![Ordinary Differential Equation (ODE) Simulations. (a) Sinc function: Input (left) and output (right) magnitude profiles for w=18\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w = 18$$\end{document}. The blue line with circled data points indicates the simulated output of the ACS, whereas the orange line shows the analytical solution. (b) Gaussian function: Input (left) and output (right) magnitude profiles for γ=48\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = 48$$\end{document}. The blue line with circled data points indicates the simulated output of the ACS, whereas the orange line shows the analytical solution. (c) Relationship between the RMSE and the parameter 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} for the Sinc function. (d) Relationship between the RMSE and the parameter 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} for the Gaussian function.](https://www.researchgate.net/publication/373214380/figure/fig3/AS:11431281206225870@1700628390719/Ordinary-Differential-Equation-ODE-Simulations-a-Sinc-function-Input-left-and_Q320.jpg)
![Partial Differential Equation (PDE) Simulations. (a) Sinc function: Input (left) and output (right) magnitude profiles for w=18\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w=18$$\end{document}. The blue line with circled data points indicates the simulated output of the ACS, whereas the orange line shows the analytical solution. (b) Gaussian function: Input (left) and output (right) magnitude profiles for γ=18\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =18$$\end{document}. The blue line with circled data points indicates the simulated output of the ACS, whereas the orange line shows the analytical solution. (c) Relationship between the RMSE and the parameter 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} for the Sinc function. (d) Relationship between the RMSE and the parameter 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} for the Gaussian function.](https://www.researchgate.net/publication/373214380/figure/fig4/AS:11431281206293336@1700628390855/Partial-Differential-Equation-PDE-Simulations-a-Sinc-function-Input-left-and_Q320.jpg)
Access to this full-text is provided by Springer Nature.
Content available from Scientific Reports
This content is subject to copyright. Terms and conditions apply.
1
Vol.:(0123456789)
Scientic Reports | (2023) 13:13471 | https://doi.org/10.1038/s41598-023-38718-1
www.nature.com/scientificreports
Solving ordinary and partial
dierential equations using
an analog computing system based
on ultrasonic metasurfaces
Robert Frederik Uy
1* & Viet Phuong Bui
2
Wave-based analog computing has recently emerged as a promising computing paradigm due to its
potential for high computational eciency and minimal crosstalk. Although low-frequency acoustic
analog computing systems exist, their bulky size makes it dicult to integrate them into chips
that are compatible with complementary metal-oxide semiconductors (CMOS). This research paper
addresses this issue by introducing a compact analog computing system (ACS) that leverages the
interactions between ultrasonic waves and metasurfaces to solve ordinary and partial dierential
equations. The results of our wave propagation simulations, conducted using MATLAB, demonstrate
the high accuracy of the ACS in solving such dierential equations. Our proposed device has the
potential to enhance the prospects of wave-based analog computing systems as the supercomputers
of tomorrow.
A plethora of electronic and mechanical analog computers have been developed in the past two millennia to solve
mathematical equations and perform mathematical operations with increased eciency1–3, but they were later
replaced by more advanced digital computers2,3. In view of the recent advancements in the eld of metamaterials,
interest in analog computing has been revived, with the focus being on wave-based analog computing1,3,8. ese
new computing systems leverage the properties of waves and metasurfaces to solve mathematical equations and
perform mathematical operations to satisfy the need for ever-greater computational eciency and capacity6,7
amidst the grim outlook for further augmentation of digital computers as Moore’s law approaches its physical
limitations2,4,5.
Due to their powerful parallel processing, high computational eciency, and minimal crosstalk, wave-based
analog computing systems have been hailed as a potential future of computing1,8,9. It was the pioneering work
of Silva etal.10 on computational metamaterials that set the stage for subsequent research into analog comput-
ing systems that perform mathematical operations and solve equations1–4,6–35,37, with a subset of these focusing
on the use of the Fourier transform (FT) to do so3,6,9,10,26–28,37. More recently, Zangeneh-Nejad etal. provided a
well-written, comprehensive overview of recent developments in this eld as a whole1.
In the realm of acoustics, Zuo etal. designed and tested an acoustic analog computing system based on laby-
rinthine metasurfaces to solve
n
th-order inhomogeneous ordinary dierential equations9. Many other studies
on acoustic analog computing systems have also been carried out, but all such systemsoperate in the kilohertz
(kHz) frequency range3,9,26–28. Even when thin planar metasurfaces are used, a physically bulky computing sys-
tem is required for analog computing at such low frequencies (long wavelengths). In this paper, we propose a
solution to this problem: a compact ultrasonic analog computing system (ACS) with a working frequency in the
gigahertz (GHz) range. Due to the relatively shorter wavelength of GHz ultrasonic waves, our proposed ACS is
far less bulky and can consequently be easily integrated into CMOS-compatible chips.
is paper is organized as follows. Firstly, we present the ACS’ architecture and elaborate on its working
principle. Next, the ability of the ACS to solve dierential equations is demonstrated, including a comprehen-
sive error and accuracy optimization analysis for each type of dierential equation and each type of function.
Following this, we discuss our study’s key ndings and conclusions, relating them to the wider context of wave-
based analog computing. Finally, we provide a comprehensive account of our research methodology—including
OPEN
1Hwa Chong Institution, 661 Bukit Timah Road, Singapore 269734, Singapore. 2Institute of High Performance
Computing (IHPC), Agency for Science, Technology and Research (A*STAR), 1 Fusionopolis Way, #16-16 Connexis,
Singapore 138632, Republic of Singapore. *email: robertfrederikduy@gmail.com
Content courtesy of Springer Nature, terms of use apply. Rights reserved
2
Vol:.(1234567890)
Scientic Reports | (2023) 13:13471 | https://doi.org/10.1038/s41598-023-38718-1
www.nature.com/scientificreports/
information on the design process for the ultrasonic metalens, the simulation of wave propagation through the
ACS, and important considerations for the selection of simulation parameters.
Architecture and working principle of the analog computing system (ACS)
Architecture. Our proposed ACS (Fig.1a) is made up of three key components: an Ultrasonic Fourier
Transform (UFT) block, a spatial ltering metasurface (SFM), and another UFT block. e pressure elds at
the input and output planes of the ACS are
PI
x,y
and
PO
x,y
, respectively. e side length of the entire ACS’
square cross-section is
L
.
Referring to Fig.1b, each UFT block consists of three main parts: a fused silica substrate layer, the ultrasonic
metalens, followed by another fused silica substrate layer. If the input pressure eld of the ultrasonic wave that
is made to pass through the UFT block is
P
, the output pressure eld obtained at the other end of the UFT block
is
PFT
, which has been shown to be proportional to
P
’s Fourier transform37. A key condition for obtaining the
UFT through this block is that boththe thickness of the substrate layers and the focal length of the metalens
must be
f
37. e metalens’ thickness is
tm
, as shown in Fig.1b.
Our proposed ACS is designed to operate at a frequency of
fwave =1.7
GHz, which is a high ultrasonic fre-
quency. is enables greater compactness, making it easier to integrate the ACS into CMOS-compatible chips.
Each substrate layer has a thickness of
f=1.0886
mm and is made of fused silica (which we chose for its isotropy
as a material). In fused silica, the speed of ultrasonic waves is
vwave =5880
m s-1, from which we can calculate
the wavelength to be
=vwave/fwave =3.46
µm.
In Fig.2a, the metalens is made up of several unit cells, eachof which has a thickness of
tm=16
µm and
a square cross-section of side length 3µm (a subwavelength feature). Each unit cell (Fig.2b) is composed of
a square cuboid made of Si with a cylindrical post made of SiO2 embedded in it. According to the theoretical
working principle of the ACS, the ultrasonic metalens ought to obey a paraboloidal phase prole
such that the pressure eld
PFT
would be proportional to the FT of
P
. Due to the limited number of distinct
unit cells available, however, discretization is required. erefore, the cylindrical post radius of each unit cell
must correspond to the interpolated phase shi at that point (aer discretization). e process of interpolation
transforms the ideal phase map (Fig.2c) into the discretized phase map (Fig.2d) that is later used for phase-
to-radius mapping.
In addition, the transmission coecient function
T
x,y
of the SFM must correspond to the transfer function
(TF)
H
k
x
,k
y
required to solve a particular ordinary or partial dierential equation.
Working principle. Uy and Bui37 have previously determined that the input
P
and the output
PFT
of the
UFT block are approximately (see Table1 for list of approximations)related by
(1)
φ
ideal
x,y
=
k
x
2
+y
2
2f
Figure1. Schematic of the ACS and the UFT block. (a) e gure features a schematic of the proposed ACS,
which consists of three main parts: a UFT block, an SFM, and another UFT block. (b) e gure features a
schematic of the UFT block, which has three key components: a substrate layer, the ultrasonic metalens, and
another substrate layer.
Content courtesy of Springer Nature, terms of use apply. Rights reserved
3
Vol.:(0123456789)
Scientic Reports | (2023) 13:13471 | https://doi.org/10.1038/s41598-023-38718-1
www.nature.com/scientificreports/
where
j
is the imaginary unit,
is the wavelength,
k
is the wavenumber, and the operator
F
denotes the FT.
Now, let f
x,y
and
g
x,y
be the input and output, respectively, of a particular ODE or PDE. It can be shown
that f
x,y
and
g
x,y
are mathematically related by the equation
where the operator
F−1
denotes the inverse FT and
H
k
x
,k
y
is the transfer function for a certain ODE or PDE.
At rst glance, Eqs.(2) and (3) might seem tosuggest that the ACS cannot compute the accurate result.
For one, the UFT block yields an output
PFT
that is only proportional to—but not actually equal to—the FT
of the input
P
. Moreover, it is essential to note that the correct output is obtained by taking the inverse FT of
H
k
x
,k
y
F
f
x,y
, whereas the second UFT block calculates the FT (not the inverse FT). However, these
concerns do not actually hinder the ACS from yielding the desired output. In fact, the mirror image of the cor-
rectoutput g
x,y
is given by the equation
(2)
P
FT (u,v)=jexp
(
−2jkf
)
f
F{P(ξ,η)}
,
(3)
g
x,y
=
F
−1
H
k
x
,k
y
F
f
x,y
,
Figure2. Ultrasonic Metalens. (a) Ultrasonic metalens – top view. (b) Unit cell – a SiO2 cylindrical post (gold)
embedded in a Si square cuboid (dark blue). (c) Ideal Phase Map. (d) Discretized Phase Map. Adapted from Ref.
37.
Table 1. List of approximations required to achieve
PFT ∝F{P}
.
Approximation Validity
|
r−r0|≈f
1+1
2
x−ξ
f
2
+1
2
y−η
f
2
Fresnel
|r−r0|≈f
Paraxial
cos (n,r−r0)≈1
Paraxial
1
|
r
−
r
0|
+jk ≈
jk
Distances much larger than
Content courtesy of Springer Nature, terms of use apply. Rights reserved
4
Vol:.(1234567890)
Scientic Reports | (2023) 13:13471 | https://doi.org/10.1038/s41598-023-38718-1
www.nature.com/scientificreports/
which is the additive inverse of the magnitude of the ACS’ output
PO
x,y
. erefore, to achieve the desired
result
g
x,y
, we simplyneed to take the mirror image of the ACS’ output
PO
x,y
and subsequently obtain its
additive inverse.
It is also important to recognize that the transmission coecient can only have amplitudes of less than or
equal to 1. If this condition is satised by
H
k
x
,k
y
, then all is well, and
H
k
x
,k
y
would also be the transmission
coecient function. However, this is not necessarily satised by all transfer functions. In such cases, we would
have to normalize the transfer function such that it satises this condition. Consequently, we would also not
obtain the exact output
g
x,y
but rather a scaled version of it. Nevertheless, we can recover the desired result
by appropriately rescaling the output.
Results
A function that is space-limited has non-zero magnitude values only within in a nite region in the space domain,
whereas a function that is bandlimited is one thathas a nite spectral width that includes all spatial frequency
components with non-zero magnitude values. It should be noted that it is not possible for a function to be both
space-limited and bandlimited38. Furthermore, functions that are space-limited but not bandlimited, such as
the rect function, are not of particular interest in the context of solving dierential equations. In this paper, we
thus consider two main kinds of functions: (1) bandlimited but not space-limited and (2) neither space-limited
nor bandlimited.
In the wave propagation simulations, we used the Sinc and Gaussian functions as archetypal examples for
each kind. e Sinc functions follow the general form
f(x)=sinc(x/w)
, where the parameter
w∈R+
is an
indication of the Sinc function’s geometric spread. e Gaussian functions, on the other hand, take the form
f(x)
=
exp
−
πx
2
/γ
2
, where the parameter
γ∈
R
+
serves as a measure of the Gaussian function’s geometric
spread. Additionally, we chose the root-mean-squared error (RMSE) aer normalization as the error metric used
to assess the accuracy of the ACS’ output, relative to the analytical solution.
Ordinary dierential equations (ODE). Mathematical basis. Generalizing the derivative property of
the FT39,
Consider a general
n
th-order inhomogeneous ODE
Taking the FT of both sides of Eq.(6), we obtain
from which we can deduce that
Note that Eq.(8) can be re-expressed as
By denition, the spatial frequencies are
kx=−2πx/f
and
ky=−2πy/f
, so we have
dkx=−
2π/f
dx
and
dky=−
2π/f
dy
. erefore, using these substitutions and then replacing
x
with
−x
and
y
with
−y
,
It is apparent from Eq.(10) that
f(x)
is the input and
is the transfer function needed to solve nth-order inhomogeneous ODEs of the form given in Eq.(6)9.
Simulation results. In the simulations, we used the ODE
(4)
g
−x,−y
=
1
(
f
)
2F
H
kx,ky
F
f
x,y
,
(5)
F
d
n
f(x)
dxn
=
jkx
n
F
f(x)
.
(6)
f
(
x
)=
cng(n)
(
x
)+
cn−
1
g(n
−
1)
(
x
)+...+
c
1
g
′(
x
)+
c
0
g
(
x
)
.
(7)
F
f(x)
=F
g(x)
n
i=0
ci
jkx
i
,
(8)
g(x)=F−1
n
i
=
0
ci
jkx
i
−1
F
f(x)
.
(9)
g(x)
=
1
4π2
∞
−∞
n
i=0
ci
jkx
i
−1
F
f(x)
exp
j
kxx
+
kyy
dkxdky
.
(10)
g
(−x)=1
(f)2
∞
−∞
n
i
=
0
ci
jkx
i
−1
F
f(x)
exp
−j
kxx+kyy
dxdy =1
(f)2F
n
i
=
0
ci
jkx
i
−1
F
f(x)
,
(11)
H
(kx)
=
n
i=0
ci
jkx
i
−1
(12)
4g′′(x)−8g′(x)+16g(x)=f(x).
Content courtesy of Springer Nature, terms of use apply. Rights reserved
5
Vol.:(0123456789)
Scientic Reports | (2023) 13:13471 | https://doi.org/10.1038/s41598-023-38718-1
www.nature.com/scientificreports/
For the rst type of function, the solution
g(x)
is a Sinc function with parameter
w=18
. e RMSE aer
normalization, based on our simulation, was 0.00737. From Fig.3a, we can observe that the analytical solution
Figure3. Ordinary Dierential Equation (ODE) Simulations. (a) Sinc function: Input (le) and output (right)
magnitude proles for
w=18
. e blue line with circled data points indicates the simulated output of the ACS,
whereas the orange line shows the analytical solution. (b) Gaussian function: Input (le) and output (right)
magnitude proles for
γ
=
48
. e blue line with circled data points indicates the simulated output of the ACS,
whereas the orange line shows the analytical solution. (c) Relationship between the RMSE and the parameter
w
for the Sinc function. (d) Relationship between the RMSE and the parameter
γ
for the Gaussian function.
Content courtesy of Springer Nature, terms of use apply. Rights reserved
6
Vol:.(1234567890)
Scientic Reports | (2023) 13:13471 | https://doi.org/10.1038/s41598-023-38718-1
www.nature.com/scientificreports/
and the ACS’ output are in excellent agreement, especially near the center. ere are some minor discrepancies
towards the edges, but these are actually expected since we know that the FT is only obtained in the paraxial
region (see approximations listed in Table1). Furthermore, metalens aberration—as a result of discretization of
the metalens’ phase prole—also contributes, albeit very minimally, to the observed deviations. It should also be
noted that there is some undersampling due to truncation of the input function
f(x)
as well as aliasing arising
from bandlimiting of the output function
g(x)
.
e simulation for the second type of function was conducted with a Gaussian function with parameter
γ=48
as the solution
g(x)
. In this case, the RMSE aer normalization was found to be 0.00975. Figure3b shows
that the output of the ACS and the analytical solution agree very well with each other. Granted, there are small
lobes towards the edges, largely due to metalens aberration as well as the paraxial approximation requirement
not being met. ere are also minor errors associated with undersampling of the input and bandlimiting of the
output. Be that as it may, the output nonetheless matches the analytical solution very well.
Optimization of accuracy. Figures3c and d show the relationship between the RMSE and the geometric spread
parameters
w
and
γ
, respectively, for solving ODEs. We can observe that the RMSE initially decreases then
increases as
w
or
γ
is increased. is two-part trend in the RMSE is mainly caused by two factors: (1) the level of
aliasing and undersampling and (2) the validity of the paraxial approximations.
Firstly, as
w
or
γ
is initially increased, there is reduced aliasing in the output plane of the rst UFT block and
reduced undersampling in the input plane of the second UFT block, resulting in a fall in the RMSE. However,
as
w
or
γ
continues to increase past a certain threshold, there is increased undersampling in the input plane of
the rst UFT block and increased aliasing in the output plane of the second UFT block that drive theobserved
rise in the RMSE.
Secondly, the paraxial approximations initially become more valid as
w
or
γ
increases, then it becomes less
valid as
w
or
γ
increases further. As
w
or
γ
initially increases, the energy becomes less highly concentrated near
the center of the rst UFT block’s input plane and less extensively spread out in the rst UFT block’s output
plane. e energy also becomes less extensively spread out in the second UFT block’s input plane and less highly
concentrated near the center of the second UFT block’s output plane. en, as
w
or
γ
increases further, the energy
becomes more extensively spread out in the rst UFT block’s input plane and more highly concentrated near
the center of the rst UFT block’s output plane. Moreover, the energy becomes more highly concentrated near
the center of the second UFT block’s input plane and more extensively spread out in the second UFT block’s
output plane.
erefore, to optimize accuracy, the input function
f(x)
can be scaled parallel to
x
such that the geometric
spread parameter
w
or
γ
takes on a moderate value. e output
PO
x,y
of the ACS can then be appropriately
rescaled back to obtain the desired solution
g(x)
.
Refer to Supplementary Figs.S2 to S5 for additional diagrams in support of the above explanation.
Partial dierential equations (PDE). Mathematical basis. Consider the partial dierential equation
Taking the FT of both sides of Eq.(13),
which can be re-arranged to get
In its integral form, Eq.(15) can be re-written as
Substituting
dkx=−
2π/f
dx
and
dky=−
2π/f
dy
and replacing
x
with
−x
and
y
with
−y
then yield
erefore, f
x,y
is the input and
is the transfer function needed to solve PDEs of the specied form28.
Simulation results. In the simulations, we used the PDE
(13)
κ1∇2
g
x,y
+
κ
2
g
x,y
=
f
x,y
.
(14)
F
f
x,y
=
−κ1
k2
x+k2
y
+κ2
F
g
x,y
,
(16)
g
x,y
=F−1
F{f(x,y)}
−
κ1
k2
x
+
k2
y
+
κ2
.
(16)
g
x,y
=1
4π2
∞
−∞
F{f(x,y)}
−
κ1
k2
x
+
k2
y
+
κ2
exp
j
kxx+kyy
dkxdky
.
(17)
g
−x,−y
=1
(f)2
∞
−∞
F{f(x,y)}
−κ1
k2
x+k2
y
+κ2
exp
−j
kxx+kyy
dxdy =1
(f)2F
F{f(x,y)}
−κ1
k2
x+k2
y
+κ2
.
(18)
H
kx,ky
=
1
−
κ1
k2
x
+
k2
y
+
κ2
Content courtesy of Springer Nature, terms of use apply. Rights reserved
7
Vol.:(0123456789)
Scientic Reports | (2023) 13:13471 | https://doi.org/10.1038/s41598-023-38718-1
www.nature.com/scientificreports/
e simulation for the rst type of function was carried out with a two-dimensional Sinc function with
parameter
w=18
as the solution
g
x,y
. e RMSE aer normalization, based on our simulation, was 0.00643.
We can observe from Fig.4a that the ACS’ output and the analytical solution are in excellent agreement, par-
ticularly near the center. ere are some small deviations towards the edges, which can be attributed to the
paraxial approximation not being met. In addition, metalens aberration also contributes to the discrepancies.
ere is also some undersampling due to truncation of the input function f
x,y
as well as aliasing arising from
bandlimiting of the output function
g
x,y
.
For the second type of function, the solution g
x,y
is a two-dimensional Gaussian function with parameter
γ=18
. In this case, the RMSE aer normalization was found to be 0.00266. Figure4b shows that the output
of the ACS and the analytical solution agree very well with each other, especially in the paraxial region for the
same reason cited above. Due to metalens aberration as well as the paraxial approximation requirement not
being met, there are some small lobes towards the edges. What is more, undersampling of the input f
x,y
and
aliasing of the output function
g
x,y
also contribute to the observed discrepancies. Nonetheless, the output for
the signicant magnitude values matches the analytical solution very well.
Optimization of accuracy. Figure4c and d show the relationship between the RMSE and the geometric spread
parameters
w
and
γ
, respectively, for solving PDEs. We can observe that as
w
or
γ
is increased, the RMSE initially
falls and subsequently rises. is trend in the RMSE can be largely ascribed to two key factors: (1) the level of
undersampling and aliasing and (2) the paraxial approximations’ validity.
Firstly, as
w
or
γ
is initially increased, there is less aliasing in the rst UFT block’s output plane and less
undersampling in the second UFT block’s input plane, so the RMSE decreases. However, as
w
or
γ
continues
to increase beyond a certain value, there is increased undersampling in the rst UFT block’s input plane and
increased aliasing in the second UFT block’s output plane, resulting in the observed increase in the RMSE.
Secondly, the paraxial approximations initially become more valid as
w
or
γ
initially increases, then it becomes
less valid as
w
or
γ
increases further. Initially, as
w
or
γ
increases, the energy becomes less highly concentrated
near the center of the input plane of the rst UFT block and less extensively spread out in the output plane of
the rst UFT block. e energy also becomes less extensively spread out in the input plane of the second UFT
block and less highly concentrated near the center of the output plane of the second UFT block. en, as
w
or
γ
increases further, the energy becomes more extensively spread out in the input plane of the rst UFT block and
more highly concentrated near the center of the output plane of the rst UFT block. Additionally, the energy
becomes more highly concentrated near the center of the input plane of the second UFT block and more exten-
sively spread out in the output plane of the second UFT block.
us, accuracy can be optimized by scaling the input
f(x)
parallel to
x
so that the geometric spread param-
eter takes on a moderate value. We can then appropriately rescale back the ACS’ output
PO
x,y
to obtain the
desired solution.
Refer to Supplementary Figs.S6 to S9 for additional diagrams supporting the above explanation.
Conclusion
is paper introduces an analog computing system (ACS) that uses ultrasonic waves and metasurfaces to solve
ordinary and partial dierential equations. rough our simulations, we have clearly demonstrated the ability
of the proposed ACS to yield highly accurate results when solving both types of dierential equations involving
both types of functions. In contrast to other studies in existing literature, a key contribution of our paper is the
exploration of how the accuracy of the ACS’ output may be optimized through the selection of an appropriate
(moderate) value of the geometric spread parameter
w
or
γ
.
Our study’s ndings are anticipated to advance the development of wave-based analog computing systems,
potentially surpassing the constraints of digital computers. is has far-reaching implications for the elds of
computing and signal processing, hopefully laying the foundation for technological breakthroughs in the future.
Methods
Ultrasonic metalens designing process. e process of designing the ultrasonic metalens is detailed
below. Refer to Supplementary Fig.S1 for a owchart summarizing the method.
To start, it is crucial to conduct unit cell simulations in order to establish the relationship between the radius
of a cylindrical post and the associated phase shi for that particular unit cell. is correlation (Phase-to-radius
Mapping) serves as a reference point for subsequent steps. Next, an array of the Ideal Phase Map can be gener-
ated. It consists of phase values at sampled points following the paraboloidal phase prole
required to achieve
PFT ∝F{P}
theoretically37. e next step is to use the MATLAB function interp1 to inter-
polate the nearest available phase value from the unit cell simulations, and this results in the Discretized Phase
Map, consisting of phase values that have a corresponding radius from the unit cell simulations. Subsequently,
the phase-to-radius mapping can be used to create the Radius Map, an array of radius values at each sampled
point. Finally, the MATLAB function viscircles can be used to generate a gure of the metalens, comprising unit
cells with cylindrical posts whose radii correspond to the radius at that point (according to the Radius Map).
(19)
∇2
g
x,y
+
4g
x,y
=
f
x,y
.
(20)
φ
ideal
x,y
=
k
x
2
+y
2
2f
Content courtesy of Springer Nature, terms of use apply. Rights reserved
8
Vol:.(1234567890)
Scientic Reports | (2023) 13:13471 | https://doi.org/10.1038/s41598-023-38718-1
www.nature.com/scientificreports/
Figure4. Partial Dierential Equation (PDE) Simulations. (a) Sinc function: Input (le) and output (right)
magnitude proles for
w=18
. e blue line with circled data points indicates the simulated output of the ACS,
whereas the orange line shows the analytical solution. (b) Gaussian function: Input (le) and output (right)
magnitude proles for
γ
=
18
. e blue line with circled data points indicates the simulated output of the ACS,
whereas the orange line shows the analytical solution. (c) Relationship between the RMSE and the parameter
w
for the Sinc function. (d) Relationship between the RMSE and the parameter
γ
for the Gaussian function.
Content courtesy of Springer Nature, terms of use apply. Rights reserved
9
Vol.:(0123456789)
Scientic Reports | (2023) 13:13471 | https://doi.org/10.1038/s41598-023-38718-1
www.nature.com/scientificreports/
Semi-analytical wave propagation simulations. Using the exact solutions of the Kirchho-Helm-
holtz Integral (before Fresnel and paraxial approximations as listed in Table1 were applied), we conducted semi-
analytical simulations of the propagation of ultrasonic waves through our proposed ACS. e output of each
simulation was then compared with the analytical solution.
To conduct these simulations, we used MATLAB to implement the code because it is far less computationally
costly as opposed to Finite Element Method (FEM) soware like COMSOL Multiphysics. is is to enable us
to carry out simulations involving arrays of signicantly larger size—key to understanding the proposed ACS’
true capabilities.
Propagation within each UFT block. rough an FFT-based convolution approach, we can obtain the pressure
eld
PM−
x,y
at the plane before the ultrasonic metalens. is approach involves convolving the zero-padded
input pressure eld array
PS(ξ,η)
with the convolution kernel
To avoid circular convolution errors, the
N×N
array
PS(ξ,η)
has to be padded by at least
N−1
zeros37,43.
According to convention, the convolution kernel array and the zero-padded pressure eld array are of the same
size37,43. e
N×N
subarray at the center of the larger array generated as the output of FFT-based convolution
is
PM−
x,y
.
Subsequently, we apply the phase shi due to the discretized metalens to obtain
N×N
array
PM+
x,y
,
which represents the pressure eld at the plane aer the metalens. is can be done by performing an element-
wise multiplication of the
N×N
array
PM−
x,y
and the
N×N
array
exp
iφ
discretized
. Note that
φdiscretized
is
the metalens’ discretized phase prole aer the interpolation step in the ultrasonic metalens designing process.
Following this, we use FFT-based convolution to convolve the zero-padded array
PM+
x,y
with the con-
volution kernel
in order to obtain the
N×N
output pressure eld array
PO(u,v)
.
e convolution kernels in Eqs.(21) and (22) were derived from exact solutions to the Kirchho-Helmholtz
Integral37.
Propagation through the SFM. e SFM theoretically applies the transfer function (TF) needed to solve a
particular ODE or PDE. To simulate ultrasonic wave propagation through the SFM, we perform element-wise
multiplication of the output array of the rst UFT block and the transmission coecient array
T
x,y
of the
SFM. e result is then the input array for the second UFT block. We subsequently repeat the same process
described above for the simulation of wave propagation through a UFT block, which ultimately produces the
output
PO
x,y
of the proposed ACS.
Simulation parameters. e process of selecting the most appropriate simulation parameters involves
three key considerations. Firstly, convolution requires that the spacing
between adjacent metalens unit cells
and that between the sampled points of the pressure elds must be the same37,43. Furthermore, an appropriate
length
L
for the cross-section of the UFT block should be chosen, keeping in mind that the geometric spread
parameter
w
or
γ
must take on a moderate value so that the signicant space and spatial frequency components
are within the sampled array bounds (appropriate truncation and bandlimiting). Finally, the focal length
f
ought
to satisfy the condition
which can be derived by considering the sampling requirements of the convolution kernels’ exponential phase
term36,37,40–44.
With these considerations in mind, the values of the parameters used in the simulations are presented in
Table2.
(21)
h
1(ξ,η)
=
jexp
−jk
√
f2+ξ2+η2
√
f2
+
ξ2
+
η2
.
(22)
h
2
x,y
=
1
√f2
+
x2
+
y2
+
jk
fexp
−jk
√
f2+x2+y2
2π(f2
+
x2
+
y2)
(23)
f
≥
2(L−�)�
2
−
(L
−
�)2
,
Table 2. Values of simulation parameters used.
Parameter Val u e
3µm
L
771µm
fwave
1.7GHz
vwave
5880m s-1
f
1.0886mm
Content courtesy of Springer Nature, terms of use apply. Rights reserved
10
Vol:.(1234567890)
Scientic Reports | (2023) 13:13471 | https://doi.org/10.1038/s41598-023-38718-1
www.nature.com/scientificreports/
Data availability
e datasets generated during and/or analyzed during the current study are available from the corresponding
author on reasonable request.
Received: 15 March 2023; Accepted: 13 July 2023
References
1. Zangeneh-Nejad, F., Sounas, D. L., Alù, A. & Fleury, R. Analogue computing with metamaterials. Nat. Rev. Mater. 6, 207–225.
https:// doi. org/ 10. 1038/ s41578- 020- 00243-2 (2020).
2. Cheng, K. et al. Optical realization of wave-based analog computing with metamaterials. Appl. Sci. 11, 141. https:// doi. org/ 10.
3390/ app11 010141 (2020).
3. Zangeneh-Nejad, F. & Fleury, R. Performing mathematical operations using high-index acoustic metamaterials. New J. Phys. 20,
073001. https:// doi. org/ 10. 1088/ 1367- 2630/ aacba1 (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. https:// doi. org/ 10. 1080/ 03081 079. 2014. 920997
(2014).
6. Cordaro, A. et al. High-index dielectric metasurfaces performing mathematical operations. Nano Lett. 19, 8418–8423. htt ps:// doi.
org/ 10. 1021/ acs. nanol ett. 9b024 77 (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. https:// doi. org/ 10. 1364/ OL. 42. 001197 (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).
9. Zuo, S., Wei, Q., Tian, Y., Cheng, Y. & Liu, X. Acoustic analog computing system based on labyrinthine metasurfaces. Sci. Rep.
https:// doi. org/ 10. 1038/ s41598- 018- 27741-2 (2018).
10. Silva, A. et al. Performing mathematical operations with metamaterials. Science 343, 160–163. https:// doi. org/ 10. 1126/ scien ce.
12428 18 (2014).
11. Yousse, A., Zangeneh-Nejad, F., Abdollahramezani, S. & Khavasi, A. Analog computing by Brewster eect. Opt. Lett. 41, 3467.
https:// doi. org/ 10. 1364/ OL. 41. 003467 (2016).
12. Barrios, G. A., Retamal, J. C., Solano, E. & Sanz, M. Analog simulator of integro-dierential equations with classical memristors.
Sci. Rep. https:// doi. org/ 10. 1038/ s41598- 019- 49204-y (2019).
13. AbdollahRamezani, S., Arik, K., Khavasi, A. & Kavehvash, Z. Analog computing using graphene-based metalines. Opt. Lett. 40,
5239. https:// doi. org/ 10. 1364/ OL. 40. 005239 (2015).
14. Sihvola, A. Enabling optical analog computing with metamaterials. Science 343, 144–145. https:// doi. org/ 10. 1126/ scien ce. 12486
59 (2014).
15. Zhou, Y. et al. Analog optical spatial dierentiators based on dielectric metasurfaces. Adv. Opt. Mater. 8, 1901523. https:// do i. org/
10. 1002/ adom. 20190 1523 (2019).
16. Kou, S. S. et al. On-chip photonic Fourier transform with surface plasmon polaritons. Light: Science & Applications 5, e16034
(2016). https:// doi. org/ 10. 1038/ lsa. 2016. 34.
17. Cauleld, H. J. & Dolev, S. Why future supercomputing requires optics. Nat. Photon. 4, 261–263. https:// doi. org/ 10. 1038/ nphot
on. 2010. 94 (2010).
18. Zhou, Y., Zheng, H., Kravchenko, I. I. & Valentine, J. Flat optics for image dierentiation. Nat. Photon. 14, 316–323. https:// doi.
org/ 10. 1038/ s41566- 020- 0591-3 (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. https:// doi. org/ 10. 1364/ OE. 22. 025084 (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. https:// doi. org/ 10. 1364/ OE. 379492 (2020).
21. Lv, Z., Ding, Y. & Pei, Y. Acoustic computational metamaterials for dispersion Fourier transform in time domain. J. Appl. Phys.
127, 123101. https:// doi. org/ 10. 1063/1. 51410 57 (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. https:// doi. org/ 10. 1103/ PhysR evLett. 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. https:// doi. org/ 10. 1063/1. 49737 05 (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. https:// doi. org/ 10. 1063/1. 50046 17 (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. Mohammadi Estakhri, N., Edwards, B. & Engheta, N. Inverse-designed metastructures that solve equations. Science 363, 1333–1338.
https:// doi. org/ 10. 1126/ scien ce. aaw24 98 (2019).
30. Zhu, T. et al. Plasmonic computing of spatial dierentiation. Nat. Commun. 8, 15391. https:// doi. org/ 10. 1038/ ncomm s15391 (2017).
31. Zangeneh-Nejad, F. & Fleury, R. Topological analog signal processing. Nat. Commun. 10, 2058. https:// doi. org/ 10. 1038/ s41467-
019- 10086-3 (2019).
32. Abdollahramezani, S., Hemmatyar, O. & Adibi, A. Meta-optics for spatial optical analog computing. Nanophotonics 9, 4075–4095.
https:// doi. org/ 10. 1515/ nanoph- 2020- 0285 (2020).
33. Solli, D. R. & Jalali, B. Analog optical computing. Nat. Photon. 9, 704–706. https:// doi. org/ 10. 1038/ nphot on. 2015. 208 (2015).
34. Guo, C., Xiao, M., Minkov, M., Shi, Y. & Fan, S. Photonic crystal slab Laplace operator for image dierentiation. Optica 5, 251–256.
https:// doi. org/ 10. 1364/ OPTICA. 5. 000251 (2018).
35. Macfaden, A. J., Gordon, G. S. D. & Wilkinson, T. D. An optical Fourier transform coprocessor with direct phase determination.
Sci. Rep. 7, 13667. https:// doi. org/ 10. 1038/ s41598- 017- 13733-1 (2017).
36. Goodman, J. W. Introduction to Fourier Optics (W. H. Freeman and Company, 2017).
37. Uy, R. F. & Bui, V. P. A metalens-based analog computing system for ultrasonic Fourier transform calculations. Sci. Rep. 12, 17124.
https:// doi. org/ 10. 1038/ s41598- 022- 21753-9 (2022).
Content courtesy of Springer Nature, terms of use apply. Rights reserved
11
Vol.:(0123456789)
Scientic Reports | (2023) 13:13471 | https://doi.org/10.1038/s41598-023-38718-1
www.nature.com/scientificreports/
38. Stark, H. Applications of Optical Fourier Transforms (Academic Press, 1982).
39. James, J. F. A student’s Guide to Fourier Transforms: With Applications in Physics and Engineering (Cambridge University Press,
2015).
40. Voelz, D. G. Computational Fourier Optics: A MATLAB Tutorial. (Spie Press, 2010).
41. Voelz, D. G. & Roggemann, M. C. Digital simulation of scalar optical diraction: Revisiting chirp function sampling criteria and
consequences. Appl. Opt. 48, 6132. https:// doi. org/ 10. 1364/ AO. 48. 006132 (2009).
42. Zhang, H., Zhang, W. & Jin, G. Adaptive-sampling angular spectrum method with full utilization of space-bandwidth product.
Opt. Lett. 45, 4416–4419. https:// doi. org/ 10. 1364/ OL. 393111 (2020).
43. 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. https:// doi. org/ 10. 1364/ JOSAA. 401908 (2020).
44. Zhang, W., Zhang, H. & Jin, G. Frequency sampling strategy for numerical diraction calculations. Opt. Express 28, 39916. https://
doi. org/ 10. 1364/ OE. 413636 (2020).
Acknowledgements
is work was supported by the A*STAR RIE2020 Advanced Manufacturing and Engineering (AME) Program-
matic Fund [A19E8b0102].
Author contributions
R.F.U. planned the research project, formulated the mathematical model for the ultrasonic Fourier transform,
conducted the simulations, analyzed the resulting data, wrote the manuscript, and digitally created the diagrams.
V.P.B. 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- 023- 38718-1.
Correspondence and requests for materials should be addressed to R.F.U.
Reprints and permissions information is available at www.nature.com/reprints.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and
institutional aliations.
Open Access is article is licensed under a Creative Commons Attribution 4.0 International
License, which permits use, sharing, adaptation, distribution and reproduction in any medium or
format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the
Creative Commons licence, and indicate if changes were made. e images or other third party material in this
article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the
material. If material is not included in the article’s Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from
the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.
© e Author(s) 2023
Content courtesy of Springer Nature, terms of use apply. Rights reserved
1.
2.
3.
4.
5.
6.
Terms and Conditions
Springer Nature journal content, brought to you courtesy of Springer Nature Customer Service Center GmbH (“Springer Nature”).
Springer Nature supports a reasonable amount of sharing of research papers by authors, subscribers and authorised users (“Users”), for small-
scale personal, non-commercial use provided that all copyright, trade and service marks and other proprietary notices are maintained. By
accessing, sharing, receiving or otherwise using the Springer Nature journal content you agree to these terms of use (“Terms”). For these
purposes, Springer Nature considers academic use (by researchers and students) to be non-commercial.
These Terms are supplementary and will apply in addition to any applicable website terms and conditions, a relevant site licence or a personal
subscription. These Terms will prevail over any conflict or ambiguity with regards to the relevant terms, a site licence or a personal subscription
(to the extent of the conflict or ambiguity only). For Creative Commons-licensed articles, the terms of the Creative Commons license used will
apply.
We collect and use personal data to provide access to the Springer Nature journal content. We may also use these personal data internally within
ResearchGate and Springer Nature and as agreed share it, in an anonymised way, for purposes of tracking, analysis and reporting. We will not
otherwise disclose your personal data outside the ResearchGate or the Springer Nature group of companies unless we have your permission as
detailed in the Privacy Policy.
While Users may use the Springer Nature journal content for small scale, personal non-commercial use, it is important to note that Users may
not:
use such content for the purpose of providing other users with access on a regular or large scale basis or as a means to circumvent access
control;
use such content where to do so would be considered a criminal or statutory offence in any jurisdiction, or gives rise to civil liability, or is
otherwise unlawful;
falsely or misleadingly imply or suggest endorsement, approval , sponsorship, or association unless explicitly agreed to by Springer Nature in
writing;
use bots or other automated methods to access the content or redirect messages
override any security feature or exclusionary protocol; or
share the content in order to create substitute for Springer Nature products or services or a systematic database of Springer Nature journal
content.
In line with the restriction against commercial use, Springer Nature does not permit the creation of a product or service that creates revenue,
royalties, rent or income from our content or its inclusion as part of a paid for service or for other commercial gain. Springer Nature journal
content cannot be used for inter-library loans and librarians may not upload Springer Nature journal content on a large scale into their, or any
other, institutional repository.
These terms of use are reviewed regularly and may be amended at any time. Springer Nature is not obligated to publish any information or
content on this website and may remove it or features or functionality at our sole discretion, at any time with or without notice. Springer Nature
may revoke this licence to you at any time and remove access to any copies of the Springer Nature journal content which have been saved.
To the fullest extent permitted by law, Springer Nature makes no warranties, representations or guarantees to Users, either express or implied
with respect to the Springer nature journal content and all parties disclaim and waive any implied warranties or warranties imposed by law,
including merchantability or fitness for any particular purpose.
Please note that these rights do not automatically extend to content, data or other material published by Springer Nature that may be licensed
from third parties.
If you would like to use or distribute our Springer Nature journal content to a wider audience or on a regular basis or in any other manner not
expressly permitted by these Terms, please contact Springer Nature at
onlineservice@springernature.com
