Currently, there is great interest in materials that refract light negatively. Negative refraction can be characterized by a light beam whose wavefront is refracted normally (in a positive direction), while its energy flux is refracted negatively. Materials that negatively refract light can enable applications such as near-field superlensing and optical cloaking.^{1} Approaches to developing negative refraction materials include the design of photonic crystals with geometries that result in desired dispersion properties. The photonic crystal approach has produced some notable successes in the development of microwave to infrared negative refraction materials.^{2}^{,}^{3} Another approach is to exploit material resonance modes to achieve negative refraction.^{4}^{,}^{5} Recent examples include the use of infrared plasma resonance in free-carrier doped InGaAs films to suport negative refraction.^{6} Magnetic resonance, in gold, has also been applied to the development of mid-IR negative refraction materials.^{7} In this letter, we propose that coupled photon-phonon resonance modes, called radiative surface phonon polaritons (RSPhP), can be exploited in the development of mid-IR negative refraction materials. To demonstrate this idea, we present theoretical results for a prototype structure consisting of a thin silicon carbide (SiC) film, bounded by very thick layers of silicon (Si) and diamond (Di). Approaches to achieving mid-IR negative refraction materials are necessary for the development of next-generation thermal management devices such as thermal rectifiers and specialized thermal emitters.^{8}^{,}^{9}

Physically, surface phonon polaritons are atomic lattice-waves coupled to transverse magnetic ($p$-polarized) electromagnetic fields that may be supported at the interface between a polar material bounded by a dielectric. For example, surface phonon polaritons can be excited at the interface between air (dielectric constant, ${\u03f5}_{0}\approx 1$) and a semi-infinite layer of SiC [a polar material with a frequency varying dielectric function ${\u03f5}_{1}(\omega )$]. The SiC/Air system is illustrated in Fig. 1(a), and the well-known dispersion relation for surface phonon polariton modes at an interface formed between a semi-infinite polar material and a dielectric is given by the expression

## (1)

$${k}_{x}=\frac{\omega}{c}\sqrt{\frac{{\u03f5}_{0}{\u03f5}_{1}(\omega )}{{\u03f5}_{0}+{\u03f5}_{1}(\omega )}},$$The wavevector representing the component of light propagating perpendicular to $i$’th interface is defined by ${k}_{zi}=\sqrt{{\u03f5}_{i}{k}^{2}-{k}_{x}^{2}}$. Therefore, surface modes with ${k}_{x}>{\u03f5}_{i}k$ cannot couple to propagating light, since ${k}_{zi}$ would be purely imaginary. Figure 2 shows that the surface phonon polariton mode for the SiC/Air structure is to the right of the light line and, thus, ${k}_{x}$ is too large for a beam of incident light to interact with the surface phonon polariton.

So, while it is possible for semi-infinite SiC to support mid-IR surface phonon polaritons, these modes are forbidden from direct coupling to propagating light. However, the situation completely changes for the case of a thin, 400 nm layer of SiC, sandwiched between semi-infinite thick layers of Si and diamond, as shown in Fig. 1(b). These materials were chosen because they are transparent in the infrared region and have high refractive indexes with ${n}_{\mathrm{Si}}=3.5$ and ${n}_{\mathrm{Di}}=2.4$. The dispersion relation, for surface phonon polaritons in the Si/SiC/Di structure, is determined by numerically finding values of ${k}_{x}(\omega )$ that result in zeros for the denominator of the three-layer, $p$-polarized, reflectance equation,^{10}

## (2)

$${R}_{210}^{p}={\left|\frac{{r}_{21}^{p}+{r}_{10}^{p}{e}^{2i{\alpha}_{1}}}{1+{r}_{21}^{p}{r}_{10}^{p}{e}^{2i{\alpha}_{1}}}\right|}^{2},$$## (3)

$${r}_{ij}^{p}=(\frac{{k}_{zi}}{{\u03f5}_{i}}-\frac{{k}_{zj}}{{\u03f5}_{j}})/(\frac{{k}_{zi}}{{\u03f5}_{i}}+\frac{{k}_{zj}}{{\u03f5}_{j}}),$$^{10}The plot also shows the dispersion relation for semi-infinite SiC bounded by diamond. While the horizontal axis range is limited in Fig. 2 for brevity, previous theoretical results suggest that for very large ${k}_{x}$, degeneracy is lifted and the symmetric and anti-symmetric modes become equivalent to the semi-infinite SiC/Diamond case. The SPhP modes are known to only exist for the $p$-polarized case. For comparison, we also plot the dispersion relation for the $s$-polarized case in Fig. 2 to show that the effects discussed in this letter only occur for $p$-polarized light.

The third branch of the Si/SiC/Di dispersion relation extends to the left of the light line, crossing it at approximately $\omega =1.65\times {10}^{14}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{rad}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$, or 11.4 *μ*m. Since this mode crosses the light line and is, thus, accessible by propagating light with a wavelength of about 11.4 *μ*m, we refer to it as a radiative surface phonon polariton (RSPhP). As will be shown below, calculation of the magnetic field shows that the mode is symmetric since the transverse component of the electric field (${E}_{z}$) is not zero inside the film [c.f. Fig. 3 and Eq. (4)]. Symmetric (and anti-symmetric) radiative surface polaritons have been predicted, before, for the case of multilayer structures. However, our calculations show that the radiative branch of the dispersion relation presented here has a negative slope. Since the group velocity is given by ${v}_{g}=\partial \omega /\partial k$, the negative slope implies a negative group velocity. Also, since the group velocity characterizes the direction of energy flow, which in this case is roughly opposite of the wave propagation direction (${k}_{x}$), negative refraction may be possible with the structure shown in Fig. 1(a). To validate that the radiative mode is a physical solution, we calculated the imaginary part of ${k}_{x}$ for the RSPhP mode as positive. Thus, the surface mode decays as it propagates along the film, radiating light into the silicon along the way.

In the remainder of this letter, we will use theoretical evidence to support the idea that incident light coupling to radiative surface phonon polariton modes can be accomplished in the Si/SiC/Di structure. We also use theoretical results to show that negative refraction is occurring in the SiC layer of the Si/SiC/Di structure. We can characterize this light coupling by examining the magnetic field in the Si/SiC/Di structure. Referring to the system shown in Fig. 1(b), we start by describing the magnetic field in each layer as

## (4)

$${H}_{y2}={e}^{i({k}_{x}x-\omega t)}{e}^{-{k}_{z2}z}\phantom{\rule[-0.0ex]{2em}{0.0ex}}z\ge d,$$## (5)

$${H}_{y1}={e}^{i({k}_{x}x-\omega t)}(A{e}^{{k}_{z1}z}+B{e}^{-{k}_{z1}z})\phantom{\rule[-0.0ex]{2em}{0.0ex}}\phantom{\rule{0.265em}{0ex}}0\le z\le d,$$## (6)

$${H}_{y0}=C{e}^{i({k}_{x}x-\omega t)}{e}^{-{k}_{z0}z}\phantom{\rule[-0.0ex]{2em}{0.0ex}}z\le 0.$$## (7)

$$A=[\frac{{\u03f5}_{1}{k}_{z0}}{{\u03f5}_{0}{k}_{z1}}+1],\phantom{\rule[-0.0ex]{1em}{0.0ex}}B=[\frac{{\u03f5}_{1}{k}_{z0}}{{\u03f5}_{0}{k}_{z1}}-1],\phantom{\rule[-0.0ex]{1em}{0.0ex}}\text{and}\phantom{\rule[-0.0ex]{1em}{0.0ex}}C=A+B.$$*μ*m light in Fig. 3, where the gray region represents the SiC layer.

Figure 3 shows the infrared light propagating through the diamond layer as an oscillating field below the gray region. The gray region is the SiC layer. Figure 3 also shows the field exponentially decaying within the SiC layer, which is characteristic of a surface phonon polariton mode. The inset of Fig. 3 shows the details of the exponential decay in the SiC layer. Finally, another oscillating field is shown in the region above the SiC region, which demonstrates that the RSPhP mode is leaking into the silicon layer. Furthermore, the results show that the RSPhP mode is not self-sustaining, but depends on infrared light incident from the diamond side.^{11}

We now, theoretically, show that the 11.4 *μ*m light energy flux is negatively refracted in the Si/SiC/Di structure. The Poynting vector describes the light energy flux as

## (10)

$${E}_{xi}=-\frac{ic}{\omega {\u03f5}_{i}}\frac{\partial {H}_{yi}}{\partial z}\phantom{\rule[-0.0ex]{1em}{0.0ex}}\text{and}\phantom{\rule[-0.0ex]{1em}{0.0ex}}{E}_{zi}=-\frac{{k}_{x}c}{\omega {\u03f5}_{i}}{H}_{yi},$$## (11)

$$\mathrm{tan}\text{\hspace{0.17em}}\theta =-\frac{i{k}_{z2}}{{k}_{x}}\phantom{\rule[-0.0ex]{2em}{0.0ex}}z>d,$$## (12)

$$\mathrm{tan}\text{\hspace{0.17em}}\theta =-\frac{i{k}_{z1}(B{e}^{{k}_{z1}z}-C{e}^{-{k}_{z1}z})}{{k}_{x}(B{e}^{{k}_{z1}z}+C{e}^{-{k}_{z1}z})}\phantom{\rule[-0.0ex]{2em}{0.0ex}}0<z<d,$$## (13)

$$\mathrm{tan}\text{\hspace{0.17em}}\theta =-\frac{i{k}_{z0}}{{k}_{x}}\phantom{\rule[-0.0ex]{2em}{0.0ex}}z<d.$$*μ*m light in Si/SiC/Di is plotted in Fig. 4.

Figure 4 shows that the energy flux has a positive incidence angle at the SiC/Di interface which we calculate from Eq. (11) to be ${\theta}_{0}=66\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{deg}$. The energy flux then, dramatically, changes its angular direction in the SiC such that it is negatively refracted as it propagates from the diamond to the Si/SiC interface. The flux angle in the SiC layer, ${\theta}_{1}$, is calculated using Eq. (12). The real part of ${\theta}_{1}$ varys from $-80\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{deg}$ to $-73\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{deg}$, and is plotted in Fig. 4. The curvature of the energy flux in the SiC layer is consistent with Fermat’s principle, since there is significant variation in refractive index from the diamond to Si layers. The energy flux emerges from the SiC/Si interface again with a positive angle calculated, with Eq. (13), to be ${\theta}_{2}=74\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{deg}$.

In conclusion, this letter, theoretically, predicts that a submicron thin film of SiC, bounded by silicon and diamond, may be used as a negative refraction material. The key mechanism is the radiative surface phonon polariton mode coupled to incident light at the SiC interfaces.

## Acknowledgments

This work was supported by AFRL through a subcontract with Universal Technology Corporation, subcontract No. 11-S567-0015-02-C6.