Ultrasonics 52 (2012) 412–416

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Ultrasonics

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Ultrasonic transmission through multiple-sublattice subwavelength holes arrays

Héctor Estrada a,b, Vicente Gómez-Lozano a, Antonio Uris a,⇑, Pilar Candelas a, Francisco Belmar a,Francisco Meseguer a,b

a Centro de Tecnologías Físicas, Unidad Asociada ICMM-CSIC/UPV, Edificio 8B Bloque K, Ciudad Politécnica de la Innovación, Universidad Politécnica de Valencia, 46022 Valencia, Spainb Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049 Madrid, Spain

a r t i c l e i n f o

Article history:Received 20 July 2011Received in revised form 16 September2011Accepted 20 September 2011Available online 2 October 2011

Keywords:SubwavelengthMultiple-sublatticeHole arrayUltrasonic transmission

0041-624X/$ - see front matter � 2011 Elsevier B.V.doi:10.1016/j.ultras.2011.09.007

⇑ Corresponding author. Tel.: +34 963877528; fax:E-mail address: auris@fis.upv.es (A. Uris).

a b s t r a c t

The ultrasonic transmission through plates perforated with 2 � 2 or 3 � 3 square array of subwavelengthholes per unit cell are studied by numerical simulations. Calculations are obtained by means of a theo-retical model under the rigid-solid assumption. It is demonstrated that when the inter-hole distancewithin the unit cell is reduced, new transmission dips appear resulting from Wood anomalies that haveinfluence on the second and the third order Fabry–Perot peak. When the inter-hole distance within theunit cell is reduced, the transmission spectrum of the multiple-sublattice holes arrays tends to the trans-mission spectrum of a plate perforated with only one hole in the unit cell.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

In recent years, the extraordinary optical transmission throughmetallic membranes perforated with subwavelength hole arrays[1] has attracted considerable attention. One important character-istic of periodic subwavelength hole arrays drilled on a metallicmembrane is that they transmit much light than expected fromBethe’s theory [2]. A lot of discussion has been raised in the litera-ture for elucidating the mechanisms involved in the extraordinaryoptical transmission. Martin-Moreno et al. [3] and Barnes et al. [4]attributed this effect to surface plasmon resonances, Cao and Lal-anne [5] to cavity resonances, Porto et al. [6] to waveguide reso-nance and Takakura [7] to dynamical diffraction. Inspired by thestudies in electromagnetic waves, research has been extended toacoustic waves, emphasizing the similarities between both cases,but taking into account the intrinsic differences between them,namely: acoustic waves can be transmitted through a single sub-wavelength hole [8] and, depending on the impedance contrast be-tween fluid and solid, can penetrate into solid [9]. The so-calledextraordinary acoustic transmission was reported experimentallyby Lu et al. [10] and Hou et al. [11] for slit and hole arrays, respec-tively, and theoretically by Christensen et al. [12]. It is widely ac-cepted that Fabry–Perot resonances are the main responsible forresonant acoustic transmission. Estrada et al. [13] have shown thatwater-immersed aluminium plates perforated with periodic sub-

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wavelength hole arrays exhibit not only full transmission peaks,but also extraordinary ultrasound screening over a frequency re-gion around Wood anomaly [14]. It was also demonstrated thatthe position and width of transmission peaks and dips can betuned by changing the filling fraction of holes [15] and the latticegeometry [16]. Estrada et al. [9] also demonstrated that Lamband Scholte–Stoneley modes are strongly coupled to Fabry–Perotand Wood anomalies in a water-immersed perforated plate.

The aim of this paper is to study theoretically the interactionbetween the different resonances when multiple subwavelengthholes arrays are arranged in a unit cell. The ultrasound transmis-sion through plates perforated is calculated when the individualunit cells comprise 2 � 2 or 3 � 3 square arrays of subwavelengthholes. The spacing between holes in unit cell is varied to examineits effect on the transmission spectra.

2. Basic theory

Consider a plane ultrasound wave incident on a rigid plate ofthickness h drilled with P cylindrical holes of radius r0 in positionsdetermined by their centres~ri, as schematically shown in Fig. 1.

Assuming an incident plane wave /0ð~rÞ ¼ eið~k0 �~r�xtÞ, the reflectedand the transmitted ultrasound pressure fields can be expressed interms of Rayleigh expansion. For simplicity, time harmonic excita-tion is assumed, thus the time component e�ixt can be omitted. Byusing the rigid-solid assumption, that is, there is no field inside thesolid, the pressure field in the three regions can be written as fol-lows [17,18]:

h

r0

z x

x y

ri

ri

rr-ri

1

2

3

z

ϕi

(r-ri)||r

(a)

(b)

Fig. 1. Schematic representation of the (a) ‘‘xz’’ plane and (b) ‘‘xy’’ plane of the unit-cell. Grey regions correspond to the rigid solid whereas the surrounding fluid is divided inthree regions as indicated by the labels in (a). The vector ð~r �~riÞjj in (b) represents the projection over the z = 0 plane of the vector defined from the centre of each hole to theappropriate points in the fluid region 2.

H. Estrada et al. / Ultrasonics 52 (2012) 412–416 413

/Ið~rÞ ¼ /0ð~rÞ þ /Rð~rÞ ¼ eið~Q0 �~rjj�q0zÞ þZZ

bþð~QÞeið~Q �~rjjþqzÞd2~Q ; ð1Þ

/IIð~rÞ ¼PPi¼1

P1m¼0

P1n¼1

JmðQimnjð~r �~riÞkjÞeimuiWi

mnðzÞ if j~r �~rij 6 r0;

0 otherwise

8><>:

ð2Þ

/IIIð~rÞ ¼ZZ

b�ð~QÞeið~Q �~rjj�qðzþhÞÞd2~Q ; ð3Þ

where ~k0 ¼ ð~Q0; q0Þ, k0 ¼ x=c, q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

0 � Q2q

, qimn ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

0 � Qi2

mn

q,

Jm(...) is the Bessel function of the first kind and order m, andWi

mnðzÞ is defined as

3x3 p=7a/30 3x3 p=3a

8a/30 7a/30 2a/10 3a/10

2x2 p=a/3 2x2 p=1

a/3 a/3 7a/24 10a/2

Fig. 2. Diagrammatic sketch of the un

WimnðzÞ ¼ aiþ

mneiqimnz þ ai�

mne�iqimnz:

As the plate is treated as a perfect rigid solid, zero normal veloc-ity at the hole walls is assumed and the polar eigenfunctions insidethe hole must satisfy J0mðQ

imnr0Þ ¼ 0.

Repeating the P holes periodically throughout the whole plate,they can be considered as a lattice basis with a unit-cell area Sand defined by the vectors ð~a1;~a2Þ. In this way, the coefficientsb�ð~QÞ can be expanded as Fourier series, giving discrete expres-sions [19] for Eqs. (1) and (3)

/Ið~rÞ ¼ 2eið~Q0 �~rjj Þ � cosðq0 � zÞ þX~G

bþ~G eið~Q~G �~rjjþq~GzÞ; ð4Þ

/10 3x3 p=a/3

a/6 a/3

0a/24 2x2 p=a/2 4 a/4 a/2

it cells of the samples considered.

Fig. 3. Transmitted ultrasound power coefficient of the 2 � 2 multiple-sublatticeholes arrays with the three different periodicities of the holes within the unit cell:(a) a/2, (b) 10a/24, (c) a/3, and (d) sample with one hole within the unit cell withperiod a and hole filling fraction 0.25, where a = 5 mm.

Fig. 4. Structure factor of the 2 � 2 multiple-sublattice[17,18] holes arrays with thethree different inter-hole distances within the unit cell: (a) a/2, (b) 10a/24, (c) a/3,and (d) sample with one hole within the unit cell with period a and hole fillingfraction 0.25, where a = 5 mm.

414 H. Estrada et al. / Ultrasonics 52 (2012) 412–416

/IIIð~rÞ ¼X~G

b�~G eið~Q~G �~rjj�q~GðzþhÞÞ; ð5Þ

where ~Q~G ¼ ~Q0 þ~G;~G is the reciprocal lattice vector of ð~a1;~a2Þ, and

q~G ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

0 � Q2~G

q.

The objective is to determine the coefficients bþ~G ; b�~G ;a

iþmn and ai�

mn

by imposing the continuity of the normal velocity at z = 0 andz = �h and imposing pressure continuity at z = 0 and z = �h. Oncea�mn and b�~G are obtained, the ultrasound power transmission coef-ficient can be calculated from the ultrasound power radiated by aninfinite plate [17,20],

s ¼ PTðxÞP0ðx; h;uÞ

¼X~G

Req~Gq0

� �jb�~G j

2: ð6Þ

3. Numerical results and discussion

The numerical calculations are made considering 2 � 2 and3 � 3 multiple-sublattice holes arrays placed in water. In each2 � 2 and 3 � 3 squared holes arrays considered, the period ofthe unit cell, a, is fixed and the inter-hole distance within the unitcell, p, is varied. The values of p considered are a/2, 10a/24, and a/3in the 2 � 2 squared holes arrays and a/3, 3a/10, and 7a/30 in the3 � 3 squared holes arrays. All the samples considered have a

thickness h = 3 mm and a fixed hole filling fraction 0.25. The periodof the unit cells, a, is 5 mm. Multiple-sublattice hole unit cells con-sidered are showed in Fig. 2. The transmitted ultrasound powercoefficient as a function of frequency, f, in the fluid at normal inci-dence of the 2 � 2 multiple-sublattice hole arrays is calculated sep-arately for samples with the inter-hole distance within the unit cella/2, 10a/24, and a/3 and are depicted in Fig. 3a–c, respectively. Thesample with the inter-hole distance within the unit cell a/2(Fig. 3a) corresponds to a square lattice holes arrays with perioda/2 = 2.5 mm. The full transmission peaks observed correspond toFabry–Perot resonances of the holes cavities and modulated bythe interaction among holes and the minimum transmission dipsat frequencies around 590 kHz and 840 kHz correspond to themanifestation of Wood anomalies. The Wood anomaly for normal

incidence is given by xc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2pm

l Þ2 þ ð2pn

l Þ2

q, where l is the array per-

iod, n,m are called Miller indices and c is the speed of ultrasound inwater. When the inter-hole distance within the unit cell is reducedto 10a/24 (Fig. 3b), new transmission dips appears resulting fromWood anomalies corresponding to the period of the unit cell. TheWood minima associated to the period of the unit cell (296 kHzand 420 kHz) have effect in the appearance of the second ordertransmission peak while the first order Fabry–Perot peak is invari-able. The calculated results for the sample with the inter-hole dis-tance within the unit cell 7a/30 are shown in Fig. 3c. The

Fig. 5. Transmitted ultrasound power coefficient of the 3 � 3 multiple-sublatticeholes arrays with the three different periodicities of the holes within the unit cell:(a) a/3, (b) 3a/10, (c) 7a/30, and (d) sample with one hole within the unit cell withperiod a and hole filling fraction 0.25, where a = 5 mm.

Fig. 6. Structure factor of the 3 � 3 multiple-sublattice holes arrays with the threedifferent inter-hole distances within the unit cell: (a) a/3, (b) 3a/10, (c) 7a/30, and(d) sample with one hole within the unit cell with period a and hole filling fraction0.25, where a = 5 mm.

H. Estrada et al. / Ultrasonics 52 (2012) 412–416 415

prevalence of the first order Fabry–Perot transmission peak is evi-dent, while the amplitude of the second order transmission peak isreduced to a half due the interplay with Wood anomaly minimaassociated with the period of the unit cell, which are more obvious.Fig. 3d shows transmitted ultrasound power coefficient of a samplewith one hole within the unit cell with period a = 5 mm and a holefilling fraction 0.25. The full transmission peak observed corre-spond to the first order Fabry–Perot holes resonances and the min-imum transmission dips correspond to the manifestation of Woodanomalies. The existence of Wood anomalies is related to the geo-metrical structure factor of the lattice. Thus, Fig. 4a–d are obtainedby applying a two-dimensional Fourier transform to the real spacelattice. It can be clearly seen that for p values of 10a/24 (Fig. 4b),a/3 (Fig. 4c), and single hole (Fig. 4d), the structure factor is thesame and only the relative amplitude between the peaks changes.

In the case of the 3 � 3 multiple-sublattice hole arrays samples,the transmitted ultrasound power coefficient as a function of fre-quency, f, at normal incidence is calculated separately for the casewhere the inter-hole distance within the unit cell is a/3, 3a/10 and7a/30 and are depicted in Fig. 5a–c, respectively. Fig. 5a showstransmitted ultrasound power coefficient of the sample with theinter-hole distance within the unit cell a/3, that corresponds to asquare lattice holes arrays with period a/3 = 1.67 mm. Like in the2 � 2 case, the full transmission peaks observed correspond to

Fabry–Perot holes resonances, but a new Fabry–Perot resonancearises due to the addition of extra holes in the unit cell. The mini-mum transmission dips at frequencies around 880 kHz and1260 kHz correspond to the manifestation of Wood anomalies.When the inter-hole distance within the unit cell are reduced to3a/10 (Fig. 5b) and to 7a/30 (Fig. 5c), new transmission dips appearresulting from Wood anomalies corresponding to the period of theunit cell. The effect of Wood minima associated with the period ofthe unit cell increase their effect in the appearance of the secondand third order transmission peaks as distance of the holes withinthe unit cell is reduced while the frequency of the first orderFabry–Perot peak remains invariable. As showed in 2 � 2 caseFig. 5d shows transmitted ultrasound power coefficient of a samplewith one hole within the unit cell with period a = 5 mm and a holefilling fraction 0.25. The geometrical structure factor for the 3 � 3multiple-sublattice hole array samples shows the same behaviourobserved in the 2 � 2 case, as can be seen in Fig. 6a–d. FromFig. 3a–d and Fig. 5a–d it can observed that the transmission spec-trum of the multiple-sublattice holes arrays tends to one with onlyone hole in the unit cell. One remarkable feature shown in Fig. 5c isthat the first order Fabry–Perot resonance peak splits into twopeaks. The transmission dip that gives rise to the splitting of thefirst order Fabry–Perot resonance peak arises from the interferencebetween holes [21]. When the holes have the same area, thecoupling between them is strong and when the phase difference

Fig. 7. Phase differences of the fields between the central and the extreme holes(black) and between the central and the middle holes (grey) on the 3 � 3 multiple-sublattice holes arrays. The distances between the holes within the unit cell are: (a)a/3, (b) 3a/10, and (c) 7a/30, where a = 5 mm.

416 H. Estrada et al. / Ultrasonics 52 (2012) 412–416

between holes in a unit cell approaches p, as shown in Fig. 7a–c,the interference between them lead a destructive interference. Inaddition to the p phase shift linked to the resonance splitting ofthe first Fabry–Perot mode, other phase shift peaks can be ob-served at higher frequencies in Fig. 7b and c. However, these shiftpeaks are highly influenced by the lattice structure factor (seeFig. 6b and c). Thus, as the structure factor for both lattices isroughly the same, the difference observed in the transmissionspectra can be attributed to the inter-hole distance p as it clearlymodifies the inter-hole interaction.

4. Conclusions

Ultrasound transmission through periodically perforated plateswith multiple-sublattice holes arrays has been studied theoreti-cally. Ultrasound transmission spectrums were calculated by usinga model in the rigid-solid limit. The results show that, in both 2 � 2and 3 � 3 square hole arrays, when the inter-hole distance withinthe unit cell is reduced, new transmission dips appear resultingfrom Wood anomalies corresponding to the period of the unit celland the multiple-sublattice inter-hole distance. The Wood minimaassociated with the period of the unit cell have effect in the appear-ance of the second order transmission peak while the first orderFabry–Perot peak is invariable. For the 3 � 3 square hole arraysthe first order Fabry–Perot resonance peak splits into two whenthe phase difference between holes in a unit cell approaches p.

As the inter-hole distance within the unit cell is reduced, the trans-mission spectrum of the multiple-sublattice holes arrays tends toone with only one hole in the unit cell. These results are expectedto have applications in ultrasonic filters.

Acknowledgments

This work has been supported by the Spanish MICINN(MAT2010-16879), Generalitat Valenciana (PROMETEO 2010/043)and Universidad Politecnica de Valencia (PAID-06-10-1839). H.E.acknowledges the support of CSIC-JAEpredoc scholarship.

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