Accepted Manuscript Bulk and surface acoustic waves in solid-fluid Fibonacci layered materials I. Quotane, E.H. El Boudouti, B. Djafari-Rouhani, Y. El Hassouani, V.R. Velasco PII: DOI: Reference:
S0041-624X(15)00071-2 http://dx.doi.org/10.1016/j.ultras.2015.03.004 ULTRAS 5021
To appear in:
Ultrasonics
Please cite this article as: I. Quotane, E.H. El Boudouti, B. Djafari-Rouhani, Y. El Hassouani, V.R. Velasco, Bulk and surface acoustic waves in solid-fluid Fibonacci layered materials, Ultrasonics (2015), doi: http://dx.doi.org/ 10.1016/j.ultras.2015.03.004
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Bulk and surface acoustic waves in solid-fluid Fibonacci layered materials I. Quotanea , E.H. El Boudoutia,b,∗, B. Djafari-Rouhanib , Y. El Hassouanic , V. R. Velascod a Laboratoire
de Dynamique et d’optique des mat´ eriaux, D´ epartement de Physique, Facult´ e des Sciences, Universit´ e Mohamed I, Oujda, Morocco b Institut d’Electronique, de Micro´ electronique et de Nanotechnologie (IEMN), UMR CNRS 8520, UFR de Physique, Universit´ e des Sciences et Technologies de Lille, 59655 Villeneuve d’Ascq, France c ESIM, Dpartement de Physique FSTE, Universit Moulay Ismail, Boutalamine B.P. 509, Errachidia, Morocco d Instituto de Ciencia de Materiales de Madrid, CSIC, Sor Juana Ins de la Cruz 3, E-28049, Madrid, Spain
Abstract We study theoretically the propagation and localization of acoustic waves in quasi-periodic structures made of solid and fluid layers arranged according to a Fibonacci sequence. We consider two types of structures: either a given Fibonacci sequence or a periodic repetition of a given sequence called Fibonacci superlattice. Various properties of these systems such as: the scaling law and the self-similarity of the transmission spectra or the power law behavior of the measure of the energy spectrum have been highlighted for waves of sagittal polarization in normal and oblique incidence. In addition to the allowed modes which propagate along the system, we study surface modes induced by the surface of the Fibonacci superlattice. In comparison with solid-solid layered structures, the solid-fluid systems exhibit transmission zeros which can break the self-similarity behavior in the transmission spectra for a given sequence or induce additional gaps other than Bragg gaps in a periodic structure.
Keywords: Surface acoustic waves; Solid-fluid layers; Quasi-periodic structure; Superlattice; Self-similarity 1. introduction Phononic crystals (PC) constituted by periodic arrangements (cells) of elastic/acoustic materials according to one (1D)[1], two (2D)[2], and three (3D)[3] dimensions, have been a subject of great interest during the last two decades because of their interesting properties in the development of new acoustical systems[4]. These systems are characterized by the presence of frequency regions where sound can propagate (bulk bands) and frequency regions where sound cannot propagate (gaps). This property has been exploited in the control and the guidance of the propagation of sound in different PCs[5]. As concerns 1D systems, different types of periodic structures such as solid-solid and solid-fluid layered materials as well as waveguides with different geometries are conducted as analogs of 2D and 3D leading to several interesting phenomena such as: omnidirectional band gaps [6, 7] and selective transmission by either guided modes[8] or interface resonance modes [9], the possibility to enhance acousto-optical interaction in hypersonic crystals[10, 11] and to realize stimulated emission of acoustic phonons[12] as well as ultrasonic metamaterials[13]. The advantage of 1D systems lies in the facilities to design different geometries and they require simple analytical and numerical calculations to understand deeply different physical phenomena observed in such systems. Besides periodic systems, quasi-periodic ones have been the subject of intensive study during the last two decades[14]. The quasi-periodic structures are generally built from two blocks A and B. Among them, the Fibonacci structure is constituted following the Fibonacci rule Sk+1 = Sk Sk−1 with S1 = A, ∗ Corresponding
author Email address: [email protected] (E.H. El Boudouti)
Preprint submitted to Elsevier
March 13, 2015
S2 = AB and k is the generation number. This leads to the Fibonacci sequences (FS): S3 = ABA, S4 = ABAAB, S5 = ABAABABA,... Merlin et al.[15] were the first to have studied such structure in semiconductor GaAs-AlAs superlattices (SLs). Since this work, much attention has been paid to observe the exotic phenomena of Fibonacci systems[16, 17] and interesting characteristics of these systems have been concluded[18] essentially by theoretical studies based on simple 1D models.it is known also that deterministic quasi-periodic systems may exhibit localization, as the Anderson localization, of sound and vibration[19]. Such phenomenon characterize any wave when the structures exhibit disorder[20]. An example of the properties of the propagation and localization of acoustic waves in Fibonacci modulated waveguides have been studied theoretically by some of us [21] and checked experimentally by King and and Cox[22]. Compared to solid-solid SLs, solid-fluid SLs have received less attention[1]. Some years ago [23, 24], we have performed an extensive study on the existence of confined and surface modes in finite and semi-infinite SLs made of a periodic repetition of solid-fluid bilayers. We have shown that besides the gaps due to the periodicity of the system these structures exhibit, other gaps due to transmission zeros. These transmission zeros can lead to new phenomena such as acoustic meta-materials[13], Fano resonances[24] as well as the possibility to mimic an acoustic transparency by squeezing a resonance between two transmission zeros[25]. The possibility of using these materials as acoustic mirrors and filters was also demonstrated [24]. Based on this work[24], a recent paper[26] has explored the tunneling effect between two different fluids via a solid-fluid SL with application to petroleum exploration. Also, we have shown the possibility of existence of surface modes in such systems depending on wether the SL is terminated by a solid or a fluid layer[23]. In the long-wavelength limit, an homogenization analysis has been developed[27] to study the effective properties of propagating modes of periodic solid layers in an ideal or viscous fluid. To our knowledge, few works have been devoted to solid-fluid quasi-periodic structures [28, 29]. In these papers finite size FS[28] and periodic FS [29] are studied by means of the transmission coefficient and the localization length deduced from the Lyapunov exponent. The Fibonacci-type structures [14] are formed from two blocks A (fluid layer) and B (solid layer). In reference [28], normal incidence waves through single FS have been considered and the fragmentation of the transmission bands as a function of the generation number has been discussed. However, the self-similarity of the transmission spectra following a scaling law has not been addressed in such systems. In reference [29], a periodic repetition of a FS was considered and the splitting of the bulk bands for normal and oblique incidence was studied by means of the localization length. However, in order to fully characterize the band gap structure of a Fibonacci SL made of a periodic repetition of a given cell where each cell is constructed by a given FS, a dispersion relation involving the Bloch wavevector k and the pulsation ω should be calculated. In this paper, we give closed form expressions of the transmission coefficient through one FS and the dispersion relation of a given FS repeated periodically. For one FS, we show at normal incidence the property of self-similarity of the transmission spectra each three generations following a scaling law. However, at oblique incidence this property no longer exists because of the existence of transmission zeros. For a Fibonacci SL, we show that when the generation number increases, the pass bands exhibit a fragmentation following a power law. Also we show the existence of different types of gaps: stable and transient gaps induced by the periodicity of the system and new gaps induced by the transmission zeros. These latter gaps are a characteristic of solid-fluid SL and do not exist in solid-solid SL. These bulk properties have not yet been addressed in the literature. In addition to bulk modes we show for the first time the possibility of existence of surface modes in solid-fluid Fibonacci SLs. These modes show different behaviors depending on whether they fall inside stable gaps or transient gaps. The rest of the paper is organized as follows: in Sec.2 we give a brief presentation of the method of calculation employed here, which is based on the Green function method. Sec.3 is devoted to the discussion of the numerical results for the transmission along a given Fibonacci sequence for normal and oblique incidence and the dispersion curves for bulk and surface modes in Fibonacci SL. The final section contains the concluding remarks.
2
2. METHOD OF THEORETICAL AND NUMERICAL CALCULATION 2.1. Interface response theory of continuous media In this paper, we consider a multilayered structure made of solid and fluid layers arranged perpendicularly to the x3 direction. The planes of the layers are contained within the (x1 , x2 ) directions. The acoustic waves propagating through such a system are polarized in the sagittal plane defined by the normal to the interfaces (x3 direction) and the wave vector k// (parallel to the interfaces). We choose k// along the x1 direction without loss of generality. We consider a non viscous fluid layer for which the viscous skin depth σ = (2η/ρω)1/2 is much smaller than the fluid layer thickness df over a very broad frequency range (η and ρ are the viscosity and the density of the fluid). This study is performed with the help of the interface response theory[30] of continuous media which permits us to calculate the Green’s function of any composite material. In what follows, we present the basic concepts and the fundamental equations of this theory [30]. Let us consider a composite system defined in its whole space domain labeled D. This system contains different subsystems i connected together by their interface domains Mi . The whole interface space of the ∪ system is labeled M = Mi . The elements of the Green’s function g(DD) of any composite material can be obtained from[30] g(DD)
=
G(DD) − G(DM )G−1 (M M )G(M D)
(1)
+G(DM )G−1 (M M )g(M M )G−1 (M M )G(M D), where G(DD) is the Green’s function of a given continuous medium and g(M M ) is the Green’s function of the composite system in its interface domain M . As we are interested in elastic waves in solid and fluid media, the corresponding Green’s functions G(DD) can be derived from the equation of motion of displacement fields as explained in Ref.[30]. The inverse [g(M, M )]−1 of g(M M ) is obtained as a superposition of the different gi−1 (Mi , Mi )[30], inverse of the gi (Mi , Mi ) for each constituent i of the composite system. The inverse of g(M M ) enables us to obtain the eigenmodes of a composite system through the relation[30] det[g −1 (M M )] = 0,
(2)
U (D) being an eigenvector of the reference system, Eq. (2) leads to the eigenvectors u(D) of the composite material as u(D) =
U (D) − U (M )G−1 (M M )G(M D) + U (M )G−1 (M M )g(M M )G−1 (M M )G(M D).
(3)
ln Eq. (3), U (D), U (M ), and u(D) are row vectors. If U (D) is a bulk wave launched in one homogeneous piece of the composite material, then Eq. (3) enables the calculation of all the waves reflected and transmitted by the interfaces, as well as the reflection and transmission coefficients of the composite system [1]. 2.2. Inverse surface Green’s functions of the elementary constituents As mentioned above, the calculation of the Green’s function of any composite material made of solid and fluid layers, within the interface response theory, requires the knowledge of the surface elements of its elementary constituents, namely, the Green’s function of an ideal fluid of thickness df , sound speed vf and mass density ρf and an elastic isotropic solid characterized by its thickness ds , longitudinal speed vℓ , transverse speed vt , and mass density ρs . In addition, the calculations of the dispersion relations and transmission coefficients (see below) can be deduced only from the knowledge of the components of the Green’s functions g −1 (M M ). The inverse Green’s function of an ideal fluid layer in the space of the two surfaces Mf = {−df /2, +df /2} of the layer is given by[23] ( ) af bf [gf (Mf Mf )]−1 = , (4) bf af 3
where af = −F
Cf F , bf = , Sf Sf
(5a)
Cf = cosh(αf df ), Sf = sinh(αf df ), ) 12 ( √ ω2 ω2 2 − k// and αf = j (j = −1), F = −ρf 2 αf vf
(5b) (5c)
and the inverse Green’s function of the elastic solid layer in the space of the two surfaces Ms = {−ds /2, +ds /2} of the layer is given by[23] ( ) as bs −1 [gs (Ms Ms )] = , (6) bs as where Ct γ β Cℓ − β , bs = + , Sℓ St Sℓ St v4 v4 2 2 γ = −ρ 2t (k// , + αt2 )2 , β = 4ρ t2 αt k// ω αℓ ω Ct = cosh(αt ds ), Cℓ = cosh(αℓ ds ), St = sinh(αt ds ), Sℓ = sinh(αℓ ds ), as = −γ
and
2 αt2 = k// −
2
(7a) (7b) (7c)
2
ω ω 2 , αℓ2 = k// − 2. vt2 vℓ
(7d)
The inverse Green’s function of a semi-infinite fluid is given by[23] gf−1 (0, 0) = −F,
(8)
where F is defined by Eq. (5c). 2.3. Transmission coefficients through one solid-fluid Fibonacci sequence The solid-fluid Fibonacci sequence of a given generation k is constructed from a finite number of layers A and B arranged according to the Fibonacci rule. The interface space M is made of all the sub-interfaces between the layers. Within this space the inverse of the matrix [gk (M M )]−1 giving all the elements of the Green’s function g is a finite tri-diagonal matrix formed by a linear superposition of the sub-matrices −1 g(i=s,f ) (Mi Mi ) (Eqs. (4),(6)). For example, in the case of a finite structure corresponding to the fourth Fibonacci generation S4 = ABAAB (Fig. 1), [g4 (M M )]−1 can be written in the following tri-diagonal matrix form: af bf 0 0 0 bf af + as bs 0 0 0 bs as + af bf 0 −1 . [g4 (M M )] = (9) 0 bf 2af bf 0 0 0 0 0 bf af + as bs 0 0 0 0 bs as The Green’s function elements on the two surfaces bounding the FS are obtained by inverting the [g4 (M M )]−1 (Eq. (9)) and keeping only the extreme elements of the truncated matrix to forme a 2×2 matrix. Then by inverting again this matrix, we obtain a 2×2 Green’s function matrix of the FS as follows: ( ) a b −1 [g4 (Me Me )] = , (10) b c where Me = {0, L} is the space of interfaces at both free extremities of the S4 generation and L is the total length of the structure. 4
The four matrix elements are real quantities functions of the different parameters of the constituent’s elements gi (Mi Mi ) (i = s, f ) [Eqs. (4),(6)]. From Eqs. (2) and (10), one can deduce the expression giving the eigenmodes of the FS, namely ac − b2 = 0. (11) When the finite composite system is bounded by two homogeneous semi-infinite fluids with [gf (0, 0)]−1 = −F (Eq. (8)), the inverse Green’s function of the whole composite system can be obtained by a superposition of the two matrices given by Eqs. (8) and (10) as: ( ) a−F b −1 [g(Me Me )] = . (12) b c−F An incident plane wave launched from the left and travelling through the structure gives rise to the transmission function t = −2F g(0, L)[24]. Therefore, we obtain t=−
2bF . ac − b2 + F 2 − F (a + c)
(13)
2.4. Dispersion relations of infinite and semi-infinite periodic sequences Let us consider an infinite SL made of a periodic repetition of a given sequence [Fig. 1(a)]. The system Green’s function is obtained by a linear juxtaposition of the 2×2 matrices (Eqs. (4),(6)) at the different interfaces. A tridiagonal matrix thus obtained. By using the Bloch theorem and Eq. (2), we obtain the following expression for dispersion relation of an infinite SL: cos(KD) = −(a + c)/2b,
(14)
∑Fk di Fk being the number of blocks where K is the Bloch wave-vector and D the period of the SL (D = i=1 in the generation k).Then, the surface modes dispersion relation of a semi-infinite SL lying in the half space x3 > 0 [Fig. 1(b)RSL], is given by ac − b2 = 0 , (15) together with the following condition to ensure that the waves decay from the surface when penetrating into the SL, b < 1. (16) a If the semi-infinite SL lies in the half space x3 < 0 [Fig. 1(b)LSL], we obtain the same dispersion relation [Eq. (15)] but with the condition b > 1. (17) a Thus we see that the surface mode is induced either by one semi-infinite SL or by its complementary one. In addition, Eq. (15) shows that the expression for the surface modes of two complementary SLs is the same one than that of the eigenmodes of one sequence [Eq. (11)]. 3. Numerical results In the following, we shall present some numerical calculations of transmission coefficients and dispersion curves of acoustic waves in solid-fluid Fibonacci structures made of Plexiglas-water. The case of periodic structures has been studied both theoretically[31] and experimentally[32, 33]. Table 1 gives the numerical values of velocities of sound and mass densities of the materials used in this work. We shall focus our attention on two different structures, namely a given FS inserted between two fluids and a periodic repetition of a given FS.
5
Plexiglas Water
ρ(g/cm3 ) 1200 1000
vt (105 cm/s) 1.38 -
vℓ (km/s) 2.7 1.49
Table 1: Velocities of sound and mass densities of Plexiglas and Water.
3.1. Transmission along one Fibonacci sequence Figure 2(a) shows the transmission spectrum as a function of the dimensionless frequency Ω = ωdf /vf for normal incident longitudinal acoustic waves in a periodic structure made of six solid-fluid bilayers inserted between two semi-infinite water media. The layer thicknesses are chosen such that df /vf = ds /vs where df (vf ) and ds (vs ) represent the thickness (longitudinal speed) of fluid and solid layers respectively. One can notice the existence of a forbidden band around the central frequencies Ωc = (2m + 1)π/2 for which the two layers behave as quarter wavelength layers (m being an integer). Let us mention that such a SL exhibits gaps only at the edge of the Brillouin zone (around π/2, 3π/2,...), however the gaps falling at the center of the Brillouin zone (around π, 2π,...) close. Fig. 2(b)-(g) show the transmission coefficients for the generations S4 (five blocks), S5 (eight blocks), S6 (13 blocks), S7 (21 blocks), S8 (34 blocks) and S9 (55 blocks), respectively. In fig. 2(b)-(g) one can distinguish the regions of frequencies where the transmission show several dips as the generation number increases. These regions correspond to the forbidden bands (or transmission gaps). In the other regions of frequencies, the transmission is more noticeable corresponding to the allowed modes (or transmission bands). As compared to the transmission spectrum in the periodic structure (fig. 2(a)), the quasi-periodic spectra undergo a fragmentation especially for high generations (fig. 2(g)). One can notice that around the central frequencies Ωc = π/2, 3π/2, ... of the gaps in the periodic structure (fig. 2(a)), the transmission spectra in the quasi-periodic structures (fig. 2(b)-(g)) show a certain recursive order which is a characteristic of Fibonacci systems. In order to characterize this self-similarity (also known as a scaling law[14]) in the transmission spectra, Kohmoto et al.[34] have introduced a scale factor F = [I + 4(1 + I)2 ]1/2 + 2(1 + I),
(18)
where I is an invariant describing the recursive procedure. Its expression has the following form[34] I = 1/4(Zf /Zs − Zs /Zf ) sin2 (ωdf /vf ),
(19)
where Zf = ρf vf and Zs = ρs vs are the impedance of the fluid and solid layers respectively. Also, it has been shown that this phenomenon occurs around Ωc where quasi-periodicity is most effective. This means that the transmission coefficient should have a self-similarity nature around the central frequency with Tj+3 = Tj (the transmission coefficient exhibits a third order periodicity) with a scale factor F[34]. When applied to the central frequency Ωc , equations (18) and (19) give I = 0.65 and F = 6.73. Indeed, we note a great resemblance between the generations S5 − S8 and S6 − S9 around the central frequency Ωc with a periodicity of three and a scaling factor F as it is illustrated in the inserts of fig. 2(b), (c) and (d) as compared to fig. 2(e), (f) and (g) respectively. This result is similar to that found experimentally by Gellermann et al.[35] in the optical multi-layers and El Boudouti et al.[36] in coaxial cables. This property of self-similarity has been interpreted[14] as a sign of localization of waves in Fibonacci systems. In the case of oblique incidence (θ = 3 ◦ ) the transmission spectrum shows a flat band around Ωc for the periodic structure (Figure 3(a)). Also a new gap appears around Ω = π with a flat band inside this gap which appears like a Fano resonance[37]. These flat bands enable the filtering of acoustic modes through the system in a narrow frequency region inside the gaps without introducing defect layers as it is usually the case[24]. The transmission spectra of the Fibonacci structure S4 (3 (b)) exhibits a transmission zero around π/2, π, 3π/2,...which transforms to a very thin gap for high generations (Figs. 3(b)-(g)). Therefore we can distinguish gaps due to transmission zeros from those induced by the quasi-periodicity of the system. Also, 6
this transmission zero breaks the self-similarity effect observed at normal incidence around Ωc , indicating that the transmission spectra are sensitive to the angle of incidence. For example, the sequences S6 and S9 in Fig 3 clearly show the absence of the self-similarity of the two spectra around Ωc due to the existence of the transmission zeros. By increasing the incidence angle, the gaps due to transmission zeros become larger and the disappearance of the self-similarity becomes further noticeable. 3.2. Band structure of Fibonacci superlattice In this section, we consider the dispersion curves of Fibonacci SLs made of periodic repetition of a given sequence Sk . The study of the dispersion curves of the bulk bands (reduced frequency Ω = ωdf /vf versus the reduced wave vector k// df ) is shown in figure 4 for different Fibonacci sequences. The shaded gray areas correspond to bulk bands; these bands are separated by gaps where the waves are prohibited from propagation. To show the sound cones of the constituting materials in the superlattice, we present the longitudinal and transverse velocities of sound in Plexiglas by dotted and straight lines and the longitudinal velocity of sound in water by the dashed line (table 1). Due to the low contrast between the acoustic parameters of Plexiglas and water, the gaps are not very wide. For the second FS S2 (Fig. 4 (a)), the first two bands are located below the transverse velocity of sound in Plexiglas and the longitudinal velocity of sound in water. These two bands are constituted by evanescent waves in these two media and tend asymptotically to the interface mode between Plexiglas and water for large values of k// df . When the generation number increases from S2 to S5 (Figs. 4 (b)-(d)), we can notice a fragmentation of the bulk bands and the existence of new gaps between them. These gaps are induced by the periodicity of the SL. The stars represent the gaps resulting from transmission zeros, the latter being a characteristic of the solid layer independently of the fluid layer[23]. The Bloch wave-vectors along the star positions go to infinity[24] which renders these frequency regions completely forbidden for acoustic waves. We have shown in Ref.[24] that these frequencies are induced exclusively by solid layers, that is why their positions are independent of the Fibonacci sequence (Figs. 4 (a)-(d)). Let us notice that the S3 generation in Fig. 4 (b) (..ABAABAABA...) is similar to the S2 generation (...ABABABAB...) but with the condition to replace the A layer by the AA layers. However, the corresponding band gap structures (Figs. 4 (a),(b)) are very different. For example the forbidden gap in Fig. 4 (a) at k// df = 0 becomes completely allowed in Fig. 4 (b) (see also Fig. 5(a) for the second and third generations). Figure 5(a) shows the distribution of the widths of the allowed and forbidden bands of a Fibonacci SL for different generation number k up to the 9t h generation, which means a unit cell with 34 A and 21 B layers (i.e., 55 building blocks), for θ = 0 ◦ . One can notice that the regions of allowed bands become narrower for the high generation number. In addition, there are stable gaps (denoted Gs ) that appear for all generations and transient gaps (denoted Gt ) that appear each three-generations (see for example the gaps around Ωc = π/2 for S2 , S5 and S8 ); this is illustrated in Fig. 5 (b) with a large zoom of the bandwidths around Ωc . In Fig. 5 the width of the transient gaps decreases as the number of generation increases, while the width of the stable gaps remains almost constant for high generation numbers. By changing slightly the incidence angle from θ = 0 ◦ to θ = 3 ◦ as illustrated in Fig. 6 (a), the stable gaps Gs remain unchanged and the transient gaps Gt shifts to the lower frequencies below Ωc whereas new gaps labeled G0 appears around Ω = 0.51π, π (see Fig. 6(a)). These gaps indicated by horizontal arrows are due to transmission zeros as mentioned above in the transmission spectrum (Fig. 2). Fig. 6(b) gives a large zoom around Ωc and clearly shows the positions of the G0 , Gs and Gt . By increasing the incidence angle to θ = 24 ◦ (Fig. 7), the gap G0 becomes wider and the transient gap Gt appears slightly above Ωc = π/2 while the band gap profile exhibits a fragmented energy spectrum for large k. The total width ∆k of the allowed energy bands (Lebesgue measure of energy spectrum [14]) decreases when the generation number increases with a power law ∆k = Fk−δ as it is shown in Fig. 8 where Fk is the Fibonacci number (number of blocks in each sequence) and δ is the diffusion constant of the spectrum. We note that this power law is valid for different values of the angle θ in the fluid and the parameter δ is not sensitive to the variation of θ. When the generation number k becomes larger, the band width ∆k tends to zero whatever the value of θ. This result which is a characteristic of quasi-periodic structures has been demonstrated for other types of excitations as well[14]. 7
3.3. Surface modes of Fibonacci superlattice In Ref.[38], we have shown that the terminations of two complementary SLs obtained from the cleavage of an infinite SL do not behave in the same way when going from generation to generation. For example, the cleavage of an infinite SL corresponding to the fourth Fibonacci generation (Fig. 1(a)) ...| ABAAB | ABAAB | ABAAB | ABAAB |... gives rise to two complementary semi-infinite SLs (Fig. 1(b)): LSL (left superlattice) and RLS (right superlattice). ...| ABAAB | ABAAB | and | ABAAB | ABAAB |... In the same way, the creation of two semi-infinite SLs obtained by the cleavage of an infinite SL corresponding to the fifth generation gives the two complementary SLs: ...| ABAABABA | ABAABABA | and | ABAABABA | ABAABABA |... One can notice from the passage from one generation to another that terminations of the LSL change from generation to generation, although they remain the same for even generations (with B block at the surface) and for odd generations (with A block at the surface). As concerns the terminations of the RSL, they remain unchanged for all generations. Figure 9 represents the surface modes for two complementary semi-infinite SLs for various generations until the ninth generation at the normal incidence (θ = 0 ◦ ). The blue circles give the surface modes of the RSL and the red triangles those of the complementary LSL. These modes are obtained from equations (14), (16) and (17). The existence of one surface mode per gap is clearly seen. This mode is induced by the surface of one of the two semi-infinite periodic structures. These results generalize those we have obtained before for a bilayer SL[23]. However, there is an exception to this rule for the S3 generation. Indeed, the cleavage of an infinite SL corresponding to the third generation gives rise to two identical semi-infinite SLs. Therefore, the surface modes do not appear neither on one SL nor on its complementary, but fall exactly at the edge of the bulk bands (see Ref.[39] for more details on this particular case). In Fig. 9 corresponding to θ = 0 ◦ , one can distinguish two types of surface modes: i) The modes falling inside the stable gaps. Here one can distinguish the modes associated to the RSL (blue circles), which fall almost at the same frequency as the endings of these SLs remain unchanged from generation to generation. On the other hand , the modes associated to the LSL (red triangles) appear at the same frequency but for one generation over two as the terminations of the LSL change in an alternating way from even to odd generations. This result is well illustrated in Fig 10 which gives a zoom of Fig. 9 around Ω = π/2 (Figs. 10(a),(b)) and Ω = π/5 (Figs. 10(c), (d)) up to the 11the generation. ii) The modes falling inside the transient gaps around the central frequency Ωc . These modes appear at the same frequency each six generations (see Figs. 10(a),(b)). In Ref. [38] we have shown that an analysis of the spatial localization of the modes falling in stable gaps present a decreasing behavior from the surface while penetrating in the SL, whereas the modes falling in transient gaps show a self-similar behavior of order six. This latter behavior is due to the quasi-periodicity of the layers A and B. Figure 11(a) gives the surface modes for the incidence angle θ = 24 ◦ . One can notice that the properties related to the surface modes discussed in the case of normal incidence (θ = 0 ◦ ) remain the same for θ = 24 ◦ . However, in the region above the central frequency there exists a surface mode which appears inside the thin band gap around Ω ≃ 0.64π (see Fig. 7(b)). These modes appear in the RSL for even generations (blue circles) and in the LSL for odd generations (red triangles) starting from the 4th generation as it is illustrated in Fig. 11(b). As we have shown analytically (see Eqs. (11) and (15)), the surface modes of two complementary SLs are exactly the stationary modes of one given FS. However, some of these modes falling at the same frequencies by increasing the generation number remain surface modes even for the finite size FS[36]. 8
Finally, it is worth pointing that the thicknesses of the layers chosen for the solid-fluid layers enabled us to discuss the main results of self-similarity, band gap fragmentation with different surface modes, around the central frequency Ωc = π/2. These results remain unchanged for any thicknesses of the layers but the main phenomena appear at different frequencies. Also increasing the contrast between the characteristic impedances of the chosen materials like in the case of Aluminium-water structure[23], give the same results as those presented here but with large Bragg gaps. 4. Conclusion In this work, we have studied sagittal acoustic waves in quasi-periodic structures of Fibonacci type. These systems are formed by two blocks A (fluid layer) and B (solid layer). Closed form expressions of the transmission coefficient and the dispersion relations are obtained. The analysis of the transmission spectra of different Fibonacci generations (sequences) allowed us to conclude that these spectra exhibit self-similarity of order three with a scaling factor F. We also showed the existence of gaps which are due to the quasi-periodicity and gaps due to transmission zeros. Also we have presented the dispersion curves of Fibonacci SLs made of a repetition of cells where each cell is formed by a given sequence and obtained the power law for the measure of the energy spectrum in these systems. We also studied the surface modes in the quasi-periodic SLs when considering two semi-infinite SLs obtained from the cleavage of an infinite one. We have generalized the rule on the existence of surface modes in the binary[23] solid-fluid SLs to the case of Fibonacci SLs. In particular, we have shown the existence of two types of surface modes: i) modes inside stable gaps whose frequencies remain constant for all generations as the corresponding SLs remain unchanged from generation to generation, ii) modes within transient gaps which appear each six generations. Finally let us mention that this study can be extended without difficulties to other quasi-periodic structures like aperiodic and Thue Morse systems[14].The experimental verification of these theoretical results should be possible using ultrasonic techniques[33, 8, 40, 41, 42]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
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10
Figure 1: (a) Schematic representation of an infinite periodic superlattice where each cell is composed of a Fibonacci generation (for example S4 ). (b) Schematic representation of two semi-infinite superlattices LSL (left superlattice) and RSL (right superlattice) obtained by the cleavage of the infinite superlattice (a).
11
1
(a) 0 1 S4 0 1
Transmission
(b)
π/2
S5
(c)
π/2
0 1
π/2
0 1
π/2
0 1
π/2
S6
(d)
S7
(e)
S8
(f)
0 1 S9 π/2
0
ο
(g) π/2
π
3π/2
2π
Reduced frequency Ω
Figure 2: Variation of the transmission coefficient as a function of the reduced frequency Ω for different structures: (a) The periodic structure,(b)-(g) different generations Sk (k = 4-9) of the Fibonacci structure at normal incidence. The inserts show the self similarity between the generations (S4 , S7 ), (S5 , S8 ) and (S6 , S9 ) around Ωc = π/2
12
1
(a) 0 1
S4
(b)
S5
(c)
S6
(d)
Transmission
0 1
0 1
0 1
S7
(e)
0 1
S8 (f) 0 1
S9 (g) 0
0
π
π/2
3π/2
Reduced Frequency Ω
2π
Figure 3: Variation of the transmission coefficient as a function of the reduced frequency Ω for different structures: (a) The periodic structure, (b)-(g) the different generations Sk (k = 4-9) of the Fibonacci structure at oblique incidence (θ = 3 ◦ ).
13
Figure 4: Band structure (i.e., Ω = ωdf /vf versus the reduced wave vector k// df ) for a Fibonacci SL made of Plexiglas and water for different generations (S2 − S5 ). The gray areas represent the allowed bands separated by gaps (white areas). The stars give the frequencies that correspond to the transmission zeros. Straight and dotted lines, respectively, give transverse and longitudinal velocities of sound in Plexiglas, whereas dashed line gives the longitudinal velocity of sound in water.
14
π (b)
(a)
Reduced Frequency Ω
Gs
Gs
π/2
Gt
Gt
π/2
Gt
Gt
Gt
Gt
Gt
Gs
Gs
0 2
3
4
5
6
7
8
2
9
3
4
5
6
7
8
9
10
11
Generation Number k
Generation Number k
Figure 5: (a) Distribution of band widths (solid lines) as function of the generation number for θ = 0 ◦ . Gs and Gt denote stable and transient gaps respectively. (b) gives a large zoom around Ωc = π/2 in Fig.(a).
15
π
(b)
G0
(a)
G0
Reduced Frequency Ω
Gs
G0
G0
Gs
Gs
π/2
G0
π/2
G0
Gt
Gt
Gt
Gt
8
9 10 11
Gs
Gs Gs
0
2
3
4
5
6
7
8
9
2
Generation Number k
3
4
5
6
7
Generation Number k
Figure 6: (a) Distribution of band widths (solid lines) as function of the generation number for θ = 3 ◦ . Gs and Gt denote stable and transient gaps respectively, whereas G0 denotes a gap due to a transmission zero indicated by an arrow. (b) gives a large zoom around Ωc = π/2 in Fig.(a).
16
π
G0
(a)
(b) G0 Gs
Reduced Frequency Ω
Gs
G0
π/2 Gt
Gt
Gt
Gt
Gt
Gt
π/2
Gt
Gs
Gs
Gs
0 2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
10 11
Generation Number k
Generation Number k
Figure 7: Same as in Fig.6 but for θ = 24 ◦
17
0.40
θ=0° δ=0.147
0.35
θ=3° δ=0.140
Log(∆ ∆k)
0.30
θ=24° δ=0.134
0.25
0.20
0.15
∆ k ≈ Fk−δ 0.10 0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Log(Fk)
Figure 8: The Log-Log curves representing the width of the allowed bands ∆k as a function of number of blocks Fk for each generation k and for different values of θ.
18
Reduced Frequency Ω
π
π/2
0 2
3
4
5
6
7
8
9
Generation Number k
. Figure 9: Distribution of the bandwidths as a function of the generation number for θ = 0 ◦ . Blue circles and red triangles correspond to the surface modes of two complementary superlattices RSL and LSL (Fig. 1(b))
19
(b)
Reduced Frequency Ω
(a)
0.55π
0.5π
0.45π
1
2
3
4
5
6
7
8
9
10
11 1 2 3 4 5 6 7 8 9 10 11 (d)
Reduced Frequency Ω
(c)
0.25π
0.2π
0.15π
1
2
3
4
5
6
7
8
9
10
11 1 2 3 4 5 6 7 8 9 10 11
Generation Number k Figure 10: Same as in Fig.9 but enlarged around Ω =
20
Generation Number k π (a), 2
(b) and Ω =
π (c), 5
(d)
π (b)
(a)
Reduced Frequency Ω
0.7π
0.6π
π/2
0.5π
0 2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9 10 11
Generation Number k
Generation Number k
Figure 11: Same as in Fig. 9 but for θ = 24 ◦
21
S0041-624X(15)00071-2 http://dx.doi.org/10.1016/j.ultras.2015.03.004 ULTRAS 5021
To appear in:
Ultrasonics
Please cite this article as: I. Quotane, E.H. El Boudouti, B. Djafari-Rouhani, Y. El Hassouani, V.R. Velasco, Bulk and surface acoustic waves in solid-fluid Fibonacci layered materials, Ultrasonics (2015), doi: http://dx.doi.org/ 10.1016/j.ultras.2015.03.004
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Bulk and surface acoustic waves in solid-fluid Fibonacci layered materials I. Quotanea , E.H. El Boudoutia,b,∗, B. Djafari-Rouhanib , Y. El Hassouanic , V. R. Velascod a Laboratoire
de Dynamique et d’optique des mat´ eriaux, D´ epartement de Physique, Facult´ e des Sciences, Universit´ e Mohamed I, Oujda, Morocco b Institut d’Electronique, de Micro´ electronique et de Nanotechnologie (IEMN), UMR CNRS 8520, UFR de Physique, Universit´ e des Sciences et Technologies de Lille, 59655 Villeneuve d’Ascq, France c ESIM, Dpartement de Physique FSTE, Universit Moulay Ismail, Boutalamine B.P. 509, Errachidia, Morocco d Instituto de Ciencia de Materiales de Madrid, CSIC, Sor Juana Ins de la Cruz 3, E-28049, Madrid, Spain
Abstract We study theoretically the propagation and localization of acoustic waves in quasi-periodic structures made of solid and fluid layers arranged according to a Fibonacci sequence. We consider two types of structures: either a given Fibonacci sequence or a periodic repetition of a given sequence called Fibonacci superlattice. Various properties of these systems such as: the scaling law and the self-similarity of the transmission spectra or the power law behavior of the measure of the energy spectrum have been highlighted for waves of sagittal polarization in normal and oblique incidence. In addition to the allowed modes which propagate along the system, we study surface modes induced by the surface of the Fibonacci superlattice. In comparison with solid-solid layered structures, the solid-fluid systems exhibit transmission zeros which can break the self-similarity behavior in the transmission spectra for a given sequence or induce additional gaps other than Bragg gaps in a periodic structure.
Keywords: Surface acoustic waves; Solid-fluid layers; Quasi-periodic structure; Superlattice; Self-similarity 1. introduction Phononic crystals (PC) constituted by periodic arrangements (cells) of elastic/acoustic materials according to one (1D)[1], two (2D)[2], and three (3D)[3] dimensions, have been a subject of great interest during the last two decades because of their interesting properties in the development of new acoustical systems[4]. These systems are characterized by the presence of frequency regions where sound can propagate (bulk bands) and frequency regions where sound cannot propagate (gaps). This property has been exploited in the control and the guidance of the propagation of sound in different PCs[5]. As concerns 1D systems, different types of periodic structures such as solid-solid and solid-fluid layered materials as well as waveguides with different geometries are conducted as analogs of 2D and 3D leading to several interesting phenomena such as: omnidirectional band gaps [6, 7] and selective transmission by either guided modes[8] or interface resonance modes [9], the possibility to enhance acousto-optical interaction in hypersonic crystals[10, 11] and to realize stimulated emission of acoustic phonons[12] as well as ultrasonic metamaterials[13]. The advantage of 1D systems lies in the facilities to design different geometries and they require simple analytical and numerical calculations to understand deeply different physical phenomena observed in such systems. Besides periodic systems, quasi-periodic ones have been the subject of intensive study during the last two decades[14]. The quasi-periodic structures are generally built from two blocks A and B. Among them, the Fibonacci structure is constituted following the Fibonacci rule Sk+1 = Sk Sk−1 with S1 = A, ∗ Corresponding
author Email address: [email protected] (E.H. El Boudouti)
Preprint submitted to Elsevier
March 13, 2015
S2 = AB and k is the generation number. This leads to the Fibonacci sequences (FS): S3 = ABA, S4 = ABAAB, S5 = ABAABABA,... Merlin et al.[15] were the first to have studied such structure in semiconductor GaAs-AlAs superlattices (SLs). Since this work, much attention has been paid to observe the exotic phenomena of Fibonacci systems[16, 17] and interesting characteristics of these systems have been concluded[18] essentially by theoretical studies based on simple 1D models.it is known also that deterministic quasi-periodic systems may exhibit localization, as the Anderson localization, of sound and vibration[19]. Such phenomenon characterize any wave when the structures exhibit disorder[20]. An example of the properties of the propagation and localization of acoustic waves in Fibonacci modulated waveguides have been studied theoretically by some of us [21] and checked experimentally by King and and Cox[22]. Compared to solid-solid SLs, solid-fluid SLs have received less attention[1]. Some years ago [23, 24], we have performed an extensive study on the existence of confined and surface modes in finite and semi-infinite SLs made of a periodic repetition of solid-fluid bilayers. We have shown that besides the gaps due to the periodicity of the system these structures exhibit, other gaps due to transmission zeros. These transmission zeros can lead to new phenomena such as acoustic meta-materials[13], Fano resonances[24] as well as the possibility to mimic an acoustic transparency by squeezing a resonance between two transmission zeros[25]. The possibility of using these materials as acoustic mirrors and filters was also demonstrated [24]. Based on this work[24], a recent paper[26] has explored the tunneling effect between two different fluids via a solid-fluid SL with application to petroleum exploration. Also, we have shown the possibility of existence of surface modes in such systems depending on wether the SL is terminated by a solid or a fluid layer[23]. In the long-wavelength limit, an homogenization analysis has been developed[27] to study the effective properties of propagating modes of periodic solid layers in an ideal or viscous fluid. To our knowledge, few works have been devoted to solid-fluid quasi-periodic structures [28, 29]. In these papers finite size FS[28] and periodic FS [29] are studied by means of the transmission coefficient and the localization length deduced from the Lyapunov exponent. The Fibonacci-type structures [14] are formed from two blocks A (fluid layer) and B (solid layer). In reference [28], normal incidence waves through single FS have been considered and the fragmentation of the transmission bands as a function of the generation number has been discussed. However, the self-similarity of the transmission spectra following a scaling law has not been addressed in such systems. In reference [29], a periodic repetition of a FS was considered and the splitting of the bulk bands for normal and oblique incidence was studied by means of the localization length. However, in order to fully characterize the band gap structure of a Fibonacci SL made of a periodic repetition of a given cell where each cell is constructed by a given FS, a dispersion relation involving the Bloch wavevector k and the pulsation ω should be calculated. In this paper, we give closed form expressions of the transmission coefficient through one FS and the dispersion relation of a given FS repeated periodically. For one FS, we show at normal incidence the property of self-similarity of the transmission spectra each three generations following a scaling law. However, at oblique incidence this property no longer exists because of the existence of transmission zeros. For a Fibonacci SL, we show that when the generation number increases, the pass bands exhibit a fragmentation following a power law. Also we show the existence of different types of gaps: stable and transient gaps induced by the periodicity of the system and new gaps induced by the transmission zeros. These latter gaps are a characteristic of solid-fluid SL and do not exist in solid-solid SL. These bulk properties have not yet been addressed in the literature. In addition to bulk modes we show for the first time the possibility of existence of surface modes in solid-fluid Fibonacci SLs. These modes show different behaviors depending on whether they fall inside stable gaps or transient gaps. The rest of the paper is organized as follows: in Sec.2 we give a brief presentation of the method of calculation employed here, which is based on the Green function method. Sec.3 is devoted to the discussion of the numerical results for the transmission along a given Fibonacci sequence for normal and oblique incidence and the dispersion curves for bulk and surface modes in Fibonacci SL. The final section contains the concluding remarks.
2
2. METHOD OF THEORETICAL AND NUMERICAL CALCULATION 2.1. Interface response theory of continuous media In this paper, we consider a multilayered structure made of solid and fluid layers arranged perpendicularly to the x3 direction. The planes of the layers are contained within the (x1 , x2 ) directions. The acoustic waves propagating through such a system are polarized in the sagittal plane defined by the normal to the interfaces (x3 direction) and the wave vector k// (parallel to the interfaces). We choose k// along the x1 direction without loss of generality. We consider a non viscous fluid layer for which the viscous skin depth σ = (2η/ρω)1/2 is much smaller than the fluid layer thickness df over a very broad frequency range (η and ρ are the viscosity and the density of the fluid). This study is performed with the help of the interface response theory[30] of continuous media which permits us to calculate the Green’s function of any composite material. In what follows, we present the basic concepts and the fundamental equations of this theory [30]. Let us consider a composite system defined in its whole space domain labeled D. This system contains different subsystems i connected together by their interface domains Mi . The whole interface space of the ∪ system is labeled M = Mi . The elements of the Green’s function g(DD) of any composite material can be obtained from[30] g(DD)
=
G(DD) − G(DM )G−1 (M M )G(M D)
(1)
+G(DM )G−1 (M M )g(M M )G−1 (M M )G(M D), where G(DD) is the Green’s function of a given continuous medium and g(M M ) is the Green’s function of the composite system in its interface domain M . As we are interested in elastic waves in solid and fluid media, the corresponding Green’s functions G(DD) can be derived from the equation of motion of displacement fields as explained in Ref.[30]. The inverse [g(M, M )]−1 of g(M M ) is obtained as a superposition of the different gi−1 (Mi , Mi )[30], inverse of the gi (Mi , Mi ) for each constituent i of the composite system. The inverse of g(M M ) enables us to obtain the eigenmodes of a composite system through the relation[30] det[g −1 (M M )] = 0,
(2)
U (D) being an eigenvector of the reference system, Eq. (2) leads to the eigenvectors u(D) of the composite material as u(D) =
U (D) − U (M )G−1 (M M )G(M D) + U (M )G−1 (M M )g(M M )G−1 (M M )G(M D).
(3)
ln Eq. (3), U (D), U (M ), and u(D) are row vectors. If U (D) is a bulk wave launched in one homogeneous piece of the composite material, then Eq. (3) enables the calculation of all the waves reflected and transmitted by the interfaces, as well as the reflection and transmission coefficients of the composite system [1]. 2.2. Inverse surface Green’s functions of the elementary constituents As mentioned above, the calculation of the Green’s function of any composite material made of solid and fluid layers, within the interface response theory, requires the knowledge of the surface elements of its elementary constituents, namely, the Green’s function of an ideal fluid of thickness df , sound speed vf and mass density ρf and an elastic isotropic solid characterized by its thickness ds , longitudinal speed vℓ , transverse speed vt , and mass density ρs . In addition, the calculations of the dispersion relations and transmission coefficients (see below) can be deduced only from the knowledge of the components of the Green’s functions g −1 (M M ). The inverse Green’s function of an ideal fluid layer in the space of the two surfaces Mf = {−df /2, +df /2} of the layer is given by[23] ( ) af bf [gf (Mf Mf )]−1 = , (4) bf af 3
where af = −F
Cf F , bf = , Sf Sf
(5a)
Cf = cosh(αf df ), Sf = sinh(αf df ), ) 12 ( √ ω2 ω2 2 − k// and αf = j (j = −1), F = −ρf 2 αf vf
(5b) (5c)
and the inverse Green’s function of the elastic solid layer in the space of the two surfaces Ms = {−ds /2, +ds /2} of the layer is given by[23] ( ) as bs −1 [gs (Ms Ms )] = , (6) bs as where Ct γ β Cℓ − β , bs = + , Sℓ St Sℓ St v4 v4 2 2 γ = −ρ 2t (k// , + αt2 )2 , β = 4ρ t2 αt k// ω αℓ ω Ct = cosh(αt ds ), Cℓ = cosh(αℓ ds ), St = sinh(αt ds ), Sℓ = sinh(αℓ ds ), as = −γ
and
2 αt2 = k// −
2
(7a) (7b) (7c)
2
ω ω 2 , αℓ2 = k// − 2. vt2 vℓ
(7d)
The inverse Green’s function of a semi-infinite fluid is given by[23] gf−1 (0, 0) = −F,
(8)
where F is defined by Eq. (5c). 2.3. Transmission coefficients through one solid-fluid Fibonacci sequence The solid-fluid Fibonacci sequence of a given generation k is constructed from a finite number of layers A and B arranged according to the Fibonacci rule. The interface space M is made of all the sub-interfaces between the layers. Within this space the inverse of the matrix [gk (M M )]−1 giving all the elements of the Green’s function g is a finite tri-diagonal matrix formed by a linear superposition of the sub-matrices −1 g(i=s,f ) (Mi Mi ) (Eqs. (4),(6)). For example, in the case of a finite structure corresponding to the fourth Fibonacci generation S4 = ABAAB (Fig. 1), [g4 (M M )]−1 can be written in the following tri-diagonal matrix form: af bf 0 0 0 bf af + as bs 0 0 0 bs as + af bf 0 −1 . [g4 (M M )] = (9) 0 bf 2af bf 0 0 0 0 0 bf af + as bs 0 0 0 0 bs as The Green’s function elements on the two surfaces bounding the FS are obtained by inverting the [g4 (M M )]−1 (Eq. (9)) and keeping only the extreme elements of the truncated matrix to forme a 2×2 matrix. Then by inverting again this matrix, we obtain a 2×2 Green’s function matrix of the FS as follows: ( ) a b −1 [g4 (Me Me )] = , (10) b c where Me = {0, L} is the space of interfaces at both free extremities of the S4 generation and L is the total length of the structure. 4
The four matrix elements are real quantities functions of the different parameters of the constituent’s elements gi (Mi Mi ) (i = s, f ) [Eqs. (4),(6)]. From Eqs. (2) and (10), one can deduce the expression giving the eigenmodes of the FS, namely ac − b2 = 0. (11) When the finite composite system is bounded by two homogeneous semi-infinite fluids with [gf (0, 0)]−1 = −F (Eq. (8)), the inverse Green’s function of the whole composite system can be obtained by a superposition of the two matrices given by Eqs. (8) and (10) as: ( ) a−F b −1 [g(Me Me )] = . (12) b c−F An incident plane wave launched from the left and travelling through the structure gives rise to the transmission function t = −2F g(0, L)[24]. Therefore, we obtain t=−
2bF . ac − b2 + F 2 − F (a + c)
(13)
2.4. Dispersion relations of infinite and semi-infinite periodic sequences Let us consider an infinite SL made of a periodic repetition of a given sequence [Fig. 1(a)]. The system Green’s function is obtained by a linear juxtaposition of the 2×2 matrices (Eqs. (4),(6)) at the different interfaces. A tridiagonal matrix thus obtained. By using the Bloch theorem and Eq. (2), we obtain the following expression for dispersion relation of an infinite SL: cos(KD) = −(a + c)/2b,
(14)
∑Fk di Fk being the number of blocks where K is the Bloch wave-vector and D the period of the SL (D = i=1 in the generation k).Then, the surface modes dispersion relation of a semi-infinite SL lying in the half space x3 > 0 [Fig. 1(b)RSL], is given by ac − b2 = 0 , (15) together with the following condition to ensure that the waves decay from the surface when penetrating into the SL, b < 1. (16) a If the semi-infinite SL lies in the half space x3 < 0 [Fig. 1(b)LSL], we obtain the same dispersion relation [Eq. (15)] but with the condition b > 1. (17) a Thus we see that the surface mode is induced either by one semi-infinite SL or by its complementary one. In addition, Eq. (15) shows that the expression for the surface modes of two complementary SLs is the same one than that of the eigenmodes of one sequence [Eq. (11)]. 3. Numerical results In the following, we shall present some numerical calculations of transmission coefficients and dispersion curves of acoustic waves in solid-fluid Fibonacci structures made of Plexiglas-water. The case of periodic structures has been studied both theoretically[31] and experimentally[32, 33]. Table 1 gives the numerical values of velocities of sound and mass densities of the materials used in this work. We shall focus our attention on two different structures, namely a given FS inserted between two fluids and a periodic repetition of a given FS.
5
Plexiglas Water
ρ(g/cm3 ) 1200 1000
vt (105 cm/s) 1.38 -
vℓ (km/s) 2.7 1.49
Table 1: Velocities of sound and mass densities of Plexiglas and Water.
3.1. Transmission along one Fibonacci sequence Figure 2(a) shows the transmission spectrum as a function of the dimensionless frequency Ω = ωdf /vf for normal incident longitudinal acoustic waves in a periodic structure made of six solid-fluid bilayers inserted between two semi-infinite water media. The layer thicknesses are chosen such that df /vf = ds /vs where df (vf ) and ds (vs ) represent the thickness (longitudinal speed) of fluid and solid layers respectively. One can notice the existence of a forbidden band around the central frequencies Ωc = (2m + 1)π/2 for which the two layers behave as quarter wavelength layers (m being an integer). Let us mention that such a SL exhibits gaps only at the edge of the Brillouin zone (around π/2, 3π/2,...), however the gaps falling at the center of the Brillouin zone (around π, 2π,...) close. Fig. 2(b)-(g) show the transmission coefficients for the generations S4 (five blocks), S5 (eight blocks), S6 (13 blocks), S7 (21 blocks), S8 (34 blocks) and S9 (55 blocks), respectively. In fig. 2(b)-(g) one can distinguish the regions of frequencies where the transmission show several dips as the generation number increases. These regions correspond to the forbidden bands (or transmission gaps). In the other regions of frequencies, the transmission is more noticeable corresponding to the allowed modes (or transmission bands). As compared to the transmission spectrum in the periodic structure (fig. 2(a)), the quasi-periodic spectra undergo a fragmentation especially for high generations (fig. 2(g)). One can notice that around the central frequencies Ωc = π/2, 3π/2, ... of the gaps in the periodic structure (fig. 2(a)), the transmission spectra in the quasi-periodic structures (fig. 2(b)-(g)) show a certain recursive order which is a characteristic of Fibonacci systems. In order to characterize this self-similarity (also known as a scaling law[14]) in the transmission spectra, Kohmoto et al.[34] have introduced a scale factor F = [I + 4(1 + I)2 ]1/2 + 2(1 + I),
(18)
where I is an invariant describing the recursive procedure. Its expression has the following form[34] I = 1/4(Zf /Zs − Zs /Zf ) sin2 (ωdf /vf ),
(19)
where Zf = ρf vf and Zs = ρs vs are the impedance of the fluid and solid layers respectively. Also, it has been shown that this phenomenon occurs around Ωc where quasi-periodicity is most effective. This means that the transmission coefficient should have a self-similarity nature around the central frequency with Tj+3 = Tj (the transmission coefficient exhibits a third order periodicity) with a scale factor F[34]. When applied to the central frequency Ωc , equations (18) and (19) give I = 0.65 and F = 6.73. Indeed, we note a great resemblance between the generations S5 − S8 and S6 − S9 around the central frequency Ωc with a periodicity of three and a scaling factor F as it is illustrated in the inserts of fig. 2(b), (c) and (d) as compared to fig. 2(e), (f) and (g) respectively. This result is similar to that found experimentally by Gellermann et al.[35] in the optical multi-layers and El Boudouti et al.[36] in coaxial cables. This property of self-similarity has been interpreted[14] as a sign of localization of waves in Fibonacci systems. In the case of oblique incidence (θ = 3 ◦ ) the transmission spectrum shows a flat band around Ωc for the periodic structure (Figure 3(a)). Also a new gap appears around Ω = π with a flat band inside this gap which appears like a Fano resonance[37]. These flat bands enable the filtering of acoustic modes through the system in a narrow frequency region inside the gaps without introducing defect layers as it is usually the case[24]. The transmission spectra of the Fibonacci structure S4 (3 (b)) exhibits a transmission zero around π/2, π, 3π/2,...which transforms to a very thin gap for high generations (Figs. 3(b)-(g)). Therefore we can distinguish gaps due to transmission zeros from those induced by the quasi-periodicity of the system. Also, 6
this transmission zero breaks the self-similarity effect observed at normal incidence around Ωc , indicating that the transmission spectra are sensitive to the angle of incidence. For example, the sequences S6 and S9 in Fig 3 clearly show the absence of the self-similarity of the two spectra around Ωc due to the existence of the transmission zeros. By increasing the incidence angle, the gaps due to transmission zeros become larger and the disappearance of the self-similarity becomes further noticeable. 3.2. Band structure of Fibonacci superlattice In this section, we consider the dispersion curves of Fibonacci SLs made of periodic repetition of a given sequence Sk . The study of the dispersion curves of the bulk bands (reduced frequency Ω = ωdf /vf versus the reduced wave vector k// df ) is shown in figure 4 for different Fibonacci sequences. The shaded gray areas correspond to bulk bands; these bands are separated by gaps where the waves are prohibited from propagation. To show the sound cones of the constituting materials in the superlattice, we present the longitudinal and transverse velocities of sound in Plexiglas by dotted and straight lines and the longitudinal velocity of sound in water by the dashed line (table 1). Due to the low contrast between the acoustic parameters of Plexiglas and water, the gaps are not very wide. For the second FS S2 (Fig. 4 (a)), the first two bands are located below the transverse velocity of sound in Plexiglas and the longitudinal velocity of sound in water. These two bands are constituted by evanescent waves in these two media and tend asymptotically to the interface mode between Plexiglas and water for large values of k// df . When the generation number increases from S2 to S5 (Figs. 4 (b)-(d)), we can notice a fragmentation of the bulk bands and the existence of new gaps between them. These gaps are induced by the periodicity of the SL. The stars represent the gaps resulting from transmission zeros, the latter being a characteristic of the solid layer independently of the fluid layer[23]. The Bloch wave-vectors along the star positions go to infinity[24] which renders these frequency regions completely forbidden for acoustic waves. We have shown in Ref.[24] that these frequencies are induced exclusively by solid layers, that is why their positions are independent of the Fibonacci sequence (Figs. 4 (a)-(d)). Let us notice that the S3 generation in Fig. 4 (b) (..ABAABAABA...) is similar to the S2 generation (...ABABABAB...) but with the condition to replace the A layer by the AA layers. However, the corresponding band gap structures (Figs. 4 (a),(b)) are very different. For example the forbidden gap in Fig. 4 (a) at k// df = 0 becomes completely allowed in Fig. 4 (b) (see also Fig. 5(a) for the second and third generations). Figure 5(a) shows the distribution of the widths of the allowed and forbidden bands of a Fibonacci SL for different generation number k up to the 9t h generation, which means a unit cell with 34 A and 21 B layers (i.e., 55 building blocks), for θ = 0 ◦ . One can notice that the regions of allowed bands become narrower for the high generation number. In addition, there are stable gaps (denoted Gs ) that appear for all generations and transient gaps (denoted Gt ) that appear each three-generations (see for example the gaps around Ωc = π/2 for S2 , S5 and S8 ); this is illustrated in Fig. 5 (b) with a large zoom of the bandwidths around Ωc . In Fig. 5 the width of the transient gaps decreases as the number of generation increases, while the width of the stable gaps remains almost constant for high generation numbers. By changing slightly the incidence angle from θ = 0 ◦ to θ = 3 ◦ as illustrated in Fig. 6 (a), the stable gaps Gs remain unchanged and the transient gaps Gt shifts to the lower frequencies below Ωc whereas new gaps labeled G0 appears around Ω = 0.51π, π (see Fig. 6(a)). These gaps indicated by horizontal arrows are due to transmission zeros as mentioned above in the transmission spectrum (Fig. 2). Fig. 6(b) gives a large zoom around Ωc and clearly shows the positions of the G0 , Gs and Gt . By increasing the incidence angle to θ = 24 ◦ (Fig. 7), the gap G0 becomes wider and the transient gap Gt appears slightly above Ωc = π/2 while the band gap profile exhibits a fragmented energy spectrum for large k. The total width ∆k of the allowed energy bands (Lebesgue measure of energy spectrum [14]) decreases when the generation number increases with a power law ∆k = Fk−δ as it is shown in Fig. 8 where Fk is the Fibonacci number (number of blocks in each sequence) and δ is the diffusion constant of the spectrum. We note that this power law is valid for different values of the angle θ in the fluid and the parameter δ is not sensitive to the variation of θ. When the generation number k becomes larger, the band width ∆k tends to zero whatever the value of θ. This result which is a characteristic of quasi-periodic structures has been demonstrated for other types of excitations as well[14]. 7
3.3. Surface modes of Fibonacci superlattice In Ref.[38], we have shown that the terminations of two complementary SLs obtained from the cleavage of an infinite SL do not behave in the same way when going from generation to generation. For example, the cleavage of an infinite SL corresponding to the fourth Fibonacci generation (Fig. 1(a)) ...| ABAAB | ABAAB | ABAAB | ABAAB |... gives rise to two complementary semi-infinite SLs (Fig. 1(b)): LSL (left superlattice) and RLS (right superlattice). ...| ABAAB | ABAAB | and | ABAAB | ABAAB |... In the same way, the creation of two semi-infinite SLs obtained by the cleavage of an infinite SL corresponding to the fifth generation gives the two complementary SLs: ...| ABAABABA | ABAABABA | and | ABAABABA | ABAABABA |... One can notice from the passage from one generation to another that terminations of the LSL change from generation to generation, although they remain the same for even generations (with B block at the surface) and for odd generations (with A block at the surface). As concerns the terminations of the RSL, they remain unchanged for all generations. Figure 9 represents the surface modes for two complementary semi-infinite SLs for various generations until the ninth generation at the normal incidence (θ = 0 ◦ ). The blue circles give the surface modes of the RSL and the red triangles those of the complementary LSL. These modes are obtained from equations (14), (16) and (17). The existence of one surface mode per gap is clearly seen. This mode is induced by the surface of one of the two semi-infinite periodic structures. These results generalize those we have obtained before for a bilayer SL[23]. However, there is an exception to this rule for the S3 generation. Indeed, the cleavage of an infinite SL corresponding to the third generation gives rise to two identical semi-infinite SLs. Therefore, the surface modes do not appear neither on one SL nor on its complementary, but fall exactly at the edge of the bulk bands (see Ref.[39] for more details on this particular case). In Fig. 9 corresponding to θ = 0 ◦ , one can distinguish two types of surface modes: i) The modes falling inside the stable gaps. Here one can distinguish the modes associated to the RSL (blue circles), which fall almost at the same frequency as the endings of these SLs remain unchanged from generation to generation. On the other hand , the modes associated to the LSL (red triangles) appear at the same frequency but for one generation over two as the terminations of the LSL change in an alternating way from even to odd generations. This result is well illustrated in Fig 10 which gives a zoom of Fig. 9 around Ω = π/2 (Figs. 10(a),(b)) and Ω = π/5 (Figs. 10(c), (d)) up to the 11the generation. ii) The modes falling inside the transient gaps around the central frequency Ωc . These modes appear at the same frequency each six generations (see Figs. 10(a),(b)). In Ref. [38] we have shown that an analysis of the spatial localization of the modes falling in stable gaps present a decreasing behavior from the surface while penetrating in the SL, whereas the modes falling in transient gaps show a self-similar behavior of order six. This latter behavior is due to the quasi-periodicity of the layers A and B. Figure 11(a) gives the surface modes for the incidence angle θ = 24 ◦ . One can notice that the properties related to the surface modes discussed in the case of normal incidence (θ = 0 ◦ ) remain the same for θ = 24 ◦ . However, in the region above the central frequency there exists a surface mode which appears inside the thin band gap around Ω ≃ 0.64π (see Fig. 7(b)). These modes appear in the RSL for even generations (blue circles) and in the LSL for odd generations (red triangles) starting from the 4th generation as it is illustrated in Fig. 11(b). As we have shown analytically (see Eqs. (11) and (15)), the surface modes of two complementary SLs are exactly the stationary modes of one given FS. However, some of these modes falling at the same frequencies by increasing the generation number remain surface modes even for the finite size FS[36]. 8
Finally, it is worth pointing that the thicknesses of the layers chosen for the solid-fluid layers enabled us to discuss the main results of self-similarity, band gap fragmentation with different surface modes, around the central frequency Ωc = π/2. These results remain unchanged for any thicknesses of the layers but the main phenomena appear at different frequencies. Also increasing the contrast between the characteristic impedances of the chosen materials like in the case of Aluminium-water structure[23], give the same results as those presented here but with large Bragg gaps. 4. Conclusion In this work, we have studied sagittal acoustic waves in quasi-periodic structures of Fibonacci type. These systems are formed by two blocks A (fluid layer) and B (solid layer). Closed form expressions of the transmission coefficient and the dispersion relations are obtained. The analysis of the transmission spectra of different Fibonacci generations (sequences) allowed us to conclude that these spectra exhibit self-similarity of order three with a scaling factor F. We also showed the existence of gaps which are due to the quasi-periodicity and gaps due to transmission zeros. Also we have presented the dispersion curves of Fibonacci SLs made of a repetition of cells where each cell is formed by a given sequence and obtained the power law for the measure of the energy spectrum in these systems. We also studied the surface modes in the quasi-periodic SLs when considering two semi-infinite SLs obtained from the cleavage of an infinite one. We have generalized the rule on the existence of surface modes in the binary[23] solid-fluid SLs to the case of Fibonacci SLs. In particular, we have shown the existence of two types of surface modes: i) modes inside stable gaps whose frequencies remain constant for all generations as the corresponding SLs remain unchanged from generation to generation, ii) modes within transient gaps which appear each six generations. Finally let us mention that this study can be extended without difficulties to other quasi-periodic structures like aperiodic and Thue Morse systems[14].The experimental verification of these theoretical results should be possible using ultrasonic techniques[33, 8, 40, 41, 42]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
E. H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, Surf. Sci. Rep. 64 (2009) 471. Y. Pennec, J. Vasseur, B. Djafari-Rouhani, L. Dobrzynski, P. A. Deymier, Surf. Sci. Rep. 65 (2010) 229. I. E. Psarobas, N. Stefanou, A. Modinos, Phys. Rev. B 62 (2000) 278. M. S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Phys. Rev. Lett 71 (1993) 2022. P. A. Deymier, Acoustic Metamaterials and Phononic Crystals, Springer Series in Solid- State Sciences Vol. 173, Springer, Berlin, 2013. D. Bria, B. Djafari-Rouhani, Phys. Rev. E 66 (2002) 056609. B. .Manzanares-Martinez, J.Sanchez-Dehesa, A.Hakansson, F.Cervera, F.Ramos-Mendieta, Appl. Phys. Lett 85 (2004) 154. R. James, S. M. Woodley, C. M. Dyer, F. Humphrey, J. Acoust. Soc. Am 97 (1995) 2401. H. Kato, Phys. Rev. B 59 (1999) 11136. M. Trigo, A. Bruchhausen, A. Fainstein, B. Jusserand, V. Thierry-Mieg, Phys. Rev. Lett 89 (2002) 227402. D. Schneider, F. Liaqat, E. H. El Boudouti, O. El Abouti, W. Tremel, H-J.Butt, B. Djafari-Rouhani, G.Fytas, Phys. Rev. Lett. 111 (2013) 164301. A. J. Kent, R. N. Kini, N. M. Stanton, M. Henini, B. A. Glavin, V. A. Kochelap, T. L. Linnik, Phys. Rev. Lett 96 (2006) 215504. N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, X. Zhang, Nature Materials 5 (2006) 452. E. L. Albuquerque, M. G. Cottam, Physics Report 376 (2003) 225. R. Merlin, K. Bajema, R. Clarke, F. Y. Juang, P. K. Bhattacharya, Phys. Rev. Lett. 55 (1985) 1768. K. Bajema, R. Merlin, Phys. Rev. B 36 (1987) 4555. D. C. Hurley, S. Tamura, J. P. Wolfe, K. Ploog, J. Nagle, Phys. Rev. B 37 (1988) 8829. E. Macia, F. Dominguez-Adame, Electrons, Phonons and excitons in Low Dimensional Aperiodic Systems, Editorial Complutense, Madrid, 2000. R. L.Weaver, Phys. Rev. B 47 (1993) 1077. L. Macon, J. P. Desideri, D. Sornette, Phys. Rev. B 44 (1991) 6755. H. Aynaou, E. H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, V. R.Velasco, J. Phys.: Condens Matter 17 (2005) 4245. P. D. C.King, T. J.Cox, J. Appl. Phys. 102 (2007) 014902. Y. El Hassouani, E. H. El Boudouti, B. Djafari-Rouhani, H. Aynaou, L. Dobrzynski, Phys. Rev. B 74 (2006) 144306. Y. El Hassouani, E. H. El Boudouti, B. Djafari-Rouhani, H. Aynaou, Phys. Rev. B 78 (2008) 174306.
9
[25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]
A. Santill´ an, S. I. Bozhevolnyi, Phys. Rev. B 84 (2011) 064304. C. Liu, X.-D. Xu, X.-J. Liu, C. Glorieux, Appl. Phys. Express 7 (2014) 067302. Y.-H. Liu, C. C. Chang, C.-Y. Kuo, Phys. Rev. B 78 (2008) 054115. E. L. Albuquerque, P. D. Session, Physica B 405 (2010) 3704. A.-L. Chen, Y.-S. Wang, C. Zhang, Physica B 407 (2012) 324. L. Dobrzynski, Surf. Sci. Rep 11 (1990) 139. M. Schoenberg, Wave Motion , 6th Edition, 1984. T. J. Plona, K. W. Winkler, M. Schoenberg, J. Acoust. Soc. Am. C. Gazanhes, J. Sageloli, Acustica 81 (1995) 221. M. Kohmoto, B. Sutherland, K. Iguchi, Phys. Rev. Lett 58 (1987) 2436. W. Gellermann, M. Kohmoto, B. Sutherland, P. C. Taylor, Phys. Rev. Lett 72 (1994) 633. E. H. El Boudouti, Y. El Hassouani, H. Aynaou, B. Djafari-Rouhani, A. Akjouj, V. R. Velasco, J. Phys.: Condens. Matter 19 (2007) 246217. U. Fano, Phys. Rev 124 (1961) 1866. Y. El Hassouani, H. Aynaou, E. H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, V. R. Velasco, Phys. Rev. B 74 (2006) 035314. E. H. El Boudouti, Y. El Hassouani, H. Aynaou, B. Djafari-Rouhani, Phys. Rev. E 76 (2007) 026607. M. Shen, W. Cao, Appl. Phys. Lett 75 (1999) 3713. A. Khaled, P. Marechal, O. Lenoir, M. E. El-Kettani, D. Chenouni, Ultrasonics 53 (2013) 642–647. H. Sanchis-Alepuz, Y. A. Kosevich, J. Sanchez-Dehesa, Phys. Rev. Lett. 98 (2007) 134301.
10
Figure 1: (a) Schematic representation of an infinite periodic superlattice where each cell is composed of a Fibonacci generation (for example S4 ). (b) Schematic representation of two semi-infinite superlattices LSL (left superlattice) and RSL (right superlattice) obtained by the cleavage of the infinite superlattice (a).
11
1
(a) 0 1 S4 0 1
Transmission
(b)
π/2
S5
(c)
π/2
0 1
π/2
0 1
π/2
0 1
π/2
S6
(d)
S7
(e)
S8
(f)
0 1 S9 π/2
0
ο
(g) π/2
π
3π/2
2π
Reduced frequency Ω
Figure 2: Variation of the transmission coefficient as a function of the reduced frequency Ω for different structures: (a) The periodic structure,(b)-(g) different generations Sk (k = 4-9) of the Fibonacci structure at normal incidence. The inserts show the self similarity between the generations (S4 , S7 ), (S5 , S8 ) and (S6 , S9 ) around Ωc = π/2
12
1
(a) 0 1
S4
(b)
S5
(c)
S6
(d)
Transmission
0 1
0 1
0 1
S7
(e)
0 1
S8 (f) 0 1
S9 (g) 0
0
π
π/2
3π/2
Reduced Frequency Ω
2π
Figure 3: Variation of the transmission coefficient as a function of the reduced frequency Ω for different structures: (a) The periodic structure, (b)-(g) the different generations Sk (k = 4-9) of the Fibonacci structure at oblique incidence (θ = 3 ◦ ).
13
Figure 4: Band structure (i.e., Ω = ωdf /vf versus the reduced wave vector k// df ) for a Fibonacci SL made of Plexiglas and water for different generations (S2 − S5 ). The gray areas represent the allowed bands separated by gaps (white areas). The stars give the frequencies that correspond to the transmission zeros. Straight and dotted lines, respectively, give transverse and longitudinal velocities of sound in Plexiglas, whereas dashed line gives the longitudinal velocity of sound in water.
14
π (b)
(a)
Reduced Frequency Ω
Gs
Gs
π/2
Gt
Gt
π/2
Gt
Gt
Gt
Gt
Gt
Gs
Gs
0 2
3
4
5
6
7
8
2
9
3
4
5
6
7
8
9
10
11
Generation Number k
Generation Number k
Figure 5: (a) Distribution of band widths (solid lines) as function of the generation number for θ = 0 ◦ . Gs and Gt denote stable and transient gaps respectively. (b) gives a large zoom around Ωc = π/2 in Fig.(a).
15
π
(b)
G0
(a)
G0
Reduced Frequency Ω
Gs
G0
G0
Gs
Gs
π/2
G0
π/2
G0
Gt
Gt
Gt
Gt
8
9 10 11
Gs
Gs Gs
0
2
3
4
5
6
7
8
9
2
Generation Number k
3
4
5
6
7
Generation Number k
Figure 6: (a) Distribution of band widths (solid lines) as function of the generation number for θ = 3 ◦ . Gs and Gt denote stable and transient gaps respectively, whereas G0 denotes a gap due to a transmission zero indicated by an arrow. (b) gives a large zoom around Ωc = π/2 in Fig.(a).
16
π
G0
(a)
(b) G0 Gs
Reduced Frequency Ω
Gs
G0
π/2 Gt
Gt
Gt
Gt
Gt
Gt
π/2
Gt
Gs
Gs
Gs
0 2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
10 11
Generation Number k
Generation Number k
Figure 7: Same as in Fig.6 but for θ = 24 ◦
17
0.40
θ=0° δ=0.147
0.35
θ=3° δ=0.140
Log(∆ ∆k)
0.30
θ=24° δ=0.134
0.25
0.20
0.15
∆ k ≈ Fk−δ 0.10 0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Log(Fk)
Figure 8: The Log-Log curves representing the width of the allowed bands ∆k as a function of number of blocks Fk for each generation k and for different values of θ.
18
Reduced Frequency Ω
π
π/2
0 2
3
4
5
6
7
8
9
Generation Number k
. Figure 9: Distribution of the bandwidths as a function of the generation number for θ = 0 ◦ . Blue circles and red triangles correspond to the surface modes of two complementary superlattices RSL and LSL (Fig. 1(b))
19
(b)
Reduced Frequency Ω
(a)
0.55π
0.5π
0.45π
1
2
3
4
5
6
7
8
9
10
11 1 2 3 4 5 6 7 8 9 10 11 (d)
Reduced Frequency Ω
(c)
0.25π
0.2π
0.15π
1
2
3
4
5
6
7
8
9
10
11 1 2 3 4 5 6 7 8 9 10 11
Generation Number k Figure 10: Same as in Fig.9 but enlarged around Ω =
20
Generation Number k π (a), 2
(b) and Ω =
π (c), 5
(d)
π (b)
(a)
Reduced Frequency Ω
0.7π
0.6π
π/2
0.5π
0 2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9 10 11
Generation Number k
Generation Number k
Figure 11: Same as in Fig. 9 but for θ = 24 ◦
21
