Prediction of interface states in liquid surface waves with one-dimensional modulation
PPrediction of interface states in liquid surface waves with one-dimensional modulation
Xi Shi,
1, 2, ∗ Yong Sun, Chunhua Xue, and Xinhua Hu † Department of Physics, Shanghai Normal University, Shanghai 200234, China MOE Key Laboratory of Advanced Micro-Structured Materials,School of Physics Science and Engineering, Tongji University, Shanghai 200092, China Department of Materials Science and Key Laboratory of Micro- and Nano-PhotonicStructures (Ministry of Education), Fudan University, Shanghai 200433, China
We theoretically studied the interface states of liquid surface waves propagating through theheterojunctions formed by a bottom with one-dimensional periodic undulations. Via consideringthe periodic structure as a homogeneous one, our systematic study shows that the signs of theeffective depth and gravitational acceleration are opposite within the band gaps no matter thestructure is symmetric or asymmetric. Those effective parameters can be used to predict the interfacestates which could amplify the amplitudes of liquid surface waves. These phenomena provide newopportunities to control the localization of water-wave energy.
I. INTRODUCTION
In recent decades the propagation of liquid surfacewaves (LSWs) over an uneven bottom with a periodicmodulation, such as rippled bottoms, periodic drilledholes as well as periodic arrays of surface scatters, hasattracted a great deal of attentions. Originating from theBragg resonances of water waves, the periodic structurewith the scale of the half-wavelength can strongly reflectthe water waves [1, 2], and thus exhibited many pecu-liar phenomena including water wave blocking [3–5], su-perlensing effects [6], self-collimation [7] and directionalradiation [8–10]. Recently, a concept of effective liquidwas developed for thoroughly understanding the inter-action of water waves with periodic structures [11, 12].Theoretical studies have shown that the effective grav-ity in water pierced by a cylinder array is larger thanthe one on the earth, which induces a new type of waterwave refraction [11]. For water with a resonator array,the effective gravity g e can be negative near resonant fre-quency, so that water waves cannot propagate throughthe array [12, 13]. More particularly, the effective grav-ity g e even can be infinite when water is covered by athick, rigid and unmovable plate, which can be used forbroadband focusing and collimation of water waves [14].These results promise a new mechanism to control thepropagation of water waves, which exhibits particular ap-plications in wave energy conversion and coastal protec-tion [15–19].Most studies concerning liquid waves have, hitherto,been focused on the effective gravity of water in periodstructures. In fact, the water-wave process is affectednot only the gravity of the earth but also the depth ofwater. Therefore, the effective theory for liquid wavesincludes not only the effective gravity g e but also theeffective depth h e . However, the latter has rarely beenpaid much attention [11, 12]. Here we further explore ∗ Corresponding author: [email protected] † Corresponding author: [email protected] the interactions of water waves with periodic structuresby utilizing the concept of the effective gravity g e and ef-fective depth h e . Our results show that the band gaps ofLSW propagating over rippled bottom can be character-ized with either a negative depth with a positive gravita-tional acceleration (negative depth bottom, NDB), or anegative gravitational acceleration with a positive depth(negative gravity bottom, NGB). In the NDB-NGB pairstructure, the interface states of LSW are realized accu-rately under the matching conditions with respect to theeffective impedance and effective phase shift. Theoreti-cal calculations show that the LSW are strongly localizedat the interface, with one order of magnitude enhance-ment, which possesses potential applications in wave en-ergy conversion. Recent results about classical wave peri-odic systems show that materials with two different singlenegative parameters (for example, electromagnetic mate-rials with negative permittivity or negative permeability)have different topological orders [20–25]. Hence, inter-face states formed at the boundary separating two pe-riodic structures having different band gap topologicalcharacteristics. Our results may provide another insightto understand the band gap properties in LSWs.The paper is organized as follows. In Sec. II, wepresent the method used for the retrieval of effectivedepth and effective gravitational acceleration for LSWs.The effective parameters for LSWs within band gaps arediscussed with the retrieval method. In Sec. III, we ex-plore the interface state existing at the boundary betweentwo periodic bottoms with different effective parameters.Here two types of paired structures are discussed: oneis composed of asymmetric unit cells, and the other issymmetric unit cells. In Sec. IV, we investigate the fieldenhancement behavior for the interface state. Finally, aconclusion is given in Sec. V. II. EFFECTIVE PARAMETERS FOR LIQUIDSURFACE WAVE IN BAND GAPS
In this section, we will explore the effective parametersof LSW system using the retrieval method. This concept a r X i v : . [ phy s i c s . c l a ss - ph ] S e p h C h d D d C h D Strucuture (CD) h A h d B d A h B Strucuture (AB) LiquidLiquid (a) (c) F r equen cy f ( H z ) q ( d A + d B ) /π h e T Frequency (Hz) F r equen cy f ( H z ) q ( d C + d D ) /π T − − g e − T − (b) (d) (e) h e Frequency (Hz) − g e −
100 6 7 8 96 7 8 910 (f)
FIG. 1. Schematic views of (a) the finite periodic rippled bottoms (AB) , (b) the finite periodic rippled bottoms (CD) . (c),(d) Band structures and transmission spectra of (AB) and (CD) . The retrieved effective gravitational acceleration g e andeffective depth h e of (AB) and (CD) within band gaps are shown in (e) and (f). has been introduced firstly by Hu et al [11]. For thelinear, inviscid, irrotational and shallow LSW over anuneven bottom, the wave equation satisfies [20, 26, 27](rigorous when kh (cid:28) (cid:18) ∇ · h ∇ + ω g (cid:19) η = 0 , (1)where η is the displacement of the liquid surface, ω isthe angular frequency, g is the gravitational acceleration,and h is the liquid depth. If the LSW system is takenas an even bottom with same reflection and transmis-sion coefficients, it has the same effective gravitationalacceleration and depth as the artificial homogeneous liq-uid. Supposing that the plane LSW normal incident intoa one dimensional (1D) structure along the x direction,and thus the surface displacement of LSW η can be writ-ten as the superposition of forward and backward waves η = Ae ikx + Be − ikx , with A and B the amplitudes ofthe forward and backward waves, respectively. Suppos-ing that the properties of the structure can be describedwith the effective parameters, namely the effective rela-tive depth h e and effective relative gravitational acceler-ation g e . The effective index and the effective impedancecan be written as n e = 1 / √ g e h e and z e = (cid:112) h e /g e ,respectively [11]. For the hypothetic artificial homoge-neous liquid, the reflection coefficient r and transmissioncoefficient t can be obtained by using transfer matrix method [28]1 t = cos( n e k d ) − i (cid:18) z e + 1 z e (cid:19) sin( n e k d ) , (2a) rt = − i (cid:18) z e − z e (cid:19) sin( n e k d ) , (2b)where k is the wave vector within background liquiddepth, d is the total thickness of structure. Through astandard retrieval procedure the effective index n e andeffective impedance of water wave z e are given as follows z e = ± (cid:115) (1 + r ) − t (1 − r ) − t (3)and e in e k d = ( z e + 1) tz e + 1 − ( z e − t , (4)where n e = k d { Im[ln( e in e k d )] + 2 mπ − i Re[ln( e in e k d )] } with m an integer. The sign on the right-hand side inEq. (3) can be determined by Re( z e ) ≥
0, Im( z e ) ≤ h e and g e can be obtainedby h e = z e /n e and g e = 1 / ( z e n e ). In this way, the 1Dstructure can act as an even bottom with the effectiverelative depth h e and the effective relative gravitationalacceleration g e .It is known that the periodic modulation of liquiddepth over the bottom would lead to the band struc-ture of LSW, and the wave propagation is forbidden inthe band gap [5]. In the following, we use the effectiveparameters to investigate the band gap of 1D periodicstructure. Two different 1D periodic structures, (AB) and (CD) , are studied, as shown in Fig. 1 (a) and (b),respectively. Here A and C represent the valleys withthe width of d A = 9 . d C = 4 . d B = 11 mmand d D = 4 . h A = h C = 6 mm and h B = h D = 1 mm, respectively.The liquid depth of the background is h = 10 mm. Theliquid is chosen as CFC-113 of the Dupont company, apopular solvent with surface tension 17.3 dyn/cm anddensity 1.48 g/cm . We use this liquid instead of watersince this liquid has a very low capillary length and smalldissipation. Thus the phenomena of LSW can easily beobserved in experiments [6, 7].The band strcture and transmission of (AB) areshown in Fig.1(c). In this figure, there are two bandgaps in this structure from low to high frequency region.The band gap from 5.92 to 7.45 Hz is our concerned re-gion and retrieved effective parameters are given in Fig.1(e). (AB) has effective single negative parameters inthe band gap, for the wave incident from right side. Theeffective parameters satisfy h e < g e > h e > g e < is given in Fig. 1(d) and (f). Clearlythere is a band gap from 5.98 to 9.18 Hz. Using theretrieval method, the effective liquid depth h e and grav-itational constant g e of (CD) is plotted. It shows that(CD) have effective single negative parameters in theband gap, for the wave incident from left side. In de- (a) (b)(c) Frequency (Hz) Frequency (Hz)0 01230240.5 T I m ( Z ) I m ( φ ) (AB) (CD) FIG. 2. (a) Transmission spectrum of the heterostructure(AB) (CD) . (b) Imaginary impedance of (AB) and (CD) (the sign of the imaginary parts of (CD) have been reversed).(c) Imaginary phase of (AB) and (CD) . Matching condi-tions are satisfied at 6.01 Hz, as indicated by the vertical graydashed line. tails, h e > g e < h e < g e > and (CD) show different behavior from lower frequency to higherin band gaps. For (AB) , it acts as a NDB at lowerfrequency and a NGB at higher frequency. For (CD) ,it shows opposite behavior in band gap. The band gapof finite periodic rippled bottoms could be characterizedby the effective negative depth and the effective negativegravitational acceleration with parameters g e h e < III. INTERFACE STATES IN LIQUID SURFACEWAVE
The results above show that the LSW period system inband gap can prevent the propagation of water wave andmay mimic two types of effective structure with param-eters h e < g e > h e > g e < z NDB ) = − Im( z NGB ) , (5a)Im( n NDB k d NDB ) = Im( n NGB k d NGB ) , (5b)where z is the effective impedance of LSW retrieved fromtransmission and reflection coefficients of correspondingstructures, and n and d denote the refractive index andthickness, respectively; Im represents the imaginary part.Therefore, for the NDB-NGB pair structure, the perfecttransmission of LSW can occur at the frequency whichmeets the matching condition.To explore the perfect transmission behavior of LSWunder the matching condition, we give the imaginaryparts of the effective impedances and the effective phaseshifts of (AB) and (CD) in Fig. 2(b) and (c). As in-dicated by the vertical dashed line, the matching condi-tions are satisfied at the frequency of f = 6 .
01 Hz, where(AB) acts as a NDB structure and (CD) acts as a NGB.Thus one can infer that there would be an interface stateof LSW at 6.01 Hz for the paired structure (AB) (CD) .The transmission of (AB) (CD) is given in Fig 2(a).Aperfect transmission peak emerges around f = 6 .
01 Hz,which is original in the band gap of (AB) and (CD) .It means the LSW tunnels through the heterostructurealthough it cannot propagate through (AB) or (CD) .The discussion above is about the periodic modu-lated bottoms composed of asymmetric unit cells. Next,we will introduce a special type of structure composedof symmetric unit cells, i.e. the structures (PQQP) ,(QPPQ) and the heterostructure (PQQP) (QPPQ) , asis shown in figure 3(a). Here P and Q correspond to val-ley and ridge with the same width 2.5 mm. The liquiddepth over valley and ridge are set to be 4 mm and 1mm, respectively. The transmission spectra of the struc-ture (PQQP) and (QPPQ) , as is illustrated in figure h e g e (a)(b) (c) Frequency (Hz)Frequency (Hz)00.50.50 051005 − T h e & g e − h e & g e I m ( Z ) I m ( φ ) (PQQP) (PQQP) (QPPQ) (PQQP) (QPPQ) (PQQP) (QPPQ) (QPPQ) FIG. 3. (a) Transmission spectrum of the heterostructure(PQQP) (QPPQ) . A schematic view of the structure isshown in the inset. (b) The retrieved effective parameters g e and h e versus frequency for (PQQP) and (QPPQ) . (c)Imaginary impedance of(PQQP) and (QPPQ) for upperpanel (the sign of the imaginary parts of (QPPQ) have beenreversed). Imaginary phase of (PQQP) and (QPPQ) forlower panel. Matching conditions are satisfied at 6.26 Hz, asindicated by the vertical dashed line. satisfy h e > g e < can mimic a NGB structure. On the con-trary, (PQQP) can mimic a NDB structure throughoutits whole band gap, i.e. h e < g e >
0. It has beenmentioned, for the 1D periodic structures composed ofasymmetric unit cells, the band gap is divide into tworegimes with respect to the retrieved effective parame-ters, which corresponds to NGB and NDB, respectively.Here, for 1D periodic structures composed of symmetricunit cells, the retrieved effective parameters throughoutthe whole band gap would only be one type, i.e. h e > g e < , and h e < g e > . Since the structure (PQQP) and (QPPQ) respectively corresponds to NGB and NDB, we can ex-pect the realization of the imaginary impedance matchingand imaginary phase matching between these two struc-tures. The imaginary impedance and imaginary phase of(PQQP) and (QPPQ) are calculated in figure 3(c). Onthe one hand, these two structures have the same imag-inary phase in band gap, which indicates the automaticsatisfaction of the imaginary phase matching. Conse-quently, the appearance of tunneling effect is only de- termined by the imaginary impedance matching, whichmeans that the interface states could be obtained moreeasily with symmetric unit cells. On the other hand,the imaginary impedance matching is satisfied at 6.26Hz, which indicates the tunneling frequency. As a result,for the heterostructure (PQQP) (QPPQ) , an interfacestate with high transmission can be observed at 6.26 Hz,which locates in the first gaps of (PQQP) and (PQQP) ,as is shown in figure 3(a). IV. FIELD DISTRIBUTION OFTRANSMISSION SPECTRA
In this section we will investigate the field enhancementof (PQQP) (QPPQ) induced by the interface state.In our calculations, the amplitude of incident LSW isnormalized. In figure 8(a), the field distributions of(PQQP) (QPPQ) are given for the frequencies from 3 to10 Hz. It shows that the LSW cannot propagate throughthe structure in the frequency region of band gaps (5Hz to 8.9 Hz). However, the fields show that LSW canclearly tunnel through the structure at 6.26 Hz, whichcorresponds to the frequency of interface state. Whenthe interface state is excited, LSW is primarily localizedat the interface and exponentially decays from the inter-face to both ends, which is in detail illustrated in figure3. Our results show that the amplitude of LSW at theinterface between (PQQP) and (QPPQ) is effectivelydriven up as large as 25 times than that of incident wave. V. CONCLUSION
In summary, we theoretically studied the propagationof LSWs over a bottom with a one-dimensional peri-odic undulation. The results reveal that the signs of theeffective depth and gravitational acceleration are oppo-site within the band gaps. Under the conditions of the F r equen cy ( H z ) (PQQP) (QPPQ) FIG. 4. The field intensity of (PQQP) (QPPQ) . Horizontalaxle is the length of structure. The amplitude of LSW isenhanced about 25 times near the interface. impedance matching and phase matching, an interfacestates can be realized at the interface between the struc-ture with negative h e and the structure with negative g e . Moreover, the LSW energy localizations at the inter-face can be enhanced over one order of magnitude thanthat of the incident LSW. Our work possesses potentialapplications in the utilization of energy in water wave.In addition, these results may pave the way for realizingmany exotic phenomena based on single negative mate-rials and zero-refractive-index materials. ACKNOWLEDGMENTS
The authors are grateful to Ang Chen for helpful dis-cussions. This work is supported by the NSFC (No.61422504, 11474221, 11234010 , 11204217 and 11704254)and financial support from the China Scholarship Coun-cil (Grant No. 201706265021). [1] X. Hu, Y. Shen, X. Liu, R. Fu, and J. Zi, Physical ReviewE , 066308 (2003).[2] X. Hu, Y. Shen, X. Liu, R. Fu, J. Zi, X. Jiang, andS. Feng, Physical Review E , 037301 (2003).[3] T. S. Jeong, J.-E. Kim, H. Y. Park, and I.-W. Lee, Ap-plied physics letters , 1645 (2004).[4] Y. Shen, X. Liu, Y. Tang, Y. Chen, and J. Zi, Journalof Physics: Condensed Matter , L287 (2005).[5] Y. Tang, Y. Shen, J. Yang, X. Liu, J. Zi, and X. Hu,Physical Review E , 035302 (2006).[6] X. Hu, Y. Shen, X. Liu, R. Fu, and J. Zi, Physical ReviewE , 030201 (2004).[7] Y. Shen, K. Chen, Y. Chen, X. Liu, and J. Zi, PhysicalReview E , 036301 (2005).[8] J. Mei, C. Qiu, J. Shi, and Z. Liu, Physics Letters A , 2948 (2009).[9] J. Mei, C. Qiu, J. Shi, and Z. Liu, Wave Motion , 131(2010).[10] Z. Wang, P. Zhang, Y. Zhang, and X. Nie, Physica B:Condensed Matter , 75 (2013).[11] X. Hu and C. Chan, Physical Review Letters , 154501(2005).[12] X. Hu, C. T. Chan, K.-M. Ho, and J. Zi, Physical ReviewLetters , 174501 (2011).[13] X. Hu, J. Yang, J. Zi, C. T. Chan, and K.-M. Ho, Sci-entific Reports , 1916 (2013).[14] C. Zhang, C.-T. Chan, and X. Hu, Scientific Reports ,6979 (2014).[15] K. Budar and J. Falnes, Nature , 478 (1975). [16] E. Callaway, Nature , 156 (2007).[17] J. Cruz, Ocean wave energy: current status and futureprespectives (Springer Science & Business Media, 2007).[18] J. Engstr¨om, M. Eriksson, J. Isberg, and M. Leijon,Journal of Applied Physics , 064512 (2009).[19] X. Garnaud and C. C. Mei, in
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