Target Acoustic Field and Transducer State Optimization using Diff-PAT

AIP/123-QED

Target Acoustic Field and Transducer State Optimization using Diff-PAT

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Tatsuki Fushimi,1, 2 Kenta Yamamoto,1, 3 and Yoichi Ochiai1, 2, 4
1) R&D Center for Digital Nature, University of Tsukuba, Tsukuba 305-8550,

Japan2) Faculty of Library, Information and Media Science, University of Tsukuba,

Tsukuba 305-8550, Japan
3) Graduate School of Comprehensive Human Sciences, University of Tsukuba,

Tsukuba 305-8550, Japan
4) Pixie Dust Technologies, Inc., Tokyo 101-0061, Japan(*Electronic mail: tfushimi@slis.tsukuba.ac.jp)(Dated: 15 November 2021)

Phased array transducers (PAT) are used in many applications, from airborne ultrasonictactile displays to acoustic levitation. Acoustic holograms play a significant role in determining the performance of these applications. Many PAT and optimizers have been developed; however, only the following have been demonstrated in the literature: “phase” and“phase and amplitude” control of transducers and “phase” and “amplitude” only control attarget points. Thus, most of the combinations of transducer state and target acoustic field
conditions are yet to be explored. Here, we explore such combinations using Diff-PAT, oneof the latest acoustic hologram optimizers. Diff-PAT is based on automatic differentiationand stochastic gradient descent. This optimizer achieves higher accuracy than conventional optimizers. We formulated multiple loss functions and wave propagators to enableeach combination of operation mode and quantitatively assessed the performance of eachcombination. The developed optimizers will offer new opportunities in the field and could
allow further simplifications in PAT applications.

1

Target Acoustic Field and Transducer State Optimization using Diff-PAT
Acoustic holograms are ubiquitous in ultrasonic applications in medicine, biology, physics,
and chemistry. Consequently, the accuracy and precision of acoustic holograms are becoming aThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
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significant point of interest. In mid-air ultrasonics, the phased array transducer (PAT) is the mostpopular method to render acoustic holograms; PAT has been designed by multiple authors1–3 .
It has been applied in many applications, such as airborne ultrasonic tactile display4 , acoustic
levitation5 , and displays purposes6,7 . Acoustic hologram optimizers for PAT has also been investigated. Long et al. first developed Eigensolver and Tikhonov regularization based method in
20148 , and Marzo et al. developed iterative back propagation (IBP) method in 20185 . Placensia
et al. developed GS-PAT and corrected version of Eigensolver and Tikhonov regularization based
method (CES) in 20209 . GS-PAT and IBP are modified versions of the Gerchberg-Saxton (GS)algorithm. Moreover, Sakiyama et al. developed Levenberg–Marquardt (LM) algorithm method in
202010 and Suzuki et al. demonstrated Greedy Algorithm with Brute-Force Search (GA) method
in 202111 . Most recently, Fushimi et al. developed a new acoustic hologram optimization method
based on stochastic gradient descent algorithm and automatic differentiation called Diff-PAT12 .
While many optimization methods and PAT have been developed, these optimizers can only
optimize for phase or amplitude at the target point. In addition, only “phase” and “phase andamplitude” control methods have been explored for the side of transducers. The currently explored
and unexplored combination of transducers and target point control methods in PAT systems are
summarized in Fig. 1; much of the combinations are yet to be explored. Thus, in this paper,
we developed an unifying methodological framework for sound-field computation that allows all
prior approaches in PAT to be modelled (e.g. IBP, GS-PAT) while also exploring other approaches
which, even if in principle applicable13–18 , had never been adapted to PATs. Achieving acoustic
holograms with "amplitude" only control of transducers may contribute to further simplificationof PAT. In addition, more advanced access to pressure amplitude and phase at the target point
could offer a new opportunity in ultrasonic applications such as acoustic manipulation5,19 , acoustic
streaming20 , and ultrasonic tactile display4,8 .
Here, Diff-PAT was selected as the basis of the generalized acoustic hologram optimizer because Diff-PAT was shown to be more effective than a conventional algorithm based on Eigensolver or GS methods. Furthermore, it is simple to implement different modes of wave propagationand loss functions in Diff-PAT. By exploring the unchartered opportunities in the field, this paper
also aims to demonstrate the flexibility of the newly developed optimizer. The basic components
for the gradient-descent based hologram optimizer are similar to the original Diff-PAT. The acous2

Target Acoustic Field and Transducer State Optimization using Diff-PAT
tic field at the target point (pa ) is calculated using linear superposition of pressure contribution
from each transducers:
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T

Pre f
D(θ )At e j(kd(xc ,xt )+φt ) ,t=0 d(xc , xt )pa (xc , xt , At , φt ) = ∑(1)

where T is the total number of transducers; d(xc , xt ) is the Euclidean distance between xc andxt ; Pre f is the reference pressure amplitude. Here, a single sided PAT with 16 × 16 (T = 256)transducers is assumed; D(θ ) = 2J1kr(krsinsinθ θ ) is the far-field directivity function for the piston source
based on the angle θ which is evaluated as the angle between the transducer normal and target
point xc ; r = 5 mm is the transducer radius, and J1 is the Bessel function of the first kind of order 1;
k = 2πc0f is the wavenumber given the acoustic frequency, f = 40 kHz, and the speed of sound c0 ;Pre f and c0 are assumed to be 1 Pa at 1 m and 346 ms−1 , respectively; pa is a complex number; theamplitude Aca = |pa | and phase φac = Arg(pa ) at the target point are calculated accordingly. The
optimization function is defined as:

minimize L (τ, Ω, xt , xc ).

(2)

τFIG. 1: Explored and unexplored combinations in PATs. The classification of the target point
control was set whether the optimizer can control the given variable to an arbitrarily set target.
3

Target Acoustic Field and Transducer State Optimization using Diff-PAT
The contents and size of optimization variable τ depend on the transducer drive mode (phase,
amplitude, P&A). For phase only control, τ = φt with size T, and At = 1 were defined for all
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transducers. For amplitude only control, τ = At with size T and φt = 0 for all transducers. For P&A
control, τ = [Ψ] with Ψ ∈ C and size T was defined. Since passing on complex number to optimizer
was considered unfavorable, the input values were separated into real and imaginary numbers;
τ pa = [ℜ(Ψ) ℑ(Ψ)] with size 2T in the actual implementation. The amplitude and complex angleof Ψ were used to determine At and φt . For both amplitude only, and P&A control, the amplitudeof transducer is limited to values between 0 and 1. In order to enforce this, the amplitude of|τ|transducer was normalized as At = max(|τ|).

Different mode of control at the target point is achieved by customizing the loss function foreach purpose. In general the loss function is defined as:

G
1 c
L =∑
(Aa cos φac − Ag cos φg )2 + (Aca sin φac − Ag sin φg )2 .
g G

(3)

This method is similar to methods proposed by Li et al.17 , who first demonstrated P&A control
at target plane using a single phase plate; equation 3 determine loss function by evaluating thedifference of real and imaginary parts separately. Similar to the optimization variable τ, target
variable Ω depends on the mode at the target point. Here, we assume G number of target points.
For phase only control at the target point, Ω = φg with size G is set, while Aa = Ag = 1. For
amplitude only control at the target point, Ω = Ag with size G is set, while φa = φg = 0. For
P&A control, Ω = [Ag , φg ] with size 2C. The methods proposed above are examples of the manydifferent implementation approaches. In all cases considered in this manuscript, the optimizers
are tasked to optimize the variables until the iteration number reached n = 1000. These loss
functions as defined above were programmed in Python 3.7.10, and the derivative of the loss
function with respect to the optimization variables τ was automatically differentiated using JAX21(ver. 0.2.17). Adam optimizer22 was used to update the current guess of τn : τn = τn−1 −α · (√vbmbn+ε) ,nmtvn
b n = 1−βbn = 1−βwhere mn , and v
n are bias-corrected first and second raw moment estimates. mn =
1
2
 
 2
β1 mn−1 + (1 − β1 ) δδLτ , and vn = β2 vn−1 + (1 − β2 ) δδLτ . The hyperparameters α, β1 , β2 , ε

are 0.1, 0.9, 0.999, and 1 × 10−8 respectively. These values are arbitrary; further improvements
may be achieved by tuning these parameters. The initial guess of τ was generated randomly. The
sets of codes necessary to fully reproduce all results and graphs in this paper are made available
as shown in the Data Availability statement23 .
The generalized acoustic hologram optimizer can combine any propagation modes and loss
4

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Target Acoustic Field and Transducer State Optimization using Diff-PAT

FIG. 2: Demonstration of phase surfer using P&A control at transducer and target point sides.
Here, 50 target points were specified with a radius of 40 mm. The target pressure amplitude was
50 Pa, and target phase was set such that it goes around the circle from 0 to 2π.

functions, and allows advanced acoustic field control. Fig. 2 is an example field when P&A control
is specified for both transducer and target point sides. This can generate phase surfer19 using PATs
and effectively transport particles at the surface of the water using the phase gradient5 . This offers
new opportunity in particle manipulation or acoustic streaming.
To further assess the effectiveness of optimizer in each combination, we assigned a set of target
positions, amplitudes and phases. Here, two cases were studied with varying number of focal
points, N = 2 and 4. For each case, 1000 sets of target points were randomly generated suchthat it lies within ROI (x = [−0.05, 0.05], y = [−0.05, 0.05], z = [0.05, 0.15] m). The pressure
amplitudes were assigned such that minimum pressure amplitude for each point is 10 Pa, and suchthat sum of each target pressure adds up to maximum acoustic pressure amplitude of the PAT(identified by averaging acoustic pressure amplitude of single focus at the vertex of ROI, in thiscase, 950 Pa). The target phases were assigned such that it lies between [0, 2π]. The performanceAof the optimizer was measured using two index; target pressure accuracy (Ra = Agc ) and phase errora(R p = cos φg − cos φac ). The result of performance assessments is shown in Fig. 3 (a) and (b). As
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Target Acoustic Field and Transducer State Optimization using Diff-PAT(a) N = 2

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1.5

N.A.

0.2
0

A

B

C

A

B

C

A

B

C

FIG. 3: Box-and-whisker plot comparing the performance of optimizer in each combination using
target pressure accuracy (Ra ) and phase error (R p ). Red line shows the median, blue box shows
Q1 and Q3, and the whiskers show 1.5 times the interquartile range. Black circles in the plotshows outliers, beyond the whisker range. 1000 sets of target points were randomly assigned; (a)shows the results when two control points (N = 2) are specified; (b) shows the results for N = 4.

it can be confusing to have phase control for target point and transducer; combination of letters andnumber will be used to specify each combination. For example, [A], [B], and [C] specify “phase”
only, “amplitude” only, and “P&A” control of transducers respectively; [i], [ii], and [iii] specify“phase” only, “amplitude” only, and “P&A” control at target points, respectively. Furthermore, A:i
is a combination of phase only transducer control with phase only control at target point, and C:iii
denotes a combination of P&A for both transducer and target point.
It is evident that [i] can be achieved well in all transducer driving modes and control number
points, as shown in Fig. 3 (a) and (b). The average phase error (R p ) for A:i, B:i, and C:i are
9.8 × 10−4 , 1.61 × 10−5 , and 9.7 × 10−6 respectively when N = 2, and 1.26 × 10−5 , 6.73 × 10−6 ,and 8.08 × 10−5 respectively when N = 4. [B] achieves good phase error in comparison with [A]6

Target Acoustic Field and Transducer State Optimization using Diff-PAT
and [C].
However, when [ii] is assigned, the performance of [B] drops significantly in comparison toThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0069182[A] and [C]. The transducer amplitude only method is considerably more difficult than phase-only
control since acoustic field can only be defined via "the relative position of the focal point andthe transducer position" and "the amplitude of each transducer". The average target pressure accuracies (Ra ) for A:ii, B:ii and C:ii are 1.00, 0.603 and 1.00, respectively, when N = 2, and 1.00,
0.859, and 1.00, respectively, when N = 4. The lack of performance by [B] is attributed to thefact that on average 45.4 and 44.0 % of the transducers in PAT were off (off state is defined whentransducer amplitude <1 %) when N = 2, and N =4, respectively (see supplementary material for
example of transducer amplitude distribution). Thus, the achievable target pressure amplitudes by
PAT drop significantly and cannot achieve high pressure amplitudes when assigned. The improved
performance of B:ii when N = 4 (in comparison to B:ii in N = 2) is attributed to the pressure assignment method requiring each target point to ‘share’ maximum pressure amplitude (thus reducing
the assigned target amplitude average). Since some transducers are already at 100% capacity (seesupplementary material), uniform scaling of the transducers to improve the performance is notpossible.
Similar trends are observed in [iii] when both phase and amplitude are specified at the target
point. The average target pressure accuracy (Ra ) for A:iii, B:iii and C:iii are 1.00, 0.501, and 1.00
when N = 2, and 1.00, 0.665, and 1.00, respectively, when N = 4. The average phase error (R p )are 8.48 × 10−4 , 0.0907, and 9.03 × 10−4 , respectively, when N = 2, and 9.66 × 10−5 , 0.0957 and1.35 × 10−4 , respectively, when N = 4.
The performances difference between A:ii to C:ii or A:iii to C:iii are not statistically significant(i.e. equivalent performance) in matching the target specifications (see supplementary material for
details on statistical analysis). Hence, changing from [A] to [C] does not significantly change theperformance, and the benefit of adding amplitude control in transducer side is not apparent solely
from Fig. 3.
On the other hand, a close inspection of the convergence graph demonstrate that in some circumstances the loss value of [A] is lower than [C] (see supplementary material). Since the solution
set of [A] is a subset of [C], [C] should be better than [A]. We believe the increased number of
optimization variables and complexity of optimizing both amplitude and phase as optimization
variables decreases the performance of the [C] optimizer. However, as demonstrated in Fig. 3; theperformance between [A] and [C] is not significant. In addition, the current performance of [C]7

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Target Acoustic Field and Transducer State Optimization using Diff-PAT

FIG. 4: Acoustic pressure field optimized by (a) A:ii and (b) C:ii. The residual ambient pressure
amplitude surrounding focal points in (a) is more visible than (b). The black cross shows the
position of the focal point. The white dashed box and circle in (a) is an example of how theambient noise was analyzed in Fig. 5

is sufficiently better than traditional optimizers in the field9,12 . Further exploration to improve [C]optimizer may not be necessary.
Fig. 4 (a)-(b) is an example of acoustic field as optimized by A:ii and C:ii; it shows that theoutput field is significantly different between (a) A:ii and (b) C:ii. The ambient noise surrounding
the focal point is qualitatively reduced in Fig. 4 (b) despite using the same loss function (i.e. theloss function only evaluates the acoustic amplitude at the target point). To assess this ambient
noise more quantitatively, we calculated the sum of pressure amplitude in the x-y plane (square
area with side length 0.03 m, and focal point at the center) for A:ii and C:ii (excluding circularregion with radius 2.5λ around the focal point, see Fig. 4) using 1000 sets of optimized field datafrom Fig. 3 (N=2). The results are as shown in Fig. 5; and we quantitatively confirm the reductionof ambient noise in C:ii (this trends are continue to be seen in N = 4, 8, 16 and 32 as shownin the supplementary material). The reasoning behind this reduction in ambient noise is unclear,and further study is required to understand the difference in results. The reduction in ambient
noise around the focal point is suitable for applications such as airborne ultrasonic tactile display
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Target Acoustic Field and Transducer State Optimization using Diff-PAT

120

A:ii
C:ii

Occurences [-]This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
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100
80
60
40
20
0

0.5

1

1.5

Sum of Ambient Noise [Pa]2
#106

FIG. 5: Histogram showing the sum of ambient noise surrounding the focal point using 1000
samples from Fig. 2 (N=2). Determined by summing the pressure amplitude around a square box
in x-y plane with side length of 3 cm whilst excluding 2.5λ radial region from the focal point.
Results for N = 4, 8, 16, and 32 are made available in the supplementary material.

(AUTD)4,8 , and Diff-PAT with [C] could be better suited for AUTD than [A].
In conclusion, we demonstrated a generalized acoustic hologram optimizer that can control
phase, amplitude, and phase and amplitude at the transducer side and target point. The optimizer
was based on Diff-PAT, and various loss functions and optimization variables were developed toachieve the target combination. The performance of the optimizer with each combination wasquantitatively assessed; we discussed the potential and limitations. We showed that [A] and [C]perform equally when assessed at the target point. However, around the focal point, [C] suppresses
the ambient noise and makes it more suitable for use in AUTD than [A]. PAT has already been
demonstrated to be a versatile and useful tool for ultrasonic applications. A more advanced control
of PAT and the insights we obtained in this study will aid further development of applications in
the field.

9

Target Acoustic Field and Transducer State Optimization using Diff-PAT
SUPPLEMENTARY MATERIAL

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See the supplementary material for transducer amplitude distribution, statistical analysis, convergence graph, and broader assessment of noise reduction in P&A.

ACKNOWLEDGEMENTS
This work was funded by Strategic Research Platform towards Digital Nature Powered by Pixie
Dust Technologies, Inc. This work was partially supported by JSPS KAKENHI Grant Number
21K14103. We would like to thank Editage (www.editage.com) for English language editing.

CONFLICT OF INTEREST
The authors have no conflicts to disclose.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available within this article and its supplementary material. The program codes and underlying dataset to fully recreate the figures in the
manuscripts are made openly available23 in Zenodo at 10.5281/zenodo.5701914.

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