Optics Communications 285 (2012) 5492–5495

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Optics Communications

0030-40

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E-m

journal homepage: www.elsevier.com/locate/optcom

Surface modes in binary metal-dielectric metamaterials

Yuanyuan Zhao

State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, People’s Republic of China

a r t i c l e i n f o

Article history:

Received 14 February 2012

Received in revised form

3 July 2012

Accepted 10 July 2012Available online 18 August 2012

Keywords:

Surface mode

Tamm-like and Shockley-like

Metamaterials

18/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.optcom.2012.07.086

ail address: zhaoyuanyuan@fudan.edu.cn

a b s t r a c t

Motivated by Nam et al. (2010) [1], we present general properties of localized surface modes, including

the Shockley-like and Tamm-like surface modes, and give their existence conditions with coupling

perturbation at the surface in semi-infinite binary metal-dielectric metamaterials in the frame of

coupled mode theory.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Recently, controlling photons in optical periodic lattices hasbeen studied theoretically, and practically. Take a specific exam-ple, in 2009, Ishizaki and Noda [2] have firstly shown that photonscan be manipulated at the surface of 3D photonic crystals, theirresults could lead to advanced photonic circuits, high-sensitivitysensors and novel photonic nanodevices. Thus, the studies onsurface states in optical lattices would open up a new route forthe manipulation of photons.

It has been studied for many years that termination of theperiodicity generates a special class of localized modes at thesurface, since the pioneering works by Tamm [3] and Shockley [4].Associated with surface states, there are many peculiar physicaland chemical phenomena. In condensed matter physics, differingin their physical origin, surface states can be categorized intoTamm states and Shockley states. Tamm surface states forelectrons appear due to strong perturbation above a certainthreshold at the surface, e.g., an asymmetric potential about thesurface, or asymmetric termination of periodical potential; whileShockley states require a different condition. Shockley states arisefrom crossing of adjacent bands, which require at least twoalternating weak and strong bondings in the lattice and termina-tion of the stronger coupling at the surface. There are a lot ofstudies on the optical analogues of the surface states in manysystems, such as photonic crystals [5–8] and waveguide arrays[9–11]. Malkova et al. [12] provided the experimental demonstra-tion of linear Shockley-like surface states in an optically inducedsemi-infinite photonic superlattice. Moreover, Ref. [1] shows thenovel properties of surface modes in one-dimensional metal-

ll rights reserved.

dielectric metamaterials. Ref. [13] shows the origins of the surfacemode formation and identifies the existence conditions, caused bythe internal couplings and surface perturbations. Metal-dielectricmultilayers, leading to deep subwavelength mode profiles, cansupport the localized mode, and thus these metamaterials pro-vide an ideal platform for the surface mode investigation.

The properties of optical modes are described with goodapproximation in the formalism of the tight-binding model[14,15], which is analogous to the model for calculation ofelectronic states in solid [16]. In this paper, we investigate howthe surface perturbations affect the formation of the surfacemodes in semi-infinite metamaterial in the frame of a tight-binding method. Different from the case in Refs. [7,13], weconsider the perturbation, based on the different couplings onthe surface, and give the threshold of the coupling perturbationfor the existence condition of the Tamm-like and Shockley-likesurface modes in the analytical way. In the following, we beginwith the general properties of the bulk modes in infinite binarymetamaterials, and then we will show the existence conditionsfor the surface modes with the surface perturbation and internalstructure.

2. General properties of band dispersion relation in binarymetamaterials

In this section, at first we present the bulk band dispersionrelation in the binary metal-dielectric metamaterial, which can betaken as an array of single-mode waveguide. Each waveguidecouples to its nearest neighbors, and the single surface plasmonpolariton (SPP) mode [17,18] will split into symmetric andantisymmetric mode, of which the propagation constant can becontrollable through adjusting the layer parameter as shown in

x

t1

t2

E1 E2 E4 E5Λ1 E3Λ2 Λ3 Λ4 Λ5

z

Fig. 1. (a) Schematic illustration of infinite 1D binary array with alternating coupling t1 and t2, the black region corresponds to the waveguide, the white region

corresponds to the coupling region. (b) Semi-infinite binary metal-dielectric metamaterial (circles represent single SPP mode), with two field amplitudes En and Ln

representing the nth odd-numbered and even-numbered SPP modes.

Y. Zhao / Optics Communications 285 (2012) 5492–5495 5493

Fig. 1, thus the corresponding coupling coefficients betweenadjacent mode can be described as alternating ones t1 and t2

[1,13], the binary metal-dielectric metamaterial array can beanalogous to an s- and p-orbital hybrid atomic chain [16].Compare with conventional dielectric waveguide arrays, in whichthe coupling coefficient is positive, here in the model we choose,the strong and weak couplings can be realized with different signs[19]. Then we will give the equations satisfying the boundarycondition. From the coupled-mode theory for the nth waveguideEnðLnÞ, we can write down the field distribution of propagatingSPP modes without boundary satisfying the following differentialequation [20]

i@

@zþan

� �Enþt1Ln�1þt2Ln ¼ 0

i@

@zþan

� �Lnþt1Enþ1þt2En ¼ 0, nZ2 ð1Þ

where En and Ln are the field amplitudes of the nth odd-numbered and even-numbered SPP modes from the terminationof the binary metamaterials, respectively, t1 and t2 are the twocoupling constants corresponding to the sequential odd-numbered and even-numbered SPP modes, and an is the linearpropagation constant of the SPP mode in an isolated unit cell. Inthe following, we consider an array of identical mode, thus an ¼ a.

To find out the existence condition for the localized surfacemodes, we consider a semi-infinite chain in our following discus-sion. When the periodicity is terminated, the amplitudes of thelocalized modes decay away from the surface, the coupled differ-ential equations for the localized modes at the surface are given as

i@

@zþa0

� �E1þtL1 ¼ 0

i@

@zþa

� �L1þt1E2þtE1 ¼ 0 ð2Þ

where a0 is the propagation constant of the surface mode, and t isthe coupling constant corresponding to the first and second SPPmodes.

For a trial solution, the field amplitudes En and Ln can take theform EnðzÞ ¼ En expðiozÞ and LnðzÞ ¼Ln expðiozÞ, where o is theBloch-wave propagation constant. Thus for the infinite binarymetamaterial, Eq. (1) can be written in the form of

KEn ¼Ln�1þZLn

KLn ¼ Enþ1þZEn ð3Þ

where K ¼ ðo�aÞ=t2 and Z¼ t1=t2 are the dimensionless coeffi-cients of the mode, respectively. First consider the bulk situation,we define a vector cn ¼ ðEn, LnÞ

T to express the usual coupleddifferential equations in a compact Hamiltonian form

Hcn ¼ Kcn ð4Þ

Take the trail solution, characterized by the wave numberkA ð�p,p�, in the form of

cnðkÞ ¼EnðkÞ

LnðkÞ

!¼

ek

lk

!expðiknÞ ð5Þ

We can get the form of the Hamiltonian as follows

HðZ,kÞ ¼0 Zþe�ik

Zþeik 0

!ð6Þ

According to Eqs. (4) and (6), we obtain the dispersion for the bulkmodes as a function of two parameters k and Z satisfying

K2¼ Z2þ2Z cos kþ1 ð7Þ

and besides

ek

lk

!¼

Zþe�ik

K

!or

ek

lk

!¼

K

Zþeik

!ð8Þ

Consider the field amplitudes at the surface, we can rewriteEq. (2) in the similar way

ðKþNÞE1 ¼ bL1

KL1 ¼ E2þbE1 ð9Þ

where N¼ ða�a0Þ=t2 and b¼ t=t2.Take the semi-infinite chain as our model, consider the

boundary condition and the bulk situation, we can take the trialsolution to our whole system in the forms of

ckð1Þ ¼ cE1

L1

!ð10Þ

ckðjÞ ¼ cek

lk

!expðikðj�2ÞÞ, jZ2 ð11Þ

where c is the normalized factor here. For the extended modesalong the chain, k must be real with kAð�p,p�, while for thelocalized modes decaying away from the surface, the condition9eik9r1 must be met. Ref. [13] discusses the case with theexistence of the surface perturbation as a0aa, here we will focuson the existence condition for the surface mode with differentvalues of t.

3. Existence conditions for the surface modes

First K¼0, which leads to Ln ¼ 0, for all n¼ 1,2,3, . . . . Andbesides, Enþ1 ¼�ZEn ðn¼ 1,2,3, . . .Þ and E2 ¼�bE1. No matterwhether the coupling perturbation between the first and secondmode exists or not, only the odd-numbered SPP modes are non-zero. Especially, eik ¼�Z, if and only if 9Z9¼ 9t1=t29o1, thelocalized surface mode exists, and the modes correspond to

Y. Zhao / Optics Communications 285 (2012) 5492–54955494

the Shockley-like states. Define k¼ krþ iki, the solution to thelocalized modes requires kr and ki satisfying: kr ¼ 0 ðt1t2o0Þ orkr ¼ p ðt1t240Þ, and ki ¼�lnð9t1=t29Þ.

Next we will discuss the case for Ka0. According to thedispersion relation equation (3), and considering the boundarycondition equation (9), we have

eik ¼Z

b2�Z2

ð12Þ

As discussed above, the localized modes correspond to 9eik9o1, sohere we can get the existence condition for the decay wave

Zb2�Z2

����������o1)

t1t2

t2�t21

����������o1

According to the relation shown above, we can easily find that whenthere is no perturbation at the surface, which leads to t-t1, therewould be no localized surface mode with non-zero K, when9t1=t29o1, the Shockley-like mode can be observed, otherwise, thereis no surface mode. However, with the appearance of the couplingperturbation between the first and the second SPP, the couplingperturbation becomes more and more decisive. Besides, K satisfies

K2¼

b2

b2�Z2þb2¼

t2

t2�t21

þt2

t22

ð13Þ

If t1t240, then t24t1ðt1þt2Þ, or t2ot1ðt1�t2Þ and t14t2. Ift1t2o0, then t24t1ðt1�t2Þ, or t2ot1ðt1þt2Þ and 9t1949t29. Here

−pi −pi/2 0 pi/2 pi

−2.5−2

−1.5−1

−0.50

0.51

1.52

2.5

k

K

−pi −pi/2 0 pi/2 pik

−2.5−2

−1.5−1

−0.50

0.51

1.52

2.5

K

Fig. 2. Dispersion curve in a semi-infinite model with Z¼ 1:5 (a and b), and Z¼�1:5

(a) and (c) strong perturbation with t¼2.5 satisfying t2 4t1ðt1þt2Þ, (b) and (d) weak pe

to color in this figure legend, the reader is referred to the web version of this article.)

the non-zero mode cannot fall into the continuum band, take t1t240for an example, since the continuum bands are symmetric about theabscissa, we consider the upper band in our discussion, the maximumand minimum propagation K of the upper continuum band aret1=t2þ1 and 9t1=t2�19, respectively. According to Eq. (13), wehave K41þt1=t2 for t24t1ðt1þt2Þ, while Ko91�t1=t29 fort2ot1ðt1�t2Þ. With k¼ krþ iki, we can get kr ¼ 0 or kr ¼ p for thesetwo cases, respectively. For t1t2o0, the results are similar to theabove discussions, just consider 9t19 and 9t29 instead. Due to t1t2o0,the maximum and minimum propagation K of the upper continuumband are 1�t1=t2 and 9t1=t2þ19, then, for t24t1ðt1�t2Þ, we haveK41�t1=t2, while for t2ot1ðt1þt2Þ, we have Ko9t1=t2þ19, on theother hand, choose k¼ krþ iki, for the above two cases, we havekr ¼ p and kr ¼ 0, respectively.

In the following, we will generalize the detailed propertiesof the modes, besides, the dispersion relations have been shownin Figs. 2 and 3, with t2 ¼ 1 and k¼ krþ iki. In these figures,we show the value of K in unit of t2, the horizontal coordi-nates corresponding to k, on the other hand, we show the decaymodes with kr. Figs. 2 and 3 correspond to Z41 and Zo1,respectively.

(1) Z40: from the above discussions, we can get the non-zeroK satisfying K2

¼ t2=ðt2�t21Þþt2=t2

2, the existence conditions forwhich are: (i) t24t1ðt1þt2Þ; (ii) t2ot1ðt1�t2Þ and t14t2. And thecorresponding kr is 0 and p, respectively. Since there is norestriction on t1 and t2 in (i), when Zo1, K¼0 can exist withkr ¼ p.

(2) Zo0: similarly, we can get the non-zero K satisfyingK2¼ t2=ðt2�t2

1Þþt2=t22, the existence conditions for which are (i0)

−pi −pi/2 0 pi/2 pi

−2.5−2

−1.5−1

−0.50

0.51

1.52

2.5

k

−pi −pi/2 0 pi/2 pik

K

−2.5−2

−1.5−1

−0.50

0.51

1.52

2.5

K

(c and d). The non-zero K corresponds to the dot (blue point online). Parameter:

rturbation with t¼0.5 satisfying t2 ot1ðt1�t2Þ. (For interpretation of the references

−pi −pi/2 0 pi/2 pi

−1.5

−1

−0.5

0

0.5

1

1.5

k

K

−pi −pi/2 0 pi/2 pi

−1.5

−1

−0.5

0

0.5

1

1.5

k

−pi −pi/2 0 pi/2 pik

−pi −pi/2 0 pi/2 pik

K

−1.5

−1

−0.5

0

0.5

1

1.5

K

−1.5

−1

−0.5

0

0.5

1

1.5

K

Fig. 3. Dispersion curve in a semi-infinite model with Z¼ 0:5 (e and g), and Z¼�0:5 (f and h). The decay propagation K, including the zero-mode and the non-zero mode

correspond to the dot (blue point online). Parameter: (e) and (f) strong perturbation with t¼1.2 satisfying t2 4t1ðt1þt2Þ, (g) and (h) weak perturbation with t¼0.2

satisfying t2 ot1ðt1�t2Þ. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Y. Zhao / Optics Communications 285 (2012) 5492–5495 5495

t24t1ðt1�t2Þ; (ii0) t2ot1ðt1þt2Þ and 9t1949t29; And the cor-responding kr here is p and 0, respectively. Since there is norestriction on t1 and t2 in (i0), when Zo1, K¼0 can exist withkr ¼ 0.

4. Conclusion

In our previous discussion, zero-mode appears with alternat-ing strong and weak coupling, no matter whether the surfaceperturbation exists or not, which corresponds to the Shockley-likemode, besides the non-zero-mode can appear with the surfaceperturbation. Strong surface perturbation can lead to Tamm-likemodes, while the Shockley-like modes can exist at the same time,with proper internal coupling structure. On the other hand, weaksurface perturbation can just lead to the Shockley-like mode.

With the perturbation, based on the different couplings on thesurface, and the internal structure of the system, we investigatethe existence conditions for the surface modes in the semi-infinite model binary metal-dielectric metamaterials. These sur-face modes in metamaterials could be used for manipulatingsurface plasmon polaritons in the nanodevices.

Acknowledgments

We thank Dr. X.Z. Yan for helpful comments. This work issupported by the National Natural Science Foundation of Chinaunder Grant No. 10847001 and National Basic Research Program

of China (973 Program) under the grant (No. 2009CB929204,No. 2011CB921803) project of China.

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