## Abstract

We describe a layered metal-dielectric waveguide, whose fundamental mode has an effective index as high as 7.35 at 1.55*μ*m, enabling subwavelength spatial confinement. The loss is found to be reasonable in relation to the confinement. The indefinite dielectric tensor of the stratified metamaterial core generally leads to multimode operation of the waveguide, exhibiting a “reversed” mode ordering contrary to conventional waveguides. The waveguide features a strong leveraging in modal index change subject to a change of index in the dielectric layers, opening the design possibilities of very compact active electro-optic devices.

© 2011 Optical Society of America

## 1. Introduction

Recent progress in integrated photonics primarily exploited the silicon-air material system, which has a relatively large refractive index contrast [1, 2]. However, one can show that at a vacuum wavelength (*λ*_{0}) of 1.55*μ*m the minimum lateral field width for a planar silicon waveguide in air is about 300nm. Thus, any attempts to increase the nanophotonics integration should surpass this value; for this purpose, it is necessary to find a successor to the current silicon nanowire technology. Such successors seem to have to rely on materials with negative *ɛ*, notably metals [3]. Metals offer a possibility for increasing the integration density in photonic circuits in two ways: (1) As surface-plasmon polaritons (SPPs) are guided by metal-dielectric interfaces, subwavelength diffraction-free waveguides can be realized, exemplified the slot waveguide [4,5]; (2) One can artificially engineer metal-based metamaterials with very large effective refractive indices, which can then be deployed in waveguides to achieve tight mode confinement. In the literature, the second method above has been comparatively less explored. Although there have been studies on waveguides based on metamaterials [6–8], they are mostly not directly intended for the photonic integration purpose.

In this paper we incorporate a layered metal-dielectric metamaterial with a very high effective index as the core of a waveguide to achieve tight mode guidance. Even with a core size of less than 0.005*μ*m^{2}, its fundamental mode exhibits an effective mode index (*n*_{eff}) as high as 7.5 at *λ*_{0} = 1.55*μ*m. The high *n*_{eff} allows in principle miniaturized optical devices with higher lateral packing density than the current Si-based approach. Also, interestingly enough, a change in the refractive index of the dielectric material leads to a corresponding change in the mode effective index, but amplified by more than 3 times. Such a leveraged increase in *n*_{eff} can help to produce, e.g. very compact modulators. We notice that there exist a number of studies on layered metal-dielectric media (e.g. [9, 10]). Especially a waveguide based on a similar geometry was briefly mentioned in [9]. However, the study in [9] was only on a slab geometry, without relying on realistic material parameters; it also lacks general discussion on properties over a wide spectral span while in reality such waveguides are inherently dispersive. Other related studies focus on different properties of the multilayered metal-dielectric structure than what we treat here. For example, in [10], it is proposed that such a metamaterial medium, with an appropriated composition of the two materials, can have its surface plasmon resonance shifted to a much longer wavelength compared to a bare metallic material.

## 2. Geometry and subwavelength confinement

The layered metal-dielectric medium is schematically shown in the inset of Fig. 1(a). It is characterized by the period *a* and the filling fraction of the metallic layer *f _{m}* =

*d*, where

_{m}/a*d*is the thickness of the metal layer. A

_{m}*slab waveguide*can be formed when one chooses

*N*periods of the metamaterial along

*y*(hence height of the core

*h*=

*Na*) and place it in a low-index cladding (air is always considered here). One can further truncate the slab to a finite width along

*x*(core width, or

*w*) to obtain a 3D

*stripe waveguide*design. In all cases wave propagation is along

*z*. Further, to make use of the plasmonic nature of the system, we consider wave propagation in the

*yz*plane with its magnetic field perpendicular to the plane, i.e. with the so-called transverse-magnetic (TM) polarization [13]. Silver is used in this study, whose permittivity is described by the Drude model as ${\varepsilon}_{\text{m}}=1-{\omega}_{p}^{2}/\left({\omega}^{2}+i\omega \gamma \right)$, with the plasma frequency

*ω*= 1.38 × 10

_{p}^{16}rad/s and the collision frequency

*γ*= 3.2258 × 10

^{13}rad/s [11]. Our investigations do not rest on a specific dielectric material, but in general use a permittivity value of

*ɛ*= 9. A practical medium with index around 3 can be silicon or gallium lanthanum sulphide (GLS) [12] etc.

_{d}If the period of the layered medium is infinitesimal, one can use the Maxwell-Garnet theory (MGT) to treat the stack as an effective medium with its effective permittivity tensor components obtained as [9, 10]

*f*= 1 –

_{d}*f*. In such a homogenized effective medium a natural

_{m}*z*-propagating mode solution is a

*y*-polarized plane wave whose wave number is solely determined by its

*ɛ*permittivity component. In other words, in this MGT limit we can, for the concerned polarization, simply obtain the medium’s effective refractive index after homogenization as $\overline{n}=\sqrt{{\varepsilon}_{y}}$. For a stack with

_{y}*f*= 0.8, we calculate its

_{m}*n̄*as shown by the black curve in Fig. 1(b). A feature significant to this study is that, beyond the resonant wavelength ∼ 0.83

*μ*m, the medium has a rather large

*n̄*, valued at 7.92 at

*λ*

_{0}= 1.55

*μ*m. Such a value far exceeds those in natural dielectric materials. The loss of the medium at

*λ*

_{0}= 1.55

*μ*m is at a reasonable level of 1.46dB/

*μ*m. Here we should point out that the portion of the black curve in Fig. 1(b) to the left of the resonant wavelength (shown as dotted line) is not meaningful for the current study since over there we have negative

*ɛ*, implying evanescent field within the layered medium.

_{y}Motivated by the finding, in the following we calculate the real part of the effective mode index *n*_{eff} and the loss of a stripe waveguide with *h* = 5.5*a* (5 bilayers and an extra silver layer) with *a* = 20nm and *w* = 40nm. *f _{m}* = 0.8 is used. The choice of the waveguide height does not affect the main conclusion of this study. Here we simply chose this value since the height is much smaller than that of a Si nanowire waveguide (∼450nm); at the same time the stack still shows properties agreeing with the MGT prediction. The width of the waveguide is kept small such that no high-order modes with a vertical (

*y*-oriented) nodal line would appear. Nevertheless, such a waveguide is always multimode with

*x*-oriented nodal lines present in mode fields, with the number of modes determined in general by the number of dielectric layers in the core stack [9]. The fundamental mode has the properties as shown by the solid red curves in Fig. 1(b) and 1(c). COMSOL Multiphysics is used for the simulation. The curves agree reasonably well with those based on the MGT predictions. The resonant wavelength is noticeably shifted to a shorter wavelength (slightly less than 0.5

*μ*m). We also present in Fig. 1(b) and 1(c) the curves (red-dashed) for the corresponding slab structure (infinitely extending along

*x*), calculated with an analytical transfer-matrix method which takes full account of the nanostructured core. Very close agreement between the slab and stripe waveguide results is noticed. We point out that here we have used

*a*= 20nm. When a smaller

*a*and more periods are used, both the

*n*

_{eff}and the loss results for the slab and stripe waveguides asymptotically approach those of the MGT results (not shown).

For the stripe waveguide, most importantly, the high *n*_{eff} value of the mode is retained. Given *n*_{eff} = 7.35 at *λ*_{0} = 1.55*μ*m, the transverse decay constant of the mode intensity in air is
$1/\left(2{k}_{0}\sqrt{{n}_{\text{eff}}^{2}-1}\right)=17\text{nm}$. Therefore a rough estimation of the mode field size is about 0.01*μ*m^{2}, about 20 times smaller than that of a Si nanowire with a crosssectional size of 450×250nm^{2} placed in air. We plot in Fig. 2 distributions of three field components and the *z*-component of the Poynting vector *S _{z}* for the fundamental mode at

*λ*

_{0}= 1.55

*μ*m. The fields plotted are the non-vanishing components for the waveguide in its slab limit. The propagation loss at this wavelength is 3.3dB/

*μ*m. Even though the propagation distance of a signal along such a guide is limited to a few micrometers, many functional devices, such as splitters, ring resonators, modulators etc, can be realized owing to the small mode size.

The intriguing characteristics of the higher-order modes merits a detailed, independent discussion, which will be presented in Section 4.

## 3. Leveraged modulation sensitivity in mode index

Due to the highly composite nature of the waveguide, its guidance properties are sensitive to a number of parameters. Among these parameters, the dielectric filling fraction *f _{d}* and the index of the dielectric medium

*n*are of special interest. The effects of these two parameters can be easily revealed by calculating the homogenized index

_{d}*n̄*of the layered medium in the MGT limit. With default parameters of

*ɛ*= 9 and

_{d}*f*= 0.2, we notice that when

_{d}*f*is increased from 0.2 to 0.25, a blueshift of the resonance [from the original 0.83

_{d}*μ*m as in Fig. 1(b)] is observed; the corresponding shift of the (

*n̄*) curve leads to a decrease in

*n̄*(from 7.915 to 6.756 at 1.55

*μ*m). When

*n*decreases from 3 to 2.5, the

_{d}*n̄*value of the medium similarly decreases, from 7.915 to 6.233 at 1.55

*μ*m.

The second sensitivity study above shows that the metamaterial exhibits a change in index over three times as large as a corresponding change in the dielectric index. Such an unusually high sensitivity of the metamaterial’s effective index to *n _{d}* can be exploited for enhanced light modulation. Based on the waveguide studied in Fig. 2, we numerically calculate the change in the fundamental mode index

*n*

_{eff}(Δ

*n*

_{eff}) when

*n*changes from 3 to 3.1. A 0.1 change in refractive index is possible in some reported materials, e.g. GLS [12] and some other chalcogenide glass [14]. In Fig. 3, we show Δ

_{d}*n*

_{eff}versus wavelength. Δ

*n*

_{eff}is in general over 3 times as large as the change in

*n*, affirming the MGT prediction. At

_{d}*λ*

_{0}= 1.55

*μ*m, we have Δ

*n*

_{eff}= 0.36. The characteristic lengh

*L*of a phase modulator is determined by the length over which a light signal experiences a

*π*phase change compared to a reference, or $L=\frac{\lambda}{2\Delta {n}_{\text{eff}}}$. Therefore, the length of a phase modulator based on the proposed waveguide can be as short as 2.2

*μ*m. Given such a length, the difference between losses in two branches of a Mach-Zehnder modulator is about 0.58dB, which can easily be corrected by controlling the power splitting in the modulator.

## 4. Reversed mode ordering

The waveguide presented in Fig. 2 has a peculiar multimode guidance property, even with a cross-sectional dimension of 116 × 40nm^{2}. Overall the waveguide guides five modes. The effective mode indices as well as their losses are shown in Figs. 4(a) and 4(b) respectively over the wavelength range of 0.5 ∼ 2*μ*m. For all higher-order modes, *x*-oriented nodal lines appear, as shown by the *H _{x}* field plots in the insets of Fig. 4. A striking feature from Fig. 4(a) is that the higher-order modes have higher

*n*

_{eff}values than the fundamental one, as opposed to the situation in all-dielectric waveguides. In general, the loss of a higher order mode is higher.

We explain the reversed mode ordering from the MGT limit by treating the waveguide as if its core material is homogeneous but with an effective permittivity tensor of the form diag(*ɛ _{x}*,

*ɛ*,

_{y}*ɛ*). To further simplify our analysis, we examine the problem in the slab waveguide limit, where the waveguide extends infinitely along

_{z}*x*. According to MGT, at the wavelength range of 0.85∼2

*μ*m, the medium has a positive

*ɛ*, while negative

_{y}*ɛ*and

_{x}*ɛ*. With such an indefinite permittivity tensor [15,16], one can analytically obtain the guided TM modes (with

_{z}*H*,

_{x}*E*,

_{y}*E*). The dispersion and loss curves (not shown) are found to be similar to those in the actual 3D waveguide case; in particular, the modes also have reversed ordering. To see the cause more clearly, one can, for the slab case, show that the wave equation in the core is

_{z}*k*

_{0}is the vacuum wavenumber. In a conventional isotropic waveguide with a positive permittivity, the expression in brackets is negative, giving a transversally oscillatory solution in the core. The higher the mode order, the lower the

*n*

_{eff}and the larger the transversal spatial frequency of the field. On the other hand, for a negative

*ɛ*, as is the case here, the opposite is true: the requirement of an oscillatory solution in the core mandates a positive value of the expression in brackets; higher-order modes with higher transversal spatial frequencies correspond to higher

_{z}*n*

_{eff}values, just as calculated.

In practical applications, one can selectively couple light to the fundamental mode. Due to the fact that the *n*_{eff} values (correspondingly the propagation constants) of higher-order modes are separated from the fundamental one by a fairly large margin, the adverse coupling to the higher-order modes can be safely ignored as long as the fabricated waveguide does not show severe irregularities.

## 5. Comparison with other waveguides and discussion

We finally compare the proposed waveguide with the slot waveguide as well as the Si waveguide at *λ*_{0} = 1.55*μ*m, all in their slab limits. According to Fig. 1(b) and 1(c), our slab waveguide has *n*_{eff} = 8.91 with a loss of 2.8dB/*μ*m. Such a high *n*_{eff} value can only be achieved in a plasmonic slot waveguide if its dielectric gap (with index *n _{d}*) is as small as 7nm with a loss of 3.2dB/

*μ*m in the case of

*n*= 3, or even smaller at 0.7nm with 3.6dB/

_{d}*μ*m in the case of an air gap. The requirement of an extremely narrow gap size will seemingly increase the mode mismatch issue with conventional waveguides. As for the sensitivity in mode index, our slab waveguide exhibits a Δ

*n*

_{eff}= 0.40 subject to a Δ

*n*= 0.1, whereas for the slot waveguide case (7nm gap filled with

_{d}*n*= 3) one has Δ

_{d}*n*

_{eff}= 0.37. Though rather similar performances in the two cases, the waveguide presented here has a larger mode size which eases the coupling issue; at the same time, from the modulation perspective, it requires only the same voltage as the slot waveguide counterpart for obtaining the same electric field in the dielectric layers through operating the metal layers as capacitors in parallel. For a 250nm-thick Si slab in air background, its guided transverse-electric mode, apart from a small

*n*

_{eff}= 2.91, only has a Δ

*n*

_{eff}= 0.10 when a 0.1 change in Si index occurs.

In Section 3, we have briefly mentioned the effect of the dielectric filling fraction *f _{d}*. A value of

*f*= 0.2 is used consistently for the current investigation. An increase of

_{d}*f*will lead to a decreased effective index

_{d}*n̄*of the layered medium, which will in turn lead to a smaller

*n*

_{eff}of a waveguide based on the medium, or equivalently a less confined mode. One can on the other hand decrease

*f*for an even higher

_{d}*n̄*. However that will lead to a thinner dielectric layer (from the current 4nm), which would pose challenges to nanofabrication processes. To a certain degree, one can adopt a larger period for the layered medium to circumvent the problem. However care has to be taken to prevent the waveguide from getting bulky or being similar to weakly-coupled slot plasmonic waveguides.

## 6. Conclusion

In conclusion, we have proposed a type of layered metal-dielectric waveguide which exhibits an extraordinarily high effective mode index for achieving subwavelength light channeling. Waveguides based on such a metamaterial can potentially enable photonic integration with a footprint surpassing that based on notably the silicon photonics technology. One special property of such a waveguide is that the effective mode index of its guided mode shows a much greater variation (more than three times) than a variation in the refractive index in the dielectric layers. Such a leveraged mode index sensitivity can be potentially used for realizing, e.g. a phase modulator as short as 2.2*μ*m at the telecommunication wavelength. The multimode operation and the reversed mode ordering found in such a waveguide are discussed and explained theoretically.

## Acknowledgments

This work is supported by the Swedish Foundation for Strategic Research (SSF) and the Swedish Research Council (VR).

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