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We present improved characteristics of the curved plasmonic waveguide which consists of a thin metal stripe with asymmetric cladding layers. It is shown that in the proposed curved asymmetric plasmonic waveguides, a balance between a radiation due to bending and a radiation due to the asymmetric claddings allows a bending with a smaller radius curvature and a lower loss compared to the waveguide with symmetric claddings. At the same time, a symmetric metal stripe waveguide’s typical trade-off between the bending characteristics and the propagation loss of a straight waveguide is relaxed with proper amount of asymmetry. With the proposed structure, a plasmonic waveguide bending whose radius is as small as 2 mm with a total loss of 1.8 dB/90° is designed. Enhanced sensitivity to the surrounding medium and its application are discussed.
©2009 Optical Society of America
1. Introduction
At the interface between metal and dielectric media, a transverse magnetic (TM) polarized guided surface wave is supported, which results from the coupling between electromagnetic waves and collective electronic oscillations along the metal surface, and which is called surface plasmon-polariton (SPP) [1]. The SPP is evanescently confined to the interface and shows a unique dispersion characteristic that the highest reachable frequency is limited no matter how large wave-vector it has. As a result of the unique dispersion characteristic, the SPP can be localized below the diffraction limit, which opens huge opportunity to subwavelength optics [2].
In addition to those nano-metallic structure based subwavelength optic devices, the SPP in rather a simple metal structure such as a thin metal film or a metal stripe also finds various practical applications as an optical waveguide. In a thin metal film embedded in a dielectric medium two SPPs in both surfaces are coupled to form a long range-SPP (LR-SPP) which shows enhanced propagation length [3,4]. The existence and the characteristics of LR-SPP based guided modes in thin (~20 nm) metal stripes have been studied numerically and experimentally [5–7]. The LR-SPP modes in metal stripes showed a rather loose confinement with mode sizes comparable to that of a single-mode fiber and a propagation length of cm scale at optical communication wavelength. Therefore, there have been numerous demonstrations of optical devices based on surface plasmonic metal stripe waveguides in the interest of optical integrated circuits [8–11]. In optical integrated circuits, waveguide bending structures are inevitable. The characteristics of the metal stripe waveguides curved in the direction of a wide side of the stripe have been studied numerically, which revealed that 90° bending loss of the metal stripe shows a minimum value at an optimal radius of curvature due to the inherent propagation loss [12]. The typical minimum 90° bending loss is rather large (~20 dB) and the optimal radius is in 10 mm range, which may limit the size and the integration density of devices.
Another very promising application of the metal stripe waveguide may be a flexible optical wiring since a thin metal stripe embedded in polymer is highly flexible and hence, suitable for such an application of mobile folder phone as mentioned in Ref [13]. In this case, bending in the direction of a thin side of the metal stripe is demanded, and its loss has been also studied numerically, which showed unaffordably large bending loss [13].
In this paper, we propose a plasmonic waveguide of a thin metal stripe embedded in asymmetric dielectric media for bending loss reduction in flexible optical wiring applications and the bending loss of the proposed waveguide structure is investigated numerically. In the proposed asymmetric structure, the balance between the bending radiation and the radiation due to the asymmetry improves the loss characteristics of the curved metal stripe waveguide so that it is suitable for practical flexible optical wiring applications. In addition, enhanced sensitivity of the bending loss to the surrounding medium and its application to a variable optical attenuator are also discussed. To analyze the guided mode of the bent waveguides, the full-vector finite difference method (FDM) in a cylindrical coordinate is used [14,15] with the perfectly matched layer absorbing boundary condition adopting the stretched coordinate concept [16,17].
2. Waveguide structure with asymmetric cladding layers
Figure 1 shows the metal stripe waveguide structure investigated in this paper. The waveguide is composed of curved metal (gold) stripe surrounded by asymmetric clad materials. The refractive index of the inner clad material is set to be n2 = 1.47 and that of the outer clad is varied as n1 < n2. In the case of n1 = n2, the structure corresponds to the symmetric waveguide considered as a reference waveguide. The height (or width) of the stripe is set to be W = 5μm and several thicknesses (t = 10 ~30 nm) are considered in this work. At the operating wavelength of λ0 = 1.55μm, the dielectric constant of εm = −131.5 + j 12.65 is used for gold [18]. As depicted in Fig. 1, the radius of curvature of bending, R, is defined as the distance from the origin to the center of the stripe in radial direction.
Fig. 1 Structure of curved asymmetric waveguide bend: (a) cross-sectional view, (b) top view, and (c) perspective view (n1 = variable, n2 = 1.47, W = 5μm, λ0 = 1.55μm).
3. Straight asymmetric waveguide analysis
Prior to investigating the curved waveguides, we analyzed the effect of the asymmetry on the propagation characteristics of the straight waveguides. For the analysis of the straight waveguides, a full-vector FDM in a rectangular coordinate system has been used. In order to check the correctness of our calculation, we also have compared the results to the calculation of the curved waveguide with a large radius, and it has been confirmed that the characteristics of the curved waveguide asymptotically converge to those of the straight waveguide as R goes infinite.
Figure 2 shows the calculated effective index and the propagation loss [dB/mm] of the LR-SPP modes of a set of straight asymmetric waveguides with different thicknesses, t = 20, 25, 30 nm as n1 is varied with n2 fixed to 1.47. For all the thicknesses, the effective indices of the modes decrease down to the vicinity of neff = 1.47 as n1 decreases to a certain values. With further decrease of n1, the modes are cut-off, which means waves are no longer guided by the metal stripe and the energy completely radiates into the higher index cladding layer [19]. The losses of the modes increase slightly for small asymmetry due to the increase of radiation into high index cladding layer, and as reaching the cut-off, they decrease very rapidly since the portion of energy confined in the metal stripe decreases rapidly and hence, almost wave does not experience the ohmic loss of the metal. In Fig. 2, one can see that guided mode can be sustained up to larger amount of asymmetry (i.e., smaller n1) for the thicker metal stripe. Another thing to note in Fig. 2(b) is the fact that the propagation loss of the metal stripe waveguide is reduced with proper amount of asymmetry.
Fig. 2 (a) Effective refractive index and (b) propagation loss [dB/mm] for the straight asymmetric waveguides with n2 = 1.47, W = 5μm and λ0 = 1.55μm.
Figure 3 shows the mode profiles of the dominant electric field (Er) in the straight waveguides of t = 20 nm. A mode field diameter can be defined as the point at which field strength decreases to 1/e. Figure 3(a) indicates that the symmetric waveguide has mode field diameters close to 8 μm in both r and z directions, which matches well with a typical single-mode fiber (SMF) and has a small coupling loss from the SMF. In the asymmetric waveguides, the modes become asymmetric mainly in r direction and the coupling loss from the SMF will increase. We estimated the additional loss in coupling with the SMF caused by the mode asymmetry by calculating overlap integral. The calculated additional losses are ~0.1 dB and ~0.5 dB for n1 = 1.469 and 1.4685, respectively. They are acceptable values and much smaller than bending loss reductions caused by the asymmetry that will be discussed in the following section.
Fig. 3 Electric field (Er) distributions of the straight waveguides (t = 20 nm, n2 = 1.47, W = 5μm and λ0 = 1.55μm): (a) n1 = 1.47 (symmetric), (b) n1 = 1.469, and (c) n1 = 1.4685.
4. Curved asymmetric waveguide analysis
Figure 4 shows the calculated propagation characteristics of the curved asymmetric waveguides as a function of R. Three asymmetric waveguides with n1 = 1.467, 1.468, and 1.469 have been considered with fixed n2 = 1.47 and t = 20 nm. For references, two symmetric waveguides (n1 = n2 = 1.47) with two different thicknesses (t = 20, 30 nm) also have been considered.
Fig. 4 (a) Effective refractive index, (b) bending loss per unit length [dB/mm], and (c) bending loss per 90° [dB/90°] for the curved asymmetric waveguide with n2 = 1.47, W = 5μm and λ0 = 1.55μm.
Let us focus on the characteristics of the curved symmetric waveguides first. (See the dashed curves in Fig. 4.) The effective indices tend to increase as R decreases, which is a general tendency of the curved waveguides as explained in Ref [12]: 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 bending loss, which includes both the propagation loss and the radiation loss due to bending, also shows a similar tendency as in Ref [12,13]. For a large R, the propagation loss of the straight waveguide dominates so that the bending loss per unit length [dB/mm] does not change as R varies and the loss for 90° bending is proportional to R. Whereas, for a small R, the radiation loss dominates, and both the bending loss per unit length and the loss for 90° bending rapidly increase as R decreases. Therefore, there exists an optimal radius, Ropt for the minimum 90° bending loss. In Fig. 3(c) shows that the minimum 90° bending losses are 42 and 38 dB with Ropt = 30, 5 mm for t = 20, 30 nm, respectively. The thicker stripe shows better bending characteristics but it suffers from a large propagation loss in the straight waveguide. (See Fig. 2(b).) Therefore, there is trade-off between the bending performances and the propagation loss of the straight waveguide [12,13]. Besides the minimum loss for t = 30 nm is too large for practical application.
Now let us see the curved asymmetric waveguide. In Fig. 4(a), the effective indices show a rather different tendency from the symmetric case; they decrease as R decrease. The decrease slope becomes larger as the increase of the asymmetry. This is because the center of mode is shifted toward the outer layer of a small index as R decreases and more portion of wave experiences the lower index cladding. In Fig. 4(b) and 4(c), the loss properties of the curved asymmetric waveguides are also characterized mainly by the asymmetric straight waveguides for a very large R, and for a very small R, they are characterized mainly by the bending radiation. So, the difference comes from the radiative (or leaky) behavior of the asymmetric straight waveguide as discussed in section 3. To illustrate this clearly, the dominant electric field (Er) distributions of the guided mode in the curved waveguide for t = 20 nm, n1 = 1.468, and n2 = 1.47 are plotted in Fig. 5 for various radii. Note that the straight waveguide of this structure is in just beneath the cut-off as seen Fig. 2, and that is why there is huge leakage toward the large index cladding in Fig. 5(a) for R = 1,000 mm. As R decreases down to 10 mm, the leakage decreases. (See Fig. 5(b), 5(c), and 5(d).) As R decreases further, the mode shows leakage toward the outer layer and finally it becomes very leaky for R = 3 mm. It seems that the leakage due to the bending catches up with the leakage due to the asymmetry at R = 10 mm. Because of this compensation of the bending radiation by the radiation due to the asymmetry, the radius for which the bending radiation becomes dominant is decreased to a smaller value compared to the symmetric waveguide case. Consequently, as seen in Fig. 4(b) and 4(c), the asymmetric waveguides have improved bending loss characteristics. For n1 = 1.468, a minimum 90° bending loss is about 20 dB for Ropt = 10 mm. For the larger amount of asymmetry, i.e. n1 = 1.467, a minimum 90° bending loss is about 17 dB for Ropt = 6 mm. However, these amounts of asymmetry are too large to sustain the guided mode for the straight waveguide. A proper choice of n1 for n2 = 1.47 and t = 20 nm seems to be 1.4685, and for this structure, a minimum 90° bending loss of 23 dB for Ropt = 12 mm is achieved, which is not shown in Fig. 4. Note that in this case the propagation loss of the straight waveguide is also slightly reduced compared to the symmetric one, so that introducing asymmetry into the metal stripe waveguide can improve bending loss characteristics without degrading strength waveguide performance.
Fig. 5 Electric field (Er) distributions of the fundamental mode in the curved waveguide with various radii (n1 = 1.468, n2 = 1.470, t = 20 nm). (a) R = 1,000 mm, (b) R = 50 mm, (c) R = 20 mm, (d) Ropt = 10 mm, (e) R = 6 mm, (f) R = 3 mm.
5. Curved waveguide with large asymmetry and other application
In the previous section, it has been shown that the curved waveguide can sustain a well confined mode even for a very large asymmetry for which a mode is not sustained in the straight waveguide. In certain applications, a waveguide bending is fixed and its cross-sectional structure can be different from that of the straight waveguide part. Therefore, a curved waveguide with a large asymmetry and its possible applications are discussed in this section.
Figure 6 shows calculated losses of curved waveguides having n1 = 1.460 and 1.463, which correspond to cut-off in straight waveguides of t = 10 and 20 nm. The curved waveguide of t = 20nm with n1 = 1.463 has a minimum 90° bending loss of 15 dB for R = 2.2 mm, which a huge improvement of a bending size but the bending loss has a room for further improvement. For an optimal bending radius, the radiations due to bending and asymmetry are canceled each other and ohmic loss of a metal stripe mainly determines the loss. So, further loss improvement is achieved by reducing a metal stripe thickness. In Fig. 6(b), the curved waveguide of t = 10 nm shows a minimum 90° bending loss of 1.8 dB for R = 2 mm. In this case, the straight waveguide is so radiative that a guided mode is not sustained for R > 3mm.
Fig. 6 (a) Bending loss per unit length [dB/mm] and (b) bending loss per 90° bending [dB/90°] for the curved large asymmetric waveguides with n2 = 1.47, W = 5μm and λ0 = 1.55μm.
In addition to the improvement of the bending loss, the introduction of the asymmetry enhances the sensitivity of waveguide characteristics to the index change of the outer cladding layer. In Fig. 6(a), the curved waveguide of t = 10 nm and R = 1.5mm shows a loss change of 6 dB/mm for a change of 3 x 10−3 in the index of the outer cladding layer, n1. It is expected that the loss change increases more rapidly for a larger n1 change. Similar behavior is observed in Fig. 4(b). The curved waveguide of t = 20 nm and R ≈10 mm shows a loss change of 3 dB/mm for a n1 change of 10−3, which is about an order of magnitude larger than that of the straight waveguide in Fig. 2(b). This enhanced loss sensitivity on a cladding index may be very useful for the optical variable attenuator based on a thermo-optic polymer and a plasmonic waveguide [20]. This suggests that the required power can be reduced to 1/10 by simply bending the structure given in Ref [20]. with proper choices of a radius and the refractive index of polymers. In this kind of application, the bending part is fixed and thus, the cross-sectional structure of the bending can be designed differently from the straight part. This implies that a large asymmetric structure may be applied only to the bending while the straight part remains symmetric. If the variable attenuator is designed in this manner, there will be an additional coupling loss due to the mode mismatch between the straight waveguide and the bending part. For example, in the case of the curved waveguide of t = 10 nm and n1 = 1.460, the coupling loss due to the mode miss match is estimated to be ~1.5 dB from overlap integral calculation. So, the total insertion loss of ~4.8 dB including the bending loss of 1.8 dB will be added in return to the operation power reduction in the thermo-optic polymer related applications.
6. Conclusion
In this paper, we investigated mode characteristics of metal (gold) stripe waveguides with asymmetric cladding layers and showed that their bending losses can be improved without degrading the performance of the straight waveguide of the same cross-sectional structure. Our results revealed that proper asymmetry makes a remarkable improvement in terms of radius of curvature and corresponding minimum loss in the curved waveguide. For the gold stripe of t = 20 nm, the introduction of small asymmetry brought about more than 20 dB bending loss reduction per right angle bending, and the radius of curvature was reduced to less than a half compared to the symmetric waveguides. With a large amount of asymmetry, we could design a compact low loss bending; a 10 nm thick gold stripe showed a bending loss of 1.8 dB/90° for R = 2mm.
Due to the asymmetric structure, the proposed waveguide structure will be good for applications in which the bending direction is predetermined as in the case of mobile folder phone. The curved asymmetric waveguides showed enhanced sensitivity to an index change of the outer cladding layer, which may be useful for operating power reduction in a variable optical attenuator based on a metal stripe waveguide and a thermo-optic polymer.
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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