When replacing a bulk negative index material (NIM) with two resonant surfaces that allow for surface plasmon polariton (SPP) propagation it is possible to recreate the same near-field imaging effects as with Pendry’s perfect lens. We show that a metallic meander structure is perfectly suited as such a resonant surface due to the tunability of the short (SRSPP) and long range surface plasmon (LRSPP) frequencies by means of geometrical variation. Furthermore, the Fano-type pass band between the SRSPP and LRSPP frequencies of a single meander sheet retains its dominant role when being stacked. Hence, the pass band frequency position, which is determined by the meander geometry, controls also the pass band of a meander stack. When building up stacks with different periodicities the pass band shifts in frequency for each sheet in a different way. We rigorously calculate the spectra of various meander designs and show that this shift can be compensated by changing the remaining geometrical parameters of each single sheet. We also present a basic idea how high- transmission stacks with different periodicities can be created to enable energy transfer at low loss over practically arbitrary distances inside such a stack. The possibility to stack meander sheets of varying periodicity might be the key to far field superlenses since a controlled transformation of evanescent modes to traveling wave modes of higher diffraction order could be enabled.
©2011 Optical Society of America
Following the theoretical prediction of Veselago , materials with both a negative permittivity ε and a negative permeability μ are able to transmit evanescent electromagnetic waves without exponential damping. Furthermore, light would bend in the “wrong direction” and hence exhibit a negative index of refraction. In 2000 Pendry  argued that a slab of such a material with n = −1, which he called superlens, would be able to restore the evanescent source fields perfectly in the image plane. However, this image is still non-magnified and cannot be used for conventional non-scanning microscopy. Among others, one approach to realize a magnifying superlens incorporates an anisotropic metamaterial crystal with hyperbolic dispersion [3–5]. This way the evanescent waves propagate along the radial direction of the layered metamaterial and the field distribution of the input plane is mapped for example onto a curved output plane. Another approach is to transform the evanescent wave modes from the object into propagating ones by adding a grating to a thin silver film, thus creating spatially magnified images with subwavelength details [6,7]. However, both approaches suffer from high absorption losses in the bulk negative index material (NIM) or metal-dielectric stack, respectively, and can only be used over a very narrow frequency range.
Another idea is to replace the bulk NIM by two resonant sheets that support surface plasmon polariton (SPP) propagation [8–11]. Due to the interaction between the resonating surface modes of the sheets, an exponential growth in the amplitude of an evanescent plane wave can be achieved within such a system.
In this contribution we investigate stacks of vertical meander structures , which basically represent such resonant surfaces with near-field imaging capabilities. Recent research has shown [12,13] that the excitation and interaction of localized surface plasmon modes on such meander structures can be controlled over a large spectral range by varying the geometrical parameters. It is important to note that a pass band of high transmission occurring between the short (SRSPP) and long range surface plasmon polariton (LRSPP) modes  can be changed in the same flexible way. The transfer of energy at low loss over practically arbitrary distances might enable a controlled mapping of evanescent modes to traveling wave modes of different orders, which is particular interesting for subwavelength imaging. We investigate the amplitude transfer between resonantly coupled meander sheets and demonstrate how well a meander stack with different grating periods can be designed by means of varying geometry. Finally, we also show that such a stack is able to transfer energy resonantly over a large distance.
2. Numerical simulation models and analysis methods
The meander structure that represents one sheet of the stack is shown in Fig. 1a . It basically resembles a binary grating with a thickness t, a meander depth D and the periodicity P x. To achieve inversion symmetry along the propagation direction of the incident light the condition W r = P x/2 - t has to be fulfilled.
When expanding this structure towards a meander stack, the distance between two single sheets is described by D spa. For stacks with geometrically varying meander sheets, the geometry parameters are additionally labeled with an integer number i from top to bottom according to the direction of the incident light, hence t i, D i, P x,i and D spa,i (Fig. 1b).
We choose silver for our study due to the low absorption losses in the visible region. The dielectric function was derived from the Drude model and should be complemented by the Johnson-Christy material data  if higher frequencies are of interest. We used the plasma frequency ω p = 1.37 × 1016 rad/s and the scattering frequency ν = 8.5 × 1013 rad/s . The structure is placed in vacuum and the incident light is always p-polarized.
For the computation we use an in-house simulation package (MicroSim), which employs the rigorous coupled-wave analysis (RCWA) improved by factorization rules . We considered 100 diffraction orders, which provide both reasonable accuracy and computation time. Finally, the results were verified with another software package using RCWA  and adaptive spatial resolution .
3. Influence of the geometric parameters on the plasmonic band structure of a single meander sheet
In this section we focus on a single meander sheet in order to understand the transmission properties of the whole stack in more detail. We first consider a thin metal film embedded in a dielectric. When excited, surface plasmon polaritons (SPPs) will propagate along both metal-dielectric interfaces and interact with each other. Two modes occur with respect to charge distribution in the metallic film, namely a symmetric and antisymmetric plasmon oscillation with low and high oscillation frequencies caused by different field confinement in the metal film. A low confinement corresponds to small damping and therefore large propagation length (long range surface plasmon polariton (LRSPP)). A large confinement on the other hand corresponds to strong damping and a short propagation length (short range surface plasmon polariton (SRSPP). They can be easily identified by lowering the thickness of the metal film, which will decrease the frequency of the SRSPP mode more strongly [13,18].
When non-conformal corrugation is added to obtain a meander structure, the SRSPP and LRSPP branches are folded back into the f-k space, enabling coupling to incident radiation (diffractive coupling) . Furthermore, Fu et al. pointed out that a Fano-type pass band  of high transmission occurs at certain geometries between the SRSPP and LRSPP frequencies (Fig. 2a ) due to the interaction of SRSPP/LRSPP oscillators, which is responsible for the direct transmission . A dispersion diagram (Fig. 2b) helps to identify these plasmon frequencies and shows that the same pass band can also exist for an oblique incidence of light (inset Fig. 2b).
In the following we investigate how the SRSPP and LRSPP frequencies and thus the pass band change by variation of the geometrical parameters. Figure 3a depicts the extinction -ln(T) of a meander structure as a function of frequency and meander thickness t for perpendicular incidence (k x = 0) retrieved with RCWA. It is evident that the SRSPP mode decreases whereas the LRSPP mode remains almost constant for lower values of t .
The increase of meander depth D at a constant thickness t and periodicity P x as shown in Fig. 3b leads to a down-shift of the SRSPP frequency, while the LRSPP mode experiences only a slight red-shift for smaller values but then stays fairly constant. Finally, Fig. 3c shows the extinction as a function of frequency and grating periodicity P x at a constant meander depth D and thickness t. In this case the SRSPP mode changes less with P x compared to the LRSPP mode, which becomes even more emphasized for deeper corrugation D.
To sum up, with the variation of the meander thickness t and meander depth D there are two degrees of freedom available to control the pass band position, which is limited by the SRSPP/LRSPP frequencies, for a grating with fixed periodicity P x. This is the key for the design of a high-transmission stack consisting of meander sheets with different periodicities anywhere in the optical domain.
4. Properties and design of meander stacks
4.1 Properties of meander stacks with two sheets having the same periodicity Px
In a stack of two meander sheets Fabry-Pérot (FP) modes occur and interact with local SPP modes. Figure 4 depicts the transmittance of a stack with i = 2 as a function of the distance between the meander sheets D spa and the frequency f for different meander depths D. The FP modes that exist in the structure can be predicted with a modified cavity equation  (black dashed lines in Fig. 4):
For shallow values of the meander depth D (Fig. 4a), FP modes prevail in the optical response of the stack as expected. For increasing D a red shift of both SRSPP and LRSPP modes is observed with a stronger shift on the SRSPP side (Fig. 4b). With increasing depth D (Fig. 4c) the coupling between FP and plasmon modes becomes stronger and exhibits highest transmittance in the frequency range where the single meander structure already shows its SRSPP/LRSPP associated pass band. With further increasing meander depth (Fig. 4d) the coupling becomes weaker again. For all meander depths D an anticrossing between local SPP (white dashed lines) and FP modes can be observed at the points of their intersection, which we interpret as cavity plasmon polaritons in analogy to cavity polaritons occurring in micro resonators . Overall, one can observe that with an optimum meander depth D the plasmon pass band of the single meander sheet also governs the transmission pass band in the double meander stack.
Within the pass band of the structure in Fig. 4c near-field imaging can be achieved as demonstrated in Fig. 5 . In this picture a TM-polarized plane wave (f = 600 THz, λ = 500 nm) is perpendicularly incident from the top onto a 100 nm thick layer of chromium with a subwavelength slit (w = 100 nm). Behind the aperture, a double meander structure with P x = 400 nm, t = 20 nm, D = 50 nm and D spa = 200 nm follows. As expected, the meander sheets behave as coupled resonant surfaces and create a near-field deep subwavelength image of the slit similar to the perfect lens. This first image has a FWHM of 126 nm, which is shown in the upper left inset of Fig. 5. However, the slit can also be imaged by a second focus with a FWHM of 346 nm almost two wavelengths behind the stack (lower right inset). This can still be considered as subwavelength imaging but occurs in a more usable distance behind the lens.
4.2 Design and properties of meander stacks with two sheets and different periodicities Px
The fact that the pass band is limited by the SRSPP and LRSPP modes of the single meander sheet gives us the opportunity to shift it in frequency by changing the geometry parameters on the single meander structure. This is even possible for a stack that consists of meander sheets with different periodicities. Figure 2c indicates that for meander sheets with a constant depth D and thickness t the SRSSP and LRSPP frequencies shift down in frequency for higher periodicities P x. To achieve high transmission in a meander stack with sheets of different periodicities P x, the remaining geometrical parameters t and D have to be altered in order to compensate for this change (compare Figs. 2a and 2b). For instance, in order to match the pass bands of two sheets with the periodicities P x,1 = 250 nm and P x,2 = 500 nm, the values for t 1 / D 1 have to be decreased / increased whereas the values for t 2 / D 2 have to be increased / decreased, respectively.
For our meander stack with P x,1 = 250 nm and P x,2 = 500 nm we are using the combination t 1 = 20 nm, D 1 = 25 nm, t 2 = 14 nm and D 2 = 40 nm to obtain a pass band around f = 820 THz (λ = 365 nm) in the final structure. The transmittance of the overall stack as a function of frequency f and D spa is shown in Fig. 6a . It is evident that the overall pass band has been shifted in frequency and momentum compared to the meander stack from the previous section (Fig. 4c). The dispersion diagram of a particular meander stack with D spa,1 = 110 nm (Fig. 6b) shows features similar to the low-periodicity meander structure (not shown) but the influence of the FP modes is clearly visible. It is also important that for a given frequency the transfer function can be extended to higher k x values to obtain resonant transmission for various directions, hence demonstrating k-filtering.
4.3 Design and properties of meander stacks consisting of four sheets with different periodicities Px
Given the possibility to stack meander sheets with different geometry parameters it might be possible to design a bulk-like material that exhibits negative refraction and spatially magnifies a source field. In this section we demonstrate a meander stack with four sheets with periodicities P x,1 = 250 nm, P x,2 = 300 nm, P x,3 = 350 nm and P x,4 = 400 nm. The pass bands of the single sheets are shown in Fig. 7a . The compensation by geometrical variation of t and D for the lowest and highest periodicity is displayed in Fig. 7b and Fig. 7c, respectively.
All four sheets possess their pass band in the same frequency range, which makes it preferable to pick a frequency within this range. For f = 650 THz (λ = 484 nm) the optimized geometry parameters can be found in Fig. 8 along with the dispersion diagrams to show the k x dependence of the pass band. Please note that due to the periodicity of the meander sheets it is possible that plasmon modes from neighboring Brillouin zones interact with the Fano-type resonance and lower the transmission in the pass band (see arrows in Fig. 8).
Figure 9a shows the transmission of the combined meander stack over the frequency f and the distance between the sheets D spa = D spa,1 = D spa,2 = D spa,3. Because of the enormous computational effort to calculate a 42 µm wide stack, which would be the least common multiple of the four different periodicities, we used a lower cell length and bore with the grating irregularities at the edges. Earlier calculations showed that this estimation is sufficient and hardly changes the results. For this particular dispersion diagram we used 250 modes and a cell width of 6 µm to achieve convergence. Within this width we fit 24 unit cells with a length of P x,1, 18 unit cells with P x,2, 16 unit cells with P x,3 and 14 unit cells with P x,4 (principle illustration in Fig. 9a).
It is obvious that the pass band is still independent from D spa as in the case of a meander stack with two sheets of same geometry (Fig. 4b) albeit with a narrower spectral width. In this case a different combination of geometrical parameters might create a broader spectral width. However, this example illustrates that it is in principle possible to resonantly transfer the plasmon field from layer to layer via a meander stack with different periodicities. The transmission behind the whole structure still accounts to 0.5 and shows that within the pass band energy can be transferred over distances larger than 3 µm as for instance at D spa = 1 µm.
From the dispersion diagram of the combined stack at D spa = 370 nm (Fig. 9b) we also find that the whole stack allows high transmission for a range of k x values, which is important for potential subwavelength applications.
Unfortunately, due to the various available FP and plasmon mode interactions within the pass band, the transmission is lowered and the shape of the pass band altered for higher k x values.
With these findings it might be possible to design a stack consisting of many more layers with only a marginal change in periodicity that slowly changes the k x values until a transition from evanescent to propagating modes occurs. For that, the last sheet requires a periodicity that is smaller than the wavelength in order to enable first-order diffraction and therefore coupling from plasmon modes with originally higher momenta to the vacuum field.
Strong interaction between local plasmon modes and FP modes was demonstrated. From the dispersion relation f(D spa) one observes anticrossing of FP and local SPP modes, which we interpret as cavity plasmon polaritons. Furthermore, we showed that the single meander SRSPP/LRSPP behavior can be changed to a large degree by variation of geometry. We demonstrated that for meander stacks the interplay of SPP and FP modes can create a pass band, which is limited by the LRSPP and SRSPP frequencies. By means of geometry variation it can be placed anywhere in the optical domain. This especially holds for stacks composed of meander sheets of varying geometry, which could be suitable for realization of k-filters.
We believe that a stack of many sheets with slowly changing geometry can be designed to show near-field to far-field transforming properties within the pass band range. As a first step in this direction we investigated a stack with four sheets of varying geometry and showed that the transmission can be kept high behind the structure. It will be interesting to further investigate such structures and their variations, such as curved meander stacks, with respect to their negative refraction and imaging properties. Other interesting properties of meander stacks include photon tunneling and filtering, which can find application in the design of epsilon near zero materials .
We acknowledge the financing by Baden-Württemberg Stiftung, Project OPTIM.
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