Foam gratings as an alternative to customized acoustic lenses

: This article describes a method of manipulating acoustic ﬁelds using transmission through foam gratings. The approach is investigated with an analytical model, a numerical model simulating full wave ultrasound propagation through the gratings, and experimental measurements. A grating is demonstrated that mimics a conventional ultrasound lens, modulating the phase of transmitted ultrasound while maximizing the transmitted amplitude. The performance of a foam grating is compared to a lens made of polydimethylsiloxane or three-dimensional printed resin. Using two gratings, independent control of amplitude and phase is demonstrated, with increased insertion loss. The primary advantages of this technique over conventional lenses are very rapid manufacture ( < 30 min), high repeatability due to the simplicity of manufacture, and the ability to control the amplitude of the transmitted ultrasound. Potential applications include generation of complex ultrasound ﬁelds for patient speciﬁc treatments in ultrasound therapy. V C 2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) . https://doi.org/10.1121/10.0016755


I. INTRODUCTION
Tissue inhomogeneity can affect the accuracy with which ultrasound can be focused in the body.This is particularly problematic in applications in which high intensity ultrasound is used, such as therapeutic ablation.][4] In existing clinical systems, any phase corrections are typically made using ultrasound arrays, but there has been recent interest in using customized acoustic lenses to make patient specific phase corrections. 5,6Acoustic lenses offer a more compact and affordable system than an equivalent array, particularly when phase corrections require a high spatial resolution and therefore a large number of elements and associated driving electronics.
An acoustic lens consists of a material with a different sound speed from the propagating medium, enabling a change in the phase of the transmitted wave.An ideal material for an acoustic lens would be easy to form into complex shapes, have low acoustic attenuation, and have a sound speed significantly different from that of water but a similar characteristic acoustic impedance to that of water to minimize reflection at the lens interface.
Recent studies investigating customized acoustic lenses using three-dimensional (3D) printing technology have used silicone, 5,7 polylactic acid (PLA), 6,8 or polyjet resins. 8,9ilicone has a sound speed of 1030 m/s and an impedance of 1.06 megarayls. 10The transmission of a silicone lens is excellent due to the similar impedance to water and attenuation of just 0.7 dB/cm at 500 kHz. 10 Construction requires casting, which slows production and may be less repeatable unless mixing and degassing are tightly controlled.PLA has a sound speed of approximately 1818 m/s and acoustic impedance of approximately 2.05 megarayls. 8PLA is normally printed using fused deposition modeling (FDM) that will likely leave voids, 11 which could lead to poor acoustic properties.Jim enez-Gamb ın et al. made lenses from PLA and did not note any transmission problems caused by voids, but they did change the lens manufacturing process in later experiments.Polyjet printers can print with a large number of polymers 12 with high accuracy.Similar to FDM, microscopic voids may be present, 13 but both Melde et al. 9 and Jim enez-Gamb ın et al. 8 achieved good results with this technology.Polyjet printers are relatively fast compared to other 3D printing technologies, but more expensive, and time is also required to finish lenses, e.g., to remove support structures. 14n this study, a grating consisting of a perforated foam sheet is proposed as an alternative to acoustic lenses, in particular, for biomedical applications.Gratings can be produced very rapidly and repeatably and may be an appropriate alternative to lenses in some applications.The method may also be of interest for other applications, such as for non-destructive testing or underwater imaging.The approach is similar to that used in lenses constructed from convoluted waveguides 15,16 and to some metamaterials utilizing periodic structures to control the index of refraction. 17aux et al. made optical measurements in a very thin conducting sheet, obtaining results with some similarity. 18In the proposed method, the sound propagates through foam waveguides, which have a much lower impedance than the water in which the sound is propagating.This makes it a) Electronic mail: eleanor.stride@eng.ox.ac.uk possible to modify the propagation constant inside each waveguide by varying its diameter rather than controlling the length.The behavior of the foam structure is modelled using both analytical and numerical methods, and the results are used to design a structure to modulate transmitted phase in a manner similar to a conventional lens.Using more complex geometries for the foam structure, a grating capable of controlling both phase and amplitude is demonstrated.

A. Grating structure
This paper investigates the feasibility of manipulating sound transmission using arrays of holes cut into an acoustically soft foam sheet and suspended in a water bath.The sound propagates in the water filling the holes, and it is assumed that there is negligible transmission into the foam itself due to the large impedance difference.Holes of varying diameter were cut into ethylene-vinyl acetate (EVA) foam sheet (Zunchuan Maoyi Ltd., Shanghai, China) using a diode laser cutter (Laser Master 2, 15 W, Ortur, Changping, Guangdong, China).A 3.4 mm centre to centre distance is used for all gratings in this paper, as this was experimentally found to maximize transmitted amplitude with 500 kHz ultrasound.Holes are placed on a hexagonal grid to maximize the number of holes at a given centre to centre spacing.Figure 3 shows an example of the hexagonal grating layout, including sections of various hole sizes.

B. Numerical model
Ultrasound propagation through the gratings was modelled using k-wave, a pseudospectral method designed for weakly heterogeneous media. 19The software was used to model a repeating cell of the grating structure, which was approximately 3.4 mm Â 5.9 mm, with the third dimension dependent on the thickness of the foam being used (Fig. 1).The cell was modelled using repeating boundary conditions in the directions planar with the foam sheet (x and y in Fig. 1), but a perfectly matched layer (PML) was used above and below the foam sheet.Simulations used a Courant-Friedrichs-Lewy (CFL) number of 0.1 and a points per wavelength (ppw) of 35.For 500 kHz ultrasound, this meant a time step of 5.5 ns and a spatial grid step size of 86 lm.The sound was simulated for 60 ls, which took approximately 60 s on a desktop personal computer (PC).
The ultrasound source was modelled as an additive sound source on a uniform plane placed k/8 ¼ 0.375 mm from the foam grating, from which a pulse was emitted.The pulse had a duration of 15 ls and was band limited to 500 kHz.Water was modelled as a fluid with a sound speed of 1500 m/s and density of 1000 kg/m 3 .The foam grating structure was modelled as a fluid with sound speed 1500 m/s and density 1.2 kg/m 3 .The actual material has a density of 100 kg/m 3 , but as there was found to be negligible through transmission of ultrasound at 500 kHz, the sound speed is not known.The simulation values were therefore chosen to model a very low impedance material into which sound would not be readily transmitted, but with a sufficiently high sound speed to provide numerical stability.
The transmitted ultrasound field was recorded on a plane also 0.375 mm past the foam grating.Transmission coefficients were determined by taking the spatial average of the recorded ultrasound field and comparing the Fourier transform of the transmitted field to the Fourier transform of the source field.

C. Analytical model
It is hypothesized that the foam holes will act as waveguides, despite their short length (1-10 mm) relative to their diameter (1.4-3.06 mm).Expressions may be obtained for ultrasound propagation inside a narrow tube, assuming the foam acts as a free surface. 18,20,21 symmetric acoustic field inside a waveguide can be described by a velocity potential /ðr; z; tÞ, from which the particle velocity is uðr; z; tÞ ¼ r/ and the pressure is p r; z; t ð Þ¼ Àq 0 ð@/=@tÞ. 21Time harmonic waves propagating in the z direction can be described by Eq. ( 1), If the field is bounded at r ¼ 0, the only solution that is allowable in the hole is a zero order Bessel function, given by Eq. ( 2), The free surface at r ¼ a enforces a boundary condition / ¼ 0, in which case the term a ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi , where a 0i is the ith zero of J 0 ðxÞ.The lowest order mode is when a 01 ¼ 2:4048, in which case Propagation only occurs for frequencies high enough that the argument in the square root is positive, or alternatively for a given frequency f , there is a critical size that the hole must exceed for sound to propagate.For the 500 kHz ultrasound used in this study, the critical radius is 1.15 mm, By using waveguides similar in size to this critical radius, the ability of waves to propagate changes dramatically.Thus, in a waveguide of fixed length, the amplitude of the transmitted wave can be controlled.In addition, when the hole size is above the critical radius, the real part of the propagation constant k z is dependent on the hole diameter.This means that the phase of the wave as it is transmitted to the end of the waveguide can also be changed.

D. Experiment
Equations ( 1)-( 4) predict that for cylindrical waveguides in foam, the phase velocity inside the waveguide is a function of the waveguide radius, rather than just the material properties of the fluid inside it.This indicates that it should be possible to manipulate sound transmission using an array of suitable waveguides of varying radius.This hypothesis is tested using the following methods.A 19 mm diameter planar cylindrical transducer (NDT V318, Olympus IMS, Waltham, MA) was used to insonate 60 mm Â 60 mm gratings placed 500 mm from the transducer (see Fig. 2) in a custom made acrylic water tank (90 cm Â 56 cm Â 50 cm).This setup ensures that the sound incident on the gratings is approximately plane, with a phase deviation of 136 and amplitude deviation of 30% in the field incident on the 60 mm Â 60 mm gratings.Each grating or lens has the same layout, with the 60 mm Â 60 mm area containing a grid of 9 or 16 sample sections (Fig. 3).Each section contains a different hole size or, in the case of polydimethylsiloxane (PDMS) or resin lenses, a different material thickness.Figure 4 shows a photograph of one of these gratings.
The transducer was driven at 500 kHz with ten cycle sinusoidal pulses of 20 V amplitude by a signal generator (33250 A, Keysight, Santa Rosa, CA) and radio frequency (RF) amplifier (325LA, Electronics & Innovation, Ltd., Rochester, NY) every 20 ms.This led to a peak negative pressure of 3.9 kPa at the location where the gratings were placed.The water in the tank was filtered and at room temperature.Degassing was not deemed necessary due to the very low sound pressures used.The transmitted field was measured over a plane 10 mm past the grating with a 400 lm needle hydrophone (HNA-0400, Onda Corp., Sunnyvale, CA).Measurements were taken on a 54 Â 54 grid at locations 1.3 mm apart (spatial extent 68.9 mm Â 68.9 mm), and waveforms were saved using a digital oscilloscope (WaveRunner 64Xi-A, LeCroy, Chestnut Ridge, NY).
On each day of experiments, a scan was made of the sound field with a 60 mm Â 60 mm aperture cut into a piece of foam, in place of a lens or grating.The aperture was replaced by the lens or grating to be measured, and the resulting sound field was measured again.

E. Data processing
The scans were processed using MATLAB (R2020A, The Mathworks, Natick, MA).The complex value for the signal at every point at 500 kHz was extracted by using a 16 384 (2 14 ) point zero padded fast Fourier transform (FFT).The angular spectrum method (ASM) was used to estimate the complex pressure at the surface of the grating from the measurements made 10 mm past the grating.As part of the backpropagation method, a spatial filter was used that reduces the visibility of grating lobes.Figure 5 shows an example of the measured data after being backpropagated to the location of the foam grating.The backpropagated data were then spatially averaged over The process was repeated for the daily control scan with no lens or grating present.The ratio of the complex pressures with a grating present compared to no grating gives the pressure transmission coefficient for all 16 sections.The measurement and data processing methods were intended to correct for diffraction after transmission through the grating, any changes to the sound source or water from day to day, and the heterogeneity of the incident sound field.
As the spacing between holes is greater than half the wavelength, grating lobes are present in the signal.Six grating lobes occur at approximately 62 from normal incidence, and each contains 3%-6% of the transmitted acoustic power.

A. Effect of foam thickness and hole size
Using the setup described in Sec.II D, transmission was measured through three foam gratings having thicknesses of 1.5, 3, and 5 mm, with the gratings containing nine sections with hole radii evenly spaced from 0.9 to 1.5 mm.Numerical simulations were performed for the same conditions.
The phase and amplitude of the transmitted ultrasound from both the experiments and simulations are shown in Fig. 6.Both Cartesian and polar plots are shown.The polar plot indicates the phase and amplitude change that can be achieved by transmission through the gratings but does not as readily show which hole radius leads to what effect.
All grating thicknesses were able to vary the amplitude of the transmitted ultrasound up to at least 75%.The thicker gratings had a slightly higher threshold radius at which transmission increased and a sharper transition to a maximum value.All three thicknesses exhibited a maximum transmission of around 80% when holes were larger than the critical radius.
All gratings caused a phase change in the transmitted sound.For small holes, each foam piece caused a phase lead, which reduced as the size of the holes increased.The size of the phase lead and the rate of change of phase with respect to hole radius were larger for thicker foam pieces.
The results of the k-wave numerical model are also shown and were in generally good agreement with the experimental results.The numerical model accurately predicted the phase in almost all cases, with significant deviations from the measurements only when the transmission amplitude was <20%.The numerical model also captured the main features of the amplitude changes but consistently overpredicted its magnitude.In one of the simulations (5 mm foam), a transmitted amplitude of 107% is reported, which appeared to be due to a small numerical inaccuracy in capturing the air-water interface and requires further investigation.The discrepancy is, however, within the calibration accuracy of the hydrophones used in the experiments ($10%), and the simulation parameters yielded acceptable agreement with the analytical result for the case of sound propagation around a small spherical void in water.
The phase results from the experiment are duplicated in Fig. 7, with the addition of the predictions from the analytical waveguide model.The analytical model also showed showing the approximately uniform sound field.On the right is the sound field when a 5 mm foam grating with holes from 0.9 to 1.5 mm is used.The green lines and markers show the centres of the square sections of different hole sizes, where the pressure for transmission through each section is sampled.
close agreement with the experimental measurements, and in most cases, the predictions were more accurate than those of the numerical model.This suggests the proposed mechanism for the phase change produced by the gratings is correct.
According to the waveguide model, when the holes are smaller than a critical radius, transmission only occurs through evanescent waves.This predicts that there is a large phase lead, which should be directly proportional to the thickness of the foam.According to Eq. ( 3), as the hole size increases, the propagation constant for waves in the waveguides becomes a small real value and eventually approaches the propagation constant in water.This predicts that the phase lead caused by the holes will reduce to zero as the hole size increases.

B. Optimizing for phase change
A grating was designed to maximize phase change and transmission amplitude.To achieve this, thicker foam was used to maximize phase change, and the range of hole sizes was restricted to above the critical radius (a crit ) to maximize transmission.
The grating was constructed from two identical pieces of 5 mm foam stacked together.Similar to the previous experiment, the grating was sectioned into 16 squares, each containing a different hole size spaced equally from 1.07 to 1.53 mm.It was necessary to construct this grating from two sheets because the laser cutter was not capable of accurately cutting through foam thicker than 5 mm.
As a comparison, two lenses were produced from PDMS (10:1, Sylgard 184, Dow, Midland, MI) and from a photopolymer resin (Visijet M2R-TN resin, using an MJP 2500 Plus printer, 3D Systems, Rock Hill, SC).A computeraided design (CAD) model of the resin lens is shown in Fig. 8, and the PDMS lens used a similar geometry.The 60 mm Â 60 mm area is divided into 16 sections in the same way for the lenses and for the foam gratings.The thickness of the lenses was chosen to give a phase change of approximately 360 when the lenses were used in water.The results from these lenses are shown in Fig. 9, alongside the results from the foam grating optimized for phase change.On the polar plot of transmission and phase, the data points for the lenses made from these two materials spiral slightly inward as their thickness is increased, and attenuation reduces the transmitted amplitude.They spiral in different directions as the sound speed of PDMS (1030 m/s 10 ) is less than that of water, and the sound speed of the resin (2500 m/s) is greater than that of water.Both lenses were able to generate any required phase change with more than 75% transmission (Fig. 9).The foam grating was able to generate phase changes of greater than 180 with 50% transmission, but transmission was reduced for larger phase changes.

C. Composite grating
A grating designed to modulate both amplitude and phase was constructed by placing a 1.5 mm foam grating on top of a 5 mm foam grating, such that the holes lined up and the foam pieces were in contact.Figure 10 shows a cross section of this design.Based on the predictions of the numerical model and earlier experiments, it was expected that thicker gratings with hole sizes from 1.09 to 1.53 mm would cause a large phase change with limited change in amplitude, while thinner gratings with smaller hole sizes of 0.77-1.47mm could alter the transmitted amplitude with a lesser effect on phase.Figure 11 shows the results for the composite grating.The structure demonstrates the feasibility of generating a range of phase changes and transmission amplitudes.The transmission amplitude is as high as 68%, and for transmission amplitude greater than 40%, this system was able to vary phase over 150 .

IV. DISCUSSION
Grating structures have been created that exploit the dispersive nature of waveguides to control the magnitude and phase of transmitted acoustic waves.Figures 6, 7, and 9 show that the foam gratings employed here can produce phase changes of more than 180 while maintaining a transmission amplitude of more than 50%.There is good agreement between experimental results and numerical predictions for the amplitude of the transmitted ultrasound and good agreement for the phase when the transmission amplitude is greater than 10%.The numerical simulations consistently underestimated the attenuation.Major sources of error are likely to have been the use of a non-physical density and sound speed to model the foam, inaccurate laser cutting of the foam material, and wetting of the surface of  The analytical model of the grating as a set of waveguides accurately predicted the phase and the approximate frequency of the gratings high pass filter effect.Sources of errors are similar to those for the numerical model, with the most likely problems being geometry differences and the assumption in the analytical model that ultrasound perfectly reflects off the foam.The agreement between the analytical model and the results suggests that the holes acting as waveguides is the mechanism for the phase change, but a more complex model including diffraction into and out of the holes may be necessary to describe the amplitude reduction.
Figure 9 shows that by careful selection of foam thickness and hole size, it is possible to modify the phase of transmitted ultrasound with a limited reduction in amplitude.In this example, hole sizes of 1.38-1.53mm in a 10 mm foam sheet caused a change in phase of >180 while keeping the pressure transmission more than 50%.The phase change was increased by using thicker foam and, hence, increasing the length of the waveguides.It may be possible to get a wide range of phase changes with smaller losses by further increasing material thickness and using larger holes, by optimizing the shape to maximize sound entering into holes, or by using a foam with a lower density and with less wetting.
Figure 11 shows a proof of concept in which a thin foam grating is placed in front of a thicker foam grating.The thinner foam uses smaller hole sizes (0.77-1.47 mm) and primarily controls the amplitude of transmitted sound, and the thicker foam uses larger holes (1.09-1.53mm) and primarily controls the phase.Independent control of transmitted amplitude and phase is demonstrated, but transmission does not exceed 60%, which may limit applications in its current form.The difference between experiment and simulations is increased for the two layer grating, which may be due to a gap between sheets in the experiment.The techniques mentioned in the previous paragraph could increase the transmitted amplitude, as could improving the interface between the thicker and thinner gratings.One way to improve the interface would be to manufacture the composite grating from a single piece of foam, but this would require a different manufacturing method.
This study has investigated the behavior of these gratings with low pressure sound waves (3.9 kPa) in water, at almost normal incidence.The maximum allowable pressure for these gratings should be investigated, as it is likely that high pressures would cause cavitation around the foam or possibly drive air out of the foam.Other topics for further investigation would be the bandwidth of the gratings and how well they behave when sound is incident away from normal.Coupling into the waveguides is likely to be compromised at angled incidence, and grating lobes might become more prominent.
Increasing the maximum transmission for these gratings may make them practical for a wider range of applications.Losses might be reduced by experimenting with different foams, such that a more ideal pressure release interface is presented to sound traveling through the waveguide.Other materials may also allow larger holes to be made for a given spacing between holes, which should increase maximum transmission amplitude and maximum phase change.Efficiency might also be gained by changing the geometry at the entrance and exit to the holes to allow more of the incident sound to diffract into the waveguide.
Another topic for future investigation is how the gratings and an ultrasound source in their current position behave when long pulses are used, of sufficient duration to reflect back to the ultrasound source.The behavior of the waveguides could also be investigated when the grating is physically attached to an ultrasound transducer.This will likely affect how the sound is able to diffract into the holes.In both these cases, the reflected sound from the grating may change the behavior of the transducer, making accurate measurements more difficult, which is why this study was limited to the case of a grating in a free field, using short tone bursts.
In its current form, this technique may be suitable for some biomedical ultrasound techniques, e.g., for correcting phase aberration due to bone or fat.The transmission magnitude reported here (<75%) will make the technique more suitable for low pressures, such as those used for neuromodulation, 22 or moderate pressures, such as those for blood-brain barrier disruption, 23 where a large focal gain is possible.An inefficient lens or grating in a high power application could lead to excessive heating in the lens or grating or the requirement for an excessively powerful transducer and amplifier.The gratings are sensitive to frequency (see the Appendix) but this would not cause a problem for most applications in therapeutic ultrasound.
If the maximum transmission of the gratings can be increased, then there may be a wider range of uses.For biomedical applications, one might imagine a disposable grating placed on a large high intensity focused ultrasound (HIFU) transducer for patient specific corrections for HIFU treatments in the abdomen 24 or the skull. 25Applications such as hyperthermia driven drug delivery 26 may also become practical with a higher maximum transmission.There could also be further uses for customized acoustic lenses or gratings in other fields, such as underwater imaging or non-destructive testing, although the limitations in terms of the bandwidth performance of these gratings would need to be investigated further.

FIG. 1 .
FIG. 1. (Color online) The k-wave simulation domain, showing the location of the foam sheet, source plane, measurement location, and PML.

FIG. 2 .
FIG. 2. (Color online)The scanning setup.A small source was placed approximately 500 mm from the grating to generate waves that were approximately uniform in phase and amplitude.A scan was made 10 mm behind the grating with a needle hydrophone.FIG. 3. A top down view of a typical grating.Circular holes (white) are on a hexagonal grid with 3.4 mm centre to centre spacing.The grating has been split into 16 sections containing different hole sizes, with smallest holes in the top right and largest in the bottom left.

FIG. 4 .
FIG. 4. (Color online) Photo of a typical grating.Foam measures 80 mm Â 80 mm, with mounting holes around the edge and the grating in the centre.

FIG. 6 .
FIG. 6. (Color online) Simulated and measured sound transmission through gratings made from foam of 1.5, 3, and 5 mm thickness.(a) and (b) show the variation in amplitude and phase of transmitted sound as the thickness increases.Phase is referenced to the arrival phase with no lens, and positive values indicate a phase lag.(c) shows the same data on a polar plot.

FIG. 8 .
FIG. 8. CAD model of the lens made on the polyjet printer.A 60 mm Â 60 mm square was divided into 16 sections, each section containing a different material thickness.The material thicknesses were spaced in 16 equal steps from 2 to 10 mm.

FIG. 10 .
FIG. 10. (Color online) Cross section of holes (waveguides) for composite grating.A 5 mm grating is stacked on top of a 1.5 mm foam grating.Each hole is described by a radius for the 5 mm grating and the 1.5 mm grating.

FIG. 12
FIG. 12. (Color online) Additional data for transmission through the 10 mm phase change lens at various frequencies.(a) Transmission magnitude; (b) transmission phase;(c) the same data using a polar plot.For holes larger than 1.5 mm, the frequency dependence of the lens is small, with a 100 kHz frequency shift causing a drop in transmission of approximately 25% and a phase change of approximately 180 .For holes smaller than 1.5 mm, the changes in phase and amplitude with frequency are greater.