#  Polarization effects in the total rate of biharmonic $\omega+3\omega$
ionization of atoms

J. Hofbrucker  Helmholtz-Institut Jena, Fröbelstieg 3, D-07743 Jena, Germany
GSI Helmholtzzentrum für Schwerionenforschung GmbH, Planckstrasse 1, D-64291
Darmstadt, Germany  S. Ramakrishna  Helmholtz-Institut Jena, Fröbelstieg 3,
D-07743 Jena, Germany  GSI Helmholtzzentrum für Schwerionenforschung GmbH,
Planckstrasse 1, D-64291 Darmstadt, Germany  Theoretisch-Physikalisches
Institut, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, D-07743 Jena,
Germany  A. V. Volotka  Department of Physics and Engineering, ITMO
University, Kronverkskiy pr. 49, 197101 St. Petersburg, Russia  S. Fritzsche
Helmholtz-Institut Jena, Fröbelstieg 3, D-07743 Jena, Germany  GSI
Helmholtzzentrum für Schwerionenforschung GmbH, Planckstrasse 1, D-64291
Darmstadt, Germany  Theoretisch-Physikalisches Institut, Friedrich-Schiller-
Universität Jena, Max-Wien-Platz 1, D-07743 Jena, Germany

###### Abstract

The total ionization rate of biharmonic ($\omega+3\omega$) ionization is
studied within the independent particle approximation and the third order
perturbation theory. Particular attention is paid to how the polarization of
the biharmonic light field affects the total rate. The ratios of the
biharmonic ionization rates for linearly and circularly polarized beams as
well as for corotating and counterrotating elliptically polarized beams are
analyzed, and how they depend on the beam parameters, such as photon frequency
or phase between $\omega$ and $3\omega$ light beams. We show that the
interference of the biharmonic ionization amplitudes determines the dominance
of a particular beam polarization over another and that it can be controlled
by an appropriate choice of beam parameters. Furthermore, we demonstrate our
findings for the ionization of neon $L$ shell electrons.

##  I Introduction

The influence of the polarization of the light beam on the total
photoionization cross sections has been a focus of research for decades. While
the total cross section for the one-photon ionization of unpolarized atoms is
independent of the polarization of the ionizing light, the multi-photon
ionization cross section of atoms is typically influenced by its polarization.
Based on rather a simple analysis of angular factors, Klarsfeld and Maquet
Klarsfeld and Maquet (1972) predicted that atoms are ionized more efficiently
by a circularly polarized light beam than by a linearly polarized beam.
Moreover, they also pointed out that the ratio of total multi-photon
ionization cross sections for circular versus linear polarization increases
rapidly with the order of the process. However, subsequent theoretical studies
Reiss (1972); Pei-Jie and Feng (2009); Buica and Nakajima (2008) showed that
multiphoton ionization by linearly polarized light beams is predominant in the
higher order interaction regime. These theoretical findings were
experimentally confirmed for low Fox et al. (1971); Cervenan and Isenor (1974)
and high Lompre et al. (1977) order ionization processes.

A number of Free Electron Laser (FEL) facilities are capable today of
producing intense circularly polarized extreme-ultraviolet (XUV) beams Allaria
et al. (2014) which, together with an additional laser, allow ionization of
atoms by a circularly polarized XUV + IR light beam. In this process, the XUV
pulse pumps the atomic system to several states with well-defined projection
of angular momentum, while the IR beam subsequently interacts with an already
polarized target. Different total ionization rates are measured and depend on
whether the IR beam is co- or counterrotating with the XUV beam. The question
of whether the ionization by two co- or counterrotating beams is more
efficient depends strongly on the frequency and intensity of the IR beam
Ilchen et al. (2017); De Silva et al. (2021). It was shown that ionization by
corotating XUV+IR beams dominates Ilchen et al. (2017); Grum-Grzhimailo et al.
(2019) at low IR beam intensities, whereas the result reverses for higher
intensities. However, due to the complexity of the process, further
investigations need to be carried out to fully understand the process.

In this paper, we will address the two questions mentioned above namely, about
the dominance of ionization by linearly or circularly polarized and co- or
counterrotating beams, for the case of biharmonic $\omega+3\omega$ ionization
of atoms. Biharmonic beams consist of two co-propagating field components with
a fixed phase difference and with frequencies that are integer multiples of
the same fundamental frequency $\omega$, i.e. $m\,\omega+n\,\omega$. In the
biharmonic multi-photon ionization of atoms by such beams, the photoelectrons
ionized by $n\times(m\,\omega)$\- or $m\times(n\,\omega)$-photons are
therefore released with the same energy. It has been shown that the
interference between the two processes gives rise to properties in
photoelectron angular distributions, such as up/down asymmetry Douguet et al.
(2017); Gryzlova et al. (2019), elliptical dichroism Fifirig (2002);
Hofbrucker et al. (2020) or circular dichroism Volotka et al. (2021), and it
can be even used for the creation and control of electron vortices Pengel et
al. (2017); Kerbstadt et al. (2019); Armstrong et al. (2019). In
$\omega+\;2\omega$ ionization of atoms, the photoelectron partial-wave states
are orthogonal to each other and, hence, the total ionization rate is
independent of interference between the two process Yin et al. (1992).
However, the interference between the two processes in $\omega+\;2\omega$
ionization of atoms is imprinted in the photoelectron angular distribution
Prince et al. (2016); Giannessi et al. (2018); Di Fraia et al. (2019). In
contrast, both photoelectron distributions as well as the total rate of
$\omega+\;3\omega$ ionization depends on the interference between the one- and
three-photon ionization processes, as discussed in the theoretical paper by
Chan, Brumer and coworkers Chan et al. (1991).

Here we address two main questions in detail. Is it more effective to ionize
an atom with a linearly or circularly polarized beams? Are the total electron
yields dominant for ionization of atoms by corotating or counterrotating
biharmonic fields? We show that the answers to these questions strongly depend
on the beam properties and the interference between the one- and three-photon
ionization processes. We demonstrate our findings for the ionization of neon
$L$ shell electrons.

This paper is structured as follows. We first introduce our fully relativistic
approach, which is based on the third-order perturbation theory and the
independent particle approximation in Sec. II. Sec. III shows our main
results, where the total biharmonic $\omega+3\omega$ ionization rates are
compared for various combinations of polarization as an example of ionization
of the $2s_{1/2}$ as well as $2p_{3/2}$ electrons of neutral neon atoms. A
summary is given in Sec. IV. A more detailed description of the first and
third-order ionization amplitudes are provided in the Appendix.

##  II Theory

The vector potential of a plane wave photon with frequency $n\omega$, wave
vector $\bm{k}$ can be written as

$\displaystyle\bm{A}^{(n\omega)}(\bm{r},t)$ $\displaystyle=$
$\displaystyle\bm{\varepsilon}^{(n\omega)}e^{-in\omega
t+i\bm{k}^{(n\omega)}\cdot\bm{r}}.$ (1)

Here, the polarization is denoted by $\bm{\varepsilon}^{(n\omega)}$ and can be
expressed in terms of ellipticity $\gamma^{(n\omega)}$ and basis vectors in
helicity representation $\bm{\varepsilon}_{\pm 1}$ as

$\bm{\varepsilon}^{(n\omega)}=\frac{\bm{\varepsilon}_{-1}[1-\gamma^{(n\omega)}]-\bm{\varepsilon}_{+1}[1+\gamma^{(n\omega)}]}{\sqrt{2[1+(\gamma^{(n\omega)})^{2}]}}.$
(2)

The ellipticity takes values in the range ${|\gamma^{(n\omega)}|\leq 1}$,
where ${\gamma^{(n\omega)}=-1}$ refers to left-circularly,
${\gamma^{(n\omega)}=0}$ to linearly and ${\gamma^{(n\omega)}=1}$ to right-
circularly polarized light.

Let us moreover consider a biharmonic $\omega+3\omega$ beam that consists of a
fundamental frequency $\omega$ and its third harmonic-order co-propagating
along the quantization axis $\bm{\hat{k}}||\bm{\hat{z}}$. The vector potential
of such a biharmonic beam can be written as

$\displaystyle\bm{A}(\bm{r},t)$ $\displaystyle=$ $\displaystyle
A_{0}^{(\omega)}\bm{A}^{(\omega)}(\bm{r},t)+e^{i\Phi}A_{0}^{(3\omega)}\bm{A}^{(3\omega)}(\bm{r},t),$
(3)

where $\Phi$ defines a constant phase shift between the two beam components.
Furthermore, $A_{0}^{(n\omega)}$ is the amplitude of the vector potential of
each component and is directly proportional to the flux of the component
${F^{(n\omega)}=(A_{0}^{(n\omega)})^{2}}$ and its intensity
${I^{(n\omega)}=n\omega F^{(n\omega)}}$.

Figure 1:  Schematic representation of the electric dipole ionization pathways
in biharmonic $\omega+3\omega$ ionization of atoms. Both the one- and three-
photon ionization always lead to at least one partial wave of the free
electron that coincides and that gives rise to the interference effects in the
total biharmonic rates.

Below, we consider the ionization of a closed-shell neutral atom in an initial
many-electron state $\left|\alpha_{i}J_{i}M_{i}\right\rangle$ by a biharmonic
$\omega+3\omega$ field as given in Eq. (3). The atomic state is characterized
by the total angular momentum $J$, its projection $M$ and further quantum
numbers $\alpha$ which are necessary to uniquely describe the atomic state.
The interaction of the biharmonic field with the atom can lead to a number of
different processes. We shall consider the below-threshold ionization
($2\omega<E_{b}$) of an atom due to the absorption of one photon with energy
$3\omega$ or three photons with energy $\omega$, as shown schematically in the
Fig. 1 for the biharmonic $\omega+3\omega$ ionization. The final state of the
system consists of a singly charged ion, a photoelectron which is denoted by
$\left|\alpha_{f}J_{f}M_{f},\bm{p}_{e}m_{e}\right\rangle$, momentum
$\bm{p}_{e}$ and the projection of spin $m_{e}$. In this work, we will
consider infinitely long biharmonic pulses, an assumption that applies well
for the biharmonic pulses produced by the current FEL’s Douguet et al. (2017);
Gryzlova et al. (2018); Di Fraia et al. (2019). We describe the one- and
three-photon ionization processes within the lowest-order perturbation theory,
whose transition amplitudes are

$M_{M_{i}M_{f}m_{e}}^{(3\omega)}=\left\langle\alpha_{f}J_{f}M_{f},\bm{p}_{e}m_{e}\left|\bm{\alpha\cdot}\bm{A}^{(3\omega)}\right|\alpha_{i}J_{i}M_{i}\right\rangle,$
(4)

$\displaystyle M_{M_{i}M_{f}m_{e}}^{(\omega)}$ $\displaystyle=$
$\displaystyle\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{\nu_{2}}\left\langle\alpha_{f}J_{f}M_{f},\bm{p}_{e}m_{e}\left|\bm{\alpha\cdot}\bm{A}^{(\omega)}\right|\alpha_{\nu_{2}}J_{\nu_{2}}M_{\nu_{2}}\right\rangle$
(13) $\displaystyle\times$
$\displaystyle\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{\nu_{1}}\left\langle\alpha_{\nu_{2}}J_{\nu_{2}}M_{\nu_{2}}\left|\bm{\alpha\cdot}\bm{A}^{(\omega)}\right|\alpha_{\nu_{1}}J_{\nu_{1}}M_{\nu_{1}}\right\rangle$
(22) $\displaystyle\times$
$\displaystyle\frac{\left\langle\alpha_{\nu_{1}}J_{\nu_{1}}M_{\nu_{1}}\left|\bm{\alpha\cdot}\bm{A}^{(\omega)}\right|\alpha_{i}J_{i}M_{i}\right\rangle}{(E_{i}+2\omega-
E_{\nu_{2}})(E_{i}+\omega-E_{\nu_{1}})},$ (23)

respectively, and where $\bm{\alpha}$ denotes the vector of Dirac matrices.
Moreover, we make use of the independent-particle approximation, where the
electron wave function is represented by a single active electron, while all
other electrons are taken into account by a screening potential in the
Hamiltonian of the Dirac equation; see Hofbrucker et al. (2016) for a detailed
description of the numerical method. However, numerical calculations for a
many-electron atomic system can be performed with atomic structure theory
codes such as JAC Fritzsche (2019). Due to the interaction of the atom with
the electromagnetic field, the active electron of the substate described by
the principal $n_{a}$, relativistic $\kappa_{a}$ quantum numbers, and the
projection of total angular momentum $m_{a}$ given by
$\left|a\right\rangle\equiv\left|n_{a}\kappa_{a}m_{a}\right\rangle$ of the
atom is promoted into the continuum, leaving a vacancy in the atomic substate.
The relativistic quantum number $\kappa_{a}$ can be obtained from the total
($j_{a}$) and orbital ($l_{a}$) angular momentum quantum numbers by
$\kappa=(-1)^{l_{a}+j_{a}+1/2}(j_{a}+1/2)$. In the second quantization, the
final many-electron state can be described by a Slater determinant wave
function with the use of the electron creation
$a^{\dagger}_{\bm{p}_{e}m_{e}}$, annihilation operators
$a_{n_{a}\kappa_{a}m_{a}}$ and the Clebsch-Gordan coefficients
$\left\langle..,..\left|\right...\right\rangle$ as

$\displaystyle\left|{\alpha_{f}J_{f}M_{f},\bm{p}_{e}m_{e}}\right\rangle$
$\displaystyle=$ $\displaystyle\sum_{m_{a}M}\left\langle
j_{a}-m_{a},J_{i}M\left|\right.J_{f}M_{f}\right\rangle$ $\displaystyle\times$
$\displaystyle(-1)^{j_{a}-m_{a}}a^{\dagger}_{\bm{p}_{e}m_{e}}a_{n_{a}\kappa_{a}m_{a}}\left|{\alpha_{i}J_{i}M}\right\rangle.$

In the independent particle approximation, the many-electron amplitudes (4)
and (13) can be simplified to amplitudes which depend only on one-electron
wave functions of the active electron,

$\displaystyle M_{M_{i}M_{f}m_{e}}^{(3\omega)}$ $\displaystyle=$
$\displaystyle\sum_{m_{a}}\left\langle
j_{a}-m_{a},J_{i}M_{i}\left|\right.J_{f}M_{f}\right\rangle(-1)^{j_{a}-m_{a}}$
$\displaystyle\times$
$\displaystyle\left\langle\bm{p}_{e}m_{e}\left|\bm{\alpha\cdot}\bm{A}^{(3\omega)}\right|a\right\rangle,$

and

$\displaystyle M_{M_{i}M_{f}m_{e}}^{(\omega)}$ $\displaystyle=$
$\displaystyle\sum_{m_{a}}\left\langle
j_{a}-m_{a},J_{i}M_{i}\left|\right.J_{f}M_{f}\right\rangle(-1)^{j_{a}-m_{a}}$
(26) $\displaystyle\times$
$\displaystyle\sum_{n_{1}n_{2}}\frac{\left\langle\bm{p}_{e}m_{e}\left|\bm{\alpha\cdot}\bm{A}^{(\omega)}\right|n_{2}\right\rangle\left\langle
n_{2}\left|\bm{\alpha\cdot}\bm{A}^{(\omega)}\right|n_{1}\right\rangle}{(E_{a}+2\omega-
E_{n_{2}})(E_{a}+\omega-E_{n_{1}})}$ $\displaystyle\times$
$\displaystyle\left\langle
n_{1}\left|\bm{\alpha\cdot}\bm{A}^{(\omega)}\right|a\right\rangle.$

To calculate the third-order transition amplitude, we perform two summations
over the complete energy spectra of single-electron intermediate states
$\left|n_{1,2}\right\rangle$. For the sake of numerical evaluation, it is
convenient to expand the transition amplitudes. To do that, we express the
photoelectron wave function into its partial-wave components

$\displaystyle\left|\bm{p}_{e}m_{e}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{E_{e}|\bm{p}_{e}|}}\sum_{jm_{j}}\sum_{lm_{l}}i^{l}e^{-i{\delta_{\kappa}}}\left\langle
lm_{l},1/2m_{e}\left|\right.jm_{j}\right\rangle$ (27) $\displaystyle\times$
$\displaystyle\left|E_{e}\kappa m_{j}\right\rangle
Y^{\ast}_{lm_{l}}(\theta,\phi),$

where the electron energy is given by $E_{e}=\sqrt{\bm{p}_{e}^{2}+1}$, the
phases of the partial waves by $\delta_{\kappa}$ and the emission direction of
each partial wave in terms of the polar $\theta$ and azimuthal $\phi$ angles
by the spherical harmonics $Y_{lm_{l}}(\theta,\phi)$. Furthermore, the vector
potential ${{\bm{A}}^{(n\omega)}}$ can be decomposed into spherical tensors
with electric ($p=1$) and magnetic ($p=0$) components of multipolarity $J$
using

${{\bm{A}}^{(n\omega)}}=4\pi\sum_{JMp}i^{J-p}[\bm{\varepsilon}^{(n\omega)}\cdot\bm{Y}^{(p)\ast}_{JM}(\hat{\bm{k}})]{\bm{a}_{JM}^{(p)}}(\bm{r}).$
(28)

The explicit form of the two transition amplitudes in Eqs. (II) and (26) are
provided in the Appendix for the first- and third-order transition amplitudes,
respectively.

The amplitudes of the one- and three-photon ionization are applied to
construct the biharmonic $\omega+3\omega$ ionization rate

$\displaystyle W(\gamma^{(\omega)},\gamma^{(3\omega)})$ $\displaystyle=$
$\displaystyle\int
d\Omega\frac{1}{[J_{i}]}\sum_{M_{i}M_{f}m_{e}}|K^{(\omega)}M_{M_{i}M_{f}m_{e}}^{(\omega)}$
(29) $\displaystyle+$ $\displaystyle
K^{(3\omega)}e^{i\Phi}M_{M_{i}M_{f}m_{e}}^{(3\omega)}|^{2},$

with $[J_{i}]=(2J_{i}+1)$ and with the prefactors for one- and two-photon
ionization $K^{(3\omega)}=\sqrt{\frac{4\alpha\pi^{2}F^{(3\omega)}}{3\omega}}$
and $K^{(\omega)}=\frac{4\pi^{2}(\alpha F^{(\omega)})^{3/2}}{\omega^{3/2}}$,
respectively. These prefactors are derived from the $S$-matrix formalism, see
e.g., Achiezer and Berestecky (1969). The equation (29) for the ionization
rate contains all relativistic effects and all multipoles of the electron-
photon interaction. In practice, however, it is generally sufficient to
account only for the dominant electric dipole transitions. Therefore, in the
calculations of our results, the electric dipole approximation was applied.

##  III Results

###  III.1 Energy dependence of the total ionization rate

Figure 2:  Total biharmonic $\omega+3\omega$ ionization rate as a function of
incident photon energy. The rates are shown for ionizing the neon $2s_{1/2}$
(left) and $2p_{3/2}$ electrons (right) by linearly (top) and circularly
(bottom) polarized biharmonic light are shown. An intensity of the fundamental
light beam $I^{(\omega)}=1.0\times 10^{14}\text{W}/\text{cm}^{2}$ was used,
while intensity of third harmonic beam was chosen to be
$I^{(3\omega)}=1.1\times 10^{11}\text{W}/\text{cm}^{2}$ for ionization of the
$2s_{1/2}$ electrons and $I^{(3\omega)}=8\times 10^{11}\text{W}/\text{cm}^{2}$
for $2p_{3/2}$ electrons. The phase difference between the fundamental and
third harmonic light beam was set to zero in the above plots.

The total ionization rate of the biharmonic $\omega+3\omega$ ionization of
atoms can be divided into the one- and three-photon ionization rates and their
interference. The interference includes all combinations of the ionization
pathways of the two processes, the angular dependence of which is determined
by the spherical harmonics; see Eq. (27). Owing to the angular integration in
Eq. (29) and orthogonality of spherical harmonics, only partial waves with the
same orbital angular momentum interfere in the total ionization rate. For
example, the one-photon ionization of an $s$ electron leads to a $s\rightarrow
p$ ionization pathway which comprises both, the $p_{1/2}$ and $p_{3/2}$
partial waves, while three-photon ionization proceeds through the
$s\rightarrow p\rightarrow s\rightarrow p$, $s\rightarrow p\rightarrow
d\rightarrow p$, $s\rightarrow p\rightarrow d\rightarrow f$ ionization
pathways with all the associated fine structure levels, as shown in Fig. 3.
The one-photon ionization pathway ($s\rightarrow p$) in this example
interferes only with the first two three-photon ionization pathways
($s\rightarrow p\rightarrow s\rightarrow p$ and $s\rightarrow p\rightarrow
d\rightarrow p$), as all of them lead to a final partial wave $p$. This
interference in the total biharmonic $\omega+3\omega$ ionization rate is,
however independent of the phases of the photoelectron partial waves
$\delta_{\kappa}$. However, the dependence on the phase difference $\Phi$
between the two beam components remains and is proportional to $\cos\Phi$.

Figure 3: Schematic representation of the electric dipole ionization pathways
in biharmonic $\omega+3\omega$ ionization of atoms for a specific case of s
electron. Both the one- and three-photon ionization pathways leads to
$s\rightarrow p$ ionization pathway which proceeds through different paths. In
this example, the one-photon ionization pathway interferes with two three-
photon ionization pathways.

Figure 2 displays the total ionization rate as a function of incident photon
energy for biharmonic $\omega+3\omega$ ionization of neon $2s_{1/2}$ (left)
and $2p_{3/2}$ (right) electrons by linearly (top) and co-rotating circularly
(bottom) polarized beam. In this figure, the intensity of the the fundamental
frequency beam is $I(\omega)=10^{14}$W/cm2 and the intensity of the third
harmonic beam was chosen to be $I(3\omega)=1.1\times 10^{11}$W/cm2 for
ionization of the $2s_{1/2}$ electrons and $I(3\omega)=8\times 10^{11}$W/cm2
for ionization of the $2p_{3/2}$ electrons, such that the ionization rates of
both processes at threshold energies are comparable. In this work, we use
intensity $I(\omega)$ of fundamental and $I(3\omega)$ third-harmonic light
field of the order of $10^{14}$W/cm2 and $10^{11}$W/cm2, respectively, which
results in comparable ionization rates for the one- and three-photon
ionization processes. Although, it is currently challenging to reach such beam
configurations, experiments with similar parameters have been recently
successfully carried out Di Fraia et al. (2019) and advancements in FEL
science will likely deliver light beams considered here in near future. The
phase difference between the two beam components was set to zero, i.e.
$\Phi=0$, which results in the maximum absolute value for the interference
term. The ionization rate for one-photon ionization follows the typical
$(3\omega)^{-7/2}$ dependence on the incident photon energy, and hence the
dynamical dependence of the biharmonic rate on the incident photon energy
pictured in Fig. 2 is dominantly determined by the three-photon ionization
process.

In the top left plot, for the biharmonic $\omega+3\omega$ ionization of the
$2s_{1/2}$ electron by linearly polarized light, a local minimum in the rate
as well as a resonance behavior appear. This minimum arises from the
destructive interference of the one- and three-photon ionization paths and
vanishes if the interference is zero, i.e. if the phase difference between the
beam components is chosen to be $\Phi=\pi/2$. The resonance in the ionization
rate at $\omega=21.5$ eV arises from the $2\omega$ energy matching the energy
difference between the ground state of neon and the $1s^{2}2s2p^{6}3s$ excited
state. No such resonant enhancement of the rate is observed for the biharmonic
$\omega+3\omega$ ionization of the $2s_{1/2}$ electron by circularly polarized
beams. This behavior can be readily understood because two circularly
polarized photons transfer two units of angular momentum projection to the
atom. Since the projection of angular momentum of the excited state is zero,
the excitation of the neon in its ${}^{1}S_{0}$ ground state to the
$1s^{2}2s2p^{6}3s$ level is forbidden for circularly polarized light. A
resonantly increasing ionization rate can be also observed for the biharmonic
$\omega+3\omega$ ionization of the $2p_{3/2}$ electron of neon due to the
transition of the ground state neon into the $1s^{2}2s^{2}2p^{5}3p$ (first
resonance) and $1s^{2}2s^{2}2p^{5}4p$ (second resonance) excited states. For
these transitions, the individual intermediate fine-structure states with
different total angular momenta of the excited atom differ by about less than
$0.5$ eV in energy, which is not resolved in our calculations. The total rates
for biharmonic $\omega+3\omega$ ionization of the $2p_{1/2}$ electron of
neutral neon are similar to those on the right side of Fig. 2, but shifted by
$\approx 0.1$ eV, which arises from the difference in the binding energies of
the $2p_{1/2}$ and $2p_{3/2}$ electrons of neon.

###  III.2 Circular versus linear polarization in $\omega+3\omega$ ionization

Figure 4: The ratio of the ionization rates for linear polarization vs.
circular polarization is plotted against the photon energy. The above plot
describes the ionization $2s_{1/2}$ (top) electrons and $2p_{3/2}$ (bottom)
for three different phase differences between the first and third harmonic
light beams. In the above plots, an intensity of fundamental light beam
$I^{(\omega)}=1.0\times 10^{14}\text{W}/\text{cm}^{2}$ was applied, while the
intensity of third harmonic beam was fixed to be $I^{(3\omega)}=1.1\times
10^{11}\text{W}/\text{cm}^{2}$ for the ionization of the $2s_{1/2}$ electrons
and to $I^{(3\omega)}=8\times 10^{11}\text{W}/\text{cm}^{2}$ for $2p_{3/2}$
electrons.

In contrast to the total one-photon ionization cross-section, early
experiments showed that the multi-photon ionization cross-section depend on
the polarization of the ionizing light beam Fox et al. (1971); Kogan et al.
(1971). For low-order ionization processes, the cross-section ratio for
ionization by circularly and linearly polarized light is well described by the
simple estimate
$\textrm{max}(\sigma_{\textrm{lin}}/\sigma_{\textrm{circ}})=N!/(2N-1)!!$ for
$N$-photon ionization by Klarsfeld and Maquet Klarsfeld and Maquet (1972).
This estimate refers to the maximum possible value while the actual ratio
might deviate significantly for specific incident photon energies, for
example, if the photon energy matches a resonance atomic transition or
nonlinear Cooper minimum Hofbrucker et al. (2019, 2020).

For the biharmonic $\omega+3\omega$ ionization of atoms, the question of
whether ionization by linearly or circularly polarized light dominates the
other is more complex than for ionization by monochromatic light, since there
are different possible combinations of polarization states that the
fundamental and third harmonic can take. In order to facilitate the
discussion, we here chose both the fundamental and the third harmonic to have
the same polarization, i.e. either they are both linearly or both circularly
polarized. The corresponding biharmonic ionization rates are, therefore,
represented by $W(0,0)$ and $W(\pm 1,\pm 1)$, respectively. Furthermore, there
is a significant difference between the biharmonic ionization of atoms by
linearly and circularly polarized light. The total ionization rate for
biharmonic $\omega+3\omega$ ionization by linearly polarized light comprises
the one-, three-photon ionization rates as well as the interference between
the corresponding processes. In contrast, the one- and three-photon ionization
by circularly polarized light does not lead to any partial wave with the same
orbital angular momentum, and hence do not interferes in the total ionization
rate; see the theory section for more details.

To evaluate the relative total biharmonic $\omega+3\omega$ ionization rate for
ionization by linearly and circularly polarized beams, the ratio
$W(0,0)/W(1,1)$ was calculated for three different values of the phase
difference between the beam components $\Phi=0,\pi/2,\pi$ and as a function of
incident photon energies. Calculations were performed for the biharmonic
$\omega+3\omega$ ionization of the $2s_{1/2}$ and $2p_{3/2}$ electrons of
neutral neon and are shown in Fig. 4. The results were obtained by using the
intensity of the fundamental light beam of $I(\omega)=1.0\times
10^{14}\text{W}/\text{cm}^{2}$ and the third harmonic $I(3\omega)=1.1\times
10^{11}\text{W}/\text{cm}^{2}$ for the ionization of $2s_{1/2}$ electrons and
$I(3\omega)=8\times 10^{11}\text{W}/\text{cm}^{2}$ for the ionization of
$2p_{3/2}$ electrons. This choice of beam intensities then results in
comparable total ionization rates between the one- and three-photon ionization
rates for all off-resonant incident beam energies.

From Fig. 4, we can see the behavior of the ratio $W(0,0)/W(1,1)$ as a
function of the incident beam energy, which is especially pronounced for the
biharmonic ionization of the neon $2s_{1/2}$ electron (upper plot). Let us
therefore start with the description of the upper plot. Apart from the
enhancement due to the $1s^{2}2s2p^{6}3s$ two-photon resonance at around
$\omega=21.5$ eV, the ratio $W(0,0)/W(1,1)$ demonstrates a very strong
dependence of the rates on the phase $\Phi$ near the ionization threshold. By
choosing the phase $\Phi=0$ (short-dashed blue), the constructive interference
of the one and three-photon ionization is maximized. Therefore, biharmonic
ionization by linearly polarized light dominates over ionization by circularly
polarized light. For $\Phi=\pi/2$ (long-dashed orange), the interference is
zero and $W(0,0)/W(1,1)$ represents purely the ratio of the one- and three-
photon ionization rates. For $\Phi=\pi$ (solid red), the interference becomes
destructive and hence, the rate corresponding to ionization by linearly
polarized light becomes significantly smaller, while rate for ionization by
circularly polarized light remains unaffected.

The sign change of the transition amplitudes near a resonance is often
difficult to observe in other processes. However, the sign change of the
three-photon ionization amplitudes near to the mentioned two-photon resonance
can be read off from the $W(0,0)/W(1,1)$ ratio in the upper plot of Fig. 4.
For $\Phi=0$, the interference between the one- and three-photon ionization
processes becomes negative, which results in a trough in the $W(0,0)/W(1,1)$
ratio. On the contrary, for $\Phi=\pi$ the resonance enhancement of the ratio
appears to be shifted to lower photon energies as a result of the constructive
interference, increasing the rates of ionization by linearly polarized light.
Similar analysis can be carried out for the ionization of the $p_{3/2}$
electrons, although the described effects become less pronounced.

Figure 5:  Ionization of neon $2s_{1/2}$ electrons by co- (solid blue) and
counterrotating (dashed-dot orange) biharmonic $\omega+3\omega$ beams. The
photon energy was chosen near to the threshold at $\omega=26.7$ eV, the
polarization of the third harmonic is chosen to be right-circular
$\gamma^{(3\omega)}=1$, while the ellipticity of the fundamental frequency
$\gamma^{(\omega)}$ is varied. The intensity of the fundamental beam
$I(\omega)=10^{14}$ W/cm2 and of the third harmonic $I(3\omega)=1.1\times
10^{11}$ W/cm2 which gives rise to the same one- and three-photon ionization
rates. Rates for ionization by biharmonic beams with $\phi=0,\pi/2,\pi$ phase
difference (top three plots, respectively) are shown as functions of the
ellipticity of the fundamental beam $\gamma^{(\omega)}$. The dichroism
parameter corresponding to these three considerations is shown at the bottom.

###  III.3 Corotating versus counterrotating beams in $\omega+3\omega$
ionization

The dominance of ionization by two-color co- or counterrotating beams often
refers to circularly polarized light. Since the total ionization rates by
circular biharmonic $\omega+3\omega$ beams do not contain any interference
terms (as explained before), we will investigate this issue for ionization by
right-circularly polarized third harmonic, i.e. $\gamma^{(3\omega)}=1$ and as
a function of the ellipticity of the fundamental frequency
$\gamma^{(\omega)}$. To enumerate the dominance of co- or counter-rotating
beams in biharmonic ionization, we define the dichroism parameter
$\Delta(\gamma^{(\omega)})$ as

$\Delta(\gamma^{(\omega)})=\frac{W(\gamma^{(\omega)},\gamma^{(3\omega)}=1)-W(-\gamma^{(\omega)},\gamma^{(3\omega)}=1)}{W(\gamma^{(\omega)},\gamma^{(3\omega)}=1)+W(-\gamma^{(\omega)},\gamma^{(3\omega)}=1)}.$
(30)

In Fig. 5, we only show the ionization rates for the biharmonic ionization of
$2s_{1/2}$ electrons of neon.

The rates for ionization of neon $2s_{1/2}$ electrons by co- and
counterrotatig biharmonic $\omega+3\omega$ beams with different phase
difference between the beam components are shown in Fig. 5 as a function of
the ellipticity of the fundamental beam together with the corresponding
dichroism parameter. This figure shows that the phase difference between the
beam components is the key parameter which determines the dominance of one
polarization setting over another. For $\Phi=0$ (first plot), the interference
between one- and three-photon ionization is constructive, leading to the
dominance of the ionization by co-rotating beams. For the phase shift
$\Phi=\pi/2$ (second plot), no interference occurs between the two processes,
and therefore the total biharmonic rates are independent of the sign of the
ellipticity of the fundamental frequency component. For $\Phi=\pi$ (third
plot), the interference between the process is destructive which leads to the
dominance of the ionization by counterrotating biharmonic beams. The fourth
plot shows the dichroism parameters for the three scenarios discussed. The
zero dichroism parameter for $\Phi=\pi/2$ reflects the equal rates of
ionization by co-rotating and counter-rotating biharmonic beams. Fig. 5 also
reveals that the dichroism arising from destructive interference for
$\Phi=\pi$ reaches higher absolute values than the dichroism for $\Phi=0$.
This can be understood from the comparison of the ionization rates. While the
magnitude of the interference is the same for both $\Phi=0$ and $\Phi=\pi$,
the latter case results in lower ionization rates which then leads to higher
dichroism values. For ionization of $2p_{3/2}$ electrons, the dependence of
the dichroism parameter on the ellipticity of the fundamental frequency beam
and the phase difference between the beam components $\Phi$ is the same for
the ionization of both $2s_{1/2}$ and differs only in magnitude.

##  IV Summary

The polarization effects in biharmonic $\omega+3\omega$ ionization of atoms
were studied within the third-order perturbation theory. In particular, the
total rates for ionization by linearly and circularly polarized beams were
compared, and the dominance of ionization by co- and counterrotating
elliptically polarized beams was analyzed. In an example of biharmonic
ionization of the neon $L$ shell, we showed that the dominance of a particular
beam polarization over another is strongly influenced by interference in the
biharmonic ionization process and can be controlled by an appropriate choice
of beam parameters. This interference can be controlled most efficiently by
varying the phase difference between the biharmonic beam components, which has
the strongest effect and has a simple dependence on $\cos(\Phi)$.

## Acknowledgments

Disclaimer: The results presented in this paper are based on work performed
before February $24^{th}$ 2022.
A.V.V. acknowledges financial support by the Government of the Russian
Federation through the ITMO Fellowship and Professorship Program and by the
Ministry of Science and Education of the Russian Federation (Project No.
075-15-2021-1349).

## Appendix

Using the multipole (28) as well as the electron partial wave expansion (27)
and carrying out the angular integration over the spatial direction
$\bm{\hat{r}}$ of the electron wave functions, the transition amplitudes (II)
and (26) can be written in the following form

$\displaystyle M_{M_{i}M_{f}m_{e}}^{(3\omega)}$
$\displaystyle=4\pi\sum_{jm_{j}}\sum_{lm_{l}}(-i)^{l}e^{i\delta_{\kappa}}\left\langle
lm_{l},1/2m_{e}\left|\right.jm_{j}\right\rangle
Y_{lm_{l}}(\bm{\hat{p}}_{e})\sum_{JMp}i^{J-p}[\bm{\hat{\varepsilon}}^{(3\omega)}\cdot\bm{Y}_{JM}^{(p)}]\sum_{m_{a}}\left\langle
jm_{j},JM\left|\right.j_{a}m_{a}\right\rangle[j]^{-1/2}$ (31)
$\displaystyle\times(-1)^{j_{a}-m_{a}}\left\langle
j_{a}-m_{a},J_{i}M_{i}\left|\right.J_{f}M_{f}\right\rangle\left\langle
j\left\|T_{J}\right\|j_{a}\right\rangle U_{\kappa}(pJ)$

and

$\displaystyle M_{M_{i}M_{f}m_{e}}^{(\omega)}$ $\displaystyle=$ $\displaystyle
16\pi^{2}\sum_{jm_{j}}\sum_{lm_{l}}(-i)^{l}e^{i\delta_{\kappa}}\left\langle
lm_{l},1/2m_{e}\left|\right.jm_{j}\right\rangle
Y_{lm_{l}}(\bm{\hat{p}}_{e})\sum_{J_{1}M_{1}p_{1}}\sum_{J_{2}M_{2}p_{2}}\sum_{J_{3}M_{3}p_{3}}i^{J_{1}-p_{1}+J_{2}-p_{2}+J_{3}-p_{3}}[\bm{\hat{\varepsilon}}^{(\omega)}\cdot\bm{Y}_{J_{1}M_{1}}^{(p_{1})}]$
(32) $\displaystyle\times$
$\displaystyle[\bm{\hat{\varepsilon}}^{(\omega)}\cdot\bm{Y}_{J_{2}M_{2}}^{(p_{2})}][\bm{\hat{\varepsilon}}^{(\omega)}\cdot\bm{Y}_{J_{3}M_{3}}^{(p_{3})}]\sum_{j_{n_{1}}l_{n_{1}}m_{n_{1}}}\sum_{j_{n_{2}}l_{n_{2}}m_{n_{2}}}[j_{n_{1}},j_{n_{2}},j]^{-1/2}\left\langle
jm_{j},J_{3}M_{3}\left|\right.j_{n_{2}}m_{n_{2}}\right\rangle$
$\displaystyle\times$ $\displaystyle\left\langle
j_{n_{2}}m_{n_{2}},J_{2}M_{2}\left|\right.j_{n_{1}}m_{n_{1}}\right\rangle\sum_{m_{a}}\left\langle
j_{n_{1}}m_{n_{1}},J_{1}M_{1}\left|\right.j_{a}m_{a}\right\rangle\left\langle
j_{a}-m_{a},J_{i}M_{i}\left|\right.J_{f}M_{f}\right\rangle$
$\displaystyle\times$ $\displaystyle(-1)^{j_{a}-m_{a}}\left\langle
j\left\|T_{J_{3}}\right\|j_{n_{2}}\right\rangle\left\langle
j_{n_{2}}\left\|T_{J_{2}}\right\|j_{n_{1}}\right\rangle\left\langle
j_{n_{1}}\left\|T_{J_{1}}\right\|j_{a}\right\rangle
U^{(\kappa_{n_{2}},\kappa_{n_{1}})}_{\kappa}(p_{1}J_{1},p_{2}J_{2},p_{3}J_{3}),$

in terms of radial transition amplitudes for one-photon

$U_{\kappa}(pJ)=R_{\kappa\kappa_{a}}(pJ)$ (33)

and three-photon ionization

$U^{(\kappa_{n_{2}},\kappa_{n_{1}})}_{\kappa}(p_{1}J_{1},p_{2}J_{2},p_{3}J_{3})=\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{n_{1}}\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{n_{2}}\frac{R_{\kappa\kappa_{n_{2}}}(p_{3}J_{3})R_{\kappa_{n_{2}}\kappa_{n_{1}}}(p_{2}J_{2})R_{\kappa_{n_{1}}\kappa_{a}}(p_{1}J_{1})}{(\epsilon_{n_{a}\kappa_{a}}+2\omega-\epsilon_{n_{n_{2}}\kappa_{n_{2}}})(\epsilon_{n_{a}\kappa_{a}}+\omega-\epsilon_{n_{n_{1}}\kappa_{n_{1}}})}.$
(34)

The transition amplitudes $U_{\kappa}(pJ)$ and
$U^{(\kappa_{n_{2}},\kappa_{n_{1}})}_{\kappa}(p_{1}J_{1},p_{2}J_{2},p_{3}J_{3})$
of course depend on the principal quantum numbers of each involved electronic
state, however, this dependence was left out from the notation for practical
purposes. The angular integration of the space coordinate is given by

$\left\langle
j_{f}\left\|T_{J}\right\|j_{i}\right\rangle=(-1)^{j_{i}+j_{f}-J+1}[j_{i}]^{1/2}\left\langle
j_{i}1/2,J0\left|\right.j_{f}1/2\right\rangle\Pi_{l_{i},l,J},$ (35)

where $\Pi_{l_{i},l,J}=1$ if $l_{i}+l+J$ is even and $\Pi_{l_{i},l,J}=0$
otherwise. In the transverse (velocity) gauge, the radial integrals are
explicitly given for the magnetic ($p=0$, or $pJ=MJ$) transitions

$\displaystyle R_{\kappa_{f}\kappa_{i}}(MJ)$ $\displaystyle=$ $\displaystyle
i\sqrt{\frac{[J](J+1)}{4J\pi}}\int_{0}^{\infty}dr\frac{\kappa_{i}+\kappa_{f}}{J+1}j_{J}(kr)$
(36) $\displaystyle\times$
$\displaystyle\big{[}P_{i}(r)Q_{f}(r)+Q_{i}(r)P_{f}(r)\big{]},$

where $j_{J}(x)$ are the spherical Bessel functions, the radial wave functions
$P(r)$ and $Q(r)$ are the large and small components of the radial Dirac wave
functions for the orbital with principal and Dirac quantum numbers $n_{i}$ and
$\kappa_{i}$, respectively. These components are obtained from single-electron
Dirac equation, with a screening potential in the Hamiltonian, which partially
accounts for the interelectronic interaction. We compared a number of
different potential models. The core-Hartree potential, which reproduces the
binding energies in good agreement with the experimental values, was used to
produce the results presented in this work. For the electric transitions
($p=1$, or $pJ=EJ$)

$\displaystyle R_{\kappa_{f}\kappa_{i}}(EJ)$ $\displaystyle=$ $\displaystyle
i\sqrt{\frac{[J](J+1)}{4J\pi}}\int_{0}^{\infty}dr\Bigg{\\{}-\frac{\kappa_{i}-\kappa_{f}}{J+1}\big{[}j_{J}^{\prime}(kr)$
$\displaystyle+$
$\displaystyle\frac{j_{J}(kr)}{kr}\big{]}\big{[}P_{i}(r)Q_{f}(r)+Q_{i}(r)P_{f}(r)\big{]}$
$\displaystyle+$
$\displaystyle{J\frac{j_{J}(kr)}{kr}\big{[}P_{i}(r)Q_{f}(r)-Q_{i}(r)P_{f}(r)\big{]}\Bigg{\\}}}.$

In the length gauge, this integral is given by

$\displaystyle R_{\kappa_{f}\kappa_{i}}(EJ)$ $\displaystyle=$ $\displaystyle
i\sqrt{\frac{[J](J+1)}{4J\pi}}\int_{0}^{\infty}dr~{}j_{J}(kr)\big{[}P_{i}(r)P_{f}(r)$
(38) $\displaystyle+$ $\displaystyle
Q_{i}(r)Q_{f}(r)\big{]}+j_{J+1}(kr)\Big{\\{}\frac{\kappa_{i}-\kappa_{f}}{J+1}$
$\displaystyle\times$
$\displaystyle\big{[}P_{i}(r)Q_{f}(r)Q_{i}(r)P_{f}(r)\big{]}+\big{[}P_{i}(r)Q_{f}(r)$
$\displaystyle-$ $\displaystyle Q_{i}(r)P_{f}(r)\big{]}\Big{\\}}.$

The presented results were calculated in the velocity gauge, however, the
calculations were performed in both velocity and length gauges to check the
consistency and accuracy of our calculations.

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11institutetext:  Institute of Software Engineering, University of Stuttgart,
Universitätsstraße 38, 70569 Stuttgart, Germany

# Towards Immediate Feedback for Security Relevant Code in Development
Environments††thanks: This work is funded by the BMBF project CRITICALMATE
(16KIS0995).

Markus Haug  0000-0001-9377-0677 Ana Cristina Franco da Silva
0000-0001-8549-350X Stefan Wagner  0000-0002-5256-8429

###### Abstract

Nowadays, the correct use of cryptography libraries is essential to ensure the
necessary information security in different kinds of applications. A common
practice in software development is the use of static application security
testing (SAST) tools to analyze code regarding security vulnerabilities. Most
of these tools are designed to run separately from development environments.
Their results are extensive lists of security notifications, which software
developers have to inspect manually in a time-consuming follow-up step. To
support developers in their tasks of developing secure code, we present an
approach for providing them with continuous immediate feedback of SAST tools
in integrated development environments (IDEs). Our approach also considers the
understandability of security notifications and aims for a user-centered
approach that leverages developers’ feedback to build an adaptive system
tailored to each individual developer.

###### Keywords:

Software development IDE Security SAST Notifications

##  1 Introduction

An essential practice for ensuring IT security in software applications
nowadays is the use of cryptography libraries, such as the Bouncy Castle
Crypto APIs111https://bouncycastle.org or the Java security APIs. For example,
in the communication among distributed services in cloud environments, the
security of information in transit is a highly important requirement [4]. Yet,
using cryptography libraries is not easy because of their complexity. They are
often used incorrectly, for example, due to misuse of parameters, which can
lead to severe security issues [7].

Already during implementation, static application security testing (SAST)
tools, such as SonarQube222https://www.sonarqube.org or CogniCrypt [5], can be
used to analyze the code in development regarding security vulnerabilities
[8]. For this, SAST tools contain comprehensive rule sets, which define, for
example, correct or incorrect API usage patterns, and are used to recognize
issues and decide when to show security notifications.

Many current SAST tools are designed to be used separately from integrated
development environments (IDEs). That is, security analyses are usually
performed as additional steps in build processes, for example, in nightly
builds. This normally leads to extensive lists of security notifications,
which software developers have to inspect manually in a follow-up step. In
addition to being time-consuming and cumbersome, the interruption and
associated loss of context increase the challenge of understanding and fixing
an issue for developers.

In this paper, we present an approach that supports software developers in
their tasks of developing secure code. Using our approach, developers are
continuously provided with immediate feedback from SAST tools in their IDEs.
Our approach considers the understandability of security notifications and
also leverages developer feedback to build an adaptive system tailored to each
individual developer. This approach has been developed as a prototype within
the BMBF funded project _Cybersecurity static analysis with immediate
feedback_ (CRITICALMATE333https://www.forschung-it-sicherheit-
kommunikationssysteme.de/projekte/criticalmate).

The remainder of this paper is structured as follows: in Section 2, we present
our approach. Section 3 discusses related work. Finally, Section 4 concludes
our paper and gives an outlook on future work.

##  2 Approach

Our approach consists of two major parts regarding notifications of security
tools: (i) their efficient integration into developer’s IDEs to provide
immediate feedback, and (ii) the understandability of security notifications.

###  2.1 Adaptive Immediate Feedback

To efficiently integrate notifications of SAST tools into development
environments, we propose an adaptive IDE plugin, which continuously receives
analysis results of a SAST tool and shows them in the IDE’s code editor as
security notifications. This approach follows the client-server pattern, where
the server is any SAST tool conducting the security analyses, and the client
is the IDE plugin receiving the results of the analyses.

An advantage of immediate feedback in IDEs is that the typical workflow
interruption while running static code analysis separately is avoided [9]. A
further advantage of such a tool support is that security issues in the code
are recognized as early as possible. This allows developers to fix issues
while they still have most of the context available.

In CRITICALMATE, we have integrated a novel static code analysis engine into
IntelliJ, Eclipse, and Visual Studio Code through plugins. Software developers
get immediate feedback about security issues, directly in the code editor of
their IDEs. Any information security issues in the code are underlined in red.
An on-demand pop-up provides additional details about each issue. Furthermore,
in-code annotations are supported to suppress notifications as desired. This
is useful to suppress notifications that developers classify as false
positives.

The analysis engine and IDE plugins have been implemented as prototypes and
successfully provide immediate feedback. In the following, we discuss
potential improvements to increase their usefulness for developers.

One opportunity for improvement is the false positive rate. In SAST tools, the
increased analysis performance required for immediate feedback frequently
comes at the cost of an elevated false positive rate [1, 2]. Generally, such
tools have two options to handle a high false positive rate: (i) they
implement more accurate analysis strategies, usually at a performance cost, or
(ii) they tighten their confidence threshold. As a consequence, they issue
fewer notifications, which however can increase their false negative rate. Due
to the severe consequences of undetected vulnerabilities, we want to ensure a
low false negative rate. That is, tightening the confidence threshold is not a
suitable option.

Furthermore, a high false positive rate reduces user trust in such tools,
which might cause adverse effects. For example, developers might disable
analyses across their whole projects instead of single notifications because
of frustrating high false positive rate. In doing so, they might also
inadvertently disable notifications indicating actual security issues in the
code.

To handle false positives, we consider an adaptive analysis approach with
incremental accuracy. For each class of issues, the analysis could choose a
suitable strategy depending on factors, such as time budget, user-defined
confidence rating, or recorded false positive rate for a certain combination
of issue and strategy. For example, for a class of notifications known to have
a low false positive rate, the analysis could choose a less detailed strategy,
which would save time for other analyses. In contrast, for a class with a low
confidence level, the analysis could choose a more accurate iterative
strategy, until it achieves a desired confidence level, as long as the time
budget permits this.

To be tailored to software developers, the proposed IDE plugin should be able
to learn from software developers’ feedback, especially about how they assess
notifications as false positives in the context of their development projects.
Based on this, the plugin should be able to adapt the classes and amount of
shown notifications. For example, the plugin could learn to suppress
notifications, which developers frequently classify as false positives.

Finally, the plugin could also learn the developer’s typical workflow, such as
if they usually fix the code as soon as notifications are shown or just before
committing or sharing their code. Based on this, the plugin could increase the
amount of shown notifications when a developer is most likely to react to
them. At other times, the shown amount would be reduced, e.g., by disabling
highlights in the editor and listing the notifications only in the IDE’s error
list. This would allow developers to perform their tasks in the way that is
most suitable for them. Reducing the amount of information when developers are
unlikely to need it, could also mitigate information overload.

###  2.2 Understandable Security Notifications

Several studies in the literature have shown that many usability issues exist
in current SAST tools with respect to understandability of security
notifications. Tahaei et al. [10] conducted an experiment aiming to understand
how helpful static analysis tools notifications
(SonarQube444https://www.sonarqube.org and
SpotBugs555https://spotbugs.github.io) are to developers. They emphasize as a
finding that developers make mistakes even if these were well-known mistakes
and had known solutions, which suggests a lack of developer awareness and/or
missing support in addressing security issues.

Smith et al. [9] conducted a usability evaluation of four SAST tools. Within
the list of found usability
issues666https://figshare.com/s/71d97832ae3b04e0ff1a, there are issues related
to the content of the notification, such as “Fix suggestions not adequately
explained / sometimes missing” or “Verbose XML output distracting”.

Figure 1: Security notification regarding misuse of Bouncy Castle Crypto API
for elliptic-curve (EC) encryption

Figure 1 shows an example code snippet employing the Bouncy Castle Crypto API
for asymmetric ElGamal encryption with an elliptic curve (EC). The encryptor
needs to be initialized by calling the init() method with suitable parameters,
i.e., the public key of the recipient, before encrypting the data. However,
the developer forgot to call the init() method in this snippet.

In Figure 1, we can also see the IDE notification that the CRITICALMATE
analysis engine generates for this example. While the notification correctly
identifies the problem, it is difficult to understand, especially for
developers who are not well-versed in the field of cryptography. Furthermore,
there is no example of how to fix the issue, which has been identified as a
problem for understandability above. Consequently, the solution for the issue
might not be clear enough because of missing information.

Figure 2: Notification format using as example an API misuse

Therefore, to support software developers in avoiding misinterpretation of
security notifications, we propose an approach for presenting security
notifications within IDEs that follows usability guidelines to increase the
understandability of the security notifications.

Figure 2 shows our proposed notification format based on the format used in
the SonarSource rule repository777https://rules.sonarsource.com. In this
format, information about the error should be provided in a concise language,
as well as code examples of how to fix the error should be provided. The
depicted security notification refers to a misuse of the Bouncy Castle Crypto
API for asymmetric ElGamal encryption.

The aforementioned notification format has been integrated into the
prototypical implementation of CRITICALMATE’s analysis engine and IDE plugins
(cf. Section 2.1). In the following, we discuss potential improvements.

The participants of the study conducted by Tahaei et al. [10] imply that the
most helpful part were the provided code examples in the notifications. On the
other side, they indicated that metadata was not relevant for them. However,
code examples can lead to the problem of mismatched examples. One solution to
tackle this problem would be to create parameterized code example templates
that can be dynamically completed to match the specific scanned code snippet
and the context of the software project.

This approach could also be taken further by automatically suggesting quick
fixes for a given notification. Developers can then choose to apply these
fixes directly in their IDE. In some cases, where a concrete fix might not be
available, a snippet with placeholders for the developer to fill in could be
suggested instead. This would help developers in fixing security issues in
their software more quickly and more accurately. Some existing static analysis
tools, such as Clippy888https://github.com/rust-lang/rust-clippy or rust-
analyzer999https://rust-analyzer.github.io/ from the Rust programming language
ecosystem, already support such features for some problems.

Finally, the feedback of software developers to a notification, for example,
the reason a notification was not helpful, can also be used to learn about the
helpfulness of notifications and adapt future notifications’ content.

##  3 Related Work

This section presents related work to SAST tools, which provide an IDE
integration and consequently immediate feedback.

Eclipse
CogniCrypt101010https://www.eclipse.org/cognicrypt/documentation/codeanalysis
is an open-source platform for static code analysis based on CrySL rules [6].
Such rules can describe different error types indicating incorrect usage of
cryptography libraries. For example, a _constraint error_ indicates that the
static analysis detected an incorrect argument in a specific method call.

Find Security Bugs (FSB)111111https://find-sec-bugs.github.io is an open-
source extension of the SpotBugs static analysis tool for security audits of
web application in Java. It can detect security issues, such as those
described in the open web application security project (OWASP) Top 10.
Furthermore, it provides plugins for its integration in several IDEs, such as
for Eclipse and IntelliJ.

The goals of our approach are very similar to both aforementioned works in
respect to supporting software developers with immediate feedback within their
IDEs. However, in addition, we aim for a user-centered approach that leverages
developers feedback to build an adaptive system.

##  4 Results and Conclusion

In this paper, we introduced an approach for supporting software developers to
address possible security issues directly in their IDEs. Through continuous
immediate feedback integrated in IDEs, developers can see notifications
regarding security issues, such as misuse of cryptography libraries, as soon
as code is typed in the code editor and the typed code has been analyzed by a
SAST tool. Within the project _Cybersecurity static analysis with immediate
feedback_ (CRITICALMATE), the University of Stuttgart and RIGS IT have jointly
developed a prototype that integrates the analysis results of RIGS IT’s
security tool continuously into different IDEs, namely IntelliJ, Eclipse IDE,
and Visual Studio Code. Preliminary measurements tell us that security
notifications are shown to developers in average under a second. However,
there are several factors that influence the response time from triggering the
static code analysis until the visualization of the found security
notifications. To get more insights about these issues, we plan to conduct
further experiments.

Furthermore, we plan to conduct a user study to get insights about the
interaction of software developers with the immediate feedback and the
adaptivity feature. One possibility is to conduct a study that also conducts
emotion recognition based on physiological signals or video input. We can gain
valuable feedback by recognizing confusion or frustration [3] of software
developers while they are working with selected security notifications for the
study.

One big challenge in the aforementioned approach is how to differentiate
notifications that are actual false positives and what developers might label
as false positives because they do not agree on the issue or perceive them as
noise. Furthermore, if there are many notifications regarding the same code
snippet, these should be prioritized to be visualized sequentially based on,
for example, their severity, to avoid cluttered visualization or missing
important notifications.

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