Solid State Communications 150 (2010) 285–288

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Solid State Communications

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

Delocalized waves in one-dimensional disordered layered systems withoblique incidence

Shi Chen a,∗, Shuyu Lin a, Zhaohong Wang b

a School of Physics and Information Technology, Shaanxi Normal University, Xi’an, 710062, Chinab Key Laboratory of Physical Electronics and Devices, Ministry of Education, Xi’an Jiaotong University, No. 28 West Xianning Road, Xi’an, Shaanxi 710049, China

a r t i c l e i n f o

Article history:Received 5 September 2009Received in revised form31 October 2009Accepted 5 November 2009 by P. ShengAvailable online 10 November 2009

Keywords:A. Disordered systemsD. Anderson localizationD. Transmission coefficients

a b s t r a c t

In this paper, we present disordered layered systems in which delocalized waves with nonzero frequencycan exist in the case of oblique incidence. The appearance of the delocalized waves is attributed totwo factors: one is the existence of the localized modes in the systems, and the other is the phasematching condition. The conclusions are not in agreementwith the traditional view, andmay provide newunderstanding to the wave phenomena in one-dimensional disordered systems. In addition, the systemsproposed here are also very useful for detecting and generating quasi-monochromatic acoustic waves.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Since Anderson localization [1] was discovered in 1958, propa-gation of waves in disordered systems has been studied widely inmany fields, such as electron waves in solids, lattice waves in dis-crete atomic chains, optical waves in dielectric media, and acousticwaves in elastic media.Due to the complexity of disordered systems, the theory of

localization of waves in these systems is not in fact conclusive. Forexample, many researchers [2–8] think that in one-dimensional(1D) random disordered systems, all the acoustic waves are lo-calized unless the frequency of the waves is zero. However, in aprevious paper [9], we designed special disordered layered sys-tems, and delocalized waves with nonzero frequency were de-tected in these systems, where the wave vectors of waves wereperpendicular to the interfaces. In other fields, such as latticewaves [10] in discrete atomic chains or electron waves [11–13]in solids, some researchers have also realized this problem, anddesigned special disordered systems with delocalized waves withnonzero frequency. Thus studying delocalized waves in all kinds ofdisordered systems has become important for understanding thesewave phenomena.

∗ Corresponding author.E-mail address: chenshi@snnu.edu.cn (S. Chen).

0038-1098/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.ssc.2009.11.005

In this paper, we present new disordered layered systems inwhich delocalized acoustic waves can exist in the case of obliqueincidence. We also find that the frequency width of passbands isextremely small, so the systemsmay be used for detecting and gen-erating quasi-monochromatic waves [14,15]. Compared with thedelocalizedwaves in the case of perpendicular incidence in Ref. [9],these delocalized waves studied heremay bemore useful for someacoustic wave devices in which oblique incidence is needed. Themechanism of the formation of the delocalized waves in this pa-per is completely different from that in Ref. [9]. When waves areobliquely cast on layered systems, partialwaves in somemedia lay-ers can be attenuated in the direction normal to the interfaces. Thisimplies that these layers have properties of metamaterials withnegative mass density. Thus, in these systems, there may be lo-calized modes, which can create delocalized waves if the phasematching condition (PMC) is satisfied. In conclusion, the necessaryand sufficient conditions for the existence of delocalized waves inthese systems are: the existence of the localized modes and thesatisfaction of the PMC.We think that this conclusion is general forsystems (periodic or disordered systems) where attenuated partialwaves (caused by oblique incidence or single-negative metamate-rials) always exist.In layered systems composed of single-negative metamateri-

als (SNMs), because partial waves in the SNMs are attenuated, solocalized modes may exist in these systems. They can lead to thetransmission enhancement of waves. In 2008, Han [16] and his col-leagues investigated the localization properties of optical waves in

286 S. Chen et al. / Solid State Communications 150 (2010) 285–288

Fig. 1. A disordered layered system consisting of media A, B, C and D.

one-dimensional stacking disordered structures composed of al-ternating ε-negative andµ-negativemetamaterials. They detectedthat there was an abnormal transmission peak in the single nega-tive gap (SNG). According to our views, the phenomenon can be ex-plained as follows: the center frequency of the transmission peakis just that of the localizedmode in the ε-negative/µ-negative two-layered structure (this point of view is not expressed definitivelyin Ref. [16]). The smaller the departure degree from the PMC is,the weaker the wave localization becomes. If the PMC is satis-fied rigidly, a passband about the frequency of the localize modeappears.Our results imply that delocalized modes can also exist in dis-

ordered layered systems composed of alternating single-negativeand generalmaterials. However, these systems are not investigatedhere since their realization is not easy. The structures studied inthis paper are similar to systems consisting of alternating single-negative and general materials, but they only contain generalmaterials.Although these results are derived from acoustic wave systems,

they are also correct in other fields, such as optical waves andelectron waves in disordered 1D systems.

2. Theory formulation

A disordered layered system is shown in Fig. 1; it consistsof N cells, i.e., (A/B/C)j for the jth cell. The whole system is:(A/B/C)1(A/B/C)2 . . . (A/B/C)N . The two semi-infinite mediumD layers act as the generator and detector of acoustic waves,respectively. The thickness of layer B is hB. In the jth cell, thethicknesses of layers A and C are hAj and hCj, respectively.The media A, B, C , D considered here are isotropic or belong to

m3m class, so the shear horizontal (SH) acoustic waves describedby the displacement u3 in the system can be decoupled from otherkinds of wave. In this paper, only SH waves are investigated, forsimplification. For an oblique incident wave with the frequency ωand incidence angle θ from the left side of the system, we want toknow what the reflection and transmission waves are.The field variables have a common factor exp(−iωt) exp(ikyy),

which is omitted hereafter. ky is the wave vector in the y-direction.The field equation and normal stress in the nth layer are [17]

c(n)∂21u(n)+(ρ(n)ω2 − c(n)k2y

)u(n) = 0, (1)

T (n) = c(n)∂1u(n). (2)

Here u(n), ρ(n) and c(n) are the elastic displacement, mass densityand elastic constant in the nth layer, respectively. c ≡ c44.Because the incident angle is θ , ky = (ω/vD) sin(θ). Here vD =√cD/ρD, cD and ρD are the velocity of waves, elastic constant andmass density in the medium D. Eq. (1) can be rewritten as follows:

c(n)∂21u(n)+ ρ ′(n)ω2u(n) = 0. (3)

Here ρ ′(n) is the effective mass density in the nth layer, and ρ ′(n) =ρ(n) − ρD sin2(θ)c(n)/cD or ρ ′(n) = c(n){1/[v(n)]2 − sin2(θ)/v2D}.v(n) =

√c(n)/ρ(n) is the velocity of waves in the nth layer.

The elastic displacement and normal stress in the nth layer canbe expressed as follows:

u(n)(x) = a(n) exp(ik(n)x

)+ b(n) exp

(−ik(n)x

), (4)

T (n)(x) = z(n)[a(n) exp

(ik(n)x

)− b(n) exp

(−ik(n)x

)]. (5)

Here, a(n) and b(n) are constants to be determined. We assume thatthe amplitude of the incident wave in the 0th layer is 1, i.e., a(0) =1. There is no reflectedwave in the (3N+1)th layer, so b(3N+1) = 0.

z(n) = c(n)ik(n), k(n) = ω√ρ ′(n)/c(n) = ω

√1/[v(n)]2 − sin2(θ)/v2D.

The transmission coefficient Tc = |a(3N+1)|2 of waves can becalculated by the standard transfer matrix method [18]. We definea state vector Γ = [u, T ]τ , where u and T denote the elasticdisplacement and normal stress, respectively. For the nth layer,the left interface is the nth interface, and the state vectors on theinterface are denoted by Γn = [un, Tn]τ ; the right interface isthe (n + 1)th interface, and the state vectors on the interface aredenoted by Γn+1 = [un+1, Tn+1]τ . The transfer matrix relates thestate vector on the (n+ 1)th interface to that on the nth interfacethrough Γn+1 = MnΓn, where the matrix Mn has the followingform:

Mn =[cosh(α(n)) sinh(α(n))/z(n)

sinh(α(n))z(n) cosh(α(n))

], (6)

where α(n) = ik(n)h(n) and z(n) = c(n)ik(n). h(n) is the thickness ofthe nth layer.The transmission coefficient Tc can be obtained by evaluating

Tc =∣∣∣∣ 2M(1, 1)+M(2, 2)−M(1, 2)zD −M(2, 1)/zD

∣∣∣∣2 . (7)

Here, zD = icDkD. M = M3NM3N−1 · · ·M1. M(1, 1), M(1, 2),M(2, 1), andM(2, 2) are the components of the matrixM .

3. Disordered layered systems with delocalized modes

The systems discussed here are shown in Fig. 1. The materialand structure parameters are chosen as follows: (1) vD is the leastamong vA, vB, vC and vD. vB is less than vA and vC . (2) We choosea proper incidence angle θ that makes kB be real, and kA and kCbe purely imaginary. Here kA ≡ k(1), kB ≡ k(2), and kC ≡ k(3).These imply that the effective mass densities of layers A and Care negative, and that those of layers D and B are positive. (3) Thethicknesses of all the medium B layers are same. (4) For the jth cellin the systems, i.e., (A/B/C)j, hAj = hj(1 + ∆)f , hCj = hjg . Heref = (kA + kC ) /kA, g = (kA + kC )/kC . hj is random. If∆ = 0, then

ikAhAj = ikChCj, (j = 1, 2, . . .N). (8)

Eq. (8) is called the phasematching condition (PMC).∆ implies thedegree of departure from the PMC.The relationship between the average transmission coefficient

and the relative frequency is shown in Figs. 2 and 3. Here, D =B = BaF2, A = Al, C = NaF. vA = 3.2519 (km/s), ρA =2695 (kg/m3); vB = 2.2934 (km/s), ρB = 4886 (kg/m3); vC =3.1735 (km/s), ρC = 2790 (kg/m3) [19]. N = 8, hj is random,and hj ∈ [1, 2, 3] (mm). The average transmission coefficient iscalculated for all cases (there are 38 different cases).From these two figures, we detect: (1) there are delocalized

waves near the center frequencies ωL = 1.9304 (MHz) and ωL =6.8336 (MHz), respectively. At the two center frequencies, thetransmission coefficients are maximal, and almost equal to unity.We will demonstrate subsequently that these center frequenciescorrespond to the localized modes in the A/B/C three-layer struc-ture. (2) The transmission peak decreases with the increase of ∆.

S. Chen et al. / Solid State Communications 150 (2010) 285–288 287

Fig. 2. The average transmission coefficients in disordered (A/B/C)j systems; hB =2.5 (mm), θ = 0.3π , ωL = 1.9304 (MHz).

Fig. 3. The average transmission coefficients in disordered (A/B/C)j systems; hB =2.5 (mm), θ = 0.3π , ωL = 6.8336 (MHz).

This implies that the PMC plays an important role in the appear-ance of these delocalized waves. (3) The relative frequency widthof the passband near the center frequency ωL = 6.8336 (MHz) isvery small, and less than 10−5. Thus the systems can be used forgenerating and detecting quasi-monochromatic acoustic waves.When the effective mass densities of layers A, B, and C are all

negative, there are no delocalizedwaves in the disordered systems.

4. The localized modes in the A/B/C three-layer structure

Because the effective mass densities of layers A and C are nega-tive, there may be localized modes in the A/B/C three-layer struc-ture, where layers A and C are semi-infinite; the thickness of layerB is hB. The elastic displacement and normal stress in layer A canbe expressed as

u = d exp(−ikAx), T = −dzA exp(−ikAx). (9)

The elastic displacement and normal stress in layer C are

u = e exp(ikCx), T = ezC exp(ikCx). (10)

The elastic displacement and normal stress in layer B can beexpressed as

u = a exp(ikBx)+ b exp(−ikBx),T = zB [a exp(ikBx)− b exp(−ikBx)] .

(11)

Here, a, b, d, and e are unknown constants.

The boundary conditions are that the elastic displacement andnormal stress are continuous at the two interfaces. The four bound-ary conditions lead to the dispersion equation of the localizedmodes as follows:

exp(2β) = (zB + zC ) (zB + zA) / [(zB − zC ) (zB − zA)] . (12)

Here, β = hBikB.If ρ ′A < 0, ρ ′C < 0, and ρ ′B > 0, where ρ ′A, ρ

′

B, and ρ′

C arethe effective mass densities in layers A, B, and C , then β and zB arepure imaginary numbers; however,zA and zC are real numbers. Themoduli of the two sides of Eq. (12) are all one, thus the number ofthe real roots of Eq. (12) is infinite. This implies that the numberof localized modes is infinite in this case. If ρ ′A < 0, ρ

′

C < 0, andρ ′B < 0, the modulus of the left-hand side of Eq. (12) is less than 1;however, that of the right-hand side of Eq. (12) is greater than 1.Thus there are no localized modes in this case.For the Al/BaF2/NaF three-layer structure, where hB =

2.5 (mm), ky = (ω/vB) sin(θ), and θ = 0.3π , the frequenciesof the lowest two localized modes are ωL = 1.9304 (MHz) andωL = 6.8336 (MHz), respectively. They are just the center fre-quencies in Figs. 2 and 3. The existence of these localized modesis the necessary condition for the appearance of delocalized wavesin disordered systems.

5. The explanation for the existence of delocalizedwaves by thetransfer matrix method

The appearance of delocalized waves can be explained by thetransfer matrix method. If ∆ = 0, the transfer matrix of the jthcell is M(j)

= MCMBMA. Here MC ,MB and MA denote the transfermatrices of layers C , B, and A, which are defined in Eq. (6).M(j) canbe expressed as

M(j)=

[η σ/zAζ zC χ

]. (13)

Here,

η = (1/4)[exp(2γ (j))p+ exp(−2γ (j))q]+ (1/2)(1− zA/zC ) coshβ,

χ = (1/4)(zC/zA)[exp(2γ (j))p+ exp(−2γ (j))q] + (1/2)(1− zC/zA) coshβ

σ = (1/4)[exp(2γ (j))p− exp(−2γ (j))q]+ (1/2)(zA/zB − zB/zC ) sinhβ,

ζ = (1/4)[exp(2γ (j))p− exp(−2γ (j))q]+ (1/2)(zB/zC − zA/zB) sinhβ,

p = (1+ zA/zC ) coshβ + (zB/zC + zA/zB) sinhβ,q = (1+ zA/zC ) coshβ − (zB/zC + zA/zB) sinhβ.

(14)

Here, β = hBikB, γ (j) = hCjikC . For simplification, only the high-frequency case is discussed, i.e., γ (j) � −1. For waves with fre-quencies ωL of the localized modes, we have q = 0, which can bederived from Eq. (12). ThusM(j) can be simplified as follows:

M(j)=12

[s coshβ −1/zAr sinhβzC r sinhβ s′ coshβ

]. (15)

Here, s = (1− zA/zC ), s′ = (1− zC/zA), r = (zB/zC − zA/zB). FromEq. (15), we find that the transfer matrix of the jth cell M(j) is notdependent on hj.For the simplification of further discussion, we assume that

media A and C are same. In this case,M(j) can be simplified furtheras follows:

M(j)=

[0 −1/zCzC 0

]. (16)

288 S. Chen et al. / Solid State Communications 150 (2010) 285–288

In the derivation of Eq. (16), r sinhβ = 2 has been used, which canbe obtained from Eq. (12).The total transfer matrix M of the whole systems is ±1 for

even N , or ±M(j) for odd N . According to Eq. (7), the transmissioncoefficient is 1 for even N , or |2/(zD/zC − zC/zD)|2 for odd N .Because zC is a real number, zD is a pure imaginary number, so|2/(zD/zC − zC/zD)|2 ≤ 1.In conclusion, if the PMC condition is satisfied, in these disor-

dered systems, there are delocalized waves about the frequenciesof the localized modes.

6. Conclusions

Whenwaves are obliquely cast on layered systems, the effectivemass densities of some layersmay become negative, which impliesthat the partialwaves in these layers are attenuated in thedirectionnormal to the interfaces. In disordered layered systems consistingof A/B/C cells, where the effective mass densities of layers A and Care negative, and that of layer B is positive, there are delocalizedwaves if the PMC is satisfied. The center frequencies of thesedelocalized waves are the same as those of the localized modes inthe A/B/C three-layer structure. These disordered systems can beused for detecting and generating quasi-monochromatic acousticwaves.

Acknowledgements

The research has been partially supported by the ChinaPostdoctoral Science Foundation (20080441161), and the NationalNature Science Foundation of China (60707011).

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