## Abstract

Currently subwavelength surface plasmon polariton (SPP) waveguides under intensive theoretical and experimental studies are mostly based on the geometrical singularity property of such waveguides. Typical examples include the metal-insulator-metal based waveguide and the metallic fiber. Both types of waveguides support a mode with divergent propagation constant as the waveguides’ geometry (metal gap distance or fiber radius) shrinks to zero. Here we study an alternative way of achieving subwavelength confinement through deploying two materials with close but opposite epsilon values. The interface between such two materials supports a near-resonant SPP. By examining the relationship between mode propagation loss and the mode field size for both planar and fiber waveguides, we show that waveguides based on near-resonant SPP can be as attractive as those solely based on geometrical tailoring. We then explicitly study a silver and silicon based waveguide with a 25nm core size at 600nm wavelength, in its properties like single-mode condition, mode loss and group velocity. It is shown that loss values of both materials have to be decreased by ~1000 times in order to have 1dB/*µ*m propagation loss. Hence we point out the necessity of novel engineering of low-loss metamaterials, or introducing gain, for practical applications of such waveguides. Due to the relatively simple geometry, the proposed near-resonant SPP waveguides can be a potential candidate for building optical circuits with a density close to the electronic counterpart.

© 2008 Optical Society of America

## 1. Introduction

Surface plasmon polaritons (SPPs) have recently attracted a great amount of attention in achieving subwavelength optical waveguiding. In such waveguides, light guidance is achieved by coupling of photons and surface plasmon at the metal-dielectric interface. Unlike in conventional dielectric waveguides, light guided by SPP waveguides does not experience a diffraction limit. Therefore mode field of such a SPP waveguide can be squeezed into an arbitrarily small size. However, it should be noticed that not all SPP waveguides reported so far have subwavelength mode field size (MFS). For example, most long-range SPP waveguides (e.g. [1, 2]) achieve relatively low propagation loss because their modal energy is largely distributed in the low-loss dielectric material. Such waveguides usually have a very large MFS. In order to deploy SPP waveguides as integrated optical circuits, one has to design true subwavelength waveguides. To date, there are several types of SPP waveguides known for achieving the goal. The first type is derived from metal-insulator-metal (MIM) structure [3]. In 1D planar limit, two vertically-placed parallel metal plates can be used to confine light in the middle dielectric layer. The even TM gap mode is free of geometrical cutoff and its effective mode index (*n*
_{eff}) diverges as the gap distance shrinks [3]. A horizontal slice of the trilayered structure can in principle squeeze light into subwavelength scale in 2D [4, 5, 6]. Another known method for realizing subwavelength SPP guiding is to use a 2D metal corner, be it either a V channel or a Λ wedge [7, 8, 9]. It has been shown that the smallest MFS of most practical corner waveguides made of Ag and air is limited to ~100nm at 633nm wavelength [9]. For an even smaller MFS, a corner angle at less than 5° and (or) a very sharp corner with a less-than-5nm tip curvature are necessary, which would impose great challenge on our current nanofabrication technology. Notice that the subwavelength field confinement capability of metal-corner waveguides, especially the Λ-wedge waveguide, roots in the tiny surface curvature at their corner tips [9]. From this viewpoint, such Λ-wedge waveguides can be regarded as a derivative from the metallic fiber-shaped waveguide. The singular behavior of the SPP mode guided by a metallic fiber as its radius decreases has been reported in [10].

The abovementioned subwavelength SPP waveguides are all based on geometrical tailoring. In fact, the peculiar guidance principle of SPP waveguides allows their mode field to be confined in a subwavelength fashion without resorting to geometrical tailoring. For a 1D metal-dielectric interface, the guided SPP mode has its neff value defined as

where *ε*
_{+} and *ε*
_{-}(|*ε*
_{-}|>*ε*
_{+}) are permitivities of the dielectric material and metal, respectively. The materials are assumed to be non-permeable. From Eq. 1, it is noticed that *n*
_{eff} can be arbitrarily large, depending on how close (*ε*
_{+}+*ε*
_{-}) is to zero. It follows that the transverse field decay constant in the cladding,
${k}_{t}={k}_{0}\sqrt{{n}_{\mathrm{eff}}^{2}-{\epsilon}_{\mathrm{clad}}}$
(*ε*
_{clad} is either *ε*
_{+} or *ε*
_{-}), can also be made arbitrarily large. This gives rise to the possibility of tightly confined field at the interface. A section of the interface can potentially confine light in nanodimensions in 2D. Such a subwavelength waveguide has the obvious advantage of being structurally very simple. A primiary reason for lack of proper study on such waveguide probably is that, in addition to the divergent propagation constant, the propagation loss will also tend to infinity as the operation is near to the *ε*
_{+}=-*ε*
_{-} resonance condition. In fact, the tradeoff between confinement and loss for SPP waveguides has been observed for a wide varieties of guiding structures (e.g. see [4, 5, 6, 8, 9]). In view of many published results on SPP waveguides, it has generally been accepted that some loss reduction technique (rather than merely geometrical optimization) has to be deployed in order to make functioning integrated optical circuits based on SPP.

In this paper, we first compare the confinement-to-loss relationships for two waveguides sharing the same geometry, with one based on the approach of structural tailoring and the other based on the approach of material variation. Waveguides in both planar and fiber shapes are considered. It will be shown that waveguides based on the two approaches experience comarable propagation loss as their mode fields decrease to subwavelength size. Owing to this factor, near-resonant SPP waveguides deserve as much attention as other types of SPP waveguides do in realizing subwavelength light channeling. From the perspective of integrated photonic circuit, we will specifically look into a realistic waveguide design based on a finite section of near-resonant silver-silicon interface.

## 2. Comparisons

#### 2.1. Planar case

Here we provide a parallel comparison between a MIM waveguide and simply a MI (metalinsulator) waveguide in their confinement versus loss relationships. Refer to the insets in Fig. 1 for illustrations of the structures. For the MIM waveguide, we decrease the width of the insulator layer (or metal gap distance) for achieving subwavelength field confinement. While for the MI waveguide, we increase the *ε*
_{+} value (assumed to be lossless) for achieving subwavelength field confinement. Silver is chosen as the negative-epsilon material for both types of waveguides. Air is used as the filling insulator for the MIM structure. Wavelength is at 600nm. According to [11] we have *ε*
_{-}=*ε*
_{Ag}=-16.08+0.4434*i*. Mode field size (MFS) for the MIM waveguide is equal to the gap distance (*d*) plus the skin depth (*d _{s}*) in two metal claddings, or MFS=

*d*+2

*d*.

_{s}*d*is defined as the distance from the core-clad interface to the position where field has decayed to its 1/

_{s}*e*in amplitude. Specifically,

*d*is related to the transverse decay constant

_{s}*k*

^{-}

_{t}in the cladding as

*d*=1/

_{s}*k*

^{-}

_{t}with ${k}_{t}^{-}={k}_{0}\sqrt{{n}_{\mathrm{eff}}^{2}-{\epsilon}_{\mathrm{Ag}}}$ . MFS of the MI waveguide is calculated as MFS=1/

*k*

^{-}

_{t}+1/

*k*

^{+}

_{t}, where

*k*

^{+}

_{t}is the transverse decay constant in the positive-epsilon material.

The *n*
_{eff}, loss and MFS of the fundamental symmetric TM mode guided by the MIM waveguide as the gap distance decreases are shown in Fig. 1(a) and (b). It is seen that MFS decreases with *d* almost linearly in the initial stage, which is due to the nearly constant *d _{s}* when

*d*is large. At the same time, we observe that loss increases with

*d*, especially when MFS is less than 100nm. The same modal parameters for the MI waveguide as its

*ε*

_{+}value increases are shown in Fig. 1(c) and (d). It is noticed that subwavelength MFS can be achieved when

*ε*

_{+}becomes close to

*ε*

_{-}in absolute value. Similar tradeoff between MFS and loss is observed. One would straightforwardly use confinement-versus-loss as the figure of merit to assess the performances of the SPP waveguides [12]. Here to make a quantitative comparison between two approaches, we eye particularly a MFS value of 100nm. At this MFS, the MIM waveguide has a loss of ~0.5dB/

*µm*, while the MI waveguide has a loss of ~1dB/

*µm*. It therefore shows that the MIM structure can achieve a subwavelength confinement at a reduced propagation loss. However, the loss reduction is only by a factor of ~2. And this loss reduction factor will decrease to ~1.33 when the air is replaced by silica (not shown). Solid filling may be required for certain applications for, e.g. avoiding contamination by water molecules etc.

#### 2.2. Fiber case

In addition to planar-shaped waveguides, a metallic fiber also serves as a perfect waveguide structure for comparing the two aproaches for achieving a subwavelength MFS. Previous analyses of such fiber have appeared in, e.g. [13]; and tapered metallic fiber was also theoretically analyzed in [10] for achieving nanofocusing. Similar to the previous study, we choose silver as the negative-epsilon material, and operating wavelength is at 600nm. Figure 2(a) and (b) show the variations of *n*
_{eff}, MFS and loss of the zero^{th}-order TM mode guided by a silver fiber as the fiber’s radius decreases from 500nm to 1nm. The fiber is placed in air background. The MFS of the SPP mode is calculated as the distance from the field maximum (at core-clad interface) to a radial position in the cladding where the field decays to its 1/*e*. The electric field in the cladding can be expressed analytically in modified Bessel function as
${K}_{0}\left({k}_{0}\sqrt{{n}_{\mathrm{eff}}^{2}-{\epsilon}_{+}r}\right)$
. Here for ease of MFS calculation, we approximate it to exp
$\left(-{k}_{0}\sqrt{{n}_{\mathrm{eff}}^{2}-{\epsilon}_{+}r}\right)$
. In turn we have MFS=1/*k _{t}*, where
${k}_{t}={k}_{0}\sqrt{{n}_{\mathrm{eff}}^{2}-{\epsilon}_{+}}$
. In Fig. 2(c) and (d), we show variations of the same SPP modal parameters for three different fiber radii as the background material permittivity

*ε*

_{+}increases from 1 to ~16 (again assumed to be lossless). From Fig. 2, we observe that the

*n*

_{eff}value tends to diverge either as the fiber’s geometry

*r*shrinks to zero [Fig. 2(a)] or as the material permittivities approach the

*ε*

_{+}=

*ε*

_{-}condition [Fig. 2(c)]. Hence, both geometrical tailoring and material engineering can be deployed independently for achieving subwavelength light confinement.

Here, still, we assess the performances of the waveguides derived from two approaches by examining their loss values when they achieve 100nm MFS. At this particular MFS, the waveguide based on geometrical tailoring experiences a loss of ~1dB/*µ*m [Fig. 2(b)], while the waveguide based on material variation experiences a loss value of ~0.7dB/*µ*m (for both *r*=100 and *r*=1000nm situations). Therefore in the case of fiber geometry, electromagnetic field can be more “cost-efficiently” confined to the metal-dielectric interface for the waveguide based on material engineering as compared to the waveguide based on geometrical tailoring.

Based on the studies presented in above two subsections, one can notice that a single-interface SPP waveguide with a proper choice of materials can achieve subwavelength confinement at a similar loss value as compared to a conventional SPP waveguide based on geometrical tailoring. In fact, one can also see from the studies that, if one solely relies on geometrical tailoring to achieve subwavelength (especially deep-subwavelength) field confinement, the mode property of the final waveguide would be very sensitive to geometrical perturbation. Current fabrication process based on lithography can easily lead to a surface roughness of a few nanometers. Therefore the performance of such waveguides may be heavily influenced by structural irregularities resulted from fabrication. In comparison, geometry is a less significant issue for a subwavelength waveguide based on a single-interface SPP waveguide. In the next section, we pay particular attention to a realistic SPP waveguide design which relies on a section of near-resonant metal-dielectric interface for achieving tight field confinement. Such nanoscaled SPP waveguides can be potentially deployed for building integrated optical circuits.

## 3. Realistic design

Although we know from Fig. 2(d) that subwavelength confinement of light can be achieved even when |*ε*
_{+}| is far smaller than |*ε*
_{-}|, however here we purposely choose |*ε*
_{+}|≈|*ε*
_{-}|, i.e., the near-resonant material parameters, in order to realize deep subwavelength MFS. Two materials with close but opposite epsilon values (in their real part *ε*″) at certain wavelengths do exist in nature, but not without loss. One example is silver (Ag) and silicon (Si). An examination of their dispersion curves tells that their epsilon values meet our requirement around the free-space wavelength of 600nm, at which *ε*
_{Ag}=-16.08+0.4434*i* and *ε*
_{Si}=15.58+0.2004*i* [14]. The imaginary part of the epsilon values (denoted as *ε*”) is directly responsible for attenuation of the guided surface mode. A single surface mode formed by the two materials at *λ*=600nm has a loss value as large as 690.7dB/*µ*m, rendering almost any waveguide built upon such a surface impractical. One of our objectives is to investigate how small the imaginary epsilon values (*ε*”) of Ag and Si should be for practical applications.

A general waveguide cross-section considered is schematically shown in Fig. 3(a). Composition choices of the left and the right cladding regions are many. In the simplest case, two claddings can be the same homogeneous material (*ε*
_{1}=*ε*
_{2}=*ε*
_{3}=*ε*
_{4}). Among other variations, the whole structure can share the same metal substrate (*ε*
_{2}=*ε*
_{4}=*ε*
_{-}), or the same dielectric superstrate (*ε*
_{1}=*ε*
_{3}=*ε*
_{+}); each cladding can also have a different metal-dielectric interface. However, the underlying principle for achieving subwavelength light guiding is the same, i.e. relying upon the near-resonant metal-dielectric interface in the center. As the waveguiding interface has finite lateral size (*w*), the structure can be used to achieve high-density photonic integration in two dimentions. In Fig. 3(b) and (c), two similar SPP waveguides working under two different conditions are shown. By comparing their major mode field, we can clearly observe the advantage of near-resonant operation in terms of confinement. Several SPP waveguides which are similar to that sketched in Fig. 3(a) have been reported (e.g. [15, 16]). However few of the studies have paid particular attention to achieving subwavelength guidance. Here, by starting from naturally available materials, we analyze the feasibility and challenges of such a nanoscaled near-resonant SPP waveguide. In the following, without loss of generality we take a representative configuration in which *ε*
_{2}=*ε*
_{4}=*ε*
_{-} (i.e., common metal substrate). In addition, we assume the claddings materials in the upper half plane (*ε*
_{1} and *ε*
_{3}) as air. Note that the near-resonant operation ensures a tight field confinement even when *ε*
_{1} and *ε*
_{3} are larger than *ε*
_{+} (see e.g. [15]).

First, to make sure the waveguide is single-mode, we calculate the geometric dispersion as a function of the core width w at *λ*=600nm (Fig. 4). Mode derivation is done in COMSOL Multiphysics (from COMSOL AB) with an electric-field- and edge-element-based finite element method. The blue curves (with dots) are the first two modes derived with *ε*”=0 for both Si and Ag materials. The red dots are calculated with *ε*” values reduced to 1% (compared to their natural values). The mode index changes little when the *ε*” values change from 0 to 1%. We will show later that when losses are higher, the waveguide is too lossy to be useful. From Fig. 4, it is seen that the waveguide is single-mode when *w*<27nm. We hence take *w*=25nm in our following analyses. The *n*
_{eff} value is ~15.6 at *w*=25nm, which ensures the mode field is highly evanescent in the cladding regions.

Mode supported by the waveguide is depicted in Fig. 5. For this particular mode derivation, material losses of Ag and Si are all reduced to 1% of their natural values. The field does not change appreciably when the material losses vary from 0 to 0.1 (in fractions of their natural values). In the cladding regions, the mode field decreases to its 1/*e* over a ~6nm distance. Therefore its MFS is approximated to be 37×12nm^{2}. The mode field has a major polarization along y direction. The z-component of the Poynting vector (*S _{z}*) shown in Fig. 5(c) confirms the highly confined energy flow in the waveguide. Notice that, although

*S*in Ag region is negative, the net energy flow is positive.

_{z}The loss of the waveguide with *w*=25nm is then computed as *ε*” values of both Ag and Si are varied. The result is shown in a contour map in Fig. 6. *ε*” values of both materials are varied from 10^{-6} to 10^{-1}, in fractions of their natural values. It is observed that the waveguide loss is almost equally sensitive to variations in each of the two *ε*” values. In practice, the requirement of propagation length depends on the application. Here, given such a tiny circuit cross-section, a loss level of 1dB/*µ*m (corresponding to a propagation length of a few micrometers) could be suitable for a wide range of purposes. A circuit with over 100 length-to-crosssection aspect ratio permits necessary waveguide bends for forming basic components (coupler, interferometer etc) and inter-connecting various ports in a high-density fashion. From Fig. 6, it is shown that both *ε*” values (or equivalently, conductivities of the two materials) have to be decreased by ~1000 times in order to have 1dB/*µ*m propagation loss. It should be noted that keeping the desired negative *ε*′ and decreasing *ε*” will, as dictated by the Kramers-Krönig relations, require either other (meta)materials than the materials employed here, or possibly low temperature operation.

It’s understood that near-resonant 1D SPP exhibits an interesting property of slow group velocity (GV) [15]. However, explicit studies on GV and also the group velocity dispersion (or GVD, which is reposnible for pulse broadening in digital communication links) for 2D SPP waveguides are often ignored in most published works. Here we numerically derive GV and GVD of the particular waveguide depicted in Fig. 5. Results obtained are shown in Table 1. Frequency-dependent *ε*
_{Ag} and *ε*
_{Si} values are taken from [11] and [14] respectively, except that the imaginary parts are kept at their 1%. The result show that the group velocity in this particular waveguide can be slowed down by over 1700 times at the near-resonant operation condition. The huge negative GVD value (notice the propagation length unit is in mm) at 600nm wavelength suggests that such waveguide may be promising for dispersion compensation applications. Further GV and GVD tailoring can be realized by using a multilayer dielectric material, in replacement of the homogeneous region denoted by *ε*
_{+} in Fig. 3 [15].

## 4. Conclusion

In conclusion, we have shown that apart from solely relying on geometrical tailoring, choosing appropriate materials can be an equally compelling approach for achieving small mode field size for an SPP waveguide. We systematically studied a nanoscaled optical waveguide based on the phenomenon of near-resonant SPP confined at a silver-silicon interface at 600nm wavelength. In particular, we show that the material losses for both silver and silicon have to be reduced by ~1000 times in order for the waveguide to achieve practical propagation length. Therefore we point out the urgency of material loss reduction for such waveguides. Decreasing environment temperature [15] and using quantum-dot-based metamaterials [17] could be two viable ways to achieving the goal. Such SPP waveguides can be potentially useful for constructing exotic miniature optical devices once the metamaterial or loss reduction/compensation technology matures.

## Acknowledgement

This work is supported by the Swedish Foundation for Strategic Research (SSF) through the INGVAR program, the SSF Strategic Research Center in Photonics, and the Swedish Research Council (VR).

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