1. Introduction

Optical multilayers of alternating thin films of metal and dielectric were traditionally applied for filtering [1], but the emergence of photonic crystals (PhCs) and plasmonic optics has revitalized interest in such configurations. Their applicability was further extended with the introduction of left handed materials [2] and their proposed use in sub-diffraction imaging to form a perfect lens [3,4]. These latter applications are founded on the Surface Plasmon Polariton (SPP) resonance effect in negative refractive index materials [5], such as metals in the optical range – where a metal-dielectric interface supports a guided surface wave.

Metal-dielectric multilayer stacks were extensively studied in the context of PhC-like structures [6–9], where the bands of traveling waves were considered, i.e. the field is evanescent only inside metal layers while propagating inside dielectric layers. In such cases the bands are formed by means of constructive interference, and plasmonic effects due to propagation of SPPs on the metal-dielectric interfaces are not considered.

Less common are studies of the metal-dielectric stack that incorporate surface waves (SPPs), i.e. the fields are evanescent in both metal and dielectric layers – some theoretical analysis of resonant tunneling of evanescent waves through such structures has already been presented in [10], and it has been shown that 100% transmission of the evanescent waves can be achieved [11], as was also supported by experiments [12,13]. In such studies the plasmonic bands are usually perceived as formed by propagation of evanescently coupled SPPs propagating along each metal-dielectric interface [10–14]. Since SPPs couple in a different fashion via metal or dielectric media (the plasmonic gap – MIM, and the plasmonic slab – IMI, have different dispersion relations), such an approach can be too simplistic to account for the variety of phenomena latent in the metal-dielectric stack’s dispersion.

The upgraded approach we propose considers the fields as if formed by evanescent coupling of propagating gap modes or slab modes. Such an approach is somewhat similar to the case of discrete diffraction in all-dielectric stacks where a guided mode in the high-index media is evanescently coupled to others like it in adjacent layers [15,16]. However, while in all-dielectric stacks one can clearly perceive the configuration as an array of coupled basic waveguides with the high index media as the core (since a low index core cannot guide light), such clarity does not exist for a metal-dielectric stack – as both a metal-dielectric configuration and its dual (MIM and IMI) guide light, meaning both types of coupling are competing and no clear choice among gap and slab for the basic waveguide exists (similar situation occurs for alternating layers of media having negative and positive index of refraction). Therefore a more careful approach is needed if one is to accurately utilize any coupling-based approach to the metal-dielectric stack to account for its dispersion and gain insight into its band structure – the introduction of such an approach is the stated goal of this paper.

In section 2 we analytically derive the dispersion relation of the metal-dielectric for all-evanescent mode profiles, and in section 3 introduce such an approach of competing coupled slabs and gaps and utilize it to account for the phenomenon of the plasmonic band intersection. Then in section 4, using the aforementioned approach, we show how the band form is determined by the dispersion curves of the gap and slab modes, while in section 5 we show that the field symmetry of the modes comprising the bands is also closely related to that of gap and slab modes. In section 6, we propose a couple of design schemes for the stack that incorporate predefined dispersion properties of both gap and slab configurations and in section 7 we note some of the effects introduced by using a finite, instead of a periodic, stack. We summarize our results in section 8.

2. The stack dispersion relation and the band-boundary curves

We consider a periodic configuration of alternating metal and dielectric layers with permittivity ε_M,D and layer thicknesses d_M,D respectively (Fig. 1 ). We assume nonmagnetic media (µ_M,D = 1) and use a lossless Drude model for the dispersion in the metal, i.e. ε_M = ε₀(1 – ω_p²/ ω²), ω_p denoting the metal’s bulk plasma frequency, ε₀ the vacuum permittivity and ω the angular frequency (we use ε_D = 1, ω_p = 1.37·10¹⁶ for an air-gold stack in numeric calculation throughout this paper).Defining the x-direction as the normal to each layer, we consider light propagation in the x-z plane (Fig. 1). Focusing on sub-wavelength periodicity, we consider only TM polarized modes and allow for evanescent transverse mode profiles for the H-field distribution in both metal and dielectric media:

 

Fig. 1 A periodic stack of alternating metal and dielectric layers of thicknesses d_D,M and permittivity ε_D,M respectively. Wave propagation through the structure via vertical evanescent coupling (x-direction) of competing propagating plasmonic gap (blue) and slab (orange) modes (z-direction) is illustrated.

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Equation (2) represents the relationship between ω, β and k_B for eigen-states (CW modes) that are allowed to propagate along the periodic stack. For all modes having the same value of k_B, it yields their common dispersion curve (ω-β relationship). These k_B-curves, when spanning k_B over [0,π/L], turn into the allowed bands.

A simple numerical examination of the resulting band structure, readily shows that the first two bands (highest β), much like the SPP dispersion curve, are characterized by a rapid increase in β as ω approaches ω_SPP = ω_p/(1 + ε_D)^½ and are mainly below the light line (Fig. 2 ) – hence we refer to these as the plasmonic bands. Higher order bands (lower β), like those of all-dielectric stacks, asymptotically converge towards the light line as ω increases and are completely above it [13,14]. This difference in convergence between the first two bands and the higher order bands, introduced by the plasmonic effect in the metal, results in a complete band gap between the plasmonic and non-plasmonic bands [14] – a unique property for a 1D periodic configuration.

 

Fig. 2 The two plasmonic bands (yellow filling) of a periodic metal-dielectric stack of layer thicknesses (a) d_D = 30nm, d_M = 40nm, and (b) d_D = 40nm, d_M = 20nm. The k_B = 0 curves – the S-curve (red) and AS-curve (green) – intersect in (b), additional k_B-curves (blue) include k_B = 0.25,0.5,0.75·π/L (order of increasing k_B is noted by blue arrows in each band) and k_B = π/L (thick black). The light-line and SPP frequency – dashed black lines. The gray shaded area is a mirror image of the negative branch of the AS-band into the positive β range – it is there for visual convenience alone while the actual band is located in the negative β range, since its k_B-curves have a negative index.

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Being mainly concerned with the plasmonic effects in the stack, we henceforth focus on the first two bands alone – referred to as the symmetric (S) and anti-symmetric (AS) bands respectively (for reasons soon to be disclosed).

As shown in Fig. 2a, in the AS-band the k_B = 0 and k_B = π/L dispersion curves constitute the lower and upper boundaries of the band respectively (referred to as boundary curves), while other k_B-curves filling the band are monotonically arranged (smaller k_B for lower curves). In the S-band the curves are arranged in an opposite fashion. Therefore, the two k_B = 0 boundary curves are central to our discussion since an intersection between those two accompanies any intersection of the plasmonic bands – as clearly shown in Fig. 2b.

It should be noted that when plotting the band structure one should be aware that k_B-curves with negative slope (grayed area in Fig. 2) do not exist, for the same excitation, together with the positively sloped dispersion curves. Their appropriate dispersion is rather the mirror image into the negative β range as shown in Fig. 2 - this is the so called negative index branch [17]. However, from this point forward we will plot the negative index branch in the first quadrant and it should be understood to mean a mirror image for a causal solution.

Substituting k_B = 0 into Eq. (2) and applying some algebraic manipulation yields the dispersion relation for the k_B = 0 boundary curves:

3. The competing coupled-gaps and coupled-slabs approach and the two frequency regimes

In this paper we aim at introducing an approach that does not only provide an understanding of the plasmonic band structure for the periodic metal-dielectric stack, but can also be used for a detailed design of such stack as to yield a band structure with a predefined set of desired properties. We begin by introducing our approach and show how it could be applied to clarify the (already hinted at) phenomenon of plasmonic band intersection (section 3.1); we then verify our results (section 3.2).

One unique characteristic of the plasmonic bands of the stack is that they can in principle intersect, provided layer thicknesses are appropriately chosen (Fig. 2b). We thus begin by showing how this phenomenon can be interpreted using a novel coupling approach, and in section 6, applied to band structure design.

3.1 The competing coupled-gaps and coupled-slabs approach and the plasmonic band intersection

When utilizing a coupling analysis for the periodic metal-dielectric stack to gain insight into its plasmonic band structure, it is important to note that its basic decomposition into an array of coupled metal-dielectric interfaces, each supporting a SPP [10,11], is insufficient and may lead to erroneous results. The reason being that SPPs couple quite differently via dielectric and metallic media – accounted for in the difference between the dispersion curves of plasmonic gap (MIM) and slab (IMI) modes respectively [17].

Hence, a better approach would regard the stack as either an array of coupled gaps or an array of coupled slabs, thereby partly addressing this difficulty while introducing another – determining which approach better fits a given stack.

Our approach is based on simultaneously perceiving the stack as both an array of coupled gaps, and an array of coupled slabs – thus retaining characteristics from both, and eliminating the aforementioned difficulty.

Meaning, that each dielectric or metal layer in the stack is regarded as a separate plasmonic gap or a slab respectively, coupled to others of its kind throughout the stack, and that at some frequencies a given stack can mostly retain ‘gap-like’ dispersion properties (and better fits a coupled gaps description), while at other frequencies the same configuration mostly retains ‘slab-like’ dispersion properties (and better fits a coupled slabs description) – distinguishing between the two extremes can be done by using an appropriate frequency dependent criterion.

Formalizing the last statement (i.e. defining such a criterion) can be done in one of two ways, which will later be shown as equivalent. The first and most intuitive way is to use the coupling strength of two SPP propagating along adjacent metal-dielectric interfaces in the stack, when coupling via a dielectric (‘gap-like’ coupling) or metal (‘slab-like’ coupling) layers:

A transition between the two regimes according to this criterion (applies separately for any given value of k_B), occurs at a frequency ω_i at which the coupling strength of SPPs via both layer types are equal (needless to say, in addition to satisfying the dispersion relation):

A second formalization, albeit somewhat less intuitive, uses the following symmetry principle of gap and slab modes [17]: in a gap the first order (higher β) mode TM₀^G has a symmetric H-field distribution (no nulls) and TM₁^G has an anti-symmetric one (one null), while in a slab it is vice versa (symmetric – TM₁^S, anti-symmetric – TM₀^S). In other words, the symmetric gap-mode’s dispersion curve is below that of the anti-symmetric one, while the opposite holds with slab-modes. In both cases the SPP dispersion curve is located in between the two.

Consequently, at any given frequency, the problem of determining which of the coupled gaps or coupled slabs description fits better becomes analogue to determining if the aforementioned symmetry principle applies to the gaps or to the slabs in the stack. Since the S-curve denotes a symmetric field distribution in both layer types, it can be regarded as both the symmetric gap-mode and the symmetric slab-mode when determining the above, and similarly for the AS-curve (more on field symmetry in section 5). Hence, substituting the S-curve and AS-curve for the dispersion curves of the symmetric and anti-symmetric modes in both gap and slab – we define the stack as ‘gap-like’ at frequencies in which the S-curve is below the AS-curve and ‘slab-like’ when the S-curve is above the AS-curve (and expect the SPP curve to be in between) – again two mutually exclusive frequency regimes.

A transition between the two regimes according to this second criterion (which applies only for k_B = 0), occurs at an intersection of the S-curve and AS-curve along the SPP curve, meaning at a frequency ω_i for which the SPP dispersion relation holds (in addition to the dispersion relation at k_B = 0):

If we assume the equivalence of the two criteria defined, then both Eqs. (6) and (7) can be easily solved for the k_B = 0 curves to yield a closed form solution for their intersection coordinates:

As an illustration, Fig. 3b depicts the S-curve and AS-curve, while Fig. 3a,c show their SPP coupling strengths in both layer types, for three stacks having the same d_M but different d_D. As indicated by the magenta lines, the S-curve is below the AS-curve exactly when SPP coupling via dielectric is more prominent and above it otherwise, with the SPP curve in between – supporting our assumption. Also, when d_D<d_M an intersection does not exist; when it does its coordinates fit Eq. (8).

 

Fig. 3 The coupling strengths c_D (orange) and c_M (purple) for (a) the AS-curve (green) and (c) S-curve (red) and (b) the curves themselves, for a metal dielectric stack of layer thicknesses d_M = 30nm and d_D = 100nm (solid), d_D = 50nm (dashed), d_D = 20nm (dotted). Intersections of the S-curve and AS-curve are noted by magenta lines.

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3.2 Verification of results and the anti-crossing characteristics

Next, we analytically validate our assumptions and results so far. First, the intersection of the S-curve and AS-curve given by Eq. (8) indeed forms an exact solution to the dispersion relations of both these curves separately; clearly seen by a substitution of Eqs. (6) and (7) into each part of Eq. (3) separately – thereby validating our main result.

Second, the assumption made regarding the equivalence of the two criteria for the frequency regimes is proved by substituting a_+,– = κ_Dε_M ± κ_Mε_D and φ_+,– = κ_Dd_D ± κ_Md_M into the dispersion relation Eq. (2) and rearranging it in the more elegant form:

We note that cosh(φ₊)>cosh(φ_–)≥1 and a_–²> a₊²≥0 since κ_D,κ_M,d_D,d_M,ε_D>0 and ε_M<0 (below ω_p), hence the denominators on both sides of Eq. (9) are non-vanishing. For the numerators to vanish and Eq. (9) to still hold we get:

Fully, Eq. (10) states that no k_B-curve can intersect the SPP curve unless it is the S-curve or AS-curve, and only when their SPP coupling strength via both layer types is equal. If we add to it that (i) two curves of the same k_B are separated by the SPP curve (they are on different bands), and that (ii) two curves of different k_B cannot intersect at all (since then both curves would have the same LHS for the dispersion relation Eq. (2) but different RHS), it follows that among all the k_B-curves of both bands, the only two that can intersect (and thus the only band intersection possible) are the S-curve and AS-curve – and only with one another. This property will henceforth be referred to as the anti-crossing characteristics.

We conclude by noting, in light of our analysis so far, that the plasmonic band intersection can indeed be interpreted as a shift in the characteristics of the stack between these of a coupled gaps and a coupled slabs type configuration – hinting at the possibility of similar phenomena in configurations in which both types of SPP coupling are competing.

4. The band form shift and the leading gap and slab modes

The anti-crossing characteristics result in another unique property of the plasmonic bands: when the intersection exists, and the configuration turns ‘slab-like’, the bulk of the AS-band (minus the AS-curve) becomes sandwiched between its upper boundary and a new effective lower boundary – the S-curve (instead of the AS-curve). Similarly the AS-curve turns an effective upper boundary for the bulk of the S-band. The whole process changes the shape of the bands drastically, as shown in both Fig. 2b and Fig. 4b – nevertheless, this could be easily explained using our approach.

 

Fig. 4 The two plasmonic bands (‘gap-like’ region – yellow, ‘slab-like’ region – purple) of a periodic metal-dielectric stack of layer thicknesses (a) d_D = 30nm, d_M = 40nm and (a) d_D = 40nm, d_M = 20nm. The dispersion curves of the symmetric (solid) and anti-symmetric (dashed) gap (orange) and slab (purple) modes ‘lead’ the bands in their corresponding regions, as seen by the magenta markings, as well as by inspecting Δe (lower subplots). Only the boundary curves in (b) are centered on the gap modes (Δe≈0 in teal) throughout all values of β – also indicated by teal markings. The intersection of the S-curve (red) and AS-curve (green) – vertical solid black line; intersections of the gap and slab modes – vertical dashed black lines.

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Since this shift is dependent upon the existence of an intersection, and the latter was already shown to be a result of ‘gap-like’ to ‘slab-like’ transition, it is only reasonable that its interpretation should follow along similar lines. Namely, that at the ‘gap-like’ region the two plasmonic bands are centered on the two gap-modes’ dispersion curves, while at the ‘slab-like’ region they are centered on the slab-modes’ dispersion curves – the shift in band form corresponds to a shift in the leading mode of each band from a gap-mode to a slab-mode, in accordance with the ‘gap-like’ to ‘slab-like’ transition.

We formalize the last statement in two steps: first, we define a band as ‘following’ a certain mode’s dispersion curve ω = f(β) (referred to as a ‘leading mode’) for a given β, if its normalized deviation from the band center Δe(β) = (f(β)-ω_mid(β))/Δω(β) is small enough (Δω(β) and ω_mid(β) being the bandwidth and band center for a given β – measured using the effective upper and lower boundaries of the band at β).

Second, since the aforementioned shift in a leading mode now involves the entire band, and not only the k_B = 0 curves, it is useful to use the more general criteria Eq. (6) to accurately define the transition of each k_B-curve from ‘gap-like’ to ‘slab-like’ – spanning a curve along the ω-β plane henceforth referred to as the transition curve.

Next, to correlate a shift in leading mode with the gap-like to slab-like transition, represented by the transition curve, we examine the dispersion relations of our candidates for leading modes – the gap and slab modes:

As easily shown (by substituting Eq. (6) into both Eq. (11a) and Eq. (11b) rendering the two equation equivalent), TM₀^G and TM₀^S intersect along the transition curve. Similarly, TM₁^G and TM₁^S also intersect along the transition curve (their dispersion relations are derived by interchanging tanh↔coth in Eqs. (11a) and (11b) respectively).

It is therefore left to show that the two mode-pairs, (TM₀^G, TM₀^S) and (TM₁^G, TM₁^S), are the leading modes for the S-band and AS-band respectively. The first in each pair is the leading mode in the ‘gap-like’ region while the second in the ‘slab-like’.

As shown in Fig. 4b, this is exactly the case – in the ‘gap-like’ region (yellow area) the gap-modes (orange) are leading (negligible Δe), while in the ‘slab-like’ region (light purple area) the slab-modes (purple) are leading, while the shift occurs precisely on the transition curve.Finally, it should be noted that close enough to the light line the configuration is always ‘gap-like’ (regardless of k_B or layer thicknesses), since κ_D→0 while κ_M ≥ω_p/c making c_D>c_M, thus when no intersection exists the configuration is solely ‘gap-like’, and the leading modes are exclusively the gap-modes (Fig. 4a).

Also, as seen in Fig. 4b, only the boundary curves in each band are always centered on the gap modes – including the ‘slab-like’ region where the effective band boundaries have changed as to follow the slab modes.

5. The gradual ‘gap-like’ to ‘slab-like’ transition and the corresponding shift in field symmetry

As is well known, when a mode is coupled to its replica (in an adjacent identical configuration) its dispersion curve splits into two separate curves – one above and one below, approximately equidistant to the original. When an infinite number of such modes couple (periodic array), a band forms centered on the original curve – hence, the previous section results imply that the different k_B-curves can be represented as coupling in different phase combinations of the leading gap or slab mode (according to the regime) in each band. In this section, we solidify this statement by carefully examining the H-field symmetry properties of modes represented by the different k_B-curves and correlating them with that of the leading gap or slab modes in each band.

Since both leading mode-pairs (TM₀^G, TM₀^S) and (TM₁^G, TM₁^S) are constructed of a symmetric and anti-symmetric mode (in regard to the appropriate layer type: gap mode – dielectric, slab mode – metal), we would expect that the ‘gap-like’ to ‘slab-like’ shift should also accompany a corresponding shift in the H-field distribution’s symmetry for k_B-curves as to follow that of the leading mode.

First, it is necessary to decompose the H-field term for modes corresponding to any given k_B-curve into its symmetric and anti-symmetric parts in each layer type – relative to the layer’s center (since H_y(x + L) = e ^jk_B^L·H_y(x) one period is sufficient):

Second, to enable focusing on the continuous process of the ‘gap-like’ to ‘slab-like’ transition we need to extend our definitions. So far we mainly referred to the transition in a discrete fashion – as k_B-curves intersect the transition curve κ_Dd_D – κ_Md_M = 0 (equal SPP coupling strength via both layer types, c_D/c_M = 1) the band becomes ‘slab-like’. We now extend to include equal coupling strength ratios (c_D/c_M = e^α): κ_Dd_D – κ_Md_M = α. The resulting α-curves denote an identical stage for all k_B-curves in the continuous progression from ‘gap-like’ to ‘slab-like’. Analogically, these can be thought of as latitudes, while the k_B-curves as longitudes, in mapping the plasmonic band structure between the ‘gap-like’ and ‘slab-like’ poles – the light-line (corresponds to α/d_M = –1) and SPP frequency (β→∞) respectively, while the transition curve (α = 0) acts as the equator.

As an illustration, Fig. 5 shows the two plasmonic bands for two metal-dielectric stacks: one with no band intersection and thus completely ‘gap-like’ (Fig. 5a) and the other with an intersection (Fig. 5b). The lower subplot in each representing the normalized R_D,M along the k_B = 0.1·π/L curve in both bands. Though we postpone detailed examination of this figure, it is evident that when no intersection exists there is no change in the field symmetry for the aforementioned k_B-curves as no ‘gap-like’ to ‘slab-like’ transition occurs. When such a transition does occur the symmetry gradually shifts in both bands – though mostly around α-curves nighboring the transition curve.Now we move to correlate the field symmetry of k_B-curves with that of the leading mode of each band and region. We examine the field symmetry coefficients r_D,M along different α-curves spanning over both region types (α<0 – ‘gap-like’, α>0 ‘slab-like’). Figure 6a shows the complex r_D (orange) and r_M (purple) for both S-band (solid) and AS-band (dashed), along both negative and positive α-curves (plotted at different radii). Along α<0 curves, r_D indicates a clear tendency toward symmetric (r_D≈1), while r_M shows no such clear symmetry over all k_B values (r_D spans over the whole half-circle range) – thus we have what we refer to as a dominant symmetry in the dielectric layers (symmetric), but no such dominant symmetry in the metal layers. Similarly, in the AS-band along α<0 curves there is also a dominant symmetry in the dielectric (anti-symmetric) but not the metal layers.

 

Fig. 5 The plasmonic bands (ω as a function of β – upper subplot) and the normalized relative symmetry coefficients in dielectric (R_D – oragne) and metal (R_M – purple) layers plotted for k_B = 0.1·π/L (as a function of β – lower subplot) of the S-band (solid lines) and AS-band (dashed lines), for a stack of (a) d_D = 30nm, d_M = 40nm and (b) d_D = 40nm, d_M = 20nm. R_D,M = 0,1 denote a completely anti-symmetric and symmetric H-field distribution respectively (in the corresponding D or M layer type). ‘Gap-like’ and ‘slab-like’ regions are marked by yellow and purple fillings respectively. The k_B-curves for k_B = 0.01,0.25,0.5,0.75,1·π/L (thin black lines) are added (direction of increasing k_B is indicated by blue arrows) – the k_B = 0.1·π/L curves are emphasized in thick black. Also, α-curves in (b) are added for α·d_M = −0.9,-0.5,0,0.5,1·k_p (dahsed red lines, k_p = ω_p/c) – the α·d_M = −1·k_p (light-line) is in dahsed black. The β-values corresponding to the intersection of the k_B = 0.1·π/L curves with the transition curve (α = 0) are marked by thick vertical dashed lines, while the intersection with the light-line by a thin vertical black line.

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Fig. 6 The real vs imaginary parts of the H-field symmetry coefficients in dielectric (r_D – orange) and metal (r_M – purple) layers for the S-band (solid lines) and AS-band (dashed lines), plotted along (a) α-curves (constant α) for α·d_M = −0.9,-0.5,0,0.5,1·k_p, and (b) k_B-curves (constant k_B) for k_B = 0.01,0.25,0.5,0.75,1·π/L. Since |r_D,M| ≡1, different plots of r_D,M are joined one alongside another by plotting them at diffAerent radii, providing a convenient comparision tool – each radius corresponding to a different α or k_B value (marked in red). Re{r_D,M} = −1 and 1 correspond to a completely anti-symmetric and symmetric H-field distribution respectively (in the corresponding D or M layer type). The direction of change in β (from 0 to ∞) for the r_D,M plots of both bands in (b), is indicated by a corresponding arrow (color coded). The stack’s layer thicknesses are d_D = 40nm, d_M = 20nm. The α-curves and k_B-curves for which r_D,M is plotted are shown in Fig. 5b.

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These dominant symmetries in dielectric layers (gaps) along ‘gap-like’ α-curves fit the symmetries of the leading gap-modes in the S-band (TM₀^G – symmetric) and AS-band (TM₁^G – anti-symmetric). In the same way, along ‘slab-like’ α-curves there is a dominant symmetry in metal layers (slabs) for the S-band (anti-symmetric, like TM₀^S) and AS-band (symmetric, like TM₁^S) that fits that of the leading slab-modes – proving a direct correlation.We note that the notion of dominant symmetry is an approximation, and the ‘gap-like’ to ‘slab-like’ transition is gradual – with correlation gradually increasing as α is further away from 0 (transition curve).

We also note that along the transition curve (α = 0), for both bands, there is no dominant symmetry in neither layer type. To better understand the implication, we examine in Fig. 7 the normalized R_D (orange) and R_M (purple) along α-curves close to the transition curve for both S-band (solid) and AS-band (dashed). We again see the aforementioned gradual increase in correlation, however the flipping of the dominant symmetry itself along the transition is sudden at α = 0 (both R_D,M change fundamentally in terms of the range they cover).

 

Fig. 7 The normalized relative symmetry coefficient in dielectric (R_D – orange) and metal (R_M – purple) layers for the (a) AS-band (dashed lines) and (b) S-band (solid lines) as a function of k_B, plotted along different α-curves (constant α) fitting α·d_M = −0.2,-0.05,-0.01,0,0.01,0.05,0.2·k_p (red arrows indicate direction of increasing α). The α·d_M = −0.2·k_p curves are thicker for better visual convinience. R_D,M = 0,1 denote a completely anti-symmetric and symmetric H-field distribution respectively (in the corresponding D or M layer type), while R_D,M = 0.5 is the exact middle where dominant symmetry is not defined. The α = 0 curves are in solid black – where the field symmetry in both bands swithces. The stack’s layer thicknesses are d_D = 40nm and d_M = 20nm.

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Furthermore, for α = 0 and k_B→0 (not k_B = 0) we get R_M = R_M = 0.5 for both bands. Meaning, that at these coordinated the H-field shifts back and forth between a completely symmetric for all layers to a completely anti-symmetric one as a function of ωt–βz – that is a π/2 phase shifted superposition of the fields of the S-curve and AS-curve (k_B = 0). This makes α = 0, k_B = 0 a point of discontinuity in regard to field symmetry of k_B-curves; for α≠0 however the limit is continuous, but only in the sense that the field symmetry converges to that of the effective band limit (the S-curve or AS-curve depending on the region). The implications of this dynamic are discussed in section 7.

So far we have shown that for one layer type the symmetry is relatively fixed on that of the leading mode, while for the other layer type it is spanned over the whole range. To prove correlation of the field symmetry of the k_B-curves to that of the leading curve (and thus support our view that they represent coupling combinations of that leading mode) we focused on the layer type with fixed field symmetry. To show what phase combination of the leading mode each k_B-curve represents, we now focus on the field symmetry in the other layer type.

Since the phase accumulated along each period in the stack (in the x-direction) for each k_B-curve is k_BL, we would expect k_BL to be the phase combination (in coupling the leading mode to itself) that each k_B-curve represents – meaning the k_B-curves denote a constant phase combination of the leading mode throughout all α-curves in the ‘gap-like’ to ‘slab-like’ transition.

If this is the case, taking k_B→0 curves in the S-band, should translate into an in-phase (k_BL→0) combination of (i) the symmetric TM₀^G (R_D = 1) for the ‘gap-like’ region via metal layers (thus symmetric there, R_M = 1), or of (ii) the anti-symmetric TM₀^S (R_M = 0) for ‘slab-like’ region via dielectric layers (thus R_D = 0). Meaning, we would expect the ‘gap-like’ to ‘slab-like’ transition to accompany a shift in both R_D,M from 1 to 0; similarly, for the AS-band, we would expect the transition to accompany a shift in both R_D,M from 0 to 1 – indeed, these are verified in the lower subplot of Fig. 5b, where R_D,M for the S-band (solid) and AS-band (dashed) shift accordingly around the transition curve.

In the same manner taking k_B = π/L curves, an anti-phase (k_BL = π) combination of the leading modes, should lead to no change in symmetry between ‘gap-like’ and ‘slab-like’ regions (S-band – R_D = 1 and R_M = 0; AS-band – R_D = 0 and R_M = 1). For all other k_B-curves the range of R_D,M should follow monotonically in between these two extremes.

This is again verified in Fig. 6b, where r_D (orange) and r_M (purple) are plotted along different k_B-curves (different radii) for the S-band (solid) and AS-band (dashed), while the direction of increasing β for each curve is noted by the corresponding color-coded arrow. While r_D,M remain unchanged for k_B = π/L as β increases, for k_B = 0 they shift completely, while for all other k_B values they range monotonically in between the two extremes.

6. Schemes for smart design of metal-dielectric stacks

Using our approach of competing gap and slab type coupling, we managed to decipher the plasmonic band structure of the metal-dielectric stack and deduced a set of governing symmetry principles. These principles can now be incorporated into intelligent design schemes centered on achieving a predefined set of desired properties out of the different possibilities for the stack’s dispersion.

When contemplating the different possible shapes for the plasmonic band structure, in accordance with our analysis so far, it is useful to keep in mind the general rule for the layout of the bands: in each region type (‘gap-like’ or ‘slab-like’) the spread between the two bands is determined by the spread between their leading modes (gap-modes or slab-modes respectively), and thus results from the choice of layer thickness for the dominant layer type (dielectric or metal respectively) – the thinner it is, the stronger the dominant coupling type and the larger the spread; The bandwidth of each band on the other hand, is a result of the splitting from the coupling in different phase combinations of the leading mode in each band to itself via the other layer type (metal or dielectric respectively), and thus depends on its thickness – the thinner it is, the stronger the non-dominant coupling type is and the wider each band will be.

Since the dominant coupling type shifts when the ‘gap-like’ to ‘slab-like’ transition occurs, it is possible to have relatively wide bands with small inter-band spread in the ‘gap-like’ region while the same bands becoming relatively narrow and far apart for the ‘slab-like’ region.

Thus a simple approach for band structure design can consider the intersection curve coordinates first (setting the d_D/d_M ratio) if the field symmetry shift is of primary concern, then the spread between the bands or bandwidths can be modified by setting either d_D or d_M – in accordance with the aforementioned outline of the bands

Another design scheme we suggest is concerned with incorporating gap or slab dispersion effects into the stack’s dispersion as follows: select a desired gap or slab dispersion effect that exists for a range of gap or slab thicknesses, which is to be incorporated into the dispersion properties of the periodic stack via a leading gap or slab mode – thereby introducing that thickness range as a constraint on either d_D or d_M respectively so it applied for the leading mode.

Then, for that gap or slab effect to apply for the k_B-curves of the periodic stack the ω-β region in which it applies for the leading gap or slab mode should correspond to a ‘gap-like’ or ‘slab-like’ region for the periodic stack respectively – thereby introducing a constraint on the ratio d_D/d_M using Eq. (6).

Both constraints combined lead to an appropriate choice of layer thicknesses so that the desired effect applies for the k_B-curve in a ‘gap-like’ or ‘slab-like’ region of the band structure, since it applies for the leading mode at that region. This process can continue in iterations as other effects from both gap and slab configurations are added to the same stack provided the emerging constraints are compatible with those of previous iterations.

As an example, in a sufficiently narrow slab configuration TM₁^S has a frequency region of negative refractive index – negative slope of the dispersion curve preceded by a local maximum, given in implicit form as a solution (if exists) to [17]:

Using such a sufficiently small d_M, this effect can be incorporated into the AS-band of a periodic stack provided the ratio d_D/d_M is chosen such that the negative index region of TM₁^S is included within the ‘slab-like’ region, in which it is the leading mode – this is done by substituting the solution for Eq. (14) in Eq. (6), for the minimal value of d_D/d_M permitted.

Similarly, in a sufficiently narrow gap configuration TM₁^G has a negative index region preceding a local minimum (implicit form given by Eq. (14), substituting D↔M and sinh→ –sinh) [17], which can be incorporated into the AS-band in the ‘gap-like’ region when TM₁^G is the leading mode.

For a gold-air stack, the first constraint is d_M<33nm thus we take d_M = 20nm, then the second constraint yields d_D≥26nm – thus we take d_D = 26nm for the local minimum of TM₁^S to coincide with the transition curve (red circle in Fig. 8a ). In this case TM₁^G also has negative index throughout the ‘gap-like’ region, therefore most of the k_B-curves in the AS-band have negative index in both ‘gap-like’ and ‘slab-like’– thereby we get negative index k_B-curves that can exhibit both types of dominant field symmetry (due to the symmetry transition outlined in section 5).

 

Fig. 8 The two plasmonic bands (‘gap-like’ region – yellow, ‘slab-like’ region – purple) for a metal-dielectric stack of (a) d_D = 26nm, d_M = 20nm and (b) d_D = 55nm, d_M = 60nm, where the AS-band has k_B-curves of mostly negative slope (negative index) or relatively flat respectively. The leading modes are added (TM₁^G - orange, TM₁^S – purple), as well as k_B-curves for k_B = 0.2,0.3,0.4,0.5,0.6·π/L (thin black lines) with the direction of increasing k_B marked by blue arrows, and the band boundaries (thick black lines). The local maximum of TM₁^S in (a) coincides with the transition curve intersection (marked by a red circle). Light-line and SPP frequency – dashed black lines.

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As a second example, taking the maximal value of d_D for which there exists a local minimum for TM₁^G, thus making its dispersion curve as flat as possible (d_D = 55nm for gold-air stack), and taking d_M>d_D (d_M = 60nm in Fig. 8b) so that no transition exists and TM₁^G is the leading mode throughout the AS-band, results in an AS-band that is as close to a flat band as possible (flatter for larger d_M).

7. Finite stack effects

Since one can only fabricate a finite number of periods out of the periodic metal-dielectric stack, we set out to examine how stack termination affects the band structure. We look at two possible configurations for an N-period stack: a metal-coated stack of N dielectric layers separated by metal layers (Fig. 9a ), and a dielectric-coated stack of N metal layers separated by dielectric layers (Fig. 9b). These were notated (M_1/2DM_1/2)^N and (D_1/2MD_1/2)^N in [11] when the existence of a transparent band and its dependence on the choice for unit cell were discussed. To extract the propagating CW modes and their H-field distributions in both stacks we used the matrix formalism outlined in [18].

 

Fig. 9 An illustration of (a) a metal coated and (b) a dielectric coated finite metal (ε_M) dielectric (ε_D) stack of N periods and layer thicknesses d_M,D respectively.

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As illustrated in Fig. 10 , all the CW modes of the finite stack (blue dotted lines) aside from the analogues for the S-curve and AS-curve (magenta dotted lines), referred to as the S-mode and AS-mode, fit within the band boundaries of the periodic stack. As expected, the S-mode and AS-mode intersect (for d_D>d_M) along the SPP curve and their intersection does converge towards the intersection of the S-curve and AS-curve in the periodic stack as N→∞ – what is less expected however is that as these modes intersect they exhibit a cut-off.

 

Fig. 10 The two plasmonic bands (‘gap-like’ region – yellow, ‘slab-like’ region – purple) of a periodic metal-dielectric stack of layer thicknesses d_D = 40nm, d_M = 20nm and the modes (dotted blue) of the analogue finite stack of N = 10 periods for (a) a metal coated stack, and (b) a dielectric coated stack. The S-mode and AS-mode are colored magenta – their cut-off at their intersection is noted by a red circle. For (b) the intersection is magnified inset. The S-curve (red) and AS-curve (red) are added for comparison. Light line – dashed black line.

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For the metal-coated stack (Fig. 10a) the S-mode and AS-mode exist only in their ‘gap-like’ region – when the AS-mode is above the S-mode, while for the dielectric-coated stack (Fig. 10b) they exist only in their ‘slab-like’ region (AS-mode below S-mode). Therefore, it is reasonable to deduce that this cut-off is a result of the choice of coating.

Our proposed explanation is that the coating issues a constraint on mode symmetry. For the metal-coated stack the two SPPs propagating along the first and last interfaces couple only via a dielectric layer, forcing ‘gap-like’ symmetry in the first and last dielectric layers – namely that a mode with a symmetric H-field distribution in these layers has to have its dispersion curve below that of a mode with anti-symmetric distribution. For the dielectric-coated stack it is vice versa.

As discussed in section 5, the H-field of all k_B≠0-curves is a π/2 phase shifted superposition of both a symmetric and an anti-symmetric component and even for α = 0 and k_B→0 these parts are equal and do not converge to the completely symmetric or anti-symmetric distribution in all layers of the S-curve and AS-curve respectively – making the latter two curves the only two that necessarily violate the aforementioned symmetry constraint in the boundary layers, and therefore must exhibit a cut-off as this violation takes place, as seen in Fig. 10.

8. Conclusion

In this paper we introduced an analysis approach for the periodic metal-dielectric stack, by simultaneously perceiving it as an array of coupled plasmonic gaps and an array of coupled plasmonic slabs – so that the different types of SPP coupling via dielectric and metal layers are accounted for.

Using this approach we were able to gain insight into the plasmonic band structure. We showed that an intersection of the bands exists and can be interpreted as a transition in the characteristics of the stack from that of a coupled gaps type (‘gap-like’) to that of a coupled slabs type (‘slab-like’), and deduced an analytic expression for that intersection.

We then showed that this intersection involves a drastic shift in the band form that corresponds to the change in the bands’ leading modes – in the ‘gap-like’ region the bands are centered on (and thus follow) the dispersion curves of gap-modes, while in ‘slab-like’ region the bands follow those of slab-modes.

Additionally, we showed that the H-field symmetry of the different modes comprising the bands also follow the symmetry of the leading gap and slab mode, and the modes comprising each band can be interpreted as different phase combinations of coupling the leading mode to itself throughout the stack.

We then suggested two design schemes that based on the aforementioned symmetry principles to yield metal-dielectric stacks that simultaneously incorporate predefined desired dispersion properties from both the gap and slab configurations.

We then concluded by noting that in a finite approximation to the periodic stack there is a phenomena of cut-off for the completely symmetric and anti-symmetric modes in the bands, that we suggest is related to a symmetry constraint introduced by the choice of cladding.