This paper presents a method to generate two- and three-dimensional periodic or quasi-periodic acoustic lattices in air using polygonal active diffraction gratings. The radiated field depends on design parameters and is well predictable in terms of a superposition of oblique plane waves, with very good agreement with simulations and experiments. Our device represents a simple and efficient approach for producing acoustic lattices with attractive advantages, such as planar single-sided geometry, simple electronics, axial steering, and broadband operation. The design is scalable and compatible with other scientific applications, such as microfluidic platforms.

In the last decade, structured acoustic fields have received increasing attention due to their numerous emerging applications, such as haptics,^{1} material manufacturing,^{2} communications^{3} and, especially, for particle manipulation,^{4–6} to name but a few.

There are different strategies for structuring an acoustic field. One of the simplest approaches is to use a single transducer and a reflector to generate standing waves along one spatial dimension (1D)^{7} or multiple faced transducers to produce 2D standing waves with orientation control.^{8} Another way is to use phased arrays of transducers operated at a fixed frequency (typically 40 kHz), which can tailor the acoustic field by controlling the phase of each transducer independently.^{9} By setting the appropriate phase of each transducer, holographic acoustic elements can produce structured acoustic fields, such as vortex beams, self-bending beams, and bottle beams in air.^{9,10} Another strategy based on phononic crystals and metasurfaces,^{11} which are artificial periodic structures of appropriate unit cells, can also engineer the acoustic wavefronts, such as ultrasonic vortices,^{12} Airy beams,^{13} and reconfigurable tightly focusing in air.^{14,15} In a similar manner, 3D-printed acoustic holograms generate arbitrary static pressure fields in air or water, limited by diffraction and the printer resolution.^{16}

Other diffractive methods are also important to produce structured pressure fields in a certain region, like acoustic Bessel beams generated with circular concentric gratings,^{17} high-order Bessel beams with spiral gratings,^{18} focused acoustic vortex beams with a spiral Fresnel lens,^{19} and recently, a subwavelength acoustic field for trapping microspheres with a quasi-periodic plate.^{20} In contrast to passive elements, which modify the incident field, Active Diffraction Gratings (ADGs) can generate, by direct emission, acoustic Bessel, and vortex beams with many advantages, such as high efficiency, broadband operating frequency, and dynamical axial steering of the field.^{21,22}

Among the different structured fields, acoustic lattices are particularly interesting for acoustic tweezers and acoustofluidics applications. These are formed by standing waves in one, two, or three spatial dimensions, which behave as periodic potentials for multi-particle trapping. From a theoretical viewpoint, Guevara Vasquez and Mauck^{23} studied the different acoustic Bravais lattices that could be formed by the trapping of particles smaller than a wavelength in inviscid liquids using a superposition of four plane waves. In fact, the same group showed that the theory based on interference of plane waves presents good agreement with experiments in a given region.^{24}

Experimentally, standing waves are commonly generated by several sound emitters in appropriate orientations and alignments to produce acoustic lattices in air^{25,26} or in liquid-filled reservoirs. In particular, these liquid chambers have been used in many biological studies. For instance, in a research on Alzheimer's disease,^{27} Cai *et al.* employed four 1 MHz faced transducers to generate lasting periodic acoustic patterns, forming 3D neurospheroids and inflammatory microenvironments. The same group used a hexagonal configuration to rotate, transport, and fuse organoids to model processes of the human brain.^{28}

On the other hand, quasi-periodic acoustic lattices are also relevant structured fields for quasicrystal applications that exhibit interesting physical properties. For example, in 1993, de Espinosa *et al.*^{29} used five transducers at a fixed frequency (5.33 MHz) in water to produce and verify, by diffraction, the generation of acoustic quasi-periodic patterns. A more recent work used an octagonal water reservoir comprised of eight transducers, operated at 1 MHz, to trap nano-spheres in a quasi-periodic pattern.^{24}

In this work, we propose and demonstrate an alternative approach for the generation of well-predictable 2D- and 3D-periodic and quasi-periodic acoustic lattices in air, whose axial position can be controlled by varying the operating frequency. Our device is a Polygonal Active Diffraction Grating (PADG), which is a single planar transducer based on a ferroelectret film with a broad frequency range of operation in air, from about 30 to 300 kHz. The appropriate design for the experiments is determined by optimizing the most relevant parameters, chosen from a theoretical analysis.

^{30,31}Consequently, considering a harmonic time dependence $ e \u2212 i 2 \pi f t$, the complex amplitude of the radiated pressure field of a PADG can be properly described using the Rayleigh diffraction integral:

^{32}

*ρ*

_{0},

*f*, and

*k*, respectively. An arbitrary point in space has the position vector $ R = ( x , y , z )$, while a point on the radiating surface of the

*L*-sided polygonal grating has the position vector $ r s = ( x s , y s )$.

*A*=

*1 if $ r 0 \u2264 x l \u2264 r f$ and $ \u2212 tan \u2009 ( \theta / 2 ) x l \u2264 y l \u2264 tan \u2009 ( \theta / 2 ) x l$, and*

*A*=

*0 otherwise. The inner and outer apothem of the polygonal grating (see Fig. 1) are*

*r*

_{0}and $ r f = r 0 + ( N \u2212 1 ) a + \Delta r$, where

*N*is the total number of closed tracks or grating grooves of width $ \Delta r$, and

*a*is the pitch or grating period. Thus, the total radiated pressure field of any PADG, setting the propagation direction along the

*z*-axis, can be found by substituting Eq. (2) in Eq. (1),

*n*accounts for the diffraction orders, $ C n = \Delta r a sin \u2009 c ( n \Delta r a )$, with $ sin \u2009 c ( x ) = sin \u2009 ( \pi x ) / \pi x$,

*c*

_{0}is the sound speed in air.

Notice that Eq. (5) represents a superposition of plane waves with wave vectors $ k l n = ( k \u22a5 n \u2009 cos \u2009 \varphi l , k \u22a5 n \u2009 sin \u2009 \varphi l , k z n )$, where $ k \u22a5 n = k \u2009 sin \u2009 \beta n$ and $ k z n = k ( 1 \u2212 k \u22a5 n 2 / 2 k 2 )$. The latter expression corresponds to the paraxial approximation of $ k z n = ( k 2 \u2212 k \u22a5 n 2 ) 1 / 2 = k \u2009 cos \u2009 \beta n$.

*n*th diffraction order of each PADG side with respect to the

*z*-axis is geometrically related to $ sin \u2009 \beta n = n \lambda / a = n c 0 / ( a f )$. Where the inward diffraction orders $ | n |$ of the

*L*sides interfere, an acoustic lattice is formed, with transverse period related to the pitch

*a*, via $ k \u22a5 n = 2 \pi n / a$. Notice that $ k \u22a5 n$ does not depend on the operating frequency, which means that the transverse pattern is preserved when the frequency is varied. The transverse region of existence is limited by $ x i = \u2212 ( r f \u2212 r 0 ) / 2$ and $ x f = \u2212 x i$, whereas the limits of the axial region [see Fig. 1(a)] are

^{21}

The *n*th diffraction order appears only if $ f \u2265 f n = n c 0 / a$, which defines the cutoff frequencies. Higher diffraction orders have the same geometrical pattern and transverse existence region ( $ 2 x i$), but with a scaled transverse period $ \Lambda \u22a5 n = \Lambda \u22a5 1 / n$ and a reduced axial existence region [ $ \Delta z ( n ) = z f ( n ) \u2212 z i ( n )$]. Since the diffraction angle with respect to the *z*-axis (*β _{n}*) increases with

*n*, higher orders arise closer to the transducer's surface.

^{21}However, a given diffraction order

*m*can be annihilated regardless of the operating frequency by choosing $ \Delta r / a = 1 / m$, since the coefficient

*C*in Eq. (3) vanishes for

_{n}*n*=

*m*.

^{21}This is also true for higher orders

*n*=

*Mm*, for any integer

*M*.

As an example, Fig. 1(a) presents a numerical result of the pressure field amplitude emitted by two linear gratings of pitch *a *=* *4 mm, whose inner edges are separated by a distance $ 2 r 0 = 100$ mm, each one having *N *=* *20 tracks of width $ \Delta r = 2$ mm, operated at $ f = 130$ kHz. The diffraction orders $ n = \u2212 1 , 0 , 1$ can be clearly identified for each grating. We recognize that there are three interference zones, corresponding to $ [ n 1 , n 2 ] =$ $ [ 0 , \u2212 1 ] , \u2009 [ 1 , \u2212 1 ]$, and $ [ 1 , 0 ]$, being $ n i = 1 , 2$ the diffraction order of the *i*th grating, but we are interested only in the symmetrical lattices formed by the inward orders, $ [ n 1 , n 2 ] = [ 1 , \u2212 1 ]$. There are no higher orders appearing because $ f 1 < f < f 2$, with $ f 1 = 114$ and $ f 2 = 228$ kHz the cutoff frequencies. Yet, even if $ f > f 2$, the second diffraction order would be absent due to the design condition $ \Delta r / a = 1 / 2$, setting annihilation for *n *=* *2. A closeup of the superposition of the inward diffraction orders $ | n | = 1$ illustrating the phase of the field is presented in Fig. 1(b), which corresponds to the limited existence region of volume $ V e ( n )$, where the wavefronts are approximately plane according to Eq. (5).

The geometry of a PADG has a relationship with the shape of the generated lattice. A triangular, square, and hexagonal PADGs, such as those shown in the top row of Figs. 1(c), 1(d), and 1(f), for instance, can be used to generate 2D-periodic lattices. While the top row of Fig. 1(e), depicts the generation of a quasi-periodic lattice using a pentagonal PADG, which is also possible for polygons with $ L \u2265 7$. The limit when $ L \u2192 \u221e$ yields a circular grating, as shown in the top row of Fig. 1(g), which produces a Bessel beam.^{17} These transverse patterns of the pressure amplitude remain approximately invariant from $ z i ( n )$ to $ z f ( n )$, while the periodicity of the lattice has a direct relationship only with the pitch for a given diffraction order.

*p*given by Eq. (5). For example, for a square PADG, taking only the first diffraction order,

_{g}*α*

_{0}the amplitude ratio of the plane wave to the lattice. By looking at the total pressure amplitude, $ | p | = ( | p c | 2 + | p g | 2 + I c g ) 1 / 2$, the interference term with axial periodicity is revealed,

*p*field, we include a filled central polygon in the PADG with an apothem

_{c}*r*[see Fig. 2(a)], whose design enables a central emission that behaves as a plane wave in a certain region.

_{g}The experimental generation of a 2D periodic acoustic lattice by a PADG is demonstrated with the design and fabrication of a square PADG with $ r 0 = 46$ mm, *N *=* *9, *a *=* *6 mm, and $ \Delta r = a / 2$, as shown in Figs. 2(a) and 2(b). The latter relationship ensures the annihilation of the second diffraction order. For the 3D-periodic pattern generation, the prototype has a filled central square with *r _{g}* = 43 mm. The design parameters are determined through an optimization process involving a comparison between the analytic [Eq. (7)] and simulated fields [Eq. (3)], by varying the pairs of control parameters (

*N*,

*r*

_{0}) and (

*α*

_{0},

*r*). The relevance of these parameters is connected with the overlapping region of

_{g}*p*and

_{g}*p*and their relative amplitudes.

_{c}The fabrication process of the ferroelectret-based-PADG is similar to previous works.^{21,22} The experimental setup is illustrated in Fig. 2(c), which consists of a PADG excited with a 200 Vpp chirped signal (frequency range: 90–250 kHz) driving a signal generator (Tektronix, Model AFG3022C), connected to a power amplifier (Falco System, Model WMA300). A 2 mm-diameter pinhole is attached to the calibrated microphone (1/8″ B&K, Type 4138) to enhance the spatial resolution of the pressure distribution measurements. A signal conditioning as well as an additional amplifier are connected between the microphone and the oscilloscope. The oscilloscope acquires the instantaneous pressure over a transverse grid of dimensions 20 × 20 mm^{2}, with grid spacing $ d x = d y = 0.4$ mm, and an axial grid of dimensions (*x* = 20 mm) × (*z* = 100 mm), with *dx *=* *0.4 mm and *dz *=* *2 mm. From a Fourier analysis of the measured instantaneous pressure, we can attain the spectral response of the grating. In particular, our following results correspond to *f *=* *141 kHz.

The fabricated square PADG produces a structured pressure field with a transverse distribution corresponding to a 2D-periodic acoustic lattice. The analytical, numerical, and experimental results for the *x–y* plane are depicted in Fig. 3(a), whereas the results for the *x–z* plane are depicted in Fig. 3(b). Comparisons among analytic, numerical, and experimental results for normalized profiles of the pressure amplitude along the *x*-axis and the line *x *=* y* are shown in Figs. 3(c) and 3(d), respectively. The agreement is clearly better in the central region, since the plane wave analytical approach neglects diffraction effects due to the finite size of the transducers, which are responsible of the field distortion in the outer region showed in the numerical and experimental results. Notice the proportionality between the grating pitch (*a *=* *6 mm) and the spatial period of the lattice along the *x* and *y* directions.

A simultaneous activation of the outer grating and the central filled square enables the generation of a 3D-periodic acoustic lattice, as illustrated in Fig. 4. In contrast with the approximately propagation-invariant field produced in the 2D case, now the obtained field has a periodic structure along the propagation axis, while preserving a square lattice pattern in the transverse pressure amplitude [see Fig. 4(a)].

However, by comparing the transverse planes of the 2D- and 3D-periodic fields, Figs. 3(a) and 4(b) respectively, some changes are apparent. Namely, even when the normalized profiles along the *x* direction illustrated in Figs. 3(c) and 4(e) are very similar, showing a period equal to the grating pitch, if we look at the diagonal profiles, it is seen that alternate maxima of Fig. 3(d), either disappear (analytically) or lower in Fig. 4(f). Again, this discrepancy is due to the difference between the ideal field calculated using the plane wave approximation and the diffraction-affected numerical and experimental results obtained with finite-size sources, which prevents total destructive interference.

On the other hand, the *x–z* plane depicted in Fig. 4(c), representing the amplitude of the acoustic pressure field, shows the axial periodicity of the distribution, whereas the (*x *=* y*)–*z* plane depicted in Fig. 4(d) reveals also an interlacing of the longitudinal pressure maxima. This means that the positions of the maxima and minima in a transverse pattern taken at an axial maximum (e.g., $ z = 140$ mm) are interchanged with respect to those of a transverse pattern taken at an axial minimum (e.g., *z *=* *154 mm). In the 3D case, the field is also properly approximated within a certain region using the analytical approach (solid line), as shown by its good agreement with numerical (dash) and experimental (markers) profiles of the normalized pressure amplitude along the *z*-axis, shown in Fig. 4(g). Yet, notice that the numerical and experimental results have a slight axial shift downward with respect to the analytical field, as can be appreciated with the horizontal dashed lines in Figs. 4(c) and 4(d). All the discrepancies are likely due to the diffraction effects discussed before.

Finally, Fig. 5 unveils two relevant advantages of our device: an axial position control of the 3D pattern by varying the operating frequency and a fine control of the position of the nodes and antinodes by varying the relative phase shift $ \phi 0$ between the grating and the central filled polygon [see Eq. (8)]. In particular, each curve in Fig. 5 shows the results of an axial tracking of a given antinode in the *x–z* plane of the pressure field amplitude as a function of *f*, either for the analytically (dashed line) or for the numerically calculated field (markers), for the indicated value of $ \phi 0$. The fact that the whole lattice shifts upward as *f* increases, while its existence volume lengthens, can be seen from the expressions for $ z i ( f )$ and $ z f ( f )$ in Eq. (6), which are valid for both the 2D- and the 3D-periodic lattices. Noteworthy, there is no change in the transverse pattern while the lattice is axially steered.

Now, for a given *f*, the existence region keeps fixed, but the position of the nodes and antinodes can be moved by varying $ \phi 0$. For instance, a change in the relative phase in the range $ \u2212 \pi \u2264 \phi 0 \u2264 \pi $ enables fine displacements of a specific transverse pattern within an entire axial period Λ_{z}, as seen in Fig. 5.

To summarize, PADGs are single planar transducers with basic electronics and broadband frequency operation, capable of producing 2D- and 3D-periodic and quasi-periodic acoustic lattices in air. The generated field has good agreement between an analytic approach based on oblique plane waves, a numerical result based on the Rayleigh diffraction integral, and the experiment. The spatial structure and the extension of the lattice are respectively related to the number of sides *L* of the polygon and the parameters *r*_{0} and *r _{f}*, whereas the transverse period of the pattern is proportional to the grating pitch. The axial position of the whole acoustic lattice can be steered by varying the operating frequency. In the 3D periodic lattice, the position of the nodes and antinodes can also be controlled by introducing a relative phase shift between the central polygon and the outer grating. Because of these characteristics, PADGs represent a very appealing alternative for multiple particle manipulation in air and other recent applications, such as topological studies of the acoustic field.

^{33}Furthermore, these kinds of gratings are scalable, which makes them suitable also for microfluidic platforms.

## SUPPLEMENTARY MATERIAL

See the supplementary material for the numerical result of the axial steering of the 3D lattice by varying the operating frequency and the axial displacement of nodes and antinodes by means of the relative phase.

This work was partially funded by DGAPA-UNAM (Grant No. IN113422). D.P.-A. acknowledges CONACYT-Mexico for the Ph.D. grant. R.D.M.-H. and J.E., respectively, acknowledge the support from the Colciencias Scholarship program (No. 727) and the Universidad del Valle intramural research project (No. CI21182).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Darby Paez Amaya:** Formal analysis (lead); Investigation (equal); Methodology (equal); Software (lead); Validation (equal); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). **Ruben Dario Muelas Hurtado:** Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (equal). **Joao Luis Ealo:** Funding acquisition (equal); Methodology (equal); Resources (equal); Validation (equal); Writing – review & editing (equal). **Karen Volke-Sepulveda:** Conceptualization (lead); Funding acquisition (equal); Methodology (equal); Resources (equal); Software (supporting); Supervision (lead); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).