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In this work, we propose an efficient approach to compensate for the commonly observed substrate-induced bianisotropy that occurs in on-wafer optical metamaterials at normal incidence. First, the consequence of placing a finite thickness substrate underneath a metamaterial is analyzed, indicating that the induced bianisotropy is a near-field effect. The properties of metamaterials sandwiched between an infinitely thick substrate and a finite-thickness superstrate with different permittivity and thickness values are then investigated. It is demonstrated from full-wave simulations that by adding an ultrathin superstrate with a judicious choice of its thickness and permittivity value, the substrate-induced bianisotropy of the system can be suppressed and even eliminated. In addition to the extracted nonlocal effective medium parameters, the induced electric and magnetic dipole moments calculated from the volumetric microscopic fields are also presented, validating that the magnetoelectric coupling compensation is a real physical phenomenon. This study will benefit future optical metamaterial design and implementation strategies as well as the corresponding fabrication and characterization methodologies.
©2013 Optical Society of America
1. Introduction
Optical metamaterials [1, 2], by virtue of their exotic electromagnetic properties, such as negative refractive index [3], zero refractive index [4], electromagnetic-induced transparency [5], frequency selective high absorption [6], broadband polarization control [7], and so on, have garnered a vast amount of research interest in recent years. In general, the optical metamaterials demonstrated to date usually consist of patterned metallodielectric composite nanostructures mounted on a supporting wafer, exhibiting controllable electric and/or magnetic resonances. A great deal of effort has been devoted to realizing self-symmetric nanostructures that more accurately approximate an ideal anisotropic/isotropic medium under illumination at normal incidence [8]. Compared with other possible building blocks such as split-ring resonators [9] and electric-LC resonators [10] that are commonly employed in the microwave range, multilayer fishnet structures with alternating metal and dielectric layers are more promising candidates for achieving backward wave [3] and/or zero-phase delay [4] phenomena at optical wavelengths. The electric response of a fishnet is determined by the cutoff wavelength of the hole waveguide, while its magnetic response can be tailored by controlling the gap surface plasmon polaritons propagating along the dielectric spacers in between the metal layers [11]. During the initial development, single-functional-layer on-substrate fishnet nanostructures were considered both numerically and experimentally, due to the limitations of nanofabrication techniques [12, 13]. However, it was soon realized that the effective medium parameters extracted from the single-functional-layer metamaterials do not represent their bulk properties, which are better manifested in nanostructures with multiple functional layers [14, 15]. Recently, multilayer fishnet structures were demonstrated in the near-infrared and even at visible wavelengths [3, 15]. It is worth mentioning that the salient optical properties of a properly designed multilayer fishnet are driven by its fundamental Bloch mode, which was observed in Ref [3], where the measured refractive index was found to be equal to the calculated index of the fundamental Bloch mode. This statement was later confirmed by numerical studies that the energy transport inside the fishnet is indeed mediated by the fundamental Bloch mode [16].
However, a lack of symmetry still exists in the optical metamaterials discussed above. First, inherent to the popular fabrication processes, such as focused-ion beam (FIB) etching or lift-off, the sidewalls of these multilayer nanostructures are often non-vertical facets. It has been numerically explained [17] and experimentally demonstrated [18, 19] that these tapered sidewalls break the symmetry of the nanostructures in the direction of wave propagation, resulting in magnetoelectric coupling. Hence, the fabricated metamaterial nanostructure becomes bianisotropic, which develops an electric and magnetic polarization that are determined by both the electric and magnetic fields. This not only complicates the effective medium description, but it also may inhibit the desired properties in the index of refraction and introduce additional reflection loss. Only very recently was the non-vertical sidewall profile issue addressed by optimizing the anisotropy etching process in the electron-beam lithography, which ensures a sidewall angle larger than 89° [4]. In addition to the tapered sidewall, it was found that the presence of the supporting substrate also induces bianisotropy due to the broken symmetry of the system, which should be taken into account during the design process and the characterization of fabricated metamaterials [20, 21]. A similar substrate-induced bianisotropy effect in plasmonic grids, which are referred to as metafilms or metasurfaces, was also revealed [22]. According to the reported studies, the magnetoelectric coupling induced by the substrate was shown to have the strongest strength at the plasmonic resonances (either electric or magnetic) of the nanostructures. In response to this effect, free-standing metamaterial films with a single functional layer have been recently realized both in the terahertz [23] and near-infrared regimes [4]. However, when peeling off the fabricated nanostructure samples from the wafer, such a delicate process could easily cause structural deformation that results in a change in their optical properties and even potentially destroy the multilayer nanostructures due to their relatively weak mechanical supports. From this point of view, on-substrate metamaterials have advantages in maintaining the nano-scaled features of the structure while being easily mounted on apertures for optical characterization and practical device implementation, at the expense of the broken symmetry of the system.
In this paper, we propose an effective way to suppress and even eliminate the substrate-induced bianisotropy (as seen by an outside observer) of the system containing a finite-thickness metamaterial. First, in addition to the previous work where the substrate was considered as a semi-infinite half space, we study the effect of a substrate with varying finite thickness values, showing that the induced bianisotropy is a near-field effect. A transmission matrix cascading technique is used to extract the scattering matrix of the metamaterial alone from the total scattering matrix of the system. Then, the effect of introducing a superstrate with various thickness and permittivity values is investigated by full-wave simulations, where the substrate is considered to be semi-infinite, closely approximating the practical situation. It is shown that by properly choosing the thickness and permittivity values of the superstrate, the substrate-induced bianisotropy of the system can be greatly suppressed or even completely cancelled out. Furthermore, to confirm that the bianisotropy compensation is a real physical effect that does not rely on the homogenizability of the structure, the induced electric dipole moment (EDM) and magnetic dipole moment (MDM) excited by approximately pure electric or magnetic fields are calculated from volumetric microscopic fields. The results exhibit a non-zero magnetoelectric coupling only for the case with a substrate, but near-zero values for the cases without substrate and with both substrate and properly designed superstrate. It should also be noted that for a metamaterial with an infinite number of layers, the substrate-induced bianisotropy will be negligible. However, considering that the current fabrication techniques used for demonstrating optical metamaterials usually produce nanostructures with less than eight functional layers, the studies presented in this paper will be particularly beneficial, especially when creating bianisotropy-free on-substrate metamaterials or metasurfaces for experimental validation as well as practical implementation.
2. On-substrate multilayer fishnet without superstrate
2.1 Transmission matrix cascading technique
Here we consider a specific example of the experimentally important fishnet nanostructure. Instead of a single functional layer, a thirteen-layer (i.e. six functional layers) fishnet structure is employed. The general setup under consideration is displayed in Fig. 1(a) , which includes a multilayer fishnet sandwiched between a superstrate and a substrate each with a certain thickness (hsup, hsub) and relative permittivity value (εsup, εsub). Below the substrate and above the superstrate are two semi-infinite half-spaces each having a relative permittivity εb and εt, respectively. A time dependence of exp(-iωt) is assumed throughout the paper. In order to retrieve the effective parameters of the fishnet, first, the scattering matrix of the metamaterial alone (SMM), which cannot be measured directly in experiment, must be extracted from the measurable total scattering matrix (Stot).
Fig. 1 (a) A multilayer fishnet metamaterial sandwiched between a superstrate and a substrate with finite thickness. Beneath the substrate and above the superstrate are the bottom and top half-spaces, respectively. (b) The unit cell geometry of the multilayer fishnet nanostructure composed of Ag and SiO2. The dimensions are px = 600, py = 600, wx = 72, wy = 336, ta = 15, td = 20 (all in nanometers). (c) Retrieved bianisotropic effective medium parameters for the free-standing fishnet displayed in (b). (d) Evolution of the real part of the effective index as a function of the number of functional layers (N) for the free-standing fishnet in (b).
The procedure starts by obtaining the total transmission matrix from the total scattering matrix (Stot) as [24]
Then, because the entire system can be regarded as several cascaded two-port networks separated by infinitesimally thin free-space layers, its total transmission matrix (Ttot) can be readily expressed as the product of all the two-port transmission matrices in free-space [25]where Tths(bhs) and Tsup(sub) are the transmission matrices for the top (bottom) half space interface and the superstrate (substrate), respectively. Hence, the transmission matrix of the metamaterial alone (TMM) isand the scattering matrix of the metamaterial alone can then be calculated from TMM byIn regards to the substrate and the superstrate layers, their transmission matrices can be written in the form of the transmission matrix of a slab with a known relative permittivity εs, and thickness d in free space, as expressed in Eq. (5):The first and the last terms represent the transmission matrices for the two interfaces of the slab, i.e. the interface from free-space into the slab and the interface from the slab into free-space, while the matrix in the middle contains the phase delay and amplitude attenuation information within the homogeneous slab. The transmission matrices for the top and bottom half-space interfaces are simply in the form of the first and the last matrices in Eq. (5), respectively, with properly chosen values for the permittivities. Following the above procedure, the scattering matrix of the metamaterial alone can be obtained, which may then be used for retrieving the corresponding effective medium parameters.2.2 Effective medium properties of free-standing fishnet
The effective medium parameters of the on-substrate multilayer fishnet nanostructure can be retrieved from its complex transmission and reflection coefficients. Due to the substrate-induced bianisotropy, the electric displacement (D) and the magnetic flux density (B) are expressed as Di = ε0εiEi + iξijHj /c and Bj = μ0μjHj – iξjiEi/c where ξij = ξji = ξ [26]. In addition, the impedances seen by waves propagating in opposite directions differ with and , resulting in different reflection coefficients on opposite sides. A bianisotropic retrieval method is adopted here [27, 28], which utilizes the reflection coefficients obtained from the two interfaces of the nanostructure, denoted as S11 and S22 (S11 ≠ S22) and the transmission coefficient S21 or S12 (S21 = S12), to retrieve the effective permittivity (ε), permeability (μ), and magnetoelectric coupling parameter (ξ). This method can also be applied for a free-standing symmetrical fishnet nanostructure, where the magnetoelectric coupling should be zero.
The geometry of the multilayer fishnet nanostructure unit cell considered here is shown is Fig. 1(b), along with its dimensions. The high frequency structure simulator (HFSS) finite element method full-wave solver was used to calculate the scattering matrix with periodic boundary conditions applied to the lateral walls of the simulation domain. A fifth order Drude-Lorentz model was employed to fit the measured permittivity of silver (Ag) [29] and the permittivity of SiO2 was chosen to be 2.25. The normally incident plane wave has its E-field polarized in the x direction. Before investigating the effects of the added substrate and/or superstrate, we first retrieve the effective medium parameters of the free-standing multilayer fishnet, as displayed in Fig. 1(c), where for clarity only the real parts are shown. A negative index band can be identified between 180 and 210 THz (i.e. 1.67 and 1.43 μm), while the magnetoelectric coupling is zero throughout the entire band due the structural symmetry. As previous studies have shown, at normal incidence, the zeroth-order Bloch mode dominates light propagation inside the fishnet nanostructure so that it can be considered effectively homogeneous [16]. An indicator of such a property is the convergence of the effective refractive index with an increasing number of functional layers [14, 30]. In Fig. 1(d), the effective index as a function of the number of functional layers (N) is presented, showing that the effective index begins to converge with N = 6, i.e. thirteen layers, which corroborates previous findings in the literature [14, 31]. In the remainder of the paper, the thirteen-layer fishnet nanostructure will be employed for studying the effect and compensation of substrate-induced bianisotropy.
2.3 Substrate effect on the multilayer fishnet
In previously reported work on substrate-induced bianisotropy in metamaterials, the substrate was considered as a semi-infinite half-space [19], which is a practical approximation because the thickness of the supporting wafer is usually on the order of more than 200 wavelengths. In Fig. 2(a) and 2(b), the scattering parameters and the retrieved effective parameters of the multilayer fishnet nanostructure on a semi-infinite substrate made of SiO2 are plotted, respectively. It is shown that both the amplitudes and phases of S11 and S22 are different, with the most significant differences occurring near the plasmonic resonances of the nanostructure due to resonance hybridization [20, 32]. While the retrieved effective permittivity and permeability exhibit similar properties compared to those of the free-standing fishnet, there is a slight frequency shift and additional small resonances due to the substrate loading. The relevant component of magnetoelectric coupling (ξ), however, becomes non-zero and shows a strong resonance around the main magnetic resonance at 180 THz (1.67 μm). In particular, when comparing Fig. 2(a) with Fig. 2(b) it is observed that the reflection phase difference and the magnetoelectric coupling parameter both have a main peak in the 185~195 THz (i.e. 1.54~1.62 μm) range and two small peaks around 130 THz (2.31 μm) and 160 THz (1.87 μm); whereas, the differences in the magnitudes of S11 and S22 can be seen as having a main peak around 160 THz (1.87 μm) and two additional smaller peaks at 130 THz (2.31 μm) and 185 THz (1.62 μm). This indicates that the property of the magnetoelectric coupling is manifested more in the difference of the phases, rather than the amplitudes of S11 and S22 as commonly stated in literature. In all, both the scattering parameters and the retrieved effective parameters corroborate previous substrate-induced bianisotropy studies [20, 21].
Fig. 2 (a) Scattering parameters of the multilayer fishnet alone on an infinite substrate. (b) Retrieved effective permittivity, permeability and magnetoelectric coupling parameter corresponding to (a). (c) Extracted scattering parameters of the multilayer fishnet alone on a 5μm thick substrate. (d) Retrieved effective permittivity, permeability and magnetoelectric coupling parameter corresponding to (c). (e) Evolution of the maximum real and imaginary parts of the retrieved magnetoelectric coupling parameter as a function of the thickness of the SiO2 substrate. The point on the right edge of the plot corresponds to the semi-infinite substrate case.
In addition to the semi-infinite substrate scenario, we further investigated the dependence of the induced bianisotropy on the thickness of the substrate. First, a 5μm thick substrate was considered, which is about 3 wavelengths thick at the frequency of the magnetic resonance. Different from the semi-infinite substrate case, a Fabry-Perot effect appears in the total scattering parameters due to the multiple reflections at the fishnet-substrate and substrate-air interfaces [15, 16] (not shown here). However, it can be seen that the scattering parameters of the metamaterial alone (see Fig. 2(c)), both in amplitudes and phases, are very similar to the case where the metamaterial is on top of a semi-infinite substrate. This results in a set of effective parameters (see Fig. 2(d)) that are similar to those of the metamaterial on the semi-infinite substrate. Hence, this implies that the induced bianisotropy caused by the substrate is a near-field effect primarily attributable to the evanescent modes, which corroborates findings on the asymmetric reflection effect due to substrate loading for chiral metamaterials [33]. In order to confirm this, the strength of the magnetoelectric coupling induced by a substrate with different thickness values was studied. In Fig. 2(e), the maximum real and imaginary parts of the retrieved magnetoelectric coupling parameter over the entire frequency range from 100 to 250 THz (i.e. 1.2 to 3.0 μm) as a function of the substrate thickness are plotted. As the substrate thickness increases from zero, the magnetoelectric coupling parameter first increases, and then starts to converge and remains at a complex value of 1.85 + i3.75. The threshold thickness is around 200nm, which is on a subwavelength scale for this specific fishnet. Hence, in terms of the strength of the induced magnetoelectric coupling, a semi-infinite substrate yields a similar effect compared to a subwavelength-thick substrate. This result is important since it will be utilized when the bianisotropy compensation technique is discussed in the following sections.
While the retrieved effective parameters reproduce the original scattering matrix, they are nonlocal in nature since they are closely related to the surface-averaged transverse fields rather than the volume-averaged fields [34, 35]. Thus, volumetric microscopic fields inside the metamaterial must be probed directly in order to confirm that the induced bianisotropy is a real physical effect. In other words, we need to quantify the electric displacement (Dx) induced by the magnetic field (Hy) and the magnetic flux density (By) induced by the electric field (Ex) over the entire volume of the multilayer fishnet. With reference to Fig. 1(a), two plane waves were used to illuminate the structure at normal incidence from both the top and bottom half-spaces simultaneously. By adjusting the magnitudes and phases of the incident waves, a standing wave pattern can be formed such that the center of the fishnet nanostructure is positioned at a zero of the magnetic or electric field [36]. In particular, a MATLAB code was employed, which takes into account the multiple reflections occurring in the system, to determine the magnitudes and phases at the two ports for the desired standing wave pattern. Due to the fact that the total thickness of the fishnet is much less than a wavelength, it is reasonable to consider that the nanostructure is excited by a pure electric or magnetic field, respectively. For each excitation, the induced EDM and MDM were evaluated using volume integrals on the microscopic fields predicted by HFSS, which are expressed by [37]
where εd and εm denote the relative permittivity of the dielectric and metal, respectively. Hence, this gives the EDM induced by the electric field (pe) or the magnetic field (pm), and the MDM induced by the electric field (me) or the magnetic field (mm). The four quantities calculated for the fishnet in free-space and the fishnet on a semi-infinite substrate, which are all normalized to the unit cell volume of the fishnet and the strength of the incident field, are shown in Fig. 3 . Hence, they are dimensionally equivalent to the polarizabilities χee, χem, χmm and χme. Comparing the free-standing and on-semi-infinite-substrate fishnet nanostructures, the quantities pe and mm show similar properties except for small frequency shifts and strength differences. The EDM induced by the electric field (pe) has its minimum value at 235 THz (1.28 μm) and 220 THz (1.36 μm) for the free-standing and on-substrate fishnets, respectively, which agree well with the ε ≈1 frequencies in the retrieved effective permittivity. The MDM induced by the magnetic field (mm) has a main peak around 195 THz (1.54 μm) and another minor peak at 130 THz (2.31 μm) for both the free-standing and on-substrate fishnets. These are also in accordance with the resonant frequencies of the retrieved effective permeability. Additionally, it can be seen that in the case of the free-standing fishnet structure, pm and me are zero throughout the entire band, meaning that no magnetoelectric coupling exists. However, with the semi-infinite substrate present, pm and me both exhibit near-zero values at low frequencies and peaks around 190 THz (1.58 μm), again in close proximity to the frequency of the magnetic resonance. Furthermore, the similarities in the line shape, peak value and peak spectral position of pm and me provide mutual confirmation of the validity of the calculated quantities because they are obtained with different excitations - one electric and one magnetic. In summary, the induced EDM and MDM calculated from the microscopic fields in the entire volume of the fishnet nanostructure clearly demonstrate that the substrate indeed induces magnetoelectric coupling even in a multilayered structure, with the strongest effect at the resonance of the plasmonic modes.Fig. 3 (a) Magnitudes of the induced electric (p) and magnetic (m) dipole moments in the free-standing multilayer fishnet under an electric (e) or magnetic (m) excitation. (b) Magnitudes of the induced electric (p) and magnetic (m) dipole moments in the multilayer fishnet on a semi-infinite substrate under an electric (e) or magnetic (m) excitation.
3. On-substrate multilayer fishnet with superstrate
In many applications, the parasitic effect of bianisotropy due to the unit cell geometry can be eliminated by employing subwavelength resonators with mirror symmetry [38, 39]. However, the bianisotropy inevitably reintroduced by the substrate is not easily removed. In this section, we consider how adding a superstrate to an on-substrate metamaterial can affect the induced bianisotropy. It will be shown that by judiciously choosing the permittivity and thickness values of the superstrate, the bianisotropy of the entire system can be greatly suppressed. First, a superstrate having the same permittivity (εr = 2.25) with that of the substrate, but different thickness values, is considered. In the extreme case when the superstrate is semi-infinite, i.e. a half space, the entire system becomes symmetrical. Under such circumstance, the reflection coefficients obtained from both sides of the metamaterial and the impedances seen by waves travelling in opposite directions are the same; thus, no magnetoelectric coupling exists for the entire system, which has been previously observed in Ref [19]. Figure 4(a) shows the maximum real and imaginary parts of the retrieved magnetoelectric coupling parameter as a function of superstrate thickness. It can be seen that both the real and imaginary parts of the magnetoelectric coupling parameter monotonically drop and converge to zero as the superstrate increases in thickness. A threshold thickness, defined as the thickness of the superstrate that causes both the Max[real(ξ)] and Max[imag(ξ)] to be below 0.02, is observed around 200nm. This corresponds well with the finite-thickness substrate study presented in previous sections, demonstrating that the strength of the bianisotropy induced by the substrate converges at a thickness of 200nm. Figure 4(b) shows the retrieved magnetoelectric coupling parameters for the fishnet on a semi-infinite substrate and the fishnet when placed under a 200nm thick superstrate. The magnetoelectric coupling parameters of these two systems exhibit approximately the same magnitudes but opposite signs, meaning that the magnetic (electric) fields induced by the incident electric (magnetic) fields of the two systems are complex conjugate to each other. For the multilayer fishnet with only the semi-infinite substrate present, in response to a y-directed incident magnetic field, uneven currents arise on the metal layers of the fishnet, which further induces a non-zero net electric dipole moment in the x-direction. At the same time, unequal currents in the x-direction induced by the incident electric field also result in a non-zero magnetic dipole in the y-direction. When the superstrate is present, the net electric dipole moment formed by the superstrate-induced uneven currents has similar amplitude but opposite direction compared to that caused by the substrate. The same behavior occurs with the magnetic dipole moment originated from the unequal currents in the x-direction produced by the superstrate alone as well. When the two systems are combined into one - a fishnet sandwiched between an infinite substrate and an appropriately designed finite superstrate - the magnetoelectric coupling of the entire system that can be seen by outside observers is cancelled out. It should be noted that at the region near the interface between the metamaterial and the substrate, the bianisotropy cannot be eliminated. However, for the entire system, the addition of the superstrate controllably introduces another source of bianisotropy to compensate that induced by the substrate. To verify that this is a real physical phenomenon, the induced EDM and MDM were calculated, as shown in Fig. 4(c). Similar to what we have observed from the retrieved effective parameters, the EDM induced by the magnetic excitation (pm) and the MDM induced by the electric excitation (me) have near-zero values throughout the entire frequency range. The MDM induced by the magnetic excitation has a main peak at around 185 THz (1.62 μm) and a small peak at 130 THz (2.31 μm), while the EDM induced by the electric excitation has a near-zero value at 200 THz (1.50 μm). Compared to the free-standing fishnet, the induced dipole moments have slight shifts in their line-shapes and variations in their values, which are attributed to the joint loading effect of the substrate and superstrate; but in all, they maintain similar properties.
Fig. 4 (a) Evolution of the maximum real and imaginary parts of the retrieved magnetoelectric coupling parameter as a function of the superstrate thickness with a permittivity of 2.25. The point on the right edge of the plot corresponds to the semi-infinite SiO2 superstrate case. (b) Retrieved magnetoelectric coupling parameter for the fishnet on a semi-infinite SiO2 substrate and the fishnet underneath a 200nm SiO2 superstrate. (c) Magnitudes of the induced electric (p) and magnetic (m) dipole moments in the multilayer fishnet sandwiched between a semi-infinite substrate and a 200nm superstrate under an electric (e) or magnetic (m) excitation.
Further studies were carried out by considering superstrates having different permittivity values from that of the substrate. In Fig. 5(a) , the maximum real and imaginary parts of the retrieved magnetoelectric coupling parameters as a function of the superstrate thickness with a permittivity value of 1.75 and 2.75 are shown. For the case where the superstrate has a permittivity of 1.75, which is smaller than that of the substrate, the magnetoelectric coupling monotonically decreases as the superstrate gets thicker, and finally converges to a certain value around 0.63 + i1.61. It is important to note that, in this case, the magnetoelectric coupling parameter is reduced, but never eliminated. This is because when the superstrate has a permittivity smaller than that of the substrate, stronger field confinement always occurs on the substrate side, regardless of the superstrate thickness. Hence, the bianisotropy and the resulting asymmetry in the reflection coefficients seen from opposite sides can only be reduced. In contrast, when the permittivity of the superstrate is 2.75, which is larger than that of the substrate, the strength of magnetoelectric coupling varies very differently with superstrate thickness. As shown in Fig. 5(a), the maximum real part of the magnetoelectric coupling parameter first drops and then grows. Finally, it converges at a value of around 0.83 + i1.58. Interestingly, when the superstrate is around 74nm thick, the magnetoelectric coupling reaches its minimum with both the real and imaginary parts less than 0.01, meaning that the magnetoelectric coupling of the system is almost eliminated, resulting in a symmetric scattering matrix of the fishnet, despite being sandwiched inside of an asymmetric electromagnetic environment. It should be noted that, in the extreme case where the superstrate becomes an infinite half-space, the magnetoelectric coupling parameter for these two cases has a different sign, which is attributed to the permittivity values either being larger or smaller compared to that of the substrate. Actually, at the optimum thickness when the superstrate permittivity is 2.75, the line shape of the magnetoelectric coupling parameter switches from a Lorentz-shaped profile to a flipped Lorentz-shaped profile. Furthermore, a superstrate with several different permittivity values larger than that of the substrate were employed to determine the optimum thickness corresponding to the elimination of the substrate-induced bianisotropy of the entire system. In Fig. 5(b), it is shown that as the permittivity value of the superstrate increases, the optimum thickness decreases. The line shape for the optimum superstrate thickness as a function of relative permittivity follows an exponentially decaying profile. It should be noted that any superstrate permittivity value greater than or equal to the substrate relative permittivity is capable of suppressing the magnetoelectric coupling magnitude down to less than 0.01, all while possessing a subwavelength thickness. This is particularly important from the experimental viewpoint because a subwavelength-thick superstrate coating can be easily fabricated on top of a metamaterial to eliminate the bianisotropy of the entire system. Moreover, as an added practical benefit, this ultrathin dielectric superstrate can also serve as a protective layer to keep the otherwise exposed top metallic layer from oxidizing and corroding over time.
Fig. 5 (a) Evolution of the maximum real and imaginary parts of the retrieved magnetoelectric coupling parameter as a function of the superstrate thickness with a permittivity of 1.75 and 2.75, respectively. The point on the right edge of the plot corresponds to the infinite substrate case. (b) Optimum superstrate thickness as a function of the permittivity value of the superstrate.
4. Conclusion
In conclusion, we have proposed an effective and practical way to fully compensate for the undesirable substrate-induced bianisotropy commonly encountered in optical metamaterials. As an example, the experimentally important multilayer fishnet nanostructure was considered. First, with the aid of a cascaded transmission matrix approach, the effect of a finite substrate with varying thickness values on the induced bianisotropy was studied, showing that substrate-induced bianisotropy is a near-field effect. Superstrates with different permittivities and thickness values were then investigated for the on-substrate metamaterial. It was shown that by properly choosing the permittivity value for the superstrate, which needs to be larger than that of the substrate, along with an appropriate subwavelength thickness, the substrate-induced bianisotropy of the entire system can be greatly suppressed or even eliminated. In addition to the retrieved effective medium parameters, induced EDM and MDM were calculated using the volumetric microscopic fields, confirming the bianisotropy compensation to be a real physical effect. The ultrathin superstrate not only provides a means for system bianisotropy suppression seen by outside observers, but also can serve as a protective coating layer. This work will be beneficial to achieving future optical metamaterial designs with bianisotropic-free properties thereby facilitating their implementation into practical devices.
Acknowledgments
This work was supported by a NSF MRSEC under Grant DMR-0820404. The authors acknowledge fruitful discussions with D. Brocker.
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