Simulation of GHz ultrasonic wave piezoelectric instrumentation for Fourier transform computation

Abstract

The recent emerging alternative to classic numerical Fast Fourier transform (FFT) computation, based on GHz ultrasonic waves generated from and detected by piezoelectric transducers for wavefront computing (WFC), is more efficient and energy-saving. In this paper, we present comprehensive studies on the modeling and simulation methods for ultrasonic WFC computation. We validate the design of the WFC system using ray-tracing, Fresnel diffraction (FD), and the full-wave finite element method (FEM). To effectively simulate the WFC system for inputs of 1-D signals and 2-D images, we verified the design parameters and focal length of an ideal plano-concave lens using the ray-tracing method. We also compared the analytical FFT solution with our Fourier transform (FT) results from 3-D and 2-D FD and novel 2-D full wave FEM simulations of a multi-level Fresnel lens with 1-D signals and 2-D images as inputs. Unlike the previously reported WFC system which catered only for 2-D images, our proposed method also can solve the 1-D FFT effectively. We validate our proposed 2-D full wave FEM simulation method by comparing our results with the theoretical FFT and Fresnel diffraction method. The FFT results from FD and FEM agree well with the digitally computed FFT, with computational complexity reduced from O(N2logN)O(N2logN)O(N^2 logN) to O(N) for 2-D FFT, and from O(NlogN) to O(N) for 1-D FFT with a large number of signal sampling points N.

Full text

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Simulation of GHz ultrasonic wave
piezoelectric instrumentation
for Fourier transform computation
Zaifeng Yang
1,2, Xing Haw Marvin Tan
1,2*, Viet Phuong Bui
1 & Ching Eng Png
1
The recent emerging alternative to classic numerical Fast Fourier transform (FFT) computation, based
on GHz ultrasonic waves generated from and detected by piezoelectric transducers for wavefront
computing (WFC), is more ecient and energy-saving. In this paper, we present comprehensive
studies on the modeling and simulation methods for ultrasonic WFC computation. We validate the
design of the WFC system using ray-tracing, Fresnel diraction (FD), and the full-wave nite element
method (FEM). To eectively simulate the WFC system for inputs of 1-D signals and 2-D images, we
veried the design parameters and focal length of an ideal plano-concave lens using the ray-tracing
method. We also compared the analytical FFT solution with our Fourier transform (FT) results from
3-D and 2-D FD and novel 2-D full wave FEM simulations of a multi-level Fresnel lens with 1-D signals
and 2-D images as inputs. Unlike the previously reported WFC system which catered only for 2-D
images, our proposed method also can solve the 1-D FFT eectively. We validate our proposed 2-D
full wave FEM simulation method by comparing our results with the theoretical FFT and Fresnel
diraction method. The FFT results from FD and FEM agree well with the digitally computed FFT, with
computational complexity reduced from
O(
N2logN
)
to O(N) for 2-D FFT, and from O(NlogN) to O(N) for
1-D FFT with a large number of signal sampling points N.
Fourier transform (FT) is commonly used in a wide variety of digital computations1, including signal and image
processing, solving dierential equations, articial intelligence (AI) models, etc. Repetitive FFT computations
could lead to considerable power consumption and prevent real-time signal/image processing, especially when
the dimension of the input data is extremely large. For example, many types of image processing are implemented
in frequency/spectral domain such as de-noising, edge detection, etc. us, FFT could transform the image from
spatial domain to the corresponding frequency counterpart2. Image processing techniques have ourished in the
recent years with the rapid development of deep learning methods, especially for those based on convolutional
neural networks (CNN)3. Repetitive CNN calculation is needed for training or running a trained model with
various inputs. In this case, FFT also can be used for convolutional calculation, given that the convolution of
two images is equivalent to the multiplication of the FFT results of the two images. For example, FFT accelera-
tion using photonics for AI is an ongoing heated topic4, and photonic integrated circuits can do the FFT physi-
cally rather than digitally5,6. e 2-D FFT for an image with
N×N
pixels has a computational complexity of
O
(
N2logN)
. Obviously, the computational complexity would be exponentially increased if the number N (the
size of the image) becomes larger. Unlike 2-D image processing, signal processing7 based on electromagnetic
waves is usually based on 1-D temporal input data. Usually, real-time frequency response is required for applica-
tions such as object detection, recognition, distance measurement, etc. In these scenarios where repetitive FFT
computations are usually needed, excessive energy will be consumed and the eciency could be also low if the
resolution of the input signal is high (large 1-D input data). Additionally, the computational complexity of the
1-D FFT for signals is O(NlogN). Similar to the 2-D FFT computation, the computational complexity would
become high if the number N (the sampling points of the signal) is large.
Instead of calculating the FFT digitally by computer using the Cooley–Tukey algorithm8, there are some alter-
native methods to implement FFT physically. At the Fourier plane of a 4f optical lens system9,10, the diraction
pattern shows the Fourier transformation of the input 2-D image, where low frequency components are located
close to the optical axis and higher frequency ones are placed farther away from the origin. Photonic integrated
circuits (PIC)5,6 are another ecient method to achieve Fourier transform. By choosing the angular locations
of the input and output waveguides, the star coupler can implement a discrete Fourier transform. However, the
OPEN
1Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR), 1
Fusionopolis Way, Connexis #16-16, Singapore 138632, Republic of Singapore. 2These authors contributed equally:
Zaifeng Yang and Xing Haw Marvin Tan. *email: marvin_tan@ihpc.a-star.edu.sg
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resolution of the Fourier transform result is not high and such a component is dicult to be integrated with the
other electronic devices.
Recently, an emerging ultrasonic wavefront computing (WFC) technique was proposed to compute the
FFT11,12. is method uses the principles of wave mechanics in the acoustic domain by implementing the Fourier
transform through ultrasonic waves propagating within Silicon. According to Patel etal.12, the computational
complexity of WFC is
O(δ)
, where
δ
is the transit time of the ultrasonic wavefront. is is because the number of
cycles consumed in the microprocessor is comparable to the transit time of the ultrasonic wavefront. As a result,
WFC can achieve a signicant speedup over CPU-computed FFT algorithms. For a WFC module with an
N×N
transducer array, the computational complexity is O(N). e WFC technique achieves a
2317×
system-level
energy-delay product and benets a simultaneous 117.69
×
speedup with 19.69
×
energy reduction, as compared to
the state-of-the-art baseline all-digital conguration12. Table1 summarizes the above mentioned physical Fourier
transform realization approaches against digital computation, in terms of the complexity and its pros and cons.
To date, the ultrasonic WFC has been investigated only for the FFT of 2D data for image processing, however,
1-D FFT for signal processing is important and how to use such a WFC system for signal processing remains
unknown. On the other hand, the WFC system using GHz ultrasonic waves11,13,14 passing through a at lens15,16
will be nally packaged into a chip by semiconductor technologies. Before fabrication and measurement, it is
important to validate the idea, investigate the main factors that which cause errors, and optimize the system
design. To this end, accurate modeling and simulation for such an ultrasonic WFC system is required. However,
most verication based on modeling and simulation for such whole system is based on the Fresnel diraction
method without considering the complex material factors such as losses and piezoelectric eects from the
transducers17.
e contributions of our work are:
• We simulate an emerging GHz ultrasonic wave piezoelectric instrument for computing Fourier transforms.
e techniques include ray tracing, Fresnel diraction and full-wave nite element method (FEM). ese
methods are used for dierent tasks: ray tracing simulation can be used to validate the design parameters of
the WFC system; Fresnel diraction simulation is ecient and can handle a larger computational domain;
full-wave FEM simulation is the most accurate but it is computationally intensive.
• To the best of our knowledge, we are the rst to implement the full-wave FEM simulation for the emerging
ultrasonic wavefront computing instrument. Unlike ray-tracing and Fresnel diraction methods which con-
sider input signals or images based on the transducer shapes, full-wave FEM simulation takes the piezoelectric
eects, losses due to the transducers and acoustic blocks, and anisotropic properties of the lens into account.
rough full-wave simulation, we can have an insightfully predict the performance of the WFC system to be
fabricated, by considering practical factors which will be present in the experiment.
• We perform novel full-wave modeling and simulation methods for both 1-D signals and 2-D images as the
input for FFT. Unlike previous reported WFC system which only cater to 2-D images11,13, our proposed
method also can solve the 1-D FFT eectively. We demonstrate multiple simulation examples which validate
our proposed simulation method, by comparing our full-wave FEM simulation results with the results from
theoretical FFT and Fresnel diraction techniques. e computational complexity is reduced from
O
(
N2logN)
to O(N) for 2-D FFT, and from O(NlogN) to O(N) for 1-D FFT with larger number of N.
Theory
Ultrasonic Fourier transform with a lens only works if the Fresnel approximation or the paraxial approximation
is adhered to. In contrast, if a lens is not used, the Fraunhofer approximation is necessary, which is valid only in
the Fraunhofer far-eld zone, as illustrated in Fig.1. e paraxial approximation assumes that the waves emitted
from the pixels and aperture of the source plane do not diverge at large angles o the normal9. In order for this
condition to be met, the aperture size must be signicantly larger than the wavelength, but much smaller than the
path length. e resonant frequency of an aluminium nitride (AlN) piezoelectric transducer, which is a crucial
Table 1. A summary of some physical Fourier transform realization approaches.
Approach Pros and Cons Complexity
Cooley–Tukey8Easy to implement with digital signals
Computational costly for large domain accuate computationa easy to be imple-
mented in digital equipment
2D:
O(N2logN)
1D: O(NlogN)
4f free-space Lens9
Extremely fast
High resolution
Dicult to integrate with electronic devices
Dicult to obtain the phase
Delicate experimental setup
2D: O(N)
Photonic integrated circuit
(PIC): Star Coupler5,6
Highly ecient
Low resolution
Dicult to integrate with electronic devices
2D: O(N)
1D: O(N)
Wavefront computing using GHz ultrasonic piezoelectric transducers11,13
Highly ecient
Easy to fabricate
Integrated with the other electronic circuits
Energy saving
Low cost
2D: O(N)
1D: O(N)
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component of ultrasonic FT instrument, is determined by its thickness and other material properties such as
its density, elasticity and compliance matrix components. e resonant frequency is the frequency at which the
transducer vibrates most eciently and produces the highest amplitude of acoustic waves. In this context, a 2
µ
m thick AlN piezoelectric transducer results in a resonant frequency between 1.6 and 1.8 GHz, which includes
our targeted resonant frequency of 1.7 GHz. e speed of sound in fused silica (SiO
2
) determines the ultra-
sonic wavelength, which is the distance between two consecutive peaks or troughs of the wave. In the present
scenario, the speed of sound in Fused Silica is c = 5900 m/s, which corresponds to an ultrasonic wavelength of
3.5
µ
m. us, an understanding of these important parameters is crucial for the design and implementation of
the ultrasonic FT instrument.
To validate the WFC system using the paraxial approximation, the physical size of the pixel

should be much
larger than the wavelength:
≥10
. Here, we choose the pixel size to be 50
µ
m
×
50
µ
m, which is compatible
with 130 nm complementary metal-oxide-semiconductor (CMOS) technology. Accordingly, each pixel in the
transmitting and receiving piezoelectric arrays has a width and length of
�=Ŵ
, where
Ŵ≈14
. A larger value
of
Ŵ
causes the aperture to produce an eectively paraxial wavefront. is consists of a 40
µ
m
×
40
µ
m AlN trans-
ducer area with a 10
µ
m gap surrounding every pixel to minimize the acoustic coupling between neighboring
pixels. e receiving piezoelectric sensor array also comprises a 50
µ
m
×
50
µ
m array of pixels.
e width of the entire aperture of the transmitting piezoelectric actuator array (along the lateral dimen-
sion) is
where N refers to the number of pixels, which also corresponds to the number of elements in the transducer array.
e length
Lth
which the wave has to travel over can be determined from both Fresnel diraction approxima-
tion, and the requirement for the distance to be suciently long to satisfy the constant phase condition (i.e. the
sampling condition of the phase term)9:
erefore,
Lth
≈
κN
, where
κ
depends on the phasing used in the transmitting piezoelectric actuator trans-
ducers, the type of lens used, and also the size of the pixels. us, the propagation length
Lth
is eectively linear
with respect to the number of array elements N9. e time taken for the ultrasonic wave to propagate in the
medium from the input plane to the output plane (Fig.1),
ttransit
, can be derived as:
where
T
=
1
f
=
0.59
ns is the period of the ultrasonic wave. e ultrasonic FT system described in this study
utilizes a thin lens, thus allowing the omission of its thickness in calculations. At the input plane, waves from each
pixel of the transmitting piezoelectric actuator array propagate to the receiving piezoelectric sensor array located
at the output plane, taking T cycles to traverse the system. Notably, the ultrasonic frequency of several GHz is
comparable to modern micro-processor clock frequencies. is warrants a comparison of the number of cycles
required for computation in the micro-processor and the transit time of the ultrasonic wavefront. Computation
complexity can be approximated as O(N), where N represents the number of cycles required by the GHz clock for
completion. Consequently, the latency of the FT computation using the wavefront computing (WFC) approach
is primarily inuenced by the time taken for the wave to travel in the substrate, which is inversely proportional
to the speed of sound in the acoustic blocks.
e computation of ultrasonic FT involves summing up k-vectors while adhering to the principles of acoustic
wave propagation. To prevent energy loss at the sides, the size of the lens must exceed that of the input aperture
(1)
Lw=�×N=ŴN
(2)
L
w
Lth
×
L
w
4≫
1
(3)
t
transit =
L
th
c
=
κN
c
=
κN
f
=κ
NT
(a) (b)
Figure1. e schematic of the WFC system in (a) free space (b) solids.
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Lw
. is ensures that all rays emitted by the transmitting piezoelectric actuator array are captured. Specically,
the lens must be larger than the covering maximum aperture, which includes the size of the aperture as well as
the side gap, and is greater than the spread
∂D
:
In conventional lenses, a spherical surface transforms an incident plane wave to a spherical wave, resulting in
an emerging spherical wave which is then focused on the focal plane. is satises the paraxial approximation9.
To achieve a thin, compact, and CMOS-compatible WFC instrument, we designed a multi-phase Fresnel lens
with a thickness in the range of 20–25
µ
m. is Fresnel lens eects a parabolic phase shi as a function of radius,
causing the focusing of ultrasonic waves to produce a Fourier transform at the output plane. It is worth noting
that the mechanism is based on diraction instead of refraction or reection. e incident waves diract around
the lens and the diracted waves interfere constructively at the designed focal length13,18. All design parameters
are listed in Table2.
Methods
Ray-tracing. Ray-tracing is a geometrical optics method used to simulate the behavior of light in optical
systems, and we apply it for the ultrasonic WFC system to validate the design parameters such as focal length. It
makes use of the Fresnel equations by dening the positions and directions of the input rays. Ray-tracing is used
to simulate the behavior of light in optical systems by tracing the path of light rays as they interact with objects
in the system. e rays interact with certain modeled objects (e.g. the lens in the WFC system). e directions
of the rays are changed due to the refractive index at the interface. e basic idea behind ray-tracing is to model
light as a series of rays that originate from a light source and travel through the optical system, interacting
with objects along the way. We applied the ray-tracing to acoustic waves. However, the characteristic of how
the acoustics waves interact with the complicated medium was not considered. Ray tracing only considers the
interaction at the boundary of dierent materials. us, the ray-tracing method cannot obtain the FT results at
the focal plane. We used an in-house MATLAB code to implement the ray-tracing method, to validate the focal
length design of the WFC system. Our Ray Tracing code in MATLAB based on the equations in Ref.19.
Fresnel diraction. Fresnel diraction originated from the eld of optics9, but has been applied to
acoustics20,21. e Fresnel approximation is derived from the Huygens–Fresnel Principle9. e basic idea behind
Fresnel diraction is that the wavefronts are divided into many small segments, each of which acts as a point
source of light. e light from each of these point sources interferes with the light from all of the other point
sources to create a diraction pattern. us, it allows us to compute the acoustic pressure eld as a complex
number aer propagating though a medium. Previous works have computed the propagation of acoustic pres-
sure eld in air or water22. Fresnel diraction provides a most eective way to model the diraction of waves.
We used the code available from Ref.23 to perform our Fresnel diraction simulations. However, the propagation
along the acoustic blocks are simplied. In particular, the piezoelectric eects, losses due to the transducers and
acoustic blocks, and anisotropic properties of the lens, cannot be captured by the Fresnel diraction model.
Full-wave FEM modeling. In FEM, a complex system is divided into smaller, simpler parts, or “elements”,
that can be modeled mathematically. ese elements are then connected to form a nite element model of the
entire system. e behavior of each element can be described using mathematical equations, and the behavior of
the entire system can be calculated by solving these equations. As long as the modeling for FEM is well discre-
tized, the simulation results are usually of high accuracy. We used the soware COMSOL Multiphysics for FEM
simulations. Perfectly matched layers were used in frequency domain to absorb the waves at the boundaries of
the simulation domain. e full-wave FEM method24 yields the most accurate results among the three simula-
tion methods due to it takes into account the wave nature (elastic wave for the WFC system) and allows for more
accurate modeling of complex optical phenomena (multi-level Fresnel lens), such as diraction and scattering.
e limitation of FEM is it’s computationally intensive nature which requires a lot of computational resources.
For this reason, we performed the FEM simulations in 2D. Due to our lack of computational resources, we were
unable to perform the FEM simulations in 3D. Other full-wave potential alternatives such as Finite-dierence
Time-domain (FDTD) or Finite-element Time-domain (FETD) method are even more computationally inef-
cient. Moreover, the FFT results cannot be directly observed in time-domain simulation.
(4)
∂D
=
f
×
(/�)
Table 2. Design parameters and the radius of the curvature in ideal lens, derived from Fresnel optics, for
dierent transducer arrays.
N by N array 4 by 4 8 by 8 32 by 32 128 by 128
Min aperture (

) (mm) 0.05 0.05 0.05 0.05
Max aperture (A) (mm) 0.2 0.4 1.6 6.4
Focal length (f) (mm) 0.36 0.72 2.86 11.43
Full length (L = 2f) (mm) 0.71 1.43 5.71 22.86
Radius of the curvature in ideal lens25 (mm) 0.12141 0.24283 0.96456 3.85486
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Relationship between Fresnel diraction and theoretical fourier transform. e wavefront
computation (WFC) can be modeled using the Fresnel diraction equation for wavefront propagation. To model
the entire wavefront propagation from the transmitters’ pixels to the receivers’ pixels, two Fresnel propagation
steps are needed. e rst Fresnel propagation step of the complex pressure eld is from the transmitters’ pixels
(also called the input plane in Fig.1a) to just in front of the lens. en, the pressure eld needs to be phase trans-
formed by the lens. Aer phase transformation by the lens, the pressure eld needs to be propagated again using
Fresnel diraction to the receivers’ pixels (also called the output plane in Fig.1a).
From equations (5–19) in Joseph Goodman’s textbook9 which are used to perform Fresnel diraction propaga-
tion via computing the Fourier transform, for the special case where d = f, where d is the distance from the object
plane to the lens, and f is the focal length, then the equation reduces an exact Fourier transform:
where
Ul(ξ,η)
is the pressure eld in the input plane, and
Uf
x,y

is the pressure eld in the output plane.
Performing the substitutions
u
=
x

f
and
v
=
y

f
, we get:
e resulting 2-D Fourier transform F(u,v) has frequency axes in the SI units of 1/m. In order to convert the
frequency axes to the SI units of meters, we need to multiply the Fourier axes by
f
, where

is the wavelength,
and f is the focal length. is will give us the pressure eld
Uf
x,y

in the image plane with spatial axes in SI
units of meters.
Results
Ray tracing simulation. Our ray-tracing simulation19 results conrm a focal length of 2.86 mm for the the
32 by 32 transducer array (Fig.2), consistent with the calculated focal length in Table2. e focal length of 11.43
mm predicted by Table2 for the 128 by 128 transducer array was also validated by ray tracing (Fig.2).
Full-wave FEM simulations and Fresnel diraction. e 1D voltage signals in Figs.3a, 6a and 7a were
numerically Fourier transformed using the Fast fourier transform package in Python (SciPy). e frequency
axes of the numerically Fourier transformed results were converted to the spatial axes as explained in under the
Method subsection “Relationship between Fresnel diraction and theoretical fourier transform”. Aer conver-
sion to spatial axes, the numerically Fourier transformed results are plotted as “FFT1D(signal)” in Figs.3c, 6c
and 7c. Similarly, the 2D voltage signals in Figs.3b, 6b and 7b were numerically Fourier transformed using the
Fast Fourier transform package in Python (SciPy). e frequency axes of the numerically Fourier transformed
results were converted to the spatial axes as explained in under the Method subsection “Relationship between
Fresnel diraction and theoretical fourier transform3.4”. Aer conversion to spatial axes, the numerically Fou-
rier transformed results are plotted as “FFT2D(image)” in Figs.3c, 6c and 7c.
We simulated Case 1 which corresponds to a 1D array of 32 piezoelectric transducers with 10 transducers
being activated by voltage signals (Fig.3a). e 2D input voltages at the input plane are depicted by 10 stripes
represented by blue and white pixels in Fig.3b. e rst order diraction peak in the normalized magnitude
of the Fourier transform computed by the full-wave simulation of a 16-phase Fresnel lens agrees well with the
(5)
U
f

x,y

=

Ul(ξ,η)exp

−j
2
π

f
ξx+ηy

dξd
η
(6)
F
(u,v)=

Ul(ξ,η)exp

−j2π(uξ+vη)

dξd
η
Figure2. Ray-tracing simulation results. (a) 32
×
32 array with focal length of 2860
µ
m aperture 100
µ
m, 100
rays. (b) 128
×
128 array with focal length of 11,430
µ
m aperture 200
µ
m, 100 rays.
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Fourier transforms computed by Fresnel diraction and 1-D and 2-D digital methods (Fig.3c). It is observed
that higher diraction orders do not match as well as the zeroth and rst order peaks.
Performance degradation due to component imperfections can be modeled by the 2-phase, 4-phase and
8-phase Fresnel lenses which do not approximate the ideal phase prole of an ideal lens as well as the 16-phase
Fresnel lens does. We performed full-wave FEM simulations for Case 1, using Fresnel lenses having dierent
number of phase steps (Fig.4). e Fourier transform computed by the 8-phase and 16-phase Fresnel lenses agree
well with the digitally computed FFT. e Fourier transform computed by the 4-phase Fresnel lens exhibits slight
deviations from the digitally computed FFT, while that computed by the 2-phase lens is drastically dierent from
the digitally computed FFT. e 4-phase Fresnel lens provides a reasonable approximation to computing the
Fourier transform. It must be noted that Fresnel lenses with more phase steps (e.g.
≥
8-phase) are more dicult
to fabricate than Fresnel lenses with fewer phase steps (e.g.
≤
4-phase).
We demonstrate for Case 1, the displacement magnitude in the Fused Silica medium, solved by full-wave
FEM simulation for a 32 transducer WFC block in which only 10 transducers are activated, with a 16-phase
Fresnel lens (Fig.5a). e geometries in Fig.5a–c are symmetric with respect to zero in the abscissa axis. e
transducers array is at the y-coordinate of
2860 µm
as shown in Fig.5b. e displacement magnitude in Fused
Silica near the transducer array is also shown. e 16-phase Fresnel lens along with the displacement magni-
tude is shown in Fig.5c. From Fig.5d, we observe that the Fresnel lens causes focusing at the focal plane with
y-coordinate of
−2860 µ
m.
(a) 10 pulses (b) 10 stripesimage
(c) Comparison of results from various methods
Figure3. Case 1 which corresponds to the array of 32 piezoelectric transducers with 10 transducers being
excited by (a) 1-D voltage signals with 10 rectangular signals. (b) Image of the input voltages at the input plane
having 10 stripes represented by blue and white pixels. (c) Validation of the WFC system simulated by the full-
wave FEM method, compared with digital FFT and Fresnel diraction results using 16-phase Fresnel lens.
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Figure4. e simulated Fourier transform results for Case 1 which corresponds to the array of 10 piezoelectric
transducers excited by 1-D voltage signals with 10 rectangular pulses, using multi-level Fresnel lenses from 2 to
16 phases, compared with theoretical 2-D FFT.
Figure5. Case 1 which corresponds to the array of 10 piezoelectric transducers excited by 1-D voltage signals
with 10 rectangular pulses. (a) e displacement magnitude solved by full-wave FEM simulation for a 10
transducer WFC block with 16-phase Fresnel lens. e displacement magnitude (b) near the transducer side by
full-wave simulation, (c) around the 16-phase Fresnel lens zone, and (d) around the designed focal plane.
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To model a scenario which better resembles experimental conditions, we simulated Case 2 which corre-
sponds to a 1D array of 32 piezoelectric transducers excited by randomized 8-bit voltage signals (Fig.6a). e
2D input voltages at the input plane are depicted by 32 stripes represented by blue and white pixels in Fig.6b.
From positions from 0 to
200 µ
m, we observe reasonable agreement between the normalized magnitude of the
Fourier transform computed by the full-wave simulation of a 16-phase Fresnel lens, and the Fourier transforms
(a) 8-bit, 32 pulses (b) 8-bit, 32 stripesimage
(c)
Comparisonof results fromvarious methods
Figure6. Case 2 which corresponds to the array of 32 piezoelectric transducers excited by (a) randomized 8-bit
voltage signals. (b) Image of the input voltages at the input plane having 32 stripes represented by colored pixels.
(c) Validation of the WFC system simulated by the full-wave FEM method, compared with digital FFT and
Fresnel diraction results using 16-phase Fresnel lens.
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computed by Fresnel diraction and 1-D and 2-D digital methods (Fig.6c). It is observed that higher diraction
orders do not match as well as the zeroth and rst order peaks.
e average error and the
L2
norm error at the expected focal plane for 2, 4, 8, and 16-phase Fresnel lens are
calculated by comparing the full-wave simulation with the 1-D and 2-D FFT results (Table3). e normalized
total displacement elds at the focal plane are simulated by full-wave FEM simulation and compared with the
1D normalized FFT results of Cases 1 and 2.
We also demonstrate Case 3 which is an intermediate between Case 1 and Case 2. Case 3 corresponds to an
array of 10 piezoelectric transducers excited by randomized 3-bit voltage signals (Fig.7a). e input voltages at
the input plane are depicted by 10 stripes represented by the shades of blue pixels in Fig.7b. Similar to Cases 1
and 2, the higher diraction orders do not match as well as the zero
th
and rst order peaks (Fig.7c).
In addition to the quantitative comparison (norm error for dierent multi-level phase Fresnel lens) given
in Table3, we also show the absolute error compared with analytical 1D and 2D FFT results for case 1, 2, and
3 in Fig.8. It is expected that case 1 has the minimum error compared to both 1D and 2D FFT results, and case
2 has more errors (within 0.2) as the input signals or image is the most complicated (random 8 bit input). Fig-
ure9 shows the absolute error compared with analytical 2D FFT results for case 1 using 2-, 4-, 8-, and 16-phase
Fresnel lens. It is observed that the absolute error reduces as the phase number increases. It is obvious 2-phase
Fresnel lens is not suitable for the implementation of the GHz wavefront computing system due to the large
error compared to analytical FFT results, while the accuray become better as the increase of the phase number
of the Fresenel lens.
To summarize the errors shown by our comparisons for Cases 1, 2, and 3 in in Figs.3c, 6c and 7c, respectively,
we plot the absolute errors in Fig.8 for the three cases. e absolute errors corresponding to Fig.4 for the 2-phase,
4-phase, 8-phase, and 16-phase Fresnel lenses as compared to the CPU-computed 2D FFT are shown in Fig.9.
Discussion
e fabrication of the dierent components will lead to non-uniformities across the pixel array of piezoelectric
transducers. e pixel array itself can have variations in pixel sizes due to lithography error. e lens pillars and
radii will be aected by lithography errors. e bonding of the lens to the fused silica (Quartz) block may lead
to bond-layer thickness variations. ese thickness variations will produce variations in phase that will aect
the phase shis and the amplitudes of the acoustic waves received at the receiving piezoelectric sensor array.
Using the full-wave FEM simulation approach, we can have very accurate modeling, but it is time consuming.
Using the Fresnel integral-based approach oers the possibility to scale up the simulation to very large array of
transducers. However, the disadvantage of this approach is that there needs to be isotropic, lossless propagation
medium. e use of the Fresnel diraction integral also requires and a lens transfer function which needs to
come from full-wave FEM simulations16.
In conclusion, we have presented the GHz ultrasonic wave piezoelectric instrumentation for Fourier trans-
form computation, which we have demonstrated to perform reasonably accurate FT calculations. Our full-wave
FEM simulations have showed the capabilities of the GHz Ultrasonic Wave Piezoelectric Instrumentation.
Our ndings are signicantly important. Performing FT computations faster than the Cooley–Tukey8 digital
FFT algorithm, our instrumentation has the potential to meet the expanding need for such computations in uses
like real-time video processing in self-driving automobiles. Our instrumentation has the potential to enhance
the performance of wave-based analog computation devices to enable super-computers of the future.
Table 3. e average error and the L norm error at the expected focal plane for 2, 4, 8, and 16-phase Fresnel
lens for case 1: 10 uniform rectangular pulse signal and 10 uniform distributed 2D stripe image; and case 2: 32
randomized 8-bit pulse and 32 randomly distributed 2D stripe image. e normalized total displacement elds
at the focal plane are simulated by full-wave simulation and compared with the 1D normalized FFT result of
case 1 and 2.
Case Lens phase
Compared with 1D FFT Compared with 2D FFT
Average error
L1
Norm error Average error
L1
Norm error
1
2 0.08393 228.376 0.08801 239.466
4 0.04009 109.083 0.04262 115.981
8 0.03649 99.2782 0.03799 103.380
16 0.02283 62.1277 0.02397 65.2107
2
2 0.10403 283.066 0.10082 274.323
4 0.06009 163.508 0.05675 154.418
8 0.04616 125.601 0.04218 114.759
16 0.04257 115.836 0.03844 104.587
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(a) 3-bit, 10 pulses (b) 3-bit, 10 stripes image
(c) Comparisonofresults fromvarious methods
Figure7. Case 3 which corresponds to the array of piezoelectric transducers excited by (a) randomized 3-bit
voltage signals. (b) Image of the input voltages at the input plane having 10 stripes represented by colored pixels.
(c) Validation of the WFC system simulated by the full-wave FEM method, compared with digital FFT and
Fresnel diraction results using 16-phase Fresnel lens.
(a) case 1 (b)
case 2
(c)
case 3
Figure8. e absolute error compared with analytical 1D and 2D FFT results for case 1, 2, and 3.
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Data availability
e data that support the ndings of this study are available from the corresponding author upon reasonable
request.
Received: 27 February 2023; Accepted: 6 September 2023
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Acknowledgements
is work is supported by the A*STAR RIE 2020 Advanced Manufacturing and Engineering (AME) Program-
matic Fund [A19E8b0102], and theA*STAR Career Development Fund (CDF) [C222812026].
Figure9. e absolute error compared with analytical 2D FFT results for case 1 using 2-, 4-, 8-, and 16-phase
Fresnel lens.
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Author contributions
Z.Y. performed the ray-tracing and full wave FEM simulations, analyzed the data, wrote the abstract, introduc-
tion and part of the methods section. X.H.M.T. performed the Fresnel diraction simulations, analyzed the data,
wrote the results section and part of the methods section. V.P.B. planned the research project, wrote the theory
section, and supervised both Z.Y. and X.H.M.T. C.E.P. initiated the research project and supervised both Z.Y.
and X.H.M.T. All authors reviewed the manuscript.
Competing interests
e authors declare no competing interests.
Additional information
Correspondence and requests for materials should be addressed to X.H.M.T.
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