## Competing coupled gaps and slabs for plasmonic metamaterial analysis |

Optics Express, Vol. 19, Issue 21, pp. 20372-20385 (2011)

http://dx.doi.org/10.1364/OE.19.020372

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### 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

2. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism From Conductors and Enhanced Nonlinear Phenomena,” IEEE Trans. Microw. Theory Tech. **47**(11), 2075–2084 (1999). [CrossRef]

4. S. Feng and J. M. Elson, “Diffraction-suppressed high-resolution imaging through metallodielectric nanofilms,” Opt. Express **14**(1), 216–221 (2006). [CrossRef] [PubMed]

5. F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range surface modes supported by thin films,” Phys. Rev. B **44**(11), 5855–5872 (1991). [CrossRef]

6. V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B Condens. Matter **50**(23), 16835–16844 (1994). [CrossRef] [PubMed]

9. M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, Metallo-Dielectric, One-Dimensional, Photonic Band-Gap Structures,” J. Appl. Phys. **83**(5), 2377–2383 (1998). [CrossRef]

10. S. Feng, J. M. Elson, and P. Overfelt, “Optical properties of multilayer metal-dielectric nanofilms with all-evanescent modes,” Opt. Express **13**(11), 4113–4124 (2005). [CrossRef] [PubMed]

11. S. Feng, J. M. Elson, and P. Overfelt, “Transparent Photonic Band in Metallodielectric Nanostructures,” Phys. Rev. B **72**(8), 085117 (2005). [CrossRef]

12. M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. **83**(5), 2377 (1998). [CrossRef]

10. S. Feng, J. M. Elson, and P. Overfelt, “Optical properties of multilayer metal-dielectric nanofilms with all-evanescent modes,” Opt. Express **13**(11), 4113–4124 (2005). [CrossRef] [PubMed]

14. J. Zhang, H. Jiang, S. Enoch, G. Tayeb, B. Gralak, and M. Lequime, “Two-Dimensional Complete Band Gaps in One-Dimensional Metlo-Dielectric Periodic Structures,” Appl. Phys. Lett. **92**(5), 053104 (2008). [CrossRef]

15. T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. **88**(9), 093901 (2002). [CrossRef] [PubMed]

16. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. **85**(9), 1863–1866 (2000). [CrossRef] [PubMed]

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

*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:where

*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:where*

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

_{B}*x*-direction. This relation is consistent with [10

10. S. Feng, J. M. Elson, and P. Overfelt, “Optical properties of multilayer metal-dielectric nanofilms with all-evanescent modes,” Opt. Express **13**(11), 4113–4124 (2005). [CrossRef] [PubMed]

14. J. Zhang, H. Jiang, S. Enoch, G. Tayeb, B. Gralak, and M. Lequime, “Two-Dimensional Complete Band Gaps in One-Dimensional Metlo-Dielectric Periodic Structures,” Appl. Phys. Lett. **92**(5), 053104 (2008). [CrossRef]

*ω*,

*β*and

*k*for eigen-states (CW modes) that are allowed to propagate along the periodic stack. For all modes having the same value of

_{B}*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.

*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

14. J. Zhang, H. Jiang, S. Enoch, G. Tayeb, B. Gralak, and M. Lequime, “Two-Dimensional Complete Band Gaps in One-Dimensional Metlo-Dielectric Periodic Structures,” Appl. Phys. Lett. **92**(5), 053104 (2008). [CrossRef]

**92**(5), 053104 (2008). [CrossRef]

*k*= 0 and

_{B}*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}*k*-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

_{B}*β*range as shown in Fig. 2 - this is the so called negative index branch [17

17. G. Rosenblatt, E. Feigenbaum, and M. Orenstein, “Circular motion of electromagnetic power shaping the dispersion of Surface Plasmon Polaritons,” Opt. Express **18**(25), 25861–25872 (2010). [CrossRef] [PubMed]

*k*= 0 into Eq. (2) and applying some algebraic manipulation yields the dispersion relation for the

_{B}*k*= 0 boundary curves:The two independent parts on the LHS of Eq. (3) are the dispersion relations for the

_{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

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

**13**(11), 4113–4124 (2005). [CrossRef] [PubMed]

11. S. Feng, J. M. Elson, and P. Overfelt, “Transparent Photonic Band in Metallodielectric Nanostructures,” Phys. Rev. B **72**(8), 085117 (2005). [CrossRef]

17. G. Rosenblatt, E. Feigenbaum, and M. Orenstein, “Circular motion of electromagnetic power shaping the dispersion of Surface Plasmon Polaritons,” Opt. Express **18**(25), 25861–25872 (2010). [CrossRef] [PubMed]

*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.

*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}*k*), occurs at a frequency

_{B}*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}17. G. Rosenblatt, E. Feigenbaum, and M. Orenstein, “Circular motion of electromagnetic power shaping the dispersion of Surface Plasmon Polaritons,” Opt. Express **18**(25), 25861–25872 (2010). [CrossRef] [PubMed]

*β*) mode

*TM*

_{0}

*has a symmetric*

^{G}*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}*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.

*k*= 0), occurs at an intersection of the S-curve and AS-curve along the SPP curve, meaning at a frequency

_{B}*ω*for which the SPP dispersion relation holds (in addition to the dispersion relation at

_{i}*k*= 0):

_{B}*k*= 0 curves to yield a closed form solution for their intersection coordinates: where

_{B}*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}*d*but different

_{M}*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

*separately*; clearly seen by a substitution of Eqs. (6) and (7) into each part of Eq. (3)

*separately*– thereby validating our main result.

*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}*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:thus proving that Eqs. (6) and (7) are equivalent (for

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

_{B}*k*-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

_{B}*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}## 4. The band form shift and the leading gap and slab modes

*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*).

*k*= 0 curves, it is useful to use the more general criteria Eq. (6) to accurately define the transition of each

_{B}*k*-curve from ‘gap-like’ to ‘slab-like’ – spanning a curve along the

_{B}*ω*-

*β*plane henceforth referred to as the transition curve.

*TM*

_{0}

*and*

^{G}*TM*

_{0}

*respectively.*

^{S}*TM*

_{0}

*and*

^{G}*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).

*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}*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*or layer thicknesses), since

_{B}*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}## 5. The gradual ‘gap-like’ to ‘slab-like’ transition and the corresponding shift in field symmetry

*k*-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

_{B}*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}*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}*H*-field term for modes corresponding to any given

*k*-curve into its symmetric and anti-symmetric parts in each layer type – relative to the layer’s center (since

_{B}*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).

*k*-curves intersect the transition curve

_{B}*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.

*R*along the

_{D,M}*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.

*α-*curves fit the symmetries of the leading gap-modes in the S-band (

*TM*

_{0}

*– symmetric) and AS-band (*

^{G}*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).

*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*(orange) and

_{D}*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}*α*= 0 and

*k*→0 (

_{B}*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.

*k*-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

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

_{B}*other*layer type.

*x*-direction) for each

*k*-curve is

_{B}*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.

*k*→0 curves in the S-band, should translate into an in-phase (

_{B}*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}*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}*r*(orange) and

_{D}*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

*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}*d*or

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

_{M}*k*-curves of the periodic stack the

_{B}*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}*k*-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.

_{B}*TM*

_{1}

*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*

^{S}**18**(25), 25861–25872 (2010). [CrossRef] [PubMed]

*d*, this effect can be incorporated into the AS-band of a periodic stack provided the ratio

_{M}*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}*TM*

_{1}

*has a negative index region preceding a local minimum (implicit form given by Eq. (14), substituting*

^{G}*D*↔

*M*and sinh→ –sinh) [17

**18**(25), 25861–25872 (2010). [CrossRef] [PubMed]

*TM*

_{1}

*is the leading mode.*

^{G}*d*<33nm thus we take

_{M}*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}*d*for which there exists a local minimum for

_{D}*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

*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

11. S. Feng, J. M. Elson, and P. Overfelt, “Transparent Photonic Band in Metallodielectric Nanostructures,” Phys. Rev. B **72**(8), 085117 (2005). [CrossRef]

*H*-field distributions in both stacks we used the matrix formalism outlined in [18].

*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.

*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.

*H*-field of all

*k*≠0-curves is a

_{B}*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

*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.

## References and links

1. | H. A. Macleod, |

2. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism From Conductors and Enhanced Nonlinear Phenomena,” IEEE Trans. Microw. Theory Tech. |

3. | S. A. Ramakrishna, “Physics of Negative Refractive Index Materials,” Rep. Prog. Phys. |

4. | S. Feng and J. M. Elson, “Diffraction-suppressed high-resolution imaging through metallodielectric nanofilms,” Opt. Express |

5. | F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range surface modes supported by thin films,” Phys. Rev. B |

6. | V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B Condens. Matter |

7. | M. M. Sigalas, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Metallic photonic band-gap materials,” Phys. Rev. B Condens. Matter |

8. | S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B Condens. Matter |

9. | M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, Metallo-Dielectric, One-Dimensional, Photonic Band-Gap Structures,” J. Appl. Phys. |

10. | S. Feng, J. M. Elson, and P. Overfelt, “Optical properties of multilayer metal-dielectric nanofilms with all-evanescent modes,” Opt. Express |

11. | S. Feng, J. M. Elson, and P. Overfelt, “Transparent Photonic Band in Metallodielectric Nanostructures,” Phys. Rev. B |

12. | M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. |

13. | M. Sarajlic, Z. Jaksic, O. Jaksic, M Maksimovic, and D. Jovanovic, “Dispersion of Propagating and Evanescent Modes in 1D Metallodielectric Photonic Crystal,” 14th Telecommunications Forum TELFOR (2006). |

14. | J. Zhang, H. Jiang, S. Enoch, G. Tayeb, B. Gralak, and M. Lequime, “Two-Dimensional Complete Band Gaps in One-Dimensional Metlo-Dielectric Periodic Structures,” Appl. Phys. Lett. |

15. | T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. |

16. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. |

17. | G. Rosenblatt, E. Feigenbaum, and M. Orenstein, “Circular motion of electromagnetic power shaping the dispersion of Surface Plasmon Polaritons,” Opt. Express |

18. | E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J Lightwave Technol. |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

**ToC Category:**

Metamaterials

**History**

Original Manuscript: July 1, 2011

Manuscript Accepted: September 17, 2011

Published: October 3, 2011

**Citation**

Gilad Rosenblatt and Meir Orenstein, "Competing coupled gaps and slabs for plasmonic metamaterial analysis," Opt. Express **19**, 20372-20385 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20372

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### References

- H. A. Macleod, Thin-Film Optical Filters, Institute of Physics Publishing (2001).
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism From Conductors and Enhanced Nonlinear Phenomena,” IEEE Trans. Microw. Theory Tech.47(11), 2075–2084 (1999). [CrossRef]
- S. A. Ramakrishna, “Physics of Negative Refractive Index Materials,” Rep. Prog. Phys.68, 3966–3969 (2000).
- S. Feng and J. M. Elson, “Diffraction-suppressed high-resolution imaging through metallodielectric nanofilms,” Opt. Express14(1), 216–221 (2006). [CrossRef] [PubMed]
- F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range surface modes supported by thin films,” Phys. Rev. B44(11), 5855–5872 (1991). [CrossRef]
- V. Kuzmiak, A. A. Maradudin, and F. Pincemin, “Photonic band structures of two-dimensional systems containing metallic components,” Phys. Rev. B Condens. Matter50(23), 16835–16844 (1994). [CrossRef] [PubMed]
- M. M. Sigalas, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Metallic photonic band-gap materials,” Phys. Rev. B Condens. Matter52(16), 11744–11751 (1995). [CrossRef] [PubMed]
- S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallodielectric photonic crystals,” Phys. Rev. B Condens. Matter54(16), 11245–11251 (1996). [CrossRef] [PubMed]
- M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, Metallo-Dielectric, One-Dimensional, Photonic Band-Gap Structures,” J. Appl. Phys.83(5), 2377–2383 (1998). [CrossRef]
- S. Feng, J. M. Elson, and P. Overfelt, “Optical properties of multilayer metal-dielectric nanofilms with all-evanescent modes,” Opt. Express13(11), 4113–4124 (2005). [CrossRef] [PubMed]
- S. Feng, J. M. Elson, and P. Overfelt, “Transparent Photonic Band in Metallodielectric Nanostructures,” Phys. Rev. B72(8), 085117 (2005). [CrossRef]
- M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys.83(5), 2377 (1998). [CrossRef]
- M. Sarajlic, Z. Jaksic, O. Jaksic, M Maksimovic, and D. Jovanovic, “Dispersion of Propagating and Evanescent Modes in 1D Metallodielectric Photonic Crystal,” 14th Telecommunications Forum TELFOR (2006).
- J. Zhang, H. Jiang, S. Enoch, G. Tayeb, B. Gralak, and M. Lequime, “Two-Dimensional Complete Band Gaps in One-Dimensional Metlo-Dielectric Periodic Structures,” Appl. Phys. Lett.92(5), 053104 (2008). [CrossRef]
- T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett.88(9), 093901 (2002). [CrossRef] [PubMed]
- H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett.85(9), 1863–1866 (2000). [CrossRef] [PubMed]
- G. Rosenblatt, E. Feigenbaum, and M. Orenstein, “Circular motion of electromagnetic power shaping the dispersion of Surface Plasmon Polaritons,” Opt. Express18(25), 25861–25872 (2010). [CrossRef] [PubMed]
- E. Anemogiannis, E. N. Glytsis, and T. K. Gaylord, “Determination of guided and leaky modes in lossless and lossy planar multilayer optical waveguides: reflection pole method and wavevector density method,” J Lightwave Technol.17(5), 929-941 (1999).

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