## Abstract

We demonstrate that strongly anisotropic planar dielectric systems can be used to create waveguides supporting modes with extremely slow group velocity. Furthermore, we show that such systems can be used for 3D imaging, with a potential for subwavelength resolution.

©2006 Optical Society of America

## 1. Introduction

Materials with negative refractive index (NIMs) have attracted significant attention in recent years [1, 2]. NIMs’ predicted potential for subwavelength resolution [3] and aberration-free imaging, along with their demonstrated realizability through modern processing technologies, spurned much interest in negative index phenomena.

Despite successful proof-of-principle experiments, significant challenges exist for implementing NIM materials, especially at optical frequencies. The existing methods to achieve negative refractive index either rely on simultaneously negative values of dielectric permittivity and magnetic permeability [4] (which for high frequencies are created via resonant response and are thus accompanied by high losses), or require nontrivial subwavelength-scale material patterning [5], which results in strong sensitivity to disorder. Due to these limitations, all practical NIM designs are currently restricted to GHz frequencies [1].

Recently, a new class of NIMs has been proposed that does not require magnetic resonance or periodic patterning. The proposed method of obtaining negative effective index relies on traveling wave solutions in a strongly anisotropic planar waveguide geometry and presents a promising platform for creating devices at optical frequencies. It was shown earlier [7, 8] that a planar metallic waveguide with sufficient anisotropy can support modes with negative group velocity and exhibit negative refractive index.

In the present paper we perform a detailed analysis of modes in a strongly anisotropic dielectric waveguide and demonstrate that it can enable slow group velocities. Furthermore, we show that when used as a slab lens, this system is capable of 3D imaging with a potential for subwavelength resolution.

## 2. Slow light in negative-index media

Slow light systems, particularly in the solid state, are actively studied due to their intriguing behavior, as well as multiple potential applications. However, all existing methods of slowing light either rely on a narrow material resonance (and thus only allow for limited tunability) [9, 10, 11], or require periodic patterning and are thus highly sensitive to disorder [12]. Here we propose an alternative approach to attaining slow group velocities, which can be implemented in a planar, non-periodic semiconductor system.

Slow light in our system results from combining materials with positive and negative effective refractive index. Negative index behavior, in turn, arises as a consequence of strong material anisotropy, which drastically affects light propagation and leads to negative group velocity [7, 8].

We consider a uniaxial system with negative permittivity along the optical axis. Light propagating in a uniaxial medium may constitute an ordinary or an extraordinary wave, depending on whether the *E*-field vector has a non-vanishing component along the optical axis. Since ordinary waves are not affected by the anisotropy, in what follows we focus on the extraordinary polarization. Taking *x̂* as the direction of the optical axis, we may characterize the extraordinary wave in a uniaxial crystal by the dispersion relation

This equation can be used to understand one key consequence of material anisotropy: the deviation of the Poynting vector *S⃗* from the direction of the wave vector *k⃗*. For sufficiently weak absorption, the direction of the Poynting vector is identical to the direction of the group velocity vector *v⃗ _{g}*=∇

*(*

_{k⃗}ω*k⃗*) [13]. This implies that S⃗ is normal to the isofrequency curves given by Eq. (1).

In Fig. 1 we draw vectors *S⃗* and *k⃗* for a lossless isotropic medium, as well as for two different cases of uniaxial anisotropy (*ε _{x}*,

*ε*>0 and

_{z}*ε*<0,

_{x}*ε*>0). In the isotropic case, the wave vector surfaces are circles, and therefore

_{z}*S⃗*∝∇

*(*

_{k⃗}ω*k⃗*)∝

*k⃗*, i.e.

*S⃗*and

*k⃗*are collinear. For

*ε*≠

_{x}*ε*,

_{z}*ε*>0, the wave vector surfaces become ellipsoidal; as a consequence, the angle between

_{x,z}*S⃗*and

*k⃗*is non-zero (increasing with stronger anisotropy). Finally, for a material with negative transverse dielectric permittivity

*ε*<0 and positive in-plane permittivity

_{x}*ε*>0, the dispersion relation (1) becomes hyperbolic. The curvature of the hyperbola is such that the signs of

_{z}*S*and

_{z}*k*are

_{z}*opposite*[Fig. 1(c)], as was pointed out by Belov in Ref. [6].

This result can be understood quantitatively: the *ẑ* component of the Poynting vector for the extraordinary wave is given by

Evidently, if *ε _{x}*<0,

*S*is negative, i.e. opposite to the direction of the wave vector component

_{z}*k*. As we shall see, this sign difference leads to the (effective) negative refractive index for refraction at an interface and for waveguiding.

_{z}In particular, when {*ε _{x}*<0,

*ε*>0} material is used as a core of a planar mirror waveguide (oriented along the

_{z}*yz*plane), negative group velocity modes arise [7]; for these modes the wave vector and the energy flux are antiparallel. Similarly, when such material is used as a core of a

*dielectric*waveguide, energy flux in the core is antiparallel to the wave vector. However, the energy flux in the waveguide cladding (made of regular, isotropic dielectric) is, as usual, collinear with the wave vector.

It has been recently suggested that balancing positive energy flux in a dielectric with negative energy flux in a medium with simultaneously negative values of dielectric permittivity and magnetic permeability can be used to effectively slow the group velocity of propagating modes [14]. While our system does not exhibit magnetic response, the general concept of balancing flux in different regions of a compound structure can still be applied, provided that one medium is characterized by negative group velocity.

In the next section we examine the case of a non-magnetic negative index waveguide. As we show, there indeed exists a value of the light frequency *ω* (and the waveguide thickness *d*) such that the negative energy flux inside the waveguide is nearly balanced by the positive energy flux outside, leading to a dramatic suppression of the signal velocity.

## 3. Guided modes of a strongly anisotropic waveguide

Planar dielectric waveguides admit two distinct polarizations – TE and TM modes. If the waveguide core possesses anisotropy (with optical axis *x* transverse to the waveguide plane *yz*), only TM modes are affected by the anisotropy (much like only the extraordinary waves are affected by the anisotropy of bulk uniaxial media). In particular, when *ε _{x}*<0, the dispersion curves for these modes acquire a qualitatively different character from those in an

*ε*>0 waveguide.

_{x,z}For guided TM waves, the electric field in the three regions of the waveguide in Fig. 2(a) can be expressed as

Requiring continuity at the boundaries and compliance with Maxwell’s equations, we can obtain the guidance condition in the form

where (*i*, *j*) ∆ {(1, 3), (3, 1)} and

${f}_{j}({k}_{x},{k}_{z};{\kappa}_{j})=\left(\frac{{k}_{x}}{{\epsilon}_{z}}\right)\left(\frac{{\epsilon}_{{d}_{j}}{k}_{x}-{\epsilon}_{z}{\kappa}_{j}\mathrm{cot}{k}_{x}d}{{\epsilon}_{z}{\kappa}_{j}+{\epsilon}_{{d}_{j}}{k}_{x}\mathrm{cot}{k}_{x}d}\right)$ .

Dispersion relations in the three regions are

For the case ${\epsilon}_{{d}_{1}}={\epsilon}_{{d}_{3}}\equiv {\epsilon}_{d}$ these equations can be combined as

while the guidance condition becomes

Finally, *k _{x}* may be expressed through

*k*and

_{z}*ω*using Eq. (5). We thereby obtain a set of transcendental equations, which may be solved graphically or numerically to yield

*ω*vs.

*k*dispersion curves.

_{z}A particular feature of an anisotropic waveguide is that propagating TM modes can exist for various *sign combinations* of *ε _{x}* and

*ε*. For

_{z}*ε*,

_{x}*ε*>0 the modes resemble those in an isotropic waveguide, while for

_{z}*ε*,

_{x}*ε*<0 propagating solutions vanish. If only

_{z}*one*of the

*ε*,

_{x}*ε*is negative, propagating solutions exist, and their behavior is strongly affected by the altered character of the dispersion relation.

_{z}The most prominent impact results in the case *ε _{x}*<0,

*ε*>0. As discussed in the previous section, this leads to

_{z}*S*=

_{z}*S*<0, i.e. negative energy flux in the waveguide core. This choice of signs for

*ε*and

_{x}*ε*has additional implications for the modes, as evident from Eq. (7). When

_{z}*ε*>0, this equation describes an ellipse. Consequently, simultaneous solutions of Eqs. (7) and (8) are possible only for a range of

_{x}*k*values up to some cut-off transverse wave vector

_{x}*k*

_{x max}(this reflects the fact that high

*k*modes of dielectric waveguides cannot meet the guidance conditions). However, as discussed in the previous section, when

_{x}*ε*<0, Eq. (7) describes a hyperbola. Arbitrarily large values of

_{x}*k*can now satisfy Eq. (7) and simultaneously solve Eq. (8). We see that this waveguide has

_{x}*no large k*(instead, there exists a minimum allowed value of

_{x}cutoff*k*). Finally, we note that no-cutoff mode solutions are also possible for the case

_{x}*ε*>0,

_{x}*ε*<0, in which case Eq. (7) is still hyperbolic, but with

_{z}*χ*as the major axis coordinate.

In Fig. 2(b) we plot dispersion curves of the guided modes resulting from solving Equations (7) and (8) (with a negative value of the transverse permittivity *ε _{x}*). For every guided mode we observe regions with both positive group velocity (most of the energy travels in the waveguide cladding) and negative group velocity (most of the energy is in the core). Furthermore, it is evident that for each mode there exists the value of the signal frequency

*ω*

_{0}corresponding to an extremely strong suppression of the group velocity.

It should be noted that Fig. 2(b) curves do not include the effects of material dispersion or losses, necessarily present in any realistic design of the waveguide structure. These effects alter the exact appearance of the dispersion curves, but preserve slow group velocity behavior around *ω*
_{0}.

## 4. Practical realizations of slow light in anisotropic waveguides

A significant challenge in designing non-magnetic negative index devices is obtaining the anisotropy necessary for their operation. Furthermore, realistic models of such devices must take into account the dispersive and lossy nature of materials.We now discuss possible realizations of {*ε _{x}*<0,

*ε*>0} materials and reexamine results of the previous section using realistic parameters of one such materials system.

_{z}The {*ε _{x}*<0,

*ε*>0} anisotropy required for the waveguide core can be found in several naturally occurring substances – such as e.g. bismuth in the THz domain and sapphire in the far IR [15]. It can also be readily achieved in artificially nanostructured systems, for instance, in a layered medium with alternating permittivities in the

_{z}*x*direction [7, 8, 16] – see Fig. 3(a). This medium consists of a sequence of “dielectric” layers (

*ε*

_{1}>0) and “conductive” layers (

*ε*

_{2}<0) [17, 18]. The effective dielectric tensor of such structure (with the volume fraction of the conducting layers

*N*) is given by [19]

_{c}$${\epsilon}_{z}=\left(1-{N}_{c}\right){\epsilon}_{1}+{N}_{c}{\epsilon}_{2}.$$

Provided *ε*
_{1}>0, *ε*
_{2}<0 these equations lead to a well-defined frequency interval with *ε _{x}*<0,

*ε*>0. Such a layered system can be fabricated using epitaxial semiconductor growth, with selective doping used to attain

_{z}*ε*

_{2}<0 in the “metallic” regions. Alternatively, these conductive layers may be made from a naturally occurring material with

*ε*<0.

In the visible spectrum, plasmon resonance results in *ε*<0 for a number of metals, which, however, suffer from substantial losses due to free carriers. One relatively low-loss plasmonic material is silver. Ag/SiO_{2} multilayer systems for imaging applications are actively being investigated both experimentally [20] and theoretically [21]. Such systems could potentially be utilized to create anisotropic waveguide structures described above for operating in the visible range.

At different wavelengths, other mechanisms can result in *ε*<0, with losses lower than those in silver. In the mid-infrared, for instance, negative permittivity occurs in a number of compounds due to phononic resonances. A low-loss material, well-suited for studying negativeindex phenomena in the mid-IR, is silicon carbide [16, 22], a wide bandgap, environmentally robust semiconductor with multiple existing and prospective applications in optoelectronics, power electronics, MEMS, and sensors. Its dielectric function is given by

where *ω*
_{LO}=972 cm^{-1}, *ω*
_{TO}=796 cm^{-1}, *ε*
_{∞}=-6.5, and *γ*=5 cm^{-1} [22, 23]. The resonant behavior results in *ε*
_{SiC}<0 for the wavelengths of 10.3–12.5 *µ*m.

We model the anisotropic waveguide core as a metamaterial composed of interleaved SiC and SiO_{2} (*ε*≃3.9) layers, with the SiC volume fraction *N _{c}*=50%. In Fig. 3(b) we plot

*ε*and

_{x}*ε*as given by Eqs. (9) and (10). Several distinct regions can be defined by the signs of

_{z}*ε*and

_{x}*ε*. For high and low wavelengths (

_{z}*λ*>12.6

*µ*m and

*λ*<10.3

*µ*m) both dielectric components are positive (

*ε*′

*≡Re[*

_{x}*ε*],

_{x}*ε*′

*≡Re[*

_{z}*ε*]>0). In the 11–12.6

_{z}*µ*m spectral band,

*ε*′

*>0,*

_{x}*ε*′

*<0. Finally, in the range of 10.3–11*

_{z}*µ*m Fig. 3(b) shows

*ε*′

*<0,*

_{x}*ε*′

*>0, the anisotropy needed to realize negative effective index and hence slow light. The center of this frequency range (≈10.6*

_{z}*µ*m) corresponds to the frequency that we denote by

*ω**.

In Fig. 4(a) we plot the guided mode dispersion curves of the air-clad (*ε _{d}*=1) waveguide with the SiC/SiO

_{2}metamaterial core. Values of

*ω*/

*ω** ≳ 1.1 (

*λ*≲10.3

*µ*m) cover the region where

*ε*′

*,*

_{x}*ε*′

*>0. The mode dispersion curves in this region correspond to the usual guided TM modes of a dielectric waveguide. Curves in the range 0.85 ≲*

_{z}*ω*/

*ω** ≲ 0.9 (

*λ*=11–12.6

*µ*m) represent the

*ε*′

*>0,*

_{x}*ε*′

*<0 modes, which we do not treat here. Finally, the spectral region 0.9 ≲*

_{z}*ω*/

*ω** ≲ 1.0 (λ=10.3–11

*µ*m) corresponds to

*ε*′

*<0,*

_{x}*ε*′

*>0 – the requirement for negative index modes.*

_{z}This region is examined in Fig. 4(b).We note that the dispersion curves of modes in the figure appear qualitatively similar to those of interface plasmon or phonon polaritons of a negative permittivity slab [24, 25]. However, the structure of the modes in our system is markedly different from that of slab-guided polaritons. Guided modes of Fig. 4(b) are essentially bulk states and, as such, their dispersion characteristics do not depend on the thicknesses of individual layers.

Frequency-dependent group velocity of a single slow mode [indicated by arrow in Fig. 4(b)] is plotted in Fig. 4(c). We obtain *v _{g}*≲0.004

*c*over a 1.1 THz frequency range. Such wide bandwidth suggests the possibility of using the proposed system as an optical buffer. Assuming operation around the point of zero second-order dispersion and restricting group velocity deviation from that point to less than 10% (shaded region in the figure), we obtain a usable data transmission bandwidth of 390 gigabits per second, with the required device length of 14.4

*µ*m for a 4-bit buffer. These parameters are comparable to operational characteristics of most currently proposed solid state slow light devices, in particular those based on electromagnetically induced transparency and coupled resonator systems [26]. The combination of large data bandwidth and compact device size exhibited by our system is similar to that of the recently proposed plasmonic slow light waveguides [24]. Like the plasmonic devices, our system is strongly limited by losses (~ 4 dB/

*µ*m). It should be noted that our device exhibits somewhat lower losses while attaining slower group velocities than the plasmonic slow light structures [24].

## 5. Negative refraction at the surface of negative index media

Interesting features of non-magnetic negative index systems based on dielectric anisotropy are not limited to the behavior of guided modes. Just like {*ε*<0, *µ*<0} materials, {*ε _{x}*<0,

*ε*>0} systems described in the previous sections support negative refraction. This property can be utilized in developing novel planar imaging systems with a potential for subwavelength resolution.

_{z}As discussed above, extraordinary waves in a uniaxial crystal exhibit a non-zero angle between the wave vector *k⃗* and the Poynting vector *S⃗*. The extraordinary waves may be excited by *P*-polarized light incident on the crystal (provided that its optical axis is in the plane of incidence). If the anisotropy is such that *ε _{x}*<0 and

*ε*>0, as in the previous section, the sign of the tangential component of the wave vector (

_{z}*k*) is

_{z}*preserved*upon transmission through the boundary (in this sense the refractive index is

*positive*), while the Poynting vector (and thus the energy flux) undergoes negative refraction [Fig. 5(a)], as pointed out in [6]. The refraction angles

*χ*and

*ϑ*, corresponding to the directions of the wave vector and the Poynting vector respectively, are given by

leading to *χ*>0, *ϑ*<0 for *ε _{x}*<0,

*ε*>0. In Fig. 5(b) we show the direct (numerical) calculation of the refraction of an optical Gaussian beam incident on the surface between air and strongly anisotropic dielectric with

_{z}*ε*<0,

_{x}*ε*>0. As expected, the direction of the beam follows the negative refraction of the Poynting vector, while the tilted wavefronts indicate positive refraction of the wave vector.

_{z}As a result of this behavior, a dielectric slab with negative transverse permittivity (i.e. exactly the planar anisotropic waveguide discussed earlier) can be used for planar imaging or beam focusing (Fig. 6), as pointed out in Ref. [27] and, independently, in Refs. [28] and [21]. A particular property of this system is the potential for imaging certain high spatial frequency Fourier components that correspond to subwavelength details of the object. While these waves undergo evanescent decay in free space, for a given frequency, a set of such waves can *resonantly couple to the slow light modes of the slab* discussed in the previous section. This mechanism is different from the operation of the “superlens” based on simultaneously negative *ε* and *µ*, which relies on the excitation of high-wavenumber surface plasmon-polariton modes for super-resolution. Superlens achieves its effect by allowing coupling to a *continuum* of surface modes with an arbitrarily high wavenumber *k _{z}* at the lensing frequency

*ω*

_{0}(provided perfect matching, i.e.

*ε*=

*µ*=-1). The planar dielectric system, on the other hand, features a

*discrete*set of high-

*k*modes. Like in the case of the superlens, however, the subwavelength resolution performance of this system is limited to the near field due to inherent materials losses [29].

_{z}## 6. Conclusion

We have demonstrated that a planar anisotropic waveguide with negative transverse permittivity (*ε _{x}*<0,

*ε*>0) supports slow light modes. Such modes are made possible by the balance of positive energy flux in the cladding and negative energy flux in the core. Resonant coupling to these slow light modes with

_{y,z}*k*>

_{z}*ω*/

*c*allows the use of this structure as a planar lens with subwavelength resolution.

## Acknowledgements

This work was partially supported by National Science Foundation grants DMR-0134736 and ECS-0400615, and by Princeton Institute for the Science and Technology of Materials (PRISM).

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