11institutetext: Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France
11email<EMAIL_ADDRESS>22institutetext: DOTA, ONERA, Université
Paris Saclay, F-91123 Palaiseau, France 33institutetext: IFREMER, Laboratoire
Detection, Capteurs et Mesures (LDCM), Centre Bretagne, ZI de la Pointe du
Diable, CS 10070, 29280, Plouzane, France 44institutetext: Université Côte
d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, France

# Variation on a Zernike wavefront sensor theme: optimal use of photons

V. Chambouleyron , 1122 O. Fauvarque 33 J-F. Sauvage, 2211 K. Dohlen 11 N.
Levraud , 2211 A. Vigan 11 M. N’Diaye 44 B. Neichel 11 T. Fusco, 2211

###### Abstract

Aims. The Zernike wavefront sensor (ZWFS) is a concept belonging to the wide
class Fourier-filtering wavefront sensor (FFWFS). The ZWFS is known for its
extremely high sensitivity while having a low dynamic range, which makes it a
unique sensor for second stage adaptive optics (AO) systems or quasi-static
aberrations calibration sensor. This sensor is composed of a focal plane mask
made of a phase shifting dot fully described by two parameters: its diameter
and depth. In this letter, we aim to improve the performance of this sensor by
changing the diameter of its phase shifting dot.

Methods. We begin with a general theoretical framework providing an analytical
description of the FFWFS properties, then we predict the expected ZWFS
sensitivity for different configurations of dot diameters and depths. The
analytical predictions are then validated with end-to-end simulations. From
this, we propose a variation of the classical ZWFS shape which exhibits
extremely appealing properties.

Results. We show that the ZWFS sensitivity can be optimized by modifying the
dot diameter and even reach the optimal theoretical limit, with a trade-off
for low spatial frequencies sensitivity. As an example, we show that a ZWFS
with a $2\ \lambda/D$ dot diameter (where $\lambda$ is the sensing wavelength
and $D$ the telescope diameter), hereafter called Z2WFS, exhibits a
sensitivity twice higher than the classical $1.06\ \lambda/D$ ZWFS for all the
phase spatial components except for tip-tilt modes. Furthermore, this gain in
sensitivity does not impact the dynamic range of the sensor, and the Z2WFS
exhibits a similar dynamical range as the classical $1.06\ \lambda/D$ ZWFS.
This study opens the path to the conception of diameter-optimized ZWFS.

###### Key Words.:

instrumentation: adaptive optics – telescopes – wavefront sensing and control

##  1 Introduction

The role of a wavefront sensor (WFS) is to encode the phase information at the
entrance of an optical system into intensities on a detector. For ground-based
astronomy, WFS are mostly used for Active or Adaptive Optics (AO), in
conjunction with a wave-front control strategy in order to compensate for
optical aberrations induced by the atmosphere or the telescope itself. In the
context of astronomy, one of the main driver for a WFS design is its
sensitivity, or in other words, its ability to provide an accurate measurement
in the presence of noise. Sensitivity is therefore a useful metric to assess
WFS performance in terms of photon noise, which is related to key quantities
in AO field: loop speed and sky-coverage. Existing WFS can be separated into
two main categories, usually defined as focal plane WFS, and for which the
measurements are done in a focal plane (like the Shack-Hartmann wavefront
sensor) and pupil plane WFS, for which the measurements are done in a pupil
plane. Among this later category, the Fourier Filtering WFS (FFWFS) represents
a wide class of sensors of particular interest thanks to their superior
sensitivity. From a general point of view, a FFWFS consists of a phase mask
located in an intermediate focal plane which performs an optical Fourier
filtering. As such, the Zernike phase mask (Zernike & Stratton, 1934; Bloemhof
& Wallace, 2003; Dohlen et al., 2006; Wallace et al., 2011) forms a FFWFS,
hereafter called Zernike WFS (ZWFS). In this case, the filtering element is a
phase shifting dot which is, for a given substrate, fully described by two
parameters: its diameter and its depth (or phase-shift). In a classical
implementation, the ZWFS phase dot has a diameter of $1.06\ \lambda/D$ (where
$\lambda$ is the sensing wavelength and $D$ the telescope diameter) and a
phase shift of $\pi/2$. This ZWFS is known to be one of the most sensitive WFS
(Guyon, 2005). Its drawback being its limited dynamic range, it has therefore
been mostly implemented as a second-stage WFS, or as a quasi-static
aberrations calibration sensor like on VLT/SPHERE (N’Diaye et al., 2016;
Vigan, A. et al., 2019). In this paper, we show that the classical
implementation of the ZWFS with a phase dot diameter of $1.06\ \lambda/D$ is
actually not optimal: by using a larger dot diameter, the sensitivity of the
ZWFS can be significantly improved at the expense of lower spatial frequencies
sensitivity, and even reach a performance close to the theoretical limit. For
this reason, Section 2 starts from a theoretical study of the ZWFS based on a
general convolutional formalism for FFWFS (Fauvarque et al., 2016). This
analytical work shows that the sensitivity of the ZWFS can be improved by
increasing its dot diameter. We then confirm the theoretical results with end-
to-end simulations in Section 3 and we show that a gain in sensitivity by a
factor two can be reached without impacting the dynamic range of the sensor.
Conclusions are given in Section 4.

##  2 Theoretical analysis of the ZWFS sensitivity with a convolutional
approach

###  2.1 Definition of a FFWFS sensitivity

Following the formalism introduced by Fauvarque et al. (2016), the raw
intensities recorded by a FFWFS are processed with a return-to-reference
operation. It simply consists in removing from the FFWFS recorded intensities
map $I(\phi)$ the one corresponding to the reference phase $I_{0}$ (usually a
flat wavefront). We also posit that all the intensities are normalized by the
number of photons. The resulting quantity is called tared intensities:

$\Delta I(\phi)=I(\phi)-I_{0}$ (1)

One of the most important performance criteria for a sensor is its behaviour
in terms of noise propagation. This criteria is encoded in a quantity called
sensitivity which depends on the energy in the columns of the Interaction
Matrix (IM - Rigaut & Gendron (1992)). For a given WFS, each column of the IM
is built as the linear response of the sensor to a given mode $\phi_{i}$ which
is usually obtained experimentally through a ”push-pull” method:

$\delta
I(\phi_{i})=\frac{I(\epsilon\phi_{i})-I(-\epsilon\phi_{i})}{2\epsilon}$ (2)

where $\epsilon$ is the amplitude of the mode. The sensitivity $s$ for a given
mode $\phi_{i}$ is then defined through the Euclidean norm:

$s(\phi_{i})=\frac{||\delta I(\phi_{i})||_{2}}{||\phi_{i}||_{2}}\,.$ (3)

For a given uniform noise distribution $\sigma_{n}$, the noise propagation
coefficient $\sigma_{WFS}$ for a mode $\phi_{i}$ is then related to the
sensitivity by the following relationship:

$\sigma^{2}_{WFS}=\sigma^{2}_{n}s(\phi_{i})^{-2}\,.$ (4)

For ground-based astronomy, where WFS are usually implemented within an AO
loop, the sensitivity is a critical metric as it describes how the system
performs in the presence of noise. Optimising the WFS sensitivity has always
been one of the main motivation in the conception of new WFS.

Finally, it should be noticed that we choose to visualize the FFWFS
sensitivity as a 2 dimensional map along the spatial frequencies of the
wavefront. It consists in calculating sensitivity with respects to the Fourier
modes $\phi_{i}$ (close from what was done in Jensen-Clem et al. (2012)) which
are simply defined by the sum of a cosine and a sine carrying a given spatial
frequency f. The following quantity then encodes the sensitivity:

$s_{\textbf{f}}=\sqrt{s\Big{(}\cos_{\textbf{f}}\Big{)}^{2}+s\Big{(}\sin_{\textbf{f}}\Big{)}^{2}}\,.$
(5)

###  2.2 A convolutional approach to compute FFWFS sensitivity

The FFWFS sensitivity can be computed based on a convolutional model, as
described in Fauvarque et al. (2019). This model assumes that the sensor can
be fully characterized by an impulse response IR that links the entrance phase
to the measured tared intensities:

$\Delta I(\phi)\approx\textbf{IR}\star\phi\,,$ (6)

where $\star$ stands for the classical convolutional product. A convenient
aspect of the convolutional approach is the fact that one can compute the
transfer function TF of a FFWFS. FFWFS can be described by two parameters:
their phase masks $m$, and their weighting functions $\omega$ which describe
the energy distribution in the focal plane during one acquisition time of the
sensor. We precise that this function is normalized to 1 in order to ensure
energy conservation. Assuming that $\omega$ is a real function and that
$\omega$ and $m$ are both centro-symmetric, which is generally the case for
most of the known FFWFS, TF is expressed through the simple following formula:

$\textbf{TF}=2\text{Im}\big{[}m\star\overline{m\omega}\big{]}\,,$ (7)

where Im is the imaginary part and the bar is the complex conjugate operator.
From the knowledge of a FFWFS transfer function, it is then possible to
compute the sensitivity with respect to spatial frequencies thanks to the
formula exposed in Fauvarque et al. (2019):

$s_{\textbf{f}}\approx\sqrt{|\textbf{TF}|^{2}\star\textbf{PSF}}\Big{|}_{\textbf{f}}\,,$
(8)

where the quantity PSF is the Point Spread Function of the system. Its energy
correspond to the incoming flux and is normalized to 1.

At this point, it is important to note that the sensitivity is bounded. Since
the mask transmission, $|m|$, cannot be greater than one ($|m|\leq 1$),
Equation 7 implies that $\textbf{TF}|_{\textbf{f}}\leq 2$. Hence, given
Equation 8, we conclude that, in the frame of our normalizations, the
sensitivity cannot be greater than 2:

$\forall\textbf{f},\ s_{\textbf{f}}\leq 2\,.$ (9)

This is an important result as it defines the theoretical limit for a FFWFS
sensitivity.

###  2.3 Application to the ZWFS

The convolutional formalism introduced in the previous section is now applied
to the ZWFS in order to find a simple formula of its sensitivity according to
the mask parameters. As it was previously mentioned, the ZWFS mask is defined
by two free parameters:

  * $\bullet$

The depth (phase shift) of the dot $\delta$. For the classical ZWFS:
$\delta=\pi/2$,

  * $\bullet$

Its diameter $p$. For the classical ZWFS: $p=1.06\ \lambda/D$.

As described in N’Diaye et al. (2013), the dot diameter value was chosen in
order to get an equivalent flux inside and outside the focal plane dot. This
configuration with $p=1.06\ \lambda/D$ also allows to get an uniform reference
intensity distribution. The purpose of this section is to demonstrate that
this choice of $p=1.06\ \lambda/D$ is actually not optimal, and that the
sensitivity can be improved with a larger dot.

For the sake of clarity, we carry this study in one dimension. As such, the
spatial frequency vector f becomes the scalar frequency $f$. We further assume
that the weighting function $\omega$, which corresponds to the PSF, can be
described as a top-hat function of diameter radius $a$. This simplified
geometry is summarized by Figure 1-(a). From this simplified geometry, we then
compute $\textbf{TF}|_{f}$ and plot this quantity in Figure 1-(b). Note that
the full derivation can be found in appendix B.

Figure 1: (a) : Simplified 1D framework for convolutional derivations: in
black, the dot diameter equals to $p$ and the phase shift equals to $\delta$.
In green, the PSF is approximated by a normalized top-hat function with a
diameter $a$. (b) : Transfer function of the ZWFS for two cases in red, the
dot size is smaller than the PSF. In blue, the dot is larger than the PSF. The
optimal case appears for $p=a$ and not $p=a/2$ as it is done for the classical
ZWFS.

From Figure 1-(b), one can distinguish two cases:

  * $\bullet$

$p\geq a$, i.e. the dot diameter is larger than the PSF characteristic size.
Frequencies above $(p+a)/2$ reach the sensitivity $2\sin(\delta)$. Frequencies
below $(p-a)/2$ are at 0.

  * $\bullet$

$p<a$: the dot diameter is smaller than the PSF size. Frequencies over $p$
have a value of $2p/a\times sin(\delta)$, which is smaller than the
theoretical limit of $2$.

From this simplified model one can first conclude that a phase shift of
$\delta=\pi/2$ will maximize the sensitivity as expected. But surprisingly, it
shows that for this phase shift, a dot radius of $p=a$ offers a sensitivity of
the optimal value for almost all the modes. At this point it is important to
remember that the value of $\textbf{TF}|_{f}$ directly sets the sensor
sensitivity through Equation 8. It is therefore possible to design a ZWFS that
reaches the theoretical sensitivity value. As a comparison, the classical ZWFS
configuration (N’Diaye et al., 2013) uses $p=a/2$, which leads to a sub-
optimal TF value of $1$ for frequencies above $p$.

Although this simplified study uses strong assumptions, it shows that the ZWFS
can be further optimized compared to its classical form. In the next section,
we demonstrate that these simplified results are actually accurate and enable
us to build the most sensitive sensor ever proposed.

##  3 Towards an optimal ZWFS

Following the results from the convolutional approach, the goal of this
section is to make use of numerical simulations to confirm the sensitivity of
the ZWFS with respect to its dot diameter, and eventually to propose an
optimal configuration. For that, we consider different configurations for a
dot diameter ranging from 0 to 5 $\lambda/D$. The phase shift is set to
$\delta=\pi/2$ for the rest of this letter.

###  3.1 Impact of the dot diameter on the ZWFS Sensitivity

As a first step, we want to illustrate the impact of the dot diameter on
sensitivity for a spatial frequency located outside of the dot (horizontal
part of the curves Figure 1-(b)). For that purpose, we arbitrary choose a
spatial frequency with 6 cycles over the pupil (left insert of Figure 2),
which is far enough from the maximum diameter dot value ($5\ \lambda/D$ i.e a
radius of $2.5\ \lambda/D$). This configuration is illustrated in right insert
of Figure 2. The sensitivity results for this spatial frequencies are shown in
Figure 3.

Figure 2: Cosine phase (left insert, linear scale) corresponding to a spatial
frequency of 6 cycles in D and its corresponding PSF speckles (right insert,
logarithmic scale, in blue). This frequency lies outside of the dot footprint,
here $5\ \lambda/D$ (in red).

As predicted by the convolutional approach, Figure 3 shows that the
sensitivity for a high spatial frequency increases with the dot diameter. This
behaviour has been discussed briefly in previous literature (Ruane et al.,
2020) without further analysis. It is here explained thanks to the
convolutional model. It is also interesting to notice that the sensitivity
growth is closely following the PSF encircled energy in the dot diameter,
confirming the analytical results presented in Figure 1. For a classical ZWFS,
with a dot diameter of 1.06 $\lambda/D$, the sensitivity is actually far from
being optimal.

Figure 3: Sensitivity evolution for a frequency outside of the dot, while
increasing dot diameter. The sensitivity is in strong accordance with the
proportion of the PSF energy located inside the dot, as predicted by the
convolutional approach.

However one cannot just increase the dot diameter inconsiderately because the
sensitivity to frequencies lying inside the dot would drop to zero (Figure 1).
We illustrate this effect in Figure 4, where we plot the sensitivity curves
with respect to a wide range of spatial frequencies for the three dot
diameters configurations $p=1.06,\leavevmode\nobreak\ 2$ and $5\lambda/D$. For
$p=5\lambda/D$, the sensitivity to high-spatial frequencies (those larger than
5 cycles per pupil) almost reaches the theoretical limit of 2, however the
sensitivity becomes close to 0 for low-spatial frequencies (those smaller than
3 cycles per pupil). There is therefore a trade-off between enhanced
sensitivity and unseen modes. In the following, we propose to choose the
configuration with $p=2\lambda/D$ and we call this specific configuration the
Z2WFS.

Figure 4: Sensitivity curves for different dot diameters. This figure has to
be compared with the convolutional approach Figure 1.

As a remark, we also emphasize that the reference intensities, i.e. the
intensity distribution for a flat wavefront, change with the value of $p$.
This is illustrated in Figure 5 for the previous three different values of
$p$. The classical ZWFS shows a flat reference illumination, while the Z2WFS
appears to be less uniform. This spatial distribution could involve practical
issues in terms of detector dynamics or for complex pupil shapes, as for
instance central obscuration or spiders. These potential practical
implementation issues are beyond the scope of this paper. In this letter, we
will only assume a full aperture pupil with monochromatic light for the sake
of clarity. It is to be noted that there is no sticking point here, the
formalism and results developed here are maintained with a central obscuration
in the pupil.

Figure 5: Reference intensities of the ZWFS for different dot diameter. (a):
$p=1.06\ \lambda/D$. (b): $p=2\ \lambda/D$. (c): $p=5\ \lambda/D$.

###  3.2 Comparison with other FFWFS

In this section, we compare the Z2WFS with other well-know FFWFS: the
classical ZWFS with $p=1.06\ \lambda/D$, the non-modulated PyWFS (Ragazzoni
(1996)), the modulated PyWFS (with here a modulation radius of $3\ \lambda/D$)
and a flattened pyramid (FPyWFS) proposed by Fauvarque et al. (2015) with a
pupils overlapping rate of $75$%. Spatial frequencies basis, i.e. Fourier
basis, is chosen for this comparison. Results are given Figure 6. First, we
retrieve well-know results as for instance the gain around a factor $2$ in
sensitivity between the classical ZWFS and the PyWFS. We can also highlight
the behaviour of the FPyWFS showing oscillating sensitivity and peaks for some
specific frequencies, as described in Fauvarque et al. (2015). (The
explanation of the PyWFS class behaviour through to the convolutional approach
is also given in appendix C.) The Z2WFS is clearly the most sensitive sensor,
except for extremely low frequencies. As expected from Figure 3, it has a
sensitivity twice better than the classical ZWFS for almost all modes and is
four times more sensitivity than the non-modulated PyWFS.

Figure 6: Sensitivity curves for different FFWFS. We can distinguish the ZWFS
class and the PyWFS class. The Z2WFS overtakes all the other sensors.

The behavior at low spatial frequencies deserves some further analysis: we
plot Figure 7 the sensitivity with respects to the tip-tilt and focus modes
(which are the lowest frequency Zernike modes) for the ZWFS class with a dot
diameter ranging from 0 to 5 $\lambda/D$. For the tip-tilt modes, the Z2WFS is
twice less sensitive than the classical ZWFS, but the Z2WFS provides better
results for the focus. Even if the Z2WFS is less sensitive for tip-tilt than
the classical ZWFS, it is important to note that it remains as sensitive as
the non-modulated PyWFS which is around 0.4. As a remark, it is interesting to
see that the sensitivity curve for the tip-tilt is following the PSF shape:
for the edge of the dot lying on a dark area of the PSF, the sensitivity drops
to 0. By taking a Z1.5WFS ($p=1.5\ \lambda/D$), one could have a better
sensitivity for all the frequencies compared to the classical ZWFS, but a
lowest gain overall compared to Z2WFS.

Figure 7: Evolution of the low order Zernike modes sensitivities with respect
to the dot diameter. We can see that the Z2WFS senstivity is lower for the
tip-tilt modes, but higher for all the other ones. The classical ZWFS is not
even optimized for the tip-tilt modes.

To conclude this section, we demonstrated that a Z2WFS significantly improves
the sensitivity, and approaches the ideal FFWFS behavior. When compared to the
classical ZWFS, the gain in sensitivity for all modes, except the Tip-Tilt, is
a factor 2. In the next section we investigate if this gain in sensitivity
costs in dynamic range.

###  3.3 Dynamic Range

To complete our study, we now compare the dynamic range of the Z2WFS with the
classical ZWFS. A drastic loss of dynamic range while increasing the dot
diameter could indeed prevent from a practical utilisation of the Z2WFS. To
calculate this quantity with respects to a given mode $\phi_{i}$, we evaluate
its capture range $C_{\phi_{i}}$. To calculate it, we look at the lowest
amplitude value (in absolute value) such that :

$\left.\frac{\text{d}\Delta I(a\phi_{i})}{\text{d}a}\right|_{a_{0}}=0\,.$ (10)

We then define the capture range as $C_{\phi_{i}}=2a_{0}$, where the factor 2
allows one to take into account negative and positive amplitudes in the
capture range calculation.

The capture range can be larger than the pure linearity regime. However, we
decided to use this definition for two reasons. First because it defines the
amplitude below which we are ensured that a closed loop system will eventually
converge. Indeed, even if the measurement is not linear anymore, there is
still a one-to-one correspondence with the input signal. Secondly, because,
the ZWFS measurements are often processed through non-linear reconstructors
(N’Diaye et al. (2013), Steeves et al. (2020)) that can be perfectly applied
to Z2WFS or other variations of the ZWFS.

Figure 8: Capture range of different FFWFS. The Z2WFS has the same capture
range as the classical ZWFS.

Capture range values for cosine phase modes at frequencies ranging from $0$ to
$5$ cycles in diameter are given Figure 8 for Zernike and Pyramid wave-front
sensors. The PyWFS class have a better capture range over all spatial
frequencies, matching the fact that sensitivity and dynamic range are
competing properties. This graph also confirms the great benefit in terms of
dynamic range provided by the modulation of the PWFS. More importantly, we see
that the Z2WFS exhibits the same capture range as the classical ZWFS for high
frequencies and is even higher for the lowest frequencies where the Z2WFS
sensitivity goes below the classical ZWFS one. The Z2WFS is therefore more
sensitive than the classical ZWFS while exhibiting the same capture range.

##  4 Conclusion

In this paper, we provided a physical description of the sensitivity behaviour
for the ZWFS class, and in particular we studied the sensitivity evolution for
different dot diameters. We showed that it was possible to significantly
improve the current sensitivity of the ZWFS at the expense of the lower
spatial frequencies, simply by increasing the dot diameter. The resulting
sensitivity can even almost reach the fundamental limit of FFWFS. We further
studied the specific case of a dot diameter of $2\ \lambda/D$ \- called Z2WFS,
which exhibits a averaged gain of sensitivity by a factor two (with a loss of
sensitivity compared to the classical ZWFS only for the tip-tilt modes),
without loss of the dynamic range with respect to the classical ZWFS. This new
sensor then becomes the most sensitive WFS available for ground-based
astronomy. It still exhibits the low dynamic range of the ZWFS, but as for the
PyWFS, modulation schemes can be imagined. For instance, one way to increase
its linearity range is to dynamically change the dot diameter during one
integration time of the sensor camera. Further studies will now investigate
its practical implementation and the impact of chromaticity on wavefront
sensing.

##  5 Acknowledgements

This work benefited from the support of the WOLF project
ANR-$18$-CE$31$-$0018$ of the French National Research Agency (ANR). It has
also been prepared as part of the activities of OPTICON H$2020$ (2017-2020)
Work Package $1$ (Calibration and test tools for AO assisted E-ELT
instruments). OPTICON is supported by the Horizon 2020 Framework Programme of
the European Commission’s (Grant number $730890$). Authors are acknowledging
the support by the Action Spécifique Haute Résolution Angulaire (ASHRA) of
CNRS/INSU co-funded by CNES. Vincent Chambouleyron PhD is co-funded by ”Région
Sud” and ONERA, in collaboration with First Light Imaging. AV acknowledges
funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (grant agreement No. 757561).
Finally, part of this work is supported by the LabEx FOCUS
ANR-$11$-LABX-$0013$, and received the support of Action Spécifique Haute
Résolution Angulaire ASHRA.

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##  Appendix A convolutional approach: general framework

In the infinite pupil approximation and assuming that the weighting function
(energy distribution at the focal plane) $\omega$ is a real centro-symmetric
function and that the focal-plane mask function $m$ is also centro-symmetric,
the transfer function of a FFWFS may be written as:

$\textbf{TF}=2\text{Im}\big{[}m\star\overline{m\omega}\big{]}\,.$ (11)

We consider in this formalism only pure phase mask for the Fourier filter
function $m$, because the global context is the search for sensitivity of WFS
so we did not amplitude mask that would result in wasting of photons.
Therefore, they are pure phase masks that can be written as $m=e^{i\Delta}$.
Moreover, we continue to carry out these mathematical developments in one
dimension. The previous equation (11) becomes:

$\textbf{TF}|_{f}=2\int_{\mathbb{R}}\text{d}\text{u}\leavevmode\nobreak\
\omega|_{\text{u}}\sin\Big{(}\Delta|_{\text{u}}-\Delta|_{f-\text{u}}\Big{)}\,,$
(12)

where $u$ is expressed in unit of $\lambda/D$. We choose to approximate the
weighting function as a rectangular and normalized function with a diameter
$a$:

$\omega|_{u}=\frac{1}{a}\mathbb{I}_{[-a/2;a/2]}\big{|}_{u}\,.$ (13)

The normalization allows one to respect the energy conservation. Furthermore,
$a$ may be seen as the characteristic size of the energy distribution at the
focal plane. $a$ is therefore the typical size of the modulation while
studying the Pyramid WFS class and corresponds to the PSF characteristic
length when modulation is inactive. In other words, the weighting function
allows us to take into account the finite size of the pupil in spite of the
”infinite pupil approximation” needed in the convolutional approach.

##  Appendix B Zernike WFS class

For the ZWFS, we consider the following mask: a centered dot with a diameter
of $p$ and a depth $\delta$. Thus, the phase of the filtering mask equals to:

$\Delta|_{\text{u}}=\delta\leavevmode\nobreak\
\mathbb{I}_{[-p/2;p/2]}\big{|}_{u}\,.$ (14)

Figure 9: In blue, ZWFS mask in 1 dimension with a uniform modulation
representing the PSF. In black, the centered mask $\Delta|u$. In red, the
shifted mask $\Delta|_{f-u}$.

The problem being symmetric, the derivation of equation (12) is only done for
positive frequencies $f\geq 0$. We can distinguish two cases: whether the size
of the dot is bigger than the PSF, i.e. $p\geq a$ or not, i.e. $p<a$.

Case 1. The dot is larger than the PSF: $p\geq a$.

Case 1.1 $f\geq\frac{p+a}{2}$

We have $\Delta|_{\text{u}}=\delta$ and $\Delta|_{f-\text{u}}=0$

$\textbf{TF}|_{f}=\frac{2}{a}\int_{-a/2}^{a/2}\text{d}\text{u}\sin\Big{(}\Delta|_{\text{u}}-\Delta|_{f-\text{u}}\Big{)}=\frac{2}{a}\int_{-a/2}^{a/2}\text{d}\text{u}=2\sin(\delta)\,.$
(15)

Case 1.2 $f<\frac{p+a}{2}$

In that case, $\Delta|_{\text{u}}-\Delta|_{f-\text{u}}$ depends on $f$. We
get:

$\displaystyle\textbf{TF}|_{f}$ $\displaystyle=$
$\displaystyle\text{max}\left(2\sin(\delta)-\frac{1}{a}(p+a-2f)\sin(\delta),0\right)$
(16) $\displaystyle=$
$\displaystyle\text{max}\left((2f+1-\frac{p}{a})\sin(\delta),0\right)\,.$ (17)

Case 2. The dot is smaller than the PSF: $p<a$.

Case 2.1: $f\geq p$

We have again $\Delta|_{\text{u}}=\delta\ \text{and}\ \Delta|_{f-\text{u}}=0$.
Consequently, the transfer function equals to:

$\displaystyle\textbf{TF}|_{f}$ $\displaystyle=$
$\displaystyle\frac{2}{a}\int_{-a/2}^{a/2}\text{d}\text{u}\sin\Big{(}\Delta|_{\text{u}}-\Delta|_{f-\text{u}}\Big{)}$
(18) $\displaystyle=$
$\displaystyle\frac{2}{a}\int_{-p/2}^{p/2}\text{d}\text{u}\sin(\delta)=\frac{2p}{a}\sin(\delta)\,.$
(19)

Case 2.2: $f<p$

$\Delta|_{\text{u}}-\Delta|_{f-\text{u}}$ still depends on $f$. Equation (12)
becomes:

$\displaystyle\textbf{TF}|_{f}$ $\displaystyle=$
$\displaystyle\frac{2}{a}\int_{-a/2}^{a/2}\text{d}\text{u}\sin\Big{(}\Delta|_{\text{u}}-\Delta|_{f-\text{u}}\Big{)}$
(20) $\displaystyle=$
$\displaystyle\frac{2}{a}\int_{-f}^{f}\text{d}\text{u}\sin(\delta)=\frac{2f}{a}\sin(\delta)\,.$
(21)

The plot corresponding to these results is given earlier in the letter, Figure
1.

##  Appendix C Pyramid WFS class

We derive once more the equation (12) for the PyWFS class. The phase of the
filtering mask now equals to:

$\Delta|_{u}=\alpha|u|\,,$ (22)

where $\alpha$ equals to $2\pi\theta/\lambda$ where $\theta$ is the pyramid
apex angle and $\lambda$ is the sensing wavelength. Equation 12 then becomes:

$\displaystyle\textbf{TF}|_{f}=2\int_{\mathbb{R}}\text{d}\text{u}\leavevmode\nobreak\
\omega|_{\text{u}}\sin\Big{(}\alpha(|u|-|f-u|)\Big{)}\,.$ (23)

In the sine, we have:

$\alpha(|u|-|f-u|)=\left\\{\begin{array}[]{ll}-\alpha f&u\in]-\infty;0]\\\
\alpha(2u-f)&u\in]0;f[\\\ \alpha f&u\in[f;\infty[\\\ \end{array}\,.\right.$
(24)

Moreover $\omega$ is centro-symmetric so it is possible to reduce the
integration interval:

$\textbf{TF}|_{f}=2\int_{-f}^{f}\text{d}\text{u}\leavevmode\nobreak\
\omega|_{\text{u}}\sin\Big{(}\alpha(|u|-|f-u|)\Big{)}\,.$ (25)

Figure 10 allows us to visualize the functions involved in previous equation.
Notably, it appears that the value of the characteristic length of the
weighting function over two $a/2$, i.e. the modulation radius when this device
is active, plays a major role in the integration and may be seen as a cutoff
frequency. Equation (25) becomes:

Figure 10: In blue (resp. green): case of a weighting function with a small
(resp. large) characteristic length. In red, function of equation (24).

$\textbf{TF}|_{f}=\frac{2}{a}\left[\int_{-\min(a/2,f)}^{0}\sin(-\alpha
f)\text{d}u\right.+\\\
\left.\int_{0}^{\min(a/2,f)}\sin\big{(}\alpha(2u-f)\big{)}\text{d}u\right]$
(26)
$=\frac{2}{a}\left[\frac{\sin(\alpha\min(a/2,f))\sin[(\min(a/2,f)-f)\alpha]}{\alpha}\right.\\\
-\min(a/2,f)\sin(\alpha f)\Big{]}\,.$ (27)

Finally, we get the transfer function of the Pyramid class depending on its
two optical parameters, namely the Pyramid apex angle and the weighting
function characteristic size.

$\textbf{TF}|_{f}=\left\\{\begin{array}[]{ll}\frac{-2f\sin(\alpha f)}{a}&f\leq
a/2\\\ \text{sinc}(\alpha a/2)\sin[(a/2-f)\alpha]-\sin(\alpha f)&f>a/2\\\
\end{array}\,.\right.$ (28)

We may now use this formula to explain the sensitivity with respects to the
spatial frequencies of the Pyramid WFS class.

Firstly, we are interested in apex angle parameter influence on the
sensitivity. In other words we study the difference between classical and
flattened pyramids. To do so, we assume that the modulation is inactive.
Consequently, the parameter $a$ is related to the PSF size. Considering a
pupil diameter of D, a sensing wavelength of $\lambda$ and a imaging focal of
$f_{\text{oc}}$, we have:

$a=\frac{\lambda f_{\text{oc}}}{D}\,.$ (29)

To distinguish between flattened or classical pyramids, we just have to
identify the limit apex angle $\theta_{\text{limit}}$ allowing to totally
separate pupil images. It can actually be linked with the pupil diameter and
the imaging system focal via the following formula:

$\theta_{\text{limit}}=\frac{D}{f_{\text{oc}}}\,.$ (30)

Consequently, if $\theta$ is below $\theta_{\text{limit}}$ there is an overlap
of the pupil images and the Pyramid is therefore a flattened one whereas a
$\theta$ above $\theta_{\text{limit}}$ implies a complete separation of the
pupil images and thus, a classical pyramid. If we use the $a$ and $\alpha$
variables, these two cases may be summarized in the following way:

$\displaystyle a\alpha<\pi$ Flattened Pyramid (31) $\displaystyle
a\alpha\geq\pi$ Classical Pyramid (32)

Such a distinction allows to explain why the sensitivity of the flattened
pyramid may be larger in absolute value than the classical pyramid one. As a
matter of fact, the sinc function in equation (28) may be significant for some
frequencies when $a\alpha$ is small, i.e. for flattened pyramids. Keeping in
mind that the sensitivity is linked with the TF via the convolution product of
Equation (8), this shows why the FPyWFS sensitivity may reach 2 when the
classical pyramid optimally attains 1. Moreover, the small $\alpha$ implies
large oscillations with respects to the spatial frequencies (see green curve
of Figure 11). This is relevant regarding to the observed sensitivity which
indeed oscillates (dotted green curve in Figure 6).

Figure 11: Graphical visualisation of transfer function, i.e. Equation (28),
for the PyWFS configurations of Figure 6: non-modulated PyWFS in yellow,
modulated PyWFS in purple and FPyWFS in green.

By contrast, if $a\alpha$ is large, i.e. if pupil images are completely
separated, the sinc function may be neglected and the transfer function can be
summarized as:

$\textbf{TF}|_{f}=\left\\{\begin{array}[]{ll}\frac{-2f\sin(\alpha f)}{a}&f\leq
a/2\\\ -\sin(\alpha f)&f>a/2\\\ \end{array}\,.\right.$ (33)

This function corresponds may be seen as the transfer function of the
classical pyramid. We notice that it oscillates more rapidly than the
flattened pyramid one (yellow and purple curves of Figure (11)). However,
these oscillations disappear when we look at the corresponding sensitivity
curves (Figure (6)). Such a peculiarity may be explained by Equation (8): to
get the sensitivity curve, the transfer function is convoluted with the PSF
which is in this case larger than oscillation period. As a result, the
transfer function is smoothed and the sensitivity follows the TF envelope.

Concerning this envelope, we may observe two regimes. The first one goes from
the null spatial frequency to the modulation radius $a/2$ ; it is linear with
$f$. The second one corresponds to spatial frequencies above the modulation
radius ; it is constant and equals to 1. We identify here the typical behavior
of the classical pyramid (modulated or not) with its two regimes, slope and
phase sensors separated by a cutoff frequency corresponding to the modulation
radius (Vérinaud 2004).

The convolutional approach therefore shows its capability to describe the
sensitivity of pyramid sensors. It indeed allows to get a unique formula which
explains both the enhanced and oscillating sensitivity of the FPyWFS and the
dual behavior slope/phase sensors of the classical modulated PyWFS.


# Time-Domain and Frequency-Domain Mappings of Voltage-to-Charge and Charge-
to-Voltage in Capacitive Devices

Anis Allagui1,2, Ahmed S. Elwakil3,4, _Senior Member IEEE_ and Chunlei Wang2,
1Dept. of Sustainable and Renewable Energy Engineering, University of Sharjah,
United Arab Emirates
2Dept. of Mechanical and Materials Engineering, Florida International
University, Miami, FL33174, USA
3Dept. of Electrical Engineering, University of Sharjah, United Arab Emirates
4Dept. of Electrical and Software Engineering, University of Calgary, Alberta,
Canada<EMAIL_ADDRESS>
###### Abstract

In this work, we aim to show that there are generally four possible mapping
functions that can be used to map the time-domain or frequency-domain
representations of an applied voltage input to the resulting time-domain or
frequency-domain electrical charge output; i.e. when the capacitive device is
voltage-charged. Alternatively, there are four more possible combinations when
the device is current-charged. The dual relationship between each pair of
functions for the case of voltage or charge input are provided in terms of
single or double Fourier transforms. All eight system functions coincide with
each other if and only if a constant time- and frequency-independent
capacitance is considered.

###### Index Terms:

Capacitve devices, Circuit theory, Capacitance, Fourier transform

##  I Introduction

Ideal electrostatic capacitors are constructed from two conductors separated
by a perfect insulator. The standard charge-voltage relation $q=cv$ represents
the linear relationship between the magnitude of the charge on each conductor
with the difference of potential applied across the two conductors. It is true
in both time and frequency domains for ideal capacitors having constant, time-
independent and frequency-independent capacitances that can be only computed,
and not measured, from voltage and charge.

However, other capacitive devices, such as electrochemical capacitors
(including electric double-layer capacitors (EDLCs) and pseudocapacitors),
cannot be characterized with a constant capacitance value [1, 2, 3]. Their
electrodes are fractal and porous in nature, and are separated by ionic
conductors leading to complex charge transports that are inherently different
from ideal capacitors. Coupled ionic and electronic transports take place in
these devices, which are ultimately controlled by diffusion dynamics and
migration in the bulk electrolyte, and through their micro/nano-porous
structured electrodes. The correlated disorder of the background electrolyte
also has an impact on the overall charge transport process which makes it
deviate from normal diffusion to subdiffusion, and hence internal dissipative
processes [4, 5]. From a system-level point of view, the spectral impedance
phase angle of most -if not all- EDLCs and other similar energy storage
devices deviate from the -90∘ value expected for ideal capacitors, even at
near-dc, sub-Hz frequencies [6, 3]. It has also been demonstrated
experimentally that an EDLC charged with two different time-domain voltage
waveforms to the same thermodynamic state of voltage and charge, $(v,q)$,
exhibits different relaxation profiles when discharged into the same resistor
[7]. This means that there are inherent memory effects in these devices [8, 7,
9], and that the dynamic (dis)charging paths should be taken into
consideration when evaluating their performance metrics [10, 11, 12]. This is
not the case with ideal capacitors whose relaxation depends only on the
initial condition. In addition, several time-domain transient measurements
conducted on EDLCs show in some form or another nonlinearities and power-law
behaviors that prohibit the extraction of a constant capacitance from the
system’s response [12, 11, 13, 14]. We note that the EDL structure for
instance is ubiquitous in many other fields of fundamental and applied
electrochemistry such as in intercalation materials, capacitive deionization
and electrosorption, colloidal suspensions, bipolar membranes, etc., and
therefore properly evaluating its electrical parameters is paramount for
meaningful progress and development in these research areas.

Unfortunately, there is a conceptual misunderstanding, even in specialized
literature, on the difference in computation of performance metrics between an
ideal capacitor and other capacitive energy storage devices [15]. A classical
example is the direct calculation of capacitance from the time-domain
expression $q=cv$, but the same formulation of charge equal to the product of
capacitance by voltage expressed in the frequency domain, and indirectly
derived from impedance spectroscopy measurements, is used to characterize the
same device, as if it was an ideal capacitor with constant capacitance (see
[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] to cite a few). This is in
direct contradiction with the convolution theorem [28], and clearly indicates
that there is an urgent need for a unified methodology of system
identification for such devices. Most recently, in [15] the authors studied
the difference between the time-domain definition $q_{t}(t)=c_{t}(t)v_{t}(t)$
and its frequency-domain counterpart defined as a convolution integral, i.e.
$Q_{t}(f)=(V_{t}\circledast C_{t})\,(f)$, and the frequency-domain definition
$Q_{f}(f)=C_{f}(f)\,V_{f}(f)$ and its time-domain equivalent
$q_{f}(t)=(c_{f}\circledast v_{f})\,(t)$. We denote by $V(f)$ the Fourier
transform of the input time-domain voltage $v(t)$ and by $Q(f)$ the Fourier
transform of the output time-domain charge $q(t)$ with the definitions
$\mathcal{F}\\{g(t)\\}=G(f)=\int_{-\infty}^{\infty}g(t)e^{-j2\pi ft}dt$ and
$\mathcal{F}^{-1}\\{G(t)\\}=g(t)=(2\pi)^{-1}\int_{-\infty}^{\infty}G(f)e^{j2\pi
ft}dt$. In [15], we showed that the extraction of capacitance functions
(rather than constant capacitance) from these two pairs of equations for the
case of dispersive mono-fractional-order capacitors (known also as constant
phase elements) subjected to sinusoidal voltage excitation do not lead to the
same results. The same has been illustrated and verified for the case of
linear voltage ramp and constant current input applied to a commercial
supercapacitor device [10]. It is therefore one pair of equations or the other
that can be used simultaneously, and the relation $q=cv$ can only be applied
on one-domain (time or frequency) but not the other. We recommended
nonetheless the use of the multiplicative capacitance function in the
frequency domain, i.e. $Q_{f}(f)=C_{f}(f)\,V_{f}(f)$, in order to be in line
with the definition of impedance of linear time-invariant systems [15, 28].

This work is a direct extension of our previous studies [15, 10], and is
primarily concerned with the system identification problem of capacitive
devices from the two measurable quantities voltage and charge. The goal is to
establish the different relationships between possible system capacitance
functions _without_ the prior assumption of any model. In Section II, we
present the theoretical framework of this work while its implications are
discussed further in Section III. It is important to note that there are no
simulations or experimental results presented here since they have been
readily presented in [15, 10] upon dealing with specific devices.

##  II Theory

Consider the schematic illustration presented in Fig. 1 relating the
input/output of a capacitive device in the time and frequency domains. It is
evident that there are four possible operators that can map an input voltage
(in the time or frequency domain) to an output charge (in the time or
frequency-domain). When we refer to the frequency-domain, the time-domain
quantity is dealt with indirectly through its Fourier frequency spectrum. We
can write from the four signal representations $v(t)$, $V(f)$ and $q(t)$,
$Q(f)$ the following relations [29, 30]:

$\displaystyle q(t)$ $\displaystyle=\mathcal{C}_{\text{tt}}\\{v(t)\\}$ (1a)
$\displaystyle Q(f)$ $\displaystyle=\mathcal{C}_{\text{ff}}\\{V(f)\\}$ (1b)
$\displaystyle q(t)$ $\displaystyle=\mathcal{C}_{\text{tf}}\\{V(f)\\}$ (1c)
$\displaystyle Q(f)$ $\displaystyle=\mathcal{C}_{\text{ft}}\\{v(t)\\}$ (1d)

where the four operators
$\mathcal{C}_{\text{ij}}\;(\\{i,j\\}=\\{\text{t},\text{f}\\})$ map or
transform the input signal in space $j=\\{\text{t},\text{f}\\}$ for time or
frequency to space $i=\\{\text{t},\text{f}\\}$. For example, the operator
$\mathcal{C}_{\text{tt}}$ in Eq. (1a) represents the mapping relationship
between the voltage input $v(t)$ (cause) and charge output $q(t)$ (effect)
both defined in the time domain, whereas the operator
$\mathcal{C}_{\text{tf}}$ in Eq. (1c) represents the mapping relationship
between the voltage input $V(f)$ defined in the frequency domain and the
charge output $q(t)$ defined in the time domain. The mapping operators
$\mathcal{C}_{\text{ij}}$ are unique, and can be in general related to
differential or integro-differential equations that govern the space-time
charge distributions in a capacitive device.

Figure 1: Schematic diagram illustrating the two time-domain and frequency-
domain representations of voltage input and the two time-domain and frequency-
domain representations of the charge output in a capacitive energy device. The
different relationships between the capacitance functions $C_{\text{ij}}$
defined in Eq. 2 are also summarized (FT: Fourier transform).

For the case of devices which are assumed to be linear time-invariant wherein
the input-output relationship is invariant under translation in time and the
superposition principle holds, we can write the following linear integral
equations of convolution [31]:

$\displaystyle q(t)$ $\displaystyle=\int C_{\text{tt}}(t-x)v(x)dx$ (2a)
$\displaystyle Q(f)$ $\displaystyle=\int C_{\text{ff}}(f-y)V(y)dy$ (2b)
$\displaystyle q(t)$ $\displaystyle=\int C_{\text{tf}}(t-y)V(y)dy$ (2c)
$\displaystyle Q(f)$ $\displaystyle=\int C_{\text{ft}}(f-x)v(x)dx$ (2d)

The four kernels $C_{\text{ij}}\;(\\{i,j\\}=\\{\text{t},\text{f}\\})$ can be
viewed as kernel system functions as proposed by Bello [29, 30]. Note that
when the limits of integration are not explicitly indicated in the equations
they are to be taken from $-\infty$ to $\infty$, and the integrals are assumed
to exist. Also, all quantities are treated as normalized and thus
dimensionless. It is important to clarify at this point that in [15] the focus
was to establish the difference between assuming the correctness of the
multiplicative equation $q=cv$ in the time domain versus assuming its
correctness in the frequency domain, which are both widely used definitions in
the literature. This led us to define four different capacitance functions
$c_{t}(t)$, $C_{t}(f)$ and $C_{f}(f)$, $c_{f}(t)$ by virtue of the convolution
theorem as explained above. In this work, we approach the problem from a
general system identification perspective as described by the set of Eqs. (2).
Therefore the functions $C_{\text{ij}}\;(\\{i,j\\}=\\{\text{t},\text{f}\\})$
should not be confused with $c_{t}(t)$, $C_{t}(f)$, $C_{f}(f)$ or $c_{f}(t)$,
which are not in conflict with the present analysis.

Equations (2a) and (2c) provide two different expressions for the same time-
domain charge $q(t)$. With the use of Parseval’s equation [32] we can write:

$\int_{-\infty}^{\infty}f(t)G(t)dt=\int_{-\infty}^{\infty}F(\omega)g(\omega)d\omega$
(3)

(under the assumption that $f(t)$ and $g(t)$ are absolutely integrable over
$-\infty<t<\infty$). The kernel functions $C_{\text{tf}}(t-y)$ and
$C_{\text{tt}}(t-x)$ constitute a Fourier transform pair with $t$ considered
as a parameter:

$\mathcal{F}\\{C_{\text{tf}}(t-y)\\}=C_{\text{tt}}(t-x)$ (4)

In the same way, we can establish from equating the frequency-domain charge
$Q(f)$ given by both Eqs. (2b) and (2d) that:

$\mathcal{F}\\{C_{\text{ff}}(f-y)\\}=C_{\text{ft}}(f-x)$ (5)

with $f$ being considered as a parameter. From Eqs. (2a), (2d) (after applying
the Fourier transform to both sides of Eq. (2a)) and Eqs. (2c), (2b) (after
applying the Fourier transform to both sides of Eq. (2c)), we can write:

$C_{\text{ft}}(f-x)=\int e^{-j2\pi ft}C_{\text{tt}}(t-x)dt$ (6)

and

$C_{\text{ff}}(f-y)=\int e^{-j2\pi ft}C_{\text{tf}}(t-y)dt$ (7)

respectively. Finally, from Eqs. (2a) and (2b) we can establish the following
relation between $C_{\text{ff}}(f-y)$ and $C_{\text{tt}}(t-x)$:

$\displaystyle C_{\text{ff}}(f-y)$ $\displaystyle=\frac{1}{2\pi}\iint
C_{\text{tt}}(t-x)e^{-j2\pi tf}e^{j2\pi yx}dt\,dx$
$\displaystyle=\frac{1}{2\pi}\iint C_{\text{tt}}(t-x)e^{j2\pi(yx-tf)}dt\,dx$
(8)

and from Eqs. (2c), (2d) we have:

$\displaystyle C_{\text{ft}}(f-x)$ $\displaystyle=\iint
C_{\text{tf}}(t-y)e^{-j2\pi tf}e^{-j2\pi xy}dt\,dy$ $\displaystyle=\iint
C_{\text{tf}}(t-y)e^{-j2\pi(tf+xy)}dt\,dy$ (9)

Figure 1 summarizes the four different forms of system functions
$C_{\text{ij}}\;(\\{i,j\\}=\\{\text{t},\text{f}\\})$ defined in Eq. (2) along
with their pair relationships either as single or double Fourier transforms.

A simple example that demonstrates the correspondence between these system
functions is the case in which the charge is a delayed copy input voltage
$v(t)$, i.e. [31]:

$q(t)=\mathcal{C}_{\text{tt}}\\{v(t)\\}=Av(t-\tau_{0})\quad(A=1)$ (10)

Thus:

$C_{\text{tt}}(t-x)=\delta(t-x-\tau_{0})$ (11)

which verifies Eq. 2a. From here the rest of the system functions in Eqs. 2b,
2c and 2d are

$\displaystyle C_{\text{ff}}(f-y)$ $\displaystyle=\delta(f-y)e^{-j2\pi
y\tau_{0}}$ (12) $\displaystyle C_{\text{tf}}(t-y)$ $\displaystyle=e^{j2\pi
y(t-\tau_{0})}$ (13) $\displaystyle C_{\text{ft}}(f-x)$
$\displaystyle=e^{-j2\pi f(x+\tau_{0})}$ (14)

respectively. These functions can be transformed into one another as indicated
in Fig. 1. Another example is when the resulting time-domain charge $q(t)$ is
a response to an input voltage $v(t)$ given by [31]:

$q(t)=\mathcal{C}_{\text{tt}}\\{v(t)\\}=v(t-\tau_{0})e^{j2\pi f_{0}t}$ (15)

The four capacitance functions are then:

$\displaystyle C_{\text{tt}}(t-x)$ $\displaystyle=\delta(t-x-\tau_{0})e^{j2\pi
f_{0}t}$ (16) $\displaystyle C_{\text{ff}}(f-y)$ $\displaystyle=\delta(f-y-
f_{0})e^{-j2\pi y\tau_{0}}$ (17) $\displaystyle C_{\text{tf}}(t-y)$
$\displaystyle=e^{2\pi jt(y+f_{0})}e^{-j2\pi y\tau_{0}}$ (18) $\displaystyle
C_{\text{ft}}(f-x)$ $\displaystyle=e^{-j2\pi(f-f_{0})(x+\tau_{0})}$ (19)

Such types of functions are not only necessary in relating charge and voltage
in capacitive devices but are also necessary in modeling communication
channels for instance. As an example, the intrabody communication channel is
predominantly capacitive and modeled by distributed networks of ideal
capacitors, as shown in [33] for an in-vehicle scenario. The availability of
frequency domain-based network analyzers forces the application of frequency-
domain modeling techniques as seen for example in the scattering matrix
results reported in [34].

Figure 2: Schematic diagram similar to Fig. 1 with the electrical charge being
the input and the voltage being the output in a capacitive energy device.

The equation pairs presented above are all related to the case when a
capacitive device is being voltage-charged. The electric charge on the device
is thus a result (output). However, when using the electrical current instead
of voltage to charge capacitive devices, this implies that the charge is
readily created by the charging power supply since $dq=\int i(t)dt$ [35]
before being applied on the device. The input in this case is the electrical
charge and the voltage measured is thus a result (output), as shown in Fig. 2.
A new set of four equations similar to Eqs. (1) can be deduced for this case
albeit the LHS must be the voltage and the RHS must be the charge, with the
mapping functions ($C^{*}_{\text{ij}}$) being inverse-capacitance functions.
When a device is reciprocal, the output charge from Fig. 1 fed as an input in
Fig. 2 should result in an output voltage from Fig. 2 identical to that
supplied as input to Fig. 1 [35]. However, this is not always necessarily true
and hence the computation of a _capacitance_ function is associated only with
voltage excitation whereas the computation of _inverse capacitance_ functions
should be done in the case of current excitations.

##  III Discussion

A capacitive device is treated here as a lumped electrical signal processing
element at the link between voltage input and charge output or charge input
and voltage output (single-input single-output device) as depicted in Figs. 1
and 2, respectively. The charge, being the time-integral of the current, can
be the result of all current contributions taking place within the device and
thus there is no real discrimination between what is effectively being used
for the purpose of energy storage for future reuse, or what is being
dissipated. In mathematical terms, the whole device can be thought of as an
operator which transforms a certain input signal into an output signal, both
of which can be described either in the time domain or frequency domain [29].
Cyclic voltammetry, constant current charge/discharge and impedance
spectroscopy are all routine techniques for time-domain and frequency-domain
measurements of capacitive energy storage devices [10, 15, 12, 36]. However,
the relations presented above are actually valid for any arbitrary excitations
as long as the linearity and time-invariance of the device are respected,
noting that there are also identification methods for time-variant and
nonlinear systems [30, 37, 38]. Furthermore, in reality the input and output
signals can have superposed noises associated with measurement errors or other
environmental conditions that should be taken into account when estimating the
corresponding mapping system functions. This requires the use of efficient
regularization methods and inverse problem computations and not simple
multiplications and divisions as commonly done using the ideal capacitor
formulæ.

In fact, from the classical treatment inherited from ideal capacitors one can
write interchangeably $q=cv$ or $v=q/c$ without worrying about which is the
input variable (voltage or charge). This is clearly incorrect for the general
case of capacitive functions. There is indeed a difference between the system
functions $C_{\text{ij}}$ and $C^{*}_{\text{ij}}$ for a linear time-invariant
capacitive device, and thus between voltage charging and current charging as
illustrated in Figs. 1 and 2. The results further highlight that computing the
capacitive (or inverse capacitive) functions is dependent on the form and type
of the applied excitation, which is different from ideal capacitors. Applying
a step function, linear ramp or sinusoidal excitation (voltage or current)
will result in different capacitive functions. Therefore, one should specify
clearly how the device is being excited and which capacitive (or inverse
capacitive) function is being computed because these functions cannot be
identical to each other except for the case of ideal capacitors (i.e.
$C_{\text{ij}}=C^{*}_{\text{ij}}=c$). We believe that this will have
significant implications on the way capacitance (and similarly inductance) is
computed in the future in an era where capacitive energy storage devices are
on high demand in electrical vehicles, IoT systems and many other
applications.

##  IV Conclusion

This work is an important companion to our recent work in [15]. We have shown
here that there are several system functions that can map an arbitrary input
voltage/charge, represented in time or frequency, to an output charge/voltage,
represented in time or frequency for a capacitive device (see Figs. 1 and 2).
From the estimation of one function (for example ${C}_{\text{tt}}$ that maps
voltage to charge when they are both defined in the time domain, one can
deduct all other functions using single or double Fourier transforms. We
emphasize that the functions $C_{\text{ij}}$ and $C^{*}_{\text{ij}}$ should be
regarded as system functions and not capacitances, which are solely
characteristics of ideal capacitors.

## Acknowledgements

This work is supported by NSF project #2126190 (C.W & A.A.)

## References

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