, we choose the VH mode to design DP devices with structural parameters l = 9 or 3 mm, h = 9.2 mm and a = 0.5d = 6 mm. Figures 3(a) , 3(b) and 3(c) show the schematic structures of a power divider, a directional coupler and a waveguide ring resonator using domino elements, theoretically proposed in the literature [24]. The vertical brown rectangles represent wave energy feeding ports which are coincident with the end of a 3cm × 2cm waveguide in experiment. Microwave absorber foam loaded with carbon particles inside was placed around the sample to reduce scattering from the edges of measurement platform mounted on a translating xy stage driven by two computer-controlled step-motors. The scanning step-length or 2D field picture resolution is fixed at 1 mm. An Anritsu vector network analyzer was used to feed the waveguide ports as source and collect local field signals through a monopole probe as detector as well. The tip of the probe with diameter = 0.8 mm moves at a fixed distance from the top surface of samples by 1 mm. The measuring frequency is set from 4 to 8 GHz coincident with the designed first VH bands of DPs.
Fig. 3 Schematics of implemented domino plasmon devices. (a) a power divider, (b) a directional coupler, and (c) a waveguide ring resonator. The vertical brown rectangles denote wave energy feeding ports which are coincident with the end of a 3cm x 2cm waveguide in experiment. In simulation all the boundaries except for the bottom plane are set by radiation boundary conditions.
To demonstrate a power divider, as shown in Fig. 3(a), we use one portion of 9 mm wide domino waveguide to feed energy and two branches of 3 mm wide domino waveguides to divide and output energy. The smallest spacing of these two branches is 3 mm, equal to their lateral widths. Their curvature radii for the curved portions are 48 mm. The measured 2D field pictures show that electromagnetic field is closely confined near waveguides and propagate and split into two well-defined energy beams in the designed frequencies. Figures 4(a) and 4(c) give the numerical and measured field patterns at an example frequency of 6.32 GHz, respectively. Numerical calculation on the waveguiding behaviors of domino devices in this work is performed using driven mode solver of AnsoftTM HFSS. Energy confinement and splitting effect is evidenced in the two figures which agree with each other very well. Impedance mismatching reflection at the interface of the two different waveguides is negligible and most energy on the wider waveguide is equally divided into the two narrower waveguides. The slight standing wave character of the beam was mainly caused by structural discontinuity at the waveguide ends which raised certain amount of reflection. The field difference of right and left splitting wings was due to sample manufacture flaws, which may also lead to attenuation of split power besides ohmic and radiation losses. Figures 4(b) and 4(d) give the numerical and measured field patterns at 8.00 GHz which is located inside the first band gap of VH mode. It is seen that wave cannot propagate along the divider at this gap frequency, rendering a low-pass filtering effect. In this case, the mode frequency determined by waveguide height is cut off at an upmost frequency while it usually has a lowest frequency for a dielectric waveguide mode.
Fig. 4 Simulation and experiment of a DP power divider. The left and right panels show the snapshots of amplitude of electric field at an example frequency of 6.32 and 8.00 GHz for simulation in (a) and (b) and experiment in (c) and (d), respectively. The field amplitude is normalized by the largest value. The fields were measured in a plane with 1 mm distance from the top surface of domino waveguides. The white strips in the figures denote the lengths of corresponding free-space wavelengths. The white dotted lines show the device profiles.
Figures 5(a) and 5(b) show the numerical and measured field pictures for a domino directional coupler at an example frequency of 6.96 GHz. The patterns in these two pictures agree with each other very well. They show that wave energy on one domino waveguide can be completely coupled onto the other after propagating certain distance, i.e., the coupling length l defined as the distance between two successive peaks appearing on neighboring waveguides. Spacing of the two domino waveguides (3 mm here) is a key factor to determine coupling efficiency and characteristic length. At current frequency the coupling length is about 72 mm as estimated from the figures. This value increases at larger wavelength. For a directional coupler based on planar light waveguide, this value can be accurately evaluated in terms of the wavevector difference between even and odd modes for a double waveguide [31]. Although these two modes indeed exist in our double domino chains [24], the same method will not work here to give a correct prediction for l due to the highly dispersive nature of domino modes and becomes meaningless at frequencies near bandgap where wave slows down and stops. Detailed investigation is required to correctly predict the coupling length in domino waveguide but is out of the scope of current paper and will not be discussed here. Nonetheless, the ability to form a single mode directional coupler with tight electromagnetic cross section is verified utilizing domino plasmons. Here we should mention a directional coupler using domino chains has been implemented in the literature [26] and similar results were observed there.
Fig. 5 Simulation and experiment of a DP directional coupler. The left and right figures show the snapshots of amplitude of electric field at an example frequency of 6.96 GHz for simulation (a) and experiment in (b), respectively.
Figures 6(a) and 6(b) show the numerical and measured field pictures for a waveguide ring resonator made of domino waveguides at 6.8 GHz. Diameter of the ring is 75.7 mm and the smallest spacing between the sides of the straight and the ring waveguides is 3 mm. Here we used the full-wave simulation to find the exact resonance frequency for a ring waveguide with given structural parameters. Thus the resonance frequency of 6.8 GHz was numerically obtained at which total electromagnetic energy passing through the output port (see figure) is minimum and maximum through the drop port. Figure 7 gives the numerical spectra of energy transmitted through the output (Io) and drop (Id) ports for the waveguide ring resonator shown in Figs. 6. There are about 10 eigenmode frequencies observed from 5.5 to 7.2 GHz at which most of inputting energy will flow out through drop port. The numerical field pattern given in Fig. 6(a) reflects the typical wave behaviors of a standard waveguide ring resonator. As the ring resonator was built here with a ring diameter of 75.7 mm which was in the same order with wavelength of 44.1 mm, such a small size is hardly achievable using a usual dielectric waveguide. The experiment part given in Fig. 6(b) grossly exhibited the function of frequency dropping behavior for a ring resonator although it was not as good quality as simulation. As estimated from the local field strength, less than 60% total input energy arrives at the drop port, while the rest part is mostly attenuated by radiation. Again the experiment performance was degraded here by manufacture flaw in positioning Al elements and can be technically improved to give better agreement with simulation.
Fig. 6 Simulation and experiment of a DP waveguide ring resonator. The left and right figures show the snapshots of amplitude of electric field at 6.8 GHz for simulation (a) and experiment in (b), respectively. The positions of input, output and drop ports are denoted in the figures.
Fig. 7 Numerical spectra of energy transmitted through the output (Io) and drop (Id) ports for the waveguide ring resonator shown in Figs. 6. The y-axis is in arbitrary unit.
The wave devices implemented here are very fundamental for beam manipulation and can be integrated to give rise to complex tight wave circuits to realize multiple wave or energy propagation functions. This is attributed to the unique features of domino plasmons that have significant field concentration ability and lateral dimension insensitive dispersions. The standing cavity modes coupled inside the slits as illustrated in Figs. 1 play the key roles in these effects. Their mode patterns can be modified to further reduce the transverse cross section of domino plasmon beam by filling dielectrics in slits or immersing waveguie in dielectric liquid, thus allowing for smaller or more compact wave devices. Different waveguiding behaviors like negative propagation constant can even be anticipated, for example by band folding effect that can happen in a compound domino waveguide with their slit cavities periodically modified either by various dielectric fillers or by changing physical dimensions [23,32].
4. Conclusion
In conclusions, recent proposed domino plasmons provide us with an efficient alternative way to control electromagnetic wave and achieve tightly confined wave beam in frequencies with no natural SPPs. We examined the full picture of possible electromagnetic modes from their band diagrams and found two kinds of different hybrid modes supported by domino chain waveguide, i.e., vertical hybrid and transverse hybrid modes. Using the vertical hybrid mode we designed three fundamental wave devices and characterized their performance both numerically and experimentally. Our results show the spoof domino plasmons are very reliable to produce versatile electromagnetic functions. The implemented devices are planar and monolithic, compatible with semiconductor procession technology. In addition to these advantages, their subwavelength cross section allows them to be assembled to build complex electromagnetic circuits/devices with multiple functions that may find important applications in the field of THz technologies.
Acknowledgments
This work was partially sponsored by The National Science Fund of China under the key project numbered by 60990322 and the Fundamental Research Funds for the Central Universities.
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