A Manufacturability‐Driven Method for Designing Metamaterial Focusing Acoustical Lenses

Herein, a method for designing and fabricating acoustic metamaterial focusing lenses for imaging applications such as nondestructive testing is proposed. First, a refractive index distribution is designed using the time delay method for a target focal length. Second, unit cell structures that can achieve the required refractive indices are found using the effective property retrieval method. At this point, factors such as fabrication cost and manufacturing limitations are considered. Finally, the pressure field from the designed lens is calculated using finite element (FE) modelling and iterated until the required focusing performance is achieved. To increase manufacturability, the unit cells necessarily become larger than required for the homogeneous regime. As an example, a lens of focal length 200 ± 10 mm at a frequency of 40 kHz is designed using cross‐shaped unit cells and fabricated using three‐dimensional printing. The design methodology leads to 2.4 mm (wavelength/3.6) unit cells which operate near the cut‐off frequency of the first mode. The lens is experimentally tested, achieving a focal length of 208 mm and a focal width of 12 mm (1.4 × wavelength), and showing good agreement with an FE model. The focusing ability of the lens is demonstrated by measuring the size of various holes in thin plates.


Introduction
Metamaterials are artificial components composed of subwavelength units that possess novel and useful material properties.[3][4] These unconventional material properties can be achieved by changing the geometry of metamaterial units.[8][9][10][11][12][13][14] AMMs have received significant attention due to their advantages of compact size and lightweight structure.Several novel geometries of metamaterial unit cells have been explored to control wave propagation.For example, Ge et al. and Xie et al. have achieved asymmetric acoustic transmission using triangular cavities and coded metasurfaces. [15,16]Zhao et al. and Lan et al.  have converted a cylindrical wave to a plane wave using AMMs. [17,18]Li et al. demonstrated nearly total transmission and beam steering performance from 60°to 80°using asymmetric unit cells. [19][24][25][26][27] Among these, the acoustic focusing lens has attracted particular interest due to its potential uses in industrial and biomedical imaging. [28]For example, Zhu et al. [29] proposed a reflective multifrequency acoustic focusing lens with a comb structure; Hladky-Hennion et al. [30] presented a foam-like metallic metamaterial lens with negative refraction to achieve a focused cylindrical beam; Song et al. [31] proposed the use of a fractal geometry to generate a flat acoustic lens for broadband (2-5 kHz) acoustic focusing; and Zigoneanu et al. [32] and Ruan et al. [33] developed cross-structured two-dimensional (2D) and three-dimensional (3D) metamaterial lenses for focusing acoustic waves in both air and underwater environments.The works mentioned above provide good descriptions of the available lens design methods, simulation results, and experimental validation.However, there is limited information available regarding the design parameter choices and methods to improve the manufacturability of the metamaterial lens.
In this article, we propose an AMM lens design method that combines the time delay method and the effective property retrieval method (EPRM), with the consideration of factors such as fabrication limitations, compactness, minimum transmission efficiency, and design tolerance.The motivation of this article is to propose a design method that utilizes low-cost 3D printing manufacturing techniques to fabricate functional acoustical lenses based on relatively large unit cells that are not necessarily in the homogeneous regime.

Methodology
The classic design process of an AMM lens is shown in Figure 1a.As shown, according to the proposed design DOI: 10.1002/adem.202301832Herein, a method for designing and fabricating acoustic metamaterial focusing lenses for imaging applications such as nondestructive testing is proposed.First, a refractive index distribution is designed using the time delay method for a target focal length.Second, unit cell structures that can achieve the required refractive indices are found using the effective property retrieval method.At this point, factors such as fabrication cost and manufacturing limitations are considered.Finally, the pressure field from the designed lens is calculated using finite element (FE) modelling and iterated until the required focusing performance is achieved.To increase manufacturability, the unit cells necessarily become larger than required for the homogeneous regime.As an example, a lens of focal length 200 AE 10 mm at a frequency of 40 kHz is designed using cross-shaped unit cells and fabricated using three-dimensional printing.The design methodology leads to 2.4 mm (wavelength/3.6) unit cells which operate near the cut-off frequency of the first mode.The lens is experimentally tested, achieving a focal length of 208 mm and a focal width of 12 mm (1.4 Â wavelength), and showing good agreement with an FE model.The focusing ability of the lens is demonstrated by measuring the size of various holes in thin plates.
requirements, a simple step-by-step design method is utilized to design an AMM lens structure.Numerical simulations and experimental measurements can then be performed to understand the performance of the resulting design.
As a lens structure design method example, Figure 1b illustrates the simple time delay method [34,35] often used for determining refractive index distribution along a lens, of thickness, l, to focus an incident plane wave on a target point at (x f , 0) in a medium with a sound speed of c 0 .The lens can be discretized into many small unit cells, and the speed of sound in the jth unit cell is c j .The refractive index of the jth unit cell should satisfy: It should be noted that n j ≥ 1 if c 0 ≥ c j .Using this method, the lens needs to be discretized into numerous unit cells, and the typical requirement is that these are each significantly less than the wavelength in size. [32,33]In this so-called homogenized regime, the unit cells have acoustical behavior resembling that of homogeneous materials.However, fabricating such small unit cells often poses significant challenges in terms of cost of manufacture and the effect of inevitable manufacturing tolerances.The motivation of this article is to propose a design method that utilizes low-cost 3D printing manufacturing techniques to fabricate functional acoustical lenses based on relatively large unit cells that are not necessarily in the homogeneous regime.
Unit cell structures such as cross-shaped structures, labyrinthine structures, [36,37] Helmholtz resonators, [38][39][40] and V-shape structures [41] have been investigated widely in recent years.In this article, cross-shaped unit cells in air are chosen as an example although the methods described are general and can be used for other unit cells.To demonstrate this generality, an additional example with a circle-shaped unit cell is shown in Section 2 of Supporting Information.The lens design method is then described, and used to design an example lens that meets the target requirements.Finally, experimental validation is performed on a fabricated lens.

EPRM
Metamaterials consist of various structures and materials and are an example of an inhomogeneous medium.The common approach to the analysis of acoustic metamaterials is to equate them to homogeneous media with classical characteristics such as refractive index and acoustic impedance, using the EPRM. [3,42,43]This allows for the estimation of effective properties based on the overall behavior of the unit cells, enabling the characterization and design of metamaterial lenses with desired acoustic properties.
In a case of an incident plane wave onto an infinitely repeating unit cell, the reflection and transmission coefficients (R & T ) of acoustic energy can be found using finite element (FE) modeling.Based on this, the refractive index n of the unit cell and the impedance ratio ξ between it and the background media (with a wave speed of c 0 and an acoustic impedance of z 0 ) can be expressed as [42] : where c unit and z unit are the wave speed and the acoustical impedance in the unit cell, respectively, k is the wavenumber, m is the branch number of the cos À1 function, and s is the lattice constant or unit cell size.

Band Structure Theory (BST)
We also compute the refractive indices of the unit cells from their dispersion curves.This enables the extraction of the band structure of the metamaterials and reveals key features such as the location of bandgaps and regions of negative velocity.Various methods exist for calculating the band structure, including, the finite-difference time-domain method, the plane wave expansion method, the multiple scattering method, [44] and FE modeling. [45,46]In this article we use FE modeling, and the unit cells are modeled with periodic boundary conditions in the frequency domain.By employing Bloch theory, [45,46] the relationship between wave vector and frequency is determined and treated as the unit cell's band structure in the first irreducible Brillouin Zone.

Performance of Cross-Shaped Unit Cells
Figure 2a shows the physical structure of a single cross-shaped unit cell, which has an overall unit size of s and a solid cross at its center with a half-height of a and thickness of b.The geometry ratio between a and s is defined as r = a/s.By changing these structure parameters it is possible to find suitable unit cells and to construct a lens to meet design requirements.
Figure 2b shows an FE model structure of a cross-shaped unit cell in air used for calculating reflection and transmission coefficients.For more information on the model process, please refer to Section 5.
Using the EPRM, Figure 3a shows that the variations of refractive indices for different cross-shaped unit cells exhibit similar trends, in which the refractive index initially increases slowly with frequency, reaches a peak value, and finally drops.For a specified cross-shaped unit cell, the frequency at which the maximum refractive index is seen corresponds to the cut-off frequency, f cut-off .The range of variation in refractive index is primarily determined by the geometry factor, r, while the cut-off frequency is determined predominantly by the unit cell size, s.
For a specified operating frequency of 40 kHz and a unit size of s = 2.4 mm (wavelength/3.6) and b = 0.083s, Figure 3b shows that the refractive index monotonically increases with a.
This indicates that the larger the solid cross, the slower the resulting wave speed becomes.Figure 3c shows that the maximum refractive index, n max , increases with r and that it rises steeply as r tends to 0.5.From a design perspective, it is preferable to employ a gentle variation of r as this means that the unit cells' properties are less sensitive to small manufacturing errors.Hence, in practice, manufacturing process limitations restrict the maximum achievable n max .Figure 3d indicates that a higher cut-off frequency can be achieved by utilizing unit cells with small s and small r.This understanding will serve as a guideline for the fabrication of the lens in subsequent sections.
Excellent agreement was observed in all cases between the refractive index obtained from both BST and EPRM, as shown in Figure 3a-d.The BST also confirms the location of the cutoff frequencies and bandgaps and is described further in Section S1, Supporting Information.

Metamaterial Lens Design Method for Cross-Shaped Unit Cells
Figure 4 shows the amended design flow chart that now includes manufacturing limitations.In this design process, manufacturing restrictions, an operating frequency, f, a desired lens width, D, and a focal length, x f , as defined in Figure 1b, are defined as essential input design requirements.For stereolithography (SLA), a typical limitation is that the size of any gaps in the printed component should exceed some threshold dimension, g.This limitation then defines the minimum separation distance between any parts within the chosen unit cells, and it is defined for our cross shape as, In the design process, the flowchart shown in Figure 4 runs from left to right and top to bottom, indicating the sequential order of steps and actions to be followed.The detailed design procedure is shown as: 1) From diagrams, as shown in Figure 3c, determine the values of n max , r max , and manufacturable minimum unit size, s min , using Equation (4).A key aspect here is to use knowledge of the manufacturing limitations, to define r max .2) Determine the maximum value of unit size, s max , using Figure 3d, r max and the desired operation frequency, f. 3) Determine the lens thickness, l, by gradually increasing the number of unit cell layers, i, until all refractive indices in the designed refractive index distribution are lower than n max .It should be noted that the proposed refractive index distributions are calculated using the time delay method and input parameters, including a desired lens width, D, and a focal length, x f , and this iterative process ensures that the lens meets the desired refractive index criteria.4) Finalizing lens structure details.The acoustical wave field of the created lens structure from step 3 is first predicted using FE modeling.If the focal length and transmission performance of the lens do not meet expectations, there are two optional processes that can be employed to improve lens performance.One option is to reduce the unit size, s, which can increase transmission efficiency.Another option is to iteratively adjust the target focal length used in the time delay method to generate various refractive index distributions.This iterative process continues until a refractive index distribution is obtained that meets the design criteria within the specified tolerance.Note that, in the second option, the initial target focal length used in the time delay method is the same as the desired focal length defined in the lens design requirements.Subsequently, in each iteration, the target focal length reduces or increases a half of the difference between the achieved and proposed focal length.
It should be noted that the overall design process is generally applicable and so can accommodate different unit cells, lens design methodologies, or design criteria.For example, the data for r max , s max , and refractive index distribution can be replaced with those for any unit cell structure.

Designed Lens
In an example, we set target requirements as: an operation frequency of f = 40 kHz, a lens width of D = 150 mm, a focusing length of x f = 200 AE 10 mm, a manufacturing restriction of g min = 0.4 mm (provided by Proto Labs Ltd, UK) and working in air with a wave speed of c 0 = 343 m s À1 and a wavelength of λ 0 = 8.6 mm.Following steps 1-4 in the design procedure described in Section 3.2.2, the parameters of a designed lens are determined sequentially as follows: s min = 2 mm; n max = 1.78; r max = 0.4; and s max = 2.4 mm. Figure 5 compares the refractive index distributions obtained from step 3 in the lens design procedure.As shown in Figure 5, the final design satisfying the n max design criteria consists of nine layers and a lens thickness of l = 21.6 mm.
Using the FE model depicted in Figure 6a and a 1 Pa input plane wave, Figure 7c shows the calculated acoustical wave pressure field for the lens with the red-dotted refractive index distribution shown in Figure 5.As shown in Figure 7a, the achieved focal length is 240 mm, and the maximum pressure in the focal zone is 1.70 Pa.As this achieved focal length does not meet the requirement, the two additional options within step 4 of the design procedure are implemented.
For option one, the unit cell size was reduced to 1.2 mm while the refractive index distribution was kept unchanged.The resulting acoustical pressure field is depicted in Figure 7b, where the For option two, the input focal length was iterated to achieve the target.Figure 7c shows the results acoustical pressure field after one iterative process with a target focal length of 200 À (240 À 200)/2 = 180 mm used in the time delay method.The achieved focal length is 193 mm and the maximum pressure is 1.77 Pa.The performance of this lens meets the requirements.

Experiment Results
Using this design methodology, the lens specified in Table 1 was manufactured and is shown in Figure 6c.Experiments were  conducted on the fabricated lens to evaluate its performance.The acoustical pressure field generated by the lens was measured to assess its focusing capability and the quality of the focal zone.
Additionally, the transmitted pressure of the focused beam passing through various defects was measured to demonstrate potential imaging applications in nondestructive testing (NDT).Figure 9a-d shows the experimentally measured pressure of the focused beam passing through various defects.In these measurements, a thin plate with various holes was used as a test sample to simulate different types of defects in thin specimens for an NDT application.The plate, as shown in Figure 6b, was positioned at the focal plane of the generated acoustical field, and a microphone was placed on the opposite side of the plate.It is expected that the size of a hole can be estimated from the measured pressure amplitude by using the 6 dB drop method (also known as full-width at half-maximum, FWHM).
The solid bottom line depicted in Figure 9a serves as a reference for defining the noise level during the experiments.Figure 9b shows the measured pressure as a function of the hole sizes Φ and microphone positions.The peak amplitudes from measurements of various holes were extracted, as shown in Figure 9c, where the measured pressure converges when the hole size exceeds 10 mm.This convergence is due to that a hole size is approximately equal to or larger than the width of the resulting focal zone, measured as W zone = 12 mm from Figure 8d, and most of the wave energy can reach the microphone.Figure 9d shows that all the sizing errors are within 3 mm (0.35 Â wavelength) of the correct value.

Conclusion
In summary, we proposed, validated, and demonstrated a design procedure for a manufacturable AMM focusing lens, which consists of the time delay method and the EPRM and considers the requirements including manufacturing restriction, an operating frequency, a desired lens width and focal length.The time delay method is used to determine refractive index distribution along a lens, while the EPRM is used to extract refractive indices of unit cells.The metamaterial design procedure was explored in an imaging example in which a 40 kHz focusing device with cross-shaped unit cells was manufactured.The fabricated lens demonstrates exceptional focusing performance experimentally, achieving a focal length of 208 mm and a focal spot width of 12 mm (1.4 Â wavelength), and a potential application in the NDT field.Using this proposed method, low-cost 3D printing manufacturing techniques can be used to fabricate functional acoustical lenses based on relatively large unit cells that are not necessarily in the homogeneous regime.This opens the possibility of the rapid manufacture of low-cost acoustic metamaterials.

Experimental Section
Numerical Simulations: In the simulation based on the EPRM, the reflection and transmission coefficients (R & T ) of cross-shaped unit cells were obtained from the model, as shown in Figure 2b.The simulation was performed using the FE method in the frequency domain pressure acoustic module (COMSOL Multiphysics 5.6, Sweden).In this model, the top and bottom boundaries are defined as periodic boundaries.A plane wave radiation condition is applied to the left and right boundaries to prevent boundary reflections.The realization of an incident plane wave is achieved by adding a background acoustical pressure.The pressure amplitudes of nodes on the two dashed lines were recoded.The measured R is the ratio between the mean of the amplitudes of the scattered wave from the nodes on the left dashed line and the amplitude of the incident wave at the same location (referred to as the reference amplitude).The measured T is the ratio between the mean of the amplitudes of the transmitted wave from the nodes on the right dashed line and the reference amplitude.In the simulation, the mass density and sound speed of the background medium are ρ 0 = 1.29 kg m À3 and c 0 = 343 m s À1 , respectively.The background pressure is a 1 Pa amplitude plane wave.Based on the measured R & T, Equation ( 2) is finally used to calculate the refractive indices.
In the simulation based on the BST, the Floquet periodic boundaries are applied to all unit cell boundaries, as depicted in Figure 2a.The parametric sweep of the wavevector is used to acquire the band structure of a unit cell.
In the lens design process, FE models, depicted in Figure 6a, are employed to predict the resulting acoustical field from the AMM lens.These models have surrounding perfectly matched layers (PMLs) around the lens structures to prevent reflections from these boundaries.The hard boundary condition is used on all interfaces between air and solid cross structures.In the simulation, a 40 kHz sinusoidal pressure of 1 Pa was applied to the top boundary of the lens.
Fabrication of AMM Lens: The acoustic metamaterial lens was fabricated using SLA technique developed by Proto Labs Ltd, UK.The chosen material is acrylonitrile butadiene styrene, which has a density of 1180 kg m À3 and a Young's Modulus of 3300 AE 400 MPa.The manufacturing tolerances are AE0.05 mm in the x and y directions, with an additional tolerance of AE0.001 mm mm À1 , and AE0.13 mm in the z direction with an additional tolerance of AE0.001 mm mm À1 .The minimum requirements for slots and gaps are 0.4 mm, and the minimum feature size is 0.13 mm.
Note that the simulations were performed using commercial FE software (COMSOL Multiphysics 5.6, Sweden), and the same parameters were then used to fabricate an example metamaterial lens.
Experimental Measurements:  In an example of a NDT application, as shown in Figure 6b, thin plates with different holes were used as inspection targets and placed at the focal plane of the lens.In the experimental measurements, the microphone was positioned behind the plate, allowing it to capture the acoustic field generated by the interaction between the incident wave and the plate.The plate moved along x axis.

Figure 1 .
Figure 1.a) Flowchart illustrating the classical design process of AMM lens and b) schematic diagram illustrating the time delay method for determining refractive index distribution along a lens.

Figure 2 .
Figure 2. a) The schematic of a cross-shaped unit cell and b) finite element model structure used to predict the transmission and reflection coefficients of a cross-shaped unit cell.

Figure 3 .
Figure 3.The comparison of the predicted performance of the modeled cross-shaped unit cells calculated using the EPRM and the BST results for: a) the refractive index, n, as a function of operating frequency for various combinations of s and r; b) n as a function of a for the unit cells with s = 2.4 mm and operating at 40 kHz; c) the maximum refractive index, n max , as a function of r for any unit cells (independent on unit-cell size s); and d) the cut-off frequency, f cut-off , as a function of s for various r.Note that b = 0.083s in all modeled unit cells.

Figure 4 .
Figure 4. Flowchart illustrating the design process of AMM lens including manufacturability considerations tailored for the utilization of cross-shaped unit cells.

Figure 5 .
Figure 5. Intermediate refractive index distribution of the lens with different layers i = 3 and i = 5 (black and red solid lines), the threshold n max = 1.78 (blue dashed line), and the final refractive index for unit cells at different positions of the proposed lens (red dots).

Figure 6 .
Figure 6.a) Schematic diagram of the FE model of the designed lens with boundary conditions.PMLs are used around the simulation domain, and the red solid line on the top of the lens indicates the input plane wave; b) schematic diagram of the experimental setup for measuring the acoustical transmission through a specimen; and c) the manufactured metamaterial lens with its support frame.

Figure 7 .
Figure 7. Predicted acoustic pressure field (for a 1 Pa plane wave input) using an FE model for the lens constructed using the cross-shaped unit cell with s=: a) 2.4 mm; b) 1.2 mm; and c) 2.4 mm.The method of reducing unit cell size was used in (b), while the approach of adjusting the target focal length used in the time delay method was applied in (c).Note that, in each figure, the red dot denotes the desired focal length of 200 mm while the red cross does the achieved focal length.

Table 1 .Figure
Figure 8a-d compares the simulated and experimentally measured acoustical pressure generated from an incident plane wave passing through the fabricated lens.It shows a good agreement and an achieved focal length of 208 mm and hence validates the proposed method for acoustical lens design.Using an amplitude threshold of À6 dB to define the focal zone size in Figure 8c,d, the length and width of the focal zone could be evaluated, which are L zone = 140 mm and W zone = 12 mm respectively.Figure9a-d shows the experimentally measured pressure of the focused beam passing through various defects.In these measurements, a thin plate with various holes was used as a test sample to simulate different types of defects in thin specimens for an NDT application.The plate, as shown in Figure6b, was positioned at the focal plane of the generated acoustical field, and a microphone was placed on the opposite side of the plate.It is expected that the size of a hole can be estimated from the measured pressure amplitude by using the 6 dB drop method (also known as full-width at half-maximum, FWHM).The solid bottom line depicted in Figure9aserves as a reference for defining the noise level during the experiments.Figure9bshows the measured pressure as a function of the hole sizes Φ and microphone positions.The peak amplitudes from measurements of various holes were extracted, as shown in Figure9c, where the measured pressure converges when the hole size exceeds 10 mm.This convergence is due to that a hole

Figure 8 .
Figure 8.The normalized pressure field generated from an incident plane wave passing through the fabricated lens obtained from: a) FE simulation and b) experimental measurements.The comparison of the simulated and experimentally measured acoustical pressure shown in (a,b), specifically at: c) x = 0 mm and d) y = 208 mm.
Figure 6b illustrates the schematic diagram of the experiment setup, where 15 ultrasonic speakers (each has a diameter of 10 mm, a center frequency of 40 kHz, model MA40S4S, fabricated by Murata Manufacturing Co., Ltd, Japan) are attached to the fabricated lens and are used as a source, and a microphone (a diameter of 0.125 inch, model 4138-A-015, fabricated by Brüel & Kjaer, Denmark) acts as a receiver and is installed on a three-axis scanning stage.The speakers are connected in parallel to the output channel of an integrated digital oscilloscope and signal-generator device (Handyscope HS5, TiePie Engineering, Netherlands), the input channel of which is connected to the output channel of the microphone.This digital device is controlled by a PC and generates a sinusoidal signal with a frequency of 40 kHz and a peak-to-peak amplitude of 20 V.During the experimental measurements, the signals generated by the digital signal-generator were used to excite the speakers, and the received signals from the microphone were recorded.To postprocess the recorded signals, a digital Gaussian filter and averaging techniques were applied to enhance the signal quality and reduce noise.

Figure 9 .
Figure 9. Experimental results from potential NDT applications for: a) the comparison of measured pressure when there is and is not a plate in the front of the microphone (red and black solid lines); b) the comparison of measured pressure from plates with holes of various sizes; c) the maximum pressure in (b) as a function of hole size; and d) the measured hole diameters against the actual ones.