## Abstract

Layered medium comprised of metal-dielectrics constituents is of much interest in the field of metamaterials. Here we introduce a novel analysis approach based on competing coupled structures of plasmonic gaps (MIM) and slabs (IMI) for the detailed comprehension of the band structure of periodic metal-dielectric stacks. This approach enables the rigorous identification of many interesting features including the intersections between plasmonic bands, flat or negative band formation, and the field symmetry of the propagating modes. Furthermore – the “gap-slab competition” concept allows us to develop design tools for incorporating desired dispersion properties of both gap and slab modes into the stack’s band structure, as well as effects of finite stack termination.

© 2011 OSA

## 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}*

_{,}

*and layer thicknesses*

_{D}*d*

_{M}_{,}

*respectively (Fig. 1 ). We assume nonmagnetic media (*

_{D}*µ*

_{M}_{,}

*= 1) and use a lossless Drude model for the dispersion in the metal, i.e.*

_{D}*ε*=

_{M}*ε*(1 –

_{0}*ω*

_{p}^{2}/

*ω*

^{2}),

*ω*denoting the metal’s bulk plasma frequency,

_{p}*ε*

_{0}the vacuum permittivity and

*ω*the angular frequency (we use

*ε*= 1,

_{D}*ω*= 1.37·10

_{p}^{16}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:

*β*denotes the parallel wave vector, corresponding to waveguide-like propagation along the

*z*-axis, and

*κ*

_{M}_{,}

_{D}^{2}=

*β*

^{2}– (

*ω*/

*c*)

^{2}

*ε*

_{M}_{,}

*are the decay coefficients in the metal and dielectric media respectively – with real and imaginary components corresponding to evanescent and sinusoidal mode profiles respectively.*

_{D}*L*=

*d*+

_{M}*d*is the stack periodicity, and the coefficients

_{D}*A*

_{D}_{,}

*,*

_{M}*B*

_{D}_{,}

*are determined by applying both Maxwell's boundary conditions and Bloch's condition for periodicity, thus arriving at the dispersion relation:*

_{M}*k*is the Bloch wave vector, corresponding to propagation along the normal

_{B}*x*-direction. This relation is consistent with [10,13,14] when considering evanescent mode profiles in both layer types.

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*, it yields their common dispersion curve (

_{B}*ω*-

*β*relationship). These

*k*-curves, when spanning

_{B}*k*over [0,

_{B}*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*/(1 +

_{p}*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.

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*-curves filling the band are monotonically arranged (smaller

_{B}*k*for lower curves). In the S-band the curves are arranged in an opposite fashion. Therefore, the two

_{B}*k*= 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.

_{B}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*= 0 boundary curves:

_{B}*k*= 0 boundary curves of the S-band (left) and AS-band (right) – henceforth referred to as the S-curve and AS-curve respectively. The

_{B}*H*-field distribution of modes represented by those two curves,

*H*and

_{y}^{S}*H*, is given by (periodic by

_{y}^{AS}*L*):

*h*(

^{S}*x*) = cosh(

*x*),

*h*(

^{AS}*x*) = sinh(

*x*),

*l*= 0 for

*H*and

_{y}^{S}*l*= 1 for

*H*. The field distribution in Eq. (4) has special

_{y}^{AS}*x*-profile symmetry: symmetric in every layer (no nulls) for the S-curve, and anti-symmetric in every layer (one null) for the AS-curve – properties of fundamental importance to our analysis of the periodic stack in the next sections (and also accounting for the choice of band names).

## 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:

*f*(

_{SPP}*x*) is the

*H*-field distribution profile of an SPP propagating along a metal (

*x*<0) dielectric (

*x*>0) interface. Then, using Eq. (5) we define the stack as ‘gap-like’ at frequencies in which

*c*>

_{D}*c*(

_{M}*κ*<

_{D}d_{D}*κ*), and ‘slab-like’ at frequencies in which

_{M}d_{M}*c*<

_{D}*c*(

_{M}*κ*>

_{D}d_{D}*κ*) – making the two frequency regimes mutually exclusive.

_{M}d_{M}A transition between the two regimes according to this criterion (applies separately for any given value of *k _{B}*), occurs at a frequency

*ω*at which the coupling strength of SPPs via both layer types are equal (needless to say, in addition to satisfying the dispersion relation):

_{i}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 *TM*_{0}* ^{G}* has a symmetric

*H*-field distribution (no nulls) and

*TM*

_{1}

*has an anti-symmetric one (one null), while in a slab it is vice versa (symmetric –*

^{G}*TM*

_{1}

*, anti-symmetric –*

^{S}*TM*

_{0}

*). 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.*

^{S}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*for which the SPP dispersion relation holds (in addition to the dispersion relation at

_{i}*k*= 0):

_{B}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:

*β*denotes the parallel wave vector of the S-curve and AS-curve at

_{i}*ω*. Since

_{i}*β*must be real it follows that such intersection exists only if

_{i}*d*>

_{D}*d*, irrespective of media permittivity. From Eq. (8a), it also follows that

_{M}*ω*≤

_{i}*ω*with equality for

_{SPP}*d*=

_{D}*d*where

_{M}*β*→∞, and from Eq. (8b), that the intersection is always below the light line.

_{i}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*. 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}*d*an intersection does not exist; when it does its coordinates fit Eq. (8).

_{M}#### 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 _{D}ε_{M}* ±

*κ*and

_{M}ε_{D}*φ*

_{+,–}=

*κ*±

_{D}d_{D}*κ*into the dispersion relation Eq. (2) and rearranging it in the more elegant form:

_{M}d_{M}We note that cosh(*φ*_{+})>cosh(*φ*_{–})≥1 and *a*_{–}^{2}> *a*_{+}^{2}≥0 since *κ _{D}*,

*κ*,

_{M}*d*,

_{D}*d*,

_{M}*ε*>0 and

_{D}*ε*<0 (below

_{M}*ω*), 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:

_{p}*k*= 0) and the frequency regimes well defined – thereby solidifying our approach.

_{B}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*are separated by the SPP curve (they are on different bands), and that (ii) two curves of different

_{B}*k*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

_{B}*k*-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.

_{B}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.

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 Δ*e*(*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*-curve from ‘gap-like’ to ‘slab-like’ – spanning a curve along the

_{B}*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:

*TM*

_{0}

*and*

^{G}*TM*

_{0}

*respectively.*

^{S}As easily shown (by substituting Eq. (6) into both Eq. (11a) and Eq. (11b) rendering the two equation equivalent), *TM*_{0}* ^{G}* and

*TM*

_{0}

*intersect along the transition curve. Similarly,*

^{S}*TM*

_{1}

*and*

^{G}*TM*

_{1}

*also intersect along the transition curve (their dispersion relations are derived by interchanging*

^{S}*tanh*↔

*coth*in Eqs. (11a) and (11b) respectively).

It is therefore left to show that the two mode-pairs, (*TM*_{0}* ^{G}*,

*TM*

_{0}

*) and (*

^{S}*TM*

_{1}

*,*

^{G}*TM*

_{1}

*), 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’.*

^{S}As shown in Fig. 4b, this is exactly the case – in the ‘gap-like’ region (yellow area) the gap-modes (orange) are leading (negligible Δ*e*), 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*→0 while

_{D}*κ*≥

_{M}*ω*/

_{p}*c*making

*c*>

_{D}*c*, thus when no intersection exists the configuration is solely ‘gap-like’, and the leading modes are exclusively the gap-modes (Fig. 4a).

_{M}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*-curves and correlating them with that of the leading gap or slab modes in each band.

_{B}Since both leading mode-pairs (*TM*_{0}* ^{G}*,

*TM*

_{0}

*) and (*

^{S}*TM*

_{1}

*,*

^{G}*TM*

_{1}

*) 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*

^{S}*H*-field distribution’s symmetry for

*k*-curves as to follow that of the leading mode.

_{B}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):

_{+,–}and M

_{+,–}are the coefficients of the symmetric and anti-symmetric field components in the dielectric and metal layers respectively. Using Eqs. (1), (2), (12), and some algebraic manipulations, one arrives at the relation:

*R*, referred to as the relative symmetry coefficient in the dielectric or metal layers respectively, 0 corresponds to a completely symmetric

_{D,M}*H*-field and ∞ to a completely anti-symmetric in the appropriate layer type. It is more visually convenient to normalize it to fit the [0,1] range using 1/(1 + |

*R*|

_{D,M}^{2}) instead, so that 0 and 1 correspond to a completely anti-symmetric and completely symmetric

*H*-field respectively. The

*r*coefficient (

_{D,M}*r*is derived from Eq. (13b) interchanging

_{M}*D*↔

*M*), referred to as the field symmetry coefficient, is complex yet its magnitude is |

*r*|≡1 – making the non-normalized

_{D,M}*R*in Eq. (13a) purely imaginary. Thus, there is a

_{D,M}*π*/2 phase difference between the symmetric and anti-symmetric components of the

*H*-field – in other words, the opposite symmetries are competing instead of just superimposing one over the other (at

*ωt–βz*values where one symmetry coefficient is null the other is at maximum magnitude and vice versa).

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*–

_{D}d_{D}*κ*= 0 (equal SPP coupling strength via both layer types,

_{M}d_{M}*c*/

_{D}*c*= 1) the band becomes ‘slab-like’. We now extend to include equal coupling strength

_{M}*ratios*(

*c*/

_{D}*c*=

_{M}*e*):

^{α}*κ*–

_{D}d_{D}*κ*=

_{M}d_{M}*α*. The resulting

*α*-curves denote an identical stage for all

*k*-curves in the continuous progression from ‘gap-like’ to ‘slab-like’. Analogically, these can be thought of as latitudes, while the

_{B}*k*-curves as longitudes, in mapping the plasmonic band structure between the ‘gap-like’ and ‘slab-like’ poles – the light-line (corresponds to

_{B}*α/d*= –1) and SPP frequency (

_{M}*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*= 0.1·

_{B}*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*-curves as no ‘gap-like’ to ‘slab-like’ transition occurs. When such a transition does occur the symmetry

_{B}*gradually*shifts in both bands – though mostly around

*α*-curves nighboring the transition curve.Now we move to correlate the field symmetry of

*k*-curves with that of the leading mode of each band and region. We examine the field symmetry coefficients

_{B}*r*along different

_{D,M}*α*-curves spanning over both region types (

*α*<0 – ‘gap-like’,

*α*>0 ‘slab-like’). Figure 6a shows the complex

*r*(orange) and

_{D}*r*(purple) for both S-band (solid) and AS-band (dashed), along both negative and positive

_{M}*α*-curves (plotted at different radii). Along

*α*<0 curves,

*r*indicates a clear tendency toward symmetric (

_{D}*r*≈1), while

_{D}*r*shows no such clear symmetry over all

_{M}*k*values (

_{B}*r*spans over the whole half-circle range) – thus we have what we refer to as a

_{D}*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.

These dominant symmetries in dielectric layers (gaps) along ‘gap-like’ *α-*curves fit the symmetries of the leading gap-modes in the S-band (*TM*_{0}* ^{G}* – symmetric) and AS-band (

*TM*

_{1}

*– anti-symmetric). In the same way, along ‘slab-like’*

^{G}*α-*curves there is a dominant symmetry in metal layers (slabs) for the S-band (anti-symmetric, like

*TM*

_{0}

*) and AS-band (symmetric, like*

^{S}*TM*

_{1}

*) 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*

^{S}*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*(purple) along

_{M}*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*change fundamentally in terms of the range they cover).

_{D,M}Furthermore, for *α* = 0 and *k _{B}*→0 (

*not k*= 0) we get

_{B}*R*=

_{M}*R*= 0.5 for both bands. Meaning, that at these coordinated the

_{M}*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*= 0). This makes

_{B}*α*= 0,

*k*= 0 a point of discontinuity in regard to field symmetry of

_{B}*k*-curves; for

_{B}*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*-curve represents, we now focus on the field symmetry in the

_{B}*other*layer type.

Since the phase accumulated along each period in the stack (in the *x*-direction) for each *k _{B}*-curve is

*k*, we would expect

_{B}L*k*to be the phase combination (in coupling the leading mode to itself) that each

_{B}L*k*-curve represents – meaning the

_{B}*k*-curves denote a constant phase combination of the leading mode throughout all

_{B}*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*→0) combination of (i) the symmetric

_{B}L*TM*

_{0}

*(*

^{G}*R*= 1) for the ‘gap-like’ region via metal layers (thus symmetric there,

_{D}*R*= 1), or of (ii) the anti-symmetric

_{M}*TM*

_{0}

*(*

^{S}*R*= 0) for ‘slab-like’ region via dielectric layers (thus

_{M}*R*= 0). Meaning, we would expect the ‘gap-like’ to ‘slab-like’ transition to accompany a shift in both

_{D}*R*from 1 to 0; similarly, for the AS-band, we would expect the transition to accompany a shift in both

_{D,M}*R*from 0 to 1 – indeed, these are verified in the lower subplot of Fig. 5b, where

_{D,M}*R*for the S-band (solid) and AS-band (dashed) shift accordingly around the transition curve.

_{D,M}In the same manner taking *k _{B}* =

*π*/

*L*curves, an anti-phase (

*k*=

_{B}L*π*) combination of the leading modes, should lead to no change in symmetry between ‘gap-like’ and ‘slab-like’ regions (S-band –

*R*= 1 and

_{D}*R*= 0; AS-band –

_{M}*R*= 0 and

_{D}*R*= 1). For all other

_{M}*k*-curves the range of

_{B}*R*should follow monotonically in between these two extremes.

_{D,M}This is again verified in Fig. 6b, where *r _{D}* (orange) and

*r*(purple) are plotted along different

_{M}*k*-curves (different radii) for the S-band (solid) and AS-band (dashed), while the direction of increasing

_{B}*β*for each curve is noted by the corresponding color-coded arrow. While

*r*remain unchanged for

_{D,M}*k*=

_{B}*π*/

*L*as

*β*increases, for

*k*=

_{B}*0*they shift completely, while for all other

*k*values they range monotonically in between the two extremes.

_{B}## 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*ratio) if the field symmetry shift is of primary concern, then the spread between the bands or bandwidths can be modified by setting either

_{M}*d*or

_{D}*d*– in accordance with the aforementioned outline of the bands

_{M}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*respectively so it applied for the leading mode.

_{M}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*using Eq. (6).

_{M}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 *TM*_{1}* ^{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*is chosen such that the negative index region of

_{M}*TM*

_{1}

*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*

^{S}*d*/

_{D}*d*permitted.

_{M}Similarly, in a sufficiently narrow gap configuration *TM*_{1}* ^{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

*TM*

_{1}

*is the leading mode.*

^{G}For a gold-air stack, the first constraint is *d _{M}*<33nm thus we take

*d*= 20nm, then the second constraint yields

_{M}*d*≥26nm – thus we take

_{D}*d*= 26nm for the local minimum of

_{D}*TM*

_{1}

*to coincide with the transition curve (red circle in Fig. 8a ). In this case*

^{S}*TM*

_{1}

*also has negative index throughout the ‘gap-like’ region, therefore most of the*

^{G}*k*-curves in the AS-band have negative index in both ‘gap-like’ and ‘slab-like’– thereby we get negative index

_{B}*k*-curves that can exhibit both types of dominant field symmetry (due to the symmetry transition outlined in section 5).

_{B}As a second example, taking the maximal value of *d _{D}* for which there exists a local minimum for

*TM*

_{1}

*, thus making its dispersion curve as flat as possible (*

^{G}*d*= 55nm for gold-air stack), and taking

_{D}*d*>

_{M}*d*(

_{D}*d*= 60nm in Fig. 8b) so that no transition exists and

_{M}*TM*

_{1}

*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*

^{G}*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_{1/2}DM_{1/2})^{N} and (D_{1/2}MD_{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].

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*) along the SPP curve and their intersection does converge towards the intersection of the S-curve and AS-curve in the periodic stack as

_{M}*N*→∞ – what is less expected however is that as these modes intersect they exhibit a cut-off.

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*→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.

_{B}## 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.

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