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Resonant electromagnetic fields in stacked complementary plasmonic crystal slabs (sc-PlCSs) are numerically explored in subwavelength dimensions. It is found that the local plasmon resonances in the sc-PlCSs are composite states of locally enhanced electric and magnetic fields. Two sc-PlCSs are analyzed in this paper and it is shown that each sc-PlCS realizes a resonant electromagnetic state suggested by one of Maxwell equations. It is moreover clarified that the local plasmons open efficient paths of Poynting flux, those result in high-contrast polarized transmission.
©2010 Optical Society of America
1. Introduction
Plasmonic resonances in simple metallic nano-structures like gratings, spheres, and rods have been extensively examined to date [1–3]. Plasmonic crystals of periodic metallic nano-structures also show intriguing features such as extraordinary transmission [4] and negative refraction [7] at optical frequencies. Before finding the appealing optical features, substantial efforts were devoted to clarify electronic states, i. e., dispersions and dielectric function responses in the superlattice structures including metals [5, 6], which were limited to simple structures such as gratings.
Most of plasmonic crystal slabs (PlCSs) reported so far are composed of single periodic layers or repeated stack of bi-layers. To explore diversity in plasmonic crystals, there are many ways. One of the ways is to design depth profile of PlCSs, for example to produce helical structures [8,9]. Another way is to study the stacked PlCSs composed of different unit cells [10–19]. In the relatively complicated PlCSs, non-trivial eigen modes are expected because of the low symmetry. One of the actual examples is stacked complementary PlCSs (sc-PlCSs), those can be fabricated in a simple procedure and show strong polarization dependence in transmission [19]. The feature in transmission spectra was experimentally and numerically demonstrated, and implied local plasmonic resonances. The physical insight in sc-PlCSs, however, has not been well revealed so far.
In this paper the electromagnetic (EM) field distributions are evaluated in a good precision by finite element method. The computation is exploited to make the linear response of plasmonic resonances clear. It is clarified that the resonant EM state resulting in high transmission forms the elaborate EM fields enhanced locally. Two of sc-PlCSs are analyzed here in details: one has butterfly structures in the unit cell, and the other has II structures. The two PlSCs are found to exhibit different subwavelength EM dynamics. The former is ascribed to one of Maxwell equation, ∇ × H = ∂ D/∂t where H is magnetic field vector and D is electric displacement vector. The latter is attributed to another Maxwell equation, ∇ × E = −∂ B/∂t where E is electric field vector and B is magnetic density-flux vector. The two sc-PlCSs thus turn out to be representatives of the typical resonant EM distributions associated with Maxwell equations. It is furthermore shown that both sc-PlCSs realize the efficient flow of EM power flux in the subwavelength dimensions.
Fig. 1. (a) Schematic drawing of sc-PlCS of butterfly-shaped structures. Gray represents metal (silver), white meas air, and pale blue stands for dielectric (EB resist). (b) The unit cell extracted from (a). The second layer of EB resist and air hole is omitted for clarity (dashed lines). (c) The unit cell prepared to execute the numerical implementation. The dimension is given in nm. The edges of metallic parts are drawn with blue lines. (d) Transmittance spectra, evaluated numerically, under normal incidence on the xy plane. Red solid line corresponds to the spectrum under y polarization and black dashed line to x polarization. (e) The ratio of transmittance, Ty/Tx.
2. sc-PlCS of butterfly shape
Figure 1(a) presents schematic drawing of sc-PlCS of butterfly-shaped structure in the unit cell. The coordinate is set as shown; periodic structures are arrayed in the xy plane, and the layers are stacked along the z axis. The sc-PlCS is composed of the three periodic layers: the first layer is perforated metallic silver films (gray), the second is perforated dielectric film (pale blue), and the third consists of the dielectric film embedded by butterfly-shaped metal. The unit cells of first and third layers are extracted in Fig. 1(b). The two layers are complementary to each other in the metallic structures. The second layer corresponding to the dashed lines is omitted for clarity. In accordance with the experimental condition [19], the dielectric film was set to be electron-beam (EB) resist with permittivity of 2.3716 at the wavelength range of present interest. The third layer is attached to quartz substrate (not drawn). The whole configuration is taken into account in the following numerical evaluations.
Figure 1(c) displays the unit cell in the numerical calculations for transmission spectra and EM fields distributions. The dimensions of the unit cell are given in nm. The edges of metallic parts are shown with blue lines. The thickness of the first and third layers is 50 nm, and that of the second layer is 150 nm. The precise sizes of the butterfly shape in the xy plane were already reported in Ref. [19].
The butterfly (or fat H) shape is similar to so-called bowtie structure. We here referred that there are many reports on the single-layered PlCSs including bowtie structure. The local electric field enhancement in the nm gap was mainly investigated [20–22] and the resonant EM fields were also reported [21, 23]. The EM distributions in more complicated unit cells are examined here.
Fig. 2. EM fields in butterfly-shaped PlCS at a moment. (a) Electric field distributions under y-polarized incident plane wave. Color plot on the slice is the intensity of electric fields at the center of the unit cell. The unit of the intensity is defined by setting the input power flux to be 1 at the input xy square. Purple cones are electric field vectors. Black arrows indicates the propagation direction of the incident plane wave on the slice. (b) Magnetic field distributions displayed in a similar way to (a). Red cones stand for magnetic field vectors. (c) Poynting flux on resonance under y polarization. The slice corresponds to those in (a) and (b). White arrows represent Poynting vectors. (d) Off-resonant Poynting flux under x polarization. The slice is taken in the way same with (c).
Figure 1(d) is calculated transmittance spectra under normal incidence on the xy plane. Red solid line corresponds to y polarization and black dashed line to x polarization. The spectra was computed by the Fourier modal method incorporating improved algorithm in numerical convergence [24,25]. The permittivity of silver was taken from literature [26] and that of quartz substrate was set to be 2.1316. Figure 1(e) is the ratio of transmittance Ty/Tx. At the wavelength range longer than 1500 nm, the extinction ratio is larger than 10 and indicates polarization-selection of transmission in the sc-PlCS.
Fig. 3. (a) Color plot of time-averaged energy-density loss in the metallic part of the third layer. Magnetic field vectors (red arrows) are plotted around the metallic part in the lower y range. (b) Schematic diagram of subwavelength EM dynamics in the unit cell of butterfly-shaped air hole and metal.
Figures 2(a) and 2(b) respectively show electric and magnetic field distributions on resonance in the sc-PlCS of Fig. 1; the wavelength is 1650 nm [Ty peak in Fig. 1(d)] and the incident plane wave is y-polarized. Incident plane wave starts at the xy square and travels from left to right. In Fig. 2, the phase of incident EM wave is defined by setting ϕ = −π/2 for Ey(ϕ) = sin(ϕ) at the input square. The definition determines the moment shown in Fig. 2. The intensities of electric and magnetic fields are shown as color plot on the central slice parallel to the xz plane in the unit cell. Numerical fluctuations were suppressed within 10−4 for electric fields and the precision was kept to be good. Purple cones in Fig. 2(a) are electric fields vectors on the slice. Figure 2(a) presents the electric field enhancement at the air hole in the first layer. On the other hand, Fig. 2(b) exhibits rotatory flow of the magnetic field (red cones) around the metal at the third layer. By the combination of the electric and magnetic field distributions, the Poynting flux (the vectors designated by white cones and the intensity by color) travels through the unit cell as shown in Fig. 2(c); the slice is same with those in Figs. 2(a) and 2(b). The ratio of the input Poynting flux to the transmitted flux is transmittance. The value is consistent with the value in Fig. 1(c).
The largest domains in Figs. 2(a) and 2(b) are the unit cells in the computation by finite element method [27]. Periodic boundary conditions are assigned on the xz and yz planes. The Poynting flux at the input square parallel to the xy plane is set to be 1 in the arbitrary units. It is assumed that the incident layer consists of air.
Figure 2(d) is the Poynting flux off resonance under x polarization at the wavelength same with Fig. 2(c). It is clearly observed that the flux heavily decreases at the unit cell. The transmittance is far smaller than that in Fig. 2(c). The result is consistent with the extinction in Fig. 1(d). It seems that the flux in Fig. 2(d) is too small, compared with the input flux that is unity. We note that the flux by color plot contains all the EM waves, that is, incident and reflected waves coexist in the incident layer; therefore, the sum of the flux can be seen to be reduced because of the cancellations of incident and reflected waves.
Figure 3(a) shows time-averaged EM energy-density loss (color plot) in the metal on the resonant condition in Figs. 2(a)–2(c). The color plot indicates negative value, which means that EM energy is getting lost. In other words, EM energy is absorbed in the metal. Magnetic field vectors (red cones) are also shown around the metal at the third layer in the y range lower than the center.
Figure 3(b) depicts the EM dynamics in the subwavelength dimensions, summing up the results of EM field distributions. Electric field (purple arrow) is locally enhanced at the aperture structure in the first layer and simultaneously magnetic field (red arrows) forms the rotatory flow around the metal, associated with the loss of EM energy at the side wall of the metal part in the third layer. The loss directly originates from one of the Maxwell equation in metallic parts:
Fig. 4. (a) Schematic drawing of the unit cell of II-shaped air holes. Extracted the unit cell in a similar way to Fig. 1(b). The second layer (dashed lines) of EB resist and II-shaped air holes is omitted for clarity. (b) The unit cell for the numerical implementation, similar to Fig. 1(c). (c) Detailed dimensions in the unit cell, shown in the top view. The unit is nm. (d) Transmittance spectra under x polarization (black dashed line) and y polarization (red solid line). Note that Ty is almost zero. (e) Ratio of transmittance, Tx/Ty.
where ε 0 is the permittivity in vacuum and ε m is relative permittivity of metal. The right-hand side of Eq. (1) has time derivative of complex value due to ε m; the imaginary part gives rise to the exponential decay of the harmonic solutions of Maxwell equations. On the basis of this absorption mechanism, we point out that the relative large absorption takes place at the narrow side walls, where the rotation of magnetic field certainly takes relatively large values in comparison with wider face parallel to the xy plane. In the case of the sc-PlCS of butterfly shape, the absorption loss is about 20%. The rest of EM waves is observed as transmission and reflection. The electric field enhancement and rotatory magnetic field are combined, and form the local plasmons under y polarization at 1650 nm.
3. sc-PlCS of II shape
Figure 4(a) is a schematic unit cell of II structure. The second layer is omitted to see the third layer, similarly to Fig. 1(b). Also metallic parts (gray) are assumed to be silver and dielectric parts (pale blue) are EB resist. Figure 4(b) is the drawing of the unit cell, that was employed in executing the numerical calculations. The edges of metal parts are represented with blue lines. The thickness of each layer is same with the sc-PlCS of butterfly structure in Fig. 1. This sc-PlCS is also assumed to be grown on the quartz substrate and the third layer is stick to the substrate. Figure 4(c) provides the detailed dimensions in the xy plane; the unit is nm.
Fig. 5. EM fields in II-shape PlCS at a moment. (a) Color slice of the electric-field intensity under x polarization. The unit of the intensity is defined similarly to Fig. 2. Purple cones are electric field vectors on the color slice. Black arrows indicates the propagation direction of the incident plane wave. (b) Magnetic fields, represented in a similar way to (a). (c) Poynting flux on resonance under x polarization. The slice is taken similarly to (a) and (b). White arrows represent Poynting vector. (d) Off-resonant Poynting flux under y polarization. The slice is taken in the way same with (c).
Figure 4(d) shows transmittance spectra under normal incidence. Black dashed line represents x-polarized transmittance and red solid one does y-polarized transmittance. There is a peak at 1680 nm only under x polarization, that strongly suggests local plasmonic resonance. On the other hand, y-polarized transmittance is very low and close to zero; almost all the incident plane waves are reflected. Numerical details in the evaluation are same to Fig. 1(d).
Extinction spectrum is shown in Fig. 4(e) and presents high-contrast polarized transmission. It is to be stressed that the extinction takes a large value more than 104 at 1650 nm. This sc-PlCS is a good candidate for plasmonic polarizers of subwavelength thickness.
Figures 5(a) and 5(b) respectively display snapshots of electric and magnetic field distributions on resonance at 1650 nm, x polarization. The incident plane wave comes from the left-hand side, travels through air layer, and sheds on the first layer of the sc-PlCS. The phase of incident EM wave is defined by setting ϕ = 0 for Ex(ϕ) = cos(ϕ) at the incident square. Apart from the metallic edges, electric field is significantly enhanced in the 100 nm slit between the metallic parts in the third layer. In addition, it is seen that electric field vectors change the sign for the z axis at the periodic boundary parallel to the yz plane. This means that non-zero rotatory flow of electric field exists. As for the magnetic field, it is obviously enhanced near the yz periodic boundary. The setup in the numerical calculation is in common with the former case in Sec. 2.
Fig. 6. Schematic diagram of the whole subwavelength EM dynamics on resonance under x polarization, summing up Fig. 5(a)–(c). Blue arrows indicate the flow of Poynting flux. Purple arrows are electric field vectors and flows. Red arrows present magnetic field vectors.
Figure 5(c) shows that local Poynting flux increases at the air slit in the first layer and that the enhanced magnetic field at the second layer contributes the efficient flow of Poynting flux; the Poynting vectors are shown with white cones. In contrast, the flow cannot go through the metallic narrow slit at the third layer; the enhanced electric field between the narrow slit hardly assists the Poynting flux.
Figure 5(d) presents the Poynting flux off resonance at 1650 nm, y polarization. Incident wave is almost stopped at the first layer and the portion of transmission is very close to zero. This corresponds to the high-contrast transmission in Fig. 4(d).
Figure 6 symbolically depicts the EM dynamics on resonance in Fig. 5. Blue arrows represent the Poynting flux, purple arrows and curves express the electric fields, and red arrows stand for the enhanced magnetic fields. The strong flux at the first layer is realized because the rectangular air slits long along the y axis serve as waveguides for x polarization. The flux efficiently propagates through the second and third layers via the enhanced magnetic fields. The enhanced magnetic field is direct consequence of another Maxwell equation:
where μ 0 is permeability in vacuum. Both rotatory flow of electric field and the spatial gradient contribute to increase the left-hand side in Eq. (2). In non-magnetic media, the relative permeability is real and unity. Equation (2) implies that magnetic field can be enhanced locally in subwavelength structures and tells that simple electric field enhancement is not enough to realize enhanced magnetic field. Indeed the enhanced electric field at the third layer is not associated with the enhancement of magnetic field.
4. Discussions
The two sc-PlCSs have been examined so far. The two show a similar optical functionality concerning polarized transmission and realize about 50% transmittance. The two are however different in the subwavelength EM dynamics: the sc-PlCS of butterfly structure has the EM dynamics connected to one of Maxwell equation [Eq. (1)], and the sc-PlCS of II structure is associated with another Maxwell equation [Eq. (2)]. Thus the two sc-PlCSs realize the representatives of subwavelength EM dynamics that Maxwell equations imply.
As for local EM field distributions, the enhancement factor is one of the criterions to estimate resonant effects in plasmonic crystals. Although the two sc-PlCSs analyzed here are not optimized to achieve enhanced fields, we evaluate the enhancement factors of both electric and magnetic fields. For comparison, the electric and magnetic fields in air were numerically computed under the condition same to Figs. 2 and 5. The incident Poynting flux per the input square was set to be unity, and then the intensities of electric and magnetic fields were 27 and 0.073, respectively.
The intensities of electric and magnetic fields are read out from Figs. 2 and 5. In the sc-PlCS of butterfly structure, the intensity of electric field is taken at the center of the air hole in the first layer where the intensity takes the maximum in Fig. 2(a), and the intensity of magnetic field is taken at the yz-wall of metallic part in the third layer where the intensity is close to the maximum in Fig. 2(b). In the sc-PlCS of II structure, the intensity of electric field is taken at the center of narrow slit between the metallic parts, and the intensity of magnetic field is taken at the yz-boundary of the unit cell where the intensity becomes maximum in Fig. 5(b). Note that the intensities taken here are not always the maximum in the unit cell. The maximum is often realized at the sharp edge of metallic parts; it is however very sensitive to the shape in the sub-nm scale and difficult to reproduce the situation in experiment. Also, to execute the precise evaluation may be beyond the limitation of Maxwell equations based on the local response of materials. We thus avoid to pursue the maximum at the sharp metallic edge. Instead, the positions where field enhancement is qualitatively confirmed in Figs. 2 and 5 are selected. The enhancement factors defined by the ratio of intensities in sc-PlCSs to those in air are listed in Table 1.
The factors in Table 1 show a-few-times field enhancement. In the sc-PlCS of II structure both electric and magnetic fields are enhanced. On the other hand, magnetic field is more enhanced in the sc-PlCS of butterfly structure. This result suggests a possibility that one can obtain enhanced electric and magnetic fields independently. As for the Poynting flux, the enhancement factors are 1.6 and 15 in Figs. 2(c) and 5(c), respectively. Local EM flux can become large by more than ten times. It will be a future issue to explore the designs of PlCSs of high enhancement factors on resonance.
At the end of discussion, it may be interesting to point out the possibility to tune resonant local-plasmon frequencies at different frequency range, for example, the visible or far infrared range. Plasmonic resonant wavelengths are not scaled in proportional to the spatial size because the permittivity of metal significantly depends on the wavelengths. It is consequently necessary to modify the unit-cell structures of PlCSs in nonlinear ways in order to move the resonant frequencies to other ranges. This is a different property from the dielectric photonic crystals which are assumed to be composed of material with constant permittivity. However, the subwavelength EM dynamics shown here are the general features coming directly from Maxwell equations; therefore, similar resonant local plasmons can be realized in the visible, far infrared, and microwaves.
5. Conclusions
EM field distributions in sc-PlCSs have been numerically examined under the resonant condition that polarization selection in transmission manifests itself. Local plasmon resonances have been described by use of the subwavelength EM diagram in the unit cell. The diagrams provide intuitive expressions of local plasmon resonances and the mechanism of the high transmission. It has been also found that the two sc-PlCSs analyzed here have different features in the EM dynamics: each is a realization of local plasmonic states implied by different Maxwell equations. Local field enhancement was evaluated on resonance. It was shown that the resonances have more than a-few-times enhancement in the field intensity and can have 15-times enhancement in the Poynting flux.
Acknowledgements
The author would like to thank A. Takahashi (SGI Japan, Ltd.) for the support in numerical implementations and H. T. Miyazaki (NIMS) for discussions. This study was also supported in part by Cyberscience Center, Tohoku University and by KAKENHI from JSPS, Japan.
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