## Abstract

A finite width dielectric-metal-dielectric (DMD) waveguide placed on a substrate is numerically investigated near the telecom wavelength *λ* = 1550 nm by the finite element method. With proper waveguide sizes, the asymmetrical DMD waveguide can support hybrid long-range surface plasmon-polariton modes which have tight field confinement (~700 nm) and long propagation lengths (*L*> 300 *μ*m) simultaneously. Compact plasmonic waveguide-ring resonators (WRRs) based on such asymmetrical DMD waveguide show high quality factors compared with dielectric-loaded surface plasmon-polariton, channel plasmon polariton, plasmonic whispering-gallery microcavity, and pure dielectric waveguide cases.

©2009 Optical Society of America

## 1. Introduction

Future photonic integrated circuits for optical information will have specific requirements on the density of integration. Surface plasmon plaritons [1] (SPPs), which can break the diffraction limit of light due to their tight field confinement to the metal surface, show great promise for applications in highly photonic integrated circuits [2,3]. Various waveguides and photonic devices based on SPPs have been reported in recent years [4–12]. Among them, long-range surface plasmon-polariton (LRSPP) modes guided by thin metal stripes embedded in infinite homogeneous background dielectrics play an importance role for their long propagation lengths up to ~mm [4,5]. However, the field confinement of LRSPP modes are very poor (effective mode sizes ~10 *μ*m [5–8]), which brings large bend loss in tight radius structures (~*μ*m), and the optimal bend radius is as large as about 10 mm [7,8]. In order to get higher densities of integration, dielectric-loaded surface plasmon-plariton (DLSPP) waveguides with strong field confinement (effective mode sizes ~1 *μ*m) were proposed and demonstrated [9–11]. Due to the tight field confinement, the optimal bend radius of DLSPP waveguides is reduced to about 5 *μ*m [10,11]. While, the propagation lengths of DLSPP modes (~45 *μ*m) [9–11] decreased significantly compared to LRSPP modes because the portion of electromagnetic field distributed in the lossy metal film increases greatly. SPP modes with both tight field confinement and long propagation lengths in a symmetrical waveguide structure have recently been presented theoretically [12]. But the difficulty to fabricate the symmetrical structure limits its real applications.

Here, by combining both merits of LRSPP and DLSPP modes, we numerically investigate a hybrid plasmonic waveguide, the finite width dielectric-metal-dielectric (DMD) waveguide placed on a substrate. Such asymmetrical DMD waveguide structure may be easily realized in experiments [13,14] and the hybrid property provides flexibility in the design of plasmonic circuits. Numerical simulations show that, with proper waveguide sizes, the hybrid LRSPP modes guided by the asymmetrical DMD waveguide exhibit tight field confinement (~700 nm) and long propagation lengths (>300 *μ*m)) simultaneously. To show the advantage of such an asymmetrical DMD waveguide, compact and high performance waveguide-ring resonators (WRRs) base on the waveguide are simulated as a demonstration.

## 2. Hybrid LRSPP mode guided by the finite width symmetrical DMD waveguide without a substrate

To illustrate the propagation characteristics of the waveguide, the finite width symmetrical DMD waveguide without a substrate is simulated first. The waveguide structure is schematically shown in Fig. 1(a)
, which consists of a thin metal strip (Au) with thickness *t* and finite width *w* symmetrically embedded inside a dielectric ridge with thickness *h* and width *w* surrounded by air. A similar symmetrical DMD structure surrounded by a low index cladding has been discussed before [15], and it has been found that the higher refractive index dielectric layers in combination with the low index cladding can achieve tight mode confinement which cannot be obtained by using either a high or low index homogeneous dielectric cladding. The main difference for the structure in Fig. 1(a) is that the low index cladding is replaced by the air, which ensures even tighter mode confinement and better bending characteristic. For such a symmetrical structure, there exists a hybrid LRSPP mode [8] which is a combination of the pure LRSPP mode and waveguide mode of the dielectric ridge. Here, we choose a high refractive index dielectric (Si_{3}N_{4}) as the ridge, so the hybrid LRSPP mode can be confined in the core of the DMD waveguide well due to the large refractive index difference between Si_{3}N_{4} and the air.

The simulations were performed using the commercial software package Comsol Multiphysics which implements the finite element method (FEM) specifically to solve optical waveguide problems and is a widely accepted method in optics [6,9–11]. The Cartesian coordinates used in the following analysis are shown in Fig. 1(a) and the light wave propagates along the z-axis with a vacuum wavelength of *λ* = 1550 nm. In the calculations, the thickness of the Au strip is set as *t* = 20 nm according to previous experimental works [5–7], and the refractive indexes of the Au strip, the Si_{3}N_{4} ridge and the air are *n*
_{1} = 0.55-11.5*i*, *n*
_{2} = 2.0 and *n*
_{3} = 1.0 [16], respectively.

In the letter, we mainly focus on the hybrid LRSPP modes for their long propagation lengths. In highly photonic integrated circuits, both the field confinement and the propagation length are important factors. To satisfy both these two aspects, simulations show that waveguide with moderate dimensions (width from about 600 to 800 nm and thickness from about 700 to 900 nm) is appropriate. As an example, with width *w* = 600 nm and thickness *h* = 800 nm, the hybrid LRSPP mode performs simultaneously a tight field confinement (effective mode sizes along the x-axis and the y-axis are *W*
_{x} = 726 nm and *W*
_{y} = 670 nm respectively) and a long propagation length (*L* = 300 *μ*m, while the propagation length of SPP along a single Au/Si_{3}N_{4} interface is only 41 *μ*m.). Here the effective mode size is defined as the distance over which the field decreases down to 1/*e* of its maximum value. The corresponding distribution of the electric field (*E*
_{y}) of the hybrid LRSPP mode is displayed in Fig. 1(b), and it is noted that the electromagnetic field is localized in the core of the DMD waveguide very well.

## 3. Hybrid LRSPP mode guided by the asymmetrical DMD waveguide with a substrate

For practical applications, the DMD waveguide needs to be placed on a substrate. Then the waveguide will become an asymmetrical system and the index-matching condition for obtaining the long propagation length will be destroyed. To maintain the index-matching condition in such an asymmetrical system, we can adjust the thickness of the Si_{3}N_{4} ridge below the Au strip (denoted by *h*
_{down}) to a proper value [17]. This scheme is schematically shown in Fig. 2(a)
where silica (*n*
_{4} = 1.444) is chosen as the substrate. The Si_{3}N_{4} ridge width and thickness above the 20-nm-thick Au strip are fixed at *w* = 600 nm and *h*
_{up} = 400 nm for comparison with previous results without the substrate.

The solid lines in Fig. 2(b) display the calculated dependences of the effective refractive index and the propagation length of the hybrid LRSPP mode on the adjustable Si_{3}N_{4} thickness below the Au strip *h*
_{down}. The propagation length reaches to its maximum (*L* = 321 *μ*m) at *h*
_{down} = 310 nm. The electromagnetic field at this point [see Fig. 2(c)] is almost symmetrically distributed with respect to the thin Au strip. This symmetrical field distribution means that by adjusting *h*
_{down} to the proper value the dielectric environments in the asymmetrical DMD waveguides can also satisfy the index-matching condition, thus the propagation length of the hybrid LRSPP mode is the longest. At this maximum point, important properties of the hybrid LRSPP mode like the field confinement (*W*
_{x} = 726 nm and *W*
_{y} = 647 nm), the effective refractive index (*n*
_{eff} = 1.63), and the propagation length (*L* = 321 *μ*m) are all very close to their values in the symmetrical DMD waveguide case. Thus, by choosing the thickness of the Si_{3}N_{4} ridge below the Au strip, the experimentally realizable asymmetrical DMD waveguide with the substrate can perform as the ideal symmetrical DMD waveguide in the air. From Fig. 2(b), it is also noticed that a small thickness deviation from the maximum point (e.g. Δ*h*
_{down} = ± 30 nm) will not bring large propagation loss, which means the requirement on the thickness *h*
_{down} to achieve long propagation lengths is not critical. To ensure single-mode operation, the propagation characteristics of the first waveguide mode (TE polarized, and it mainly distributed in the bottom part of the Si_{3}N_{4} ridge) are also calculated. The cutoff thickness for the first waveguide mode is *h*
_{down} = 594 nm [see the dashed lines in Fig. 2(b)] when *w* and *h*
_{up} are fixed at 600 nm and 400 nm respectively. The cutoff width for the high order SPP mode (first order hybrid LRSPP) is *w* = 1011 nm when the thicknesses of the Si_{3}N_{4} ridge are fixed at *h*
_{up} = 400 nm and *h*
_{down} = 310 nm. Both the two dimensions of the Si_{3}N_{4} ridge for supporting the first waveguide mode and high order SPP mode are far beyond the maximum point of the hybrid LRSPP mode.

## 4. WRRs of high performance based on the asymmetrical DMD waveguide

The most important features of the asymmetrical DMD waveguide are compared with the optimal DLSPP waveguide. The lateral effective mode size (*W*
_{x} = 726 nm) is only about 80% of that of the DLSPP waveguide and the propagation length (*L* = 321 *μ*m) is as long as about 7 times of that of the latter [9–11]. When DLSPP waveguide use the same high index material as our asymmetrical DMD waveguide, the smallest lateral effective mode size (*W*
_{x} = 536 *μ*m at the ridge width *w* = 300 nm and height *h* = 650 nm) is about 3/4 of the hybrid LRSPP case, but the propagation length is reduced to *L* = 30 *μ*m, which is only about 1/11 of that of the hybrid LRSPP. For the simultaneously obtained tight field confinement and long propagation length, the asymmetrical DMD waveguide may possess significant applications in highly integrated photonic devices, such as S-bends, Y-splitters, Mach-Zehnder interferometers, WRRs and so on. Here, compact and high performance plasmonic WRRs are simulated as an example, which is schematically shown in Fig. 3(a)
with a ring radius of *r*, and the cross section of the ring waveguide chooses the previous parameters at the maximum point. The unloaded quality factor *Q*
_{spp} of the plasmonic WRR is evaluated by *Q*
_{spp} = *f*
_{Re}/2*f*
_{Im}, where *f* = *f*
_{Re} + *if*
_{Im} represents the complex-valued eigenfrequency of the hybrid LRSPP mode [18,19]. The resulting *Q*
_{spp} [see the square dots in Fig. 3(b)] increases from about two hundreds to about three thousands with increasing *r* due to the decreasing radiation loss. Because of the tight field confinement and long propagation length of the hybrid LRSPP mode, the absolute value of *Q*
_{spp} is very high compared with other compact plasmonic WRRs. For instance, for typical compact WRRs with *r* = 5 *μ*m, the *Q* factor of the asymmetrical DMD waveguide based WRR is as high as about 1380, which is one order of magnitude greater than those of the DLSPP and channel plasmon polariton (CPP) cases [11,20]. The highest fundamental SPP *Q* factor in the plasmonic whispering-gallery microcavity (silver coating) is reported to be about 1800 [19], while the hybrid LRSPP *Q* factor of our finite width DMD resonator is about 2700 at the same radius (*r* = 10 *μ*m). For traditional LRSPP waveguides of symmetrical [7] and asymmetrical structures [21], although they have long propagation lengths in the order of mm, they cannot be made so compact because their optimal bend radius is as large as about 10 mm owing to the weak field confinement and large radiation loss in the tight radius structures [7,8]. Therefore, the asymmetrical DMD waveguide has better bending characteristic than DLSPP, CPP, plasmonic whispering-gallery microcavity, and the traditional LRSPP in the tight radius region due to the strong field confinement and the low radiation loss in the WRR.

To further clarify the origin of the high performance of the asymmetrical DMD waveguide based WRR, the unloaded quality factor *Q*
_{die} of a dielectric WRR is also calculated for comparison. The dielectric WRR is based on a pure dielectric waveguide with the same dielectric dimension (600 × 710 nm^{2} Si_{3}N_{4} placed on the silica substrate, without metal strips) as the previous asymmetrical DMD waveguide. This pure dielectric waveguide supports two waveguide modes with different polarizations. The corresponding *Q*
_{die} of the two modes are also calculated and the higher one is plotted in Fig. 3(b) (circle dots). At a large radius (*r*≥7 *μ*m), *Q*
_{spp} is less than *Q*
_{die} since the pure dielectric waveguide has no propagation loss. At a small radius (*r*≤6 *μ*m), *Q*
_{spp} is greater than *Q*
_{die} due to that the asymmetrical DMD waveguide has tighter field confinement and lower radiation loss because of the existence of the metal strip. The ratio *Q*
_{spp}/ *Q*
_{die} displayed in Fig. 3(b) (down-triangle dots) shows this effect clearly. As we see, *Q*
_{spp}/ *Q*
_{die} is greater than 1 when *r*≤6 *μ*m. At *r* = 3 *μ*m, *Q*
_{spp} can reach up to 470, which is about 1.8 times of *Q*
_{die}. Thus we can conclude that although the asymmetrical DMD waveguide has additional propagation loss (ohmic loss due to the metal strip), it possesses better bending characteristic than the pure dielectric waveguide at the tight radius region in the WRR where the radiation loss is dominant. And it is due to the existence of the metal that results in the stronger field confinement and lower radiation loss. In fact, the above effect is more evident for silver strips which have less ohmic loss. Simulations show that at *r* = 6 *μ*m, if we replace the gold strip by silver strip, *Q*
_{spp} can reach up to 4100 which is about 2.5 times of *Q*
_{die}. Moreover, *Q*
_{spp} can reach up to about 9200 at radius *r* = 10 *μ*m, which is 5.1 times of the highest one of the plasmonic whispering-gallery microcavity reported in reference [19].

Adding 20 nm Si_{3}N_{4} film to the pure dielectric waveguide to achieve exactly the same cross section of ridges doesn’t affect the conclusion above. For example, at *r* = 3 *μ*m, the hybrid LRSPP resonator has *Q*
_{spp} = 470, when changing the metal film to Si_{3}N_{4} film, the pure dielectric waveguide resonator has *Q*
_{die} = 287, and *Q*
_{spp}/*Q*
_{die} = 1.64, which is very close to 1.8 as calculated in Fig. 3(b).

Next, let us further clarify the role of the metal stripe in improving the bending characteristic of the waveguide. When the size of the dielectric ridge is large, the electromagnetic field can be well confined in the dielectric ridge, so adding a metal stripe to this ridge cannot improve its lateral confinement and bending characteristics. However, in order to improve density of integration and avoid the multimode, waveguide with small ridge size (<1000 nm) is much more important in applications. In this case, the effective refractive index of the pure dielectric waveguide is still much greater than the refractive index of the air, but it is close to the refractive index of the substrate. So the electromagnetic field distributed in the substrate increases with decreasing the dimension of the dielectric ridge, which can result in large radiation loss in tight radius structures. While, as we known, SPP has larger effective refractive index and stronger field confinement, so adding a metal stripe to support hybrid LRSPP can decease the proportion of the electromagnetic field distributed in the substrate and decrease the radiation loss significantly. That’s why the hybrid LRSPP based WRR possesses higher *Q* factor than the pure dielectric waveguide case [as shown in Fig. 3(b)] despite of the SPP ohmic loss. The above result implies that metal structure and SPP can improve the device performance in the small structure size region. And since strong lateral confinement and tight radius structures both can improve the density of integration, the asymmetrical DMD waveguide may have wide and important applications in the field of highly integrated optics.

## 5. Conclusion

In summary, we have numerically studied the finite width asymmetrical DMD waveguides placed on the silica substrates. The asymmetrical nature of the waveguides makes them experimentally realizable. Calculations show that, by adjusting the thickness of dielectric ridge to satisfy the index-matching condition, the hybrid LRSPP mode guided by the asymmetrical DMD waveguide can have long propagation length (*L* = 321 *μ*m). Tight field confinement (*W*
_{x} = 726 nm and *W*
_{y} = 647 nm) can also be achieved simultaneously due to the finite width of the DMD waveguide and the large refractive index difference between the dielectric Si_{3}N_{4} and air. The *Q* factor of a compact WRR based on such asymmetrical DMD waveguide is one order of magnitude greater than those of DLSPP and CPP based WRRs of the same radius, and it is also greater than the highest one in the plasmonic whispering-gallery microcavity at the same radius. Compared with pure dielectric waveguide based WRR, the *Q* factor is also higher for the asymmetrical DMD waveguide case in small scales (<*λ*/2) of the ridge size and at the tight radius region (*r*≤6 *μ*m) because of the stronger field confinement and lower radiation loss owing to the existence of the metal stripe. Thus the excellent performance, easy and flexible fabrications imply that such asymmetrical DMD waveguides will have wide and important applications in the field of highly integrated optics.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 10821062 and 10804004), the National Basic Research Program of China (Grant Nos. 2007CB307001), and the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 200800011023).

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