  The evolution from classical 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 matter waves. In this study, focusing of helium atoms by an electrically
biased FZP is investigated numerically. The $n$th opaque zone of the FZP is
subject to electrostatic biasing using three ways: (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. The effect of biasing the FZP on the transmission coefficient ($T_c$),
focal length ($f$), size of the focused wave packet ($\sigma_F$), transverse
intensity profile, and focusing efficiency ($\eta$) is investigated. The study
reveals that the 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 a better efficiency of $20.3\%$ is observed compared to the other cases.
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.
