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We propose a novel waveguide structure using a hetero-metal film which is composed of two metals of different plasma frequencies. In the proposed waveguide, a long-range surface plasmon-polariton (LR-SPP) mode in the central metal film of a higher plasma frequency are horizontally confined since no propagation mode is allowed in the outer films of a lower plasma frequency for a certain frequency range which is dubbed a plasmonic mode-gap (PMG). The propagation characteristics of the proposed PMG waveguide are numerically analyzed. The proposed waveguide shows tight horizontal confinement close to the diffraction limit and notable suppression of radiation loss in bendings due to the PMG effect. It seems that the PMG guiding can improve integration densities of optical devices based on the LR-SPPs without exacerbating their propagation losses.
©2010 Optical Society of America
1. Introduction
Surface plasmon-polariton (SPP) guided at the interface between dielectric medium and metal has attracted a lot of interest due to their confinement of light beyond the diffraction limit [1]. The SPP shows a unique dispersion characteristic that the highest reachable frequency is limited no matter how large wave-vector it has. This enables the localization of the SPP beyond the diffraction limit and opens huge opportunity to subwavelength optics [2]. The confinement of the SPP at the dielectric-metal interface can be enhanced further or loosened from coupling of two SPP modes in metal-dielectric-metal (MDM) and dielectric-metal-dielectric (DMD) structures [3–5]. Especially, two types of super-modes of even and odd symmetries formed in the DMD structure show opposite characteristics. The odd super-mode shows enhanced confinement as the metal layer gets thinner. The confinement of the even super-mode gets looser for the thinner metal layer and hence, its propagation length increases. Therefore, the even super-mode is called a long-range SPP (LR-SPP).
Since those planar SPP-supporting structures composed of dielectric-metal, MDM, and DMD provide field confinement only in one dimension, an additional means to confine light in the direction parallel to the metal surface is needed to implement two-dimensional confinement. Various SPP-based waveguide structures have been proposed with different additional confinement means. In waveguide structures such as metal ridge waveguides [6], metal heterowaveguides [7], and MDM waveguides with high-index cores [8], the additional confinement means are based on index-guiding, that is, light is confined in a high-effective-index region. There have been proposed other approaches such as thin dielectric film loaded-SPP waveguides [9] and MDM waveguides with low-index cores [10], in which light is confined in a low-effective-index region. In these waveguides, the additional confinement relies on the difference of the dispersion characteristics of SPP modes between the low- and the high-index regions. The high-index regions are designed not to have propagation modes so that the addition confinement is quite strong. In some other proposed waveguide structures, the additional confinement is achieved by modifying metal geometry itself, which include metal grooves [11] and metal-stripe waveguides [12,13].
In general, SPP-based waveguides show a trade-off between the field confinement and the propagation loss: a waveguide of the tighter confinement have the higher propagation loss due to Ohmic loss of metal. In terms of the propagation loss, waveguides based on the LR-SPP such as the metal-stripe waveguides are preferred. However, the horizontal confinement of the metal-stripe waveguide is not so strong requiring rather large radii of bending [14]. In planar integrated optical devices, the density is mainly limited by horizontal confinement, that is, a minimum obtainable bending radius. Therefore, LR-SPP-based waveguide structures with enhanced horizontal confinement are needed not to compromise the propagation loss for the density of the integrated optical devices.
In this paper, we propose a LR-SPP-based novel waveguide structure using a hetero-metal film which consists of two types of metals of different plasma frequencies. The horizontal confinement in the proposed waveguide is based on ‘plasmonic mode-gap’ (PMG) concept which relies on the plasma frequency difference of the two metals. The PMG concept is similar to the principle of the horizontal confinement in those low-index-guiding structures [9,10] and is discussed in section 3. The characteristics of the guided modes of the proposed structure are numerically analyzed for both straight and curved waveguides and compared to the metal-stripe waveguides. The proposed waveguide shows enhanced horizontal confinement almost with no impairment of the low propagation loss advantage of the LR-SPP. In this work, the numerical analysis of the guided modes are carried out by using the full-vector finite difference method (FDM) [15,16] with the perfectly matched layer (PML) absorbing boundary condition adopting the stretched coordinate concept [17,18].
2. Permitivities of metals
From Drude model, complex relative permittivities of noble metals such as gold, silver and copper are given by
where ω is an excitation frequency, ωp is a plasma frequency, and ωcol is a collision frequency. For all calculations in this paper, we employ Drude model with parameters of ωp = 1.366x1016 rad/s, ωcol = 4.08x1013 rad/s for gold [19], and ωp = 5x1015 rad/s, ωcol = 5x1013 rad/s for copper [9]. The concept of the proposed PMG waveguide is based on a difference of plasma frequencies of two metals and does not rely on the accuracy of Drude model, so that the use of Drude model does not impair the validity of the demonstrated PMG guiding concept.3. Plasmonic mode-gap guiding in hetero-metal films
Figure 1(a) and 1(b) show the proposed waveguide structure based on PMG in a hetero-metal film and the metal-stripe waveguide, respectively. While vertical confinement of fields in the proposed waveguide is based on the LR-SPP as in the metal-stripe waveguide, the PMG effect confines fields horizontally. The PMG effect can be understood from dispersion curves of infinitely wide metal films embedded in a dielectric medium as shown in Fig. 1(c). These ω − k dispersion curves are obtained by solving Maxwell’s equations for TM polarized wave with appropriate boundary conditions assuming lossless metals [5]. The ignorance of the losses of the metals is for clear illustration of the PMG concept. In the later analysis of waveguide characteristics, the losses of the metals are included. In Fig. 1(c), frequency, ω and momentum vector, k are normalized by ωo = 2πc/λo (λo = 0.633μm) and ko = 2π/λo, respectively. Blue dashed and solid curves represent LR-SPP modes in infinite Gold films of t = 10 nm and 20 nm, respectively. Red dashed and solid curves represent LR-SPP modes in infinite copper films of t = 10 nm and 20 nm, respectively. The LR-SPP modes have even symmetry of Ey with respect to y = 0 plane. For references, dispersion curves (dotted) for SPPs the semi-infinite metals are also plotted. Let us first consider the dispersion curves (solid) for the films of t = 20nm. Due to the aforementioned unique characteristic of the SPP (or LR-SPP) and the difference in ωp of the two metals, there exists a frequency range where only the gold film has a propagation mode. This frequency range denoted as ‘PMG’ in Fig. 1(c) is from ω/ωo = 0.625 (λ = 1.01 μm) to ω/ωo = 1.28 (λ = 0.495 μm). Since for the light of this frequency range, propagation in the copper film is not allowed, the light should be confined horizontally in the gold film region in the hetero-metal film shown in Fig. 1(a). This phenomenon is dubbed PMG guiding in this work, which is analogous to the photonic band-gap guiding in two-dimensional photonic crystal waveguides [20]. For the hetero-metal film of t = 10 nm, a PMG lies in the frequency range from ω/ωo = 0.774 (λ = 0.817 μm) to ω/ωo = 1.55 (λ = 0.408 μm).
Fig. 1 (a) Cross-sectional view of a plasmonic mode-gap waveguide using a hetero-metal film. (b) Cross-sectional view of a metal stripe waveguide. (c) Dispersion curves for fundamental modes of infinite metal films. Blue and red curves are for films of gold and copper, respectively. The dashed and the solid curves are for the films of t = 10 nm and 20 nm, respectively. The dotted curves are dispersion curves of SPPs guided by semi-infinite metals for references.
The PMG guiding concept applies to any combinations of metals which have considerable differences in ωp although in this paper only gold-copper hetero-metal film is treated. For a given metal combination, the PMG frequency range is varied by the choice of the surrounding dielectric medium. In this work, the medium of n = 3.5 is used for a PMG around λo = 0.633μm.
In the metal films, guided modes of odd symmetry of Ey in vertical direction also exist and their propagation loss is much higher than the LR-SPP modes by several orders of magnitude. The PMG guiding concept can also be used for horizontal confinement of the odd mode. In this paper, however, our study is limited to the LR-SPP modes in the interest of low-loss waveguides.
4. Propagation characteristics of straight plasmonic mode-gap waveguides
In this section, we analyze propagation characteristics of the straight PMG waveguides depicted in Fig. 1(a) by using the full-vector FDM in a Cartesian coordinate system [15]. Figure 2 shows the calculated effective indices and propagation losses of PMG waveguides of W = 0.4 μm and t = 10 nm, 20 nm for different excitation wavelengths. Distributions of dominant field component Ey in the waveguides of W = 0.4μm and t = 20 nm for λ = 0.633 μm and 0.8 μm are plotted in Fig. 3 . For comparison, metal-stripe waveguides of the same dimensions are also considered and plotted in Fig. 2 and Fig. 3. Both types of waveguides show similar dependencies of effective indices and losses on the excitation wavelength. The decrease of the effective indices and the losses for the longer excitation wavelength results from the fact that the portions of the confined fields in the metal get smaller for the longer wavelength as seen in Fig. 3. As seen in Fig. 2(b), the PMG waveguides show slightly larger losses than the metal-stripe waveguides for long excitation wavelengths. This is because of tighter horizontal confinement of the PMG waveguides than the metal-stripe waveguides and penetration of small portion of fields into copper regions as seen in Fig. 3. It is apparent that for a given W the horizontal confinement of the PMG guiding effect becomes relatively stronger than the metal-stripe waveguide as λ gets longer.
Fig. 2 (a) Effective refractive index and (b) propagation loss [dB/μm] vs. wavelength for two types of straight waveguides (W = 0.4μm).
Fig. 3 Electric field (Ey) distributions in the two types of straight waveguides of t = 20nm, W = 0.4μm. Mode-gap waveguide for (a) λ = 0.633μm and (b) λ = 0.8μm. Metal-stripe waveguide for (c) λ = 0.633μm and (d) λ = 0.8μm.
The field penetration into the copper region in the PMG waveguides is a sign of imperfectness of the PMG guiding which stems from a quasi-bound mode in the copper film. When losses of metals are taken into account, the maximum momentum vector of the SPP becomes finite and there exists the quasi-bound mode in the frequency range right above the bound SPP mode [21]. The existence of the quasi-bound mode in the copper film regions makes the PMG guiding effect imperfect. However, the quasi-bound mode exists only in a limited frequency range near the lower limit of the PMG and the losses of the quasi-bound modes are higher than the bound SPP modes by 2 ~3 orders of magnitude. Therefore, degradation of the mode-gap effect is not so severe. This can be confirmed by the fact that the difference between the propagation losses of the mode-gap and metal-stripe waveguides tends quickly to zero as λ decreases.
Another thing to note is that the PMG waveguides exhibit cutoff behaviors for the excitation wavelength longer than a certain value unlike the metal-stripe waveguides. The occurrence of the mode cutoff can be diagnosed by the effective index of the mode. At the cutoff, effective indices of the mode becomes the same as the index of the surrounding dielectric medium. In Fig. 2(a), one can see that the effective indices of the PMG waveguides of W = 0.4 μm become very close to n = 3.5 at λ = 0.75 μm and 0.95 μm for t = 10 nm and 20 nm, respectively. These wavelengths are very close to cutoff wavelengths. The exact cutoff wavelength is difficult to determine directly since mode calculation is impossible at the cutoff.
To understand the origin of the cutoff, the dependence of the cutoff behavior on W is investigated. The guided modes are calculated for the PMG waveguides of various W with a fixed thickness of t = 20 nm for three excitation wavelengths λ = 0.633 μm, 0.8 μm, and 0.9μm. For comparison, the guided modes of the metal-stripe waveguides are also calculated. Figure 4 shows the calculated effective indices and the propagation losses. For the PMG waveguides, there exists a minimum width to support a guided mode for a given λ as seen in Fig. 4(a). For the excitation wavelengths of λ = 0.633 μm, 0.8 μm, and 0.9 μm, the minimum width values seems to be very close to W = 0.085 μm, 0.22 μm, and 0.34 μm, respectively. Although the exact minimum widths are difficult to determine, it is obvious that the PMG waveguide of the larger W has the longer cutoff wavelength. This implies in the PMG waveguide, the cutoff occurs because a wave of a wavelength longer than a certain value cannot fit into the high ωp metal (gold in our structure) region of a finite width. This is more clearly illustrated in the field (Ey) distributions in the mode-gap waveguides of W = 0.2 μm and 0.085 μm for λ = 0.633 μm shown in Fig. 5 . As W decreases, the mode profile of the PMG waveguide is squashed in the horizontal (x) direction and spreads out in the vertical (y) direction. The mode profile for W = 0.085 μm shown in 5(b) is close to cutoff. If W is decreased further, the most of the field is pushed into the dielectric medium and finally the guided mode is no longer sustained. This cutoff behavior is another evidence of PMG effect in spite of the degradation due to the quasi-bound mode. Whereas, the metal-stripe waveguides do not show cutoff behaviors as W decreases. As seen in Fig. 5(d), the mode profile of the metal-stripe waveguide for W = 0.03 μm spreads out in both x and y directions in a balanced way, so that still guided mode is supported. Therefore, no mode cutoff is observed in Fig. 4(a) as W decreases.
Fig. 4 (a) Effective refractive index and (b) propagation loss [dB/μm] of vs. width of core metal (Au) for two types of straight waveguides (t = 20nm, λ = 0.633, 0.8, and 0.9 μm)
Fig. 5 Electric field (Ey) distributions in mode-gap waveguides of (a) W = 0.2μm and (b) W = 0.085μm. Electric field (Ey) distributions in metal-stripe waveguides of (c) W = 0.2μm and (d) W = 0.03μm. In all the cases, the metal film thickness is t = 20nm and the excitation wavelength is λ = 0.633 μm.
It should be noted that there is another cutoff mechanism for the PMG waveguide of a large W. If wavelength approaches to the lower limit of the PMG region in Fig. 1(c), the mode cutoff also occurs no matter how large W is. In this cutoff case, the wave will radiates along the copper film via LR-SPP mode.
In Fig. 4(b), as W decrease, the propagation loss of the PMG waveguide first increases slowly due to gradual enhancement of the horizontal confinement and drops abruptly near the cutoff. This implies that the guided mode in the PMG waveguide is sustained down to W very close to cutoff. For example, the PMG waveguide of W = 0.085 μm still supports the well confined mode for λ = 0.633 μm as seen in Fig. 5(b). Therefore, the proposed PMG waveguide can provide tight confinement close to the diffraction limit in a horizontal direction.
For more quantitative comparison between two types of waveguides, modal sizes and figures of merit (FOM) are calculated and plotted as functions of W for waveguides of t = 20 nm at λ = 0.633 μm in Fig. 6 . The modal size is defined as the square root of the area in which the power density is larger than 1/e2 times its maximum value [22]. The FOM is defined as FOM = (propagation length in μm)/(modal size in μm) in a similar way as in Ref [22], where the propagation length is a distance over which a field intensity drops to 1/e times its initial value. As seen in Fig. 6(a), the PMG waveguide has a slightly smaller modal size than the metal-stripe waveguide, but the different is not so notable. The FOMs of the two types of waveguides are also almost the same. Therefore, the PMG effect is not profound in terms of the modal size or the FOM. However, is seems that the strength of wave confinement is not completely represented by the modal size. For example, according to the mode profiles in Fig. 5(c) and 5(d), the metal-stripe waveguide of W = 0.2 μm and t = 20 nm apparently shows stronger confinement than that of W = 0.03 μm and t = 20 nm in that the former requires a smaller spacing between adjacent waveguides for negligibly small evanescent coupling than the latter. Besides, in conventional two-dimensional optical layouts, the density of integration is mostly limited by a minimum allowed bending radius of the waveguide. Therefore, another practically important metric to measure the confinement strength of the waveguide is a bending loss characteristic. The PMG effect on the bending loss characteristic is discussed in detail in the following section.
Fig. 6 (a) Modal sizes and (b) figures of merit of plasmonic mode-gap and metal-stripe waveguides as functions of W for λ = 0.633 μm.
The guided modes of the two types of the waveguides considered in this paper can be easily excited from conventional waveguides such as an optical fiber and a dielectric ridge waveguide via butt-coupling since they have even symmetries in both x and y directions. For a high coupling efficiency, however, a modal size conversion scheme or a taper structure may be needed because of the small modal size close to the diffraction limit. Since the two types of waveguides have almost same modal sizes, their coupling efficiencies will be about the same.
5. Bending loss of curved plasmonic mode-gap waveguides
Figure 7 shows structures of curved PMG and metal-stripe waveguides. To analyze propagation characteristics of these curved waveguides, guided modes are calculated by using the full-vector FDM in a cylindrical coordinate system [16] with the PML absorbing boundary condition [17,18]. Note that the coordinate system in this section is different from the one in the previous section so that a dominant electric field component in the guided mode is Ez. A radius of curvature of the curved waveguide, R is defined as a distance from the origin to the center of the waveguide along r axis.
Fig. 7 Structures of two types of curved waveguides: (a) plasmonic mode-gap waveguide and (b) metal-strip waveguide. For analysis of the curved waveguides, a cylindrical coordinate system is used.
First, the propagation characteristics of the curved waveguides of t = 20nm and W = 0.4μm are investigated for three excitation wavelengths of λ = 0.633 μm, 0.8 μm, and 0.9 μm. The calculated results are plotted as functions of R in Fig. 8 . The effective indices of the curved PMG waveguide tend to increase as R decreases in a similar way of those of the metal-stripe waveguide. This is a general tendency of the curved waveguides as explained in Ref [14,23]: when one assumes the mode propagates along curve corresponding to a specific angle, ϕ, effective index must increase if the phase along the r = R is to be the same as the phase of the center of mode. The right angle (90°) bending losses of the mode-gap waveguide plotted in Fig. 8(b), which include both the propagation losses and the radiation losses due to bending, show somewhat different tendency from those of the metal-stripe waveguide. 90° bending loss of the metal-stripe waveguide shows typically a ‘V’ shape function of R having a minimum loss for an optimal radius R opt, which is because the radiation loss dominates over the propagation loss for a small R (<R opt) [14,23]. As seen in Fig. 8(b), the 90° bending losses of the PMG waveguide show a very slow increase rate or an almost constant value as R decreases from R opt. This implies that the radiation in a radial direction is suppressed by the PMG effect. The suppression of radiation is more conspicuous for a shorter wavelength since the PMG effect is less degraded by the quasi-bound mode in copper films. Due to the suppressed radiation, both optimal bending radius and minimum 90° bending loss of the PMG waveguide are reduced compared to those of the metal-stripe waveguide for all excitation wavelengths. Figure 9 shows field (Ez) distributions of the curved waveguides of t = 20 nm and W = 0.4 μm for λ = 0.633 μm. The relative suppression of the radiation in the curved PMG waveguide is clearly illustrated in the field profiles. It should be stressed that the PMG effect improves the bending loss characteristics almost without degradation of the propagation loss of the straight waveguide. In Fig. 2(b), one can see that the PMG waveguide of t = 20 nm and W = 0.4 μm shows almost the same propagation loss as the metal-stripe waveguide for λ < ~0.75 μm.
Fig. 8 (a) Effective refractive index and (b) bending loss [dB/90°] vs. radius of curvature for two types of curved waveguides (t = 20nm, W = 0.4μm, λ = 0.633, 0.8, and 0.9μm)
Fig. 9 Electric field (Ez) distributions of two types of curved waveguides of t = 20nm, W = 0.4μm for λ = 0.633μm; Mode-gap waveguides of (a) Ropt = 1μm and (b) R = 0.5μm. Metal-stripe waveguides of (c) Ropt = 2μm and (d) R = 0.5μm.
Bending loss characteristics of the curved PMG waveguides of different W are investigated. Figure 10 shows the calculated results for the two types of waveguides of W = 0.1 μm, 0.2 μm and 0.4 μm, where the waveguide thickness and the excitation wavelength are fixed to t = 20 nm and λ = 0.633 μm. For all W, the PMG waveguides show improved performance in both minimum loss and optimal bending radius compared to the metal-stripe waveguides. The radiation loss characteristics are almost the same in the PMG waveguides of W = 0.2 μm and 0.4 μm. The PMG waveguide of W = 0.1 μm shows a larger bending loss than the wider waveguides due to the looser vertical confinement. In the PMG waveguides of small widths close to cutoff, the portion of field in the dielectric regions increases and consequently, the radiation suppression by the PMG guiding gets weaker. The PMG waveguide of a width of W = 0.1 μm and t = 20 nm shows an optimal radius of R opt = ~1 μm and 90° bending loss of ~3.5 dB, which is still better performance compared to the metal-stripe waveguide. Figure 11 shows field distributions in the two type of waveguide of W = 0.1 μm, t = 20 nm, and of R = 1 μm for λ = 0.633 μm.
Fig. 10 (a) Effective refractive index and (b) bending loss [dB/90°] vs. radius of curvature for two types of curved waveguides of different core widths (W = 0.1, 0.2, and 0.4μm, t = 20nm, λ = 0.633μm)
Fig. 11 Electric field (Ez) distributions in two types of curved waveguides of t = 20nm, W = 0.1μm, and R = 1μm for λ = 0.633μm. (a) Mode-gap waveguide and (b) metal-stripe waveguide.
Figure 12 shows the calculated bending loss characteristics of a PMG waveguide of a reduced thickness of t = 10 nm and W = 0.4 μm for λ = 0.633 μm. Although the PMG waveguide of t = 10 nm shows better performance than the metal-stripe waveguide of the same dimension, its suppression of radiation is weaker than the PMG waveguide of t = 20 nm. This is due to the looser vertical confinement of the thinner metal film. The PMG waveguide shows the same dependency of the bending loss on a waveguide thickness as the metal-stripe waveguide [14]. Figure 13 shows field distributions in the two type of the curved waveguides of t = 10 nm, W = 0.4 μm, R = 10 μm for λ = 0.633 μm. Comparing to Fig. 9, one can see that mode size in vertical direction is increased as a result of the looser vertical confinement, which weakens the PMG effect.
Fig. 12 (a) Effective refractive index, and (b) bending loss [dB/90°] vs. radius of curvature for two types of curved waveguides of different metal thicknesses (t = 10 and 20nm, W = 0.4μm, λ = 0.633μm)
Fig. 13 Electric field (Ez) distributions in two types of curved waveguides of t = 10nm, W = 0.4μm, and R = 10μm for λ = 0.633μm; (a) Mode-gap waveguide and (b) metal-stripe waveguide.
6. Conclusion
In this paper, a novel plasmonic waveguide structure using a hetero-metal film has been proposed and its propagation characteristics have been numerically analyzed. The hetero-metal film is composed of two metal films of different plasma frequencies. In the proposed waveguide, the LR-SPP mode of the central metal film is horizontally confined by the PMG effect and strong horizontal confinement is sustained even for a horizontal dimension close to the diffraction limit. For example, the proposed PMG waveguide composed of gold-copper hetero-metal of t = 20 nm supports a horizontally tight confined mode for a width down to W = 0.085 μm for λ = 0.633 μm. Unlike a metal-stripe waveguide, the PMG waveguide shows a cutoff behavior as the width of the central metal film decreases for a given excitation wavelength. The PMG guiding effect is somewhat degraded by the existence of the quasi-bound modes in the outer films only for the excitation wavelengths close to the lower limit of the PMG and the propagation loss of the PMG waveguide is almost same as that of the metal-stripe waveguide in the short wavelength range of the PMG. The strong horizontal confinement of the PMG waveguide suppresses the radiation loss in bending notably and allows a tight bending radius. For example, the PMG waveguide of W = 0.1 μm and t = 20 nm shows an optimal bending radius of Ropt = ~1μm for λ = 0.633 μm and a minimum 90° bending loss of ~3.5dB. The PMG guiding effect is apparently affected by the thickness of the hetero-metal film and it gets weaker as the thickness decreases.
In summary, the newly proposed PMG waveguide shows stronger horizontal confinement and improves bending characteristics without the low-loss advantage of the LR-SPPs impaired. Therefore, we believe that the PMG waveguides can improve integration densities of optical devices based on the LR-SPPs.
Acknowledgement
This work was supported by National Research Foundation of Korea Grant (KRF-2009-0058569), the Korea Research Foundation Grant (KRF-2007-412-J04002), and the Korea Science and Engineering Foundation grant (R11-2008-095-01000-0) funded by the Korean Government (MEST).
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