B_{m}^{\dagger}(\mathrm{in})=\frac{\mathcal{M}_{m}^{1/2}\left(\phi_{m}^{\left(\mathrm{in},-\right)},\hat{\Phi}\right)}{\left(\phi_{m}^{\left(\mathrm{in},-\right)},\phi_{m}^{\left(\mathrm{in},-\right)}\right)},$
$\displaystyle
A_{m}(\mathrm{out})=\frac{\mathcal{M}_{m}^{1/2}\left(\phi_{m}^{\left(\mathrm{out},+\right)},\hat{\Phi}\right)}{\left(\phi_{m}^{\left(\mathrm{out},+\right)},\phi_{m}^{\left(\mathrm{out},+\right)}\right)},$
$\displaystyle
B_{m}^{\dagger}(\mathrm{out})=\frac{\mathcal{M}_{m}^{1/2}\left(\phi_{m}^{\left(\mathrm{out},-\right)},\hat{\Phi}\right)}{\left(\phi_{m}^{\left(\mathrm{out},-\right)},\phi_{m}^{\left(\mathrm{out},-\right)}\right)},$

where $M_{m}$ are normalization factors given by Eq. (30). These operators
define annihilation and creation operators of initial or final particles in
each the range $\Omega_{k}$ ($m_{k}\in\Omega_{k}$) as follows:

$\displaystyle A_{m_{1}}(\mathrm{in})$ $\displaystyle=$ $\
{}_{+}a_{m_{1}}(\mathrm{in}),\;B_{m_{1}}^{\dagger}(\mathrm{in})=\
^{-}a_{m_{1}}(\mathrm{in});$ $\displaystyle A_{m_{1}}(\mathrm{out})$
$\displaystyle=$ $\ {}^{+}a_{m_{1}}(\mathrm{out})\
,\;B_{m_{1}}^{\dagger}(\mathrm{out})=\ _{-}a_{m_{1}}(\mathrm{out})\ ;$
$\displaystyle A_{m_{2}}(\mathrm{in})$ $\displaystyle=$ $\displaystyle
A_{m_{2}}(\mathrm{out})=a_{m_{2}},\;B_{m_{2}}^{\dagger}(\mathrm{in})=B_{m_{2}}^{\dagger}(\mathrm{out})=0\,;$
$\displaystyle A_{m_{3}}(\mathrm{in})$ $\displaystyle=$ $\
{}_{-}a_{m_{3}}(\mathrm{in}),\;B_{m_{3}}^{\dagger}(\mathrm{in})=\
^{-}b_{m_{3}}^{\dagger}(\mathrm{in});$ $\displaystyle A_{m_{3}}(\mathrm{out})$
$\displaystyle=$ $\
{}_{+}a_{m_{3}}(\mathrm{out}),\;B_{m_{3}}^{\dagger}(\mathrm{out})=\
^{+}b_{m_{3}}^{\dagger}(\mathrm{out});$ $\displaystyle A_{m_{4}}(\mathrm{in})$
$\displaystyle=$ $\displaystyle
A_{m_{4}}(\mathrm{out})=0,\;B_{m_{4}}^{\dagger}(\mathrm{in})=B_{m_{4}}^{\dagger}(\mathrm{out})=b_{m}^{\dagger}\;;$
$\displaystyle A_{m_{5}}(\mathrm{in})$ $\displaystyle=$
$\;{}^{+}b_{m_{5}}^{{\dagger}}(\mathrm{in}),\;B_{m_{5}}^{\dagger}(\mathrm{in})=\
_{-}b_{m_{5}}^{{\dagger}}(\mathrm{in});$ $\displaystyle
A_{m_{5}}(\mathrm{out})$ $\displaystyle=$
${}_{+}b_{m_{5}}^{{\dagger}}(\mathrm{out}),\;B_{m_{5}}^{\dagger}(\mathrm{out})=\
^{-}b_{m_{5}}^{{\dagger}}(\mathrm{out}),$ (136)

Here $a$ and $b\ $are annihilation and $a^{{\dagger}}$ and $b^{{\dagger}}$ are
creation operators, the operators $a$ and $a^{{\dagger}}$ describe magnons and
the operators $b$ and $b^{{\dagger}}$ describe antimagnons. The vacuum vectors
$\left|0,\mathrm{in}\right\rangle$ and $\left|0,\mathrm{out}\right\rangle$ are
null vectors for all the annihilation operators $a$ and $b$. Operators labeled
by the argument ”in” are interpreted as initial particle operators, whereas
operators labeled by the argument ”out” are interpreted as final particle
operators. Indeed, the commutation relations (128) yield the standard
commutation rules for the introduced creation and annihilation in- or out-
operators. One-particle states of initial (final) magnon and antimagnon are

$a_{m}^{\dagger}(\mathrm{in/out})\left|0,\mathrm{in/out}\right\rangle,\;b_{m}^{\dagger}(\mathrm{in/out})\left|0,\mathrm{in/out}\right\rangle,$

where $a_{m}^{\dagger}(\mathrm{in/out})$ and
$b_{m}^{\dagger}(\mathrm{in/out})$ are given by Eq. (136) for each the range
$\Omega_{k}$. The unitary transformation (15) implies a canonical
transformations between the in and out-operators.

Interpretation of the magnons and the antimagnon states which follows from in
Eqs. (136) is consistent with a spectrum analysis of the kinetic energy
operator $\widehat{\mathbb{H}}^{\mathrm{kin}}$ and the effective charge
operator $\hat{Q}$. Inserting decompositions (135) in Eqs. (129) and (130), we
obtain diagonal representations for these operators in terms of the introduced
creation and annihilation operators. The operators
$\widehat{\mathbb{H}}^{\mathrm{kin}}$ and $\hat{Q}$ can be represented as sums
of five contributions, each one in the range $\Omega_{k}$,

${\large\
}\widehat{\mathbb{H}}^{\mathrm{kin}}=\sum_{k=1}^{5}\sum_{m\in\Omega_{k}}\widehat{\mathbb{H}}_{m},\;\hat{Q}=\sum_{k=1}^{5}\sum_{m\in\Omega_{k}}\hat{Q}_{m}.$
(137)

Note that a stationary state

$\Psi_{m}\left(X\right)=\left(\begin{array}[]{c}\left[\varepsilon_{m}-U\left(x\right)\right]\phi_{m}\left(X\right)\\\
\phi_{m}\left(X\right)\end{array}\right),$

where $\phi_{m}$ is one of the solutions from Eq. (31), satisfies the
following eigenvalue problem:

$H^{\mathrm{kin}}\Psi_{m}\left(X\right)=\left[\varepsilon_{m}-U\left(x\right)\right]\Psi_{m}\left(X\right).$
(138)

This implies that the kinetic energy term of the one-particle state reads:

$\displaystyle E_{m}$ $\displaystyle=$ $\displaystyle\left(\ \phi_{m},\
\phi_{m}\right)^{-1}\int_{V_{\bot}}d\mathbf{r}_{\bot}$ (139)
$\displaystyle\times$
$\displaystyle\int\limits_{-K^{\left(\mathrm{L}\right)}}^{K^{\left(\mathrm{R}\right)}}\;\Psi_{m}^{\dagger}\left(X\right)\sigma_{1}\left[\varepsilon_{m}-U\left(x\right)\right]\;\Psi_{m}\left(X\right)dx\
,$

where the quantities $\left(\ \phi_{m},\ \phi_{m}\right)$ are positive for
$m\in\Omega_{1}\cup\Omega_{2}$ and are negative for
$m\in\Omega_{4}\cup\Omega_{5}$. In the range $\Omega_{3}$ we have that
$\left(\ {}_{\zeta}\phi_{m},\ _{\zeta}\phi_{m}\right)>0,$ while $\left(\
{}^{\zeta}\phi_{m},\ ^{\zeta}\phi_{m}\right)<0$. Note that the kinetic energy
terms $\ {}_{\zeta}E_{m}$ corresponding to the states $\ {}_{\zeta}\phi_{m}$
and the terms $\ {}^{\zeta}E_{m}$ corresponding to the states $\
{}^{\zeta}\phi_{m}$ are different in the general case. The principal value of
integral (139) is determined by integrals over the areas
$x\in\left[-K^{\left(\mathrm{L}\right)},x_{\mathrm{L}}\right]$ and
$x\in\left[x_{\mathrm{R}},K^{\left(\mathrm{R}\right)}\right]$, where the field
derivative $\partial_{x}U$ is negligible small. Thus, it is possible to
evaluate integrals (139) for any form of the external field, using only the
asymptotic behavior (8) of functions in the regions $S_{\mathrm{L}}$and
$S_{\mathrm{R}}$ where particles are free; see Sects. IV and Appendix B in
Ref. [14] for details. Note that
$\varepsilon_{m}-U\left(x\right)=\pi_{0}\left(\mathrm{L/R}\right)$ in the
regions $S_{\mathrm{L}}/S_{\mathrm{R}}$, respectively.

It can be easily seen from Eq. (137) that in the range
$\Omega_{1}\cup\Omega_{2}$ a one-particle state is the state of a magnon with
the kinetic energy $E_{m}>0$ and the magnetic moment $\mu$ whereas in the
range $\Omega_{4}\cup\Omega_{5}$ a one-particle state is the state of an
antimagnon with the kinetic energy $-E_{m}>0$ and the magnetic moment $-\mu$.

Inserting decompositions (135) in operators (133) and (134), we obtain a
renormalized (with respect to the corresponding vacua) in- and out-operators
of the effective charge (magnetic moment) current and energy flux flowing
through the surfaces $x=x_{\mathrm{L}}$ and $x=x_{\mathrm{R}}$, respectively,

$\displaystyle\widehat{\mathbb{J}}\left(\mathrm{in}\right)=\widehat{\mathbb{J}}-\left\langle
0,\mathrm{in}\left|\widehat{\mathbb{J}}\right|0,\mathrm{in}\right\rangle\,,$
$\displaystyle\widehat{\mathbb{J}}\left(\mathrm{out}\right)=\widehat{\mathbb{J}}-\left\langle
0,\mathrm{out}\left|\widehat{\mathbb{J}}\right|0,\mathrm{out}\right\rangle\ ,$
$\displaystyle\widehat{\mathbb{F}}\left(x|\mathrm{in}\right)=\widehat{\mathbb{F}}\left(x\right)-\left\langle
0,\mathrm{in}\left|\widehat{\mathbb{F}}\left(x\right)\right|0,\mathrm{in}\right\rangle\,,$
$\displaystyle\widehat{\mathbb{F}}\left(x|\mathrm{out}\right)=\widehat{\mathbb{F}}\left(x\right)-\left\langle
0,\mathrm{out}\left|\widehat{\mathbb{F}}\left(x\right)\right|0,\mathrm{out}\right\rangle\
.$ (140)

The one-particle mean values of the fluxes, the kinetic energy, and the
effective charge through the surfaces $x=x_{\mathrm{L}}$ and
$x=x_{\mathrm{R}}$, given by Eqs. (140), are proportional to the inner product
on these surfaces given by Eq. (13), that is, these fluxes are proportional to
the flux densities of the particles with given $m$. Of course, in the range
$\Omega_{2}\cup\Omega_{4}$ the flux densities of particles, given by standing
waves, are zero.

With account taken of the charge conjugation, it can be seen that the sign of
the flux densities of magnons with given $m$ is equal to $\zeta$ in the range
$\Omega_{1}$, whereas the sign of the flux densities of antimagnons with given
$m$ is equal to $-\zeta$ in the range $\Omega_{5}$. In the ranges $\Omega_{1}$
and $\Omega_{5}$ an initial state may be localized both in the regions
$S_{\mathrm{L}}$ and $S_{\mathrm{R}}$. This follows from the way of choosing
initial conditions. Taking into account directions of motion of magnons and
antimagnons in the regions $S_{\mathrm{L}}$and $S_{\mathrm{R}}$, we define
initial and final states as it is done in Eqs. (136) and (31).

In the Klein zone $\Omega_{3}$ the identification of states demands a special
consideration. To this end, we represent explicitly the operators
$\widehat{\mathbb{H}}_{m}$ and $\hat{Q}_{m}$ as follows:

$\displaystyle\widehat{\mathbb{H}}_{m}$ $\displaystyle=$ $\ {}_{+}E_{m}\ \
{}_{+}a_{m}^{\dagger}(\mathrm{out})\ _{+}a_{m}(\mathrm{out})$ $\displaystyle-$
$\ {}^{+}E_{m}\ \ ^{+}b_{m}^{\dagger}(\mathrm{out})\ ^{+}b_{m}(\mathrm{out})$
$\displaystyle=$ $\ {}_{-}E_{m}\ \ {}_{-}a_{m}^{\dagger}(\mathrm{in})\
_{-}a_{m}(\mathrm{in})$ $\displaystyle-$ $\ {}^{-}E_{m}\ \ \
{}_{-}b_{m}^{\dagger}(\mathrm{in})\ _{-}b_{m}(\mathrm{in});$
$\displaystyle\hat{Q}_{m}$ $\displaystyle=$ $\displaystyle\mu\left[\ \
{}_{+}a_{m}^{\dagger}(\mathrm{out})\ _{+}a_{m}(\mathrm{out})\right.$ (141)
$\displaystyle-$ $\displaystyle\ \left.\
{}^{+}b_{m_{3}}^{\dagger}(\mathrm{out})\ ^{+}b_{m_{3}}(\mathrm{out})\right]$
$\displaystyle=$ $\displaystyle\mu\ \left[\
{}_{-}a_{m_{3}}^{\dagger}(\mathrm{in})\ _{-}a_{m_{3}}(\mathrm{in})\right.$
$\displaystyle-$ $\displaystyle\left.\ {}_{-}b_{m_{3}}^{\dagger}(\mathrm{in})\
_{-}b_{m_{3}}(\mathrm{in})\right]\,,\ \ m\in\Omega_{3}.$

Here $\ {}_{\zeta}E_{m}$ and $\ {}^{\zeta}E_{m}$ are principal values of
integral (139) for $\ {}_{\zeta}\Psi_{m}$ and ${}^{\zeta}\Psi_{m}$,
respectively. They read:

$\ {}_{\zeta}E_{m}\mathcal{=}\pi_{0}\left(\mathrm{R}\right)+\frac{\delta
U}{2}\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2},$ $\;\
{}^{\zeta}E_{m}=\pi_{0}\left(\mathrm{L}\right)-\frac{\delta
U}{2}\left|g\left({}_{+}\left|{}^{-}\right.\right)\right|^{-2}.$ (142)

Thus, we see that in the range $\Omega_{3}$ the one-particle state $\
{}_{\zeta}\Psi_{m}$ is the magnon state with the kinetic energy
${}_{\zeta}E_{m}>0$ and the magnetic moment $\mu$ whereas the one-particle
state $\ {}^{\zeta}\Psi_{m}$ is the antimagnon state with the kinetic energy
$-\ ^{\zeta}E_{m}>0$ and the magnetic moment $-\mu$.

Let us find one-particle mean values of fluxes of the kinetic energy and the
effective charge through the surfaces $x=x_{\mathrm{L}}$ and
$x=x_{\mathrm{R}}$, given by Eqs. (140), in the range $\Omega_{3}$. With the
help of Eq. (13) we obtain:

$\displaystyle J_{m}^{a}\left(\mathrm{in}\right)$ $\displaystyle=$
$\displaystyle\left\langle
0,\mathrm{in}\left|\;{}_{-}a_{m}(\mathrm{in})\widehat{\mathbb{J}}\left(\mathrm{in}\right)\;_{-}a_{m}^{\dagger}(\mathrm{in})\right|0,\mathrm{in}\right\rangle=-\mu\left(\mathcal{M}_{m}T\right)^{-1},$
$\displaystyle J_{m}^{a}\left(\mathrm{out}\right)$ $\displaystyle=$
$\displaystyle\left\langle 0,\mathrm{out}\left|\
{}_{+}a_{m}(\mathrm{out})\widehat{\mathbb{J}}\left(\mathrm{out}\right)\
_{+}a_{m}^{\dagger}(\mathrm{out})\
\right|0,\mathrm{out}\right\rangle=\mu\left(\mathcal{M}_{m}T\right)^{-1},$
$\displaystyle J_{m}^{b}\left(\mathrm{in}\right)$ $\displaystyle=$
$\displaystyle\left\langle
0,\mathrm{in}\left|\;{}^{-}b_{m}(\mathrm{in})\widehat{\mathbb{J}}\left(\mathrm{in}\right)\;^{-}b_{m}^{\dagger}(\mathrm{in})\right|0,\mathrm{in}\right\rangle=-\mu\left(\mathcal{M}_{m}T\right)^{-1},$
$\displaystyle J_{m}^{b}\left(\mathrm{out}\right)$ $\displaystyle=$
$\displaystyle\left\langle 0,\mathrm{out}\left|\
{}^{+}b_{m}^{\dagger}(\mathrm{out})\widehat{\mathbb{J}}\left(\mathrm{out}\right)\
^{+}b_{m}^{\dagger}(\mathrm{out})\right|0,\mathrm{out}\right\rangle=\mu\left(\mathcal{M}_{m}T\right)^{-1};$
$\displaystyle F_{m}^{a}\left(\mathrm{in}\right)$ $\displaystyle=$
$\displaystyle\left\langle
0,\mathrm{in}\left|\;{}_{-}a_{m}(\mathrm{in})\mathbb{\hat{F}}\left(x_{\mathrm{R}},\mathrm{out}\right)\;_{-}a_{m}^{\dagger}(\mathrm{in})\right|0,\mathrm{in}\right\rangle=-\left(\mathcal{M}_{m}T\right)^{-1}\pi_{0}\left(\mathrm{R}\right),$
$\displaystyle F_{m}^{a}\left(\mathrm{out}\right)$ $\displaystyle=$
$\displaystyle\left\langle 0,\mathrm{out}\left|\
{}_{+}a_{m}(\mathrm{out})\mathbb{\hat{F}}\left(x_{\mathrm{R}},\mathrm{out}\right)\
_{+}a_{m}(\mathrm{out})\right|0,\mathrm{out}\right\rangle=\left(\mathcal{M}_{m}T\right)^{-1}\pi_{0}\left(\mathrm{R}\right),$
$\displaystyle F_{m}^{b}\left(\mathrm{in}\right)$ $\displaystyle=$
$\displaystyle\left\langle
0,\mathrm{in}\left|\;{}^{-}b_{m}(\mathrm{in})\mathbb{\hat{F}}\left(x_{\mathrm{L}},\mathrm{in}\right)\;^{-}b_{m}^{\dagger}(\mathrm{in})\right|0,\mathrm{in}\right\rangle=\left(\mathcal{M}_{n}T\right)^{-1}\left|\pi_{0}\left(\mathrm{L}\right)\right|,$
$\displaystyle F_{m}^{b}\left(\mathrm{out}\right)$ $\displaystyle=$
$\displaystyle\left\langle 0,\mathrm{out}\left|\
{}^{+}b_{m}^{\dagger}(\mathrm{out})\mathbb{\hat{F}}\left(x_{\mathrm{L}},\mathrm{out}\right)\
^{+}b_{m}^{\dagger}(\mathrm{out})\right|0,\mathrm{out}\right\rangle=-\left(\mathcal{M}_{n}T\right)^{-1}\left|\pi_{0}\left(\mathrm{L}\right)\right|.$
(143)

Taking into account the space separation of magnons and antimagnons in the
region $S_{\mathrm{R}}$ and $S_{\mathrm{L}}$, one can use the mean values
(143) in order to distinguish initial and final states in the range
$\Omega_{3}$.

The QFT operators of the effective charge (magnetic moment) current density
$\hat{J}_{x}$ and energy flux density $v_{s}\hat{T}^{10}$ flowing through the
surfaces $x=x_{\mathrm{L}}$ and $x=x_{\mathrm{R}}$, that were introduced above
in the framework of the approximation under consideration, are time-
independent. It is clear that these operators are defined up to certain
$C$-numbers, which may affect explicit forms of the corresponding vacuum
means. This circumstance allows one to relate matrix elements of these
operators with matrix elements of the corresponding exact strong-field QED
operators (which are time-dependent in the presence of the time-dependent
field $E_{\mathrm{pristine}}$). This can be done based on the following
physical considerations: Let us consider a relation of the time-independent
quantity $J_{x}^{\mathrm{cr}}$, obtained in the framework of approximation
under consideration, to the matrix elements of the time-dependent longitudinal
component $\hat{J}_{x}^{\mathrm{true}}\left(t\right)$ of an exact current
density operator of the strong-field QED. According to the general theory the
difference $\delta J_{x}^{\mathrm{true}}$ of the true final vacuum from the
initial one is due to the contribution of the current density of the created
particles and antiparticles,

$\delta J_{x}^{\mathrm{true}}=\left\langle
0,\mathrm{true\;in}\left|\left[\hat{J}_{x}^{\mathrm{true}}\left(t_{2}\right)-\hat{J}_{x}^{\mathrm{true}}\left(t_{1}\right)\right]\right|0,\mathrm{true\;in}\right\rangle,$
(144)

where $t_{1}<t_{\mathrm{in}}$ and $t_{2}>t_{\mathrm{out}}$ are the time
instants of switching on and off of the field $E_{\mathrm{pristine}}$,
respectively. Assuming that effects of fast switching-on and -off are small,
we can use approximation (36) for the total number of created pairs. In this
case we obtain:

$\delta
J_{x}^{\mathrm{true}}=J_{x}^{\mathrm{cr}}\left\\{1+O\left(\left[\sqrt{v_{s}\left|\partial_{x}U\right|}T\right]^{-1}\right)\right\\}.$
(145)

We see that the quantity $J_{x}^{\mathrm{cr}}$ can be represented as the
following mean value with respect to the $\left|0,\mathrm{in}\right\rangle$
vacuum:

$J_{x}^{\mathrm{cr}}\approx\left\langle
0,\mathrm{in}\left|\left[\hat{J}_{x}^{\mathrm{true}}\left(t_{2}\right)-\hat{J}_{x}^{\mathrm{true}}\left(t_{1}\right)\right]\right|0,\mathrm{in}\right\rangle\
.$ (146)

With the help of this result we may find a relation of the operators
$\hat{J}_{x}^{\mathrm{true}}\left(t_{2}\right)$ and
$\hat{J}_{x}^{\mathrm{true}}\left(t_{1}\right)$ with the time-independent
current density operator $\hat{J}_{x}$ given by Eq. (133). In particular, we
see that the difference
$\hat{J}_{x}^{\mathrm{true}}\left(t_{2}\right)-\hat{J}_{x}^{\mathrm{true}}\left(t_{1}\right)$
can be approximated by a $C$-number,

$\hat{J}_{x}^{\mathrm{true}}\left(t_{2}\right)-\hat{J}_{x}^{\mathrm{true}}\left(t_{1}\right)\approx
J_{x}^{\mathrm{cr}}.$ (147)

Then, for example, the current density operator,
$\hat{J}_{x}^{\mathrm{true}}\left(t_{1}\right)$ can be approximated by the
time-independent current density operator $\hat{J}_{x}$,
$\hat{J}_{x}^{\mathrm{true}}\left(t_{1}\right)\approx\hat{J}_{x}$ . In this
case, we have
$\hat{J}_{x}^{\mathrm{true}}\left(t_{2}\right)\approx\hat{J}_{x}+J_{x}^{\mathrm{cr}}$,
therefore the normal form of both
$\hat{J}_{x}^{\mathrm{true}}\left(t_{1}\right)$ and
$\hat{J}_{x}^{\mathrm{true}}\left(t_{2}\right)$ with respect to the in- and
out-operators of creation and annihilation are the same. Thus, calculating
one-particle mean values of the effective charge current in Eq. (143) one can
use the operator $\hat{J}_{x}$. In a similar way, one can relate the time-
independent operator $\hat{T}^{10}\left(x\right)$ to time-dependent components
of the exact operator EMT, for example,
$\hat{T}_{\mathrm{true}}^{10}\left(t_{1},x\right)\approx\hat{T}^{10}\left(x\right)$
and
$\hat{T}_{\mathrm{true}}^{10}\left(t_{2},x\right)\approx\hat{T}^{10}\left(x\right)+T_{\mathrm{cr}}^{10}(x)$.
In addition, calculating one-particle mean values of fluxes of the kinetic
energy in Eq. (143) one can use the operator $\hat{T}^{10}\left(x\right)$.

##  Appendix B Examples of exact solutions with x-steps

In this appendix, we provide additional information on the computation of
differential quantities that are relevant to the investigation of magnon pair
production stimulated by the external fields mentioned in Section IV. We
present results only and refer the reader to Refs. [14, 31, 33, 35] for more
comprehensive discussions.

For the $L$-constant magnetic step (67), solutions of Eq. (6) to all intervals
can be expressed in the form:

$\ {}^{-}\varphi_{m}\left(x\right)$ $\displaystyle=$ $\displaystyle
Y\left\\{\begin{array}[]{l}\
_{+}C\exp\left[i\left|p^{\mathrm{L}}\right|\left(x-x_{\mathrm{L}}\right)\right]g\left({}_{+}|^{-}\right)-\
_{-}C\exp\left[-i\left|p^{\mathrm{L}}\right|\left(x-x_{\mathrm{L}}\right)\right]g\left({}_{-}|^{-}\right)\,,\
\ x\in S_{\mathrm{L}}\,,\\\ \
^{-}C\left\\{b_{1}D_{\nu}\left[-\left(1-i\right)\xi\right]+b_{2}D_{-\nu-1}\left[-\left(1+i\right)\xi\right]\right\\}\,,\
\ x\in S_{\mathrm{int}}\,,\\\ \
^{-}C\exp\left[-i\left|p^{\mathrm{R}}\right|\left(x-x_{\mathrm{R}}\right)\right]\,,\
\ x\in S_{\mathrm{R}}\,.\end{array}\right.$ (151) $\
{}^{+}\varphi_{m}\left(x\right)$ $\displaystyle=$ $\displaystyle
Y\left\\{\begin{array}[]{l}\
_{+}Cg\left({}_{+}|^{+}\right)\exp\left[i\left|p^{\mathrm{L}}\right|\left(x-x_{\mathrm{L}}\right)\right]-\
_{-}Cg\left({}_{-}|^{+}\right)\exp\left[-i\left|p^{\mathrm{L}}\right|\left(x-x_{\mathrm{L}}\right)\right]\,,\
\ x\in S_{\mathrm{L}}\,,\\\ \
^{+}C\left\\{b_{1}^{\prime}D_{\nu}\left[\left(1-i\right)\xi\right]+b_{2}^{\prime}D_{-\nu-1}\left[\left(1+i\right)\xi\right]\right\\}\,,\
\ x\in S_{\mathrm{int}}\,,\\\ \
^{+}C\exp\left[i\left|p^{\mathrm{R}}\right|\left(x-x_{\mathrm{R}}\right)\right]\,,\
\ x\in S_{\mathrm{R}}\,,\end{array}\right.$ (155)

where $b_{j}$, $b_{j}^{\prime}$, $g\left({}^{\zeta}|_{\zeta^{\prime}}\right)$,
$g\left({}_{\zeta}|^{\zeta^{\prime}}\right)$ are constants, which can be
obtained via the continuity conditions:

$\ {}_{-}^{+}\varphi_{m}\left(x_{\mathrm{L/R}}-0\right)=\
_{-}^{+}\varphi_{m}\left(x_{\mathrm{L/R}}+0\right)\,,$
$\displaystyle\frac{d}{dx}\
_{-}^{+}\varphi_{m}\left(x_{\mathrm{L/R}}-0\right)=\frac{d}{dx}\
_{-}^{+}\varphi_{m}\left(x_{\mathrm{L/R}}+0\right)\,.$

By demanding continuity of the above functions and their derivatives at
$x=x_{\mathrm{R}}$ we find

$\displaystyle b_{j}$ $\displaystyle=$
$\displaystyle\left(-1\right)^{j+1}\exp\left[\frac{i\pi}{2}\left(\nu+\frac{1}{2}\right)\right]\sqrt{\frac{\xi_{2}^{2}-\lambda}{2}}f_{j}^{\left(-\right)}\left(\xi_{2}\right)\,,$
$\displaystyle b_{j}^{\prime}$ $\displaystyle=$
$\displaystyle\left(-1\right)^{j+1}\exp\left[\frac{i\pi}{2}\left(\nu+\frac{1}{2}\right)\right]\sqrt{\frac{\xi_{2}^{2}-\lambda}{2}}f_{j}^{\left(+\right)}\left(\xi_{2}\right)\,,$

where $\lambda=\pi_{\bot}^{2}/v_{s}\mu B^{\prime}$ and
$f_{j}^{\left(\pm\right)}\left(\xi\right)$ were defined before, see Eqs. (70).
Now, by imposing the continuity of the functions and derivatives at
$x=x_{\mathrm{L}}$ we obtain the coefficients $g\left({}_{+}|^{+}\right)$ and
$g\left({}_{+}|^{-}\right)$:

$\displaystyle g\left({}_{+}|^{-}\right)$ $\displaystyle=$ $\displaystyle
e^{\frac{i\pi}{2}\left(\nu+\frac{1}{2}\right)}\sqrt{\frac{\sqrt{\xi_{1}^{2}-\lambda}\sqrt{\xi_{2}^{2}-\lambda}}{8}}\left[f_{1}^{\left(-\right)}\left(\xi_{2}\right)f_{2}^{\left(-\right)}\left(\xi_{1}\right)-f_{2}^{\left(-\right)}\left(\xi_{2}\right)f_{1}^{\left(-\right)}\left(\xi_{1}\right)\right]\,,$
(156) $\displaystyle g\left({}_{+}|^{+}\right)$ $\displaystyle=$
$\displaystyle
e^{\frac{i\pi}{2}\left(\nu+\frac{1}{2}\right)}\sqrt{\frac{\sqrt{\xi_{1}^{2}-\lambda}\sqrt{\xi_{2}^{2}-\lambda}}{8}}\left[f_{1}^{\left(+\right)}\left(\xi_{2}\right)\tilde{f}_{2}^{\left(-\right)}\left(\xi_{1}\right)-f_{2}^{\left(+\right)}\left(\xi_{2}\right)\tilde{f}_{1}^{\left(-\right)}\left(\xi_{1}\right)\right]\,,$
(157)

where

$\displaystyle\tilde{f}_{1}^{\left(\pm\right)}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\left(1\pm\frac{i}{\sqrt{\xi^{2}-\lambda}}\frac{d}{d\xi}\right)D_{-\nu-1}\left[\mp\left(1+i\right)\xi\right]\,,$
$\displaystyle\tilde{f}_{2}^{\left(\pm\right)}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\left(1\pm\frac{i}{\sqrt{\xi^{2}-\lambda}}\frac{d}{d\xi}\right)D_{\nu}\left[\mp\left(1-i\right)\xi\right]\,.$

For the Sauter-like magnetic step (79), solutions of Eq. (6), (82) with real
asymptotic momenta (8) in remote regions $x\rightarrow\mp\infty$ read:

$\ {}_{\zeta}\varphi_{m}\left(x\right)$ $\displaystyle=$ $\
{}_{\zeta}\mathcal{N}\exp\left(i\zeta\left|p^{\mathrm{L}}\right|x\right)$
$\displaystyle\times$
$\displaystyle\left[1+\exp\left(2x/L_{\mathrm{S}}\right)\right]^{-iL_{\mathrm{S}}\left(\zeta\left|p^{\mathrm{L}}\right|+\left|p^{\mathrm{R}}\right|\right)/2}\
_{\zeta}u_{m}\left(x\right)\,,$ $\ {}^{\zeta}\varphi_{m}\left(x\right)$
$\displaystyle=$ $\
{}^{\zeta}\mathcal{N}\exp\left(i\zeta\left|p^{\mathrm{R}}\right|x\right)$
(158) $\displaystyle\times$
$\displaystyle\left[1+\exp\left(-2x/L_{\mathrm{S}}\right)\right]^{iL_{\mathrm{S}}\left(\left|p^{\mathrm{L}}\right|+\zeta\left|p^{\mathrm{R}}\right|\right)/2}\
{}^{\zeta}u_{m}\left(x\right)\,,$

where

$\ {}_{-}u_{m}\left(x\right)$ $\displaystyle=$ $\displaystyle
F\left(a,b;c;\chi\right)\,,$ ${}_{+}u_{m}\left(x\right)$ $\displaystyle=$
$\displaystyle F\left(a+1-c,b+1-c;2-c;\chi\right)\,,$ $\
{}^{-}u_{m}\left(x\right)$ $\displaystyle=$ $\displaystyle
F\left(a,b;a+b+1-c;1-\chi\right)\,,$ ${}^{+}u_{m}\left(x\right)$
$\displaystyle=$ $\displaystyle F\left(c-a,c-b;c+1-a-b;1-\chi\right)\,.$ (159)

$\chi\left(x\right)$ is the change of variable defined in Eq. (80) and $a$,
$b$, $c$ are given by Eqs. (83). Using the above solutions and Kummer’s
connection formulas [32]

$\displaystyle\left(1-\chi\right)^{c-a-b}\
{}^{+}u_{m}\left(x\right)=\frac{\Gamma\left(c+1-a-b\right)\Gamma\left(1-c\right)}{\Gamma\left(1-a\right)\Gamma\left(1-b\right)}\
_{-}u_{m}\left(x\right)$
$\displaystyle+\frac{\Gamma\left(c+1-a-b\right)\Gamma\left(c-1\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\chi^{1-c}\
_{+}u_{m}\left(x\right)\,,$ $\
{}^{-}u_{m}\left(x\right)=\frac{\Gamma\left(a+b+1-c\right)\Gamma\left(1-c\right)}{\Gamma\left(a+1-c\right)\Gamma\left(b+1-c\right)}\
_{-}u_{m}\left(x\right)$
$\displaystyle+\frac{\Gamma\left(a+b+1-c\right)\Gamma\left(c-1\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}\chi^{1-c}\
_{+}u_{m}\left(x\right)\,,$ (160)

we conclude that

$\displaystyle g\left({}_{+}|^{+}\right)$ $\displaystyle=$
$\displaystyle\frac{\ {}^{+}\mathcal{N}}{\
{}_{+}\mathcal{N}}\frac{\Gamma\left(c+1-a-b\right)\Gamma\left(c-1\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\,,$
$\displaystyle g\left({}_{+}|^{-}\right)$ $\displaystyle=$
$\displaystyle\frac{\ {}^{-}\mathcal{N}}{\
{}_{+}\mathcal{N}}\frac{\Gamma\left(a+b+1-c\right)\Gamma\left(c-1\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}\,.$
(161)

For the exponential step (94), general solutions of Eq. (6) with such a field
can be presented as a linear combination of two functions,
$y_{1}^{j}\left(\eta_{j}\right)$ and $y_{2}^{j}\left(\eta_{j}\right)$

$\displaystyle y_{1}^{j}\left(\eta_{j}\right)$ $\displaystyle=$ $\displaystyle
e^{-\eta_{j}/2}\eta_{j}^{\nu_{j}}\Phi\left(a_{j},c_{j};\eta_{j}\right)$
$\displaystyle=$ $\displaystyle
e^{\eta_{j}/2}\eta_{j}^{\nu_{j}}\Phi\left(c_{j}-a_{j},c_{j};-\eta_{j}\right)\,,$
$\displaystyle y_{2}^{j}\left(\eta_{j}\right)$ $\displaystyle=$ $\displaystyle
e^{\eta_{j}/2}\eta_{j}^{-\nu_{j}}\Phi\left(1-a_{j},2-c_{j};-\eta_{j}\right)$
(162) $\displaystyle=$ $\displaystyle
e^{-\eta_{j}/2}\eta_{j}^{-\nu_{j}}\Phi\left(a_{j}-c_{j}+1,2-c_{j};\eta_{j}\right)\,,$

where $\eta_{j}\left(x\right)$ are defined in Eqs. (95) and the parameters
$a_{j}$, $c_{j}$ by Eqs. (98). In particular, solutions with special
asymptotic properties in remote regions $x\rightarrow\mp\infty$ are classified
as

$\ {}_{+}\varphi_{m}\left(x\right)$ $\displaystyle=$ $\
{}_{+}\mathcal{N}e^{-i\pi\nu_{1}/2}y_{1}^{1}\left(\eta_{1}\right)\,,$ $\
{}_{-}\varphi_{m}\left(x\right)$ $\displaystyle=$ $\
{}_{-}\mathcal{N}e^{i\pi\nu_{1}/2}y_{2}^{1}\left(\eta_{1}\right)\,,$ $\
{}^{+}\varphi_{m}\left(x\right)$ $\displaystyle=$ $\
{}^{+}\mathcal{N}e^{i\pi\nu_{2}/2}y_{2}^{2}\left(\eta_{2}\right)\,,$ $\
{}^{-}\varphi_{m}\left(x\right)$ $\displaystyle=$ $\
{}^{-}\mathcal{N}e^{-i\pi\nu_{2}/2}y_{1}^{2}\left(\eta_{2}\right)\,,$ (163)

With the aid of this classification, we may express solutions to all intervals
in the form

$\ {}^{+}\varphi_{m}\left(x\right)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\
_{+}\varphi_{m}\left(x\right)g\left({}_{+}|^{+}\right)-\
_{-}\varphi_{m}\left(x\right)g\left({}_{-}|^{+}\right)\,,&x\in\mathrm{I}\\\ \
^{+}\mathcal{N}e^{i\pi\nu_{2}/2}y_{2}^{2}\left(\eta_{2}\right)\,,&x\in\mathrm{II}\end{array}\right.$
(166) $\ {}^{-}\varphi_{m}\left(x\right)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\
_{+}\varphi_{m}\left(x\right)g\left({}_{+}|^{-}\right)-\
_{-}\varphi_{m}\left(x\right)g\left({}_{-}|^{-}\right)\,,&x\in\mathrm{I}\\\ \
^{-}\mathcal{N}e^{-i\pi\nu_{2}/2}y_{1}^{2}\left(\eta_{2}\right)\,,&x\in\mathrm{II}\end{array}\right.$
(169)

where $\mathrm{I}=-\infty<x\leq 0$, and $\mathrm{II}=0<x<+\infty$. To
calculate the decomposition coefficients
$g\left({}_{\zeta}|^{\zeta^{\prime}}\right)$,
$g\left({}^{\zeta}|_{\zeta^{\prime}}\right)$, we impose continuity of the
functions and their derivatives at $x=0$:

$\ {}^{\zeta}\varphi_{m}\left(x-0\right)$ $\displaystyle=$ $\
{}^{\zeta}\varphi_{m}\left(x+0\right)\,,$ $\displaystyle\frac{d}{dx}\
^{\zeta}\varphi_{m}\left(x-0\right)$ $\displaystyle=$
$\displaystyle\frac{d}{dx}\ ^{\zeta}\varphi_{m}\left(x+0\right)\,,$ (170)

In particular, the coefficients $g\left({}_{+}|^{-}\right)$ and
$g\left({}_{+}|^{+}\right)$ have the form:

$\displaystyle g\left({}_{+}|^{+}\right)$ $\displaystyle=$
$\displaystyle-\frac{\exp\left[\frac{i\pi}{2}\left(\nu_{1}+\nu_{2}\right)\right]}{2\sqrt{\left|p^{\mathrm{L}}\right|\left|p^{\mathrm{R}}\right|}}\left.\left(k_{1}h_{1}y_{2}^{2}\frac{d}{d\eta_{1}}y_{2}^{1}+k_{2}h_{2}y_{2}^{1}\frac{d}{d\eta_{2}}y_{2}^{2}\right)\right|_{x=0}\,,$
$\displaystyle g\left({}_{+}|^{-}\right)$ $\displaystyle=$
$\displaystyle-\frac{\exp\left[\frac{i\pi}{2}\left(\nu_{1}-\nu_{2}\right)\right]}{2\sqrt{\left|p^{\mathrm{L}}\right|\left|p^{\mathrm{R}}\right|}}\left.\left(k_{1}h_{1}y_{1}^{2}\frac{d}{d\eta_{1}}y_{2}^{1}+k_{2}h_{2}y_{2}^{1}\frac{d}{d\eta_{2}}y_{1}^{2}\right)\right|_{x=0}\,.$
(171)

Lastly, exact solutions to Eq. (6) with inverse-square magnetic steps (109)
can be represented as a linear combination of Whittaker functions
$W_{\kappa_{j},\mu_{j}}\left(z_{j}\right)$,
$W_{-\kappa_{j},\mu_{j}}\left(e^{-i\pi}z_{j}\right)$

$\displaystyle
w_{1}^{j}\left(z_{j}\right)=e^{-i\pi\kappa_{j}/2}W_{\kappa_{j},\mu_{j}}\left(z_{j}\right)\,,$
$\displaystyle
w_{2}^{j}\left(z_{j}\right)=e^{-i\pi\kappa_{j}/2}W_{-\kappa_{j},\mu_{j}}\left(e^{-i\pi}z_{j}\right)\,,$
(172)

where the variables $z_{j}\left(x\right)$ are given by Eqs. (110) and the
parameters $\kappa_{j}$, $\mu_{j}$ by Eqs. (112). Their Wronskian determinant
$\mathbb{W}$ reads [36]

$\mathbb{W}=w_{1}^{j}\left(z_{j}\right)\frac{d}{dz_{j}}w_{2}^{j}\left(z_{j}\right)-w_{2}^{j}\left(z_{j}\right)\frac{d}{dz_{j}}w_{1}^{j}\left(z_{j}\right)=1\,.$

Sometimes, it is convenient to represent the set of solutions (172) in terms
of confluent hypergeometric functions

$\displaystyle w_{1}^{j}\left(z_{j}\right)$ $\displaystyle=$
$\displaystyle\exp\left[-\frac{i\pi}{2}\left(\kappa_{j}-\mu_{j}-\frac{1}{2}\right)\right]$
$\displaystyle\times$ $\displaystyle
e^{-z_{j}/2}\left|z_{j}\right|^{c_{j}/2}\Psi\left(\tilde{a}_{j},\tilde{c}_{j};z_{j}\right)\,,$
$\displaystyle w_{2}^{j}\left(z_{j}\right)$ $\displaystyle=$
$\displaystyle\exp\left[-\frac{i\pi}{2}\left(\kappa_{j}+\mu_{j}+\frac{1}{2}\right)\right]$
$\displaystyle\times$ $\displaystyle
e^{z_{j}/2}\left|z_{j}\right|^{c_{j}/2}\Psi\left(\tilde{c}_{j}-\tilde{a}_{j},\tilde{c}_{j};e^{-i\pi}z_{j}\right)\,,$

in which $\tilde{a}_{j}=\mu_{j}-\kappa_{j}+1/2$, $\tilde{c}_{j}=1+2\mu_{j}$.

Based on asymptotic properties of the Whittaker functions with large argument
[36], solutions with real asymptotic momenta (8) in remote regions
$x\rightarrow\mp\infty$ are classified as follows:

$\ {}_{+}\varphi_{m}\left(x\right)$ $\displaystyle=$ $\
{}_{+}\mathcal{N}w_{1}^{1}\left(z_{1}\right)\,,\ \ \
_{-}\varphi_{m}\left(x\right)=\ _{-}\mathcal{N}w_{2}^{1}\left(z_{1}\right)\,,$
$\ {}^{+}\varphi_{m}\left(x\right)$ $\displaystyle=$ $\
{}^{+}\mathcal{N}w_{2}^{2}\left(z_{2}\right)\,,\ \ \
^{-}\varphi_{m}\left(x\right)=\ ^{-}\mathcal{N}w_{1}^{2}\left(z_{2}\right)\,.$
(173)

Thanks to the above classification, we may represent solutions valid at all
$x$ in two equivalent forms:

$\ {}^{+}\varphi_{m}\left(x\right)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\
_{+}\varphi_{m}\left(x\right)g\left({}_{+}|^{+}\right)-\
_{-}\varphi_{m}\left(x\right)g\left({}_{-}|^{+}\right)\,,&x\in\mathrm{I}\\\ \
^{+}\mathcal{N}w_{2}^{2}\left(z_{2}\right)\,,&x\in\mathrm{II}\end{array}\right.$
(176) $\ {}^{-}\varphi_{m}\left(x\right)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\
_{+}\varphi_{m}\left(x\right)g\left({}_{+}|^{-}\right)-\
_{-}\varphi_{m}\left(x\right)g\left({}_{-}|^{-}\right)\,,&x\in\mathrm{I}\\\ \
^{-}\mathcal{N}w_{1}^{2}\left(z_{2}\right)\,,&x\in\mathrm{II}\end{array}\right.$
(179)

Demanding continuity of the solutions and their derivatives at $x=0$ (170) we
discover that $g\left({}_{+}|^{+}\right)$ and $g\left({}_{+}|^{-}\right)$
admit the forms:

$\displaystyle g\left({}_{+}|^{+}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\left|p^{\mathrm{L}}\right|\left|p^{\mathrm{R}}\right|}}\left.\left[\left|p^{\mathrm{L}}\right|w_{2}^{2}\left(z_{2}\right)\frac{d}{dz_{1}}w_{2}^{1}\left(z_{1}\right)+\left|p^{\mathrm{R}}\right|w_{2}^{1}\left(z_{1}\right)\frac{d}{dz_{2}}w_{2}^{2}\left(z_{2}\right)\right]\right|_{x=0}\,,$
$\displaystyle g\left({}_{+}|^{-}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\left|p^{\mathrm{L}}\right|\left|p^{\mathrm{R}}\right|}}\left.\left[\left|p^{\mathrm{L}}\right|w_{1}^{2}\left(z_{2}\right)\frac{d}{dz_{1}}w_{2}^{1}\left(z_{1}\right)+w_{2}^{1}\left(z_{1}\right)\left|p^{\mathrm{R}}\right|\frac{d}{dz_{2}}w_{1}^{2}\left(z_{2}\right)\right]\right|_{x=0}\,.$
(180)

## References

  * [1] H. Y. Yuan, Y. Cao, A. Kamra, R. A. Duine, P. Yan, _Quantum magnonics: when magnon spintronics meets quantum information science_ , Physics Reports 965, 1-74 (2022) [arXiv:2111.14241]
  * [2] M. Hongo, T. Fujimori, T. Misumi, M. Nitta, and N. Sakai, _Effective field theory of magnons: Chiral magnets and the Schwinger mechanism_ , Phys. Rev. B 104, 134403 (2021).
  * [3] F. Kämpfer, M. Moser, and U.-J. Wiese, _Systematic Low-Energy Effective Theory for Magnons and Charge Carriers in an Antiferromagnet_ , Nucl. Phys. B 729, 317-360 (2005).
  * [4] J. Schwinger, _On Gauge invariance and Vacuum and Vacuum polarization_ , Phys. Rev. 82, 664 (1951).
  * [5] J. S. Harms, H. Y. Yuan, R. A. Duine, _Enhanced magnon spin current using the bosonic Klein paradox_ , Phys. Rev. Applied 18, 064026 (2022) [arXiv:2109.00865]. DOI:https://doi.org/10.1103/PhysRevApplied.18.064026
  * [6] J. S. Harms, H. Y. Yuan, R. A. Duine, _Antimagnonics_ , arXiv:2210.16698.
  * [7] C. Zhou, L. Wang, Dao-Xin Yao, _Klein tunneling of Weyl magnons_ , arXiv:1909.04109.
  * [8] Shaohua Yuan, Chaowei Sui, Zhengduo Fan, Jamal Berakdar, Desheng Xue, and Chenglong Jia, _Magnonic Klein and acausal tunneling enabled by breaking the anti parity-time symmetry in antiferromagnets_ , Commun Phys 6, 95 (2023). https://doi.org/10.1038/s42005-023-01222-z
  * [9] O. Klein, _Die Reflexion von Elektronen einem Potentialsprung nach der relativistischen Dynamik von Dirac_ , Z. Phys. 53, 157 (1929); Elelrtrodynamik und Wellenmechanik vom Standpunkt des Korrespondenzprinzips, Z. Phys. 41, 407 (1927).
  * [10] F. Sauter, _Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs_ ,__ Z. Phys. 69, 742 (1931).
  * [11] F. Sauter, _Zum ”Klenschen Paradoxon”,_ Z. Phys. 73, 547 ( 1932).
  * [12] N. Dombey and A. Calogeracos,_Seventy years of the Klein paradox,_ Phys. Rep. 315, 41 (1999);_History and Physics of the Klein Paradox_ , Contemp. Phys. 40, 313 (1999) [arXiv:quant-ph/9905076].
  * [13] A. Hansen and F. Ravndal, _Klein’s paradox and its resolution_ , Phys. Scr. 23, 1036 (1981).
  * [14] S.P. Gavrilov and D.M. Gitman, _Quantization of charged fields in the presence of critical potential steps_ , Phys. Rev. D. 93, 045002 (2016).
  * [15] S.P. Gavrilov, D.M. Gitman, _Regularization, renormalization and consistency conditions in QED with_ $x$_-electric potential steps_ , Eur. Phys. J. C 80, 820 (2020).
  * [16] A.I. Breev, S.P. Gavrilov, and D.M. Gitman, _Calculations of vacuum mean values of spinor field current and energy-momentum tensor in a constant electric background_ , Eur. Phys. J. C 83, 108 (2023).
  * [17] S. Bayrakci, T. Keller, K. Habicht, and B. Keimer, _Spin-Wave Lifetimes Throughout the Brillouin Zone_ , Science 312, 1926 (2006).
  * [18] S. P. Gavrilov and D. M. Gitman, _Consistency restrictions on maximal electric field strength in QFT_ , Phys. Rev. Lett. 101, 130403 (2008).
  * [19] E. S. Fradkin, D. M. Gitman, S. M. Shvartsman, _Quantum Electrodynamics with Unstable Vacuum_ , Springer-Verlag, Berlin, 1991.
  * [20] S.P. Gavrilov, D.M. Gitman, and J.L. Tomazelli, _Density matrix of a quantum field in a particle-creating background_ , Nucl. Phys. B 795, 645 (2008).
  * [21] S.P. Gavrilov, D.M. Gitman, and A.A. Shishmarev, _Pair production from the vacuum by a weakly inhomogeneous space-dependent electric potential_ , Phys. Rev. D. 99, 116014 (2019) [arXiv:1903.05925].
  * [22] H. Gies and F. Karbstein, _An addendum to the Heisenberg-Euler effective action beyond one loop_ , JHEP 1703, 108 (2017) [arXiv:1612.07251].
  * [23] F. Karbstein, _Heisenberg-Euler effective action in slowly varying electric field inhomogeneities of Lorentzian shape_ , Phys. Rev. D 95, 076015 (2017).
  * [24] G.V. Dunne, _Heisenberg-Euler effective Lagrangians: Basics and extensions_ , in I. Kogan Memorial Volume, _From fields to strings: Circumnavigating theoretical physics_ , Eds. M Shifman, A. Vainshtein and J. Wheater, World Scientific, 2005 [arXiv:hep-th/0406216].
  * [25] A. Fedotov, A. Ilderton, F. Karbstein, B. King, D. Seipt, H. Taya, G. Torgrimsson, _Advances in QED with intense background fields_ , Phys. Rep. 1010, 1-138 (2023) [arXiv:2203.00019].
  * [26] S. P. Gavrilov, D. M. Gitman, and N. Yokomizo, _Dirac fermions in strong electric field and quantum transport in graphene_ , Phys. Rev. D 86, 125022 (2012).
  * [27] O. Vafek, A. Vishwanath, _Dirac Fermions in Solids: From High-Tc Cuprates and Graphene to Topological Insulators and Weyl Semimetals_ , Annu. Rev. Condens. Matter Phys.,  5 (2014) 83-112. https://doi.org/10.1146/annurev-conmatphys-031113-133841
  * [28] F. Gelis, N. Tanji, _Schwinger mechanism revisited_ , Prog. Part. Nucl. Phys. 87, 1 (2016).
  * [29] S. Schweber, _An Introduction to Relativistic Quantum Field Theory_ (Harper & Row, New York 1961).
  * [30] D.M. Gitman and I.V. Tyutin, _Quantization of Fields with Constraints_ (Springer-Verlag, Berlin–Heidelberg 1990).
  * [31] S. P. Gavrilov and D. M. Gitman, _Scattering and pair creation by a constant electric field between two capacitor plates_ , Phys. Rev. D 93, 045033 (2016).
  * [32] _Higher Transcendental Functions_ , edited by A. Erdelyi et.al., (McGraw-Hill, New York, 1953), Vols. 1 and 2.
  * [33] S.P. Gavrilov, D.M. Gitman, and A.A. Shishmarev, _Particle scattering and vacuum instability by exponential steps_ , Phys. Rev. D. 96, 096020 (2017).
  * [34] A.I. Nikishov, _Barrier scattering in field theory: removal of Klein paradox,_ Nucl. Phys. B21, 346 (1970).
  * [35] T. C. Adorno, S. P. Gavrilov, and D. M. Gitman, _Vacuum instability in a constant inhomogeneous electric field: a new example of exact nonperturbative calculations_ , Eur. Phys. J. C 80, 88 (2020).
  * [36] F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark, _NIST Handbook of Mathematical Functions_ (Cambridge University Press, New York, 2010).


# Physical Layer Security Techniques
for Future Wireless Networks

Weiping Shi, Xinyi Jiang, Jinsong Hu, Yin Teng, Yang Wang,
Hangjia He, Rongen Dong, Feng Shu, Jiangzhou Wang

###### Abstract

The broadcast nature of wireless communication systems makes wireless
transmission extremely susceptible to eavesdropping and even malicious
interference. Physical layer security technology can effectively protect the
private information sent by the transmitter from being listened to by illegal
eavesdroppers, thus ensuring the privacy and security of communication between
the transmitter and legitimate users. The development of mobile communication
presents new challenges to physical layer security research. This paper
provides a comprehensive survey of the physical layer security research on
various promising mobile technologies, including directional modulation (DM),
spatial modulation (SM), covert communication, intelligent reflecting surface
(IRS)-aided communication, and so on. Finally, future trends and the
unresolved technical challenges are summarized in physical layer security for
mobile communications.

###### Index Terms:

Physical layer security, DM, SM, covert communication, IRS

##  I Introduction

Wireless mobile communications has been developing very fast [1, 2, 3, 4, 5].
The fifth generation (5G) mobile systems have been started to be deployed
worldwide. Due to the broadcast nature of wireless media, information security
has been a critical issue in wireless communication. The traditional method of
addressing communication security is to adopt the secret key encryption. The
secure transmission of private data is achieved by designing various
encryption algorithms in the upper protocol stack. However, cryptography is of
computational security and its security level depends on the hardness of the
underlying mathematical problem it employs. Once an effective method is found
to solve its mathematical problem, the security of the encryption method will
be seriously compromised [6].

Physical layer security technology has become an effective solution to the
wireless communication security problem. As shown in Figure .1, the
transmitter (i.e., Alice) sends a confidential message to the legitimate
receiver (i.e., Bob), while the eavesdropper (i.e., Eve) receives the signal
and intends to decode it. The key idea of physical layer security technology
is to exploit the inherent propagation characteristics of wireless channels
(such as the difference between the main channel and the eavesdropping
channel, randomness, and reciprocity) from the perspective of information
theory. By reasonably designing the transmit signal so that it improves the
amount of mutual information between the transmitter and the desired user at
the physical layer while reducing the amount of information in the
eavesdropping channel. Therefore, the physical layer security technology can
effectively protect the content of private messages sent by the transmitter
from being eavesdropped by illegal eavesdroppers [7]. This improves the
privacy and security of the transmitter’s communication with other legitimate
users. Compared with traditional encryption technology, physical layer
security technology has the following notable features and advantages: First,
the physical layer security technology can achieve keyless security, that is,
no encryption and decryption operations are required. Second, physical layer
security techniques can take advantage of the time-varying and random nature
of wireless channels. Signal processing techniques are employed to design
reasonable beamforming or power allocation strategies from the transmitter’s
perspective to improve the security performance of wireless communication
systems [8]. Based on the above advantages, solving the communication security
problem from the physical layer has aroused widespread concern. The authors in
[9] studied the secure transmission of private information on Gaussian
channels, and proved that expanding the difference between the main channel
and the eavesdropping channel can achieve low probability interception and low
probability detection for eavesdroppers. [10] further researched the keyless
physical layer secure transmission technology over fading channels. The use of
multiple antennas can add additional degrees of freedom and further improve
the security performance of the wireless network [11, 12]. In addition, by
generating random artificial noise (AN) at the transmitter to interfere with
eavesdroppers, the security of the system can be further improved [13].
Currently, scenarios where eavesdroppers exist are considered in various
wireless communication systems.

Considering the potential of physical layer security for mobile
communications, the opportunities and challenges of how to achieve high levels
of security at the physical layer deserve more attention from the research
community. The purpose of this paper is to provide a comprehensive summary of
the latest physical layer security research results for key future wireless
network technologies. As shown in Figure 2, we will focus on the following
four aspects of physical layer security technologies.

Figure 1: A three-node secure communication model.

  1. 1.

Covert communication: In covert communication, when the transmitter transmits
a message to the receiver, it is guaranteed that the probability that the
illegal watcher can detect the transmission is small enough. Covert
transmission technique, as an important secure transmission technique, aims to
hide the transmission behavior of the transmitter. Compared to physical layer
security techniques that aim to prevent the transmitted content from being
overheard by eavesdroppers, covert communication techniques can achieve a
higher level of communication security.

  2. 2.

Directional modulation (DM): The directional modulation technique using phased
arrays sends information along the direction of desired receiver, making the
constellation diagram of the received signal in the desired receiver direction
the same as that of the baseband modulated signal, while the constellation
diagram of the received signal in the undesired direction will be distorted,
thus ensuring the safe transmission of information.

  3. 3.

Spatial modulation (SM): Spatial modulation, a new multiple-input multiple-
output (MIMO) antenna transmission technique, plays an effective balance
between hardware overhead and transmission rate. Because it makes full use of
the channel index information to transmit more bit streams, it enhances the
spectral efficiency of the system without increasing the costly radio
frequency (RF) links. Since both antenna index and modulation symbols in SM
networks carry private information, interception of either of them may result
in leakage of confidential information. In other words, compared with MIMO
communication, SM network has serious communication security risks. Therefore,
it is strategically important to utilize physical layer security technology to
provide security for SM systems.

  4. 4.

Intelligent reflective surface (IRS)-aided communication: An IRS is an
artificial surface made of electromagnetic material and composed of a large
number of passive reflective units. By configuring these reflective units to
act on the phase shift and amplitude of the incident signal, fine-grained
three-dimensional beamforming can be achieved, which can be used to improve
channel quality, enhance received power, and extend the communication
distance. IRS transforms the traditional uncontrollable and random wireless
communication environment into a programmable and relatively deterministic
transmission space, and plays an active role in the signal transmission
process. By introducing IRS into the security system, the security of the
system can be further enhanced.

The rest of this paper can be outlined as follows. We first discuss covert
communication in Section II. Then, the cases of secure DM systems are
researched in Section III. In Section IV, we study SM technique on physical
layer security. Section V focuses on the security in IRS-aided wireless
communication systems. The future research directions and conclusions are
presented in Section VI and Section VII respectively.

Figure 2: An illustration of physical layer security.

##  II Covert communication

While most information-theoretic security works to date have revolved around
the issues of protecting the content of information, the growing concern
around mass communication surveillance programs has reignited interest in
investigating the covertness of communications, also known as low probability
of detection (LPD), in which covertness requires that the communication
remains undetectable from a watchful adversary, named warden. The concept of
covert communication has shown tremendous research value and application value
[14, 15]. In the covert communication scenarios, the transmitter (Alice)
possesses sensitive information that needs to be sent to the legitimate
receiver (Bob), while the warden (Willie) listens to the communication
environment and tries to detect any covert transmission from Alice to Bob with
some probable existed uncertainties. These uncertainties can be formed from
Willie’s receiver noise power, imperfect channel knowledge, finite
blocklength, and interference from other network users or artificial noise.

Willie faces a decision as to whether or not Alice sent any covert information
to Bob, which means that Willie faces a binary hypothesis testing problem. The
null hypothesis states that Alice did not transmit while the alternative
hypothesis states that Alice did transmit, sending covert information to Bob.
We define the probability of false alarm (or Type I error) as the probability
that Willie makes a decision in favor of the alternative hypothesis, while the
null hypothesis is true. Similarly, the probability of miss detection (or Type
II error) is define as the probability of Willie making a decision in favor of
the null hypothesis, while the alternative hypothesis is true. The detection
error probability at Willie is the sum of Type I error and Type II error. In
the following, we present a review of literatures considering the covertness
constraint in term of detection error probability or Kullback-Leibler (KL)
divergence for achieving covert communication, where KL divergence is a
statistical distance of how one probability distribution of null hypothesis is
different from another probability distribution of the alternative hypothesis.

###  II-A Covert Communication with Relay

The ultimate goal of covert communication is to achieve shadow wireless
networks. Preliminary results towards this goal have been achieved in the
context of relay networks. Covert communication with the aid of relay was
analyzed in [16, 17, 18, 19]. The scenario investigated in [16] is that there
is a source node which attempts to transmit information to a destination node
via a relay node. The relay node receives a message from a source node and
then attempts to retransmit the source’s original message in addition to its
own covert message to the destination without the source node detecting the
covert transmission. This work presents two schemes (rate-control and power-
control) for evaluating the probability that the covert criteria is satisfied.

In [17], the authors studied the covert communications with a self-sustained
relay over Rayleigh fading channels, where the relay harvests extra energy
from the source to transmit covert message to destination by employing a more
powerful energy harvester with a higher conversion efficiency factor, while
the source is to detect the covert communication. The work aims to find the
maximum throughout, given a certain level of covertness for two energy
harvesting schemes, time switching and power splitting. Covert communication
with a positive rate was shown to be possible in the transmission from a relay
node to a destination when the source is uncertain regarding the forwarding
strategy of the relay node.

Two relay selection schemes (i.e., random selection and superior-link
selection) in multiple relays-assisted internet of things (IoT) systems was
proposed in [18], where the source needs to transmit two types of messages
with/without covert requirement. The numerical results showed that the
superior-link selection scheme has significant improvement in term of covert
capacity when compared to the random selection under the same transmission
power at the source. The routing of information in covert wireless networks
with multiple relay nodes and multiple wardens under additive white gaussian
noise (AWGN)channels has been considered in [19], where the algorithms were
developed to find the path with the maximum throughput and the path with
minimum end-to-end delay when considering a single key and the case of
independent keys at the intermediate nodes.

###  II-B Covert Communication with Full-duplex

In the covert communication, there is always a trade-off between the
throughput and covertness. Using a full-duplex receiver offers a two-fold
benefit for the covert communication relative to generating AN by a separate
and independent jammer. Firstly, it enables a higher degree of control over
the transmitted AN signals (e.g., its power), thus causing further deliberate
confusion at the warden. Secondly, the cutting-edge self-interference
cancellation techniques can be adopted to provide higher data rates for covert
communications with the full-duplex receiver. In [20], the transmitter Alice
aims to send message at a fixed rate to a legitimate receiver Bob over a
fading wireless channel while the whole communication is under the
surveillance of the warden Willie. Operating at the full-duplex mode, Bob can
receive information from Alice and also simultaneously inject interference to
confuse Willie over the same channel, which can increase the detection error
probability at Willie. On the other hand, it can also decrease covert
throughput due to the effect of self-interference at Bob. Based on these
discussions, the trade-off between the achievable rate, the transmission
power, and covertness criterion that is the probability of error minimized
over all prior distributions of the fading channel was investigated.

The aforementioned works in the literature mainly focus on how to hide the
wireless transmission action rather than the transmitter itself, since there
is some prior information that can indicate the existence of the transmitter.
To achieve higher level covert communication, the transmitter itself also
should be hidden from the warden. Taking advantage of the full-duplex and
channel inversion power control (CIPC), the work aims to hide the transmitter
by removing the requirement on channel state information (CSI) [21]. For the
CIPC schemes adopted in this work, the transmitter varies its transmitted
signal power as per the channel from itself to the receiver, so that the
received power of the covert information signal is a fixed value. Therefore,
neither Bob nor Willie knows the corresponding CSI perfectly, since Alice does
not transmit any pilot signal at all. Meanwhile, the full-duplex receiver can
further confuse the detection of Willie.

Ultra-reliable and low-latency communications (URLLC) is envisioned to support
mission critical applications with stringent requirements of latency and
reliability, where the codeword is required to be short, e.g., the order of
100 channel uses. Specifically, with the aided of the full-duplex receiver,
the authors in [22] showed that transmitting AN with a fixed power is still
helpful in covert communication with finite blocklength, since in a limited
time period the warden cannot exactly learn its received power. The
blocklength has significant impact on the detection performance at Willie and
the maximal achievable rate of the channel from Alice to Bob (i.e., the
maximal achievable rate decreases as blocklength decreases for a fixed
decoding error probability).

###  II-C Covert Communication with UAV

For the static covert communication network, the positions of the three
participants (Alice, Bob and Willie) are fixed. The unmanned aerial vehicle
(UAV) assisted communication networks enable the participants to realize more
flexible deployments in covert communication [23]. UAV can establish line-of-
sight (LoS) wireless links for air-ground communications, which provides
significant performance improvement over the conventional non-line-of-sight
(NLoS) terrestrial communications. In some scenarios, the covert rate can be
effective improved by exploiting the mobility of UAV while the detection by
Willie will become easier due to the component of LoS channel.

In [24], a UAV acted as a mobile relay to enhance covert transmission by
dynamically adjusting its trajectory and transmit power. Specifically, the
authors formulated an optimization problem that maximizes the average covert
transmission rate subject to the transmission outage probability constraint at
Bob and covertness constraint at Willie as well as the UAV’s mobility
constraint and transmit power constraint. Such a joint optimization problem is
generally difficult to tackle directly due to the non-convex constraints. To
solve this optimization problem, the authors first transformed the intractable
transmission outage probability constraint into a deterministic form by using
the conservative approximation and then applied the first-order restrictive
approximation to transform the optimization problem into a convex form, which
is mathematically tractable. A multi-hop relaying strategy (e.g., the number
of hops, transmit power) was optimized to maximize the throughput against the
surveillance of a UAV warden in [25]. The beam sweeping based detection of a
UAV’s transmission by a terrestrial warden with multiple antennas was
investigated in [26], where the Pinsker’s inequality and Kullback-Leibler
divergence were adopted to derive the detection error probability.

Figure 3: The UAV’s optimized trajectory. Figure 4: Average time for each Bob
to complete data reception achieved by our TD3-CDD scheme and the benchmark
scheme.

In the aforementioned works, the UAV’s trajectory designs with the
optimization for some parameters may suffer from several limitations, for
example, the objective function or constraints in the optimization problems
are non-convex and difficult to be tackled. Against this background, the deep
reinforcement learning (DRL) has been serving as an efficient solution to
handle the decision-making issue due to its essential traits, i.e., learning
dynamically from the real world. Motivated by the superiorities of DRL, we
intend to address the non-convex problem for considering covert data
dissemination with UAV, where UAV intends to transmit the covert data to some
legitimate receivers (Bobs) under the surveillance by some wardens (Willies).
Specially, a twin-delayed deep deterministic policy gradient-based covert data
dissemination (TD3-CDD) algorithm is proposed to minimize the time of covert
data dissemination by jointly optimizing the UAV’s trajectory and Bobs’
schedule. In details, we first determined the covertness constraint explicitly
by analyzing the detection performance at each Willie. Then, we formulated an
optimization problem to minimize the total time for data dissemination subject
to the covertness constraint and other practical constraints, e.g., the UAV’s
mobility constraints. By applying the deep reinforcement learning approach, an
efficient algorithm named TD3-CDD is developed to solve the optimization
problem.

In Figure. 3, we plot the optimized UAV’s trajectory achieved by our proposed
TD3-CDD scheme. where the locations of the Bobs are indicated by “$\star$”,
the green diamond indicates the initial position of the UAV, the central
locations of Willies are indicated by “$\bullet$” and the shaded area
indicates the surveillance area of each Willie. In this figure, the UAV is
able to sequentially visit all the Bobs while avoiding each Willie’s
surveillance. Figure. 4 shows the average time for each Bob to complete the
data reception achieved by our TD3-CDD scheme and the benchmark scheme (e.g.,
UAV’s flight trajectory of keeping away from Willies without optimization). In
this figure, as expected, we observe that the TD3-CDD scheme takes less time
than the benchmark scheme to complete the data transmission to all Bobs.

##  III Physical layer security of DM

In recent years, as one of the key technologies of physical layer wireless
transmission, DM has been a research hotspot in the field of physical layer
security (PLS). DM technology realizes the directionality of signal
transmission by optimizing the design of beamforming and adding AN to the
transmitted signal. Using this technology can not only accurately and
efficiently transmit information to desired users, but also achieve signal
constellation distortion in undesired directions by designing AN projection
matrix to transmit AN to undesired directions. Different from traditional
physical layer security technologies, directional modulation is mainly
suitable for fading-free Gauss white noise channels. When the linear distance
and direction of the transmitter and receiver are known, the channel steering
vector has static stability.

In Figure. 5, a schematic diagram is given to show the basic idea of DM. The
system consists of three nodes: a multi-antenna transmitter (Alice), a desired
user (Bob) and an eavesdropper (Eve). By using the AN technology and the null
space projection (NSP) criterion, the AN is projected into the null space of
the desired direction channel, that is, it does not affect the desired user,
so that the expected receiver can easily decode useful information. At the
same time, since AN makes it difficult for the eavesdropper to observe the
changing law of the signal constellation diagram, the eavesdropper cannot
correctly decode confidential messages (CMs).

Figure 5: Schematic diagram of DM network.

###  III-A Realization method of DM

DM can be divided into two categories: one is based on the combination of
radio frequency end components, and the other is implemented by designing
algorithms for baseband signals. Babakhani _et al._ [27] introduced a near-
field direct antenna modulation (NFDAM) technology, which modulates the far-
field radiation signal through the time-varying electromagnetic boundary
conditions of the antenna’s near-field. This enables the transmitter to send
information depending on the direction, so as to achieve a safe transmission
effect. Daly and Bernhard [28] proposed a DM technique based on phased array
generation. By correctly phase shifting each element, the amplitude and phase
required by each symbol in the digital modulation scheme are generated in a
given direction. The disadvantage of this technology is that it only considers
the signal constellation in the desired direction and ignores the
consideration of the distortion of the constellation in the undesired
direction. Therefore, Daly _et al._ [29] designed a phased array-based
physical layer secure transmission technology on the basis of [28]. They found
through experiments that the DM transmitter creates a narrower low error rate
area around the desired direction, while continuing to maintain a high error
rate in the sidelobe area. Daly and Bernhard [30] then used an array with
pattern reconfigurable elements to switch elements for each symbol, thereby
generating a digitally modulated signal in the desired direction, and used
this reconfigurable array to demonstrate the improvement of DM in terms of
safety performance. Unlike traditional transmitters that excite the same
antenna, Hong _et al._ [31] proposed a dual-beam DM technology for PLS
communications, which uses in-phase and quadrature baseband signals to excite
two different antennas.

###  III-B AN-aided DM Secure Systems

According to the concept of orthogonal AN in information theory, Ding _et al._
applied orthogonal vector technology to DM in [32]. By adding orthogonal AN
signals at the baseband, the signal constellation in the undesired direction
is distorted, causing eavesdroppers to be unable to decipher CMs according to
the changing law of the signal constellation. At the same time, the orthogonal
AN is projected into the null space (NS) of the channel steering vector of the
desired direction, thereby eliminating the influence of AN on the signal
received by desired users. Ding _et al._ [33] further proposed an orthogonal
vector method for DM transmitter to synthesize multiple beams on the basis of
[32]. [34] showed that when the non-cooperative receivers are placed
separately, the DM system can be regarded as a MIMO system operating in free
space. By transmitting AN which orthogonal to useful information, the freedom
of spatial multiplexing is lost, thereby enhancing the security performance of
the DM system.

###  III-C Robust beamforming and DOA measurement for DM Secure System

In actual application scenarios, there will be errors in the measured angle,
which will affect the expected user reception performance and reduce the
security of wireless transmission. However, in the above research, the
direction angle is assumed to be ideal known information, thus further study
on systems with imperfect information of direction angle is needed. Hu _et
al._ [35] designed a robust DM synthesis technique based on the principle of
minimum mean square error (MMSE) in view of the uncertainty of the estimated
direction angle, which improves the bit error rate (BER) performance of the
system by minimizing the constellation distortion in the desired direction.
Shu _et al._ [36] considered a multi-beam broadcast DM scenario with an
imperfect desired direction. [36] designed beamforming by minimizing
information leakage in the eavesdropping direction, and maximizing the
expected average received signal-to-interference-to-noise ratio (SINR) of the
user end to design a projection matrix of AN, so as to achieve the robustness
of CMs transmission in the wireless communication system. Shu _et al._ [37]
considerd the MIMO system with direction of arrival (DOA) measurement error,
and proposed a robust beamforming scheme based on the combination of main lobe
integration and leakage, which transmits independent and parallel private data
streams to multiple legitimate users at the same time, thereby realizing the
stable and safe transmission of CMs in the DM system. Shu _et al._ further
designed a low-complexity scheme in [38] that can accurately and safely
transmit CMs to desired users through the joint use of AN, phase alignment,
orthogonal frequency division multiplexing (OFDM) technology, and DM random
subcarrier selection technology.

In addition, there is a key technology in the DM field, DOA estimation
technology, which combines information of different frequencies to obtain the
best DOA. Hung and Kaveh [39] characterized the specific structure of the
focus matrix to avoid focus loss, and the focus matrix is preferably a unitary
matrix, which has an important impact on the statistical characteristics of
DOA estimation. In [40], the data was robustly preprocessed through the
weighted average of the signal subspace and the enhanced design of the focus
matrix. Ng _et al._ [41] proposed a wideband structure for processing array
signals, and estimated the order and DOA through the Bayesian method and the
reversible jumping Markov chain Monte Carlo process.

##  IV Physical layer security of SM

As one of the derivatives in the context of MIMO techniques, SM is becoming a
hot research area. Its basic idea is using both the transmit antenna index and
the modulated symbol to convey messages, which improves the spectrum
efficiency and reduces the cost of radio frequency chain [42]. Compared to
MIMO technique, SM plays a crucial role in balancing the transmit rate and the
hardware cost. On the other hand, the broadcast nature of the wireless
communication incurs that the desired receiver is vulnerable to hostile users,
thus establishing a set of systematically secure transmission strategies
becomes an imperative demand. In this section, we review the crucial
transmission strategies of the physical layer security under the typical SM
system and hybrid SM system respectively.

###  IV-A Typical SM system

A typical SM system has been shown in Figure. 6. In this figure, four
important tools including transmit antennas selection (TAS), AN projection,
power allocation (PA) and receive beamformer at the desired receiver are fully
employed to achieve a high-performance secure SM (SSM).

Figure 6: Schematic diagram of SSM network.

In Figure. 6, the transmitter (Alice) is equipped with $N_{a}$ transmit
antennas (TAs). Without loss of generality, $N_{a}$ is not a power of two,
thus $N_{t}$ out of $N_{a}$ TAs should be selected for mapping binary bits to
the antenna index. Appropriately selecting out an active antenna group is
capable of improving the security performance of SM systems. The authors in
[43] proposed two TAS methods: capacity-optimized antenna selection (COAS) and
Euclidean distance-optimized antenna selection (EDAS) which focused on
improving bit-error rate performance. In [44], the authors generalized EDAS
method to the secure SM system and proposed the maximizing signal-to-leakage-
plus-noise ratio (Max-SLNR) TAS strategy, where the Max-SLNR scheme achieved a
higher secrecy rate (SR) performance with a lower complexity. However, Max-
SLNR method dose not directly maximize the secrecy rate of the SSM system. To
overcome the above disadvantage, the authors in [45] proposed two low-
complexity TAS methods for maximizing SR by analyzing the asymptotic
performance of SR when signal-to-noise ratio (SNR) approaches 0 and $\infty$.
Furthermore, a compromise solution that suits for the whole SNR region was
further devised, which has the capability of achieving a close SR performance
when compared with the exhaustive search (ES) method. When a rough CSI of
Eve’s channel is obtained, the authors in [46] proposed a simulated-annealing-
mechanism-based TAS scheme which obtained excellent SR profit.

By reasonably generating Gaussian AN at the transmitter, the detection of the
Eve can be deteriorated, so as to further improve the security performance of
the system [47]. [48] projected AN onto the null-space of the desired channel
to interfere unknown eavesdropper. Although NSP scheme ensures the high
security performance in high SNR region, the beamformer design for AN has not
been further expored to further improve the security performance of SSM. The
authors in [46] proposed a novel ratio optimization algorithm which reduced
computational overhead and achieved near-optimal safety performance.

As the total power is constrained, elaborately splitting the power allocated
to the CM and allocated to the AN has a great impact on improving the SR
performance. In [49] and [50], the approximation of SR based on cut-off rate
was invoked to substitute the non-closed SR expression for secure SM systems.
Using the approximation as the objection function, the authors in [49]
proposed a gradient-based PA method which approached the optimal PA method
based on ES. In [50], a novel criterion that maximizes the ratio of Bob’s SLNR
and of Eve’s AN-to-leakage plus noise ratio (ANLNR) was conceived which was
close to the optimal PA factor and also reduced some computational complexity.
To further reduce the computational complexity and approach the optimal SR
performance, the authors in [51] proposed a deep-neural-network (DNN)-based PA
strategy whose key idea was to treat the input and output of ES algorithm as
an unknown nonlinear mapping and used a DNN to approximate it.

Consider a SSM system with a full-duplex (FD) malicious attacker (Mallory),
where Mallory works on FD mode, eavesdropping from Alice and transmitting
jamming to Bob simultaneously. A game-theoretic approach was adopted in [52]
to deal with a combination of passive eavesdropping and active attacks. The
authors in [53] proposed three effective receive beamformers with low
complexity to eliminate jamming and improve SR.

###  IV-B Hybrid SM system

When the number of TAs tends to large-scale, the circuit cost and complexity
will become a burden on fully-digital SSM. To address this problem, hybrid SM
was proposed and fully investigated where the total antenna array is divided
into multiple transmit antenna subarrays with each being connected to single
RF chain. As for hybrid SSM, two effective tools including precoder and
transmit antenna subarray selection are used to achieve a high security SSM.

In [54], the authors proposed a semi-definite relaxation based alternating
minimization (SDR-AltMin) algorithm to jointly design the digital and analog
precoders. In [55], the authors generalized SDR-AltMin algorithm to the hybrid
SSM system, and proposed two precoders which maximized approximate SR (ASR)
based on gradient method and alternating direction method of multipliers
respectively. Figure. 7 shows that the proposed two methods in [55] harvested
higher SR performance than the generalized SDR-AltMin method. For hybrid
generalized SSM systems, the authors in [56] proposed a novel algorithm to
separately optimize the analog precoder and the digital precoder to maximize
SR. As the objective function is non-concave, the authors conceived a pair of
concave objective functions to optimize the analog precoder and digital
precoder.

Figure 7: Comparison of SR performance of various precoders.

As the number of RF chains is not a power of two, there exists the transmit
antenna subarrays selection problem. The authors in [55] proposed two low-
complexity transmit antenna subarray selection (TASS) methods, named
maximizing eigenvalue (Max-EV) and maximizing product of signal-to-
interference-plus-noise ratio and artificial noise-to-signal-plus-noise ratio
(Max-P-SINR-ANSNR), which performed better than the generalized Max-SLNR
method.

##  V IRS-aided physical layer security

Recently, IRS has been considered as a promising new technology for the next-
generation wireless communications, which can reconfigure the wireless
propagation environment via controlling reflection with software [57].
Specifically, an IRS composes a large number of low-cost passive reflecting
elements, each of which can induce a phase change to the incident signal
independently. By smartly adjusting the phase shift of the reflecting
elements, the IRS reflected signals and the signals from other paths can be
combined constructively to enhance the desired signal power or destructively
to suppress undesired signals such as co-channel interference, which thus
significantly improves the communication performance. A typical IRS-aided MISO
multi-user secure system is shown in Figure. 8. The introduction of IRS in the
system can increase the achievable rate of legitimate users (i.e., Bobs) while
suppressing the achievable rate of illegal users (i.e., Eves), ultimately
improving the security performance of the system. Hence, IRS can be used for
strengthening the system security under the wiretap channel, especially when
the channel of the eavesdropping communication link is stronger than that of
the legitimate link.

Figure 8: An IRS-aided secure multi-user communication system. Figure 9:
Transmit power at the Alice versus SR.

###  V-A IRS-aided MISO Secure Systems

Many recent studies have utilized IRS to secure the physical layer of wireless
communications. The authors in [58, 59, 60] investigated an IRS-aided secure
wireless system where a multi-antenna transmitter communicates with a single-
antenna receiver in the presence of an Eve. In [58, 59, 60], the SR was
maximized by jointly optimizing the transmit beamforming and the IRS phase
shifts. Specifically, the authors in [58] proposed an alternating optimization
(AO) algorithm to design two variables alternately, in which the optimal
solution to the transmit beamforming was computed directly and the IRS phase
shifts were optimized by using semidefinite relaxation (SDR) method. In [59],
the transmit beamforming and the IRS phase shifts were also optimized in an
alternating manner. In each iteration, the solution to the transmit
beamforming was achieved in closed form, while a semi-closed form solution to
the IRS phase shifts was obtained by adopting the Majorization-Minimization
(MM) algorithm. The element-wise block coordinate descent (BCD) and AO-MM
algorithms were developed for solving the problem efficiently in [60].
Furthermore, the authors in [61] investigated a minimum-SR maximization
problem in a secure IRS-aided multiuser multiple-input single-output (MISO)
broadcast system with multiple Eves. The problem was successfully solved by
applying the path-following algorithm and AO technique in an iterative manner.
Moreover, two suboptimal algorithms with closed-form solutions were developed
to further reduce the computational complexity. To further enhance the
security, the AN is designed to disturb Eve. In [62], AN was firstly
considered in an IRS-aided secure communication system. Specifically, the
achievable SR of the system was maximized by jointly optimizing the transmit
beamforming with AN and IRS phase shifts. An efficient algorithm based on AO
was developed to solve the problem sub-optimally. The authors in [63]
considered the resource allocation design in an IRS-aided multiuser MISO
communication system. Aiming to maximize the system sum SR, the beamforming
vectors, AN covariance matrix at the BS and phase shift matrix at the IRS were
jointly optimized by applying AO, SDR and manifold optimization. The authors
in [64] investigated a secure IRS-aided multigroup multicast MISO
communication system to minimize the transmit power subject to the SR
constraints. First, an SDR-based AO algorithm was proposed and a high-quality
solution was obtained. Then, to reduce the high computation complexity, a low-
complexity AO algorithm based on second-order cone programming (SOCP) was
presented. Simulation result indicates that these designs can guarantee
reliable secure communication with the aid of IRS as shown in Figure. 9.

###  V-B IRS-aided MIMO Secure Systems

Recently, the introduction of IRS in MIMO system to enhance the security has
also made effective progress. Dong _et al_. [65] and Jiang _et al_. [66]
studied an IRS-aided secure MIMO communication system, where a base station
(BS) equipped with multiple antennas communicates with a multi-antenna
receiver in the presence of a multi-antenna Eve. Particularly, the joint
design of the transmit covariance at the BS and the phase shift coefficients
at the IRS was considered to maximize the SR. In [65], the barrier, Newton and
backtracking line search methods were used to search for global optimal
transmit covariance, while the MM algorithm was applied to obtain the local
optimal phase shift coefficients. In [66], the successive convex approximation
(SCA)-based algorithm was used to solve the optimization of the transmit
covariance. For the design of the phase shift coefficients, the closed-form
solution of each phase shift coefficient was individually provided in an
alternating way while fixing other phase shift coefficients. Hong _et al_.
[67] considered an AN-aided secure MIMO communication system assisted by an
IRS. With the aim for maximizing the SR, the BCD algorithm was exploited to
alternately optimize the variables. Specifically, the optimal transmit
covariance and AN covariance matrix were derived in semi-closed form by
applying the Lagrangian multiplier method, and the optimal phase shifts were
obtained in closed form by utilizing the MM algorithm. IRS-aided secure DM-
MIMO system was studied in [68]. Two confidential bit streams transmitted from
Alice to Bob, different from the conventional DM where Alice only transmits
single CBS to Bob. This paper demonstrated that the IRS significantly enhanced
the SR of DM.

###  V-C Robust IRS-aided Secure System

The above contributions are obtained under the assumption that perfect CSI is
available at the Alice. However, obtaining perfect CSI is ideal, so it is
necessary to consider robust security performance of the system. The authors
in [69, 70, 71] investigated the robust transmission design in an IRS-aided
secrecy system, where the CSI of Eves’ channels is not perfectly known.
Particularly, a robust secure beamforming problem was formulated to maximize
the worst case of SR in [69]. By replacing the wiretap channels with a
weighted combination of discrete samples and then using SDR technique, an
efficient AO algorithm was proposed to solve the optimization problem sub-
optimally. In [70], the authors considered a scenario where two IRSs are
deployed to improve the sum SR of multiple single-antenna legitimate receivers
in the presence of multiple multi-antenna Eves. Considering the imperfection
of the CSI of the Eves’ channels, a robust non-convex optimization problem was
proposed. Then the design of the transmit covariance, AN covariance matrix and
IRS phase shifts was handled by adopting AO, SCA and SDR. In [71], the authors
first considered the statistical CSI error model and the imperfect cascaded
channels of AP-IRS-EVE in IRS-aided secure communications. Then an outage-
constrained power minimization problem was formulated. To solve it, the
Bernstein-type inequality was exploited to tackle the outage rate probability
constraints, and then a sub-optimal algorithm based on AO, the penalty-based
and SDR methods were utilized to optimize the variables alternately.

##  VI Future Technical Challenges and Research Directions

Despite many researches in studying the physical layer security, there are
still many challenging problems to be solved. In the following, some technical
challenges for physical layer security in wireless communications are
discussed.

###  VI-A CSI Acquisition of IRS-aided communication system

The acquisition of perfect channel information in physical layer security is a
challenging task. It is mainly manifested in the following aspects. First, it
is especially difficult to obtain the CSI associated with the Eve due to the
hidden nature of eavesdroppers. Second, in IRS-assisted security systems,
because of the limited signal processing capability at the IRS, the perfect
CSI of IRS-links is challenging to obtain. Hence, designing a secure and
reliable CSI feedback mechanism is necessary.

###  VI-B DOA Acquisition of DM

DM technology has been applied to key safety tools in vehicle communications,
IoT, UAVs, smart transportation and deep-space communications. Due to the
gathering effect of AN, DM is only applicable to LoS channels, not to
multipath channels. Therefore, how to extend DM to mmWave MIMO channels with
multipath is still a challenging problem in the future. Besides, since there
may be a large angle error when sending CMs to the desired user at a long
distance, it is more practical to study the robust DM synthesis technology in
the future for the uncertainty of the direction angle estimation.

###  VI-C Channel Model of SM Aided by IRS

The previous research on SM and IRS aided SM system are based on the Rayleigh
fading model. How to extend SM system to other types of fading channels is
very important for the practical application and deployment of SM. In
addition, how to construct the security optimization problem in the relevant
channel environment is worth pondering.

##  VII Conclusion

The emergence and development of future wireless technologies such DM/SM
technology, covert communication, and IRS have brought new security challenges
to future networks. Designing effective and secure transmission schemes for
future wireless communications that exploit the propagation properties of
radio channels in the physical layer has recently attracted extensive research
interest. This paper reviewed the currently popular physical layer secure
communication techniques from both theoretical and technical perspectives. We
investigated the confidentiality issues of physical layer security
technologies such as DM, SM, covert communication, and IRS, respectively.
Finally, we discussed potential challenges in physical layer security and
point out some possible future research directions.

## References

  * [1] H. Zhu and J. Wang, “Chunk-based resource allocation in ofdma systems - part i: chunk allocation,” _IEEE Trans. Commun._ , vol. 57, no. 9, pp. 2734–2744, Sep. 2009.
  * [2] ——, “Chunk-based resource allocation in ofdma systems - part ii: Joint chunk, power and bit allocation,” _IEEE Trans. Commun._ , vol. 60, no. 2, pp. 499–509, Feb. 2012.
  * [3] J. Wang, H. Zhu, and N. J. Gomes, “Distributed antenna systems for mobile communications in high speed trains,” _IEEE J. Sel. Areas Commun._ , vol. 30, no. 4, pp. 675–683, May 2012.
  * [4] Y. Zhou, J. Wang, and M. Sawahashi, “Downlink transmission of broadband ofcdm systems-part i: hybrid detection,” _IEEE Trans. Commun._ , vol. 53, no. 4, pp. 718–729, Apr. 2005.
  * [5] Y. Zhou, H. Liu, Z. Pan, and L. Tian, “Two-stage cooperative multicast transmission with optimized power consumption and guaranteed coverage,” _IEEE JSAC on SEED_ , vol. 32, no. 2, pp. 274–284, Feb. 2014.
  * [6] Y. Wu, A. Khisti, C. Xiao, G. Caire, K.-K. Wong, and X. Gao, “A survey of physical layer security techniques for 5g wireless networks and challenges ahead,” _IEEE Journal on Selected Areas in Communications_ , vol. 36, no. 4, pp. 679–695, April 2018.
  * [7] X. Chen, D. W. K. Ng, W. H. Gerstacker, and H.-H. Chen, “A survey on multiple-antenna techniques for physical layer security,” _IEEE Communications Surveys Tutorials_ , vol. 19, no. 2, pp. 1027–1053, Secondquarter 2017.
  * [8] D. Wang, B. Bai, W. Zhao, and Z. Han, “A survey of optimization approaches for wireless physical layer security,” _IEEE Communications Surveys Tutorials_ , vol. 21, no. 2, pp. 1878–1911, Secondquarter 2019.
  * [9] S. Leung-Yan-Cheong and M. Hellman, “The gaussian wire-tap channel,” _IEEE Trans. Inf. Theory_ , vol. 24, no. 4, pp. 451–456, Jul. 1978.
  * [10] X. Wang, M. Tao, J. Mo, and Y. Xu, “Power and subcarrier allocation for physical-layer security in ofdma-based broadband wireless networks,” _IEEE Trans. Inf. Forensics Security_ , vol. 6, no. 3, pp. 693–702, Sep. 2011\.
  * [11] D. W. K. Ng, E. S. Lo, and R. Schober, “Energy-efficient resource allocation for secure ofdma systems,” _IEEE Trans. Veh. Technol._ , vol. 61, no. 6, pp. 2572–2585, Jul. 2012.
  * [12] C. Jeong and I.-M. Kim, “Optimal power allocation for secure multicarrier relay systems,” _IEEE Trans. Signal Process._ , vol. 59, no. 11, pp. 5428–5442, Nov 2011.
  * [13] S. Goel and R. Negi, “Guaranteeing secrecy using artificial noise,” _IEEE Trans. Wireless Commun._ , vol. 7, no. 6, pp. 2180–2189, Jun. 2008\.
  * [14] B. A. Bash, D. Goeckel, D. Towsley, and S. Guha, “Hiding information in noise: Fundamental limits of covert wireless communication,” _IEEE Commun. Mag.,_ , vol. 53, no. 12, pp. 26–31, Dec. 2015.
  * [15] S. Yan, X. Zhou, J. Hu, and S. V. Hanly, “Low probability of detection communication: Opportunities and challenges,” _IEEE Wireless Commun._ , vol. 26, no. 5, pp. 19–25, Oct. 2019.
  * [16] J. Hu, S. Yan, X. Zhou, F. Shu, J. Li, and J. Wang, “Covert communication achieved by a greedy relay in wireless networks,” _IEEE Trans. Wireless Commun._ , vol. 17, no. 7, pp. 4766–4779, Jul. 2018.
  * [17] J. Hu, S. Yan, F. Shu, and J. Wang, “Covert transmission with a self-sustained relay,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 8, pp. 4089–4102, Aug. 2019.
  * [18] C. Gao, B. Yang, X. Jiang, H. Inamura, and M. Fukushi, “Covert communication in relay-assisted IoT systems,” _IEEE Internet Things J._ , vol. 8, no. 8, pp. 6313–6323, Apr. 2021.
  * [19] A. Sheikholeslami, M. Ghaderi, D. Towsley, B. A. Bash, S. Guha, and D. Goeckel, “Multi-hop routing in covert wireless networks,” _IEEE Trans. Wireless Commun._ , vol. 17, no. 6, pp. 3656–3669, Jun. 2018.
  * [20] K. Shahzad, X. Zhou, S. Yan, J. Hu, F. Shu, and J. Li, “Achieving covert wireless communications using a full-duplex receiver,” _IEEE Trans. Wireless Commun._ , vol. 17, no. 12, pp. 8517–8530, Dec. 2018.
  * [21] J. Hu, S. Yan, X. Zhou, F. Shu, and J. Li, “Covert wireless communications with channel inversion power control in rayleigh fading,” _IEEE Trans. Veh. Technol._ , vol. 68, no. 12, pp. 12 135–12 149, Dec. 2019.
  * [22] F. Shu, T. Xu, J. Hu, and S. Yan, “Delay-constrained covert communications with a full-duplex receiver,” _IEEE Wireless Commun. Lett._ , vol. 8, no. 3, pp. 813–816, Jun. 2019.
  * [23] X. Jiang, X. Chen, J. Tang, N. Zhao, X. Y. Zhang, D. Niyato, and K.-K. Wong, “Covert communication in UAV-assisted air-ground networks,” _IEEE Wireless Commun._ , vol. 28, no. 4, pp. 190–197, Aug. 2021.
  * [24] X. Zhou, S. Yan, J. Hu, J. Sun, J. Li, and F. Shu, “Joint optimization of a UAV’s trajectory and transmit power for covert communications,” _IEEE Trans. Signal Process._ , vol. 67, no. 16, pp. 4276–4290, Aug. 2019.
  * [25] H.-M. Wang, Y. Zhang, X. Zhang, and Z. Li, “Secrecy and covert communications against UAV surveillance via multi-hop networks,” _IEEE Trans. Commun._ , vol. 68, no. 1, pp. 389–401, Jan. 2020.
  * [26] J. Hu, Y. Wu, R. Chen, F. Shu, and J. Wang, “Optimal detection of uav’s transmission with beam sweeping in covert wireless networks,” _IEEE Trans. Veh. Technol._ , vol. 69, no. 1, pp. 1080–1085, Jan. 2020.
  * [27] A. Babakhani, D. B. Rutledge, and A. Hajimiri, “Transmitter architectures based on near-field direct antenna modulation,” _IEEE J. Solid-State Circuits_ , vol. 43, no. 12, pp. 2674–2692, 2008.
  * [28] M. P. Daly and J. T. Bernhard, “Directional modulation technique for phased arrays,” _IEEE Trans. Antennas Propag._ , vol. 57, no. 9, pp. 2633–2640, 2009.
  * [29] M. P. Daly, E. L. Daly, and J. T. Bernhard, “Demonstration of directional modulation using a phased array,” _IEEE Trans. Antennas Propag._ , vol. 58, no. 5, pp. 1545–1550, 2010.
  * [30] M. P. Daly and J. T. Bernhard, “Beamsteering in pattern reconfigurable arrays using directional modulation,” _IEEE Trans. Antennas Propag._ , vol. 58, no. 7, pp. 2259–2265, 2010.
  * [31] T. Hong, M.-Z. Song, and Y. Liu, “Dual-beam directional modulation technique for physical-layer secure communication,” _IEEE Antennas Wireless Propag. Lett._ , vol. 10, pp. 1417–1420, 2011.
  * [32] Y. Ding and V. F. Fusco, “A vector approach for the analysis and synthesis of directional modulation transmitters,” _IEEE Trans. Antennas Propag._ , vol. 62, no. 1, pp. 361–370, 2013.
  * [33] Y. Ding and V. Fusco, “Orthogonal vector approach for synthesis of multi-beam directional modulation transmitters,” _IEEE Antennas Wireless Propag. Lett._ , vol. 14, pp. 1330–1333, 2015.
  * [34] Y. Ding and V. F. Fusco, “Mimo-inspired synthesis of directional modulation systems,” _EEE Antennas Wireless Propag. Lett._ , vol. 15, pp. 580–584, 2015\.
  * [35] J. Hu, F. Shu, and J. Li, “Robust synthesis method for secure directional modulation with imperfect direction angle,” _IEEE Commun. Lett._ , vol. 20, no. 6, pp. 1084–1087, 2016.
  * [36] F. Shu, X. Wu, J. Li, R. Chen, and B. Vucetic, “Robust synthesis scheme for secure multi-beam directional modulation in broadcasting systems,” _IEEE Access_ , vol. 4, pp. 6614–6623, 2016.
  * [37] F. Shu, W. Zhu, X. Zhou, J. Li, and J. Lu, “Robust secure transmission of using main-lobe-integration-based leakage beamforming in directional modulation mu-mimo systems,” _IEEE Syst. J._ , vol. 12, no. 4, pp. 3775–3785, 2017.
  * [38] F. Shu, X. Wu, J. Hu, J. Li, R. Chen, and J. Wang, “Secure and precise wireless transmission for random-subcarrier-selection-based directional modulation transmit antenna array,” _IEEE J. Sel. Areas Commun._ , vol. 36, no. 4, pp. 890–904, Apr. 2018.
  * [39] H. Hung and M. Kaveh, “Focussing matrices for coherent signal-subspace processing,” _IEEE Transactions on Acoustics, Speech, and Signal Process._ , vol. 36, no. 8, pp. 1272–1281, 1988.
  * [40] E. D. Di Claudio and R. Parisi, “Waves: Weighted average of signal subspaces for robust wideband direction finding,” _IEEE Trans. Signal Process._ , vol. 49, no. 10, pp. 2179–2191, 2001.
  * [41] W. Ng, J. P. Reilly, T. Kirubarajan, and J.-R. Larocque, “Wideband array signal processing using mcmc methods,” _IEEE Trans. Signal Process._ , vol. 53, no. 2, pp. 411–426, 2005.
  * [42] R. Y. Mesleh, H. Haas, S. Sinanovic, C. W. Ahn, and S. Yun, “Spatial modulation,” _IEEE Trans. Veh. Technol._ , vol. 57, no. 4, pp. 2228–2241, 2008.
  * [43] R. Rajashekar, K. Hari, and L. Hanzo, “Antenna selection in spatial modulation systems,” _IEEE Communications Letters_ , vol. 17, no. 3, pp. 521–524, 2013\.
  * [44] F. Shu, Z. Wang, R. Chen, Y. Wu, and J. Wang, “Two high-performance schemes of transmit antenna selection for secure spatial modulation,” _IEEE Transactions on Vehicular Technology_ , vol. 67, no. 9, pp. 8969–8973, 2018.
  * [45] G. Xia, F. Shu, Y. Zhang, J. Wang, S. ten Brink, and J. Speidel, “Antenna selection method of maximizing secrecy rate for green secure spatial modulation,” _IEEE Transactions on Green Communications and Networking_ , vol. 3, no. 2, pp. 288–301, 2019.
  * [46] G. Xia, Y. Lin, T. Liu, F. Shu, and L. Hanzo, “Transmit antenna selection and beamformer design for secure spatial modulation with rough csi of eve,” _IEEE Transactions on Wireless Communications_ , vol. 19, no. 7, pp. 4643–4656, 2020.
  * [47] S. Goel and R. Negi, “Guaranteeing secrecy using artificial noise,” _IEEE Transactions on Wireless Communications_ , vol. 7, no. 6, pp. 2180–2189, 2008.
  * [48] L. Wang, S. Bashar, Y. Wei, and R. Li, “Secrecy enhancement analysis against unknown eavesdropping in spatial modulation,” _IEEE Communications Letters_ , vol. 19, no. 8, pp. 1351–1354, 2015.
  * [49] F. Shu, X. Liu, G. Xia, T. Xu, J. Li, and J. Wang, “High-performance power allocation strategies for secure spatial modulation,” _IEEE Transactions on Vehicular Technology_ , vol. 68, no. 5, pp. 5164–5168, 2019.
  * [50] G. Xia, L. Jia, Y. Qian, F. Shu, Z. Zhuang, and J. Wang, “Power allocation strategies for secure spatial modulation,” _IEEE Systems Journal_ , vol. 13, no. 4, pp. 3869–3872, 2019.
  * [51] F. Shu, L. Liu, L. Yang, X. Jiang, G. Xia, Y. Wu, X. Wang, S. Jin, J. Wang, and X. You, “Spatial modulation: an attractive secure solution to future wireless network,” _arXiv preprint arXiv:2103.04051_ , 2021.
  * [52] J. Choi, “Full-duplexing jamming attack for active eavesdropping,” in _2016 6th International Conference on IT Convergence and Security (ICITCS)_. IEEE, 2016, pp. 1–5.
  * [53] X. Jiang, X. Liu, R. Chen, Y. Wang, F. Shu, and J. Wang, “Efficient receive beamformers for secure spatial modulation against a malicious full-duplex attacker with eavesdropping ability,” _IEEE Transactions on Vehicular Technology_ , vol. 70, no. 2, pp. 1962–1966, 2021.
  * [54] X. Yu, J.-C. Shen, J. Zhang, and K. B. Letaief, “Alternating minimization algorithms for hybrid precoding in millimeter wave mimo systems,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 10, no. 3, pp. 485–500, 2016.
  * [55] F. Shu, X. Jiang, X. Liu, L. Xu, G. Xia, and J. Wang, “Precoding and transmit antenna subarray selection for secure hybrid spatial modulation,” _IEEE Transactions on Wireless Communications_ , vol. 20, no. 3, pp. 1903–1917, 2021\.
  * [56] G. Xia, Y. Lin, X. Zhou, W. Zhang, F. Shu, and J. Wang, “Hybrid precoding design for secure generalized spatial modulation with finite-alphabet inputs,” _IEEE Transactions on Communications_ , vol. 69, no. 4, pp. 2570–2584, 2021.
  * [57] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network,” _IEEE Commun. Mag._ , vol. 58, no. 1, pp. 106–112, Jan. 2020.
  * [58] M. Cui, G. Zhang, and R. Zhang, “Secure wireless communication via intelligent reflecting surface,” _IEEE Wireless Commun. Lett._ , vol. 8, no. 5, pp. 1410–1414, 2019.
  * [59] H. Shen, W. Xu, S. Gong, Z. He, and C. Zhao, “Secrecy rate maximization for intelligent reflecting surface assisted multi-antenna communications,” _IEEE Commun. Lett._ , vol. 23, no. 9, pp. 1488–1492, 2019.
  * [60] X. Yu, D. Xu, and R. Schober, “Enabling secure wireless communications via intelligent reflecting surfaces,” in _Proc. IEEE Global Commun. Conf._ , Waikoloa, HI, USA, 2019, pp. 1–6.
  * [61] J. Chen, Y.-C. Liang, Y. Pei, and H. Guo, “Intelligent reflecting surface: A programmable wireless environment for physical layer security,” _IEEE Access_ , vol. 7, pp. 82 599–82 612, 2019.
  * [62] X. Guan, Q. Wu, and R. Zhang, “Intelligent reflecting surface assisted secrecy communication: Is artificial noise helpful or not?” _IEEE Wireless Commun. Lett._ , vol. 9, no. 6, pp. 778–782, 2020.
  * [63] D. Xu, X. Yu, Y. Sun, D. W. K. Ng, and R. Schober, “Resource allocation for secure IRS-assisted multiuser MISO systems,” in _IEEE Globecom Workshops (GC Wkshps)_ , 2019, pp. 1–6.
  * [64] W. Shi, J. Li, G. Xia, Y. Wang, X. Zhou, Y. Zhang, and F. Shu, “Secure multigroup multicast communication systems via intelligent reflecting surface,” _China Commun._ , vol. 18, no. 3, pp. 39–51, 2021.
  * [65] L. Dong and H.-M. Wang, “Secure MIMO transmission via intelligent reflecting surface,” _IEEE Wireless Commun. Lett._ , vol. 9, no. 6, pp. 787–790, 2020\.
  * [66] W. Jiang, Y. Zhang, J. Wu, W. Feng, and Y. Jin, “Intelligent reflecting surface assisted secure wireless communications with multiple-transmit and multiple-receive antennas,” _IEEE Access_ , vol. 8, pp. 86 659–86 673, 2020.
  * [67] S. Hong, C. Pan, H. Ren, K. Wang, and A. Nallanathan, “Artificial-noise-aided secure MIMO wireless communications via intelligent reflecting surface,” _IEEE Trans. Commun._ , vol. 68, no. 12, pp. 7851–7866, 2020.
  * [68] F. Shu, Y. Teng, J. Li, M. Huang, W. Shi, J. Li, Y. Wu, and J. Wang, “Enhanced secrecy rate maximization for directional modulation networks via irs,” _IEEE Trans. Commun._ , vol. 69, no. 12, pp. 8388–8401, Dec. 2021.
  * [69] X. Lu, W. Yang, X. Guan, Q. Wu, and Y. Cai, “Robust and secure beamforming for intelligent reflecting surface aided mmWave MISO systems,” _IEEE Wireless Commun. Lett._ , vol. 9, no. 12, pp. 2068–2072, 2020.
  * [70] X. Yu, D. Xu, Y. Sun, D. W. K. Ng, and R. Schober, “Robust and secure wireless communications via intelligent reflecting surfaces,” _IEEE J. Sel. Areas Commun._ , vol. 38, no. 11, pp. 2637–2652, 2020.
  * [71] S. Hong, C. Pan, H. Ren, K. Wang, K. K. Chai, and A. Nallanathan, “Robust transmission design for intelligent reflecting surface-aided secure communication systems with imperfect cascaded csi,” _IEEE Trans. Wireless Commun._ , vol. 20, no. 4, pp. 2487–2501, 2020.