Two-Scale Sparse Spiral Array Design for 3D Ultrasound Imaging in Air

Sparse array designs are a promising approach to improve the beam pattern and imaging quality, especially for applications, where hardware resources are severely limited. In particular, spiral sunflower arrays become increasingly popular due to their excellent point-spread-function (PSF) characteristics and their simple, deterministic and scalable design. Therefore, several sunflower modifications for further improvement have been investigated, e.g. density tapering based on window functions adapted from apodization techniques. In this article, we introduce a two-scale spiral array design concept, which exploits the specific PSF structure of the sunflower geometry, instead of relying on window functions. The modification proposed combines two nested sunflower sub-arrays featuring two different spatial element densities such that the locations of their respective main, side and grating lobe zones differ, resulting in a balanced and improved composite one-way PSF in terms of main lobe width (MLW) and maximum side lobe level (MSLL) under far-field and narrow-band conditions. First, we provide an analysis of the unmodified classic sunflower geometry, describe its PSF zones and show how their locations in the PSF can be estimated based on the array design parameters, which finally leads to the two-scale concept. Second, we examine a specific well-matching combination of nested sub-arrays to discuss the advantages and limitations of the resulting PSF. Third, we benchmark the respective optimum arrays of the classic sunflower and density tapering strategies with the two-scale method, where the latter shows an improved performance of the one-way PSF in terms of MLW and MSLL. Fourth, the two-scale design strategy is validated using a real-world 64-element prototype for narrow-band ultrasound imaging in air. We conduct two experiments to analyze the resulting PSF and angular resolution. Overall, the results demonstrate that the proposed flexible four-parameter concept is particularly valuable for high frame rate imaging as well as for transmit-only and receive-only applications.


I. INTRODUCTION
O NE of the major challenges of ultrasonic 3D imaging systems, as used for non-destructive testing, environmental perception and medical diagnostics, is to maintain image contrast while ensuring a high angular resolution and fast volume rates. Bringing these competing requirements together is particularly difficult for emerging application fields, where mobility is crucial and hardware resources are severely limited. These advancing technologies range from sonar systems, which provide a complementary perception sensor modality for upcoming mobile autonomous robotics [1], [2], [3], up to portable medical point-of-care scanners, which enable life-saving diagnoses outside the hospital, e.g. directly at the scene of an accident or for medics in combat [4], [5], [6].
In both application examples, the 3D images are generated using beamforming, so that contrast and angular resolution are typically quantified based on the maximum secondary lobe level (MSLL) and the main lobe width (MLW) obtained from the point spread function (PSF). The PSF itself is dependent on the beamforming technique and the array geometry employed, both of which provide a respective computational and physical starting point for improvement, where the latter is the focus of this article.
In order to realize and enhance volumetric high-frame-rate imaging on hardware-limited devices, a promising approach is to combine low-complexity conventional beamforming with sparse array designs. Unlike traditional fully-populated arrays, the aperiodic element positioning of sparse arrays prevents severe MSLL degradation due to high grating lobes even if the inter-element spacings exceed half wavelength (λ/2) [7]. This way, large-aperture 2D arrays can be created for improving the MLW without requiring a drastic increase in the number of elements, system complexity and cost.
Sparse array synthesis is grouped into two categories, i.e. stochastic and deterministic designs [8], [9]. Stochastic sparse arrays rely on a randomized element positioning paired with optimization schemes, e.g. genetic algorithms [10], [11], [12], [13] or simulated annealing [14], [15], [16], [17], in order to optimize for pre-defined quality metrics. In contrast, deterministic methods provide a parametric approach, that is easily controllable and scalable, allowing a rapid and flexible customization to satisfy application constraints [18], [19], [20], [21]. Additionally, the resulting geometry can be used as a seed for a further optimization procedure as it already features decent characteristics from the start.
In [56], the authors presented a spiral sunflower design modification based on two-way line-by-line beamforming, where the transmit and receive array geometries differ. By excluding specific adjacent spiral arms in the transmit and receive array pairs, the positions of the secondary lobe maxima and minima in the respective PSFs can be manipulated such that they cancel out. Although this method can effectively reduce the overall side lobe level, relying on twoway line-by-line beamforming is impractical for volumetric imaging applications that require a minimum number of firing events to achieve high frame rates. Therefore, we focus on concepts for improving the one-way PSF characteristics, which are also valuable for transmit-or receive-only applications apart from imaging.
A method for improving the general one-way PSF by modifying the spiral sunflower array geometry, referred to as density tapering, has been proposed for antenna arrays in [25]. Density tapering adapts the idea of weighting the element sensitivities based on a spatial window function to reduce the side lobe level, also known as apodization [57]. However, instead of weighting the sensitivities themselves, the spatial density of the element distribution is altered depending on the window function. This way, the MLW and MSLL can be fine-tuned without sacrificing overall sensitivity. In [48], density tapered spiral arrays have been examined for ultrasound imaging including a comparison of multiple window functions. Additionally, the authors have created two 256-element CMUT and PZT array prototypes [58], [59], [60] based on a Blackman window taper and experimentally evaluated various medical ultrasound imaging applications [49], [50], [51], [52], [53]. In [61], Sarradj investigated another density tapering approach based on a parametric window function and a non-linear least squares method for element positioning, enabling a flexible one-parameter control of the taper.
In this article, we introduce a two-scale sparse spiral array design concept, which exploits the specific PSF structure of sunflower arrays, instead of relying on window functions for density tapering. The modification proposed combines two nested sunflower sub-arrays featuring two different spatial element densities such that the locations of their respective main lobe, side lobes and grating lobes, referred to as PSF zones, differ and combine favorably. As a result, the composite array geometry has a balanced and improved one-way narrow-band PSF in the far-field in terms of MSLL and MLW compared to the previous approaches.
The main contributions are grouped into four categories. First, we provide an analysis of the PSF characteristics of the unmodified classic sunflower array for different aperture diameters and number of elements in order to introduce a concept for estimating the PSF zone locations requiring only the basic array design parameters. Second, we elaborate the two-scale array design and extend the PSF zone estimation for its sub-arrays. In addition, we investigate a specific wellmatching sub-array combination for highlighting its advantages and limitations. Third, we benchmark the respective 64-element and 256-element arrays of the classic sunflower and density tapering strategies with the two-scale method and examine multiple optimum sub-array combinations in more detail. Fourth, the two-scale design strategy is validated using a real-world prototype for ultrasound imaging in air, which consists of 64 MEMS microphones and one piezoelectric ultrasonic transducer. Based on this prototype, we conduct two experiments to analyze the resulting PSF and angular resolution.
The remainder of this article is organized as follows. Section II includes the model for generating the beam patterns and PSF, as well as the analysis of the classic sunflower geometry and its PSF zone estimation. Based on this, the two-scale array design is introduced along with the evaluation of a well-matching sub-array combination. Section III covers the benchmarking and the optimum two-scale sub-array combinations. Section IV includes the real-world prototype and experiments for validation. Finally, we conclude and discuss further ideas for improvement, as well as remaining questions in Section V.

II. METHODS
Here, we describe the classic sunflower array geometry and analyze the resulting PSF behavior in terms of MLW and MSLL for variable aperture sizes and number of elements. In addition, we introduce the different PSF zones observed and methods to estimate their locations which finally lead to the two-scale array design concept.

A. BEAM PATTERN AND POINT SPREAD FUNCTION MODEL
The key factors for the following evaluations are the far-field beam pattern and one-way PSF, i.e. the beam pattern with a single centered point source. The model used for their generation is based on the normalized and discretized well-known Rayleigh integral, where we assume point elements and point sources, as well as far-field and narrow-band conditions. Therefore, the model allows an application-independent analysis focused on the array element positions, as the effects of a particular element size, focal distance, or specific bandwidth are not included. The superimposed magnitude p for the beam pattern scanning direction (u, v) = (sin(θ) cos(ϕ), sin(ϕ)) is given by where θ and ϕ are the azimuth and elevation angles, L is the number of point sources, l is the corresponding source index, (u l , v l ) and s l are the l-th source direction and magnitude, respectively, and a(u, v) ∈ C M ×1 is the array-specific steering vector, whose m-th entry is given by Here, M is the number of elements and (x m , y m ) is the position of the m-th array element. The term p el (u l , v l ) includes the directivity of the elements themselves and is negligible if point elements are assumed, i.e. p el = 1. Otherwise, if elements with extended sizes are considered, p el can be calculated by sampling the element aperture, as explained in [40], that is where K is the number of sample points, k the sample point index and (x k , y k ) is the sample point position. The PSF is generated by evaluating (1) for the directions in the complete hemisphere, i.e.
√ u 2 + v 2 ≤ 1 (if not stated otherwise), and for a single centered point source, such that L = 1, s l = 1 and (u l , v l ) = (0, 0). The beam patterns and PSFs are generally normalized to their maximum value.

B. CHARACTERISTICS OF THE CLASSIC SUNFLOWER SPIRAL ARRAY
The position of the m-th element of the planar classic sunflower array r m is determined by discretely sampling the Fermat spiral based on the models in [22] and [61] with the design parameter V = 5, that is , and (4) where R m is the corresponding radius of the m-th element to the aperture center, φ m is the corresponding angle and R ap = D ap /2 is the maximum aperture radius. The model defines the element radii to the center R m such that the area of the ring spanned by two successive radii is constant and equivalent to the M -th part of the total aperture area, that is The design parameter V controls the number and positions of the spiral arms created by altering the angular distance between two successive elements. With the parameter V = 5, this angular distance corresponds to the Golden Angle, which in conjunction with the constant ring area A m , results in an approximately uniform spatial element density, a main characteristic of the classic sunflower array [25]. Therefore, the sunflower geometry depends only on two parameters, i.e. the total number of elements M and the aperture diameter D ap . In the following, the resulting one-way PSFs for different aperture diameters and three typical fixed total numbers of elements are used to investigate the corresponding MSLLs and the MLW at −6 dB (MLW 6 ), both being widely used metrics for contrast and angular resolution (Fig. 1). The MLWs monotonously decrease with increasing aperture diameter, whereas being independent of the observed number of elements. In contrast, the MSLLs transition from a lower to a higher plateau, where there is no further significant increase. The level of the plateaus as well as the diameter at which the plateau transition occurs are both dependent on the number of elements. Therefore, if a low MSLL is desired, there is an optimal aperture diameter for each M , where the MLW is narrowest, which is just before the transition to the higher plateau.
In order to clarify the plateau-like increase in MSLL, we examine the PSF of a 64-element array for two different aperture diameters (Fig. 2). The typical one-way PSF of the classic sunflower array can be categorized into three basic zones, i.e. 1) the main lobe zone (MLZ), 2) the side lobe zone (SLZ), where low secondary lobes are formed, 3) and the grating lobe zone (GLZ), where the side lobe level rises significantly. The three zones are clearly evident in the radial view [ Fig. 2(c)], showing the respective maximum side lobe level for each concentric ring of radius R uv = √ u 2 + v 2 centered on the PSF origin (u, v) = (0, 0). We define the transition radius from MLZ to SLZ (R MLZ ) at the first minimum of the main lobe. The transition radius from SLZ to GLZ (R GLZ ) is specified at the side lobe level that exceeds the first, and typically highest, secondary lobe in the SLZ. We found that the MLZ transition R MLZ is mainly dependent on the aperture diameter, whereas the GLZ transition (R GLZ ) depends on the inter-element spacings. Therefore, enlarging or reducing the aperture diameter with a fixed number of elements will narrow or widen the PSF zones, respectively. By maintaining a sufficiently small aperture, the GLZ can be forced out of the PSF, so that the MSLL decreases significantly [ Fig. 2(b)]. This way, the lower MSLL plateau in Fig. 1 is reached, although with the drawback of main lobe widening.
The estimation of the positions of the three zones is of major interest for the array design. We found that for the classic sunflower array, the transition from MLZ to SLZ R MLZ can be estimated with the well-known first-minimumapproximation for circular apertures [62], that is The estimation of the SLZ to GLZ transition (R GLZ ) requires analyzing the inter-element spacings. Since most of the inter-element spacings of the classic sunflower array are different to each other, we use Delaunay triangulation [63], [64] to obtain the specific inter-element spacing between each neighboring element [ Fig. 3(a)]. In [56], the authors use the most prominent inter-element spacingd in the corresponding histogram [ Fig. 3 with the highest number of occurences, to estimate the position of the radius, where the highest grating lobes are located. However, we are interested in the SLZ to GLZ transition and observed that the mean inter-element spacingd gives a good estimate, that is Based on this relation, we found that the classic sunflower array geometry allows to directly estimate R GLZ with the basic design parameters (D ap , M ) as follows. We analyze the area associated to each element A cell,m using Voronoi tessellation and found that it is in good agreement with the M -th part of the total area just as with the circular ring area A m between two successive elements in (6). The Voronoi cell area of each element can be approximated by a circular disk with a diameter corresponding to the mean inter-element

spacing, resulting in the relation
Therefore, the SLZ to GLZ transition R GLZ can be estimated by combining (8) and (9), that is We validate the approximations by determining the true respective transition radii of the classic sunflower arrays for different aperture diameters and number of elements [ Fig. 3 In order to prevent the formation of the GLZ in the PSF, as well as in beam patterns, where a source can be located off-center in a specific field-of-view within R fov = √ u 2 + v 2 , the transition to the GLZ must be chosen to be R GLZ ≥ 1 + R fov . For example, if a source can be located in a field-of-view spanning the full hemisphere (±90 • ), i.e. R fov = 1, the GLZ transition must be at least R GLZ ≥ 2 for GLZ prevention. Therefore, the aperture diameter is required to be D ap ≤ 0.5 √ M λ, so that the mean inter-element spacing isd ≤ 0.5 λ, just as for the half-wavelength criterion of periodic dense arrays.
In summary, we have analyzed the MLW and the plateau-like MSLL behavior of the classic sunflower array resulting from its three PSF zones, whose locations can be estimated prior to field simulation with the aperture diameter and number of elements.

C. TWO-SCALE SPIRAL ARRAY DESIGN
The key idea leading to the proposed two-scale array design is to exploit the specific PSF zone structure of the classic sunflower array by combining two sub-arrays featuring two different spatial element densities and aperture sizes, resulting in different PSF zone locations (Fig. 4). The combination of PSF zones enables to improve and flexibly balance the MLW and MSLL, similar to density tapering, without being constrained by almost-discrete MSLL plateaus. Since we focus on the one-way beam patterns, both combined sub-arrays are used only for transmitting or only for receiving, depending on the use case. Therefore, a combination of PSF zones corresponds to a complex addition, instead of a multiplication as in the two-way case. Although, two dedicated classic sunflower arrays can be used to combine their specific farfield PSFs, nesting an inner and outer sub-array allows for a more compact design. Assigning the sub-arrays into an inner area and outer ring ensures that their element densities can be separately defined for one-way beamforming, which is more complicated for two nested sub-arrays, that both start from the center and cover a shared area, since the respective element distances are influenced by each other. Therefore, the two-scale spiral array is fully defined with four design parameters, that is where R in = D in /2 and R ap = D ap /2 are the inner and total aperture radius, M in and M are the inner and total number of elements. The corresponding element position angles φ m and the transformation into Cartesian coordinates are equivalent to (4) and (5).
In order to estimate R MLZ and R GLZ for both sub-arrays of the two-scale geometry, we analyze their corresponding design models. First, the radii of the inner sub-array are consistent with the classic sunflower design as in (4) using the respective inner aperture diameter D in and inner number of elements M in . Second, the model for the outer sub-array radii is based on the similar design rule as in (6) of the classic VOLUME 3, 2023 sunflower geometry, i.e. the area of the ring spanned by two successive radii A m,out = π R 2 m − R 2 m−1 is constant and equivalent to the area of the outer sub-array divided by the number of outer elements. Therefore, the inner and outer ring areas are given by As a result, we obtain a composite array with only two different element densities, which are constant within each sub-array. Therefore, the two-scale array design allows to estimate the transition from SLZ to GLZ for the inner R GLZ, in and outer sub-array R GLZ, out in the same way as shown in Section II-B, that is The estimation of R MLZ, in and R MLZ, out of both sub-arrays is equivalent to (7) with the respective inner and total aperture diameters.
In summary, the ability to estimate the PSF zone locations using the basic design parameters is a major advantage of the two-scale design, since a desired PSF zone combination can be specified prior to field simulation.

D. CHARACTERISTICS OF THE TWO-SCALE SPIRAL ARRAY
Next, we examine a 64-element two-scale array to demonstrate the MLW and MSLL balancing and improvement. We focus on a geometry that exploits a specific PSF zone combination, although other advantageous combinations are possible, as shown in Section III. In particular, a GLZ-free inner sub-array with a wide MLZ is positioned within an outer sparser sub-array (Fig. 4), such that a) the high GLZ of the outer sparse sub-array is combined with the low SLZ of the inner sub-array, and b) the low SLZ of the outer sparse sub-array is paired with the MLZ and first secondary lobe of the inner sub-array, resulting in two effects. First, the main lobes of both sub-arrays accumulate to an overall higher level and form a combined main lobe with a narrow peak and broad base. Second, the overall side lobe level is more balanced, such that there are no pronounced differences between the SLZ and GLZ, which otherwise typically consist of relatively low and high side lobe levels. Both characteristics lead to an effective reduction of the MSLL, whereas a narrow peak of the combined MLW is preserved. As a result, a higher imaging resolution with reduced artifact formation compared to the classic sunflower array is expected. Nevertheless, there is another characteristic to consider, namely the main lobe base level (approx. −11 dB in the example), where the narrow peak transitions into the broad base, whose effects are pointed out next. In order to emphasize the advantages and drawbacks of this particular two-scale spiral array, we examine the resulting beam patterns of two adjacent horizontally positioned point-sources at varying angular spacings. The beam patterns of the two-scale array and classic sunflower array are then compared, whereas both geometries are selected to have an equal −6 dB MLW (MLW 6 ) in their respective PSFs (Fig. 5). The angular spacing θ s between the two adjacent positioned equal-strength point-sources is gradually increased from 0 • to 20 • in steps of 0.5 • . For each spacing θ s , we generate the resulting beam patterns of the two-scale and classic sunflower array using the multi-source model in (1) with L = 2, where the corresponding source locations are u 0 = + arcsin (θ s /2) and u 1 = − arcsin (θ s /2). Based on these beam patterns, we evaluate the angular resolution and the minimum level between the two sources. If the minimum inter-source-level [ Fig. 5(c)] drops below 0 dB, two distinct maxima are formed, such that both sources are separable in the beam pattern and the corresponding spacing defines the angular resolution.
Although both MLW 6 are equal, the two-scale array can separate the two sources at a smaller angular spacing (6 • ) compared to the classic sunflower array (8.5 • ), thus providing a higher effective angular resolution. In addition, the MSLL of the two-scale array is significantly lower (−9.86 dB vs. Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. the two-scale approach are more evenly distributed as in its PSF, without particularly high or low levels, so that the MSLL is kept low. However, the higher resolution and lower MSLL of the two-scale array come at a cost. Due to the wide main lobe base of the two-scale array, the minimum inter-sourcelevel has a flatter, plateau-like roll-off with increasing source spacing compared to the classic sunflower array. Consequently, although even closely spaced sources are separable, a high separation contrast requires a larger source spacing compared to the classic sunflower array. The roll-off of the minimum inter-source level is mainly determined by the main lobe base level of the corresponding two-scale array PSF. Therefore, the main lobe base level must be considered in the two-scale array design. The re-increase of the minimum inter-source level for the classic sunflower array at 15 • and above 20 • angular spacing arises due to the accumulation of side lobes, which is expected behavior.
In summary, compared to the classic sunflower array with the same MLW 6 , the two-scale array provides a higher angular resolution and a more effective artifact suppression due to the lower MSLL at the expense of reduced contrast between closely spaced but separable sources.

III. BENCHMARKING, RESULTS AND DISCUSSION
In this section, we benchmark the two-scale array geometries with the classic sunflower arrays and with density tapering approaches for modifying the sunflower spiral geometry. We implement two methods of density tapering for reference. One is based on a fixed window function introduced in [25] and [48], where the element positions are determined iteratively. Here, we consider a Blackman window [ Fig. 6(a2)], which has been utilized in multiple previous works, e.g. [58], [59], and [58]. The other method is described by Sarradj in The main difference of the density tapering methods to the two-scale array approach [ Fig. 6(a3)] is that the latter does not rely on window functions, originating from amplitude apodization, but rather aims for an advantageous PSF zone combination.
In order to highlight the similarities and differences of the design methods, we compare their density window functions with respect to the non-tapered classic sunflower case. Therefore, we derive the equivalent density window function for the two-scale array, enabling to recreate the geometry introduced in this article using the density tapering method described in [25] and [48], based on the following considerations. The tapering method defines the radii R m such that the ring area spanned by two successive radii weighted by the density window function f (R) is constant (K ) [25], [48], that is where K is defined as the M -th part of the effective total aperture area K = 2π/M R ap 0 f (R)R dR. For example, the equivalent density window function of the classic sunflower array is f (R) = 1, since the ring areas A m are already defined to be constant (6), such that where πR 2 ap /M = A m = K . Since the ring areas of the two-scale geometry ( A m,in , A m,out ) are constant within each sub-array as shown in (12) and (13), its equivalent density window function relative to the non-tapered classic VOLUME 3, 2023 sunflower case with equal aperture diameter is given by Therefore, the equivalent density window function of the two-scale array consists of two discrete density levels in contrast to the Blackman and SAR tapering methods, which feature a smooth characteristic [ Fig. 6 [61]. Subsequently, the corresponding PSF for each array geometry is formed from which the performance metrics, i.e. MLW 6 and MSLL, are automatically extracted. We use the MSLL as it reflects the worst-case metric in the formation of side lobe artifacts. In order to consider the side lobes that are created by an off-center point-source located within the full-hemisphere field-of-view as well, we evaluate the PSF within R uv = √ u 2 + v 2 ≤ 2 instead of ≤ 1. For clear comparison, we select and show only the optimum arrays with the lowest MSLL per MLW 6 of each approach. Here, we do not consider two-scale arrays with an main lobe base level above −9 dB. Otherwise, the comparison would be clearly in favor of two-scale arrays with a narrow MLW 6 but poor separation contrast for closely spaced sources (Section II-D). The MLW 6 and MSLL of the selected arrays of all approaches are then compared (Fig. 7). First, we conside the array geometries using 64 elements. The classic sunflower arrays provide the reference baseline for the comparison. Their characteristic MSLL plateaus at approximately −8.8 dB and −16.5 dB including their steep transition are clearly noticeable, similar as in (Fig. 1).
The Blackman window density tapering approach has, to some extent, a lower MSLL for the same MLW 6 , particularly between 14 • and 21 • , where a MSLL improvement from −8.8 dB down to −11.5 dB is observed. In addition, the MSLL can be further reduced below the lower plateau of the classic sunflower approach for MLWs wider than 32 • . The SAR arrays generally outperform the classic sunflower and Blackman window method as expected, since they are a subset of the SAR geometries. The variation of the H parameter enables a more flexible balancing between MLW and MSLL, e.g. with peripheral density tapering. This way, the performance gaps of the Blackman approach are overcome.
Finally, the two-scale array design provides the lowest MSLLs for most MLWs compared to the previous approaches, although the MSLLs for rather narrow or wide MLWs (e.g. < 7 • and > 31 • ) are similar to the SAR density tapering method. In general, similar results are obtained for the 256-element array geometries, where the classic sunflower method shows two distinct MSLL plateaus, the SAR technique outperforms the Blackman approach, and the two-scale geometries provide lower or similar MSLL per MLW compared to the SAR method. However, for wide MLWs above 15 • , the two-scale approach reaches a plateau and stops to improve the MSLL at approximately−27 dB, whereas it can be further lowered by the Blackman and SAR technique.
Clearly, there are specific PSF zone combinations that lead to particularly effective improvements, while others perform only similarly or worse compared to previous density tapering approaches. For example, one of the greatest improvements for the 64-element geometries occurs for an MLW of 14 • , where the MSLL of the two-scale approach (−13.4 dB) is significantly lower than for the classic sunflower (−8.8 dB) or SAR (−11.1 dB) method. In fact, this particular two-scale array utilizes the PSF zone combination (Type III) analyzed in Section II-D. Overall, five different types (I to V) of advantageous PSF zone combinations are observed in the set of optimum 64-element two-scale arrays, which are examined next.

B. OPTIMUM TWO-SCALE ARRAY COMBINATION TYPES
The first type [ Fig. 8(I)], consists of a large-aperture outer sub-array which includes most of the available elements, and, therefore, predominantly determines the PSF. In contrast, the contribution of the inner sparser sub-array is only supportive by adding to the relative main lobe level and by positioning its first secondary lobe (SL) minimum close to the first high side lobe of the outer sub-array.
Type two [ Fig. 8(II)], features a more balanced distribution of the inner and outer number of elements, although the latter is still dominant. Most importantly, the inner and outer aperture diameters D in and D ap differ only slightly. This way, a denser outer ring sub-array is formed filled by a sparser inner sub-array. As a result, both MLWs are similar to each other and accumulate without significant widening. The low SLZ of the sparse inner sub-array is positioned near the highest side lobes of the outer array, equalizing the overall side lobe level. In general, this type is similar to the peripheral density tapering of the SAR approach, e.g. for H = −2 [ Fig. 6(b3)].
The third type [ Fig. 8(III)] is the basis of the original idea for the two-scale array design, outlined in the previous sections. Here, the distribution of the number of elements is balanced as well, with the inner sub-array being denser and more populated. Basically, the densities of the inner and outer sub-arrays are inverted compared to type II, resulting in a different PSF zone combination, as depicted in Section II-D. The MSLL is significantly reduced compared to the other approaches given the same MLW 6 . Still, there is the drawback of reduced contrast between closely spaced sources. The optimum 256-element two-scale array geometries, which provide improved results compared to the previous approaches, consist exclusively of this type III.
In type four [ Fig. 8(IV)], the PSF is primarily determined by the inner denser sub-array, while few outer sparse elements contribute only as support for balancing out the side lobe level, forming the counterpart to type I.
Finally, in type five [ Fig. 8(V)], the sub-arrays differ only slightly in spatial density and therefore resemble the classic sunflower array more closely compared to the previous types. Nevertheless, even the small differences cause the side lobe minima and maxima to balance each other out, resulting in a VOLUME 3, 2023 FIGURE 8. Five different types of advantageous sub-array combinations observed in the set of optimum two-scale arrays (Fig. 7). Type I and IV feature either a dominant outer or inner sub-array, whereas the opposite sub-array is only supportive. In II and III, the ratio between inner and outer elements is more balanced and the sub-arrays are of different spatial densities for exploiting the PSF zone combinations. In particular, type III shows significant improvements compared to other design methods as examined in the previous sections. The sub-arrays of type V have only minor differences in spatial densities, which nevertheless result in an effective positioning of the side lobe minima and maxima for a more balanced overall side lobe level. 122 VOLUME 3, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
significantly lower MSLL compared to the classic sunflower approach. However, compared to the previous density tapering approaches, type V provides only similar or worse results, particularly for the 256-element arrays.
In summary, considering the progression from wide to narrow MLWs, the optimum two-scale arrays feature an increasing overall aperture, as well as a shift from type I to V (M = 64) or type III to V (M = 256), respectively. For all optimum combinations, the corresponding sub-array main lobes accumulate to a higher level, whereas the overall side lobe level becomes more balanced compared to the classic sunflower approach. The most effective improvements over the existing density tapering approaches are achieved, where a dense inner sub-array is combined with a sparse outer subarray, with a balanced number of inner and outer elements, as in type III. In contrast, for rather small apertures, where both sub-arrays are dense and grating lobe zones do not form (type V), the two-scale PSF zone combination is less effective. In this case, although significant improvements over the classic sunflower array are achieved, the two-scale results can be similar or even inferior compared to the Blackman and SAR method.

C. BENCHMARK WITH EXTENDED ELEMENT APERTURES
In the preceding analysis, unidirectional point elements with infinitely small aperture sizes are assumed to provide a generic technology-independent comparison. In order to highlight the impact of extended element apertures, in this section, we compare the array design methods, assuming circular elements with a diameter of D ap,el = λ/2, as an example. Here, we consider two resulting effects. First, the minimum IES is limited to d = D ap,el , such that non-realizable array geometries featuring overlapping elements are discarded. Second, the element directivity itself leads to a degradation of the MSLL, when spatial filtering for peripheral directions. Regarding the latter, we consider spatial filtering and source positioning in the entire hemisphere, i.e. within a field of view of ±90 • , as in the preceding comparison (Fig. 7). The element directivity is modeled as described in section II-A. The design parameter space investigated is identical as described in section III-A and two different numbers of elements are observed, i.e. M = 64 and M = 256.
As expected, the resulting MLWs show an upper limit because the array apertures can not become arbitrarily small due to the extended elements (Fig. 9). The worst-case MSLLs of the optimum array geometries are approximately 3 dB higher compared to the analysis based on point elements due to MSLL degradation resulting from the element directivity and the considered field-of-view. Furthermore, an increase in MLW partly leads to a degradation of the MSLL, which is shown as an example for the classic spiral and Blackman geometries. The reason is that for these non-optimum array geometries, the highest side lobes are located near the center, as the main lobe is steered to the periphery and, thus, attenuated by the element directivity. All in all, the comparison highlights that the SAR and two-scale geometries perform comparably well when assuming extended element apertures, whereas the latter design method provides the lowest MSLL per MLW, as in the comparison assuming point elements (Fig. 7). The optimum two-scale array geometries observed correspond to the types II and III, but feature an overall larger total aperture compared to the example in (Fig.8), so that the IES of the outer or inner denser sub-arrays are consistently greater than 0.5 λ.

D. SCOPE AND LIMITATIONS OF THE STUDY
In this final section of the theoretical study, we highlight the scope and limitations of the analysis and results. As explained in Section II, the analysis is performed using a generic application-independent model, assuming far-field and narrow-band conditions, i.e. continuous wave or temporally long bursts, such that the comparison is based on worst-case assumptions. However, in the nearfield case, the results obtained with focused beams may differ from the far-field results provided, depending on the application-specific focal distance and region of interest. Further deviations from the comparison results are likely to occur, if high-bandwidth transducer technologies and temporally short pulses are involved, that enable to exploit true-time-delay and broad-band beamforming methods for reducing side lobe levels. In addition, even larger element sizes than the example size considered can further constrain the realizable array geometries, so that the outcomes of the comparison may also vary for this reason. Consequently, the results presented are not directly valid for all applications, particularly if the base conditions in terms of element size and shape, bandwidth, focal distance and region of interest differ significantly from the model assumptions used.
Apart from the limitations due to the generic model, we emphasize that further modifications and optimization procedures can certainly lead to improved solutions compared to those investigated within the parameter space of the study. For example, adding more than two sub-arrays with different element densities can potentially provide further improvements at the expense of a more complicated array design including more parameters. Moreover, the well-performing deterministic two-scale solutions identified can be used as initial seeds for further stochastic optimization methods. Nevertheless, these improvement strategies are beyond the scope of this article and are worthy ideas for future work.

IV. REAL-WORLD EXPERIMENTS
In this section, we validate the two-scale array design strategy with a real-world prototype and two experiments in order to investigate the PSF and the resulting angular resolution.
For this, we created an air-coupled two-scale array based on type III, consisting of 64 small MEMS microphones (Knowles SPH0641LU4H-1) featuring a digital interface. In addition, we utilize one piezoelectric bending-plate transducer (ProWave 328ST160) with a diameter of 16 mm and a relatively low resonance frequency of 32.8 kHz to avoid strong attenuation by the medium air (Fig. 10). The frequency response of the transducer is narrow-band (1.4 kHz, 4.3%), so that a temporally long excitation signal (40 cycles, 70 V pp , bipolar square-wave) is required, which, however, enables the pulse to reach a relatively high sound pressure level of 123 dB at a distance of 30 cm. Therefore, the narrow-band assumption holds for this prototype and the experiments. The microphones are broadband (10 Hz to 80 kHz), compact (3.5 × 2.6 × 1 mm 3 ) and approximately unidirectional due to the small aperture diameter of 0.325 mm (0.03 λ). The ports of the microphones are located on the bottom side and are guided through the PCB with a thickness of 1.6 mm by using vias with a diameter of 0.6 mm. The microphones each provide a 4-MHz 1-bit PDM signal, all of which are converted into 125-KHz 16-bit signals by an FPGA (Intel MAX10) using 64 sinc-3 filters. The system architecture is based on the design described in detail in our previous work [3]. The type III two-scale array geometry is composed of the inner denser sub-array with 40 elements and an aperture diameter of 55 mm, while the outer sub-array consists of 24 elements and spans a total aperture of 190 mm. The implementation of signal pre-processing, narrow-band receive beamforming and image formation is described in detail in [65].
The experiments are conducted in an anechoic chamber, where the two-scale array is positioned in front of a linear axis equipped with a movable slide. In order to measure the PSF, we use a hollow steel sphere ( 10 cm) as target, which is mounted on a sound-absorbing fixture and positioned on the slide at a distance of 2 m, i.e. in the far-field, centered to the array. The pulse is transmitted to the sphere, its echo is received by all microphones and the PSF is formed at the pulse maximum within the field-of-view R uv = √ u 2 + v 2 ≤ 1 (Fig. 11). This measurement is repeated 32 times to obtain the average value and the standard deviation. The latter is consistently below 0.1 dB and is therefore excluded for clarity. Furthermore, the PSF measured is compared to the ideal simulation based on the model in (1). The measured and simulated PSFs are in excellent agreement, with the measurement featuring a slightly higher MSLL (−12.1 dB vs. −12.6 dB) and a wider MLW (8.05 • vs. 7.47 • ) compared to the simulation. Overall, both, the balanced side lobe level, as well as the distinctive shape of the main lobe due to the combination of the two sub-arrays, are evident, as expected.
In the next experiment, the achievable angular resolution is investigated, comparable to the simulation in (Fig. 5). For this, a horizontal mounting is positioned on the slide, onto which two laterally adjacent hollow spheres ( 5 cm) are attached, whose spacing between each other can be variably adjusted (Fig. 12). The sphere fixtures and the horizontal mounting are covered with sound absorbers as well. The slide is positioned at a fixed distance of 1 m (far-field) and the spacing between the spheres is symmetrically reduced from 20 cm to 8 cm in steps of 2 cm. At each spacing, an image is formed to evaluate the separability of the two sphere reflections. The separability is achieved if the two sphere reflections form two distinct local maxima in the image. As an example, four ultrasound images with different sphere spacings are provided, as well as their horizontal sectional view at v = 0 (Fig. 12).
As the angular spacing between the spheres decreases, the local minimum between the two reflections increases in good agreement to the simulation (Fig.5). The MSLL is between −10 dB (at 11.4 • ) and −12.25 dB (at 4.6 • ) for all spacings considered. Due to the narrow main lobe peak, the closest separable angular distance is 5.7 • , which corresponds to a spacing of 10 cm between the spheres. In total, the real-world experiments using the two-scale prototype created confirm the results of the simulation and demonstrate that the two-scale design provides a favorable trade-off between the separation of closely spaced objects and a low MSLL.

V. CONCLUSION AND OUTLOOK
The two-scale sparse spiral array is based on a deterministic and flexible 4-parameter design method for improving the one-way PSF performance in terms of MLW and MSLL compared to previous sunflower geometry modification approaches, i.e. density tapering using window function. The improvements are achieved by exploiting the sunflower-specific PSF structures of two nested sub-arrays featuring two different spatial element densities which are constant within each sub-array. This way, the PSF zone locations can be estimated prior to field simulation using the basic array design parameters, enabling to narrow the search for well-matched array configurations within pre-defined design constraints. Due to the excellent one-way PSF characteristics, the two-scale method proposed is particularly valuable for imaging applications, where high frame rates are of great importance, as well as for transmit-and receive-only applications. Future work addresses the impact of additional density tapering on the two-scale sub-arrays, potentially enabling further improvement. In addition, we evaluate whether the two-scale array achieves favorable results in the two-way beamforming mode as well, e.g. by using the inner and outer sub-arrays separately for transmitting and receiving. Moreover, we extend the prototype in terms of transducer technology and firing schemes in synergy with the two-scale design strategy to achieve further improvements for high-frame rate imaging in air and conduct additional experimental evaluations.