1. Introduction

Surface plasmon polaritons (SPP) are electromagnetic waves coherently coupled to electron oscillations and propagating at the interface between a dielectric and a conductor, evanescently confined in the perpendicular direction[1]. Recently, surface plasmon polariton waveguides have been received considerable attention for their ability to simultaneously guiding and controlling light in optical waveguides. A variety of integrated optical devices based on surface plasmon waveguides, such as optical attenuators[2,3], modulators[4], switches[5], Bragg gratings[6], etc., have been demonstrated.

Since the SPP waveguides are the key elements to build these optical components, many kinds of SPP waveguides geometries have been analyzed theoretically and experimentally. For two dimensional (2D) planar SPP waveguides, the insulator-metal-insulator (IMI) or the metal-insulator-metal (MIM) heterosturctures were investigated a long time ago. In symmetric IMI heterostructures, when the metal thickness is thin enough, the SPP waves guided by the two metal-dielectric interfaces can coupled efficiently and two kinds of SPP modes can arise: the long-range SPP (LRSPP) mode and the short-range SPP (SRSPP) mode. Due to the limitation of confinement only in one dimension for these 2D SPP waveguides, the three dimensional (3D) SPP waveguides which can confine the SPP wave in two dimensions while propagating are investigated greatly. The two typical SPP waveguide geometries are the metal strip waveguides and the channel plasmon polariton (CPP) waveguides. Theoretical studies show that the metal strip SPP waveguides can support LRSPP modes with two dimensional confinement and a very low propagation loss[7–9]. Recently, a few low loss metal strip waveguides have been realized and propagation loss of < 2dB/cm (propagation length of a few centimeters) in telecom wavelength (1550nm) was achieved[10–13]. Although with the relative low loss, both simulation and experimental results show that the mode sizes of the metal strip SPP waveguides are on the order of a few micrometers which do not suit for high photonic integration. This is because most optical fields are penetrated into the dielectric cladding and substrate. The merits of the metal strip SPP waveguides compared to the conventional dielectric waveguides are their capability of guiding and controlling light with the same metal strip simultaneously. While the CPP waveguides are very promising for high density integration because they can confine the light in sub-wavelength scale. Recently, various types of CPP geometries have been investigated theoretically and experimentally, which including the metal grooves[14–16], metal gap[18,19], and meal hole[20,21]. Generally, the losses of these CPP waveguides are high because more optic fields reside in the metal. It is obviously that the SPP confinement is achieved primarily by decreasing the SPP field into the dielectric, thereby increasing the SPP power being absorbed by metal, so there is a trade-off between SPP loss and SPP mode confinement. In order to optimize this trade-off, the dielectric-loaded surface plasmon-polariton (DLSPP) waveguide with reduced mode size based on high index contrast dielectric waveguide on metal film was demonstrated recently. Sub-wavelength confinement (915nm) and moderate propagation length (a few tens of micrometers) at telecom wavelength (1550nm) was achieved[22–25].

In this work, a symmetric dielectric-loaded surface-polariton (SDLSPP) waveguide is presented. Two types of bound modes (the symmetric mode and the asymmetric mode) supported by this structure are analyzed. The mode effective indices, propagation lengths, mode sizes and mode shapes of the bound modes are obtained using a fininte-element method (FEM). In the FEM method, the region of interest is subdivided into enough small triangle segments, and the partial differential equations are solved for the propagation constant β and magnetic field components. The boundary condition used at the edge of computational window is that of a perfect electric conductor. And at the interfaces between the core, cladding and substrate, the continuities of the tangential components of magnetic and electric fields and normal components of the electric and magnetic flux densities are used [24]. In the following section, the bound modes’ characteristics of SDLSPP waveguides are obtained first. Then some comparisons with DLSPP waveguides are presented. Finally, a discussion on the advantages of our suggested configuration will be summarized. The results show that the symmetric mode (LRSPP mode) with both of sub-wavelength confinement and relative low loss (Propagation length of a few hundred micrometers) at telecom wavelength (1550nm) can be realized.

2. Bound modes’ characteristics of SDLSPP waveguides

 

Fig. 1. The cross section of SDLSPP waveguide (a) and the cross section of DLSPP waveguide (b)

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A schematic of the proposed SDLSPP waveguide structures is presented in Fig. 1(a). Also the DLSPP waveguide cross section is given in Fig. 1(b) for comparison. The refractive indices of core, cladding, substrate, and metal are n_c, n_cl, n_s and n_m, respectively. The metal film thickness is d, the width of core is w and height of the core is h. Also the coordinate is given where x, y and z are the transverse, lateral, and propagation direction respectively. Compare to the DLSPP waveguide, the SDLSPP waveguide has two main different: one is that the SDLSPP has two dielectric core with high refractive index allocated at upper and bottom sides of a thin metal film symmetrically; The other is their dielectric refractive index distribution: the refractive indices relation of n_c > n_s = n_cl is fulfilled in the SDLSPP waveguide while the relation of n_s > n_c > n_cl in the DLSPP waveguide. The merits of SDLSPP structure is obviously: both the LRSPP modes (symmetric modes) with low loss and the asymmetric modes can be supported by the symmetric refractive index distributions in region 1, 2 and 3. Also by using the effective index method (EIM)[17], the effective index in region 2 is larger than that of region 1 and region 3, so the SPP field can also be confined in the transverse direction. While in DLSPP, only the asymmetric mode with high loss (propagation length of a few tens of micrometers) is supported due to the high refractive index difference between the cladding and substrate [10, 13]. Besides, the LRSPP mode size can be drastically reduced in the lateral direction by employing the core with high refractive index just like the conventional dielectric waveguides. With these modifications, both the mode size and propagation length can be optimized. In the simulation results presented hereafter, a excitation wavelength λ = 1550nm, refractive index of cladding, substrate n_cl = n_s = 1.46, core n_c = 1.65, gold n_m = 0.55 + 11.5i[24], the width and height of dielectric core w = 600nm, h = 300nm, and metal film thickness of d = 20nm, are used when simulating the SDLSPP bound mode characteristics unless noted other wise. These refractive indices can be realized by polymer materials easily [26, 27].

 

Fig. 2. The real parts of symmetric (LRSPP) and asymmetric mode effective indices with various metal film thicknesses (a). The propagation length of symmetric (LRSPP) mode with various metal thicknesses (b). The propagation length of asymmetric mode with various metal thicknesses (c).

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Because the metal film thickness has great effect on the LRSPP mode loss, the mode effective indices and the propagation lengths with different metal film thickness d are analyzed. The results of the symmetric mode (LRSPP) and asymmetric mode supported by the SDLSPP waveguide and the planar IMI waveguide are show in Fig. (2). The structure of planar IMI waveguide is also shown in the inset in Fig. 2(b). And the refractive indices of insulator of IMI(A) and IMI(B) structure are 1.46 and 1.65 respectively, which correspond to the n_cl and n_c value of SDLSPP waveguide. In Fig. 2(a), it is clear that the real parts of mode effective indices (Re(n_eff)) of symmetric and asymmetric of SDLSPP waveguide hold the same trends as that of IMI structures: The real parts of effective refractive indices of all symmetric modes increase with increasing metal thickness while asymmetric modes have the contrary trend. Also it is very reasonable that the Re(n_eff) of SDLSPP waveguide is between that of IMI(A) and IMI(B) structure because the three regions of SDLSPP can be regarded to IMI(A) and IMI(B) structures respectively. The propagation length is related to the image part of mode effective index (Im(n_eff)) according to $L_{\mathrm{prop}}=\frac{\lambda }{\left(4\pi \bullet \mathrm{Im}\left(n_{\mathrm{eff}}\right)\right)}$[24]. By using this relation, in Fig. 2(b) and Fig. 2(c), it is obvious that the propagation lengths of symmetric and asymmetric modes of SDLSPP waveguide also share the same trends as IMI structures, where the propagation lengths of symmetric modes are much longer than that of asymmetric modes. Besides, it is very reasonable that the propagation length of SDLSPP waveguide is shorter than those of both IMI(A) and IMI(B) structures due to more SPP wave are resided in metal film which bring the ohmic loss. This is because the SDLSPP waveguide adds the confinement of the SPP wave in transverse (x) direction and also enhance the confinement in lateral (y) direction due to the high refractive index core, which the IMI structures do not have. Although the symmetric mode propagation length of SDLSPP waveguide is shorter than that of IMI structures, it is still much larger (a few hundreds micrometers) than that of DLSPP waveguides (a few tens micrometers) [24] when the metal film is thin enough.

 

Fig. 3. The mode size of symmetric mode of SDLSPP waveguide VS the core width (a); The mode size of asymmetric mode of SDLSPP waveguide VS the core width (b); The propagation length of symmetric mode and asymmetric mode of SDLSPP waveguide VS the core width (c); Three mode shapes in both x and y directions under core with w=200, 500 and 800nm (d).

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From Fig. 2(b), it is obvious that the symmetric mode propagation length of SDLSPP waveguide increases rapidly when the metal film thickness is lower than 40nm. In addition to achieve mode confinement, the mode size and corresponding propagation length of symmetric mode and asymmetric mode under various core widths w is investaged. The results are show in Fig. (3) and the mode sizes (including the mode height and mode width) are obtained by measuring the height and width at 1/e of the normalized maximum main electric field ∣E_y∣. In Fig. 3(a) and Fig. 3(b), the mode height, mode width of symmetric mode and asymmetric mode can be decreased by increasing the core width. Also the mode sizes of these modes approach the minimum values when the core width is large enough. Even enlarge the core width, the mode sizes of these bound modes can be increased [24]. And from Fig. 3(c), the propagation length of symmetric mode decreases with increasing core width, because more SPP wave can be confined in the core region where metal loss is enhanced. Besides, the propagation length of symmetric mode is much larger than that of asymmetric mode. Since compare to the symmetric mode, the asymmetric SPP mode penetrates much deeper to the metal film, where greate omhic loss can be arisen. Also the smaller mode size relative to the symmetric mode in Fig. 3(b) proves this. Figure 3(d) shows a few mode shapes of symmetric modes with various core width: w = 200,500,800nm. It is clear the mode shapes is very Gauss like except when the core width is small enough. According to the above results, a trade off between the mode size and propagation length with the core width exists, which is the essential restrict of surface plasmon[1]. So the core width of w= 600nm is chosen in the following optimization.

Not only the geometric parameters can alter the mode size, but also the refractive index difference between the core and cladding can affect the SPP mode size and propagation length greatly just like the dielectric waveguide. By altering the cladding and substrate refractive index n_cl = n_s, the mode sizes and propagation lengths are analyzed with the core index n_c = 1.65. As shown in Fig. 4(a), the mode heights, widths of symmetric and asymmetric modes are both decreased with decreasing the cladding refractive index. This is very physical reasonable because the mode confinement capability of the core region is enhanced by increasing the refractive index difference between the core and cladding. And because this high confinement, more SPP field are attenuated by the metal film, which cause the propagation length decreases rapidly as in Fig. 4(c). Also it is very nice that the mode height and width of symmetric mode can be almost equal when choosing the cladding and substrate refractive index n_c = n_s = 1.4, which can be physical realized by polymers. With these parameters, the symmetric mode size can be very close in the lateral and transverse direction, which induces a Gauss-like symmetric mode shape as show in Fig. 4(d). The mode height, width in Fig. 4(a) is 1452 nm, 1454nm respectively and the corresponding propagation length in Fig. 4(c) is about 850μm. The sub-wavelength confinement in two dimensions and a relative long propagation length are achieved simultaneously. Besides, it is very clear that the propagation length of asymmetric mode (a few tens micrometers) is much shorter than that of symmetric mode.

 

Fig. 4. The mode size of symmetric mode with various cladding index (a); The mode size of asymmetric mode with various cladding index (b); The propagation length according to different cladding index (c); The normalized ∣E_y∣ field distribution (mode shape) of symmetric mode with n_cl = n_s = 1.4 (d).

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3. Comparison with DLSPP waveguides

As shown in Fig. (1), the DLSPP waveguide is actually buildup by the asymmetric IMI structures, while the SDLSPP waveguide is composed of several symmetric IMI structures. Due to the large refractive index differences between the core and substrate, only the asymmetric mode is supported for the DLSPP waveguide. On the contrary, the SDLSPP waveguide can support both symmetric mode and asymmetric mode. The longer propagation length can be achieved by the symmetric mode because comparing to the asymmetric mode, the SPP wave inside the metal film is greatly reduced. In order to compare the mode shapes of DLSPP and SDLSPP waveguide, a same core size of (600nm × 600nm) is chosen for simulation. The results in Figs. 5(a) and Fig. 5(b) shown the mode size of DLSPP waveguide is smaller than that of SDLSPP waveguide especially in the lateral direction, which is because the more SPP field can be resided in the metal film according to the asymmetric mode characteristics. Also due to the asymmetry of the core region with respect to the metal film, the mode shape is highly asymmetric. So the another merits of SDLSPP waveguide is that the Gauss-like symmetric mode shape can be realized while the DLSPP waveguide can not achieve. Comparing to the DLSPP waveguide, the mode size of SDLSPP waveguide is larger than that of DLSPP waveguide. Also the bend loss of SDLSPP waveguide will be a bit larger than that of the DLSPP waveguide. But still the sub-wavelength and relative long propagation length (a few hundreds micrometers) can be achieved simultaneously by the SDLSPP waveguide.

 

Fig. 5. The ∣E_y∣ field distribution (mode shape) of DLSPP waveguide with metal thickness of 100nm, n_cl = n_s = 1.46, n_c = 1.65, w = h = 600nm (a); The normalized symmetric mode ∣E_y∣ field distribution (mode shape) of SDLSPP waveguide with metal thickness of 20nm, n_cl = n_s = 1.46 and n_c = 1.65, w = 600nm, h = 300nm (b)

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4. Conclusion

In this paper, the detail characteristics of bound modes supported by the symmetric dielectric loaded surface plasmon polariton waveguide are analyzed. The simulation results show that the characters of bound modes are just like those of IMI structures. And by the symmetric geometric and refractive index distribution, the low loss symmetric mode (LRSPP) is supported by the SDLSPP waveguide. By introducing the symmetric high refractive index dielectric cores, the LRSPP can be confined in the lateral and transverse directions very well, also a much longer propagation length than that of DLSPP waveguide can be achieved because of the LRSPP mode characteristics. Based on these results, the sub-wavelength confinement (about 1.45μm) and long propagation length (820μm) are realized simultaneously by optimization. The proposed SDLSPP waveguide can be used to made passive optical components for high density photonic integration.