1. Introduction

Dyakonov surface waves (DSWs), as one kind of surface waves propagating at an interface between an isotropic dielectric and a positive uniaxial medium, were first theoretically proposed by Dyakonov in 1988 [1, 2]. In contrast to surface plasmon polaritons (SPPs) [3, 4], DSWs are non-dissipative surface waves so that they can be exploited for various applications such as optical sensing and long-range guiding [5]. Later, the supporting media of DSWs were generalized to hyperbolic [6] or biaxial materials [7,8], and even to the case when both of the two media are anisotropic [9–12]. However, DSWs usually can only propagate in a limited range of direction related to the anisotropy of the involved media. As for most natural materials possess weak birefringence, DSWs were experimentally observed until 1999 by using potassium titanyl phosphate (KTP) biaxial crystals [13]. The recent progress in nanostructured materials, on the other hand, provides a new platform for realizing DSWs owing to their tunable and large anisotropy [14–19].

One of the simplest anisotropic nanostructures is the stratified structures, which can usually be homogenized as effective uniaxial media with their optical axis normal to the stacked interface in the long-wavelength limit [20, 21]. Via the effective medium approximation (EMA), the principal permittivities of the effective media can be easily estimated, and these parameters can be used to characterize the stratified structures in the theoretical analysis for simplicity. However, stratified structures show negative birefrigence when both of the consituent materials are dielectrics. Therefore, in order to acquire positive birefrigence, stratified structures should contain components with negative permittivity, such as metals or graphene. As the metal is adopted, the metal-dielectric multilayered (MDM) structures exhibit either hyperbolic or elliptic dispersions depending on the operating wavelength due to the metallic dispersion. However, since the excitation of SPPs at the metal-dielectric interface will cause the nonlocal effect, the EMA may fail to predict the effective parameters of the MDM structures precisely [22,23]. Such nonlocal effect is closely related to the period of the MDM structure, and many different methods have been proposed to take it into account, modifying the effective principal permittivities as functions of the wave vector [22–29]. Nevertheless, such modified EMA still has some limitations. An additional extraordinary wave in the MDM structure is revealed [24,30], which causes the MDM structure cannot be simply treated as a uniaxial medium. When dealing with more complex structures such as the combination of the MDM structure and another arbitrary medium, the calculation utilizing the modified EMA may not be able to predict certain properties of the structure. As a consequence, a rigorous electromagnetic computation is required for studying these structures.

In this paper, we study Dyakonov-like surface waves (DLSW) at the interface of an MDM structure serving as an elliptic medium and an isotropic dielectric by utilizing our in-house developed finite element method (FEM) [31]. Better than the method adopted by the previous studies [17, 18, 32], this rigorous method is beneficial to precisely determine the dissipative surface wave by directly calculating its complex-valued propagation constant, k, with an arbitrary propagation direction. The effective index of such surface wave, N_eff, is then obtained through k normalized to k₀ which is the wavenumber in free space. Different from the conventional DSWs, we find this kind of DLSW exhibiting a leaky property, which has not been given proper attention previously. Such property results in an additional non-negligible propagation loss and cannot be predicted by an often-used simple model based on the EMA. Moreover, as one elliptic-like dispersion can also be obtained under certain conditions when the MDM structure is regarded as a hyperbolic metamaterial (HMM) [32], we will discuss this situation for completeness.

The rest of this paper is organized as follows. In Section II, the characteristics are described thoroughly for the MDM structure serving as an elliptic medium. Section III investigates the DLSWs propagating along the surface of the semi-infinite MDM structure. In Section IV, the corresponding phenomenon is discussed when the MDM structure is changed to serve as an HMM. Finally, Section V gives the conclusion.

2. The MDM structure serving as an elliptic medium

Figure 1(b) depicts the MDM structure composed of SiO₂ and silver (Ag) stacked alternatively along the x direction. The refractive index of SiO₂ is 1.5 and that of Ag is modeled via the Drude dispersion with a plasma frequency of ω_p = 1.15 × 10¹⁶s⁻¹ and a collision frequency of ν = 8.3 × 10¹³s⁻¹ [30]. In order to diminish the loss from the metal, we adopt a small filling factor of f = 0.25, which is defined as the thickness of Ag divided by the period of the MDM structure, p. Since we are interested in the regime when the MDM structure serves as an elliptic medium with both the ordinary and extraordinary permittivities (ϵ_o and ϵ_e, respectively) being positive, we focus on the wavelength, λ, from 217 nm to 456 nm according to the prediction based on the EMA.

 

Fig. 1 (a) Schematic of the semi-infinite MDM surface. (b) Schematic of the MDM structure. (c) Index ellipsoid of the effective medium for the structure (b).

Download Full Size | PDF

We first analyze the structure without consideration of metallic absorption, i.e., ν = 0. As shown in Fig. 1(c), the optical axis of the effective medium is along the x direction for the MDM structure. Thus, through the waves propagating along the z direction, the effective ordinary and extraordinary refractive indexes (n_o and n_e, respectively) can be obtained from the calculated effective indexes of such waves, N_z. Figure 2(a) shows the dispersions of N_z with p = 18 nm (green curves), and the corresponding results based on the EMA (black curves) are displayed for comparison. The ordinary and extraordinary waves (o- and e-waves, respectively) are represented by the dashed and solid curves, respectively. In addition, one high-order wave (HO-wave) is shown by the dotted green curve. This HO-wave is missing under the EMA and its N_z increases significantly when λ is approached to λ₀ = 295 nm which corresponds to the surface plasma frequency of SPPs at the Ag/SiO₂ interface. Moreover, we notice that the dispersion contours of the e- and HO-waves together behave similar to those of the insulator/metal/insulator (IMI) waveguide [3] as shown in the inset of Fig. 2(a). Thus, some characteristics of IMI waveguides can be applied to this HO-wave. Figures 3(a) and 3(b) show the mode-field profiles (Re[E_x], Im[E_z], and Re[H_y]) of the HO-waves at λ = 250 nm and 400 nm, respectively. One can find the HO-waves at λ < λ₀ and λ > λ₀ correspond respectively to the odd and even modes of the IMI waveguide with respect to E_z field. The non-uniform field distribution can also be observed for these HO-waves, which reveals the reason why these waves cannot be resolved by the EMA. Then, taking the metallic absorption into account, Fig. 2(b) shows the losses (Im[N_z] versus λ) of waves in the MDM structure with the amplitude indicating the amount of loss and the positive or negative sign referring to the direction of the group velocity, V_g. Here, we put the emphasis on the property of the HO-wave. As can be seen, its loss is much larger than those of the o- and e-waves, especially near λ ≈ λ₀. Moreover, an apparent difference can be observed particularly at the two sides of λ₀, the opposite signs of Im[N_z]. Referring to waves in the IMI waveguide [33], we can find the corresponding HO-wave for λ > λ₀ is a forward wave while that for λ < λ₀ is a backward wave.

 

Fig. 2 Characteristics of the waves propagating along the z direction in the Ag/SiO₂ MDM structure with p = 18 nm and the EMA. (a) Dispersions when the metallic loss is absent (ν = 0) for which N_z is real. (b) Im [N_z] versus λ for ν, 0, representing the propagation loss along the z direction. The inset of (a) displays the dispersions of the IMI waveguide for comparison.

Download Full Size | PDF

 

Fig. 3 Mode-field profiles of the HO-waves propagating along the z direction in the Ag/SiO₂ MDM structure with p = 18 nm at (a) λ = 250 nm and (b) λ = 400 nm. The fields are shown in a period of the MDM structure.

Download Full Size | PDF

Next, the isofrequency contours of the MDM structure are shown for different p ’s at λ = 250 nm and 400 nm in Figs. 4(a) and 4(b), respectively, where k_x and k_yz are the wave vector components in the x direction and in the yz-plane, respectively. As can be seen at both wavelengths, the o- and e-waves (dashed and solid curves, respectively) exhibit the predicted circular and elliptic-like dispersions, respectively, while the HO-wave (dotted curve) particularly shows a hyperbolic-like dispersion with larger Re[k_yz]. Compared with the o- and e-waves, one can notice that the HO-wave is much sensitive to p with its Re[k_yz] decreasing significantly when p gets larger.

 

Fig. 4 Isofrequency contours of the Ag/SiO₂ MDM structure. (a) λ = 250 nm. (b) λ = 400 nm.

Download Full Size | PDF

At last, one should notice that there are still more high-order waves supported in the MDM structure with more complicated field distributions. However, since they are hardly to be excited and propagate with huge losses, we omit these waves in our discussion.

3. Interface between the MDM structure and an isotropic medium

In this section, we discuss the DLSWs supported by adding one isotropic dielectric above the aforementioned MDM structure, as depicted in Fig. 1(a). The results calculated by our rigorous FEM are compared with those obtained from the effective model where we treat the involved MDM structure as an effective medium with the correction of nonlocal effects by the method proposed in [29]. For a given wave vector (k_x, k_yz), the effective n_o and n_e are acquired as $n_o=\sqrt{(1+\gamma )}(k_x/k_0)$ and $n_e=\sqrt{(1+{\gamma }^{-1})}(k_{yz}/k_0)$ with $\gamma =-\left(\frac{d{k}_x}{d{k}_{yz}}\right)\left(\frac{k_{yz}}{k_x}\right)$.

First, we consider the DLSW at λ = 250 nm. According to Fig. 4(a), we choose SiO₂ as the isotropic dielectric with its refractive index being n = 1.5 to satisfy the existing condition of the DLSW, n_e > n > n_o [1]. The isofrequency contour and propagation loss of the DLSW propagating in the xz-plane are shown with p = 18 nm in Figs. 5(a) and 5(b), respectively, where the rigorously-calculated results (green solid curves) and the effective results (green dashed curves) are displayed. k_x and k_z are the wave vector components in the x and z directions, respectively, and the angle, θ, represents the propagation direction with respect to the z-axis. In addition, in Fig. 5(a), the isofrequency contours of SiO₂ and the involved MDM structure are also shown by the black and blue curves, respectively, with the abscissa being Re[k_yz]/k₀. We can clearly observe the DLSW propagating along a range of direction, θ_min < θ < θ_max, where θ_min and θ_max indicate the cutoffs for the DLSW coupling to the continuous spectra of the e-wave in the MDM structure and the SiO₂, respectively. Even though the isofrequency contour of the DLSW only has a slight difference between the rigorously-calculated result and the effective result, one can find in Fig. 5(b) that the rigorously-calculated DLSW exhibits a more severe loss compared to that approximated from the effective model, which cannot be ignored and will seriously restrict the application of the DLSW in reality.

 

Fig. 5 Characteristics of the DLSW propagating at the interface between SiO₂ and the Ag/SiO₂ MDM structure, i.e., xz-plane, with p = 18 nm at λ = 250 nm. The results are obtained from the rigorous FEM and the effective model. (a) Isofrequency contours, where the abscissa refers to Re[k_z]/k₀ for the DLSW or Re[k_yz]/k₀ for waves in the SiO₂ and in the MDM structure. (b) Losses relative to θ.

Download Full Size | PDF

In order to explore the reason causing this discrepancy, Fig. 6(a) shows the rigorously-calculated magnetic mode-field profiles of the DLSW at θ = 40°, i.e., point A in Fig. 5(a). We notice an unexpected small field oscillation (more observable in the |H z| profile) in the semi-infinite MDM structure, i.e., y < 0. If we ignore the metallic absorption, the corresponding DLSW shown in Fig. 6(b) accompanies a component leaking into the semi-infinite MDM structure and eventually being absorbed by the perfect matched layer (PML) which is an auxiliary numerical design to truncate the computational region for problems with open boundaries. For clarity, Fig. 7(a) plots the |H_z| and Re[H_z] profiles at the center of the metal layer (x = 0) for this DLSW by the green and blue curves, respectively, and the corresponding Re[H_z] profile from the effective model is also shown by the red curve for comparison. The rigorously-calculated Re[H_z] can be apparently viewed as the combination of the confined field from the effective model and an unknown leaky field with the spatial frequency of its oscillation denoted as Re[k_y,_leaky], such that Re[k_y,_leaky] ≈ 10.89k₀ in Fig. 7(a). Along the −y direction, as the confined field decays, the field oscillation in |H_z| becomes weaker and the leaky constituent can be observed alone in the region far away from the interface such as y = −0.6 µm ∼ −0.9 µm. In addition, one should notice that such leaky property will result in an additional channel of loss. This explains why the DLSW calculated from the rigorous FEM is more dissipative than that obtained from the effective model where the leaky property is missing.

 

Fig. 6 Rigorously-calculated magnetic mode-field profiles of the DLSWs with p = 18 nm at λ = 250 nm and θ = 40°. (a) Real structure (ν ≠ 0). (b) Ignoring the metallic absorption (ν = 0). The red frame indicates the region where the field is close to include only the leaky constituent.

Download Full Size | PDF

 

Fig. 7 (a) |H_z| and Re[H_z] profiles at the center of the metal layer (x = 0) in Fig. 6(b). The corresponding Re[H_z] from the effective model is displayed for comparison. (b) |H_z| profile in (a) compared to the corresponding results with p = 9 nm and 36 nm.

Download Full Size | PDF

We notice that the leaky constituent in the red frame of Fig. 6(b) and the HO-wave in Fig. 3(a) have similar magnetic mode-field profiles with their fields predominantly localized in the metallic region. To examine the relationship between the leaky behavior of DLSWs and the HO-wave, we consider one HO-wave whose real parts of wave vector components in the x and z directions, Re[k_x,_HO] and Re[k_z,_HO], are matched with those of the DLSW, Re[k_x,_DLSW] and Re[k_z,_DLSW], respectively. Then, from Fig. 5(a), its real part of the wave vector component in the y direction can be roughly derived as $\mathrm{Re}[k_{y,HO}]=\sqrt{{\left(\mathrm{Re}[k_{yz,HO}]\right)}^2-{\left(\mathrm{Re}[k_{z,DLSW}]\right)}^2}$ with k_yz,_HO indicating the corresponding wave vector component in the yz-plane of this HO-wave. At point A, Re[k_y,_HO] is estimated to be 10.91k₀ and is found to be roughly consistent with Re[k_y,_leaky] as shown in Fig. 7(a). Furthermore, according to Fig. 4(a), Re[k_yz] of the HO-wave decreases when p gets larger, resulting in a reduction of the estimated Re[k_y,_HO]. In Fig. 7(b), the |H_z| profile in Fig. 7(a) is compared to the corresponding results with p = 9 nm and 36 nm, and the slower oscillation, i.e., smaller Re[k_y,_leaky], can be observed for the DLSW with larger p, which is coincident with the tendency of Re[k_y,_HO] when p is changed. Thus, we can confirm that the leaky component originates from the HO-wave of the MDM structure. This also reveals the reason why the leaky property cannot be well predicted by the effective model which considers merely the o- and e-waves of the MDM structure. In addition, since the HO-wave is much dissipative when considering the metallic absorption as mentioned in Section 2, the leaky component cannot propagate a long distance. Therefore, the DLSW with ν ≠ 0 in Fig. 6(a) is still confined well near the interface it is propagating along.

Furthermore, in Fig. 7(b), we notice that the amplitude of the leaky component gets larger when p increases, resulting in the larger leaky loss as shown in Fig. 8 where the leaky losses with different p ’s are displayed by the dotted curves. In addition, this leaky loss is observed to get smaller at the two sides of the propagation direction range. This is related to the decreasing leaky component in the DLSW when θ is getting close to θ_min (θ_max), where the e-wave component in the MDM structure (the field in the SiO₂) is getting more predominant. Then, we append the losses of the DLSWs to Fig. 8 with the results from the rigorous FEM (solid curves) and the effective modal (dashed curves). As can be seen, the leaky loss is much sensitive to p and becomes even more dominant than the loss from the effective model when p is larger. Thus, to reduce the loss of the DLSW, we should design the structure with smaller p to suppress the leaky component.

 

Fig. 8 Losses of the DLSWs relative to θ by different methods for p = 9, 18, and 36 nm at λ = 250 nm.

Download Full Size | PDF

On the other hand, for the DLSW at λ = 400 nm, we take air as the isotropic dielectric instead. Figure 9 shows the rigorously-calculated magnetic mode-field profiles of the DLSW with p = 36 nm at θ = 40° without considering the metallic absorption. Similar to the DLSW at λ = 250 nm, the leaky property is also found for the DLSW at this wavelength. However, due to the distinct modal properties of the HO-waves at these two wavelengths, one can see that |H_z| of the leaky component at λ = 250 nm is more localized in the metallic region in Fig. 6(b) while that at λ = 400 nm is more localized around the metal-dielectric interfaces in Fig. 9. In addition, as discussed in Section 2 that the HO-waves at λ = 250 nm and 400 nm are backward and forward waves, respectively, it should be particularly noticed that the corresponding leaky components propagate with the opposite propagation directions in the xz-plane.

 

Fig. 9 Magnetic mode-field profiles of the DLSW ignoring the metallic absorption with p = 36 nm at λ = 400 nm and θ = 40°.

Download Full Size | PDF

4. Elliptic-like dispersion of the MDM structure when serving as an HMM

When the MDM structure is recognized as an HMM according to the EMA, it is revealed that one elliptic-like dispersion may arise due to the nonlocal effect [32]. Thus, the DLSW is then discussed with this elliptic-like dispersion for completeness.

We consider the MDM structure composed of alternating Ag and TiO₂ thin films with f = 0.25. The refractive index of TiO₂ is adopted as 2.5 [34] and Ag is modeled as adopted in the previous section. According to the EMA, this MDM structure behaves as an HMM with its dispersion being a hyperboloid of one sheet, i.e., ϵ_o < 0 and ϵ_e > 0, when λ > 730 nm. Thus, we discuss the case at λ = 800 nm, and the light source can be easily available with the Ti:sapphire laser in the experiment. The isofrequency contours of the MDM structure are shown with different p ’s in Fig. 10(a), where the dashed, solid, and dotted curves represent the o-, e-, and HO-waves, respectively. As expected, we find an elliptic-like dispersion appears for the proper periods, i.e., p = 140 nm (green) and 144 nm (blue), and then the isofrequency contours of these MDM structures behave similar to Fig. 4(a) with one hyperbolic-like dispersion aside to the elliptic-like dispersion. Thus, the DLSW based on this elliptic-like dispersion may also exhibit a leaky property from the aside hyperbolic-like wave. However, it is worth to mention that, contrary to the cases discussed in Section 2, the elliptic-like wave here corresponds to the HO-wave while the hyperbolic-like wave is the e-wave of the MDM structure. Thus, compared to the elliptic-like wave, the hyperbolic-like wave exhibits the more homogeneous field distribution and is excited more easily. In addition, Fig. 10(b) shows the loss of the hyperbolic-like wave with different p’s. One can notice that this wave propagates with a much smaller loss compared with those discussed in Section 2 as shown in the inset.

 

Fig. 10 Characteristics of the Ag/TiO₂ MDM structure with different p’s at λ = 800 nm. (a) Isofrequency contours. (b) The losses of the hyperbolic-like waves (e-waves) relative to θ. For comparison, the inset shows the losses of the hyperbolic-like waves (HO-waves) for the Ag/SiO₂ MDM structure in Section 2.

Download Full Size | PDF

Next, we take this MDM structure with p = 140 nm for further discussion, investigating the DLSW based on the elliptic-like dispersion at a surface of the semi-infinite MDM structure in the air. The isofrequency contour of the DLSW is shown by the green curve in Fig. 11(a) with the black and blue curves representing the isofrequency contours of the air and the MDM structure, respectively. In particular, we find this DLSW is allowed to propagate along a wider direction range with θ_min = 0°. Its propagation loss (the solid curve) is shown at the inset of Fig. 11(a) and is observed that the leaky loss (the dashed curve), resulting from the hyperbolic-like wave in the MDM structure as discussed in Section 3, can be significant. Such leaky loss is less when θ is close to 0° and is largest around θ = 22°. On the other hand in Fig. 11(b), we examine the mode-field profiles of this DLSW, taking the wave at θ = 30°, i.e., point A in Fig. 11(a). As can be seen, this DLSW is poor confined to its propagating interface even in the presence of the metallic absorption. Its leaky component can leak far away from such interface since the corresponding hyperbolic-like wave is less dissipative. We can also notice that the amplitude of the leaky component increases along the −y direction in Fig. 11(b), which is close to the feature of leaky waves in the lossless material [35]. Therefore, even though this kind of DLSW possesses a wide range of propagation direction, its insufficient confinement to the surface due to its leaky property should be taken into account for the applications of surface waves.

 

Fig. 11 The DLSW at the interface between air and the Ag/TiO₂ MDM structure with p = 140 nm at λ = 800 nm. (a) Isofrequency contours, where the abscissa refers to Re[k_z]/k₀ for the DLSW or Re[k_yz]/k₀ for waves in the air and in the MDM structure. The inset shows the losses of the DLSW. (b) Magnetic mode-field profiles at the point A in (a).

Download Full Size | PDF

5. Conclusion

We have investigated the DLSWs at the interface of an isotropic dielectric and an MDM structure serving as an elliptic medium based on the EMA. By applying our in-house developed FEM, this kind of DLSW was determined precisely, showing a significant difference from the conventional DSWs. We found this DLSW exhibits a leaky property as well as a non-negligible leaky loss due to the existence of an additional hyperbolic-like wave supported in the MDM structure. As this hyperbolic-like wave is missing under the EMA, this leaky property cannot be well predicted by an often-used effective model based on the EMA. Moreover, such leaky component was found to be sensitive to the period of the MDM structure. Therefore, to greatly reduce the leaky loss, one can suppress the unwanted leaky component by designing the MDM structure with a smaller period.

On the other hand, when the MDM structure is regarded as an HMM, we can also obtain an additional elliptic-like dispersion from a high-order wave under certain conditions. Based on this dispersion, we found that the supported DLSW also possesses a leaky property, similar to the case discussed above. However, some differences can be observed. This DLSW propagates at a wider range of direction, but suffers from a weak confinement to the interface it is propagating along since its leaky component is less dissipative.