ICSV23 full paper Tuo Liu - The International Institute of …€¦ · · 2016-07-03demonstration of acoustic rainbow ... only the fundamental mode is considered in the long wavelength

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23I SVC23rd International Congress on Sound & Vibration

10-14 July 2016Athens, Greece

ACOUSTIC RAINBOW TRAPPING THROUGH PERFORATED STRUCTURES Tuo Liu and Jie Zhu The Hong Kong Polytechnic University – Department of Mechanical Engineering, 11 Yuk Choi Rd, Hung Hom, Kowloon, Hong Kong email: [email protected]

Acoustic rainbow trapping (ART) metamaterials are a type of artificial materials that allows in-tensive energy trapping and spatial-spectral modulation of broadband sound waves, by provid-ing strong acoustic wave dispersions lacking in natural materials. However, previous studies on ART metamaterials are all based on 2D unit cells and thus require the structure size in the third dimension to be much larger than the wavelength, which limits their further applications. Here we would like to present an ART model that is constructed using 3D hole structures. Full wave numerical simulation result shows that incident plane waves of different frequencies are effec-tively converted into a highly-dispersive structure-induced surface waves (SSAWs) mode and trapped at different positions, inducing greatly enhanced sound fields, which is in good agree-ment with our theoretical prediction. Such ART metamaterials with 3D unit cells can be easily developed into practical use and may also be extended multi-dimensional case.

1. Introduction

Studies on slowing down and trapping of optical waves have received lots of interests throughout the last decades since they provide possibilities for precise control of optical delay, temporary stor-ages of light, and strong light-matter interactions [1]-[7], which may potentially applied to many areas such as nonlinear optics, quantum optics, all-optical memories, etc [8]. Yet, the intrinsic con-flict between high delay and wide bandwidth within resonance structures restrict the practical use of traditional slow-light systems. To solve this problem, the concept of “trapped rainbow” was pro-posed and thoroughly investigated by researchers [9]-[19]. It allows various spectral components of wide-band optical waves to be slowed down and trapped at different positions, and the high delay-bandwidth product is thus realized.

Contrary to the significant advances in optical waves, until recently sound deceleration and trap-ping had not been achieved due to the fact that most natural materials are generally treated as non-dispersive medium for acoustic waves. Fortunately, the prosperity of artificial materials with unusu-al acoustic effective parameters reveal promising approaches in manipulating sound waves and the wave-matter interaction [20]-[33]. Based on this, several dispersive phononic crystals and acoustic metamaterials were successfully proposed to obtain slow sound [34]-[37]. The first experimental demonstration of acoustic rainbow trapping (ART) was realized through periodic slit array with graded lengths [38]. Such structure functions as a novel gradient-index anisotropic metamaterial with strong dispersion that allows broadband trapping and spatial-spectral splitting of sound waves [39], [40], mimicking the phenomenon of optical “trapped rainbow”. Space-coiling and micro Mie resonance-based structures are also introduced to obtain higher refractive index and deeper sub-wavelength scale [41], [42]. Another recent advance of dispersive acoustic metamaterial is the acoustic prism structure, in which both the strong dispersion based on transmission line and the im-pedance match as a result of the unique physical behaviour of acoustic leaky-wave radiation are

The 23rd International Congress on Sound and Vibration

2 ICSV23, Athens (Greece), 10-14 July 2016

achieved [43]. Besides acoustic waves, there is also a report on the Lamb wave trapping [44]. These dispersive phononic crystals and acoustic metamaterials may enable a series of applications ranging from artificial cochlea to high performance acoustic sensing and filtering.

However, those solutions to obtaining dispersive or ART effect are mostly based on 2D unit cells that, in ideal cases, require infinite size in the other direction. This unavoidable limitation restricts their further introduction to practical applications. Therefore, this article presents an improved ART model, in which 3D hole structures are adopted as unit cells, constructing a perforation surface with graded hole depths along the wave propagation direction. We firstly present the analytical model of the ART structure by discussing the dispersion relation of the highly-dispersive structure-induced surface acoustic waves (SSAWs) supported by the structure. Then, full wave numerical simulation is performed to examine the effectiveness of the model. The simulation result shows that incident plane waves of different frequencies are effectively converted into the SSAW mode and trapped at different positions, forming greatly enhanced sound fields, which agrees well with our analytical prediction. With this 3D unit cells, the ART metamaterials are more easily applied to practical use.

2. Analytical model

As shown in Fig. 1, the ART model is a rigid surface perforated with an array of square holes, in which the unit cell is a 3D subwavelength structure with period , side length and depth . The hole depth increases linearly along the wave propagating direction and distributes uniformly along the transverse direction . For general perforating structures, a surface mode exists as a result of the periodic spatial modulation, whose propagation characteristics are governed by the geometry of the unit cell [45]-[48]. It should be noted that this surface mode, namely, the so-called SSAWs, is different from surface waves at a free surface or an interfaces between two mediums due to the su-perposition of at least two types of bulk waves. Here in our model, the graded hole depth in the direction actually provides both impedance match and a gradient refractive index distribution. Inci-dent plane waves can be effectively converted into the SSAW mode travelling with gradually de-creased group velocity. For ideal effective medium case (infinite small unit cell), the SSAWs is trapped near the position where group velocity drops to zero.

In this section, we will start with a simple derivation of the dispersion relation of the SSAWs propagating along a rigid surface perforated with holes of uniform depth and then use this result to construct the ART model.

Figure 1: Schematic illustration of (a) the ART model, (b) the unit cell (square hole), and (c) cross-section of the -plane. In both analytical calculation and full wave numerical simulation, the side length of the square

hole and the period of the unit cell are 3mm and 5mm, and the background medium is air with density 1.21kg/m and sound speed 343m/s. The depth of each hole varies from 0.5mm to

25mm with a step of 0.25mm, corresponding to the position along the direction from 10mm to 500mm with a spacing of 5mm.


ICSV23, Athens (Greece), 10-14 July 2016 3

2.1 Dispersion relation of the SSAWs Consider an infinite large surface perforated with square holes of uniform depth where the

SSAWs travel along the direction. The evanescent pressure field above the opening of holes 0 takes the form

nzn jq x yk zn

n

P C e e

, (1)

where 2 ⁄ are the wave vector components along the propagation direction of the

nth order diffracted wave and the corresponding coefficients. are the transverse momenta along the x direction, in which is the wave number of the background medium. is the Bloch wave vector in the first Brillouin zone. As the propagating acoustic modes are tightly con-fined at the surface area, the acoustic field outside the metamaterial region is transversely evanes-cent, satisfying the condition . For low-frequency approximation, the 0 and -1 order modes dominate (propagating along the and – directions, respectively), and all higher order diffracted modes can thus be neglected. Eq. (1) is then simplified as

0 1

z zjq x y jq x yk z k zP C e e C e e . (2)

Inside each hole, only the fundamental mode is considered in the long wavelength limits (wave-length ≫ ) and the corresponding standing wave field can be expressed as

0 0jk z jk zhp Ae Be . (3)

As the bottom of the hole is rigid , we have 02 jk hA Be . (4)

At the hole opening area 0, the pressure and volume velocity should be continuous, which would yield

0 0d d

hole hole

hz zS S

P S p S , (5)

0 , 0d d

unit hole

z zz zh

S S

V S v S . (6)

Here and , are the components of particle velocity above and inside the hole. and indicate the area of the unit cell and the opening. By substituting Eqs. (2)~(4) into the boundary conditions Eqs. (5) and (6), one may easily obtain the dispersion relation, which is given by

2 2 2

002

0

sinc( 2)tan

sinc( 2)

q k a qak h

k d qd

. (7)

Further, when the period of the unit cell → 0, the perforation surface can be treated as an ideal effective medium with dispersion relation given by

2 2 2

002

0

tanq k a

k hk d

. (8)

Fig. 2(a) presents a typical dispersion curve of the SSAWs. It can be seen that the result based on effective medium approximation (Eq. (8)) overlaps with the analytical results from Eq. (7) at low frequency range and starts to deviate as the wave vector approaching the edge of the first Brillouin zone. This is due to the fact that the effective medium approximation becomes invalid as the effec-tive wavelength of the SSAWs is comparable to the unit cell size at high frequencies. A finite ele-ment method (FEM) simulation is also conducted, which agrees well with the result obtained from Eq. (7) (scatter/line plot in Fig. 2(a)). It is worth mentioning that the cut-off frequency correspond-ing to the edge of the first Brillouin zone where the curve flattens is close to but lower than the cavi-ty mode (quarter-wavelength resonance) of a single square hole. This phenomena can be well un-derstood since the flat region is a result of the balanced interaction between the local resonance in-side individual hole and the propagating evanescent mode due to coupling among neighbouring cavities.



Figure 2: (a) Typical dispersion relation of the airborne SSAWs, with 3mm, 5mm, and 20mm. The solid line represents the analytical result using Eq. (7), the dashed line the effective medium

result using Eq. (8), and the scatter plot the FEM result. The dotted line indicates the cavity mode frequency, which corresponds to the quarter-wavelength resonance. (b) Group velocity distribution along the propaga-tion direction. Three different line colours and styles indicate three frequencies 4000, 5500, and 7000Hz.

2.2 ART model When the hole depths of the array distribute in a graded way as shown in Fig. 1, the near-surface

incident plane wave travelling along the direction interacts with the structure and is converted into the SSAWs. We may directly derive the corresponding group velocity from the dispersion relation obtained above in Eq. (8),

2 2 22 20

0 02 2 222

0 2

1

d / d tan 11 tan

cot

g

cv

q k ha ak h k h

d d ak h

d

, (9)

where is the angular frequency and wavenumber of the background medium ⁄ . The hole depth can be replaced by the horizontal position and tapering angle , namely, ( 5°). From this equation, one may associate the group velocity with both frequency and spatial position. As shown in Fig. 2(b), the group velocity distribution indicates that acoustic wave with frequency gradually decelerates during the propagation process until eventually comes to a standstill at a spe-cific position where group velocity approaches zero. Importantly, the slowing down process and the stop position depend on the given frequency. In other words, different frequency components of broadband sound waves can be spatially separated and localized, through which the so-called “rain-bow trapping” effect is realized.

As we have mentioned above, the effective medium model breaks down when the propagating momentum becomes large near the edge of the first Brillouin zone (the unit cells are no longer sub-wavelength in size any more). Thus, to precisely predict the trapping position for any given frequencies, we recall the dispersion relation given by Eq. (7), which is obtained from the exact ART structures. Since the maximum momentum of SSAWs along the direction is ⁄ instead of ∞ as a result of the finite size of a unit cell, the trapping equation can be expressed as

220

0

0

tansin

2

kdd

kaa

kd

x

(10)



Here the term tan in the right-hand side defines the localized acoustic field inside indi-vidual holes. We will show later that, it interacts with the mutual near-field coupling through the SSAWs to form a standing wave field along the propagation direction, resulting in the “trapping” phenomenon.

3. Numerical simulation

Figure 3: Simulated absolute acoustic pressure fields of three different frequencies 4000Hz, 5500Hz, and 7000Hz. The pressure value in each sub-figure is normalized to the maximum pressure of the whole sound field with the corresponding frequency. The inset is a partial enlarged view of the absolute pressure field of

4000Hz after rescaling the color bar from 0~1 to 0~0.2. Since only component of the wave propagation is considered and the ART structure is repeated

along the transverse direction, we choose one column of unit cells and apply solid boundaries on both sides to mimic the infinite large model in the simulation for simplicity. In such circumstances, the simulation model becomes a single column of graded holes mounted on a waveguide with width equal to the period of the unit cell. The inlet, outlet and infinitely extended upper half-space are employed with perfect-matched layer (PML) to mitigate unwanted reflections. The dimensions of the structure and other parameters are the same as those in Fig. 1. A plane wave of unity amplitude is generated at the inlet boundary and travels along direction.

The simulated absolute acoustic pressure fields of three representative frequencies are given in Fig. 3. Here we select the pressure distribution of cut-plane - at 0, which is also the sym-metry plane of the model. During the propagation process, the plane wave is gradually converted to the SSAWs, which is testified by the energy distribution that becomes confined at the surface as approaching the trapping point (this can be observed by adjusting the range of the color bar, as shown in the inset of Fig. 3). This conversion is without obvious reflection and equivalent to the impedance match as a result of the gradient increase of hole depth along the propagation direction. Neighbouring holes coupling with each other through the SSAWs and reach a balance with the local



resonance inside individual holes, leading to the “trapping” phenomenon. Incident plane waves of different frequencies are clearly trapped at different positions, forming drastically enhanced sound field. This resonance-induced enhancement could be extremely huge without taken into considera-tion the viscous and thermal losses.

Fig. 4(a) are the frequency responses at the bottom of the holes for three different horizontal po-sitions. The highest peak of one particular curve corresponds to the local resonance of the hole and other peaks and valleys are due to the interference with the reflections from trapping positions. With the increase of frequency, the acoustic pressure within an individual hole rises in an oscillating manner and reach its maximum before being followed by a sharp drop. This cut-off indicates the trapping frequency of a particular hole. It should be noted that the amplitude of the peak is several tens of times of the incident plane wave.

We can then use this to extract the correlation between trapping position and frequency, which is illustrated in Fig. 4(b). This correlation is in good agreement with the result predicted by our analyt-ical model. Result based on effective medium model is also included, which is obviously deviated from the FEM simulation. We may expect that these three curves would superposition if the unit cell is infinitely small.

Figure 4: (a) Frequency responses at the bottom of the holes for three different horizontal positions. The solid, dashed and dotted lines represent the frequency responses at horizontal positions 240, 300, and 360mm, respectively. (b) Trapping position versus frequency. The solid and dashed curves are results ob-

tained from Eqs. (7) and (8), and scatter plot is obtained from full wave FEM simulation.

4. Conclusion

In conclusion, this paper have demonstrated that square hole can be used as unit cell to replace previous 2D groove to construct a type of dispersive metamaterial and achieve the so-called ART effect. Broadband incident acoustic waves propagating along the surface of such metamaterials are converted into the SSAWs, which are gradually slowed down until finally stopped at a specific trapping position that is determined by the frequency, resulting in the spectral-spatial splitting and intensive resonance-induced enhancement.

Such unit cell is with 3D structure, which means that it provide a more flexible way to design multi-dimensional dispersive metamaterials and is more suitable for practical applications.

REFERENCES

1 Noda, S., Chutinan, A. and Imada, M. Trapping and emission of photons by a single defect in a photonic bandgap structure, Nature, 407 (6804), 608-610, (2000).

2 Liu, C., Dutton, Z., Behroozi, C. H. and Hau, L. V. Observation of coherent optical information storage in an atomic medium using halted light pulses, Nature, 409 (6819), 490-493, (2001).



3 Julsgaard, B., Sherson, J., Cirac, J. I., Fiurášek, J. and Polzik, E. S. Experimental demonstration of quantum memory for light, Nature, 432 (7016), 482-486, (2004).

4 Vlasov, Y. A., O'Boyle, M., Hamann, H. F. and McNab, S. J. Active control of slow light on a chip with photonic crystal waveguides, Nature, 438 (7064), 65-69, (2005).

5 Gersen, H., Karle, T. J., Engelen, R. J. P., Bogaerts, W., Korterik, J. P., Van Hulst, N. F., Krauss, T.F. and Kuipers, L. Real-space observation of ultraslow light in photonic crystal waveguides, Physical review letters, 94 (7), 073903, (2005).

6 Xia, F., Sekaric, L. and Vlasov, Y. Ultracompact optical buffers on a silicon chip, Nature photonics, 1 (1), 65-71, (2007).

7 Fiore, V., Yang, Y., Kuzyk, M. C., Barbour, R., Tian, L. and Wang, H. Storing optical information as a mechani-cal excitation in a silica optomechanical resonator, Physical review letters, 107 (13), 133601, (2011).

8 Krauss, T. F. Why do we need slow light?, Nature Photonics, 2 (8), 448-450, (2008).

9 Tsakmakidis, K. L., Boardman, A. D. and Hess, O. “Trapped rainbow” storage of light in metamateri-als, Nature, 450 (7168), 397-401, (2007).

10 Williams, C. R., Andrews, S. R., Maier, S. A., Fernandez-Dominguez, A. I., Martín-Moreno, L. and Garcia-Vidal, F. J. Highly confined guiding of terahertz surface plasmon polaritons on structured metal surfaces, Nature Photon-ics, 2 (3), 175-179, (2008).

11 Gan, Q., Ding, Y. J. and Bartoli, F. J. “Rainbow” trapping and releasing at telecommunication wave-lengths, Physical Review Letters, 102 (5), 056801, (2009).

12 Chen, L., Wang, G. P., Gan, Q. and Bartoli, F. J. Trapping of surface-plasmon polaritons in a graded Bragg struc-ture: Frequency-dependent spatially separated localization of the visible spectrum modes, Physical Review B, 80 (16), 161106, (2009).

13 Park, J., Kim, K. Y., Lee, I. M., Na, H., Lee, S. Y., and Lee, B. Trapping light in plasmonic waveguides, Optics express, 18 (2), 598-623, (2010).

14 Gan, Q., Gao, Y., Wagner, K., Vezenov, D., Ding, Y. J. and Bartoli, F. J. Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings, Proceedings of the National Academy of Sciences, 108 (13), 5169-5173, (2011).

15 Jang, M. S. and Atwater, H. Plasmonic rainbow trapping structures for light localization and spectrum split-ting, Physical Review Letters, 107 (20), 207401, (2011).

16 Kats, M. A., Woolf, D., Blanchard, R., Yu, N. and Capasso, F. Spoof plasmon analogue of metal-insulator-metal waveguides. Optics express, 19 (16), 14860-14870, (2011).

17 Gan, Q. and Bartoli, F. J. Surface dispersion engineering of planar plasmonic chirped grating for complete visible rainbow trapping, Applied Physics Letters, 98 (25), 251103, (2011).

18 He, S., He, Y. and Jin, Y. Revealing the truth about ‘trapped rainbow’ storage of light in metamaterials, Scientific reports, 2, 583, (2012).

19 Hu, H., Ji, D., Zeng, X., Liu, K. and Gan, Q. Rainbow trapping in hyperbolic metamaterial waveguide, Scientific reports, 3, (2013).

20 Liu, Z., Zhang, X., Mao, Y., Zhu, Y. Y., Yang, Z., Chan, C. T. and Sheng, P. Locally resonant sonic materi-als, Science, 289 (5485), 1734-1736, (2000).

21 Li, J. and Chan, C. T. Double-negative acoustic metamaterial, Physical Review E, 70 (5), 055602, (2004).

22 Fang, N., Xi, D., Xu, J., Ambati, M., Srituravanich, W., Sun, C. and Zhang, X. Ultrasonic metamaterials with neg-ative modulus, Nature materials, 5 (6), 452-456, (2006).

23 Christensen, J., Fernandez-Dominguez, A. I., de Leon-Perez, F., Martin-Moreno, L. and Garcia-Vidal, F. J. Colli-mation of sound assisted by acoustic surface waves, Nature Physics, 3 (12), 851-852, (2007).

24 Lu, M. H., Liu, X. K., Feng, L., Li, J., Huang, C. P., Chen, Y. F., Zhu, Y.Y., Zhu, S.N. and Ming, N. B. Extraor-dinary acoustic transmission through a 1D grating with very narrow apertures, Physical review letters, 99 (17), 174301, (2007).

25 Chen, H., and Chan, C. T. Acoustic cloaking in three dimensions using acoustic metamaterials, Applied physics letters, 91 (18), 183518, (2007).



26 Li, J., Fok, L., Yin, X., Bartal, G., and Zhang, X. Experimental demonstration of an acoustic magnifying hyper-lens, Nature materials, 8 (12), 931-934, (2009).

27 Liang, B., Guo, X. S., Tu, J., Zhang, D. and Cheng, J. C. An acoustic rectifier. Nature materials, 9 (12), 989-992, (2010).

28 Zhu, J., Christensen, J., Jung, J., Martin-Moreno, L., Yin, X., Fok, L., Zhang, X. and Garcia-Vidal, F. J. A holey-structured metamaterial for acoustic deep-subwavelength imaging, Nature physics, 7 (1), 52-55, (2011).

29 Liang, Z., and Li, J. Extreme acoustic metamaterial by coiling up space, Physical review letters, 108 (11), 114301, (2012).

30 Christensen, J. and de Abajo, F. J. G. Anisotropic metamaterials for full control of acoustic waves, Physical review letters, 108 (12), 124301, (2012).

31 Zhu, X., Ramezani, H., Shi, C., Zhu, J. and Zhang, X. PT-Symmetric Acoustics, Physical Review X, 4 (3), 031042, (2014).

32 Fleury, R., Sounas, D. L., Sieck, C. F., Haberman, M. R. and Alù, A. Sound isolation and giant linear nonreciproc-ity in a compact acoustic circulator, Science, 343 (6170), 516-519, (2014).

33 Yang, Z., Gao, F., Shi, X., Lin, X., Gao, Z., Chong, Y. and Zhang, B. Topological acoustics, Physical review let-ters, 114 (11), 114301, (2015).

34 Robertson, W. M., Baker, C. and Bennett, C. B. Slow group velocity propagation of sound via defect coupling in a one-dimensional acoustic band gap array, American Journal of Physics, 72 (2), 255-257, (2004).

35 Christensen, J., Huidobro, P. A., Martin-Moreno, L. and Garcia-Vidal, F. J. Confining and slowing airborne sound with a corrugated metawire, Applied Physics Letters, 93 (8), 083502, (2008).

36 Laude, V., Beugnot, J. C., Benchabane, S., Pennec, Y., Djafari-Rouhani, B., Papanikolaou, N., Escalante, J.M. and Martinez, A. Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs, Optics express, 19 (10), 9690-9698, (2011).

37 Santillán, A. and Bozhevolnyi, S. I. Demonstration of slow sound propagation and acoustic transparency with a series of detuned resonators, Physical Review B, 89 (18), 184301, (2014).

38 Zhu, J., Chen, Y., Zhu, X., Garcia-Vidal, F. J., Yin, X., Zhang, W. and Zhang, X. Acoustic rainbow trap-ping, Scientific reports, 3, (2013).

39 Chen, Y., Liu, H., Reilly, M., Bae, H. and Yu, M. Enhanced acoustic sensing through wave compression and pres-sure amplification in anisotropic metamaterials. Nature communications, 5, (2014).

40 Jia, H., Lu, M., Ni, X., Bao, M. and Li, X. Spatial separation of spoof surface acoustic waves on the graded groove grating, Journal of Applied Physics, 116 (12), 124504, (2014).

41 Ni, X., Wu, Y., Chen, Z. G., Zheng, L. Y., Xu, Y. L., Nayar, P., Liu, X.P., Lu, M.H. and Chen, Y. F. Acoustic rainbow trapping by coiling up space, Scientific reports, 4, (2014).

42 Zhou, C., Yuan, B., Cheng, Y. and Liu, X. Precise rainbow trapping for low-frequency acoustic waves with micro Mie resonance-based structures, Applied Physics Letters, 108 (6), 063501, (2016).

43 Esfahlani, H., Karkar, S., Lissek, H. and Mosig, J. R. Acoustic dispersive prism, Scientific reports, 6, (2016).

44 Zhao, D. G., Li, Y. and Zhu, X. F. Broadband lamb wave trapping in cellular metamaterial plates with multiple local resonances, Scientific reports, 5, (2015).

45 Pendry, J. B., Martin-Moreno, L. and Garcia-Vidal, F. J. Mimicking surface plasmons with structured surfac-es, Science, 305(5685), 847-848, (2004).

46 Kelders, L., Allard, J. F. and Lauriks, W. Ultrasonic surface waves above rectangular-groove gratings, The Journal of the Acoustical Society of America, 103 (5), 2730-2733, (1998).

47 He, Z., Jia, H., Qiu, C., Ye, Y., Hao, R., Ke, M. and Liu, Z. Nonleaky surface acoustic waves on a textured rigid surface, Physical Review B, 83 (13), 132101, (2011).

48 Quan, L., Qian, F., Liu, X., Gong, X. and Johnson, P. A. Mimicking surface plasmons in acoustics at low frequen-cy, Physical Review B, 92 (10), 104105, (2015).

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ICSV23 full paper Tuo Liu - The International Institute of …€¦ · · 2016-07-03demonstration of acoustic rainbow ... only the fundamental mode is considered in the long wavelength