The optical properties of plasmonic crystals consisting of triangular lattices are theoretically investigated using rigorous coupled-wave analysis. Two types of structures were analyzed, one composed of an array of short cylindrical pillars on a flat metal surface and the other composed of an array of shallow cylindrical holes formed in a flat metal surface. The dispersion relations and radiation properties of the second and the third bands around the Γ point in the first Brillouin zone were investigated. We found these properties to be highly dependent on the radii of the cylindrical pillars and holes relative to the lattice constant. We also examined the influence on the dispersion relations and radiation properties of the deviation of the cross-section of the pillars and holes from a perfect circle.
© 2012 Optical Society of America
Plasmonic crystals is the name of surface-relief structures two-dimensionally and periodically formed on a metal surface. Surface plasmons have their energy confined in the direction normal to metal–dielectric interfaces but propagate along the interfaces. However, surface plasmons can be confined also in the lateral direction with plasmonic crystal structures . Kitson et al.  experimentally confirmed the existence of a full plasmonic band gap in which no surface plasmon is allowed to exist. At the bandgap edges, surface plasmons do not propagate but remain as standing waves. Since the dispersion curves of the surface plasmon modes around the first (lowest-energy) band gap are located outside the light cone, the surface plasmons do not couple to free-space radiation. Bozhevolnyi et al.  employed this band-gap effect for plasmonic waveguides. Besides this first plasmonic band gap, higher plasmonic band gaps also exist. The second plasmonic band gap plays an important role, especially in light-emitting devices [4–8]. Because the surface plasmon modes around the second band gap lie inside the light cone, they couple to free-space radiation.
The dispersion relations and radiation properties of one-dimensional plasmonic crystals have been investigated in depth [9–18]. However, few papers have been published regarding theoretical investigation of two-dimensional plasmonic crystals. Kretschmann and Maradudin  calculated the dispersion relations of plasmonic crystals composed of a two-dimensional array of metallic semiellipsoidal bumps on a metal surface, using the Rayleigh-hypothesis approximation. Baudrion et al.  also calculated the dispersion relations of plasmonic crystals composed of two-dimensional arrays of metallic parallelepipeds on metal surfaces, using a differential method. Both groups calculated dispersion relations outside the light cone, where surface plasmons do not couple to free-space radiation. They observed the first plasmonic band gap; however, they did not observe the second or higher band gaps, which lie inside the light cone. The dispersion relation behavior of plasmonic crystals inside the light cone is quite important, as mentioned above. Investigation of the optical properties of surface plasmons in this region is required for applications to light extraction in organic light-emitting diodes and plasmonic band-gap lasers . In an organic light-emitting device, the energy of excitons, which are generated by current injection, is directly transferred to the surface plasmons supported in a metallic cathode and can be extracted by using a plasmonic crystal structure [4, 6–8]. In a plasmonic band-gap laser, the coupling efficiency between surface plasmons and free-space radiation is directly related to the loss of the lasing mode, and the slope of the dispersion curves is also important as it gives the mode density [5, 17, 18].
In this paper, we investigate the dispersion relations of two-dimensional plasmonic crystals with triangular lattice structures and the coupling characteristics between the surface plasmons supported by the plasmonic crystals and free-space radiation by using a numerical coupled-wave analysis. Then, we discuss some criteria for designing a plasmonic crystal structure for organic light-emitting devices and lasers.
2. Model and analysis method
The plasmonic crystals we investigated are illustrated in Figs. 1(a) and 1(b). Cylindrical metal pillars or cylindrical air holes compose a triangular lattice on a flat metal substrate of semi-infinite thickness. The metal pillars and the metal substrate are of the same material. We call the former structure shown in Fig. 1(a) a positive plasmonic crystal, and the latter shown in Fig. 1(b) a negative plasmonic crystal. The radius and the height of the cylinders are denoted by r and d, respectively. The lattice constant Λ is defined as shown in Fig. 1. Thus, the center to center separation of the cylinders is given by in all three directions. A linearly-polarized monochromatic plane-wave is incident on the surface of the plasmonic crystal. The polar angle and the azimuthal angle of incidence are θ and ϕ, respectively.
Figure 1(c) shows the dispersion relation of a plasmonic crystal with a triangular lattice structure whose modulation depth is infinitesimal. In this dispersion relation, we assumed an ideal Drude metal with plasma frequency ωp and reciprocal lattice vector K = 2π/Λ = c/2ωp, where c is the speed of light in vacuum. In the case of the infinitesimal modulation depth, only the coupling between radiation and surface plasmons takes place, so that no band gaps are obtained. On the other hand, when the modulation depth has a finite value, the coupling between surface plasmons and surface plasmons additionally takes place. The latter coupling causes plasmonic band gaps.
We investigated the dispersion relation and coupling characteristics of plasmonic crystals with finite modulation depth in the shaded region shown in Fig. 1(c). For the calculation, we used rigorous coupled-wave analysis (RCWA, also known as the Fourier modal method), which is well suited for analyzing binary metallic gratings. Initially, RCWA for two-dimensional gratings was just extended from that for one-dimensional gratings [21, 22] but suffered from convergence problems. Later, Li  found the correct rules for Fourier factorization, which has greatly improved the convergence speed. Thus, we used the formulation of Li .
Silver was our material of choice since it exhibits the smallest absorption loss in the visible region. The dielectric function of silver was obtained from Johnson and Christy . We fixed the lattice constant and the modulation depth at Λ = 600 nm and d = 20 nm, respectively. The dielectric constant of the ambient was assumed to be unity.
The number of diffraction orders taken into account in the calculations was 1681, corresponding to 41 diffraction orders (between −20 and +20) in each direction of two reciprocal lattice vectors. This was found to be enough to provide both reasonable accuracy and short calculation times. Even when we included more than 1681 diffraction orders, the qualitative behavior of the dispersion relation did not change.
3. Results and discussion
We obtained the dispersion relations of surface plasmon modes by calculating the absorptance as a function of the energy E and the in-plane wave vector k|| of the incident plane wave. Figure 2 shows the calculated absorptance for positive plasmonic crystals with various pillar radii. Figure 3 shows the calculated absorptance for negative plasmonic crystals with various hole radii. In both figures, red represents p-polarized incidence, and blue represents s-polarized incidence. The absorptance is normalized in each figure. Higher brightness indicates higher absorptance. The dispersion relation is represented by the locus of the absorptance maxima.
In the small-modulation-depth limit, there are apparently four modes in the Γ-M direction and three modes in the Γ-K direction in the calculated region shown in Fig 1(c). In Fig. 2 and Fig. 3, however, there are six modes in both directions, because the finite modulation depth resolves the degeneracy. In the Γ-M direction, four modes couple to p-polarized radiation, whereas two modes couple to s-polarized radiation. On the other hand, in the Γ-K direction, three modes couple to p-polarized radiation, whereas three modes couple to s-polarized radiation. In order to identify these modes, we assign symbols P1–P7 to the modes that couple to p-polarized radiation and symbols S1–S5 to the modes that couple to s-polarized radiation, as shown in Fig. 2. The pairs (P2, S1), (P3, S2), (P5, S3), (P6, S4), and (P7, S5), which are degenerate in the small-modulation-depth limit, are resolved in these figures. One mode of each pair always couples to p-polarized radiation, whereas the other always couples to s-polarized radiation. The reason for resolving the degeneracy and for the coupling property between surface plasmons and radiation can be easily explained with wave-vector diagrams, as shown in Fig. 4.
Figure 4(a) shows a wave-vector diagram in the case where the in-plane wave vector k of the radiation is parallel to the Γ-M direction. Vectors K1 and K2 are the reciprocal lattice vectors of the plasmonic crystal. The hexagon indicates the first Brillouin zone. The surface plasmon of wave vector ki (i = 1, 2,..., 6) couples to the radiation when the following equation is satisfied:Figure 4(b) shows a wave-vector diagram in the case where the in-plane wave vector of the radiation is parallel to the Γ-K direction. In this case, three pairs are degenerate when the grating amplitude is infinitely small, since |k1| = |k2|, |k3| = |k6|, and |k4| = |k5|. As the amplitude of the grating increases, each of these plasmon-mode pairs splits into an odd mode and an even mode for the same reason as above, and the degeneracy is resolved.
Next, let us consider coupling between free-space radiation and surface plasmons. At the Γ point, i.e. k = 0 in Figs. 2 and 3, some modes exhibit high absorptance, whereas other modes exhibit almost no absorption. Due to the reciprocity, this indicates that some surface plasmon modes do not couple to radiation, i.e. are nonradiative, at the Γ point. For the positive plasmonic crystals with metallic pillars of small radii, e.g., r/Λ = 0.200, the upper two modes are radiative, whereas the other four modes are nonradiative. When the radius is relatively large, e.g., r/Λ = 0.400, the bottom two modes are radiative. When the radius is r/Λ = 0.289 or 0.305, the middle two modes are radiative. In the negative plasmonic crystals, the dependency of the coupling on the radius of air holes is almost opposite to that in the positive plasmonic crystals. For small radii, the bottom two modes are radiative, whereas for large radii, the upper two modes are radiative. In one-dimensional plasmonic crystals, the coupling properties around k = 0 are easily explained [12,15,17]. A similar theory might be applicable to two-dimensional plasmonic crystals to explain their coupling properties, although it might be much more complicated. Currently we have not yet succeeded in explaining all of the coupling properties. Further study is required.
It is difficult to observe the behavior of the plasmonic band gap from Figs. 2 and 3 because the dispersion curves for some modes are missing around the Γ point due to their nonradiative properties. Thus, we employed the Kretschmann geometry and calculated the absorption of internally reflected waves outside the light cone to reveal the dispersion curves. Figure 5(a) shows a schematic diagram of the assumed configuration. The assumed maximum thickness of the silver film is 60 nm. The back side of the silver film was a plane, on which a semiinfinite-thick glass substrate whose refractive index is assumed to be n = 1.5 is attached. A plane wave was incident from the glass substrate side. Figures 5(b) and (c) show the calculated absorptance mapped onto a two-dimensional space spanned by energy and cylinder radius for the positive and the negative plasmonic crystals, respectively. In the calculations, the in-plane wave vector of the incident plane wave was fixed at k|| = (2π/Λ, 0), where the dispersion relations of the plasmonic crystals are identical to those at k|| = 0. When the radius of the cylinder is r/Λ = 0.05, there is apparently only one mode in both the positive and negative plasmonic crystals. As the radius increases, the mode splits into three modes. Two of the three modes couple to both p- and s-polarized radiation, whereas the other mode couples only to the p-polarized radiation. No modes couple to s-polarization for small cylinder radius. In the negative plasmonic crystals, the energies of the modes are less sensitive to the radius than those in the positive plasmonic crystals. However, the dependences of the mode energy on the radius for both plasmonic crystals are qualitatively the same, when flipped with respect to a horizontal line. For the positive plasmonic crystals, the widest band gap opens at r/Λ ∼ 0.2, whereas for the negative ones, the widest band gap opens at r/Λ ∼ 0.45.
The fundamental band structure of plasmonic crystals may be identical to that of two-dimensional photonic crystals or photonic crystal slabs, when the modulation depth is infinitely small. Thus, it is worth comparing the band structures of plasmonic crystals with those of two-dimensional photonic crystals. In two-dimensional photonics crystals, both TE and TM modes exist, whereas in plasmonic crystals, only TM modes survive. The band structures of both crystals look similar to each other. However, we found one difference between them: At the Γ point, there are generally four different resonant energies in the calculated energy region both for TE and TM modes in two-dimensional triangular photonic crystals [25, 26]. In plasmonic crystals, on the other hand, there are apparently only three different resonant energies, as shown in Fig. 5.
Finally, we investigated the dispersion relations and the radiation properties of plasmonic crystals composed of a non-circular positive pillar array. We assumed elliptic pillars whose major axis is tilted by 15 degree from the x-axis, as shown in Fig. 6. The long axis is , and the ratio of the short axis to the long axis is b/a = 0.5. The crystal structure has no symmetric axis. The calculated absorptance maps for various planes of incidence are also shown in Fig. 6. Maps for ϕ = 0°, 60°, and 120° correspond to the Γ-M direction, whereas maps for ϕ = 30°, 90°, and 150° correspond to the Γ-K direction. Both the band structures and the radiation properties depend on the azimuthal angle of the plane of incidence, even in the same Γ-M or Γ-K directions. For plasmonic crystals with circular pillars, each mode exclusively couples to p- or s-polarized radiation when the modulation depth is finite. In the plasmonic crystals with tilted elliptic pillars, on the other hand, some modes simultaneously couple to both p- and s-polarized radiation.
We have proposed two applications using two-dimensional plasmonic crystals so far. One is the plasmonic band-gap laser, in which surface plasmons are amplified and radiated as laser light by the assistance of a gain medium . The other is organic light-emitting diodes (OLEDs). The energy of the surface plasmons in the metallic cathode surface, which are directly excited by electrically excited excitons, is extracted as radiation using a plasmonic crystal structure constructed on the cathode, resulting in improved external quantum efficiency [7, 8]. For these applications the dispersion relations at the Γ point are crucial.
For lasers, the mode density, which is inversely proportional to the group velocity, must be high to lower the lasing threshold. At the same time, the radiation must be minimized to reduce the loss. Taking account of these requirements, the pillar radius must be in the range of 0.289 ≤ r/Λ ≤ 0.352 for the positive plasmonic crystals, where the upper three modes are degenerate and nonradiative, and the hole radius must be r/Λ ≥ 0.433 for the negative plasmonic crystals, where the lower three modes are degenerate and nonradiative. Multiple degeneracy also increases the mode density. In order to maximize the mode density, the cross-section of the pillars must be circular or at least have C6 symmetry. Holographic lithography techniques are often used to fabricate plasmonic crystals. To make a triangular lattice, the double-exposure technique combined with the above technique is used [5, 7, 27]. In the process, samples are rotated 60° around their normal axis between two successive exposures. Using this technique, however, the shape of the obtained pillars is not circular. Triple-exposure may be required to fabricate arrays of approximately circular pillars.
On the other hand, surface plasmons must be radiative in order to improve the light extraction efficiency in OLEDs. The larger the number of radiative modes is, the higher the light-extraction efficiency becomes. Based on this requirement, we conclude that the best radius is r/Λ = 0.330 for the positive plasmonic crystals and r/Λ = 0.352 for the negative plasmonic crystals. Under these conditions, three modes are radiative.
We have investigated the band structures and the radiation properties of two-dimensional plasmonic crystals with triangular lattice structures composed of cylindrical pillars and air holes. We revealed the complex band behavior and the radiation properties, which depend on the relative cylinder-radius with respect to the grating pitch, especially around the Γ point. There are always three apparent modes, some of which are degenerate at the Γ point. The number of apparent modes at the Γ point is different from that in photonic crystal slabs with the same lattice structure, which exhibit four apparent modes at that point. All three apparent modes couple to p-polarized radiation, whereas only two of the three apparent modes couple to s-polarized radiation. We also revealed the effect of the shape of the cylinders that compose the lattice of the plasmonic crystals. Deviation of the cylinder cross-section from a perfect circle makes the dispersion relation more complicated and reduces the mode density.
The calculations were performed by using the RIKEN Integrated Cluster of Clusters (RICC) facility. This research was partially supported by the Ministry of Education, Culture, Sports, Science, and Technology, Grant-in-Aid for Scientific Research (B), 17360033, 2005.
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