1. Introduction

Negative index materials (NIMs), which have the significant property of ‘negative refraction’, have attracted much attention, and promise to be very important for the technology of light control. So far, NIMs are all artificial structures not found in nature. The first concept of NIMs was proposed by Veselago [1] and experimentally verified by Shelby et al at microwave frequencies by using periodic arrays of metal split ring resonators (SRRs) and dipoles [2]. For the purpose of light control, these artificial structures have to be made smaller and smaller in scale in order to work at visible or infrared wavelengths [3–8], but it is increasingly difficult to make these elements smaller because of fabrication limitations. On the other hand, metal-dielectric (or semiconductor-dielectric) layered structures are also known to exhibit negative refraction for TM polarized waves [9–11], because the permittivity of metal (and some kinds of semiconductor) layers is negative at visible and/or infrared frequencies. Such layered structures are useful, for example for super-resolution imaging systems [10–12] which may eliminate the diffraction limit associated with normal optical imaging instruments. In these layered structures, the surface wave at the interface of the metal and dielectric layers makes important contribution. More flexible in fabrication, are surface wave-based planar structures, such as some kind of planar waveguide containing metal layers which have attracted much interest [12–15]. Smolyaninov et al [12] use a single interface of metal and dielectric to produce negative refraction and verified the theory of the hyperlens [11], while arguments exist such as Ref. 16 which indicates negative refraction cannot exist at single metal-dielectric interface. Later, Lezec et al experimentally showed negative refraction at visible frequencies [15] by using the surface mode of MIM (metal-insulator-metal) structures, but the phenomenon responsible was not very clear; Dionne et al solved the dispersion relations for such three layered structures to explain the phenomenon [17]. As indicated in Ref. [17], with specific configurations, three layer systems (MIM, IIM and IMI) can show a negative refraction property, although only the MIM structure supports a single propagating negative index mode. Taking the MIM structure for example, Shin et al explain in Ref. [13]. that negative refraction is caused by the negative slope (i.e. negative group velocity) of the second band of the dispersion curves (see Fig. 1 in Ref. [13].). In Refs. [17]. and [18], the authors notice that when considering the loss in the metal layers, the dispersion curves should be obtained by ensuring that the imaginary parts of the wave vectors were positive; this explanation is reasonable, but there still results in some ambiguity and contradictions between the dispersion curves for the lossless and lossy cases in these papers.

In this paper, we study the MIM, IIM, and IMI structures again, specifically comparing the lossy case with the lossless case, to illustrate the link between these two cases. We show how the dispersion curves with loss evolve from the lossless case, and indicate how the correct dispersion curves of such structures should appear, especially for the lossless case.

2. Causality and equations

The structures of three-layer systems (MIM, IIM and IMI) are setup as shown in Fig. 1(a) with the thickness of the middle layer being d, and the substrate and cover are infinite. Since negative refraction is caused by surface plasmon polaritons, and the surface wave is always bounded to interfaces, we can treat these structures as plasmon waveguides. Given the symmetry of the structure, the field distribution is the same in the x and y directions, thus we can confine our discussion to the x-z plane, and consider surface modes of the TM polarization (i.e. only field components E_x, E_z and H_y are nonzero).

 

Fig. 1 (a) Schematic of an incident wave impinging on a three-layer structure from a normal medium. The direction of the incident wave is confined to the x-y plane. (b) and (c) are schematics where (b) shows the correct phase and energy velocities with negative refraction for normal and oblique incidence, while case (c) violates causality (see direction of energy flow).

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In many published papers, the dispersion relations for the three-layer lossless case are always shown like those black curves in Fig. 2 (see for example those in Refs. [13,14,17,18,20]). We can see that these curves show a negative slope in some frequency ranges, which corresponds to a negative group velocity, indicating the phase and energy velocities are anti-parallel. When an incident plane wave passes through the interface between a normal medium and such a structure, this anti-parallel property corresponds to negative index and hence refraction [1].

 

Fig. 2 Dispersion relations of lossless Drude model for three types of layered structures: (a) MIM, (b) IIM (here we use air as the cover layer), and (c) IMI. The black curves are typical of those published by others. The red curves are the branches consistent with causality, which we discuss in this paper.

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When negative refraction occurs, the causality condition must be satisfied [16,17,19]. When a plane wave is incident into a NIM from a normal medium, the energy flow should be towards the interface from the incident media and then away from the interface on the other side (the refractive side) and in this way, energy will not accumulate at the interface. On the other hand, if the wave vector is complex in the negative index region (caused by loss in the material), the imaginary part should point in the positive direction and lead to field decay as the wave propagates rather than enhancement which will violate causality. When negative refraction occurs and considering this anti-parallel property associated with negative index, we plot the possible directions of phase and energy flow in Fig. 1(b) and 1(c). Based on the principle of causality, we can readily see that the case in Fig. 1(c) is impossible. At the interface, there is an energy accumulation, and the conservation of transverse wave vector does not hold. For the case in Fig. 1(b), the directions of the phase and energy of the refractive wave are anti-parallel and causality is maintained; the energy flow goes into the interface and leaves from the other side, and the transverse wave vectors are conserved. However, Fig. 1(b) indicates that the phase directions will be negative which will lead to different dispersion curves (the red curves in Fig. 2) from those traditionally taken (see black curves with negative slope). After verifying causality, we study negative refraction in three-layer structures and find that the relations between the lossless and lossy cases correct the black curves in Fig. 2.

For the three types of three-layer system in Fig. 2, we suppose the wave propagation to be in the x direction and we just need to study the ω-k_x dispersion curves, where ω is the angular frequency of the incident wave and k_x is the x component of wave vector in the structure. We can find the dispersion relation for such structures from Eq. (1) [21]:

In the three distinct areas (numbered in Fig. 2), the wave vectors k_i (i = 1, 2, 3) should fulfill:

Taking ε_∞ = 1, ω_p = 9eV, and Г = 0.2687 eV, we approximately represent the permittivity of silver). We note that the value of Г is a little larger than usual to make the dispersion curves clear but this does not affect the results. Taking a smaller value, the curves of the lossless and lossy cases are too close to each other to be able to distinguish them. The materials used here are non-magnetic, thus permeability μ_i = 1 (i = 1, 2, 3).

The time averaged Poynting vector ‹S› is also derived to study the energy transport in these layered structures. As we mainly study the bound surface modes, we just need to study the x component of ‹S› (represented by ‹S_x›), since ‹S_x› can be used to describe the propagation of energy flow in the structures. As in Eq. (4) ‹S_x› is composed of three parts, the first one (‹S_1x›) represents the energy flow in layer 1, and the last two (‹S_2x› and ‹S_3x›) are for the substrate and cover, respectively:

We solve the nonlinear Eq. (1) using numerical methods to get the ω-k_x curves. The mode refractive index is decided by:

Thus, negative (positive) k_x indicates negative (positive) index in the structures. Generally, for a certain frequency, there is always an infinite number of solutions that solve Eq. (1) because of the periodic character of the formula and both the real and imaginary parts of k_x can be positive or negative. Mathematically, these solutions all satisfy Eq. (1), thus we must choose the solutions which are physically correct. Ref. [17]. uses the imaginary part of k_x, where imag(k_x) must always be positive, to decide which solution is reasonable. This is reliable when the metal is lossy and k_x always has a non-zero imaginary part. In contrast, for the lossless case imag(k_x) of the propagating mode is always zero of course, in which case this approach fails leading to a contradiction between Fig. 1 and Figs. 2 to 4 in Ref. [17]. It is therefore necessary to study the implicit relationship between the lossless and lossy cases in order to establish how the negative index property originates. To do this we calculate the time averaged Poynting vector ‹S_x› to study the energy flow directions, which is more reliable and in compliance with causality. From this we can find out the relationship between lossless and lossy cases, and clearly see how negative refraction arises in these layered structures.

When we consider the loss in metal, the propagation constant k_x is always complex. A large imaginary part of k_x means heavy damping when propagating along x direction, thus we will mainly study those modes, the imaginary parts (imag(k_x)) of which are small or comparable with the real part (real(k_x)).

3. The MIM case

For the MIM structure, the middle layer should be thin enough to obtain the phenomenon of negative refraction [13]. Here we take d = 20nm, and ε₁ = ε_metal , ε₂ = ε₃ = 4.

Because the MIM structure is symmetric, the field distribution (here we use H_y for clarity) will be either symmetric or anti-symmetric along the z direction. As the dispersion curves of these two situations are different, they will be discussed separately. We first consider the symmetric mode. The dispersion relations for both lossless and lossy cases are shown in Fig. 3 where (a) shows the real parts of k_x; (b) shows the imaginary parts of k_x; and (c) shows the corresponding time averaged Poynting vectors ‹S_x› calculated by Eq. (6) (normalized for convenience). Note that the curves in each of these figures correspond to each other in color and line-style. The horizontal dotted lines indicate${\omega }_{sp}={\omega }_p/\sqrt{(1+{\epsilon }_d)}$ hereafter, and these roughly (we will show more details later) divide the entire area into two parts. For the lossless Drude model (Г = 0), we can plot the possible ω-k curves of propagating modes (imag(k_x) = 0) as black solid and dashed curves in Fig. 3(a); in other words, both of the curves satisfy Eq. (1). It is clear that the term k_x² in Eq. (2) leads to ± k_x being solutions. For these two cases, we have imag(k_x) = 0, which means we cannot pick out the correct one from imag(k_x)>0. The more efficient and reliable way is to study the value of ‹S_x›. We need ‹S_x› > 0 to avoid the energy accumulation at the interface mentioned above and which ensures that the energy is consistent. In Fig. 3(c), we can see that for the lossless case, ‹S_x› > 0 only when real(k_x)>0 (black solid curve). In contrast, ‹S_x› will be negative when real(k_x)<0, which means this set of solutions are non physical, and these pseudo solutions are plotted as dashed curves hereafter. The correct branch for the lossless case is then decided. As a general point, when loss in the Drude model increases, the curves for the lossless and lossy cases should not behave differently. When Г = 0.2687 eV, we plot the dispersion curves in red in Fig. 3. We can see that curves for the lossy case are very close to the lossless case at lower frequencies, and follow the black curve as asymptotes. Moreover, imag(k_x) and ‹S_x› are both positive, which is consistent with causality; the other set of solutions having imag(k_x)<0 and ‹S_x›<0 is ignored. For higher frequencies, as there is no solution for the fundamental symmetric mode for the lossless case, the asymptote for the lossy case corresponds to the higher order mode of lossless case. This is indicated by the green dash-dot curve and we can see from Figs. 3(a) and 3(b) that both the real and imaginary parts of k_x for the lossless and lossy cases are close and have similar trends at frequencies above ω_sp. For high order modes and lossless case, we have ‹S_x› = 0 which means this mode is non-propagating. For the lossy case, imag(k_x) is large comparing with real(k_x) at higher frequencies, thus the fields will decay rapidly along the x direction; at the same time, ‹S_x› is very small for higher frequencies but we do not show these high order curves here, as they are not the salient for this paper. It is these features that indicate the similarity between the lossless and lossy cases.

 

Fig. 3 Dispersion relations for the symmetric mode of MIM structure, for both lossless (Г = 0) and lossy (Г = 0.2687 eV) cases. (a) and (b) show the curves of the real and imaginary parts of the wave vector, respectively; (c) shows the curves of frequency vs. x component of time averaged Poynting vector. The horizontal dotted curve stands for ω = ω_sp. The curves are colored consistently for all of three figures. The insets show details for clarity.

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For the anti-symmetric mode and lossless case, we can also get two sets of solutions at higher frequencies, with real(k_x)>0 for one set and real(k_x) < 0 for the other (black solid and dashed curves in Fig. (4) , respectively). For both curves, k_x and ‹S_x› are anti-parallel which indicates the negative index property but causality dictates that the only physical curve is the solid one, which is different from that published in many papers. According to Eq. (6), we know that the black solid curve is the negative index case. For lower frequencies there is also a pair of solutions (axially symmetric and shown by the dash-dot curves) but with positive and large imag(k_x) which means a strong exponential decay, while ‹S_x› for these solutions are both zero. Further study shows that these two curves are both physical, the reason being as follows: we can obtain a pair of solutions for the lossy case (in red and blue, and not axially symmetric anymore) both with imag(k_x) > 0 and ‹S_x› > 0, which means this case is causal. For real(k_x), we can see that these two curves are close and follow the dash-dot curves asymptotically. The red and blue curves are on different sides of the dash-dot curves; i.e. red and blue curves are respectively on the left of each dash-dot curves and are ‘separated’ by the dash-dot curves; for imag(k_x) the separation is more clear in Fig. 4(b). The red curve shows the negative index property (as real(k_x) < 0), while the blue one corresponds to a positive index (as real(k_x)>0). The positive index branch (blue) is decaying significantly when propagating because of its large imag(k_x), and for the same reason, ‹S_x› will decrease rapidly as x increases. We know this from Eq. (4) since there is an exponential coefficient related x before the integral and when imag(k_x) is large, ‹S_x› will decay exponentially.

 

Fig. 4 Dispersion relations for the anti-symmetric mode of the MIM structure, for both lossless (Г = 0) and lossy (Г = 0.2687 eV) cases. (a) and (b) show the curves of the real and imaginary parts of the wave vector, respectively; (c) shows the ω-‹S_x› curves. The curves are consistent in color for all of three figures.

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Based on the discussion of symmetric and anti-symmetric modes, for both lossless and lossy cases, we know that the positive index property is dominant at lower frequencies, while the negative index property is dominant at higher frequencies, which is similar to that described in Ref. [17].

There is an interesting area that should be further discussed for the anti-symmetric mode. In Fig. 4(a), although ω = ω_sp asymptotically, as real(k_x) increases, the black solid and dashed curves do not tend to it directly but first cross ω = ω_sp and then tend to the asymptote. The ω-k_x and ω-‹S_x› curves are shown in Fig. 5(a) and 5(b), respectively. Based on causality, the physical solutions are plotted in red solid (for negative index property) and light green solid (for positive index property) curves, and the dashed curves are pseudo or non-physical solutions (corresponding negative ‹S_x›) while the dash-dot curves are non-propagating modes like that in Fig. 4. This further indicates that negative index modes and positive modes should be treated separately, which also applies to the IIM and IMI cases.

 

Fig. 5 Details of the dispersion curves of the lossless case shown in Fig. 4. Solid curves are for the propagating modes, and solutions which violate causality are shown as dashed curves. The dash-dot curves indicate non-propagating modes (the imaginary part shown in the same picture in blue). The curves are consistent in color.

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4. The IIM and IMI cases

In this section we discuss the IIM and IMI cases. For the lossless case and when negative refraction occurs, the dispersion curves for IIM and IMI structures can always be plotted like those black curves in Fig. 2(b) and 2(c) (see also Fig. 1 in Ref [17], Fig. 2.23 and 2.25 in Ref [20].). These curves always appear as one continuous and smooth curve with parts having both positive and negative slopes, which are different from those for the MIM structure. Note that for the MIM structure, when the dielectric layer is thin enough, the positive and negative slope bands are always separated completely by the line ${\omega }_{sp}={\omega }_p/\sqrt{(1+{\epsilon }_d)}$ as in Fig. 2(a). For our study, we argue that these curves for lossless IIM and IMI cases are ambiguous and lead to self-contradictory interpretations between the lossless and lossy cases. In the following paragraph of this section, we show how they can be represented and indicate that as a result of our correction there is consistency between lossless and lossy cases.

For the IIM case, we use air as the cover medium (ε₂ = 1) and ε₁ = 4, ε₃ = ε_metal, and let d = 10nm since the negative index property will be more evident when d is thin. For the lossless case, the ω-k_x curves are shown in Fig. 6(a) and for clarity, we do not show the curves in same figure when k_x is large, instead, we will show these details in Fig. 7 . The red (blue) solid curves illustrate negative (positive) propagating mode directions; the dashed curves are for pseudo-solutions. We can obtain two pairs of non-propagating solutions (the dash-dot curves) for higher and lower frequencies (see the insets for details), which are axially symmetric and have large imag(k_x) and ‹S_x› = 0. Similar to the anti-symmetric mode case in the MIM structure, these non-propagating curves will be asymptotes for curves of the lossy case and the curves for different modes also lie on different sides of the dash-dot curves. Based on the value of imag(k_x), we find that the IIM structure shows positive index properties at lower frequencies, both positive and negative index properties over the middle frequency range (as imag(k_x) is smaller in this frequency range), while decaying at higher frequencies. For propagating modes, the ω-k_x curves are no longer continuous, which means the negative index mode and the positive index mode are never coincident and therefore should be treated separately. For the lossy case, we find physical solutions based on causality and plot them in light green (negative index mode) and light blue (positive index mode). We can see that the curves for the lossy case have those of the lossless case as asymptotes. The orange curve is a high order solution of the lossy case to further prove that the dash-dot curves are asymptotes of the lossy case. At these higher frequencies, there is another high order non-propagating mode of the lossless case which will be asymptotic to the orange curve, but we do not plot them here as it is not pertinent to this paper.

 

Fig. 6 Dispersion relations of the IIM structure for both lossless and lossy cases. (a) and (b) show the real and imaginary parts of the wave vector, respectively; (c) shows the ω-‹S_x› curves. For clarity, we do not show the curves of the lossless case when k_x is large. The curves are consistent in color.

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Fig. 7 Details of the dispersion curves for the lossless case shown in Fig. 6. Solid curves indicate the propagating modes, and solutions which violate causality are shown by dashed curves. The dash-dot curves indicate the non-propagating modes (the imaginary part shown in the same picture in blue). Based on causality, we plot positive ‹S_x› for the propagating modes in (b). The curves are consistent in color.

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For larger k_x and the lossless case, details are shown in Fig. 7. The situation is similar to the anti-symmetric mode of the MIM structure: the curves also tend asymptotically to ω = ω_sp, but first dip down, cross and then rise up to the asymptote. The right branches are consistent with causality in Fig. 7(b) and here we just plot the curves with ‹S_x›>0 and do not show the curves with negative value of ‹S_x› which correspond to the dashed curves and are axially symmetric with the solid curves in (b).

For the IMI case and because of the symmetry of the structure, there exist two modes, the symmetric and anti-symmetric mode (we still use H_y for definition as in MIM structure). The anti-symmetric mode shows no negative index property and imag(k_x) is always large and so we just study the symmetric mode here. We set ε₁ = ε_metal, ε₂ = ε₃ = 4 and d = 10 nm. The ω-k_x curves of the IMI system are very similar to those of IIM structure, and we just plot the physical curves in Fig. 8 in which the branches are picked based on consistency with causality and the case for large k_x is shown in the upper left inset. Similarly, the negative and positive index modes should be treated separately for both lossless and lossy cases. For the lossless case, the red curve indicates the propagating negative index mode and the blue one is for positive; the dash-dot curves represent the non-propagating modes. For the lossy case, the light green curve represents the negative index mode and light blue represents positive and since the curves are very close to each other, the upper right inset shows more details. Considering the imaginary parts of k_x for which a smaller value corresponds to a longer propagation length, we can see that the IMI structure shows negative and positive index properties simultaneously in certain frequency ranges.

 

Fig. 8 The physically meaningful dispersion curves of the IMI structure, all curves corresponding to causal solutions and the pseudo solutions are not shown. The details of large real(k_x) are shown in the inset upper left in (a). All the curves are consistent in color.

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5. The case of actual metal data

In this section we will use the experimental permittivity data of silver to verify the discussion in the previous sections. Here, the actual permittivity of silver is taken from Ref. [22], and the other parameters are still the same as used previously. The dispersion relation curves are shown in Fig. 9 . For the lossless case for which we just take the real parts of these data, the black solid curves indicate the propagating modes and the black dash-dot curves indicate the non-propagating modes (with the green dash-dot curves corresponding to the imaginary parts). Note that these solutions are all causal and the sign of real(k_x) decides the negative (real(k_x)<0) or positive (real(k_x)>0) indices of these modes. For the lossy case, the red (blue) solid curves represent the negative (positive) index modes, which have the lossless curves as asymptotes. We can see that the curves in Fig. 9 are very similar to those cases based on the Drude model in previous sections, which further supports the validity of our discussion.

 

Fig. 9 Dispersion relations for (a) MIM, (b) IIM and (c) IMI structures, when the permittivities of metal layers take the actual parameters for silver. All the curves are causal, and the pseudo solutions are dropped. For the lossless case, the black solid curves show the propagating modes; the dash-dot curves correspond to non-propagating modes. For the lossy case, the red (blue) solid curves indicate the negative (positive) index modes, and the dashed curves are the corresponding imaginary parts. The curves are consistent in color.

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6. Conclusion

In this paper, we have studied the dispersion relations of MIM, IIM and IMI structures, and plot the ω-k_x and ω-‹S_x› curves for both lossless and lossy cases. We initially used the lossy Drude model for the permittivity of metal layers, and then we used real data for silver for verification. The results are in agreement for both cases. Using the principle of causality and in particular that the energy flow is along the positive x direction, we were able to pick out the correct branches corresponding to solutions with physical meaning from the multiple mathematical solutions of Maxwell’s equations for these structures. We showed the relationship between the dispersion curves for the lossless and lossy cases, and explained how the curves vary as the loss increases. Based on this, for both the lossless and lossy cases, we see that the branches of negative and positive index modes are independent of each other, and should be treated separately. In particular for the lossless case, our results are different from previously published results. Real(k_x) of the propagating modes, where negative refraction occurs, will be negative and for IIM and IMI structures, the curves will not be continuous anymore; this is illustrated by the red curves in Fig. 2 for clarity. This result explains previously published dispersion curves for propagating modes were confused with non-physical modes. To the best of our knowledge, there are no publications showing such physically reasonable and self-consistent results regarding the dispersion relations of MIM, IIM, and IMI structures for both the lossy and lossless cases when negative refraction occurs. The results presented here show evident differences from those previously published papers which are ambiguous and/or self-contradictory.