![Schematic of the UFT-ACS. The figure shows a schematic of the UFT-ACS, which consists of five main parts: the input pressure field Ps(ξ,η)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${P}_{s}(\xi ,\eta )$$\end{document} at the source plane, a substrate layer (light blue), the ultrasonic metalens (dark blue), another substrate layer (light blue), and the output pressure field Po(u,v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${P}_{o}(u,v)$$\end{document} at the observation plane.](https://www.researchgate.net/publication/364319843/figure/fig1/AS:11431281089642805@1665630444239/Schematic-of-the-UFT-ACS-The-figure-shows-a-schematic-of-the-UFT-ACS-which-consists-of_Q320.jpg)
![Accuracy optimization: zero padding, truncation, and bandlimiting. (a) The graph shows the dependence of the RMSE on the side length w\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w$$\end{document} of the input square, which is inversely related to the amount of zero padding. (b) The graph shows the dependence of the RMSE on γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document}. The blue dots represent the RMSE of the simulations involving an ideal metalens. The orange vertical line indicates the value of γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document} corresponding to the case wherein the sampled array just contains 98% of the total spectral power.](https://www.researchgate.net/publication/364319843/figure/fig4/AS:11431281089642807@1665630446586/Accuracy-optimization-zero-padding-truncation-and-bandlimiting-a-The-graph-shows_Q320.jpg)
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A metalens‑based analog
computing system for ultrasonic
Fourier transform calculations
Robert Frederik Uy1* & Viet Phuong Bui2
Wave‑based analog computing is a new computing paradigm heralded as a potentially superior
alternative to existing digital computers. Currently, there are optical and low‑frequency acoustic
analog Fourier transformers. However, the former suers from phase retrieval issues, and the latter
is too physically bulky for integration into CMOS‑compatible chips. This paper presents a solution to
these problems: the Ultrasonic Fourier Transform Analog Computing System (UFT‑ACS), a metalens‑
based analog computer that utilizes ultrasonic waves to perform Fourier transform calculations.
Through wave propagation simulations on MATLAB, the UFT‑ACS has been shown to calculate the
Fourier transform of various input functions with a high degree of accuracy. Moreover, the optimal
selection of parameters through sucient zero padding and appropriate truncation and bandlimiting
to minimize errors is also discussed.
e rst known analog computer is the Antikythera mechanism, invented by the ancient Greeks1. Since then,
many other mechanical and electronic analog computers have been devised to perform mathematical operations
more eciently1–3. Subsequently, with the development of semiconductor technology and integrated circuits,
the sheer speed and reliability of digital computers eventually led to a tectonic shi in the twentieth century2,3.
When performing complex computational tasks, however, digital computers are computationally inecient
and consume a lot of energy2. Unfortunately, there is little opportunity for further improvements as Moore’s
law approaches its physical limits2,4,5. With the rising demand for ever-increasing computational capacity and
eciency6,7 and the recent breakthroughs in the eld of metamaterials3,8, a new computing paradigm with very
promising prospects has emerged: wave-based analog computing.
Wave-based analog computing leverages waves to perform analog computing. It has been heralded as a
potential future of computing because of its high computational eciency, low crosstalk, and powerful paral-
lel processing1,8,9. Silva etal.’s seminal paper10 on computational metamaterials laid the foundation for other
researchers to conduct studies into both optical and acoustic analog computing systems performing mathematical
operations1–4,6–35, with some making use of the Fourier transform to do so3,6,9,10,26–28.
The Fourier transform (FT) is a mathematical operation that maps a function in one variable to the
spectral space of its conjugate variable14,36–38. It is a powerful tool with wide-ranging applications in myriad
disciplines4,22–24,36–43. Currently, the two-dimensional Fast Fourier Transform (FFT) algorithm has a computa-
tional complexity of
O
(
N2logN)
, which is not ecient enough for certain applications, such as real-time image
processing in autonomous systems4,22–24.
Capitalizing on the Fourier transforming property of thin lenses36, researchers have developed a new, analog
method of performing FT calculations: the optical Fourier transform (OFT)35,36. e OFT, which has an apparent
computational complexity of only
O(N)
, is signicantly faster than the electronic FFT algorithm22,35. However,
due to the limitations of phase modulation and phase retrieval methods22, it would be impractical to capture
phase data when performing the OFT. Researchers thus turned to acoustic waves. Although this method is
comparatively slower than the OFT, it is nevertheless faster than the FFT, and it allows for the retrieval of phase
information, unlike the OFT.
Aiming to replicate the success of optical analog computing systems in acoustics, Zuo etal. developed an
acoustic analog computing (AAC) system that performs FT-based spatial dierentiation, integration, and
convolution26. Several other studies on AAC systems have been conducted, but they all have an operating fre-
quency in the kilohertz range3,9,26–28.
Unfortunately, performing acoustic FT at such low frequencies requires a physically bulky computing system
even with the use of thin, planar metasurfaces. erefore, researchers have sought to use ultrasonic waves instead
OPEN
1Hwa Chong Institution, Singapore 269734, Singapore. 2Electronics and Photonics Department, A*STAR Institute
of High Performance Computing, Singapore 138632, Singapore. *email: robertfrederikduy@gmail.com
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to perform ultrasonic Fourier transform (UFT). e shorter wavelength of ultrasonic waves allows for a more
compact analog computing system that is easily integrable into CMOS-compatible chips. Liu etal. developed an
ultrasonic FT system without any lens22. is, however, would require a relatively large system as, in the absence
of a lens, the UFT will only be achieved in the far eld. Subsequently, Hwang, Kuo, and Lal worked on realizing
the UFT with an acoustic Fresnel lens23, aer which they used a metalens to allow for a more compact UFT
computing system24. Besides compactness, other reasons for using a metalens include the CMOS-compatibility
of materials used and ease of fabrication.
Despite the considerable work that has been done on the UFT22–24, there has yet to be a comprehensive
study on the accuracy of the Ultrasonic Fourier Transform Analog Computing System (UFT-ACS). us, this
study aims to ll this gap in the existing literature. Firstly, this study aims to determine how accurate the UFT’s
magnitude and phase are compared with those of the analytical FT for all three types of functions, specied in
Sec. 4. Unlike previous studies22–24, this study also takes into account the UFT’s phase, not just its magnitude.
Secondly, this study also seeks to determine how to optimize the UFT calculation for space-limited functions by
examining how the accuracy is aected by the level of zero padding. irdly, another objective of this study is to
investigate the eects of truncation and bandlimiting of functions that are not space-limited and/or bandlimited
on the UFT-ACS’ accuracy to allow for the optimal selection of parameters.
Results
The Ultrasonic Fourier Transform Analog Computing System (UFT‑ACS). Referring to Fig.1, the
Ultrasonic Fourier Transform Analog Computing System (UFT-ACS) consists of ve main parts: the source
plane, a substrate layer, the ultrasonic metalens, another substrate layer, and the observation plane. e entire
UFT-ACS has a square cross-section with side length
L
. e focal length of the metalens is
f
, which is also the
thickness of both substrate layers—this is a key condition for obtaining the UFT expression. e thickness of
the metalens is
tm
.
Using concepts in acoustic wave propagation—in particular, Fresnel diraction and lenses’ Fourier transform-
ing property—and some approximations (see Table1), it can be shown that the output
Po(u,v)
is proportional
to the Fourier transform of the input
Ps(ξ,η)
:
where the operator
F
denotes the FT,
j=√−1
,
k
is the wavenumber, and
is the wavelength. Hence, multiply-
ing the pressure eld at the observation plane by the correction factor
(1)
P
O(u,v)
=
jexp
(
−2jkf
)
f
F
{
PS(ξ,η)
},
(2)
α=−
jfexp
2jkf
Figure1. Schematic of the UFT-ACS. e gure shows a schematic of the UFT-ACS, which consists of ve
main parts: the input pressure eld
Ps(ξ,η)
at the source plane, a substrate layer (light blue), the ultrasonic
metalens (dark blue), another substrate layer (light blue), and the output pressure eld
Po(u,v)
at the
observation plane.
Table 1. Summary of the approximations required to derive the UFT expression.
No Approximation Validity
1
|
r−r0|≈f
1+1
2
x−ξ
f
2
+1
2
y−η
f
2
Fresnel
2
|r−r
0
|≈f
Paraxial
3
cos(n,r−r0)≈1
Paraxial
4
1
|
r
−
r
0|
+jk ≈
jk
Distances much larger than
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theoretically yields the exact FT of the pressure eld at the source plane. Hereinaer, the corrected output pres-
sure eld will be referred to as the UFT of the input.
Design of the UFT‑ACS. e operating frequency of the UFT-ACS was chosen as
fwave =1.7
GHz, a high
ultrasonic frequency that allows for a more compact system that would be easier to integrate into CMOS-compati-
ble chips. Both substrate layers, made of fused silica due to the material’s isotropy, have a thickness of
f
=
1.0886
mm. e speed of ultrasonic waves is
vwave =5880
m s−1 in fused silica, which implies
=vwave/fwave =3.46
µm.
e metalens, shown in Fig.2a, consists of many unit cells with a square cross-section of side length 3µm
(a subwavelength feature) and a thickness of
tm=16
µm. In Fig.2b, each unit cell consists of a cylindrical post
made of SiO2 embedded in Si. eoretically, the ultrasonic metalens should have a paraboloidal phase prole
to obtain the UFT. However, discretization is needed as there is a limited number of distinct unit cells. us, the
metalens’ unit cells should be arranged such that the unit cell at each point has a cylindrical post with a radius
corresponding to that point’s interpolated phase shi. Aer interpolation, the ideal phase map becomes the
discretized phase map, shown in Fig.2c.
UFT of various input functions. A space-limited function is one whose non-zero values are contained in
a nite region in the space domain, while a bandlimited function has a nite spectral width containing all spatial
frequency components with non-zero magnitude values. In the context of the FT, there are three main types of
functions: space-limited but not bandlimited (Type I), bandlimited but not space-limited (Type II) and neither
space-limited nor bandlimited (Type III). A function cannot be both space-limited and bandlimited39. A square
input, dened by
f(ξ,η)=rect(ξ/w)rect(η/w)
with
w=135
, was used as a sample Type I input. For Type II
functions, a two-dimensional sinc function, specically
f(ξ,η)=sinc(ξ/12)sinc(η/12)
, was used as a sample
input. Lastly, the two-dimensional Gaussian f(ξ,η)
=
exp
−
π
(ξ /γ )
2+
(η/γ )
2
with
γ=30
was used as
a sample Type III input. Referring to Fig.3, the simulation results demonstrate that the UFT-ACS is indeed an
accurate Fourier transformer.
For the sample square input, the root-mean-squared error (RMSE) aer normalization is 0.9%. e UFT’s
magnitude prole is in excellent agreement with the analytical FT’s magnitude prole (Fig.3a), with only minor
deviations towards the edges since the UFT expression is derived only aer Fresnel and paraxial approximations
are made (see Table1). us, they are expected to only agree in the central region. e phase proles (Fig.3b)
of the UFT and analytical FT only match well at the center for the same reason. Aberration due to the metalens’
discretized phase prole also contributes to the observed discrepancies in both the magnitude and phase proles.
Moreover, there is some aliasing due to bandlimiting as the square’s FT is not bandlimited.
For the sample sinc input, the RMSE aer normalization is 4.6%. Figure3c shows that there is a somewhat
good agreement between the magnitude proles of the UFT and the analytical FT. As explained by Gibbs’
phenomenon49 (see Supplementary Information), ripple artifacts can be observed in the UFT’s magnitude prole
as a result of truncating the input sinc function—an inevitable consequence of the UFT-ACS’ nite size. Aber-
ration due to the discretized phase prole of the metalens exacerbated the ripples and caused small lobes to be
observed at the sides (see Supplementary Information). Nonetheless, the overall shape of the UFT’s magnitude
prole still resembles that of the analytical FT. Figure3d shows that the phase of the UFT and the analytical FT
(3)
φ
ideal
x,y
=
k
x
2
+y
2
2f
Figure2. Ultrasonic metalens. (a) Top view of the ultrasonic metalens. (b) Unit cell made up of a SiO2
cylindrical post (light blue) embedded in a Si square cuboid (dark blue). (c) Discretized phase map of the
metalens.
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only match at the center, where the magnitude is signicant. is can be attributed to the fact that the UFT is
only achieved in the paraxial region and to the fact that the magnitude is supposedly zero but there are ripple
artifacts due to truncation. e latter implies that a small error in the real and imaginary parts of the complex
pressure eld results in non-negligible error in the phase. Metalens aberration further contributes to the error.
For the sample Gaussian input, the RMSE aer normalization is 0.4%. It is evident from Fig.3e that the mag-
nitude proles of the UFT and the analytical FT agree well with each other. Granted, there are some small lobes
towards the edges of the magnitude prole, which can be attributed to aberration due to the discretized phase
prole of the metalens (see Supplementary Information). Be that as it may, the signicant magnitude values are
accurate, and the overall shape is preserved. Moreover, it is also evident from Fig.3f that the phase prole of the
UFT coincides with that of the analytical FT only at the center. is is because of the approximations required
to obtain the UFT expression and aberration due to the discretized phase prole of the metalens.
Overall, for all three types of input functions, the UFT satisfactorily matches the analytical FT in both mag-
nitude and phase (see Table2 for summary of RMSEs). e UFT’s magnitude prole has somewhat noticeable
errors towards the edges, and the UFT’s phase prole, as can be observed from Fig.3, matches the analytical
Figure3. Ultrasonic Fourier Transform (UFT) Simulations. (a) Square: Magnitude proles of the UFT and
the analytical FT. (b) Square: Phase proles of the UFT and the analytical FT. (c) Sinc: Magnitude proles of the
UFT and the analytical FT. (d) Sinc: Phase proles of the UFT and the analytical FT. (e) Gaussian: Magnitude
proles of the UFT and the analytical FT. (f) Gaussian: Phase proles of the UFT and the analytical FT.
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FT’s phase prole at the center and when the magnitude is signicant. is is acceptable as, in general, only the
signicant magnitude values and their corresponding phase values are of interest in FT applications. Further-
more, the overall shapes of the UFT and analytical FT’s magnitude proles resemble each other well, which is
usually sucient for applications.
Optimization of accuracy. Having examined the accuracy of the UFT for all three types of inputs, this
section explores how the accuracy can be optimized through appropriate zero padding, truncation, and ban-
dlimiting.
Zero padding. Besides the zero padding done to the pressure eld array at the source plane to avoid circular
convolution errors associated with FFT-based convolution46–48, space-limited functions must also be zero-pad-
ded within the sampled array bounds
ξ,η∈ [−(L−�)/2, (L−�)/2]
.
To study the eect of zero padding on the UFT-ACS’ accuracy, MATLAB was used to simulate the UFT calcu-
lation for numerous square input functions of dierent side lengths. e side length
w
was varied incrementally
from 3 to 765, inclusive. Both the UFT and the analytical FT were normalized with the maximum value of the
analytical FT as 1. is is done so that a fair comparison of the errors can be made. Aer normalization, the
root-mean-squared error (RMSE) between the magnitude pattern of the UFT and that of the analytical FT was
calculated, using MATLAB’s mse and sqrt functions, for all
w
.
Figure4a shows how the RMSE varies with the side length
w
of the input square function, which is inversely
related to the amount of zero padding. It can be observed that the error initially decreases as the width
w
increases
or as the amount of zero padding decreases. is occurs because as
w
increases, the eective bandwidth of the
input function decreases, resulting in a decrease in error due to reduced aliasing in the spatial frequency domain.
e error eventually starts to increase as
w
continues increasing. is can be attributed to the fact that the UFT
is only achieved in the paraxial region. Refer to Supplementary Information for additional explanatory diagrams
supporting the above analysis.
us, a moderate number of zeros—neither too little nor too much—must be used to pad space-limited input
functions in order to accurately calculate the FT.
Table 2. RMSE for sample input functions.
Type of Input RMSE (%)
I 0.9
II 4.6
III 0.4
Figure4. Accuracy optimization: zero padding, truncation, and bandlimiting. (a) e graph shows the
dependence of the RMSE on the side length
w
of the input square, which is inversely related to the amount of
zero padding. (b) e graph shows the dependence of the RMSE on
γ
. e blue dots represent the RMSE of the
simulations involving an ideal metalens. e orange vertical line indicates the value of
γ
corresponding to the
case wherein the sampled array just contains 98% of the total spectral power.
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Truncation and bandlimiting. Truncation, otherwise known as windowing, refers to limiting the spatial extent
of a function that is not space-limited. Analogously, bandlimiting refers to limiting the bandwidth of a function
that is not bandlimited.
To study their impact on accuracy, UFT-ACS simulations were carried out for Gaussians with
γ
(the param-
eter aecting a Gaussian’s width) varied incrementally from 4 to 640, inclusive. Both the UFT (when using an
ideal metalens) and the analytical FT were normalized. Aer normalization, the RMSE was calculated for the
results of each value of
γ
. e UFT with an ideal metalens was used for comparison with the analytical FT so as
to eliminate the errors caused by aberration due to the discretized phase prole of the metalens and, therefore,
ascertain that the error is indeed ascribed to truncation and bandlimiting.
Figure4b shows how the RMSE varies with
γ
. Initially, the RMSE decreases as
γ
increases. Notably, referring
to the inset in Fig.4b, the initial decrease is very steep until the orange line, which corresponds to the value of
γ
for which the eective bandwidth is just contained within the sampled array bounds. However, this decreasing
trend is only observed until a certain critical point, beyond which the RMSE starts to increase again.
e initial decrease occurs because as
γ
increases, the eective bandwidth of the input function decreases,
resulting in a greater percentage of the total spectral power being captured within the sampled array bounds.
erefore, there is a decrease in error due to reduced aliasing in the spatial frequency domain. Moreover, the
paraxial approximations become more valid as
γ
initially increases because if
γ
is too small, the energy, which is
initially highly concentrated at the centre of the space domain, becomes very spread out in the spatial frequency
domain. Eventually, the RMSE stops decreasing and starts rising as
γ
increases further. ere are two reasons
for this. Firstly, for a higher
γ
, the energy is spread out more in the space domain, resulting in more signicant
magnitudes lying outside the sampled array bounds. us, there is an increase in error due to undersampling
in the space domain, outweighing the decrease in error due to reduced aliasing in the spatial frequency domain
associated with a higher
γ
. Secondly, the approximations are less valid when
γ
is too large because the energy,
initially spread out in the space domain, becomes highly concentrated at the center of the spatial frequency
domain. Supplementary Information provides additional graphs supporting the above reasoning.
Hence, to optimize the accuracy of the UFT, the parameter
γ
should be neither too small nor too large such
that the input function and its FT are both of moderate width and the approximations are more valid.
Discussion
In summary, this paper presents the Ultrasonic Fourier Transform Analog Computing System (UFT-ACS), which
has been demonstrated to perform FT calculations for all three types of functions to a relatively high degree of
accuracy. e simulations in this study have shown the true capabilities, appropriately qualied by the limita-
tions, of the UFT-ACS—addressing this knowledge gap in the existing literature. Optimizing the UFT’s accuracy
was also explored and better understood by studying the eect of zero padding, truncation and bandlimiting.
is study’s ndings are of considerable signicance. Performing FT calculations faster than the electronic
FFT algorithm, the UFT-ACS satises the growing demand for such capabilities in some applications like real-
time image processing in autonomous vehicles. Existing analog computing systems also make use of the FT to
perform mathematical operations, such as spatial dierentiation, integration, and convolution. us, the UFT-
ACS can also impact the broader eld of wave-based analog computing. It can improve the prospects of wave-
based analog computers as potential supercomputers in the future, possibly surpassing the current limitations
of today’s electronic computers.
Methods
Ultrasonic metalens designing process. Referring to Supplementary Fig.S3, designing the metalens
involves a few simple steps. Firstly, carry out unit cell simulations in order to obtain a relationship between the
phase shi due to a particular unit cell and the radius of that unit cell’s cylindrical post. Secondly, obtain an array
of the ideal phase map consisting of phase values at sampled points following the theoretical paraboloidal phase
prole
required to obtain the UFT expression. irdly, perform interpolation to the nearest available phase value from
the unit cell simulations using the MATLAB function interp1. us, at this juncture, we have obtained the
discretized phase map consisting of phase values which have a corresponding radius from the unit cell simula-
tions. Subsequently, use the phase-to-radius mapping to obtain a radius map—an array of radius values at each
sampled point. Finally, using the MATLAB function viscircles, generate a gure of the metalens comprising
unit cells whose cylindrical posts have a radius corresponding to the radius at that point as per the radius map
previously obtained.
Wave propagation simulations. Exact solutions—that is, before Fresnel and paraxial approximations
were applied—of the Kirchho-Helmholtz Integral were used to numerically simulate the propagation of ultra-
sonic waves through the UFT-ACS, and the results were compared with the analytical FT. e code was imple-
mented using MATLAB.
As opposed to Finite Element Method (FEM), this semi-analytical approach is much less computationally
costly and, therefore, allows for simulations involving considerably larger arrays—key to understanding the
UFT-ACS’ true capabilities.
e pressure eld
PM−
(
x,y)
right in front of the metalens can be obtained by using an FFT-based convolution
approach to convolve the zero-padded input pressure eld array
PS(ξ,η)
with the convolution kernel
(4)
φ
ideal
x,y
=
k
x
2
+y
2
2f
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Note that the
N×N
array
PS(ξ,η)
must be padded by at least
N−1
zeros to avoid circular convolution
errors47. By convention, the convolution kernel array is the same size as the zero-padded pressure eld array47.
PM−(x,y)
is then the
N×N
subarray at the center of the larger array produced by FFT-based convolution. e
second part of the simulation involves applying the phase shi due to the discretized metalens to obtain the
N×N
pressure eld
PM+(x,y)
right behind the metalens. It can be obtained by the element-wise multiplication
of the
N×N
array
PM−
x,y
and the
N×N
array
exp
iφ
discretized
, where
φdiscretized
is the discretized phase
prole of the metalens aer interpolating each phase shi value to the closest available phase data from the unit
cell simulations. Subsequently, the
N×N
output pressure eld array
PO(u,v)
is obtained by using FFT-based
convolution to convolve the zero-padded array
PM+(x,y)
with the convolution kernel
Finally, the UFT result is obtained by multiplying the
N×N
pressure eld array
Po(u,v)
at the observation
plane by the correction factor
α=−
jfexp
2jkf
.
See Supplementary Information for the full derivation, which is partly the independent work of the authors.
Simulation parameters. Choosing the appropriate values of the simulation parameters is important as
this aects the accuracy of the UFT as well as the design of the physical system to be fabricated.
e selection of simulation parameters is a three-step process. Firstly, as required by convolution47, the spac-
ing
between the sampled points of the pressure elds in the source, metalens and observation planes must be
the same as the spacing between adjacent metalens unit cells. Secondly, depending on the specic application
for which the UFT-ACS is being used, an appropriate length
L
for the source, metalens and observation planes
must be chosen. Zero padding must be moderate; truncation and bandlimiting must be done appropriately such
that the signicant space and spatial frequency components are within the sampled array bounds. irdly, the
focal length
f
must satisfy
which is derived from a consideration of the sampling requirements of the convolution kernels’ exponential
phase term36,44–48. A detailed explanation for each step is oered in the Supplementary Information. Table3
summarizes the values of the simulations’ parameters.
Data availability
e datasets generated during and/or analyzed during the current study are available from the corresponding
author on reasonable request.
Received: 18 June 2022; Accepted: 30 September 2022
References
1. Zangeneh-Nejad, F., Sounas, D. L., Alù, A. & Fleury, R. Analogue computing with metamaterials. Nat. Rev. Mater. 6, 207–225
(2020).
2. Cheng, K. et al. Optical realization of wave-based analog computing with metamaterials. Appl. Sci. 11, 141 (2020).
3. Zangeneh-Nejad, F. & Fleury, R. Performing mathematical operations using high-index acoustic metamaterials. New J. Phys. 20,
073001 (2018).
4. Hwang, J., Davaji, B., Kuo, J. & Lal, A. Focusing proles of planar Si-SiO2 metamaterial ghz frequency ultrasonic lens. In 2021
IEEE International Ultrasonics Symposium (IUS) 1–4 (IEEE, 2021). https:// doi. org/ 10. 1109/ IUS52 206. 2021. 95935 77.
5. MacLennan, B. J. e promise of analog computation. Int. J. Gen Syst 43, 682–696 (2014).
6. Cordaro, A. et al. High-index dielectric metasurfaces performing mathematical operations. Nano Lett. 19, 8418–8423 (2019).
7. Abdollahramezani, S., Chizari, A., Dorche, A. E., Jamali, M. V. & Salehi, J. A. Dielectric metasurfaces solve dierential and integro-
dierential equations. Opt. Lett. 42, 1197 (2017).
8. Rajabalipanah, H., Momeni, A., Rahmanzadeh, M., Abdolali, A. & Fleury, R. A single metagrating metastructure for wave-based
parallel analog computing. arXiv: 2110. 07473 [physics] (2021).
(5)
h
1(ξ,η)
=
jexp
−jk
√
f2+ξ2+η2
√
f2
+
ξ2
+
η2
.
(6)
h
2
x,y
=
1
√f2
+
x2
+
y2
+
jk
fexp
−jk
√
f2+x2+y2
2π(f2
+
x2
+
y2)
.
(7)
f
≥
2(L−�)�
2
−
(L
−
�)2
,
Table 3. Summary of parameter values.
Parameter Val u e
3µm
L
771µm
fwave
1.7GHz
vwave
5880m s-1
f
1088.6µm
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9. Zuo, S., Wei, Q., Tian, Y., Cheng, Y. & Liu, X. Acoustic analog computing system based on labyrinthine metasurfaces. Sci. Rep. 8,
1 (2018).
10. Silva, A. et al. Performing mathematical operations with metamaterials. Science 343, 160–163 (2014).
11. Yousse, A., Zangeneh-Nejad, F., Abdollahramezani, S. & Khavasi, A. Analog computing by Brewster eect. Opt. Lett. 41, 3467
(2016).
12. Barrios, G. A., Retamal, J. C., Solano, E. & Sanz, M. Analog simulator of integro-dierential equations with classical memristors.
Sci. Rep. 9, 1 (2019).
13. AbdollahRamezani, S., Arik, K., Khavasi, A. & Kavehvash, Z. Analog computing using graphene-based metalines. Opt. Lett. 40,
5239 (2015).
14. Sihvola, A. Enabling optical analog computing with metamaterials. Science 343, 144–145 (2014).
15. Zhou, Y. et al. Analog optical spatial dierentiators based on dielectric metasurfaces. Adv. Opt. Mater. 8, 1901523 (2019).
16. Kou, S. S. et al. On-chip photonic Fourier transform with surface plasmon polaritons. Light Sci. Appl. 5, 16034 (2016).
17. Cauleld, H. J. & Dolev, S. Why future supercomputing requires optics. Nat. Photon. 4, 261–263 (2010).
18. Zhou, Y., Zheng, H., Kravchenko, I. I. & Valentine, J. Flat optics for image dierentiation. Nat. Photon. 14, 316–323 (2020).
19. Bykov, D. A., Doskolovich, L. L., Bezus, E. A. & Soifer, V. A. Optical computation of the Laplace operator using phase-shied Bragg
grating. Opt. Express 22, 25084 (2014).
20. Karimi, P., Khavasi, A. & Mousavi Khaleghi, S. S. Fundamental limit for gain and resolution in analog optical edge detection. Opt.
Express 28, 898 (2020).
21. Lv, Z., Ding, Y. & Pei, Y. Acoustic computational metamaterials for dispersion Fourier transform in time domain. J. Appl. Phys.
127, 123101 (2020).
22. Liu, Y., Kuo, J., Abdelmejeed, M. & Lal, A. Optical measurement of ultrasonic fourier transforms. In 2018 IEEE International
Ultrasonics Symposium (IUS) 1–9 (2018). https:// doi. org/ 10. 1109/ ULTSYM. 2018. 85799 38.
23. Hwang, J., Kuo, J. & Lal, A. Planar GHz ultrasonic lens for fourier ultrasonics. In 2019 IEEE International Ultrasonics Symposium
(IUS) 1735–1738 (2019). https:// doi. org/ 10. 1109/ ULTSYM. 2019. 89256 62.
24. Hwang, J., Davaji, B., Kuo, J. & Lal, A. Planar lens for GHz fourier ultrasonics. In 2020 IEEE International Ultrasonics Symposium
(IUS) 1–4 (2020). https:// doi. org/ 10. 1109/ IUS46 767. 2020. 92516 14.
25. Kwon, H., Sounas, D., Cordaro, A., Polman, A. & Alù, A. Nonlocal metasurfaces for optical signal processing. Phys. Rev. Lett. 121,
173004 (2018).
26. Zuo, S.-Y., Wei, Q., Cheng, Y. & Liu, X.-J. Mathematical operations for acoustic signals based on layered labyrinthine metasurfaces.
Appl. Phys. Lett. 110, 011904 (2017).
27. Zuo, S.-Y., Tian, Y., Wei, Q., Cheng, Y. & Liu, X.-J. Acoustic analog computing based on a reective metasurface with decoupled
modulation of phase and amplitude. J. Appl. Phys. 123, 091704 (2018).
28. Lv, Z., Liu, P., Ding, Y., Li, H. & Pei, Y. Implementing fractional Fourier transform and solving partial dierential equations using
acoustic computational metamaterials in space domain. Acta. Mech. Sin. https:// doi. org/ 10. 1007/ s10409- 021- 01139-2 (2021).
29. MohammadiEstakhri, N., Edwards, B. & Engheta, N. Inverse-designed metastructures that solve equations. Science 363, 1333–1338
(2019).
30. Zhu, T. et al. Plasmonic computing of spatial dierentiation. Nature Commun. 8, 1 (2017).
31. Zangeneh-Nejad, F. & Fleur y, R. Topological analog signal processing. Nature Commun. 10, 1 (2019).
32. Abdollahramezani, S., Hemmatyar, O. & Adibi, A. Meta-optics for spatial optical analog computing. Nanophotonics 9, 4075–4095
(2020).
33. Solli, D. R. & Jalali, B. Analog optical computing. Nat. Photon. 9, 704–706 (2015).
34. Guo, C., Xiao, M., Minkov, M., Shi, Y. & Fan, S. Photonic crystal slab Laplace operator for image dierentiation. Optica 5, 251
(2018).
35. Macfaden, A. J., Gordon, G. S. D. & Wilkinson, T. D. An optical Fourier transform coprocessor with direct phase determination.
Sci. Rep. 7, 1 (2017).
36. Goodman, J. W. Introduction to Fourier Optics (W. H. Freeman and Company, Hoboken, 2017).
37. James, J. F. A Student’s Guide to Fourier Transforms: With Applications in Physics and Engineering (Cambridge University Press,
Cambridge, 2015).
38. Oran Brigham, E. e Fast Fourier Transform and its Applications (Prentice Hall, Hoboken, 1988).
39. Stark, H. Applications of Optical Fourier Transforms (Academic Press, Cambridge, 1982).
40. Juvells, I., Vallmitjana, S., Carnicer, A. & Campos, J. e role of amplitude and phase of the Fourier transform in the digital image
processing. Am. J. Phys. 59, 744–748 (1991).
41. Gonzalez, R. C. & Woods, R. E. Digital Image Processing (Pearson, London, 2018).
42. Dueux, P. M. e Fourier Transform and its Applications to Optics (Wiley, Hoboken, 1983).
43. Beekes, M., Lasch, P. & Naumann, D. Analytical applications of Fourier transform-infrared (FT-IR) spectroscopy in microbiology
and prion research. Vet. Microbiol. 123, 305–319 (2007).
44. Voelz, D. G. Computational Fourier Optics: A MATLAB Tutorial (Spie Press, Bellingham, 2010).
45. Voelz, D. G. & Roggemann, M. C. Digital simulation of scalar optical diraction: Revisiting chirp function sampling criteria and
consequences. Appl. Opt. 48, 6132 (2009).
46. Zhang, H., Zhang, W. & Jin, G. Adaptive-sampling angular spectrum method with full utilization of space-bandwidth product.
Opt. Lett. 45, 4416–4419 (2020).
47. Zhang, W., Zhang, H., Sheppard, C. J. R. & Jin, G. Analysis of numerical diraction calculation methods: From the perspective of
phase space optics and the sampling theorem. J. Opt. Soc. Am. A 37, 1748 (2020).
48. Zhang, W., Zhang, H. & Jin, G. Frequency sampling strategy for numerical diraction calculations. Opt. Express 28, 39916 (2020).
49. Riley, K. F. & Hobson, M. P. Mathematical Methods for Physics and Engineering: A Comprehensive Guide (Cambridge University
Press, Cambridge, 2008).
Author contributions
R.F.U. planned the research project, mathematically modeled the ultrasonic Fourier transform, carried out the
simulations, and analyzed the data. B.V.P. initiated the project and supervised R.F.U. Both authors reviewed the
manuscript.
Competing interests
e authors declare no competing interests.
Additional information
Supplementary Information e online version contains supplementary material available at https:// doi. org/
10. 1038/ s41598- 022- 21753-9.
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