Acoustical routing based on diffraction inhibition in two-dimensional sonic crystal

Routing and guiding acoustic waves without diffraction broadening and backscattering losses is of great interest to the acoustic community. Here, we propose a diffraction-immune acoustical waveguide based on diffraction inhibition in 2D sonic crystals (SCs). Due to the flat equal-frequency contour, the propagating acoustic waves can be highly localized between two neighboring rows of SCs. A few integrated sonic circuit building blocks including arbitrary angle bends and power splitters are further designed and theoretically realized. The proposed SCs open up possibilities for the flexible control of acoustic waves and lead to applications in integrated acoustical devices.

Routing and guiding acoustic waves without diffraction broadening and backscattering losses is of great interest to the acoustic community.Here, we propose a diffraction-immune acoustical waveguide based on diffraction inhibition in 2D sonic crystals (SCs).Due to the flat equal-frequency contour, the propagating acoustic waves can be highly localized between two neighboring rows of SCs.A few integrated sonic circuit building blocks including arbitrary angle bends and power splitters are further designed and theoretically realized.The proposed SCs open up possibilities for the flexible control of acoustic waves and lead to applications in integrated acoustical devices.© 2023 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd A coustic waves naturally broaden in space during diffraction, which strongly limits the development of acoustical function devices.Methods to overcome the often-undesirable consequences of this effect have long been pursued.One approach is to utilize a routed defect in 2D sonic crystals (SCs) based on the total stop-band effect of the SCs. 1) Acoustic waves within the stop-band cannot propagate in the SC along any direction.Thus, a designed line defect becomes the only route for the acoustic waves.][4] Vasseur et al. introduced a supercell plane wave expansion method to calculate the band structure of a line-defect-included SC plate, which also produces a waveguiding effect for acoustic plate waves. 5)icek et al. extended this effect to 3D scenarios to guide acoustic waves in a flexible manner. 6)he sub-diffraction (or self-collimation) effect, namely suppressing the diffraction of light, has been carried out in 2D photonic crystals, 7) which also exist in 2D SCs.Víctor et al. first demonstrated the self-collimation propagation of sound beams in a 2D SC. 8) The self-collimation effect of SCs means that the acoustic wave of certain frequencies will be modulated by the periodic structure of the SCs and propagates along a straight line without diffraction broadening in a specific direction.Due to its great potential in acoustical applications, numerous researches on acoustical self-collimation have been proposed. 9)Li et al. proposed an all-angle and wide-band 90°-bending waveguide for self-collimated sound waves by truncating the 2D SCs, 10) and a systematically controllable beam splitter of self-collimated acoustic waves by introducing line defects into the SC. 11)A series of acoustical functional devices based on self-collimation have been realized.Zhang et al. proposed acoustic logic gates based on the linear interference of self-collimated beams in 2D SCs with line defects. 12)Zhan et al. designed a sonic demultiplexer consisting of multiple tunable Mach-Zehnder interference structures based on the self-collimated beams in 2D SCs. 13)Other interesting acoustical processes, such as one-way transmission, 14) acoustic sensing, 15) acoustic focusing and imaging 16) and acoustical cloaking, 17) together with acoustic functional devices, [18][19][20] have significantly extended the application of self-collimated sound beams.
Another method to guide the sound waves without diffraction broadening is the "topological edge state" due to the Hall effect in 2D SCs. 21)][24][25][26] Zhang et al. proposed a one-way transmission of acoustic waves based on the topological states in a flow-free metamaterial lattice. 27)ombining topological edge states and line defects, Zhou et al. proposed on-chip unidirectional waveguiding for surface waves in an SC with a triangular lattice. 28)ue to inner scattering, defect-based and topological-based waveguiding usually suffer from wave distortion.Selfcollimation is only valid for a rather wide incident beam with the distribution of wave vectors restricted to the local flat portion of the equal frequency contours (EFCs).Other methods such as graded arranged SC lattice 29) and acoustical metamaterials 30) have also been adopted for acoustic wave routing.However, most of these methods require vast periods of lattices or structures to restrict diffraction and power leakage, which severely limits the practical applications.
In this letter, we propose a nearly perfect self-collimation (diffraction inhibition) effect in 2D SCs.Since the group velocity v g is normal to the EFC, the SCs we use here exhibit flat EFCs across the entire Brillouin zone in a broad frequency band.Moreover, a series of fundamental building blocks including arbitrary angle bends and power splitters for flexible routing of the diffraction-inhibited beams are numerically realized.These high-transmission and high-compactness acoustical structures are excellent candidates for acoustical wave manipulation such as acoustic signal processing and engineering ultrasonic computation for underwater communication and clinical diagnosis.
Here, we consider a rectangular lattice 2D SC composed of epoxy resin rods immersed in mercury, as shown by the inset in Fig. 1 As a result, most of the power of the sound waves will be localized in a few lower-order Brillouin zones, giving rise to diffraction inhibition of narrow beams, in particular, a point source.
To verify the analytical prediction, we simulated the sound field distribution of a á b 50 6 SC block, as shown in Fig. 2(a).In the simulation, the point source of the frequency = f c a 0.82 / is located at the left side of the SC block, which is surrounded by a perfect-match layer of width a.It is worth mentioning that the point source contains all the k vectors, where the previously reported self-collimation would no longer be maintained due to the partially flat EFC.However, in Fig. 2(a) it can be seen that the sound waves launched by the point source are localized in the neighboring two rows of rods, yielding a nearly non-diffraction propagation of the sound beam along the x direction.In contrast, we also plot the sound field distribution of the same point source immersed in mercury, as shown in Fig. 2(b), which presents a regular concentric circle wave front with minimal disturbance caused by the perfect-match layers.The energy transmittance for both the above conditions is calculated as the ratio of the sound energy on the right-side boundary and the half radiation power of the point source, as shown in Fig. 2(c).From Fig. 2(c), it can be seen that the sound wave transmittance of the á b 50 6 SC block is above 80% within the frequency band ~c a 0.813 0.875 .m / Meanwhile, the transmittance in free space is only 8% due to diffraction.Thus, diffraction inhibition for broadband sound waves is realized.
As mentioned above, as a result of the diffraction inhibition, the sound beam will be localized between two neighboring rows of scattering rods.To evaluate the efficiency of diffraction suppression, we calculated the transmittance of the SC block with different numbers of lattices rows N on each side of the point source.A point source of frequency c a 0.82 m / is placed on the left side of the SCs, and the input power P i is set as the transmitted power at position 10a away from the point source when N = 5.The transmittance changes with the distances from 10a to 60a of the sound beam propagated are shown in Fig. 3(a).From Fig. 3(a) it can be seen that the transmitted power is decreasing with the distance, which indicates that there is still minimal diffraction during propagation.However, the transmittance is higher than 85% after 60 propagation periods, even reaching 93% for the single row of rods (N = 1) case, as shown by the inset.The transmittance after 40 periods for N = 1 is notably higher than in other cases.We suggest that the power leakage is caused by the coupling effect between the neighboring channels for the > N 1 cases.Thus, it is possible to confine sound waves with only two lattice rows.When N 2,  the transmittance is quite stable, as shown by the four almost overlapped curves for = N 2, 3, 4, 5 in Fig. 3(a).This property makes it possible to guide several closely located  6 SC block.The simulated sound pressure distribution shows that the sound beams are trapped in each excited channel over a long distance with negligible crosstalk.This effect provides a convenient method to construct efficient and compact acoustic functional blocks, such as bends and beam splitters.
Arbitrary angle bends: To build a highly compact integrated acoustical device, sound beam bending is one of the essential blocks.For typical self-collimated beams, some simple and effective bending concepts for 90°bending have been reported.Another way to bend self-collimated sound beams is to shift the band structure of the SCs by introducing fluid that circulates around each scatter rod.Thus, it is difficult to realize arbitrary angle bending in traditional SCs since selfcollimation only occurs for sound beams along one direction, and as a result, the guiding track must follow the lattice orientation of the SC structure.Here, we prove that even two rows of SCs can restrict the diffraction broadening of sound waves launched by a point source.Arbitrary angle bends can be achieved by connecting two SC structures with physical boundaries.Figures 4(a)~4(c) show the simulated sound field distributions for bend angle q =    45 , 90 , 135 , respectively.Both SC structures are rows and located with the desired bend angle.The circular arc profiles are tangent to the lattice rows to restrict reflecting and scattering losses.From Fig. 4, it can be seen that the sound beam travels smoothly at the interfaces in between the SC structure and curve boundaries, and the shape of the fields is a little distorted while propagating along the bends.The transmittance for    45 , 90 , 135 are 90%, 92% and 93%, respectively.Thus, the arbitrary angle bends are realized with ultra-thin SC structures.
Beam splitter: Another important routing mechanism for integrated acoustical applications is power splitting, which allows the division of a sound beam into multiple beams or, conversely, for a combination of individual beams into one beam.In this section, we present two designs for one-to-two and one-to-three splitters for a point source and a wide Gaussian beam, as shown in Figs.5(a) and 5(b), respectively.From Fig. 5(a), it can be seen that the splitter is basically a combination of two 90°waveguide bends, and all the input and output channels are composed of four rows of SC rods.Considering the complex construction of the junction, extra rods are placed at each corner of the splitter.Because of the splitting, the power of the two transmitted beams is equally divided, as shown by the sound field distribution.The total transmittance of the two output terminals is 86% due to the radiation loss caused by the abrupt directional change near the junction.Here, we also study a beam splitter for wide sound source, as shown in Fig. 5(b).The structure is composed of four rows of SC rods in the beginning, then they split into three individual waveguides.Each waveguide is also a two-row ultra-thin structure.A line sound source with a width of b 3 is located in the left port of the structure.The sound distribution shows that most of the sound wave launched by the line source is propagated along the left multichannel, then through the complex splitting region with minimal distortion, and it finally divides into three individual channels.Since the line source is outside the structure, it can be seen that part of the sound wave leaks to free space from the two terminals of the source.Thus, the sound energy of the middle channel is slightly higher than that of the bilateral channels, which can be overcome in a systemized integrated acoustical device.
In summary, we propose an SC structure with flat EFCs across the whole Brillouin zone, which gives rise to a diffraction inhibition effect in a considerably wide frequency band.Due to this property, even sound beams launched by a point source can propagate along one direction without diffraction broadening and evident sound power leakage.Based on this effect, we designed and numerically demonstrated several acoustical routing blocks such as arbitrary angle bends and beam splitters.These blocks do not just   work for a point sound source, but also perform well with a wide line source or several point sources with short intervals.It is worth mentioning that even two rows of the SC structure can constrain a sound beam for diffraction, which makes it possible to build a complex integrated acoustical circuit with ultra-compact SC structures.This diffraction inhibition SC shows great potential for future acoustical applications such as acoustic communication, acoustic signal processing and sensing, especially for underwater conditions.017001-4 © 2023 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd

- 1 1 © 1
(a).The lattice constant is a along the propagation direction x and = b a 2 along the y direction.The radius of the epoxy resin rods is = r a 0.38 , yielding a filling ratio of 0.227.The density and acoustic longitudinal speed are r = for epoxy, and Content from this work may be used under the terms of the Creative Commons Attribution 4.0 license.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.017001-2023 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd Applied Physics Express 17, 017001 (2024) LETTER https://doi.org/10.35848/1882-0786/ad0cd7for mercury, respectively.The first five dispersion surfaces of the SC are numerically calculated using COMSOL Multiphysics software, as shown in Fig. 1(a).It is observed that the fourth surface exhibits a large flat region without overlaps between the lower or higher bands.Thus, sound beams of this frequency band should propagate along the x direction without diffraction broadening.To directly analyze this diffraction inhibition effect, we plot the EFCs of the fourth band structure, as shown in Fig. 1(b).From Fig. 1(b), it can be seen that the EFCs of normalized frequency = f c a 0.813 0.875 m / present almost straight vertical lines across the entire first Brillouin zone.

Fig. 1 .
Fig. 1.Band structure of the SC.(a) First five band surfaces of the proposed SC; the flat region of the fourth band is sliced out by two horizontal faces.Inset shows the lattice parameters of the SC.(b) Corresponding EFC of the fourth band; the flat region of the band surface presents as almost flat lines in the first Brillouin zone.

Fig. 2 . 2 ©
Fig. 2. Broadband diffraction inhibition.(a) and (b) are the sound field distributions of a point sound source placed in the SC structure and free space, respectively.(c) Total transmittance of a point source in two cases.

Fig. 3 .
Fig. 3. (a) Transmittance evolution with the row number N of the SC structure; zoom out shows the overlapped transmission curves for the = N 2, 3, 4, 5 cases.Inset is the sound field distribution of a point source for = N 1.(b) Sound field distribution of three-point sources launched with an interval of one channel.Fig. 4. Sound field distributions for waveguide bends with angles of (a) q =  45 , (b) q =  90 and (c) q =  135 , respectively.

Fig. 4 .
Fig. 3. (a) Transmittance evolution with the row number N of the SC structure; zoom out shows the overlapped transmission curves for the = N 2, 3, 4, 5 cases.Inset is the sound field distribution of a point source for = N 1.(b) Sound field distribution of three-point sources launched with an interval of one channel.Fig. 4. Sound field distributions for waveguide bends with angles of (a) q =  45 , (b) q =  90 and (c) q =  135 , respectively.

Fig. 5 .
Fig. 5. Sound field distributions of sound beam splitters.(a) One-to-two beam splitter for a point source.(b) One-to-three beam splitter for a wide line source.