# Electrically controlled focusing of de Broglie matter waves by Fresnel zone
plate

Sushanta Barman<EMAIL_ADDRESS>Sudeep Bhattacharjee<EMAIL_ADDRESS>
###### Abstract

The evolution from classical optics to quantum matter wave optics has been
influenced by transformative optical devices. Fresnel zone plates (FZP),
initially designed for light manipulation, have now found expanded
applications in the manipulation of matter waves. In this study, focusing of
helium atoms by an electrically biased FZP, made of chromium (Cr), is
investigated through numerical solutions of the time-dependent Schrödinger
equation (TDSE) using the split-step Fourier method. The $n$th opaque zone of
the FZP is subject to electrostatic biasing using three methods: (i)
$V_{n}=V_{1}$, where $V_{1}$ is the biasing voltage applied to the central
zone, (ii) $V_{n}=V_{1}\sqrt{n}$, and (iii) $V_{n}=V_{1}\sin(k_{E}n)$, with
$k_{E}$ being the radial modulation factor in the biasing voltage. The effect
of biasing the FZP on the distribution of the transmitted wave packet,
transmission coefficient ($T_{c}$), focal length ($f$), size of the focused
wave packet ($\sigma_{F}$), transverse intensity profile at the focal planes,
and focusing efficiency ($\eta$) of the FZP is investigated. The study reavels
that electrical biasing of the FZP modulates the diffractive focusing of
neutral atoms by altering the atom-surface interaction with induced
polarization potential. It is observed that biasing with $V_{n}=V_{1}$ induces
multi-focusing of the FZP, reducing wave packet transmission and focusing
efficiency. Biasing with $V_{n}=V_{1}\sqrt{n}$ significantly enhances the
transmission coefficient by 23.7%, increases the focal length $f$ by 103%, and
improves the focusing efficiency from 10% to 20.17%, indicating enhanced
focusing performance. Biasing with $V_{n}=V_{1}\sin(k_{E}n)$ offers increased
controllability in focusing matter waves through the parameters $k_{E}$ and
$V_{1}$. In this case, a highly intense focused wave packet with better
efficiency of 20.3% is observed compared to the other cases. This
investigation explores various possibilities of electrically controlled
manipulation of matter waves and offers new methods for focusing matter waves
with higher efficiencies. The findings will be helpful in various emerging
applications of atom optics, such as improving the performance of helium
microscopes, enabling control in cold atom trapping on atom chips, and high-
precision atom lithography for quantum electronic devices.

##  1 Introduction

The historical emergence of Fresnel zone plates (FZP) can be traced back to
the early 19th century when the field of wave optics was pioneered by
Augustin-Jean Fresnel [1]. The zone plates are designed with alternating
transparent and opaque concentric rings. The width of each ring is precisely
adjusted to delay the wave passing through it by half a wavelength [2]. This
innovative notion of dividing wavefronts into concentric zones that induce
constructive interference at specific focal points, led to a revolution in
wave optics. The breakthrough led to achieving finer resolution and intricate
focusing capabilities that surpassed the limitations set by conventional
lenses [3, 4, 5, 6, 7, 8]. Over the period of time, the manifold applications
of FZP have contributed to the enhancement in various emerging fields in
optics, including microscopy [9, 10, 11, 12], imaging [13, 14], and
spectroscopy [15, 16, 17, 18, 19]. Recently, FZPs have been employed in a
range of cutting-edge technologies in optics, such as focusing of acoustic
waves [20], spin-waves [21, 4, 22, 23], phase manipulation of electromagnetic
waves with metasurfaces [24, 25], optical tweezers [26], terahertz imaging
[27, 28], astrophotonics [29], nano and integrated photonics [30, 31, 32].

Similarly, in atom optics, neutral atomic matter waves have been focused using
nano-structured FZPs, applying principles similar to those used in classical
optics [33, 34]. The FZP provides universal focusing, applicable even to
fragile, reactive, and excited atomic species [34]. The focusing occurs
without the physical interaction of particles with the zone plate structure as
they pass through open annuli, a consequence of quantum mechanical principles
and the wave-like properties of atoms.

Commercial particle microscopes that use high-energy charged particles for
imaging, cause sample charging if the sample is non-conductive and bring about
damage due to energy transfer from the particles to the layer [35]. Hence,
imaging fragile and insulating samples like polymeric nanostructures and
various biological specimens becomes challenging. In certain cases,
alternatives such as atomic force microscopy [36] or scanning near-field
optical microscopy [37] are considered, but these methods are not suitable for
samples featuring high aspect ratio structures [38]. Therefore, the
suitability of microscopy for fragile and insulating samples can be enhanced
by using low-energy neutral beams.

In this context, helium atoms serve as a potential probe for surface science,
offering minimal surface damage due to their inertness and non-penetrating
nature in the low energy range ($\leq 100$ meV). Furthermore, it is easy to
generate intense and neutral helium beams through supersonic expansion [34].
They closely match the momentum and energy of single-crystal surfaces,
enabling efficient probing and microscopy. Owing to these advantages, a
significant experimental effort has been devoted to focusing low-energy He
beams to develop a neutral atom microscope [33, 34, 35, 39, 40, 41, 42].

In 1991, the initial focusing of neutral He atoms was accomplished using a
negative FZP (opaque central zone) by Carnal et al. [33]. In this pioneering
experiment, metastable helium atoms with various de Broglie wavelengths
($\lambda_{dB}=0.5-2.5$ Å) traversed tiny openings (the source) and were
imaged using a 210 $\mu$m wide zone plate. This study unveiled the potential
to image two-dimensional sub-micron structures, representing a crucial
milestone towards the development of an atomic microscope. Subsequently,
helium atomic beams have been focused to a size of $\leq 2.0$ $\mu$m with a
demagnification factor of 0.4 using the FZP [34, 40], and submicron focusing
has been achieved through subsequent developments [35]. The first microscopy
image was captured in 2008 using neutral He atoms as the imaging probe [39],
with FZP employed for focusing. Furthermore, the use of FZPs has been extended
to focus deuterium molecules [41] and the Bose-Einstein condensates of alkali
atoms [42], which finds applications in erasable atom lithography for quantum
electronic devices.

Figure 1: A schematic diagram of a Fresnel zone plate (FZP) made of chromium
(Cr) having width $d$ ($=50$ nm), is employed to focus helium atoms. The odd
zones (odd values of $n$) are opaque, and the even zones (even values of $n$)
are transparent, defining it as a negative zone plate with the central zone
($n=1$) being opaque. The focal points, labeled as F1 and F2, correspond to
the first- and second-order focusing, respectively. The radii of the
individual zones are determined by $r_{n}=\sqrt{n\lambda_{dB}f_{cal}}$ [2,
33], where $n\in\mathbb{Z}^{+}$, $\lambda_{dB}$ and $f_{cal}$ represent the de
Broglie wavelength of helium atoms and the focal length of the FZP at F1. In
the simulations, the dynamics of a He atom is described using a Gaussian wave
packet. The unfocused part of the transmitted wave packet is referred to as
the zeroth-order, while the defocused part is characterized by the negative
orders.

The ultimate resolution of an atom microscope is limited by the width of the
outermost zone [34, 43]. Fabricating a FZP on a substrate is impractical due
to hindrance from low-energy atom beams. Atom sieves, resembling photon sieves
with numerous pinholes, are employed to overcome these challenges [43]. The
pinholes, arranged in a suitable configuration relative to the Fresnel zones,
are easier to fabricate, thereby reducing the fabrication limit to 3-5 nm
[43]. Moreover, a challenge is posed by the unfocused zeroth-order beam,
resulting in reduced signal contrast. Measurements at low beam intensities are
enabled by an experimentally demonstrated zeroth-order filter, also known as
an order-sorting aperture [44]. Therefore, it becomes crucial to optimize the
microscope design to maximize intensity for a given resolution and outermost
zone width. A model for optimizing the helium zone plate microscope has
recently been developed, demonstrating that a resolution of the order of 10 nm
is achievable with current state-of-the-art detector technology [45].

Studies so far suggest that the manipulation of low-energy neutral atoms by
FZP is applicable not only in surface science, where it is utilized for
imaging fragile and insulating surfaces but also in cutting-edge technologies.
This includes high-precision atom lithography in quantum electronic devices
[42, 46] and trapping of cold atoms in micron-sized atom chips [47]. Hence, it
is imperative to investigate novel methods for manipulating neutral atoms
using FZP and improving their focusing efficiency under various conditions to
facilitate their application in the mentioned areas. Recent experiments
demonstrate electric field control of quantum reflection for matter waves
using periodically microstructured surfaces [48]. External biasing voltage
alters the interaction between the surface and matter waves, revealing a
potential method for electrically tunable reflective diffraction elements
[48]. However, to date, no studies have been conducted on the manipulation of
matter waves by an electrostatically biased FZP and how such biasing
influences the focusing properties of matter waves is unknown, such knowledge
may pave the way for new applications in future quantum technologies.

In this study, the focusing of cold helium atoms using an electrically biased
Fresnel zone plate (FZP) is investigated. The zone plate, constructed from
chromium (Cr) metal, is of negative type, featuring an opaque central zone
($n=1$) and opaque odd $n$-indexed zones. The odd $n$th opaque zone is
subjected to DC biasing in three ways: (i) $V_{n}=V_{1}$, where $V_{1}$ is the
biasing voltage applied to the central zone. This ensures all odd zone
segments have the same voltage $V_{1}$. (ii) $V_{n}=V_{1}\sqrt{n}$, where the
voltage scales as $\sqrt{n}$ of $V_{1}$, for odd $n$. (iii) The voltage
applied to different opaque zones is spatially modulated as
$V_{n}=V_{1}\sin(k_{E}n)$, with $k_{E}$ being the modulation factor. To
investigate the focusing properties of the FZP, numerical simulations are
conducted with a two-dimensional Gaussian wave packet of a helium atom. The
simulation considers the effects of atom-surface interaction, the geometrical
potential of the zone plate, and the induced polarization potential resulting
from the biasing. The distribution of the diffracted wave packet is obtained
by solving the time-dependent Schrödinger’s equation (TDSE) using the split-
step Fourier method [49, 50, 51]. The effects of biasing the FZP on the
distribution of the transmitted wave packet, transmission coefficient
($T_{c}$), focal length ($f$), size of the focused wave packet, transverse
intensity profile at the focal planes, and focusing efficiency of the FZP are
investigated. The results demonstrate that electrically biasing the FZP
modulates neutral atom diffractive focusing by altering atom-surface
interactions. Biasing with $V_{n}=V_{1}$ induces multi-focusing, reducing
transmission and focusing efficiency. Biasing with $V_{n}=V_{1}\sqrt{n}$
enhances transmission by 23.7%, increases focal length by 103%, and improves
efficiency from 10% to 20.17%. Biasing with $V_{n}=V_{1}\sin(k_{E}n)$ enhances
control in focusing matter waves, yielding a highly intense wave packet with
20.3% efficiency. Moreover, the influence of temperature ($T_{a}$) on particle
transmission through the FZP is investigated. It is observed that the
transmission coefficient increases with the increase in $T_{a}$ and reaches a
saturation value of $26.1\%$ at $T_{a}=0.642$ mK. This study will be helpful
in manipulating the transmission and focal properties of a FZP used in matter
wave optics through external electrical biasing.

The structure of the article is as follows: In Section 2, the theoretical
methods utilized in this study are explained. In Section 3, the simulation
results of focusing He wave packets by an electrically biased FZP are
presented and discussed. Finally, in Section 4, conclusions are drawn.

##  2  Theoretical Methods

The Hamiltonian for a He atom interacting with an electrically biased metallic
FZP can be expressed as:

$\displaystyle\textbf{H}=-\frac{\hbar^{2}}{2m}\bm{\nabla}^{2}+V_{g}(\textbf{r})+V_{int}(\textbf{r})+V_{induced}(\textbf{r}),$
(1)

where $m$ and $\hbar$ are the mass of a helium atom and reduced Planck’s
constant, respectively. $V_{g}(\textbf{r})$ is the geometrical potential
energy arising from the opaque zones of the FZP. $V_{\text{int}}(\mathbf{r})$
describes the potential energy arising from the long-range Casimir-van der
Waals interaction between the He atom and the metallic surface of the FZP.
Finally, $V_{\text{induced}}(\mathbf{r})$ represents the induced potential
energy resulting from the polarization of He atoms in the presence of an
electric field. Here, the electric field arises from the biasing of the FZP.

###  2.1 The geometrical potential energy ($V_{g}(\textbf{r})$)

In this study, a negative FZP is considered, as employed in previous studies
[33, 34, 35, 44] for better mechanical stability. The alternating opaque and
transparent zones are depicted in Fig. 1. A negative FZP starts with an opaque
zone at the center (taken as the first zone), and all other odd $n$th zones
are made opaque. The radius of the $n$th zone, denoted as $r_{n}$, is the
distance from the center of the FZP to its outer edge, as depicted in Fig. 1
and given by $r_{n}=\sqrt{n\lambda_{dB}f}$ [2, 33], where
$n\in\mathbb{Z}^{+}$, $\lambda_{dB}$ represents the de Broglie wavelength of
the particle, and $f$ denotes the focal length of the FZP. The geometric
function for a negative FZP can be expressed as [33],

$\xi_{g}(\textbf{r})=\begin{cases}0,&\text{if }r_{2n-1}<\textbf{r}\leq
r_{2n}\text{ (Transparent zone)},\\\ 1,&\text{if }r_{2n}<\textbf{r}\leq
r_{2n+1}\text{ (Opaque zone)}.\\\ \end{cases}$ (2)

Therefore, the geometrical potential energy of the FZP can be obtained as,

$V_{g}(\textbf{r})=V_{g0}\xi_{g}(\textbf{r}),$ (3)

where $V_{g0}$ corresponds to the barrier height associated with the
geometrical potential energy. The value of $V_{g0}$ is taken as $\sim
30E_{k0}$, which is significantly higher than the incident particle energy
($E_{k0}$ ) to minimize quantum tunneling through the opaque regions of the
FZP [52, 53, 54].

###  2.2 Atom-surface interaction energy ($V_{int}(\textbf{r})$)

The atom-surface interaction potential energy for the zone plate can be
written as,

$\displaystyle V_{int}(\textbf{r})=U_{cv}(\chi)\xi(\textbf{r}),$ (4)

where, $\chi$ ($=|x-x_{1}|$ if $x\leq-d/2$ or $|x-x_{2}|$ if $x\geq d/2$)
corresponds to the distance of the atom from the surface of the FZP, as shown
in Fig. 1. $U_{cv}(\chi)$ corresponds to the long-range Casimir-van der Waals
interaction energy that varies in the direction normal to the surface, which
can be expressed as [55, 56, 48, 57],

$U_{cv}(\chi)=-\frac{C_{4}}{(\chi+l)\chi^{3}},\\\ $ (5)

where $l$ ($=9.3$ nm) is a characteristic length determining the transition
from the van der Waals ($\propto 1/\chi^{3}$ at $\chi\ll l$) to the Casimir
regime ($\propto 1/\chi^{4}$ at $\chi\gg l$) [58, 59, 60]. $C_{4}$ ($=C_{3}l$
with $C_{3}$ being the van der Waals adsorption coefficient) is the
interaction coefficient of He atoms with Cr surface. The value of $C_{4}$ is
taken as $2.5\times 10^{-50}$ J m3 [61].

To avoid the singularity in Eq. (5), at $\chi=0$ (on the surface of FZP), a
regularization of $U_{cv}(\chi)$ is introduced, following a similar approach
as outlined in Refs. [62, 63]. The interaction potential energy $U_{cv}(\chi)$
is cut at $\chi=\Lambda$, and for $0\leq\chi\leq\Lambda$, it is suitably
continued as a parabolic curve with the vertex at $\chi=0$. All the parameters
related to this regularization are obtained by imposing the continuity of
$U_{cv}(\chi)$ and $\partial U_{cv}/\partial\chi$ at $\chi=\Lambda$, as
proposed in Ref. [62]. The regularized form of $U_{cv}(\chi)$ is obtained as,

$\displaystyle
U_{cv}(\chi)=\begin{cases}-\frac{C_{4}}{(\chi+l)\chi^{3}},&\text{ if
$\chi>\Lambda$},\\\
\frac{C_{4}}{2\Lambda^{5}(\Lambda+l)^{2}}\left[(4\Lambda+3l)\chi^{2}-(6\Lambda+5l)\Lambda^{2}\right],&\text{if
$\Lambda\geq\chi>0$},\\\
-\frac{C_{4}}{2\Lambda^{3}}\frac{(6\Lambda+5l)}{(\Lambda+l)^{2}},&\text{if
$\chi=0$}.\\\ \end{cases}$ (6)

The plot in Fig. 2(a) illustrates the variation of the regularized
$U_{cv}(\chi)$ with respect to $\chi$, as obtained in Eq. (6). It is evident
that $U_{cv}(\chi)$ follows a parabolic curve within the region
$0\leq\chi\leq\Lambda$.

Figure 2: Variation of regularized Casimir-van der Waals potential energy
$U_{cv}(\chi)$ with the distance $\chi$ from the surface of the Fresnel zone
plate, as derived in Eq. (6). Here, the value of $\Lambda$ is ta taken as 10
nm [62].

###  2.3 The induced polarization energy ($V_{induced}(\textbf{r})$)

The FZP is biased with an electrostatic potential $V_{b}$, producing an
electric field $\textbf{E}(\textbf{r})=-\bm{\textbf{}\nabla}\phi(\textbf{r})$,
where $\phi(\textbf{r})$ is the spatial distribution of electric potential.
The spatial distribution of $\phi(\textbf{r})$ is governed by Laplace’s
equation, $\bm{\textbf{}\nabla}^{2}\phi(\textbf{r})=0$, subject to appropriate
boundary conditions determined by the geometric function $\xi_{g}(\textbf{r})$
of the FZP and the biasing voltage $V_{b}$. The induced potential energy of a
polarizable particle in presence $\textbf{E}(\textbf{r})$ is given by [63, 48,
64],

$\displaystyle
V_{induced}(\textbf{r})=-\frac{\alpha}{2}|\textbf{E}(\textbf{r})|^{2},$ (7)

where $\alpha$ ($=1.383746$ $a^{3}_{0}$ with $a_{0}$ being the Bohr radius) is
the static polarizability of He atoms [65]. Prior to determining the
distribution of $V_{\text{induced}}(\textbf{r})$, the solution for
$\phi(\textbf{r})$ is obtained numerically by solving Laplace’s equation using
the successive over-relaxation method [66, 67], as detailed in Ref. [53].

Table 1:  Values of the parameters used in the simulation. Parameters | Symbols | Values
---|---|---
Width of the initial wave packet | ($\sigma_{x}$, $\sigma_{y}$) | (50 nm, 600 nm)
Central position of the initial wave packet | ($X_{0}$, $Y_{0}$) | ($-f_{cal}/2$, $0$)
de Broglie Wavelength | $\lambda_{dB}$ | 49.84 nm
Velocity of the incident He atom | v |  $2$ m/s
Radius of the first zone | $r_{1}$ | 200 nm
Focal length of the zone plate | $f_{cal}$ |  $803.21$ nm
Thickness of the zone plate | $d$ | 50 nm
Geometrical potential barrier height | $V_{g0}$ | 24.39 meV
Scaling factor for energy | $V_{0}$ | 0.21 $\mu$eV
Scaling factor for length | $\beta$ | 5 nm
Scaling factor for time | $\tau$ |  $3.152$ ns
Biasing voltage applied to the central zone | $V_{1}$ |  $0-30$ V
Radial modulation length of the biasing voltage | $\lambda_{E}$ | 15.71 - 628.32
Wave number associated with the radial modulation of the biasing voltage |  $k_{E}$ ($=2\pi/\lambda_{E}$) | $0.01-0.4$

###  2.4 Numerical solution of TDSE

The dynamics of the matter wave associated with a He atom interacting with a
FZP can be obtained by solving the time-dependent Schrödinger equation (TDSE)
in two-dimensional space ($x$, $y$). In order to enhance the computational
efficiency in simulations, the spatial and temporal domains are transformed
into arbitrary units of space and time as
$\textbf{r}(x,y)=\beta\textbf{R}(X,Y)$ and $t=\tau T$, where
$\beta=\sqrt{\hbar^{2}/2mV_{0}}$ and $\tau=2m\beta^{2}/\hbar$ are the scaling
factors of length and time, respectively. The corresponding scaling factor for
energy can be obtained as $V_{0}=\hbar^{2}/2m\beta^{2}$. Hence, the 2D
Laplacian in the ($X$, $Y$) domain can be expressed as
$\bm{\nabla}^{2}=(1/\beta^{2})\bm{\nabla}_{\beta}^{2}=(1/\beta^{2})(\partial^{2}/\partial
X^{2}+\partial^{2}/\partial Y^{2})$. Similarly, the potential energies are
scaled as $W_{g}(\textbf{R})=(1/V_{0})V_{g}(\textbf{r})$,
$W_{int}(\textbf{R})=(1/V_{0})V_{int}(\textbf{r})$, and
$W_{induced}(\textbf{R})=(1/V_{0})V_{induced}(\textbf{r})$. Therefore, the
Hamiltonian H in Eq. (1) can be expressed in the (R, $T$) space as,

$\displaystyle\textbf{H}_{\beta}=-\bm{\nabla}^{2}_{\beta}+W_{g}(\textbf{R})+W_{int}(\textbf{R})+W_{induced}(\textbf{R}).$
(8)

The nondimensionalized form of the TDSE in arbitrary space (R) and time ($T$)
can be obtained as,

$\displaystyle i\frac{\partial\psi(\textbf{R},T)}{\partial
T}=\textbf{H}_{\beta}\psi(\textbf{R},T),$ (9)

where $i=\sqrt{-1}$ and $\psi(\textbf{R},T)$ is the complex wave function
associated with the particle.

The TDSE (Eq. (9)) is solved numerically using the split-step Fourier method
[68, 69] for an initial wave function $\psi(\textbf{R},T=0)$. The time
evolution of the wave function from $T$ to $T+\Delta T$ is carried out by [68,
69]

$\displaystyle\psi(\textbf{R},T+\Delta
T)=\bigg{[}\exp{\bigg{(}-\frac{i\mathbf{H_{pot}}\Delta
T}{2}}\bigg{)}\bm{\mathcal{F}}^{-1}\bigg{\\{}\exp{\bigg{(}-i\mathbf{H_{kin}}\Delta
T}\bigg{)}\mathcal{F}\bigg{(}\exp{\bigg{(}-\frac{i\mathbf{H_{pot}}\Delta
T}{2}\bigg{)}}\psi(\textbf{R},T)\bigg{)}\bigg{\\}}\bigg{]},$ (10)

where $\Delta T$ is the temporal step size.
$\mathbf{H_{kin}}=-\bm{\nabla}^{2}_{\beta}$ and
$\mathbf{H_{pot}}=W_{g}(\textbf{R})+W_{int}(\textbf{R})+W_{induced}(\textbf{R})$
are the kinetic and potential energy part of $\mathbf{H}_{\beta}$
($=\mathbf{H_{kin}}+\mathbf{H_{pot}}$), respectively, in Eq. (8).
$\mathcal{F}$ and ${\mathcal{F}}^{-1}$ denote forward and inverse Fourier
Transforms, respectively. The evolution operator $\mathbf{H_{\beta}}$
($=\mathbf{H_{kin}}+\mathbf{H_{pot}}$) is split using the second-order version
of the Baker-Campbell-Hausdorff formula [70, 68, 71]. This splitting is
performed with an error of $O(\Delta T^{3})$ to obtain the time evolution of
$\psi(\textbf{R},T=0)$ [68].

The initial wave packet is taken as a two-dimensional Gaussian wave packet
given by,

$\displaystyle\psi(\textbf{R},$ $\displaystyle
T=0)=N\exp{\bigg{[}-\frac{1}{2}\bigg{\\{}\frac{|(\textbf{R}-\textbf{R}_{0})\cdot\hat{\textbf{X}}|^{2}}{\sigma_{X}^{2}}+\frac{|(\textbf{R}-\textbf{R}_{0})\cdot\hat{\textbf{Y}}|^{2}}{\sigma_{Y}^{2}}\bigg{\\}}+i\textbf{K}_{0}\cdot(\textbf{R}-\textbf{R}_{0})\bigg{]}},$
(11)

where $N$ is the normalization constant and $\textbf{R}_{0}(X_{0},Y_{0})$ is
the center of the wave packet at $T=0$. $\sigma_{X}$ ($=\sigma_{x}/\beta$) and
$\sigma_{Y}$ ($=\sigma_{y}/\beta$) are the width of the wave packet along
$\hat{X}$ and $\hat{Y}$ directions, respectively. $\textbf{K}_{0}$
($=\hat{X}2\pi/\lambda^{{}^{\prime}}_{dB}$) is the wave number of the initial
wave packet moving along the $\hat{X}$ direction. The values of $\sigma_{X}$,
$\sigma_{Y}$, and $\lambda^{{}^{\prime}}_{dB}$ ($=\lambda_{dB}/\beta$) are
normalized with respect to the spatial scaling factor $\beta$.

In the simulations, artificial reflections occur when the wave packet hits the
boundary of the numerical box. Therefore, absorptive boundary conditions, as
implemented in Refs. [54, 75], are employed to prevent unwanted reflections
from the boundaries. At each time step, the wave function is multiplied by a
damping function $\gamma_{L}(X)$ given by [72],

$\displaystyle\gamma_{L}(X)=\frac{1}{1+\exp\bigg{(}-\frac{X-X_{ref}}{\mu}\bigg{)}},$
(12)

where the parameters $\mu$ ($=5$) and $X_{ref}$ ($=0$) control the sharpness
and position of the damping function at the left boundary. The function
$\gamma_{L}(X)$ maintains a unit value for all $X$ except near the left
boundary, where suppression is applied to minimize artificial reflections. For
the right boundary, appropriate values of $\mu$ ($=-5$) and $X_{ref}$
($=X_{\text{max}}$) are chosen to determine the damping function
$\gamma_{R}(X)$. Similar damping functions $\gamma_{U}(X)$ and $\gamma_{D}(X)$
are implemented in the simulations for the upper and lower boundaries,
respectively.

A 2D region ($X$, $Y$) with square grids ($\Delta X=\Delta Y$) having
dimensions $701\times 701$ is considered in the simulations. Spatial and
temporal scaling factors are chosen as $\beta=5$ nm and $\tau=3.152$ ns,
respectively. All the simulation parameters are listed in Table 1. The
numerical calculations, initiated with a normalized initial Gaussian wave
packet (Eq. (11)), are implemented using the Python programming language.
Standard modules such as NumPy and SciPy are employed in the calculations.
Convergence in the simulation is ensured by preserving the unit norm of the
wave packet $\psi(\textbf{R},T)$ at each time step.

Figure 3: (a) Distribution of the integrated probability density
$I_{X-\text{optical}}$, as defined in Eq. (14), for a Gaussian wave packet
transmitted through a thin Fresnel zone plate with a thickness of $d=10$ nm
along the optical axis ($X$-axis). The distribution of the probability density
$\rho(X/f_{cal},Y/r_{1},T)$ of the transmitted wave packet is depicted at two
specific time instances: (b) $T=1350$, corresponding to its arrival at the
third focal point F3, and (c) $T=1975$, corresponding to its arrival at the
first focal point F1. Here, the distributions of
$\rho(X/f_{\text{cal}},Y/r_{1},T)$, shown in both the panels (b) and (c), have
been normalized relative to the maximum value $\rho_{\text{max}}$ of the
first-order peak. Figure 4: (a) Distribution of the probability density
$\rho(X/f_{cal},Y/r_{1},0)=|\psi(X/f_{cal},Y/r_{1},0)|^{2}$ of the initial
Gaussian wave packet moving along the $+X$ direction. The width $w_{0}$ of the
initial wave packet is defined at the waist (AB and CD dotted lines), where
the probability density $\rho(X/f_{cal},Y/r_{1},0)$ is reduced to $1/e^{2}$
($\sim 13.5\%$) of its maximum value. (b) Distribution of
$\rho(X/f_{cal},Y/r_{1},T)=|\psi(X/f_{cal},Y/r_{1},T)|^{2}$ of the transmitted
wave packet at $T_{1}=2025$, when the wave packet reached at the focal point
F1 ($X=f_{1}$).

##  3 Results and discussions

###  3.1 Benchmarking of the simulation

To validate the simulations, an initial investigation of a thin Fresnel zone
plate (thickness $d\sim 10$ nm) was conducted. The obtained results were
subsequently compared with analytically derived findings reported earlier [73,
74]. For this analysis, a straightforward scenario was considered, excluding
the Casimir-van der Waals interaction potential energy $V_{int}(\text{{r}})$
and without the application of any electrical bias.

In the simulation, an initial Gaussian wave packet was considered, as defined
by Eq. (11). The wave packet undergoes temporal evolution and gets diffracted
by the FZP as it propagates along the $+X$ direction. Upon interaction with
the FZP, the incident wave packet experiences partial reflection and
transmission. The transmission coefficient $T_{c}$ for the FZP was calculated
by,

$\displaystyle
T_{c}=\int_{Y=-\infty}^{+\infty}dY\int_{X=0}^{+\infty}dX|\psi(\textbf{R},T)|^{2}.$
(13)

A transmission coefficient of $T_{c}=0.358$ has been obtained for the thin
FZP. A similar value of the transmission coefficient has been reported earlier
for a standard FZP in Ref. [74].

To determine the positions of the focal points, the probability density along
the optical axis was recorded at each time step [8]. Subsequently, a time-
integrated intensity profile along the optical axis ($X$-axis) was obtained
as,

$\displaystyle
I_{X-optical}(X,Y=0)=\int_{T=0}^{T_{max}}|\psi(X,Y=0,T)|^{2}dT,$ (14)

where $T_{max}$ is the total simulation time.

The simulation result, illustrating the distribution of $I_{X-optical}$ along
the $X$-axis for the thin FZP, is shown in Fig. 3(a). Multi-focusing behavior
is observed in this case, with distinct focal points, namely F1 and F3, as
depicted in Fig. 3(a). Furthermore, it is observed that the first focal length
$f_{1}$ is approximately equal to the calculated focal length,
$f_{\text{cal}}$ ($=r^{2}_{1}/\lambda_{dB}$). Additionally, the focal length
for F3 is found to be $f_{3}\approx f_{1}/3$. This confirms the odd-order
multi-focusing by the FZP, in agreement with the findings reported earlier
[73]. Moreover, the intensity at F3 is observed to be significantly lower
($\sim 16\%$) in comparison to the intensity at F1, as shown in Fig. 3(a).
This aspect is further clarified by analyzing the probability density
distributions at both focal points (F3 and F1), as shown in Fig. 3(b) and (c),
respectively.

Thereafter, the focusing efficiency $\eta_{j}$ at the $j$-th order focal point
was calculated as [73],

$\displaystyle\eta_{j}=\frac{I_{j}}{I_{initial}},$ (15)

where, $I_{initial}$ and $I_{j}$ are the intensity of the incident wave packet
and the intensity at $j$-th focal point. In this case, the values of
$\eta_{1}$ and $\eta_{3}$ are obtained as $9.92\%$ and $0.84\%$, respectively.
A similar value of $\eta_{1}$ ( $=1/\pi^{2}\approx 10\%$) has been previously
reported for a standard FZP [73, 74]. These observations validate the accuracy
of the simulation. In all subsequent cases, the actual system, including all
interaction potential energies, has been simulated.

###  3.2 Diffractive focusing of matter waves by FZP

In this case, the diffractive focusing by a metallic zone plate with a
thickness of $d=50$ nm has been investigated. All the other simulation
parameters are listed in Table 1. The distribution of the probability density
$\rho(X,Y,T=0)$ of the initial wave packet is shown in Fig. 4(a). The width
($w_{0}$) of the initial wave packet is determined at the waist of the
Gaussian distribution, as indicated by the dotted lines AB (at
$Y/r_{1}=Y_{1}/r_{1}=-4.225$) and CD (at $Y/r_{1}=Y_{2}/r_{1}=4.225$) in Fig.
4(a). At the waist, the probability density $\rho(X,Y,0)$ diminishes to
$1/e^{2}$ ($\sim 13.5\%$) of its peak value ($\rho_{max}=2.627\times 10^{-4}$)
[75, 76].

In this case, a transmission of $26.1\%$ is observed for the wave packet
through the FZP, which is relatively smaller than the transmission observed in
a thin grating (35.8%). This is due to the increased thickness of the FZP. The
spatial distribution of the transmitted wave packet $\rho(X,Y,T)$ after the
FZP at time $T=2025$ is shown in Fig. 4(b). The wave packet is observed to be
focused at the first focal point (F1) with a focal length $f_{1}$ ($\approx
740$ nm). Notably, it is observed that the focal length $f_{1}$ is
approximately $92.5\%$ of the calculated focal length $f_{\text{cal}}$ (=
$800$ nm). The reduction in the focal length is attributed to the finite width
of the zone plate ($d=50$ nm) employed in this study.

In Fig. 4(b), it can be observed that, in addition to the central peak at F1,
multiple off-axis peaks are present with reduced intensity. These peaks,
between the dotted lines AB and CD in Fig. 4(b), primarily originate from
zeroth-order diffraction, representing the unfocused part of the wave packet
[35, 44]. The presence of these unfocused parts of the wave packet decreases
the quality of focusing. In the later sections, the intensity of the unfocused
part has been investigated with different biasing voltages.

Figure 5: Variation of the de Broglie wavelength ($\lambda_{dB}$) of He atom
with temperature is shown on the left y-axis. The variation of the
transmission coefficient ($T_{c}$) with temperature, as it passes through the
Fresnel zone plate, is shown on the right y-axis. As temperature increases,
the initial kinetic energy of particles also increases, resulting in an
increase in $T_{c}$. The saturation point of the transmission coefficient
($T_{c}$) occurs at a temperature of 0.642 mK, with a value reaching
approximately $26.1\%$. Figure 6: Distributions of the biasing potentials
applied to the $n$th opaque zone are illustrated for different scenarios: (a)
$V_{n}=30$ V, (d) $V_{n}=2.5\sqrt{n}$ V, (g) $V_{n}=4\sqrt{n}$ V, (j)
$V_{n}=15\sin(0.05n)$ V, and (m) $V_{n}=30\sin(0.3n)$ V. The corresponding
distributions of the induced potential
$V_{\text{induced}}(X/f_{\text{cal}},Y/r_{1})$ arising from the polarization
of a helium atom in the presence of electric fields are shown in panels (c),
(f), (i), (l), and (o), respectively. The panels (b), (e), (h), (k), and (n)
depict the variation of $V_{\text{induced}}(0,Y/r_{1})$ in each case,
respectively.

###  3.3 Effect of particle temperature on matter wave focusing

Simulations were performed for various temperatures $T_{a}$ of the incident
particle to explore its effect on wave packet transmission through the FZP.
Here, the particles are assumed to emerge from a source maintained at
temperature $T_{a}$. The de Broglie wavelength $\lambda_{dB}$ varies with
$T_{a}$ as,

$\displaystyle\lambda_{dB}=\frac{h}{\sqrt{3mk_{B}T_{a}}},$ (16)

where $k_{B}$ is the Boltzmann constant.

The variation of $\lambda_{dB}$ with $T_{a}$ is shown on the left y-axis of
Fig. 5. From the figure, it is evident that $\lambda_{dB}$ decreases
monotonically as $T_{a}$ increases, as expected. The right y-axis of Fig. 5
depicts the variation of the $T_{c}$ with $T_{a}$. It is observed that $T_{c}$
increases with the increase in $T_{a}$. Moreover, the transmission coefficient
saturates at $T_{a}=0.642$ mK, reaching a value of $T_{c}=0.261$. This occurs
due to the dominance of quantum reflection from the diffracting surface at a
lower temperature[77]. The wave packet of the impinging particle gets
reflected at a significant distance from the surface because of the long-range
Casimir-van der Waals interaction $U_{cv}(\chi)$, leading to diminished
contributions to the transmission at lower temperatures. In the subsequent
cases, simulations were carried out at a constant temperature of 0.642 mK,
corresponding to $\lambda_{dB}=49.84$ nm.

###  3.4 Electrostatic biasing of FZP

In the presence of electric fields, a neutral He atom experiences induced
potential $V_{\text{induced}}(\textbf{r})$ due to its polarization, which
enables controlled external manipulation of the wave packet. Three distinct
biasing configurations are employed: (i) uniform biasing of all zones with a
constant potential, denoted as $V_{n}=V_{1}$, where $V_{1}$ represents the
applied voltage to the central zone; (ii) biasing of the $n$th zone as
$V_{n}=V_{1}\sqrt{n}$; and (iii) the voltage applied to the $n$th zone is
radially modulated as $V_{n}=V_{1}\sin(k_{E}n)$, with $k_{E}$ being the
modulation factor. The biasing voltages applied to different zones for all
three cases are depicted in Figs. 6(a), (d), (g), (j), and (m).

####  3.4.1 $V_{n}=V_{1}$

In this case, a simple biasing configuration has been considered to
investigate its effect on the focusing properties of a FZP, where all the
opaque zones of the FZP are biased with the same voltage $V_{1}$ applied to
the central zone ($n=1$). The biasing voltage $V_{1}$ varies from 0 to 30 V.
Fig. 6(a) illustrates the distribution of the biasing voltage to all the
opaque zones for $V_{1}=30$ V.

The distribution of the induced potential energy
$V_{\text{induced}}(\textbf{r})$, resulting from biasing the zone plate with a
voltage of $V_{1}=30$ V, is depicted in Fig. 6(c). Furthermore, the profile of
$V_{\text{induced}}(0,Y)$ along the FZP, as indicated by the yellow colored
dotted line AB in Fig. 6(c), is shown in Fig. 6(b). From Fig. 6(c), it is
evident that $V_{\text{induced}}(0,Y)$ is nearly zero inside the transparent
zones, whereas it has an enhanced magnitude near the edges of the opaque
zones. This indicates that the effective width ($=r_{n}-r_{n-1}$) of different
zones remains unchanged after applying a biasing voltage to the FZP as
$V_{n}=V_{1}$. Instead, it is seen in Fig. 6(c) that the biasing in this
manner effectively alters the cross-sectional shape of the zones from
rectangular to circular, as previously reported for a diffraction grating
[53]. Furthermore, the induced potential energy gradually decreases with the
normal distance $\chi$ away from the zone plate (along $X$-axis), as
illustrated in Fig. 6(c). Being an attractive potential in nature, the induced
polarization potential having such a distribution will accelerate the incident
wave packet as it approaches the FZP and decelerate the outgoing transmitted
wave packet. Therefore, by biasing the FZP, the diffraction of the wave packet
can be modulated, thereby allowing for controlled adjustments to the focusing
properties of the transmitted wave packet.

Figure 7: Distributions of the integrated probability density
$I_{X-\text{optical}}$ along the optical axis ($X$-axis) are shown when the
$n$th zone of the Fresnel zone plate (FZP) is biased as $V_{n}=V_{1}$, for
different biasing voltages: (a) $V_{1}=0$ V, (b) $V_{1}=5$ V, (c) $V_{1}=10$
V, (d) $V_{1}=15$ V, (e) $V_{1}=20$ V, (f) $V_{1}=25$ V, and (g) $V_{1}=30$ V.
Panel (h) shows the variation of the transmission coefficient ($T_{c}$),
whereas panel (i) illustrates the variation of the focusing efficiency
($\eta_{j}$) at the focal point Fj with the biasing voltage $V_{1}$.

####  3.4.2 $V_{n}=V_{1}\sqrt{n}$

In this case, the $n$th opaque zones of the FZP are subjected to biasing with
voltages denoted as $V_{n}=V_{1}\sqrt{n}$, as depicted in Figs. 6(d) and (g)
for $V_{1}=2.5$ V and 4 V, respectively. The biasing voltage of the central
zone, $V_{1}$, is varied from 0 to 5 V.

Since $V_{\text{induced}}(\mathbf{r})$ is attractive in nature, the
transmission of the wave packet through the transparent zones will be
proportional to the magnitude of the induced potential inside the transparent
zones. The variation of the induced potential energy inside the transparent
zones along the $Y$-axis can be obtained as,

$\displaystyle
V_{induced}(0,Y)=-\frac{\alpha}{2}\bigg{(}-\frac{dV_{n}}{dn}\frac{dn}{dr}\bigg{)}^{2}=-\frac{\alpha
V_{1}^{2}}{2\lambda_{dB}f_{cal}},$ (17)

where $dY$ is replaced by $dr$ because $dY=dr$. It can be noticed that
$V_{\text{induced}}(0,Y)$ is independent of $n$ and varies quadratically with
$V_{1}^{2}$. Therefore, the value of $V_{\text{induced}}(0,Y)$ will be the
same inside all the transparent zones. Moreover, it can be expected that the
transmission must vary quadratically with $V_{1}$.

The distributions of $V_{induced}(\text{{r}})$ for $V_{1}=2.5$ V and $4$ V are
illustrated in Fig. 6(f) and 6(i), respectively. It can be seen that larger
values of $V_{\text{induced}}(\textbf{r})$ are attained inside the transparent
zones compared to the exterior of the FZP. This is also evident from Figs.
6(e) and (h), where the distributions of $V_{\text{induced}}(0,Y)$ are shown
for $V_{1}=2.5$ V and $4$ V, respectively. In Figs. 6(e) and (h), it can be
seen that the magnitude of $V_{\text{induced}}(0,Y)$ is almost constant inside
the higher order transperent zones. Moreover, the cross-sectional shape of the
opaque zones is altered by this, which differs from the effects observed in
the first case ($V_{n}=V_{1}$). Therefore, by biasing the FZP in this manner,
the transmission of the wave packet can be modulated differently than in the
first case, thereby allowing for controlled manipulation of the focusing
properties of the transmitted wave packet.

####  3.4.3 $V_{n}=V_{1}\sin(k_{E}n)$

In this case, the $n$th opaque zones of the FZP are subjected to biasing with
voltages denoted as $V_{n}=V_{1}\sin(k_{E}n)$, with $k_{E}$ being the
modulation factor. Here, $k_{E}$ is the additional controlling factor that
offers increased flexibility in manipulating matter waves by the FZP. In this
case, the values of $V_{1}$ are varied from 0 to 30 V, and the values of
$k_{E}$ are varied from $0.01$ to $0.4$.

As obtained in the earlier case, the variation of $V_{\text{induced}}(0,Y)$
can be derived for this case as,

$\displaystyle V_{induced}(0,Y)=-\frac{2\alpha
V_{1}^{2}k_{E}^{2}}{\lambda_{dB}f_{cal}}n\cos^{2}(k_{E}n).$ (18)

Here, it can be observed that $V_{\text{induced}}(0,Y)$ varies sinusoidally
with both $k_{E}$ and $n$. Therefore, it is expected that the transmission
coefficient will also vary in a similar manner as $V_{\text{induced}}(0,Y)$
with $k_{E}$ and $n$.

Figs. 6(j) and (m) depict the distribution of the biasing voltage to the
opaque zones for ($V_{1}=15$ V, $k_{E}=0.05$) and ($V_{1}=30$ V, $k_{E}=0.3$),
respectively. It is observed that, for higher values of $k_{E}$ ($=0.3$), the
sinusoidal modulation in the biasing voltage applied to different zones
becomes more prominent (see Fig. 6(m)), particularly near the central part of
the FZP, where the wave packet primarily interacts. Conversely, for lower
values of $k_{E}$ ($=0.05$), the modulation becomes almost linear (see Fig.
6(j)), as expected.

The distributions of $V_{\text{induced}}(\text{{r}})$ for ($V_{1}=15$ V,
$k_{E}=0.05$) and ($V_{1}=30$ V, $k_{E}=0.3$) are shown in Figs. 6(l) and (o),
respectively. In Fig. 6(o), it can be observed that
$V_{\text{induced}}(\text{{r}})$ is notably small in the proximity of the
central zone ($n=1$) and undergoes modulation as one moves towards the higher-
order zones. Additionally, a decrease in $V_{\text{induced}}(\textbf{r})$ is
noted in the exterior region compared to inside the transparent zones. Figs.
6(k) and (n) illustrate the distributions of $V_{\text{induced}}(0,Y)$ for
($V_{1}=15$ V, $k_{E}=0.05$) and ($V_{1}=30$ V, $k_{E}=0.3$), respectively. In
Fig. 6(n), it is observed that the variation of the amplitude of
$V_{\text{induced}}(0,Y)$ inside different zones for ($V_{1}=30$ V,
$k_{E}=0.3$) agrees well with the results obtained using Eq. (18). For smaller
values of $k\leq 0.05$, it can be shown that $V_{\text{induced}}(0,Y)=-(\alpha
V_{1}^{2}k_{E}^{2}/\lambda_{dB}f_{\text{cal}})n$, implying a linear dependence
on $n$. This is also evident from Fig. 6(k).

Moreover, a periodic modulation in the cross-sectional shape of the opaque
zones is also evident from Fig. 6(o), in contrast to the other two cases
mentioned earlier. Therefore, biasing the FZP sinusoidally can lead to a
different modulation of the diffraction of the wave packet compared to the
other two cases.

Figure 8: The probability density distribution
$\rho(X/f_{\text{cal}},Y/r_{1},T)=|\psi(X/f_{\text{cal}},Y/r_{1},T)|^{2}$ of
the transmitted wave packet under different biasing voltages are shown in
panels: (a) $V_{1}=5$ V at $T=2050$, (b) $V_{1}=10$ V at $T=2025$, and (c)
$V_{1}=15$ V at $T=1950$, and (d) $V_{1}=30$ V at $T=1825$. Here, all panels
depict the results of biasing the $n$th zone of the FZP with $V_{n}=V_{1}$ and
at an instance of time when the wave packet arrives at the first focal point
(F1). In all panels, the distribution of $\rho(X/f_{\text{cal}},Y/r_{1},T)$
has been normalized with respect to the maximum value ($\rho_{\text{max}}$) of
the central peak. Panel (e) shows the transverse intensity profile on a screen
placed at F1, obtained by
$I=\int_{T=0}^{T_{\text{max}}}|\psi(f_{1}/f_{\text{cal}},Y/r_{1},T)|^{2}dT$.

###  3.5 Focusing properties of a biased FZP

Here, the effects of biasing the FZP on the distribution of the transmitted
wave packet $\rho(X,Y,T)$, transmission coefficient ($T_{c}$), and focusing
properties, such as focal length ($f$), size of the focused wave packet
($\sigma_{f}$), transverse intensity profile at the focal planes, and focusing
efficiency ($\eta$), are demonstrated.

####  3.5.1 $V_{n}=V_{1}$

It is important to investigate how the focal points of a biased FZP vary with
the biasing voltage ($V_{1}$). To determine the focal points, the
distributions of the integrated probability density $I_{X-\text{optical}}$
(Eq. (14)) along the optical axis ($X$-axis) are obtained for different values
of $V_{1}$ and are shown in Figs. 7(a)-(g). In Fig. 7(a), where the
distribution of $I_{X-\text{optical}}$ is shown for $V_{1}=0$, two focal
points (F1 and F2) are observed, similar to the case of a thin grating (see
Fig. 3(a)). However, the intensity at F2 is significantly lower than at F1.
Additionally, the first focal length $f_{1}$ is found to be smaller ($\sim
89\%$) than the calculated focal length $f_{\text{cal}}$
($=r_{1}^{2}/\lambda_{dB}$). This difference may be attributed to the finite
width $d$ ($=50$ nm) of the FZP, a factor that is typically not considered in
analytical calculations. More importantly, it can be observed that with the
increase in $V_{1}$ from 0 V to 30 V, the number of focal points increases,
and the intensity at F1 decreases, as depicted in Fig. 7(a)-(g). Furthermore,
the first focal length increases to $118.1\%$ of $f_{\text{cal}}$ at
$V_{1}=30$ V. Therefore, biasing the FZP as $V_{n}=V_{1}$ can control the
number of focal points, their intensities, and focal lengths.

Thereafter, the transmission coefficient $T_{c}$ of the wave packet through
the FZP has been obtained using Eq. (13). Fig. 7(h) shows the variation of
$T_{c}$ with the biasing voltage $V_{1}$. It is observed that $T_{c}$
decreases from 0.261 to 0.102 as $V_{1}$ increases from 0 to 30 V. This occurs
because the induced polarization energy $V_{\text{induced}}(\mathbf{r})$
increases on both sides of the FZP with the increase in biasing voltage
$V_{1}$, as depicted in Fig. 6(c). Hence, $V_{\text{induced}}(\mathbf{r})$
begins to dominate the long-range atom-surface interaction
$V_{\text{int}}(\mathbf{r})$ (see Fig. 2), resulting in the modulation of the
overall interaction between the helium atom and the FZP. This modulation
enhances the quantum reflection probability of He atoms by the FZP,
consequently leading to a decrease in the transmission probability with the
increase in $V_{1}$.

The focusing efficiency at the $j$th order focal point Fj has been calculated
using Eq. (15). The variation of $\eta_{1}$ with $V_{1}$ is shown in Fig.
7(i). It is observed that similar to the transmission coefficient, $\eta_{1}$
also decreases from 8.71% to 2.11% with the increase in the biasing voltage
$V_{1}$. Moreover, the values of $\eta_{j}$ at higher-order ($j\geq 2$) focal
points are found to vary under 2.5% for all the biasing voltages, as shown in
Fig. 7(i).

The distributions of the probability density of the transmitted wave packet
have been obtained as $\rho(X,Y,T)=|\psi(X,Y,T)|^{2}$. Figs. 8(a)-(d) depict
the distribution of $\rho(X,Y,T)$ for the biasing voltages $V_{1}=5$, 10, 15,
and 30 V, respectively. From the figures, it is observed that the focusing
time at F1 for $V_{1}=5$ V is $T=2050$, and it decreases with the increase in
$V_{1}$ to $T=1825$ for $V_{1}=30$ V. This occurs because the wave packet
undergoes acceleration caused by the expansion of the induced potential energy
$V_{\text{induced}}(\mathbf{r})$ on both side of the FZP, as depicted in Fig.
6(c). This expansion in $V_{\text{induced}}(\mathbf{r})$ increases with the
rise in $V_{1}$, resulting in more accelerations and, consequently, a decrease
in the focusing time.

The intensity profile in the transverse direction at a screen placed in the
first focal plane is shown in Fig. 8(e) for $V_{1}=0$, 15, and 30 V. It is
observed that with the increase in $V_{1}$, the intensity at the central peak
(at F1) decreases. This is attributed to the decrease in the transmission of
the wave packet through the FZP at higher biasing voltage, as shown in Fig.
7(h). Moreover, in Fig. 8(e), it is observed that a significant portion of the
transmitted wave packet remains unfocused.

The variation of the first focal length $f_{1}$ with $V_{1}$ is shown in Fig.
9. It is observed that $f_{1}$ increases from $0.89f_{\text{cal}}$ to
$1.18f_{\text{cal}}$ as the biasing voltage increases from 0 to 30 V. The
increase in $f_{1}$ is due to more acceleration at higher values of $V_{1}$
caused by the induced potential. However, the width $\sigma_{F1}$ (full width
at half maximum) of the focused wave packet at F1 remains almost constant at
1.7% of the width of the initial wave packet $\sigma_{Y}$, as illustrated on
the right $y$-axis in Fig. 9.

Figure 9: Variation of the first focal length ($f_{1}$) with the biasing
voltage $V_{1}$ is shown on the left $y$-axis. The results are obtained by
biasing the $n$th zone of the Fresnel zone plate (FZP) as $V_{n}=V_{1}$. The
variation of the width ($\sigma_{F1}$) of the focused wave packet at the first
focal point (F1) with $V_{1}$ is depicted on the right $y$-axis. Here, the
values of $f_{1}$ and $\sigma_{F1}$ are normalized by the theoretical focal
length $f_{\text{cal}}$ and the width of the initial wave packet
($\sigma_{Y}$), respectively.

Therefore, biasing the zones of the FZP as $V_{s}=V_{1}$ induces multiple
focal points at higher voltages. Moreover, the transmission coefficient and
the focusing efficiency decrease with the increase in $V_{1}$. However, the
first focal length $f_{1}$ increases slightly, and the size of the focused
wave packet at F1 remains almost constant while increasing the biasing
voltage.

Figure 10: Distributions of the integrated probability density
$I_{X-\text{optical}}$ along the optical axis ($X$-axis) are presented, with
the $n$th zone of the zone plate biased as $V_{n}=V_{1}\sqrt{n}$, for various
biasing voltages: (a) $V_{1}=0$ V, (b) $V_{1}=1$ V, (c) $V_{1}=2$ V, (d)
$V_{1}=3$ V, (e) $V_{1}=4$ V, and (f) $V_{1}=5$ V. Panel (g) depicts the
variation of the transmission coefficient ($T_{c}$), while panel (h)
illustrates the variation of the focusing efficiency ($\eta_{j}$) with the
biasing voltage $V_{1}$.

####  3.5.2 $V_{n}=V_{1}\sqrt{n}$

The positions of the focal points at different biasing voltages $V_{1}$ are
shown in Figs. 10(a)-(f), where the distributions of $I_{X-\text{optical}}$
along the optical axis are presented. In Figs. 10(a)-(f), it can be observed
that similar to the previous case ($V_{n}=V_{1}$), the number of focal points
depends on the biasing voltage. There are two focal points for $V_{1}\leq 1.5$
V, and its the number increases to four at $V_{1}=5$ V. The intensity at the
higher-order focal points is significantly lower than at F1. More
interestingly, it can be seen that the peak intensity at F1 for $V_{1}=4$ V is
larger compared to the other cases. Additionally, the first focal length
$f_{1}$ is found to increase with the increase in $V_{1}$. Furthermore, the
value of $f_{1}$ increases from 0.89 to 1.81 times the calculated focal length
$f_{\text{cal}}$ at $V_{1}=5$ V. Therefore, similar to the earlier case
($V_{n}=V_{1}$), biasing the FZP as $V_{n}=V_{1}\sqrt{n}$ can control the
number of focal points, intensity at the focal points, and focal lengths.

The variation of the transmission coefficient $T_{c}$ is shown in Fig. 10(g).
It is observed that $T_{c}$ varies quadratically with $V_{1}$, as mentioned
earlier, and increases from 0.261 to 0.323 as $V_{1}$ increases from 0 to 5 V.
This can be understood through the distribution of
$V_{\text{induced}}(\mathbf{r})$ resulting from the biasing of the FZP as
$V_{n}=V_{1}\sqrt{n}$. In Figs. 6(f) and (i), it is observed that the
magnitude of $V_{\text{induced}}(\mathbf{r})$ is nearly smaller near the
central zone and it increases inside the higher-order transparent zones. This
effectively enlarges the width of the higher-order transparent zones, thereby
increasing their effective area. Hence, there is an increase in wave packet
transmission with increase in $V_{1}$. Furthermore, unlike the previous case
($V_{n}=V_{1}$), $V_{\text{induced}}(\mathbf{r})$ has a small magnitude on
either side of the FZP (see Figs. 6(f) and (i)) due to the application of a
relatively smaller biasing voltage $V_{1}$ (maximum voltage 5 V) to the
central zone. Hence, the reflection probability is not enhanced as observed in
the previous case.

Figure 11: Distributions of the probability density
$\rho(X/f_{\text{cal}},Y/r_{1},T)=|\psi(X/f_{\text{cal}},Y/r_{1},T)|^{2}$ of
the transmitted wave packet, with a biasing potential applied to the $n$th
zone of the Fresnel zone plate (FZP) given by $V_{n}=V_{1}\sqrt{n}$, are
presented. The panels depict the distribution of
$\rho(X/f_{\text{cal}},Y/r_{1},T)$ for various values of $V_{1}$ and time $T$:
(a) $V_{1}=1$ V at $T=2075$, (b) $V_{1}=3$ V at $T=2400$, (c) $V_{1}=4$ V at
$T=2675$, and (d) $V_{1}=5$ V at $T=2900$. All the panels depict the
distributions at the instance when the wave packet reaches the respective
first focal point (F1). In each case, the distributions of
$\rho(X/f_{\text{cal}},Y/r_{1},T)$ have been normalized with respect to the
maximum value ($\rho_{\text{max}}$) of the respective central peak. Panel (e)
shows the time-integrated transverse intensity profile along the $Y$-axis on a
screen placed at F1 ($X=f_{1}$), obtained as
$I=\int_{T=0}^{T_{\text{max}}}|\psi(f_{1}/f_{\text{cal}},Y/r_{1},T)|^{2}dT$.

The variation of $\eta_{1}$ with $V_{1}$ is shown in Fig. 10(h). It is
observed that the value of $\eta_{1}$ strongly depends on $V_{1}$. Initially,
it decreases from 8.71% to 5.37% as $V_{1}$ increases from 0 to 2.5 V.
Thereafter, $\eta_{1}$ attains a maximum value of 20.17% at $V_{1}=4$ V. The
focusing efficiency at higher-order focal points ($\eta_{j}$, $j>2$) also
depends on $V_{1}$, but it remains within 5%.

The distributions of the probability density $\rho(X,Y,T)$ are shown in Figs.
11(a)-(d) when the wave packet reaches the respective F1 for the biasing
voltages $V_{1}=1$, 3, 4, and 5 V, respectively. In the figures, it is
observed that the focusing time at F1 increases from $T=2075$ to $T=2900$ as
$V_{1}$ increases from 1 to 5 V. This occurs because the wavefront coming from
the higher-order zones takes longer to achieve constructive interference at F1
due to the gradual increase in the magnitude of
$V_{\text{induced}}(\mathbf{r})$ near the higher-order transparent zones (see
Figs. 6(f) and (i)).

The transverse intensity profile at a screen placed in the F1 plane is shown
in Fig. 11(e) for $V_{1}=0$, 3, 4, and 5 V. In Fig. 11(e), it is observed that
the peak intensity at F1 depends on $V_{1}$, as noted in the previous case
($V_{n}=V_{1}$). However, in this case, the central peak intensity becomes
maximum at $V_{1}=4$ compared to all other biasing voltages. This is
attributed to the enhanced focusing efficiency $\eta_{1}$ at $V_{1}=4$ V, as
depicted in Fig. 10(h). More importantly, the distribution of the transverse
intensity profile indicates a significant reduction in the unfocused part at
the focal planes compared to the previous case ($V_{n}=V_{1}$).

The variation of $f_{1}$ with the biasing voltage $V_{1}$ is shown in Fig. 12
(left $y$-axis). It can be seen that $f_{1}$ increases from
$0.89f_{\text{cal}}$ to $1.81f_{\text{cal}}$ as the biasing voltage increases
from 0 to 5 V. This results from the increase in the effective radius of the
transparent zones with the increase in $V_{1}$, as depicted in Figs. 6(f) and
(i). As the first focal length $f_{1}$ is proportional to $r^{2}_{n}/n$, in
contrast to the previous case, the focal length $f_{1}$ increases with the
increase in $V_{1}$.

The variation of the width of the focused wave packet $\sigma_{F1}$ with
$V_{1}$ is shown in Fig. 12 (right $y$-axis). It is observed that
$\sigma_{F1}$ increases slightly as $V_{1}$ increases from 0 to 5 V, but it
remains within 4% of the width of the initial wave packet $\sigma_{Y}$. This
increase in $\sigma_{f}$ is a consequence of the enlarged effective size of
the higher-order transparent zones with $V_{1}$, as the size of the focused
wave packet is limited by the width of the outermost zone [34, 43].

The FZP biasing as $V_{n}=V_{1}\sqrt{n}$ results in an increase in the
transmission coefficient with the increase in $V_{1}$. Furthermore, a maximum
focusing efficiency of $\eta_{1}=20.17\%$ is observed for the first time in He
atom focusing using a biased FZP, which is twice the efficiency observed
earlier using a Fresnel-Soret zone plate [73, 74]. Moreover, the first focal
length ($f_{1}$) increases to $1.81f_{cal}$ while keeping the width of the
focused wave packet $\sigma_{F1}$ almost constant (within 4%). These suggest
improved controllability and enhanced efficiency in matter wave focusing by
the FZP employing a biasing voltage that varies as $V_{n}=V_{1}\sqrt{n}$.

Figure 12: The left $y$-axis shows the variation of the first focal length
($f_{1}$) with the biasing voltage $V_{1}$. These results are obtained by
biasing the $n$th zone of the Fresnel zone plate (FZP) as
$V_{n}=V_{1}\sqrt{n}$. The variation of the width ($\sigma_{F1}$) of the
focused wave packet at the first focal point (F1) with $V_{1}$ is illustrated
on the right $y$-axis. The values of $f_{1}$ and $\sigma_{F1}$ are normalized
by the theoretical focal length $f_{\text{cal}}$ and the width of the initial
wave packet ($\sigma_{Y}$), respectively. Figure 13: Variation of (a)
transmission coefficient $T_{c}$ and (b) efficiency $\eta_{1}$ for the first-
order focusing with the biasing voltage $V_{1}$ and the radial modulation
factor $k_{E}$. Here, the $n$th zone of the Fresnel zone plate (FZP) is biased
as $V_{n}=V_{1}\sin(k_{E}n)$. (c) Variation of the time-integrated intensity
along the optical axis is shown for ($V_{1}=0$, $k_{E}=0$) (black colored
dotted curve) and for ($V_{1}=15$, $k_{E}=0.05$) (red colored solid curve).

####  3.5.3 $V_{n}=V_{1}\sin(k_{E}n)$

In this case, the radial modulation factor $k_{E}$ in the biasing voltage,
along with its magnitude $V_{1}$, enhances the flexibility in the manipulation
of wave packet by the FZP. The factor $k_{E}$ induces a periodic modulation in
the polarization energy (Eq. (17)), which aids in controlling the transmission
and phase of the wavefront emerging from the different zones. The biasing
voltage $V_{1}$ varies from 0 to 30 V, and $k_{E}$ varies from $0.01$ to
$0.4$.

The variation of the transmission coefficient $T_{c}$ with $k_{E}$ and $V_{1}$
is depicted as a contour plot in Fig. 13(a). It can be observed that, for
lower values of $V_{1}$, the wave packet transmission remains almost
independent of $k_{E}$, whereas for higher values of $V_{1}$ (e.g., $V_{1}=30$
V), $T_{c}$ exhibits a strong dependence on $k_{E}$. This is because, at
smaller biasing voltages, the amplitude of the induced potential energy is so
small that it does not have any significant effect on the wave packet
transmission. On the other hand, for $k_{E}\leq 0.05$, the effect of
sinusoidal modulation in the biasing voltage becomes negligible, leading to a
reduced induced potential energy for all values of $V_{1}$. Therefore, $T_{c}$
becomes almost independent of $V_{1}$ for $k_{E}\leq 0.05$, as shown in Fig.
13(a). It reaches a maximum value of $T_{c}=0.316$ at ($V_{1}=30$ V and
$k_{E}=0.1$) and a minimum value of $T_{c}=0.121$ at ($V_{1}=30$ V and
$k_{E}=0.4$).

The variation of the first order focusing efficiency $\eta_{1}$ with $k_{E}$
and $V_{1}$ is shown in Fig. 13(b). It can be observed that the focusing
efficiency strongly depends upon both $V_{1}$ and $k_{E}$. A maximum value of
$\eta_{1}=20.3\%$ is observed at ($V_{1}=15$ V and $k_{E}=0.05$), marking the
highest achieved focusing efficiency among all other cases. Furthermore, to
assess the intensity at F1 for this case, the distribution of
$I_{X-\text{optical}}$ is shown in Fig. 13(c) and compared with the
$I_{X-\text{optical}}$ obtained without biasing voltage ($V_{1}=0$). The peak
intensity at F1 for this case ($V_{1}=15$ V and $k_{E}=0.05$) is observed to
be $4.1\times 10^{-2}$, which is the highest among all the cases.

The first focal length $f_{1}$ is found to vary from $0.89f_{cal}$ to
$1.7f_{cal}$, and the size of the focused wave packet $\sigma_{F1}$ remains
within 5% of the width of the initial wave packet $\sigma_{Y}$.

Therefore, controllability in wave packet manipulation is enhanced by the
radial modulation factor $k_{E}$ and biasing voltage $V_{1}$, while the FZP is
biased as $V_{n}=V_{1}\sin(k_{E}n)$. The induced periodic modulation in
polarization energy allows control over wave packet transmission. The
transmission coefficient $T_{c}$ strongly depends on $V_{1}$ for higher
values, reaching a maximum of $T_{c}=0.316$. The first-order focusing
efficiency $\eta_{1}$ reaches a maximum of 20.3% at ($V_{1}=15$ V,
$k_{E}=0.05$), exhibiting the highest achieved efficiency. These findings
suggest enhanced performance in matter wave focusing when employing a biasing
voltage that varies as $V_{n}=V_{1}\sin(k_{E}n)$.

##  4 Conclusion

In conclusion, the focusing of neutral helium atoms by a metallic Fresnel zone
plate (FZP) is investigated by numerically solving the time-dependent
Schrödinger equation (TDSE) using the split-step Fourier method. The
transmission coefficient and focusing properties of the FZP are modulated
through electrically controlled interactions between the atoms and the
metallic FZP. The zone plate, made of chromium (Cr), undergoes electrostatic
biasing in three ways: (i) $V_{n}=V_{1}$, where $V_{1}$ represents the biasing
voltage applied to the central zone, (ii) $V_{n}=V_{1}\sqrt{n}$, and (iii)
$V_{n}=V_{1}\sin(k_{E}n)$. The simulation takes into account the effects of
atom-surface interaction through the long-range Casimir-van der Waals
interaction, the geometrical potential of the zone plate, and the induced
polarization potential resulting from biasing. The study investigates the
impact of biasing the FZP on various aspects, including the distribution of
the focused wave packet, transmission coefficient ($T_{c}$), focal length
($f$), size of the focused wave packet ($\sigma_{F}$), transverse intensity
profile at the focal planes, and focusing efficiency ($\eta$) of the FZP.
Moreover, the effect of the temperature of the incident particles on $T_{c}$
has been investigated.

The study has revealed the following characteristics in the electrically
controlled focusing of matter waves by a FZP:

  1. 1.

The wave packet transmission through the FZP increases as the temperature
($T_{a}$) of the incident particle increases and it attains a saturated value
at $T_{a}\sim 0.642$ mK.

  2. 2.

Electrical biasing modifies the interaction between the polarizable He atoms
and the metallic FZP by inducing polarization energy due to the electric
field. As a result, the diffractive focusing of neutral atoms undergoes
modulation.

  3. 3.

Biasing the FZP as $V_{n}=V_{1}$ induces multi-focusing at higher values of
$V_{1}$. Moreover, it reduces the wave packet transmission and diminishes the
focusing efficiency, thereby not contributing to improving the focusing
performance of the FZP.

  4. 4.

Biasing the FZP as $V_{n}=V_{1}\sqrt{n}$ significantly enhances the
transmission coefficient by 23.7% due to the enlargement of the higher $n$th
transparent zones as $V_{1}$ increases, influenced by the distribution of
$V_{\text{induced}}(\mathbf{r})$. Additionally, the focal length $f$ increases
by 103%, and the focusing efficiency gets enhanced from 10% to 20.17%,
indicating improved focusing performance.

  5. 5.

Biasing the FZP as $V_{n}=V_{1}\sin(k_{E}n)$ offers extra controllability to
the focusing of matter waves through the parameters $k_{E}$ and $V_{1}$,
whereas in the previous cases, the focusing performance is controlled by
$V_{1}$ alone. In this case, a highly intense focused wave packet with a
better efficiency of 20.3% is observed. Therefore, this method of biasing
demonstrates better performance in focusing.

The finding of this study will be helpful in manipulating the transmission and
focal properties of a FZP employed in matter wave optics through external
electrical biasing. The externally controlled focusing can enhance the
efficiency of neutral helium-based microscopes employed in imaging sensitive
surfaces. Furthermore, it introduces novel approaches for enhanced control in
trapping cold atoms on atom chips and facilitates high-precision atom
lithography for quantum electronic devices.

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## Acknowledgements

We gratefully acknowledge the financial support from the funding agency SERB
(Department of Science and Technology), Government of India, Grant No.
CRG/2022/000112. The simulations were performed at the High-Performance
Computing Facility (HPC2013) at IIT Kanpur.

## Author contributions statement

Both authors, S. B. (Sushanta Barman) and S. Bh. (Sudeep Bhattacharjee),
contributed to the conception of the study. S. Bh. provided the facilities and
supervised the research. S. B. conducted simulations and analyzed data. Both
authors contributed to writing the manuscript, data visualization, and
manuscript revision. All authors reviewed the manuscript.

## Additional information

Competing interests: The authors declare no competing interests.
Correspondence and requests for materials should be addressed to S. B. or S.
Bh.