## Abstract

The guided modes of a negative refractive index channel waveguide have been numerically investigated. It has been found that the modes exhibit a number of unusual properties that differ considerably from those of a conventional waveguide. In particular, it has been shown that these waveguides can exhibit low or negative group velocity as well as extraordinarily large group velocity dispersion. Calculation of the Poynting vector reveals that it is possible to support a mode with a zero energy flux motivating a simple design for an optical trap.

©2003 Optical Society of America

## 1. Introduction

Negative index materials (NIM) offer a unique possibility to extend the experimental domain and investigate novel physical phenomena. Such materials, which possess simultaneously negative values of the dielectric permittivity *ε* and magnetic permeability *µ*, were theoretically investigated by Veselago in 1968 [1] where he concluded that they would have dramatically different propagation characteristics. In particular, these NIMs can exhibit extraordinary properties such as negative refraction, antiparallel group and phase velocities (backwards waves) and negative energy fluxes (radiation tension). Despite the physical significance of his analysis, the results appeared to be of limited practical application due to the absence of naturally occurring NIMs. However, motivated by earlier investigations [2, 3], in 2000 the first NIM was demonstrated by Smith *et al.* in a composite material consisting of periodic regions of negative *ε* and negative *µ* [4].

Initial experimental investigations of negative refraction phenomena have been conducted in the microwave regime where the fabrication of such composite materials is possible [4, 5]. These experiments confirmed that the electromagnetic waves behave as predicted by the theory. It is, however, unlikely that these composite materials will scale to optical frequencies and instead photonic crystals (PC) have often been suggested as an alternative to extend the effects of negative refraction into the optical regime [6]. Indeed, in 2000 Notomi performed a detailed theoretical and numerical investigation of light propagation in a PC which showed that as the effective refractive index is determined by the photonic band structure it can in fact be less than unity or even negative [7]. Although these PCs may have positive *ε* and *µ* throughout, they have nonetheless been shown to exhibit similar anomalous light behavior to the composite negative *ε* and *µ* materials [8]. Recently a PC exhibiting negative refraction was observed experimentally [9] and despite the fact that this experiment was still conducted in the microwave regime, by using electrically poled crystals [10] more complex structures can be fabricated that should scale to the optical regime.

In this paper, we numerically investigate the properties of the guided modes in a channel waveguide with a negative index (negative *ε* and *µ*) core. Similar analysis has already been performed for a planar waveguide [11]. We show that, as in the planar case, the guided modes differ considerably from those in a conventional positive index waveguide. Typical features of these waveguides include the absence of the fundamental mode, possible double degeneracy of modes and backwards propagating waves with negative energy flux. By calculating the dispersion curves we find that the group velocity can become very low, or negative, whilst the group velocity dispersion can be many orders of magnitude larger than conventional materials. Furthermore, because the low group velocity should lead to reduced nonlinear thresholds, by combining the large dispersions with large nonlinearities it should also be possible to observe novel nonlinear phenomena in extremely short device lengths. We expect these waveguides to find wide applications in many optical technologies including optical data storage, quantum computing, dispersion management and optical soliton formation.

## 2. Guided mode solutions and their properties

We consider a symmetric channel waveguide with the geometry and parameters given in Fig. 1. The dielectric permittivity and the magnetic permeability in the cladding (*i*=1) and the core (*i*=2) are related to their vacuum values via : *ε* (*x,y*)=*ε*
_{0}
*ε*
_{i} and *µ* (*x,y*)=*µ*
_{0}
*µ*
_{i}, respectively. We assume that the cladding has a positive index so that both *ε*
_{1} and *µ*
_{1} are positive, and set the core to have *ε*
_{2} and *µ*
_{2} negative. The guided modes will be stationary solutions to Maxwell’s equations of the form,

where *ω* is the angular frequency of the field, *β* is the propagation constant and E(*x,y*) and H(*x,y*) are the spatially localized transverse mode profiles of the electric and magnetic fields, respectively.

To obtain a qualitative understanding of the these modes, our first analysis is based on Marcatili’s method for 2D optical waveguides [12]. We choose the electric field to be polarized in the *x* direction, i.e., the ${E}_{\mathit{\text{pq}}}^{x}$
mode, which has *E*_{x}
and *H*_{y}
as the principal field components. Here *p* and *q* are integers which correspond to the number of peaks of the optical power in the *x* and *y* directions, respectively. It follows from Maxwell’s equations that the wave equation in this representation is [13]:

where *E*_{x}
is related to *H*_{y}
via,

We then assume that the electric and magnetic fields are confined to the core so that they decay exponentially in the cladding and are negligible in the shaded regions of Fig. 1. Solving for the field in each of the unshaded regions and applying the appropriate boundary conditions at the interfaces we obtain the following dispersion equations :

where the transverse wavenumbers, *k*_{x}
and *k*_{y}
, are related to *γ*_{x}
and *γ*_{y}
via,

The propagation constant *β* is then obtained from,

The dispersion relations of Eqs. (5) and (6) can be solved together with Eqs. (7) and (8) using standard techniques (see Ref. [13]). Previous analysis of negative index planar waveguides has shown that it is possible to obtain solutions where the transverse wavenumber *k* becomes purely imaginary [11]. These “slow wave” solutions occur when *β* exceeds a critical value and have been likened to surface waves in metal films [14]. Thus we extend the parameter planes (*k*_{x}*L,γ*_{x}*L*) and (*k*_{y}*L,γ*_{y}*L*) to include imaginary values of *k*_{x}
and *k*_{y}
by defining : *κ*_{x}
=i*k*_{x}
and *κ*_{y}
=i*k*_{y}
. Fig. 2 shows typical solutions for the *x* and *y* components of the field. The solid lines are the right-hand-sides of Eqs. (5) and (6) and the dashed lines are obtained from the right-hand-sides of Eqs. (7) and (8). Here the 3 dashed lines correspond to waveguides with the same ratio *ε*
_{1}/*ε*
_{2}, but of differing width *L*. The points of intersection indicate the existence of guided modes. From these intersections we can construct six different solutions and to illustrate this examples of the mode profiles can be seen in Fig. 3. The possible ${H}_{2,1}^{y}$ modes are plotted in the top row where the (a,*α*) mode has an imaginary *k*_{x}
but real *k*_{y}
and the (b,*β*) mode has both *k*_{x}
and *k*_{y}
real. The middle row shows the strongly (c,*δ*) and weakly (d,*δ*) localized ${H}_{3,1}^{y}$ modes. Similarly, the bottom row shows the strongly (c,*η*) and weakly (d,*η*) localized ${H}_{3,2}^{y}$ modes, however, in this instance the weakly localized mode has an imaginary *β* so that it decays exponentially as it propagates in *z*. We note that a similar analysis for the ${E}_{\mathit{\text{pq}}}^{y}$
mode (electric field polarized in the *y* direction) leads to *H*_{x}
solutions with forms such as those shown in Fig. 2, but with the *x* and *y* components interchanged. In addition, as in Fig. 2 the solutions are still functions of *L*, it is clear that we will find similar solutions for rectangular guides.

Figures 2 and 3 illustrate some of the important properties of negative index channel waveguides. Firstly, the guided modes can only be supported in high-index waveguides (i.e., *ε*
_{2}
*µ*
_{2} >*ε*
_{1}
*µ*
_{1}). This is in contrast to negative index slab waveguides [11] and is a consequence of the dispersion relations of the *y* component of the field. In addition, as the dispersion relations do not allow for an imaginary *k*_{y}
, only the *x* component of the field can exhibit surface wave effects. Secondly, however, in accordance with the observations in negative index planar waveguides we again find that the conventional hierarchy of the fast modes disappears. In particular, (i) for *p*=0 the right-hand-side of Eq. (5) is negative so that the fundamental mode, ${H}_{1,1}^{y}$, does not exist; (ii) for a given width *L*, solutions of *γ*_{x}*L* associated with the first order ${H}_{2,1}^{y}$ mode only exist for a particular range of *ω* greater than a critical value; (iii) for certain modes, as the *x* solutions in Fig. 2 decrease monotonically with *k*_{x}*L* at different rates, there can be two solutions to Eq. (5) so that it is possible for two modes with the same number of nodes to coexist in a waveguide, as illustrated by solutions (c,*δ*) and (d,*δ*).

To investigate the frequency dispersion of the guided waves we need to consider the frequency dependence of both *ε*
_{2} and *µ*
_{2} in Eq. (9). Although the specific form of the refractive index of a photonic crystal depends of the band structure [7], here we will simply use the forms proposed by Veselago [1] as employed by other authors [4, 6, 11]:

Here *ω*_{p,i}
are the electronic plasma frequencies, *ω*_{0,i}
are the magnetic resonance frequencies and *F* is a constant dependent on the material structure. Based on earlier analysis [4, 11], we choose *ω*_{p,2}
/2*π*=10GHz, *ω*
_{0,2}/2*π*=4GHz and *F*=0.56. In this case, the region of simultaneously negative *ε*
_{2} and *µ*
_{2} ranges from 4GHz to 6GHz. The values of *ω*
_{p,1}/2*π*=2GHz and *ω*
_{0,1}/2*π*=1GHz were then chosen so that *ε*
_{1} and *µ*
_{1} are always positive and *ε*
_{2}
*µ*
_{2}>*ε*
_{1}
*µ*
_{1} in this range. The dispersion curves for the two ${H}_{3,1}^{y}$ modes of Fig. 3, as found from Eq. (9), are plotted in Fig. 4(a) with *L*=2cm. The solid line corresponds to the strongly localized mode (c,*δ*) and the dashed line is the weakly localized mode (d,*δ*). From this we see that as we increase the frequency the two solutions converge until they reach a cutoff frequency *ν*_{c}
associated with their intersection.We also note that although the strongly localized mode exists for frequencies below those plotted here, the weakly localized mode does not as the right-hand-side of Eq. (5) crosses the x axis. Despite the similar appearance of the two modes in Fig. 3, clearly the opposite slopes of β indicate there will be a significant difference in the dispersion properties. Indeed, on calculating the group velocity, *ν*_{g}
=1/*β*
_{1}=d*ω*/d*β*, Fig. 4(b) shows that whilst for the weakly guided mode *ν*_{g}
is positive, for the strongly guided mode *v*_{g}
is negative, corresponding to backward wave propagation [15]. In both cases, however, as the frequency approaches *ν*_{c}*, v*_{g}
approaches zero so that the mode propagation will be slowed considerably. Thus this waveguide offers an alternative to other methods of generating “slow” light (*ν*_{g}
≪*c*), “fast” light (*ν*_{g}
is negative) and perhaps to even trap light. The possibility to slow or trap light has many potential applications such as optical data storage, optical memories and quantum computing. Furthermore, as the light-matter interaction is enhanced for low *ν*_{g}
, slow light can be used to observe nonlinear processes such as harmonic generation and four-wave mixing in even weakly nonlinear materials [16].

The group velocity dispersion (GVD) of the guided modes is calculated via : *β*
_{2}=d^{2}
*β*/d*ω*
^{2}. As seen in Fig. 4(c), as the frequency approaches *ν*
_{c} (the region of low *ν*_{g}
) the waveguide exhibits large anomalous (c,*δ*) or normal (d,*δ*) dispersion. This is typical behavior of the GVD at the band edges of PCs [17]. For these particular modes, the dispersion can be around 7 orders of magnitude larger than that of conventional materials such as silica fibers (20ps^{2}km^{-1}). This makes them idea for dispersion management and particularly for use in integrated circuits where short device lengths are favored. In addition, by exploiting the reduced nonlinear threshold/large GVD combination it should be possible to investigate nonlinear effects such as optical soliton formation so that negative index channel waveguides could offer a simple alternative to devices such as side-coupled integrated space of resonators (SCISSOR) [16].

We have also calculated the energy flux of the guided modes in Fig. 4 which is characterized by the *z* component of the Poynting vector : **S**=Re[**E**×**H***] [13]. Since for backwards waves the Poynting vector and the wave vector point in opposite directions, we expect that the energy flux of the modes will also have opposite signs [1]. The total power flux through the core and cladding regions of the waveguide are calculated as,

For both modes we find that the power flux inside the core is opposite to that in the cladding (see middle row of Fig. 3). However, on calculating the normalized energy flux defined as *P*=(*P*_{core}
+*P*_{clad}
)/(|*P*_{core}
|+|*P*_{clad}
|) [11], Fig. 5 shows that total energy flows in a positive direction for the weakly localized mode and a negative direction for the strongly localized mode, in agreement with the signs of *ν*_{g}
. We note that |*P*| <1 and *P*→1 as the mode becomes poorly confined and *P*→-1 as the mode becomes tightly confined. The significant feature of this result is that as the solutions converge (at *ν*_{c}
), the energy fluxes inside and outside the guide exactly cancel so that the total energy flux vanishes. Importantly, in their analysis for a negative index planar waveguide, Shadrivov *et al.* showed that at *P*=0 the energy flowed in a double-vortex structure so that most of the energy remained localized inside the wave packet [11]. Thus as the energy flux goes to zero the guided modes do not disintegrate and we can expect an analogous result for the modes in a channel waveguide.

In Figs. 4 and 5 we have examined the dispersion and energy flux of the modes as a function of frequency. Alternatively, we can consider the dependence of the mode properties on the waveguide width *L*. Figure 6 shows (a) the propagation constant and (b) the normalized energy flux for a fixed frequency, *ω*/2*π*=5GHz, where again the solid line corresponds to the strongly localized mode (c,*δ*) and the dashed line is the weakly localized mode (d,*δ*). As expected, these have similar forms to the previous curves for varying frequency except this time we see that the two solutions converge as we decrease *L* until they reach a cutoff length *L*_{c}
. However, significantly these results suggest that we can slow the propagating mode simply by adiabatically decreasing the waveguide width. Furthermore, by decreasing the width to the critical length *L*_{c}
it should be possible to stop the light completely. Thus we expect that a simple waveguide structure such as that shown in the inset of Fig. 6 should act as an optical trap where the frequency of light that can be trapped is determined by the range of the waveguide width.

## 3. Discussion and conclusions

Although the analysis described here pertains specifically to channel waveguides with a negative index (negative *ε* and *µ*) core, due to similarities in the geometries and the behavior of light in negative effective index PCs, we expect to obtain analogous results in an optical fiber which exhibits negative refraction. With the rapid developments in the fabrication of photonic crystal fibers (PCFs) [18], it should be possible to design fibers in the future that have photonic band structures such that the core has an effective index which is negative. This is of particular relevance to devices requiring large nonlinearities as these are readily obtained in small core PCFs [19].

In conclusion, we have numerically investigated the properties of the guided modes in a negative index channel waveguide. Our analysis has extended previous studies of negative index planar waveguides showing that, due to the nature of the dispersion relations, the 2D geometry places additional restrictions on the guided modes altering their properties. The results have demonstrated a number of unusual properties that differ considerably from those of a conventional waveguide, such as low or negative group velocity as well as extraordinarily large group velocity dispersion. By calculating the Poynting vector we have also found that it is possible to obtain solutions with a zero energy flux and this has lead us to suggest a possible design for an optical trap. Owing to the large number of exotic properties of these waveguides, we expect that they will find wide application in many areas of optical technology both in the linear and nonlinear regimes.

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