This is a consequence of the next lemma, a variant of \cite[Lemma~\ref{iso1}]{ADHfgh}.
``very small derivation'' see~\cite[Sections~1.1 and~7.3]{ADH2}.

\begin{lemma}\label{iso1 variant, 1}
Let $K$ be a pre-$H$-field with very small derivation, archimedean residue field, and $\operatorname{trdeg}(K|C)\le \aleph_0$. Let
$L$ be a closed $\eta_1$-ordered $H$-field with small derivation and $C_L=\R$.
Then $K$ embeds into~$L$.
\end{lemma}
\begin{proof}
%We arrange that $\res(K)$ is an ordered subfield of $\R$.
% By \cite[Lemma~\ref{eta}]{ADHfgh} we may arrange $C_L=\R$.
Passing to $H(K)$ we arrange that $K$ is an $H$-field. Without loss, $C_K$ is an ordered subfield of $\R$, and then adjoining new constants if necessary, we arrange~$C_K=\R$.
Take a closed $H$-field $\hat K$ extending $K$. Next, take  a countable set~$S\subseteq K$ such that~$K=\R\langle S\rangle$ and then
a countable closed $H$-subfield $K_0\supseteq S$ of~$\hat K$.   Let $E$ be a copy  of the prime model  of the theory of
closed $H$-fields with small derivation  inside $K_0$.
Applying  \cite[Lemma~\ref{iso1}]{ADHfgh} to an $H$-field embedding~$E\to L$  with $K_0$ in place of $K$
yields an   $H$-field embedding~$i\colon K_0\to L$. Then~$i$ is the identity on $C_{K_0}\subseteq \R$, and then   [ADH, 10.5.15, 10.5.16]  yield an extension
of $i$ to an $H$-field embedding $K_0(\R) \to L$ that is the
identity on~$\R$, and the restriction of this embedding to $K$ is a pre-$H$-field embedding $K\to L$.
\end{proof}

%From \cite[Lemma~7.3.8]{ADH2} we obtain a  closed $H$-field $\hat K$ with small derivation extending $K$.
%The proof of~[ADH, 16.4.8] in combination with  [ADH, 14.5.10, 16.4.1] shows that
%$\hat K$ has a closed $H$-subfield extending $K$ whose residue field is algebraic over $\res K$ and hence archimedean,
%and replacing $\hat K$ by such a closed $H$-subfield of~$\hat K$ we arrange that
%$C_{\hat K}$ is archimedean.
%As in the proof of \cite[Lemma~\ref{iso1}]{ADHfgh} we identify~$C_{\hat K}$ with a subfield of $\R$.
%Next take  a countable subset $S$ of $K$ such that~$K=C\langle S\rangle$ and then
%a countable closed $H$-subfield $K_0$ of $\hat K$ containing the countable pre-$H$-subfield~$\Q\langle S\rangle$   of $K$.   Let $E$ be a copy  of the prime model  of the theory of
%  closed $H$-fields with small derivation  inside $K_0$.
% Applying  \cite[Lemma~\ref{iso1}]{ADHfgh} to an $H$-field embedding~$E\to L$  and with $K_0$ in place of $K$
%     yields an   $H$-field embedding~$i\colon K_0\to L$.
%Then~$i$ is the identity on $C_{K_0}\subseteq\R$, and    [ADH, 10.5.15, 10.5.16]  yield an extension
%of $i$ to an $H$-field embedding $K_0(C_{\hat K}) \to L$ that is the
%identity on~$C_{\hat K}\subseteq \R$, and the restriction of this embedding to $K$ is a pre-$H$-field embedding $K\to L$.

\noindent
Using also Theorem~\ref{aneta} and its smooth version we obtain from Lemma~\ref{iso1 variant, 1}:

\begin{cor}
Let  $K$ be as in Lemma~\ref{iso1 variant, 1} and let $M$ be a maximal Hardy field. Then~$K$  embeds into~$M$.
Likewise if $M$ is a maximal analytic Hardy field or a maximal smooth Hardy field.
\end{cor}

\noindent
The following immediate consequence of the last corollary is worth recording:

\begin{cor}
Every Hardy field of countable transcendence degree over its constant field is
isomorphic to an analytic Hardy field.
\end{cor}

\noindent
The next corollary strengthens \cite[Corollary~7.1.4]{ADH2}:

\begin{cor}\label{cor:inftyomegaelemsub}
Let $M$ be a maximal analytic or maximal smooth Hardy field, and let $N$ be a maximal Hardy field with $M\subseteq N$. Then $M \preceq_{\infty\omega} N$.
%Likewise if $M$ is a maximal  smooth Hardy field.
\end{cor}
\begin{proof}
By Theorem~\ref{aneta} and its smooth version, $M$ is $\eta_1$, and by Theorem~\ref{mt} of [8], $N$~is~$\eta_1$. It remains to use \cite[Lemma~10.5]{ADHfgh}.
\end{proof}

\noindent
At the heart of the proof of \cite[Lem\-ma~\ref{iso1}]{ADHfgh} is [ADH, 16.2.3] of which we now give a version
with the cofinality hypothesis replaced by a shortness assumption:

\begin{prop}\label{prop:16.2.3 short}
Let $E$ be an $\upo$-free $H$-field and $K$ be a closed short $H$-field extending~$E$ such that~$C_{E}=C_K$.
Let $i\colon E\to L$ be an embedding where $L$ is a closed $\eta_1$-ordered $H$-field. Then $i$ extends to an embedding~$K\to L$.
\end{prop}
\begin{proof}
Suppose $E\neq K$;
it is enough to show that $i$ extends to an embedding of some $\upo$-free $H$-subfield $F$ of $K$ into $L$, where $F$ properly contains $E$.

Consider first the case~$\Gamma_E^<$ is not cofinal in $\Gamma^<$.
Then we have $y\in K^>$ such that~$\Gamma_E^< < vy < 0$.
Now $E$ is short, so we have
$y^*\in L^>$ such that~$\Gamma^{<}_{iE} < vy^* < 0$.
As in the proof of [ADH, 16.2.3] we then obtain an $\upo$-free $H$-subfield $F$ of $K$ with~$F\supseteq  E\langle y\rangle$ and
an extension of $i$ to  an embedding~$F\to L$.

For the rest of the proof we assume $\Gamma_E^<$ is cofinal in $\Gamma^<$. Then every differential subfield of $K$ containing $E$ is
an $\upo$-free $H$-subfield of $K$.

\subcase[1]{$E$ is not closed.}
This goes like Subcase~1 in the proof of~[ADH, 16.2.3].

\subcase[2]{$E$ is closed, and $E\langle y\rangle$ is an immediate extension of $E$ for some $y\in K\setminus E$.}
For such $y$, Lemma~\ref{lem:L->M} yields an extension of $i$ to an embedding $E\langle y\rangle\to L$.

\subcase[3]{$E$ is closed, and there is no $y\in K\setminus E$ such that  $E\langle y\rangle$ is an immediate extension of $E$.}
Take any $f\in K\setminus E$.
Since $E$ is short  and $L$ is $\eta_1$ we have
$g\in L$ such that for all $a\in E$, $a<f\Leftrightarrow i(a)<g$. Now [ADH, 16.1.5]
%(cf.~also \cite[Proposition~\ref{prbcon}]{ADHfgh})
gives  an $H$-field embedding $E\langle f\rangle\to L$ extending $i$ which sends $f$ to $g$.
\end{proof}

\begin{cor}\label{cor:16.2.3 short}
Let $E$, $K$, $L$, $i$ be as in Proposition~\ref{prop:16.2.3 short}, with ``$C_E=C_K$'' replaced by
``$C_K$ is archimedean and $C_L=\R$".  Then $i$ extends to an embedding~$K\to L$.
\end{cor}
\begin{proof}
The ordered field embedding $i|_{C_E}\colon C_E\to C_L=\R$ extends uniquely to an ordered field embedding
$j\colon C_K\to C_L$. Now argue as in the proof of [ADH, 16.2.4], using Proposition~\ref{prop:16.2.3 short} in place of [ADH, 16.2.3].
\end{proof}

\noindent
Proposition~\ref{prop:16.2.3 short} leads to a version of   Lemma~\ref{iso1 variant, 1} for short closed $H$-fields:

\begin{lemma}\label{lem:embed short H-fields}
Let $K$ be a  closed short $H$-field with small derivation and archimedean constant field, and $L$  a closed $\eta_1$-ordered $H$-field with small derivation and $C_L=\R$. Then $K$ embeds into $L$.
\end{lemma}
\begin{proof}
Take $x\in K$ with $x'=1$. Now $K$ has small derivation,  so~${x\succ 1}$,  the $H$-field~$C_K(x)$ is grounded, and we have an
embedding $i\colon C_K(x)\to L$ extending the unique ordered field embedding $C_K\to C_L$.  By  [ADH, 10.6.23]  we have a Liouville closure $E$ of~$C_K(x)$ in $K$ and $i$ extends to an embedding $E\to L$. Moreover, $E$ is $\upo$-free, by \cite[Lemma~1.4.18]{ADH2}, so we can use
Proposition~\ref{prop:16.2.3 short}.
\end{proof}

\noindent
With $K=\T$ and $L$ a maximal analytic Hardy field in Lemma~\ref{lem:embed short H-fields}  we conclude:

\begin{cor}\label{tan}
The ordered differential field $\T$ is isomorphic over $\R$ to an analytic Hardy field containing $\R$.
\end{cor}

\noindent
We upgrade this as follows:

\begin{cor}\label{cor:embed T}
Let $E$ be a pre-$H$-subfield of $\T$, $M$ be a maximal Hardy field, and~${i\colon E\to M}$ be an embedding.
Then $i$ extends to an embedding $\T\to M$. Likewise with ``maximal analytic'' and with ``maximal smooth'' instead of ``maximal''.
\end{cor}
\begin{proof}
Expand $E$, $M$ (uniquely) to pre-$\HLO$-fields $\mathbf E$, $\mathbf M$, respectively, such that $i$ is an embedding $\mathbf E\to\mathbf M$, and expand $\T$ (uniquely) to a pre-$\HLO$-field $\mathbf T$. Then~${\mathbf E\subseteq\mathbf T}$ by \cite[Lemma~7.1.1, Corollary~7.1.16]{ADH2}. Now [ADH, 16.4.1] yields an $\upo$-free $\HLO$-field~$\mathbf E^*$ with~$\mathbf E\subseteq\mathbf E^*\subseteq\mathbf T$ and an extension of $i$ to an embedding $E^*\to M$,
which in turn extends  to an embedding $\T\to M$ by
Corollary~\ref{cor:16.2.3 short}.
\end{proof}

\begin{remark} If $\hat\T$ is an immediate $H$-field extension of~$\T$, then any
embedding of $\T$ into a maximal Hardy field $M$ extends
to an embedding~$\hat\T\to M$, by Theorem~\ref{thm:embedd imm}.  Likewise for $M$ a maximal smooth or maximal analytic Hardy field. With $\No$ in place of $M$ we can also take strong additivity into account.
To see this   recall from~\cite[Proposition~5.1 and subsequent remarks]{ADH1+}  that the unique strongly additive embedding~$\iota\colon\T\to\No$ over $\R$ of exponential ordered fields which sends $x\in\T$ to~$\omega\in\No$
is also an embedding of differential fields, with~${\iota(\T)\subseteq\No(\omega_1)}$ by
\cite[Pro\-po\-si\-tion~5.2(3)]{ADH1+}.
By \cite[Proposition~5.2(1)]{ADH1+}, $\iota(G^{\operatorname{LE}})=\fM\cap\iota(\T)$, where~$G^{\operatorname{LE}}$ is the group of LE-monomials (cf.~[ADH, p.~718]) and
$\fM$ is the class of monomials in~$\No$ (cf.~\cite[\S{}1]{ADH1+}), hence
$\iota$ extends uniquely to a strongly additive ordered field embedding~$\hat\iota\colon\R[[G^{\operatorname{LE}}]]\to\No$. The derivation of $\No$ is strongly additive, so~${\hat\iota(\R[[G^{\operatorname{LE}}]])=\R[[\iota(G^{\operatorname{LE}})]]}$ is
a differential subfield of~$\No$. The derivation on~$\R[[G^{\operatorname{LE}}]]$
that makes~$\hat\iota$ a differential field embedding is
then the  unique strongly additive  derivation on~$\R[[G^{\operatorname{LE}}]]$ extending the derivation of~$\T$. It also makes $\R[[G^{\operatorname{LE}}]]$ a spherically complete immediate $H$-field extension of $\T$, so the result stated at the beginning of this extended remark applies to the $H$-field $\R[[G^{\operatorname{LE}}]]$ in the role of $\hat\T$.
\end{remark}

%The next question is suggested by the proof of Proposition~\ref{prop:16.2.3 short}:

%For each   $H$-subfield $E$ of $\T$ such that $\R\subseteq E$ and~${y\in\T\setminus E}$, is
%the $\R$-linear space
%$\R\Gamma_{E\langle y\rangle}/\R\Gamma_E$ finite-dimensional?
%(See \cite[Proposition~3.13]{AD2} for a finiteness property of differential sub\-fields~${E\supseteq\R}$ of $\T$
%which are finitely generated over $\R$.)

\section{Some Set-Theoretic Issues}

\noindent
We finish with some questions of a set-theoretic nature that others might be better prepared to answer. We assume our base theory ZFC is consistent, and these are questions about relative consistency with ZFC.

\begin{enumerate}
\item Is it consistent that there are non-isomorphic maximal Hardy fields?
\item Is it consistent that no maximal Hardy field is isomorphic to $\No(\omega_1)$?
\item Is it consistent that there is a complete maximal Hardy field?
% equipped with the derivation $\der_{BM}$?
\end{enumerate}
Positive answers would mean (at least) that we cannot drop the assumption CH in some results we proved under this hypothesis.
Note also that with CH we have~$\cf(H)=\ci(H^{>\R})=\omega_1$ for all maximal Hardy fields. This suggests:

\begin{enumerate}
\item[(4)]   Is it consistent that $\cf(H_1)\ne \cf(H_2)$ for some maximal Hardy fields~$H_1$,~$H_2$?
Same with $\ci(H_i^{>\R})$ instead of $\cf(H_i)$.
\item[(5)]  Is it consistent that $\cf(H)\ne \ci(H^{>\R})$ for some maximal Hardy field $H$?
\item[(6)] Is it consistent that there is a maximal Hardy field $H$ and a gap in $H$
of character $(\alpha,\beta^*)$ with $\alpha,\beta\geq\omega$, not equal to one of
$(\omega,\kappa^*)$, $(\kappa,\omega^*)$,   $(\kappa,\kappa^*)$, $(\lambda,\lambda^*)$,
where $\kappa:=\ci(H^{>\R})$, $\lambda:=\cf(H)$?
\end{enumerate}
One can also ask these questions for maximal analytic Hardy fields and maximal smooth Hardy fields instead of maximal Hardy fields. We can even ask them for maximal Hausdorff fields (containing at least $\R$, say) instead of maximal Hardy fields.
As with Corollary~\ref{CHcof}, might some weaker assumption like $\mathfrak{b}=\mathfrak{d}$ be enough for some results where we assumed CH?

If $H$ is a Hardy field with $H^{>\R}$ closed under compositional inversion, then
$$h\mapsto h^{\inv}\colon H^{>\R}\to H^{>\R}$$ is a strictly decreasing bijection, so $\operatorname{cf}(H)=\operatorname{ci}(H^{>\R})$. However, we don't know if there is a maximal Hardy field $H$ with $H^{>\R}$ closed under
compositional inversion.

% Next a more model-theoretic issue: \begin{enumerate}
%\item[(5)] Specify a Scott sentence for maximal Hardy fields.
%We finish with a problem specific to the present paper: \begin{enumerate}
%\item[(6)] Are all or some maximal analytic Hardy fields also maximal Hardy fields?
%Again, there are variants, for example, are all or some maximal Hardy fields also maximal as Hausdorff fields?

\addappendix[A Proof of Whitney's Approximation Theorem]

\noindent
For the convenience of the reader, we include here a proof of Theorem~\ref{thm:WAP dim 1, general},
adapting the exposition in \cite[\S{}1.6]{Nara}.
Throughout this appendix
$r\in\N\cup\{\infty\}$ and~$a,b\in \R$.

\medskip
\noindent
Recall that the support $\supp f$ of a function~$f\colon \R\to \R$ is the closure in
$\R$ of the set~${\big\{t\in \R:\, f(t)\ne0\big\}}$.
We begin with two lemmas,
where $f\in\Cc^m(\R)$ is such that~$\supp f$ is bounded; let also $\lambda$ range over $\R^>$.
From the Gaussian integral~$\int_{-\infty}^\infty \ex^{-s^2}\,ds=\pi^{1/2}$
we get~$(\lambda/\pi)^{1/2}\int_{-\infty}^\infty \ex^{-\lambda s^2}\,ds=1$.
Consider~${f_\lambda\colon\R\to\R}$ given by
\begin{equation}\label{eq:flambda}
f_\lambda(t) := (\lambda/\pi)^{1/2}\int_{-\infty}^\infty f(s)\ex^{-\lambda (s-t)^2}\,ds.
\end{equation}
Note that  we could have replaced here the bounds $-\infty$, $\infty$ in this integral by any~$a$,~$b$    such that $\supp(f)\subseteq [a,b]$. A change of variables gives
$$f_\lambda(t)\  =\  (\lambda/\pi)^{1/2}\int_{-\infty}^\infty f(t-s)\ex^{-\lambda s^2}\,ds.$$
As in \cite[(8.12), Exercise~2(b)]{D} one   obtains that
$f_\lambda\in\Cc^\infty(\R)$   and
for $k\leq m$:
(\lambda/\pi)^{1/2}\int_{-\infty}^\infty f^{(k)}(s) \ex^{-\lambda (s-t)^2}\,ds=
(\lambda/\pi)^{1/2}\int_{-\infty}^\infty f^{(k)}(t-s) \ex^{-\lambda s^2}\,ds.$$

\begin{lemma}\label{lem:WAP aux 1}
$f_\lambda$ extends to an entire function; in particular,
\end{lemma}
\begin{proof}
Take $a< b$  such that $\supp f\subseteq[a,b]$ and consider $g\colon [a,b]\times\mathbb C\to \mathbb C$
given by~$g(s,z):=f(s)\ex^{-\lambda(s-z)^2}$. Then $g$ is continuous,  for each $s\in [a,b]$ the function~$g(s,{-})\colon\mathbb C\to\mathbb C$ is analytic,  and $\partial g/\partial z\colon [a,b]\times\mathbb C\to\mathbb C$ is
continuous. Hence~$z\mapsto \int_a^b g(s,z)\,ds\colon\mathbb C\to\mathbb C$ is analytic by \cite[(9.10), Exercise~3]{D}.
\end{proof}

\begin{lemma}\label{lem:WAP aux 2}
$\dabs{f_\lambda-f}_m\to 0$ as $\lambda\to\infty$.
\end{lemma}
\begin{proof}
For $k\leq m$ we have
\begin{align*}
f_\lambda^{(k)}(t)-f^{(k)}(t)\  &=\  (\lambda/\pi)^{1/2}\int_{-\infty}^\infty \big(f^{(k)}(t-s)-f^{(k)}(t)\big)\ex^{-\lambda s^2}\,ds \\
&=\ (\lambda/\pi)^{1/2}\int_{-\infty}^\infty \big(f^{(k)}(s)-f^{(k)}(t)\big)\ex^{-\lambda (s-t)^2}\,ds.
\end{align*}
Let $\varepsilon\in\R^>$ be given, and choose $\delta>0$ such that
$$\abs{f^{(k)}(s)-f^{(k)}(t)} \leq \varepsilon/2\quad\text{whenever $\abs{s-t}\leq \delta$ and $k\leq m$.}$$
For $s\le t-\delta$ and for $s\ge t+\delta$ we have  $\ex^{-\lambda(s-t)^2}\le \ex^{-(\lambda/2)\delta^2} \ex^{-(\lambda/2)(s-t)^2}$, so
\begin{align*}
\int_{-\infty}^{t-\delta} \ex^{-\lambda(s-t)^2}\,ds + \int_{t+\delta}^{\infty} \ex^{-\lambda(s-t)^2}\,ds &\ \leq\
\ex^{-(\lambda/2)\delta^2}\int_{-\infty}^\infty  \ex^{-(\lambda/2)(s-t)^2}\,ds \\ &\ =\ \ex^{-(\lambda/2)\delta^2} (2\pi/\lambda)^{1/2}.
\end{align*}
Set $M:=\dabs{f}_{m}\in\R^\geq$. For $k\leq m$ we have
\int_{-\infty}^\infty \big(f^{(k)}(s)-f^{(k)}(t)\big)\ex^{-\lambda (s-t)^2}\,ds =
\int_{-\infty}^{t-\delta}(\dots)\,ds +
\int_{t-\delta}^{t+\delta} (\dots)\,ds +
\int^{\infty}_{t+\delta}  (\dots)\,ds,$$
\begin{align*}
\abs{f_\lambda^{(k)}(t)-f^{(k)}(t)} &\ \leq\  (\lambda/\pi)^{1/2} \left(
M \int_{-\infty}^{t-\delta} \ex^{-\lambda(s-t)^2}\,ds + {} \right. \\
&\qquad\qquad\qquad\quad \left.  (\varepsilon/2)\int_{-\infty}^\infty \ex^{-\lambda(s-t)^2}\,ds + M \int^{\infty}_{t+\delta} \ex^{-\lambda(s-t)^2}\,ds\right) \\
&\ \leq \ (\varepsilon/2)+\sqrt{2} M \ex^{-(\lambda/2)\delta^2}.
\end{align*}
Thus if $\lambda$ is so large that $\sqrt{2} M \ex^{-(\lambda/2)\delta^2}\leq \varepsilon/2$, then $\dabs{f_\lambda-f}_m\leq\varepsilon$.
\end{proof}

\noindent
In the next lemma we let   $U\subseteq\R$ be nonempty and open and let
$K$ range over nonempty compact subsets of $U$ and $m$ over the natural numbers~$\leq r$.

\begin{lemma}\label{lem:WAP, cauchy}
Let  $(f_n)$ be a sequence in $\Cc^r(U)$ which,
for all $K$, $m$,  is a cauchy sequence with respect to $\dabs{\,\cdot\,}_{K;\,m}$. Then there exists $f\in\Cc^r(U)$ such that for all~$K$,~$m$
we have $\dabs{f_n-f}_{K;\,m}\to 0$ as $n\to \infty$.
\end{lemma}
\begin{proof}
For all $K$, $m$, $(f_n^{(m)})$ is a cauchy sequence with respect to $\dabs{\,\cdot\,}_K$. Hence for each $m$ we obtain an $f^m\in\Cc(U)$ such that for all $K$, $\dabs{f_n^{(m)}-f^m}_K\to 0$ as~${n\to\infty}$; cf.~\cite[(7.2.1)]{D}.
Set $f:=f^0$. By induction on $m\leq r$ we show that~${f\in\Cc^m(U)}$ and~$f^{(m)}=f^m$. This is clear for $m=0$,
so suppose $0<m\leq r$ and $f\in\Cc^{m-1}(U)$, $f^{(m-1)}=f^{m-1}$.
Let $a\in U$, and take~$\varepsilon>0$   such that~$K:=[a-\varepsilon,a+\varepsilon]\subseteq U$.
Let~$t\in K\setminus\{a\}$.
Then for each $n$ we have $s_n$ with $\abs{a-s_n}\leq \abs{a-t}$   such that
$$f_n^{(m-1)}(t)-f_n^{(m-1)}(a)\ =\ f_n^{(m)}(s_n)\cdot (t-a).$$
Take a subsequence $(s_{n_k})$ of $(s_n)$ and $s=s(t)$ with~$\lim\limits_{k\to\infty} s_{n_k} = s$.
Then~${\abs{a-s}}\leq\abs{a-t}\le \varepsilon$ and
$$\lim_{k\to\infty} \big(f_{n_k}^{(m-1)}(t)-f_{n_k}^{(m-1)}(a)\big) = f^{m-1}(t)-f^{m-1}(a) = f^{(m-1)}(t)-f^{(m-1)}(a)$$
and $\lim\limits_{k\to\infty} f_{n_k}^{(m)}(s_{n_k})=f^m(s)$, since $\lim\limits_{n\to\infty}\dabs{f_n^{(m)}-f^m}_{K} = 0$. Hence
$$f^{(m-1)}(t)-f^{(m-1)}(a)=f^m(s)\cdot (t-a)$$
where $f^m\big(s(t)\big)\to f^m(a)$   as $t\to a$, since $f^m$ is continuous at $a$.
\end{proof}

\noindent
We   now prove Theorem~\ref{thm:WAP dim 1, general}.
Let $(a_n)$, $(b_n)$, $(\varepsilon_n)$ be sequences in $\R$ and $(r_n)$
in~$\N$ such that~$a_0= b_0$, $(a_n)$ is strictly decreasing, $(b_n)$ is strictly increasing, and $\varepsilon_n>0$, $r_n\le r$ for all~$n$. Set $I:=\bigcup_n K_n$, where $K_n:=[a_n, b_n]$, and let $f\in \Cc^r(I)$.
\textit{We need to show the existence of
a~$g\in\Cc^\omega(I)$ such that
$\dabs{{f-g}}_{K_{n+1}\setminus K_n;\,r_n}<\varepsilon_n$ for each~$n$.}\/
Replacing  $\varepsilon_n$ by $\min\{\varepsilon_n, \frac{1}{n+1}\}$ and  $r_n$ by $\max\{r_0,\dots,r_n\}$
we first arrange that $\varepsilon_n\to 0$
as~$n\to\infty$ and  $r_n\leq r_{n+1}$ for all $n$. Set
%Let $K_{-1}:=[a_{-1},b_{-1}]$ where $a_0<a_{-1}<b_{-1}<b_0$ are arbitrary and
$$L_n\ :=\ K_{n+1}\setminus K_n\ =\ [a_{n+1},a_n)\cup(b_n,b_{n+1}],$$
and take $\varphi_n\in\Cc^\infty(\R)$   such that $\varphi_n=0$ on a neighborhood of $K_{n-1}$ (satisfied automatically for $n=0$, by convention),
$\varphi_n=1$ on a neighborhood of~$\operatorname{cl}(L_n)=[a_{n+1},a_n]\cup[b_n,b_{n+1}]$, and~$\supp\varphi_n\subseteq K_{n+2}$. For example, for $n\ge 1$, $\alpha_{a,b}\in\Cc^\infty(\R)$  as in
\cite[\eqref{eq:alphaab}]{ADHfgh},  and sufficiently small positive~$\varepsilon=\varepsilon(n)$, set
\alpha_{a_{n+2}+\varepsilon,a_{n+1}-\varepsilon}(t) &\text{if $t\leq a_n$,} \\
1-\alpha_{a_{n}+\varepsilon,a_{n-1}-\varepsilon}(t) &\text{otherwise.}
\end{cases}$$
\alpha_{b_{n-1}+\varepsilon,b_n-\varepsilon}(t) &\text{if $t\leq b_n$,} \\
1-\alpha_{b_{n+1}+\varepsilon,b_{n+2}-\varepsilon}(t) &\text{otherwise.}
\end{cases}$$
and put $\varphi_n:=\alpha_n+\beta_n$.  (See Figure~\ref{fig:beta_n}.)

\begin{figure}[ht]
\begin{tikzpicture}
\def\anmone{1.5};\def\an{3.5};\def\anplone{5.4};\def\anpltwo{6.8};\def\epsl{0.5};
\begin{axis} [axis lines=center, xmin=-0.2, xmax=8, ymin = -0.01, ymax = 0.15, width=0.9\textwidth, height = 0.4\textwidth, xlabel={$t$}, xtick={\anmone,\an,\anplone,\anpltwo}, ytick=\empty, xticklabels={\strut $b_{n-1}$, \strut $b_n$, \strut $b_{n+1}$, \strut $b_{n+2}$},
\addplot [domain=0.25:\anmone+\epsl, smooth, very thick] {0};

\draw[dotted] (axis cs: \anmone+\epsl,-0.25) -- (axis cs: \anmone+\epsl,2);
\draw[dotted] (axis cs: \an-\epsl,-0.25) -- (axis cs: \an-\epsl,2);
\draw[dotted] (axis cs: \anplone+\epsl,-0.25) -- (axis cs: \anplone+\epsl,2);
\draw[dotted] (axis cs: \anpltwo-\epsl,-0.25) -- (axis cs: \anpltwo-\epsl,2);

\addplot [domain=\an-\epsl:\anplone+\epsl, smooth, very thick] {0.1};
\addplot [domain=\anplone+2*\epsl:7.5, smooth, very thick] {0};

\def\transanmonean{(6.5*(((2*(x-\an+\epsl)/((\an)-(\anmone)-2*\epsl))+1)))};
\def\transanplone{(6.5*(((2*(x-\anpltwo+\epsl)/((\anpltwo)-(\anplone)-2*\epsl))+1)))};

\addplot [domain=\anmone+\epsl:\an-\epsl, smooth, very thick] { 0.1/(1+exp(-\transanmonean)) };
\addplot [domain=\anplone+\epsl:\anplone+2*\epsl, smooth, very thick] { 0.1/(1+exp(\transanplone)) };
\node[right] at (axis cs:\an+0.3,0.11) {$\beta_n$};

\end{axis}
\end{tikzpicture}
\caption{The hump function $\beta_n$}\label{fig:beta_n}
\end{figure}

\medskip
\noindent
With $M_n:=1+2^{r_n}\dabs{\varphi_n}_{r_n}$,   choose~$\delta_n\in\R^>$ so that for all $n$,
\begin{equation}\label{eq:WAP, 0}
2\delta_{n+1} \leq \delta_n,\qquad \sum_{m = n}^\infty \delta_m M_{m+1}\leq \varepsilon_n/4
\end{equation}
Given $g\in\Cc(\R)$ with  bounded support and $\lambda\in\R^>$,  let $I_\lambda(g):=g_\lambda$
be as in \eqref{eq:flambda}, with $g$ in place of $f$. Next, let $f\in \Cc^r(I)$ be given. Then
we inductively define   sequences $(\lambda_n)$ in~$\R^>$ and~$(g_n)$ in~$\Cc^\omega(\R)$ as follows: Let $\lambda_m\in \R^{>}$ and $g_m\in \Cc^{\omega}$ for~$m<n$; then
consider the function $h_n\in\Cc^r(\R)$ given by
$$h_n(t)\ :=\ \begin{cases} \varphi_n(t)\cdot \big(f(t)-\big(g_0(t)+\cdots+g_{n-1}(t)\big)\big)
& \text{if $t\in I$,} \\
0 & \text{otherwise.}\end{cases}$$
Thus    $\supp h_n\subseteq\supp\varphi_n$ is bounded. Put
$g_n:=I_{\lambda_n}(h_n)\in\Cc^\omega(\R)$ where we take~$\lambda_n\in\R^>$ such that~$\dabs{g_n-h_n}_{r_n}<\delta_n$ (any sufficiently large $\lambda_n$ will do, by Lemma~\ref{lem:WAP aux 2}).
So $\dabs{g_{n+1}-h_{n+1}}_{K_{n};\,r_{n+1}}<\delta_{n+1}$, and since $\varphi_{n+1}$ and thus also $h_{n+1}$ vanish on a neighborhood of~$K_{n}$, this yields
\begin{equation}\label{eq:WAP, 1}
\dabs{g_{n+1}}_{K_{n};\, r_{n+1}}<\delta_{n+1}.
\end{equation}
Likewise, since $\varphi_n=1$ on a neighborhood of $\operatorname{cl}(L_n)$,
\begin{equation}\label{eq:WAP, 2}
\dabs{f-(g_0+\cdots+g_n)}_{L_n;\, r_n}<\delta_n.
\end{equation}
Also $$\dabs{g_{n+1}-h_{n+1}}_{L_n;\,r_{n}}\leq\dabs{g_{n+1}-h_{n+1}}_{r_{n+1}}<\delta_{n+1},$$ and thus by \eqref{eq:prod norm} and
\eqref{eq:WAP, 2}:
\begin{align*}
\dabs{g_{n+1}}_{L_n;\,r_{n}}\ 	&\ \leq\ \dabs{g_{n+1}-h_{n+1}}_{L_n;\,r_{n}}+\dabs{\varphi_{n+1}\cdot(f-(g_0+\cdots+g_n))}_{L_n;\,r_{n}}
\\
&\ \leq\  \delta_{n+1} + 2^{r_n}\dabs{\varphi_{n+1}}_{L_n;\,r_{n}} \cdot \dabs{f-(g_0+\cdots+g_n)}_{L_n;\,r_{n}} \\
&\ \leq\ \delta_{n+1}+ M_{n+1}\delta_n.
\end{align*}
Moreover, by \eqref{eq:WAP, 1} and $r_n\leq r_{n+1}$ we have $\dabs{g_{n+1}}_{K_n;\, r_n}<\delta_{n+1}$.
Hence by \eqref{eq:WAP, 0}:
\begin{equation}\label{eq:WAP, 3}
\dabs{g_{n+1}}_{K_{n+1};\,r_n}\ \leq\ \delta_{n+1}+ M_{n+1}\delta_n+\delta_{n+1}\ \leq\ M_{n+1}\delta_n+\delta_n\ \leq\ 2\delta_n M_{n+1}.
\end{equation}
Let $K\subseteq I$ be nonempty and compact, and let $m\leq r_n$ for some $n$.
We  claim that~$({g_0+\cdots+g_i})$ is a cauchy sequence with respect to $\dabs{\,\cdot\,}_{K;\,m}$.
To see this, let~$\varepsilon\in\R^>$ be given, and
take~$n$ such that $K\subseteq K_{n+1}$ and $m\leq r_n$.
% and $\varepsilon_i<\varepsilon$ for all $i\ge n$.
Then by
\eqref{eq:WAP, 0} and
\eqref{eq:WAP, 3} we have  for~$j > i\geq n$:
\begin{align*}
\dabs{g_{i+1}+\cdots+g_j}_{K;\,m} &\ \leq\
\dabs{g_{i+1}}_{K;\,m}+\cdots+\dabs{g_j}_{K;\,m} \\
&\ \leq\ \dabs{g_{i+1}}_{K_{i+1};\,r_i}+\cdots+\dabs{g_j}_{K_{j};\,r_{j-1}} \\
&\ \leq\ 2\delta_iM_{i+1}+\cdots+2\delta_{j-1}M_j\ \leq\ \varepsilon_i/2.
\end{align*}
So Lemma~\ref{lem:WAP, cauchy}  yields a function $g\colon I\to\R$ such that
$g(t)=\sum\limits_{i=0}^\infty g_i(t)$ for all~$t\in I$ and $g\in \Cc^{r_n}(I)$ for all $n$.
In the same way, using \eqref{eq:WAP, 0} and \eqref{eq:WAP, 3} and denoting the restriction of $g_i$ to $I$ also by $g_i$, we obtain
$$\dabs{g-(g_0+\cdots+g_n)}_{L_n;\,r_n}\ =\ \left\| \sum_{i=n+1}^\infty g_i\right\|_{L_n;\,r_n}\ \leq\ \varepsilon_n/2$$
and hence by \eqref{eq:WAP, 0} and \eqref{eq:WAP, 2}:
\begin{align*}
\dabs{f-g}_{L_n;\,r_n} &\ \leq\ \dabs{f-(g_0+\cdots+g_n)}_{L_n;\,r_n} +
\dabs{g-(g_0+\cdots+g_n)}_{L_n;\,r_n}\\
&\ \leq\ \delta_n+\textstyle\frac{1}{2}\varepsilon_n\ <\ \varepsilon_n.
\end{align*}
To complete the proof we are going to choose sequences $(g_n)$ and~$(\lambda_n)$ as above
so that $g$ is analytic.
Now for $t\in\R$ we have
$$g_n(t)\ =\ (\lambda_n/\pi)^{1/2}\int_{-\infty}^{\infty} h_n(s)\ex^{-\lambda_n(s-t)^2}ds\ =\ (\lambda_n/\pi)^{1/2}\int_{a_{n+2}}^{b_{n+2}} h_n(s)\ex^{-\lambda_n(s-t)^2}ds$$
and $g_n$ is the restriction to $\R$ of the entire function $\hat g_n$ given by
$$\hat g_n(z)\ =\ (\lambda_n/\pi)^{1/2}\int_{a_{n+2}}^{b_{n+2}} h_n(s)\ex^{-\lambda_n(s-z)^2}ds\qquad (z\in\mathbb C).$$ %(\lambda_n/\pi)^{1/2}\int_{a_{n+2}}^{b_{n+2}}  \varphi_n(s)\cdot \big(f(s)-\big(g_0(s)+\cdots+g_{n-1}(s)\big)\big)\ex^{-\lambda(s-z)^2}ds.
(See the proof of Lemma~\ref{lem:WAP aux 1}.)
Put $$\rho_n\ :=\ \textstyle\frac{1}{2}\min\!\big\{ (a_n-a_{n+1})^2,(b_{n+1}-b_{n})^2\big\}\in\R^>$$ and
$$U_n\ :=\  \big\{z\in \mathbb C:\, a_{n+1}<\Re z<b_{n+1},\
\Re\!\big((z-a_{n+1})^2\big),\  \Re\!\big((z-b_{n+1})^2\big)>\rho_n
\big\},$$
an open subset of $\mathbb C$ containing $K_n$ such that $\Re\!\big((s-z)^2\big)>\rho_n$
for all $z\in U_n$ and~$s\in \R\setminus K_{n+1}$. (Cf.~Figure~\ref{fig:Un}.)

\begin{figure}[ht]
\begin{tikzpicture}
\def\anplone{1.5};\def\an{3.5};\def\bn{5.4};\def\bnplone{6.8};\def\rhon{((1.4^2)/2)};
\begin{axis} [axis lines=center, xmin=-0.2, xmax=8, ymin = -3.15, ymax = 3.15, width=0.9\textwidth, height = 0.5\textwidth, xtick={\anplone,\an,\bn,\bnplone}, ytick=\empty, xticklabels={\strut $a_{n+1}$, \strut $a_{n}$, \strut $b_{n}$, \strut $b_{n+1}$}, xticklabel style = {xshift=0.04\textwidth, yshift=-0.125\textwidth, xlabel={$\Re$}, ylabel={$\Im$}}
\draw[dotted] (axis cs: \anplone,-3.15) -- (axis cs: \anplone,3.15);
\draw[dotted] (axis cs: \an,-3.15) -- (axis cs: \an,3.15);
\draw[dotted] (axis cs: \bn,-3.15) -- (axis cs: \bn,3.15);
\draw[dotted] (axis cs: \bnplone,-3.15) -- (axis cs: \bnplone,3.15);

\addplot [name path=A, domain=\anplone:((\anplone+\bnplone)/2), smooth, dashed, thick] {  ( (x-\anplone)^2 - \rhon )^(1/2) };
\addplot [name path=B, domain=\anplone:((\anplone+\bnplone)/2), smooth, dashed, thick] { - ( (x-\anplone)^2 - \rhon )^(1/2) };

\addplot [name path=C, domain=((\anplone+\bnplone)/2):\bnplone, smooth,  dashed, thick] {   ( (x-\bnplone)^2 - \rhon )^(1/2) };
\addplot [name path=D, domain=((\anplone+\bnplone)/2):\bnplone, smooth, dashed, thick] { - ( (x-\bnplone)^2 - \rhon )^(1/2) };

\addplot[pattern=north west lines] fill between[of=A and B];
\addplot[pattern=north west lines] fill between[of=C and D];

\node[right] at (axis cs:\bnplone+0.3,2.75) {$\mathbb C$};
\node[right] at (axis cs:\anplone+1.5,2.11) {$U_n$};
\end{axis}
\end{tikzpicture}
\caption{The domain $U_n$}\label{fig:Un}
\end{figure}

\medskip
\noindent
We also set
$$H_m\ :=\ 2(\lambda_m/\pi)^{1/2}\,\dabs{h_m}_{K_{m+2}} \,(b_{m+2}-a_{m+2})\in\R^\ge.$$
Recall that $h_m$ only depends on the $g_j$ with $j<m$. Fix a sequence $(c_m)$ of positive reals such that $\sum_m c_m <\infty$. Then we can and do choose the sequences $(g_m)$, $(\lambda_m)$ so that in addition
$$H_m \exp(-\lambda_m/m)\ \leq\  c_m\quad\text{for all $m\ge 1$.}$$
%Take $(c_n)$ ; e.g., $c_n=1/(n+1)^2$.
\begin{equation}\label{eq:cn}
\sum_m H_m\exp(-\lambda_m\rho)\ <\infty\ \qquad\text{for all~${\rho\in\R^>}$.}
\end{equation}
It is enough  that
%with the ensuing  choice of $(g_n)$, $(\lambda_n)$,
for each~$n$ the series~$\sum_m \hat g_m$ converges uniformly on
compact subsets of $U_n$, because then by \cite[(9.12.1)]{D} we have a holomorphic function
$$z\mapsto \sum_m \hat g_m(z)\ :\  U:=\bigcup_n U_n\to\mathbb C$$
whose restriction to $I$ is $g$.
To prove such convergence, fix $n$ and
let $m\geq n+2$. Then $\supp h_m\subseteq   K_{m+2}\setminus K_{m-1}\subseteq
K_{m+2}\setminus K_{n+1}$. Hence $\abs{\hat g_m(z)} \leq H_m\ex^{-\lambda_m\rho_n}$
% for $z\in\mathbb C$:
%$$\hat g_m(z)=
%(\lambda_m/\pi)^{1/2}\left(\int_{a_{m+2}}^{a_{n+1}} h_m(s)\ex^{-\lambda_m(s-z)^2}ds+
% \int_{b_{n+1}}^{b_{m+2}} h_m(s)\ex^{-\lambda_m(s-z)^2}ds\right).$$
for~${z\in U_n}$.
Together with \eqref{eq:cn}
%and the Weierstrass $M$-Test (see, e.g., \cite[p.~134]{D}),
this now yields that $\sum_m \hat g_m$ converges uniformly on compact subsets of $U_n$. \qed

\medskip

\noindent
In the remainder of this appendix we discuss how to control the domain of the holomorphic function~$\hat g$
in the proof of Theorem~\ref{thm:WAP dim 1, general}; this leads to   improvements of Corollaries~\ref{cor:WAP dim 1} and~\ref{cor:WAP dim 1, 1} which might be useful elsewhere: Corollaries~\ref{cor:WAP dim 1, controlled} and~\ref{cor:WAP dim 1, 1, controlled} below. %(These remarks are  not used later.)
For the next corollary we are in the setting of that theorem and~$f\in \Cc^r(I)$. With~$\alpha\in \R\cup\{-\infty\}$ and~$\beta\in \R\cup\{+\infty\}$ such that~$I=(\alpha,\beta)$,   put
$$V\ :=\ \big\{z\in\mathbb C:\  \Re(z)\in I,\ \abs{\Im z} < \Re(z)-\alpha,\,\beta-\Re(z)\big\},$$
an open subset of $\mathbb C$ containing $I$.

\begin{cor}\label{cor:WAP dim 1, general, controlled}
Suppose   $a_n-a_{n+1}\to 0$ and $ b_{n+1}-b_n\to 0$ as $n\to \infty$.
%$\limsup\limits_{n\to\infty}\operatorname{dist}(K_{n+1},K_n) = 0$.
Then there is a holomorphic $\hat g\colon V\to\mathbb C$, real-valued on $\R$,
such that $g:=\hat g|_I\in\C^\omega(I)$ satisfies
$$\dabs{{f-g}}_{K_{n+1}\setminus K_n;\,r_n}<\varepsilon_n, \text{  for all~$n$}.$$
\end{cor}
\begin{proof}
It suffices to show that the open set $U\subseteq\mathbb C$ in   the proof of Theorem~\ref{thm:WAP dim 1, general} contains $V$. Note that $\rho_n\to 0$ as $n\to \infty$.
%Note that $\rho_n=\frac{1}{2}\operatorname{dist}(K_{n+1},K_n)^2$.
Let $z=x+y\imag\in V$ ($x,y\in\R$). Then
%in $\R_{\pm\infty}$:
$$ (x-a_{n+1})^2-y^2-\rho_n \to (x-\alpha)^2-y^2 > 0 \text{ as }n\to \infty,$$
%$$\liminf_{n\to+\infty} \big( (x-a_{n+1})^2-y^2-\rho_n \big) \geq (x-\alpha)^2-y^2 > 0,$$
and thus $\Re\!\big((z-a_{n+1})^2\big)=(x-a_{n+1})^2-y^2 > \rho_n$ for all sufficiently large $n$. Likewise, ${\Re\!\big((z-b_{n+1})^2\big) > \rho_n}$  for all sufficiently large $n$. Therefore $z\in U_n$ for sufficiently large $n$.
\end{proof}

\begin{cor}\label{cor:Carleman}
Suppose $r\in\N$, $f\in\C^r(\R)$, $\varepsilon\in\C(\R)$, and $\varepsilon>0$ on $\R$. Then
there is an entire function $g\colon\mathbb C\to\mathbb C$ such that~${\abs{(f-g)^{(k)}}\le\varepsilon}$ on $\R$ for all $k\leq r$.
\end{cor}
\begin{proof}
Set $b_n:=\log(n+1)$, $a_n:=-b_n$, and $K_n:=[a_n,b_n]$. Then $\bigcup_n K_n=\R$ and~$a_n-a_{n+1}\to 0$ and $ b_{n+1}-b_n\to 0$ as $n\to \infty$. Set $\varepsilon_n:=\min\big\{\varepsilon(t): t\in K_{n+1}\big\}$ and~$r_n:=r$. Then $V=\mathbb C$ and we apply Corollary~\ref{cor:WAP dim 1, general, controlled}.
\end{proof}

\begin{remark}
Corollary~\ref{cor:Carleman} is due to Carleman~[17] for $r=0$,
to Kaplan~[33] for~${r=1}$, and to Hoischen~\cite[Satz~2]{Hoischen} in general; see~\cite[Chapter~VIII, pp.~273--276, 291]{Burckel}. In a similar way, Corollary~\ref{cor:WAP dim 1, general, controlled}  also yields the $\C^\infty$-version of Corollary~\ref{cor:Carleman}  in
\cite[Satz~1]{Hoischen}. For a multivariate version of these facts, see [1].
\end{remark}

\noindent
Given any $a$  we now consider the open sector $V_a$ in the complex plane given by
$$V_a\ :=\ \big\{z\in\mathbb C: \abs{\Im (z)} < \Re(z)-a\big\}\ =\ a+\big\{z\in\mathbb C^\times: -\textstyle\frac{\pi}{4}<\arg z<\frac{\pi}{4}  \big\}.$$

\begin{cor} \label{cor:WAP dim 1, controlled}
Let  $f$, $(b_n)$, $(\varepsilon_n)$,  $(r_n)$ be as in  Corollary~\ref{cor:WAP dim 1}. Then there are $a<~b$ and a holomorphic
function $\hat g\colon V_a\to\mathbb C$, real-valued on $\R$,
such that~$g:=\hat g|_{\R^{\geq b}}\in\C^\omega_b$ satisfies~$\dabs{{f-g}}_{[b_n,b_{n+1}];\,r_n}<\varepsilon_n$ for all~$n$.
\end{cor}
\begin{proof}
We first arrange that  $b_{n+1}-b_n\to 0$ as $n\to\infty$.  For this, let  $(b_m^*)$ be the strictly increasing sequence in $\R$ such that
$$\{b_0^*,b_1^*,\dots\}\ =\ \{b_0,b_1,\dots\}\cup\big\{b+\log 1,b+\log 2,\dots\big\},$$
for each $m$, set $\varepsilon_m^*:=\varepsilon_n$, $r_m^*:=r_n$ with $n$ such that $[b_{m}^*,b_{m+1}^*]\subseteq [b_n,b_{n+1}]$, and replace $(b_n)$, $(\varepsilon_n)$,  $(r_n)$  by $(b_m^*)$, $(\varepsilon_m^*)$,  $(r_m^*)$.
Now argue as in the proof of Corollary~\ref{cor:WAP dim 1}, using
Corollary~\ref{cor:WAP dim 1, general, controlled} instead of Theorem~\ref{thm:WAP dim 1, general}.
\end{proof}

\noindent
Now the proof of  Corollary~\ref{cor:WAP dim 1, 1}, using
Corollary~\ref{cor:WAP dim 1, controlled} instead of
Corollary~\ref{cor:WAP dim 1}, gives:

\begin{cor} \label{cor:WAP dim 1, 1, controlled}
Let     $f$, $\varepsilon$ be as in Corollary~\ref{cor:WAP dim 1, 1}.
Then there are $a<b$ and a holomorphic~$\hat g\colon V_a\to\mathbb C$, real-valued on $\R$,
such that~$g:=\hat g|_{\R^{\geq b}}\in\C^\omega_b$
satisfies~${\abs{(f-g)^{(k)}(t)}} < \varepsilon(t)$ for all $t\ge b$ and $k\leq \min\!\big\{r,1/\varepsilon(t)\big\}$.
\end{cor}

\newlength\templinewidth
\setlength{\templinewidth}{\textwidth}
\addtolength{\templinewidth}{-2.25em}

\patchcmd{\thebibliography}{\list}{\printremarkbeforebib\list}{}{}

\let\oldaddcontentsline\addcontentsline% Store \addcontentsline
\renewcommand{\addcontentsline}[3]{\oldaddcontentsline{toc}{section}{References}}

\def\printremarkbeforebib{\bigskip\hskip1em The citation [ADH] refers to our book \\

\hskip1em\parbox{\templinewidth}{
M. Aschenbrenner, L. van den Dries, J. van der Hoeven,
\textit{Asymptotic Differential Algebra and Model Theory of Transseries,} Annals of Mathematics Studies, vol.~195, Princeton University Press, Princeton, NJ, 2017.

\bigskip

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\end{document}


# Phase control of Schwinger pair production by colliding laser pulses

Chitradip Banerjee111Corresponding author.
E-mail address<EMAIL_ADDRESS>(C. Banerjee). Manoranjan P. Singh
Theory and Simulations Lab, HRDS, Raja Ramanna Centre for Advanced Technology,
Indore-452013, India  Homi Bhaba National Institute, Training School Complex,
Anushakti Nagar, Mumbai 400094, India  Alexander M. Fedotov  National Research
Nuclear University MEPhI, Moscow 115409, Russia  Laboratory of Quantum Theory
of Intense Fields, Tomsk State University, Lenin Prospekt 36, 634050, Tomsk,
Russia

(August 27, 2024)

###### Abstract

We study the Schwinger electron-positron pair production by a strong
electromagnetic field of two colliding e-polarized laser pulses with a
relative phase shift $\Psi$. The spatio-temporal distribution of created pairs
is very sensitive to this phase shift and to polarization of the pulses. We
study this dependence in detail and demonstrate how it can be explained in
terms of the underlying invariant field structure of the counterpropagating
focused pulses.

$e^{+}e^{-}$ pair production, electromagnetic field, Lorentz invariants,
electron and positron ultrashort bunches

###### pacs:

12.20.Ds

##  I Introduction

The $e^{+}e^{-}$ pair production from vacuum by strong electromagnetic (EM)
field is a fundamental prediction of quantum electrodynamics (QED)
Berestetskii et al. (2012); Dirac (1930); Ritus and Nikishov (1979); Di Piazza
et al. (2012). Although the process was foreseen theoretically several decades
ago Sauter (1931); Heisenberg and Euler (1936); Schwinger (1951); Volkov
(1935); Narozhnyi and Nikishov (1970); Nikishov (1970a); Brezin and Itzykson
(1970), its experimental verification is still missing because of the
unavailability of an electric field strength comparable to the Schwinger limit
$E_{S}=1.32\times 10^{18}\>\textnormal{V/m}$. Since the probability
$P_{e^{+}e^{-}}$ of pair production from vacuum by a strong electric field of
strength $E_{peak}$ is proportional to $\exp(-\pi E_{S}/E_{peak})$, the
process is exponentially suppressed for $E_{peak}\ll E_{S}$. Exploration of
graphene Allor et al. (2008); Fillion-Gourdeau and MacLean (2015); Fillion-
Gourdeau et al. (2016), an isomorphic $2D$-system in condensed matter physics,
was proposed as a workaround to validate the Schwinger mechanism
experimentally.

The available electric field strength for the present-day laser systems is of
the order of $E_{peak}\sim 10^{13}-10^{14}\>\textnormal{V/m}$ Mourou et al.
(2006); Yanovsky et al. (2008), considerably below the critical field limit
$E_{S}$. However, recent advances in technologies of ultrashort and
ultraintense laser pulse generation Tajima and Mourou (2002) raise the hopes
that in a foreseeable future the available laser intensity may closer approach
the elusive threshold of pair production. Moreover, such nonlinear QED effects
as $e^{+}e^{-}$-pair photoproduction by a hard photon Burke et al. (1997);
Bell and Kirk (2008); Nerush et al. (2011) and the nonlinear (multiphoton)
Compton scattering have been already observed experimentally at laser
intensity $I=10^{22}\textnormal{W/m}^{2}$ Bula et al. (1996). These
developments renewed interest in theoretical studies of pair production by
intense optical lasers.

On the other hand, using the realistic focused field models, e.g. a weakly
focused field in paraxial approximation Narozhny and Fofanov (2000), tightly
focused field models Fedotov (2009); Salamin et al. (2002), and the optimally
focused field model of e-dipole pulses Su et al. (2012a, b), it was
demonstrated that pair production can take place even at intensities
substantially lower than the critical intensity
$I_{S}=\frac{c}{4\pi}E_{S}^{2}$ (here $c$ denotes the light velocity in
vacuum). Superposition of laser pulses in a counterpropagating configuration
has been shown Bulanov et al. (2006); Bulanov et al. (2010a); Hebenstreit et
al. (2010); Bulanov et al. (2005); Bulanov (2004); Wöllert et al. (2015);
Gonoskov et al. (2012) to lower the threshold value of the required field
strength considerably Bulanov et al. (2006). Such beam configurations were
extensively used to study various aspects of pair production, including the
dynamics of post production Kim and Page (2002); Kim and Lee (2007); Kim and
Page (2006); Dunne (2009); Kim et al. (2008); Hwang and Kim (2009);
Hebenstreit and Fillion-Gourdeau (2014); Chervyakov and Kleinert (2009); Gies
and Klingmüller (2005); Kleinert et al. (2008); Gonoskov et al. (2012); Su et
al. (2012a); Abdukerim et al. (2013); Hebenstreit et al. (2009); Dumlu and
Dunne (2010); Gies and Torgrimsson (2016); Dumlu (2010); Wöllert et al.
(2015); Kohlfürst et al. (2013); Bulanov et al. (2005); Banerjee and Singh
(2017, 2018).

To the best of our knowledge, previous studies of pair production by
counterpropagating laser pulses in realistic 3D setup almost never considered
the effect of phase shift between the colliding pulses. However, it has been
shown recently in Ref. Banerjee and Singh (2018) that for a focused linearly
polarized standing wave the invariant electric field distribution, and hence
also pair production, are sensitive to the carrier envelope phase (CEP)
$\tilde{\varphi}$. Here we study the spatio-temporal distribution of
$e^{+}e^{-}$ pairs created via the Schwinger mechanism at a focal region of
the colliding laser pulses described by the Narozhny-Fofanov model Narozhny
and Fofanov (2000), assuming that the pulses are in addition mutually phase
shifted. As we demonstrate, the phase shift $\Psi$ considerably affects the
longitudinal spatial (here, z-) coordinate and time distributions of the
resulting EM field, especially for ultrashort (few cycle) laser pulses.
Furthermore, we study the dependence of the invariant field structure and of
the distribution of the created pairs on polarization, relative sense of
rotation (for circular polarization), and CEP of the counterpropagating
pulses.

The paper is organized as follows: in Sec. II we briefly discuss the basic
theory: the Schwinger formula for average pair production and the structure of
the invariant electric and magnetic fields of the coherently superposed
counterpropagating focused laser pulses. Next we present the differential pair
production rates in spatiotemporal coordinates and explanation of their
features in terms of the invariant electric field distribution in Sec. III,
finally concluding in Sec. IV. A technically useful simplification of the
envelope of counterpropagating pulses is discussed in Appendix A.

##  II Theory

###  II.1 Methodology for calculating $e^{+}e^{-}$ pair production

Assuming the validity of a locally constant field approximation, the average
number of created pairs per unit time and volume can be calculated using the
Nikishov formula Bulanov et al. (2010b); Nikishov (1970b):

$w_{e^{-}e^{+}}=\frac{d^{2}N_{e^{-}e^{+}}}{dVdt}=\frac{e^{2}E_{S}^{2}}{4\pi^{2}\hbar^{2}c}\epsilon\eta\coth\left(\frac{\pi\eta}{\epsilon}\right)\exp\left(-\frac{\pi}{\epsilon}\right),$
(1)

where $e$ is the magnitude of the electron charge, and
$\epsilon,\,\eta=\sqrt{\sqrt{{\mathcal{F}}^{2}+{\mathcal{G}}^{2}}\pm\mathcal{F}}$
[with $\mathcal{F}=\frac{1}{2}\Big{(}\textbf{E}^{2}-\textbf{H}^{2}\Big{)}$ and
$\mathcal{G}=\textbf{E}\cdot\textbf{H}$] are the normalized (by $E_{S}$)
invariant electric and magnetic field strengths, i.e. the magnitudes of the
electric and magnetic field strengths in a reference frame where they are
locally either zero or parallel. Eq. (1) is valid in a locally constant field
approximation based on an assumption that the characteristic length and time
scales of the $e^{+}e^{-}$ pair production process (the Compton length
$\hbar/{m_{e}c}$ and time $\hbar/{m_{e}c^{2}}$ scales) are much smaller than
the carrier wavelength ($\lambda\approx 1\mu\textnormal{m}$) and the period
($\lambda/c\approx 3\textnormal{fs}$) of the laser field, respectively Bulanov
et al. (2010b). In particular, pair production is negligible if $\epsilon$ is
small or vanishing in a focal region, while in the opposite case of nearly
vanishing $\eta$ Eq. (1) reduces to

$w_{e^{-}e^{+}}=\frac{d^{2}N_{e^{-}e^{+}}}{dVdt}\approx\frac{e^{2}E_{S}^{2}}{4\pi^{3}\hbar^{2}c}{\epsilon}^{2}\exp\left(-\frac{\pi}{\epsilon}\right).$
(2)

As we will see later on, these special cases are realized for the magnetic and
electric regimes in the focal region for collision of linearly polarized laser
pulses. Hence we use Eq. (2) for presenting the numerical results of
differential particle production rates in spatiotemporal coordinates for
linearly polarized laser pulses and Eq. (1) otherwise. To obtain a temporal
particle distribution we integrate $w_{e^{+}e^{-}}$ over the spatial
coordinates, and to obtain the longitudinal spatial distribution of particle
production we integrate the production rate $w_{e^{+}e^{-}}$ over the
transverse spatial coordinates and time. The actual distribution of the
invariant fields in a focal region of colliding pulses strongly depends on
their polarization and is discussed below. In all numerical calculations, we
use the exact expressions for the EM fields and assume for definiteness the
amplitude $E_{0}=0.0565$, carrier wavelength $\lambda=1\mu m$, focusing
parameter $\Delta=0.1$, and pulse duration $\tau=10fs$ for each of the
counterpropagating pulses. However, to easier interpret the results, in the
rest of the section we also derive the approximate analytical expressions for
field invariants near the focus.

###  II.2 Invariant fields (linear polarization)

Let us start with a field configuration of linearly polarized
counterpropagating focused laser pulses based on the Narozhny-Fofanov field
model Narozhny and Fofanov (2000). We assume the normalized (by $E_{S}$)
electric fields of the pulses propagating in a forward ($+z$) and backward
($-z$) directions of the form Bulanov et al. (2006)

$\textbf{E}_{f}=iE_{0}e^{-i\omega(t-z/c)-i\tilde{\varphi}}g\Bigg{[}\hat{\textbf{e}}_{x}(F_{1}-F_{2}\cos{2\phi})-\hat{\textbf{e}}_{y}F_{2}\sin{2\phi}\Bigg{]},$
(3)

and

$\textbf{E}_{b}=iE_{0}e^{-i\omega(t+z/c)-i\tilde{\varphi}-i\Psi}g\Bigg{[}\hat{\textbf{e}}_{x}(F_{1}^{*}-F_{2}^{*}\cos{2\phi})-\hat{\textbf{e}}_{y}F_{2}^{*}\sin{2\phi}\Bigg{]},$
(4)

respectively, where $E_{0}$ is the normalized (by $E_{S}$) peak electric field
strength of the laser pulse; $\omega=2\pi c/\lambda$ is the central frequency
of the pulse; $\lambda$ is the laser carrier wavelength; $F_{1},\>F_{2}$ are
the Gaussian-like functions of the form Narozhny and Fofanov (2000)

$\begin{split}F_{1}=\frac{1}{(1+2i\chi)^{2}}\left(1-\frac{\xi^{2}}{1+2i\chi}\right)\exp\left(-\frac{\xi^{2}}{1+2i\chi}\right),\quad
F_{2}=-\frac{\xi^{2}}{(1+2i\chi)^{3}}\exp\left(-\frac{\xi^{2}}{1+2i\chi}\right),\end{split}$

$F_{1}^{*}$ and $F_{2}^{*}$ are their complex conjugates; $\xi=\rho/R$ is the
normalized radial variable with $\rho=\sqrt{x^{2}+y^{2}}$ at the transverse
Cartesian spatial coordinates $x$, $y$; $R$ is the focal radius;
$\phi=\arctan(y/x)$ is the azimuthal angle; $\chi=z/L$ is the normalized
longitudinal coordinate with $L=R/\Delta$ being the Rayleigh length for a
focusing aperture parameter $\Delta=c/{\omega R}$. The envelope function $g$
accounts for temporal finiteness of the laser pulses. In this paper we take
$g=\exp(-4t^{2}/\tau^{2}-4z^{2}/{c^{2}\tau^{2}})$ Banerjee and Singh (2017,
2018) (a detailed explanation of our method of introducing $g$ is given in
Appendix A). Finally, $\tilde{\varphi}$ and $\Psi$ are CEP and the phase shift
of the backward propagating pulse, respectively.

The corresponding expressions for the normalized magnetic field of the forward
and backward propagating pulses are Bulanov et al. (2006)

$\textbf{H}_{f}=iE_{0}e^{-i\omega(t-z/c)-i\tilde{\varphi}}g\Bigg{[}\left(1-i\Delta^{2}\frac{\partial}{\partial\chi}\right)\Big{\\{}\hat{\textbf{e}}_{x}F_{2}\sin{2\phi}-\hat{\textbf{e}}_{y}(F_{1}-F_{2}\cos{2\phi})\Big{\\}}+2i\Delta\sin{\phi}\frac{\partial
F_{1}}{\partial\xi}\hat{\textbf{e}}_{z}\Bigg{]},$ (5)

and

$\textbf{H}_{b}=-iE_{0}e^{-i\omega(t+z/c)-i\tilde{\varphi}-i\Psi}g\Bigg{[}\left(1+i\Delta^{2}\frac{\partial}{\partial\chi}\right)\Big{\\{}\hat{\textbf{e}}_{x}F_{2}^{*}\sin{2\phi}-\hat{\textbf{e}}_{y}(F_{1}^{*}-F_{2}^{*}\cos{2\phi})\Big{\\}}+2i\Delta\sin{\phi}\frac{\partial
F_{1}^{*}}{\partial\xi}\hat{\textbf{e}}_{z}\Bigg{]}.$ (6)

Following the procedure of Ref. Banerjee and Singh (2018), the expressions for
the Lorentz invariants of the resultant EM field
$\textbf{E}=\textbf{E}_{f}+\textbf{E}_{b}$,
$\textbf{H}=\textbf{H}_{f}+\textbf{H}_{b}$ of the superposed
counterpropagating pulses can be derived,

$\begin{split}\mathcal{F}=\frac{1}{2}\Big{(}{Re\textbf{E}}^{2}-{Re\textbf{H}}^{2}\Big{)}\approx\frac{2E_{0}^{2}g^{2}e^{-\frac{2\xi^{2}}{{1+4\chi^{2}}}}}{(1+4\chi^{2})^{2}}\Bigg{[}\sin^{2}(\omega
t+\tilde{\varphi}+\Psi/2)-\sin^{2}(\omega z/c+\Psi/2)\Bigg{]},\end{split}$ (7)

and

$\begin{split}\mathcal{G}=Re\textbf{E}\cdot
Re\textbf{H}^{e}\approx\frac{2E_{0}^{2}g^{2}\xi^{2}e^{-\frac{2\xi^{2}}{{1+4\chi^{2}}}}}{(1+4\chi^{2})^{5/2}}\sin{(2\phi)}\sin{[2(\omega
t+\tilde{\varphi}+\Psi/2)]}\sin{[2(\omega z/c+\Psi/2)]},\end{split}$ (8)

where we retain only the leading order terms in $\Delta$, $\xi$, and $\chi$,
as justified in the focal region in a weak focusing limit. Since
$\mathcal{G}=\mathcal{O}\left(\xi^{2}\right)$ is negligibly small there, one
of the invariant fields (depending on the sign of $\mathcal{F}$) is
vanishingly small. For $\mathcal{F}>0$ we have so-called electric regime
Bashmakov et al. (2014)

$\begin{split}\epsilon_{elec}\approx\frac{2E_{0}ge^{-\frac{\xi^{2}}{{1+4\chi^{2}}}}}{(1+4\chi^{2})}\Bigg{[}\sin^{2}(\omega
t+\tilde{\varphi}+\Psi/2)-\sin^{2}(\omega
z/c+\Psi/2)\Bigg{]}^{1/2},\>\>\>\textnormal{and}\>\>\eta_{elec}\approx
0,\end{split}$ (9)

whereas, for a magnetic regime $\mathcal{F}<0$ Bashmakov et al. (2014)

$\begin{split}\epsilon_{mag}\approx
0,\>\>\>\textnormal{and}\>\>\eta_{mag}\approx\frac{2E_{0}ge^{-\frac{\xi^{2}}{{1+4\chi^{2}}}}}{(1+4\chi^{2})}\Bigg{[}\sin^{2}(\omega
z/c+\Psi/2)-\sin^{2}(\omega
t+\tilde{\varphi}+\Psi/2)\Bigg{]}^{1/2}.\end{split}$ (10)

Clearly, pairs are created solely during an electric regime, and the phases
$\Psi$ and $\tilde{\varphi}$ control toggling between the electric and
magnetic regimes at given point and time, thereby controlling also the pair
production. As is seen from the obtained approximate expressions (and in fact
is also true for the exact ones), it is enough to restrict phases by
$0\leq\tilde{\varphi}<\pi$ and $0\leq\Psi<2\pi$.

Spatiotemporal distributions of the invariant field $\epsilon^{e}$ for few
representative values $\Psi=0,\;\pi/2,\;\pi$ and $\tilde{\varphi}=0,\;\pi/2$
are presented in Fig. 1, where the rhombic structure corresponds to the
aforementioned separation into the alternating electric (color) and magnetic
(dark) regimes. It is clear from the figure, as well as from the above
equations, that the maxima are shifted with respect to the origin $t=z=0$, and
that their shift is determined by the phases. If the maxima are remote from
the origin (which is at the center of the envelope) then their magnitudes are
reduced, in this way they are also indirectly controlled by the phases.

(a) $\leavevmode\nobreak\ \tilde{\varphi}=0,\leavevmode\nobreak\ \Psi=0$ (b)
$\leavevmode\nobreak\ \tilde{\varphi}=0,\leavevmode\nobreak\ \Psi=\pi/2$ (c)
$\leavevmode\nobreak\ \tilde{\varphi}=0,\leavevmode\nobreak\ \Psi=\pi$ (d)
$\leavevmode\nobreak\ \tilde{\varphi}=\pi/2,\leavevmode\nobreak\ \Psi=0$ (e)
$\leavevmode\nobreak\ \tilde{\varphi}=\pi/2,\leavevmode\nobreak\ \Psi=\pi/2$
(f) $\leavevmode\nobreak\ \tilde{\varphi}=\pi/2,\leavevmode\nobreak\ \Psi=\pi$

Figure 1: Spatiotemporal distributions of the invariant electric field
$\epsilon(x=0,y=0,z,t)$ for linearly e-polarized Gaussian laser pulses
colliding with different values of the phases $\tilde{\varphi}$ and $\Psi$.
The laser parameters are $E_{0}=0.0565$, $\Delta=0.1$, $\tau=10fs$, and
$\lambda=1\mu m$.

The figure illustrates a variety of possible opportunities: the maxima can be
located symmetrically about the origin, with either one [see Fig. 1(d)] or a
gap [Figs. 1(a,c,f)] seating at the origin (in the latter case there can be
either four or two maxima closest to it: if there are two then they can be
lined up along either axis); or off centered [like in Figs. 1(b,e)] – in the
latter case the magnitudes of the maxima are lined up according to their
distance from the origin. Qualitatively, the pair production rate behaves the
same way [see Eq. (1) or (2)], hence, as we will see below, this variety of
opportunities precisely corresponds to peculiarities of distribution of
created pairs that we observe in our calculations.

###  II.3 Invariant fields (circular polarization)

Now consider circularly polarized forward and backward propagating laser
pulses with a relative phase difference $\Psi$. We follow closely the steps
discussed in Ref. Banerjee and Singh (2018, 2017). For a forward propagating
(along $+z$) laser pulse, the electric and magnetic fields are given by

$\textbf{E}_{f}=iE_{0}e^{-i\omega(t-z/c)-i\tilde{\varphi}}g\Big{[}F_{1}(\hat{\textbf{e}}_{x}\pm
i\hat{\textbf{e}}_{y})-F_{2}e^{\pm 2i\phi}(\hat{\textbf{e}}_{x}\mp
i\hat{\textbf{e}}_{y})\Big{]},$ (11)

and

$\textbf{H}_{f}=\pm
E_{0}e^{-i\omega(t-z/c)-i\tilde{\varphi}}g\Bigg{[}\left(1-i\Delta^{2}\frac{\partial}{\partial\chi}\right)\Big{[}F_{1}(\hat{\textbf{e}}_{x}\pm
i\hat{\textbf{e}}_{y})+F_{2}e^{\pm 2i\phi}(\hat{\textbf{e}}_{x}\mp
i\hat{\textbf{e}}_{y})\Big{]}+2i\Delta e^{\pm i\phi}\frac{\partial
F_{1}}{\partial\xi}\hat{\textbf{e}}_{z}\Bigg{]},$ (12)

respectively. Here the signs correspond to the right ($+$)- and left ($-$)-
handed rotation of the electric field vector with respect to propagation
direction. For a backward propagating (along $-z$) laser pulse with a relative
phase shift $\Psi$ the expressions for the electric and magnetic fields are
given by

$\textbf{E}_{b}=iE_{0}e^{-i\omega(t+z/c)-i\tilde{\varphi}-i\Psi}g\Big{[}F_{1}^{*}(\hat{\textbf{e}}_{x}\pm
i\hat{\textbf{e}}_{y})-F_{2}^{*}e^{\mp 2i\phi}(\hat{\textbf{e}}_{x}\mp
i\hat{\textbf{e}}_{y})\Big{]},$ (13)

and

$\textbf{H}_{b}=\mp
E_{0}e^{-i\omega(t+z/c)-i\tilde{\varphi}-i\Psi}g\Bigg{[}\left(1+i\Delta^{2}\frac{\partial}{\partial\chi}\right)\Bigg{[}F_{1}^{*}(\hat{\textbf{e}}_{x}\pm
i\hat{\textbf{e}}_{y})+F_{2}^{*}e^{\mp 2i\phi}(\hat{\textbf{e}}_{x}\mp
i\hat{\textbf{e}}_{y})\Big{]}+2i\Delta e^{\mp i\phi}\frac{\partial
F_{1}^{*}}{\partial\xi}\hat{\textbf{e}}_{z}\Bigg{]}.$ (14)

For a pair of counterpropagating circularly polarized pulses, we have two
alternatives: either both pulses have the same polarization (for definiteness
right handed, hereafter referred to as the RR configuration), or opposite
polarizations (for definiteness we assume that the forward propagating pulse
has the right handed polarization and the backward propagating one has the
left handed polarization, hereafter referred to as the RL configuration).

####  II.3.1 RR Configuration

For the RR configuration the Lorentz invariants near the focus ($\xi,\chi\ll
1$) are given by

$\begin{split}\mathcal{F}_{RR}\approx\frac{2E_{0}^{2}g^{2}e^{-\frac{2\xi^{2}}{{1+4\chi^{2}}}}}{(1+4\chi^{2})^{2}}\cos{[2(\omega
z/c+\Psi/2)]},\quad\mathcal{G}_{RR}\approx\frac{2E_{0}^{2}g^{2}e^{-\frac{2\xi^{2}}{1+4\chi^{2}}}}{(1+4\chi^{2})^{2}}\sin{[2(\omega
z/c+\Psi/2)]},\end{split}$ (15)

so that the invariant electric and magnetic fields are

$\begin{split}\epsilon_{RR}\approx\frac{2E_{0}ge^{-\frac{\xi^{2}}{{1+4\chi^{2}}}}}{(1+4\chi^{2})}|\cos{(\omega
z/c+\Psi/2)}|,\quad\eta_{RR}\approx\frac{2E_{0}ge^{-\frac{\xi^{2}}{{1+4\chi^{2}}}}}{(1+4\chi^{2})}|\sin{(\omega
z/c+\Psi/2)}|.\end{split}$ (16)

One can see that, unlike the case of linearly polarized configuration, now
their phases are time-independent and are solely controlled by single phase
$\Psi$. The overall smooth temporal dependence on a time scale $\tau$ remains
only due to the pulse envelope function $g$. The oscillatory dependence on
longitudinal coordinate $\chi$ is shown in Fig. 2. It is clear that as $\Psi$
is growing from zero, the highest central spike becomes off-centered, and
eventually at $\Psi=\pi$ is replaced with two spikes of equal height located
symmetrically about the center.

Figure 2: Dependence of the invariant electric field $\epsilon$ at
$\xi=\phi=0$ on the longitudinal coordinate $\chi$ for counterpropagating
circularly e-polarized focused Gaussian laser pulses in RR configuration with
relative phases for $\Psi=\>0,\>\pi/4,\>\pi/2,$ and $\pi$. Laser parameters
are the same as in Fig. 1.

####  II.3.2 RL Configuration

Proceeding the same way for the RL configuration, we obtain the leading order
expressions of the Lorentz invariants

$\mathcal{F}_{RL}\approx-2E_{0}^{2}g^{2}\frac{e^{-\frac{2\xi^{2}}{1+4\chi^{2}}}}{(1+4\chi^{2})^{2}}\cos{[2(\omega
t+\tilde{\varphi}+\Psi/2)]},\quad\mathcal{G}_{RL}\approx
2E_{0}^{2}g^{2}\frac{e^{-\frac{2\xi^{2}}{1+4\chi^{2}}}}{(1+4\chi^{2})^{2}}\sin{[2(\omega
t+\tilde{\varphi}+\Psi/2)]},$ (17)

and the invariant electric and magnetic fields read as follows:

$\begin{split}\epsilon_{RL}\approx
2E_{0}g\frac{e^{-\frac{\xi^{2}}{1+4\chi^{2}}}}{(1+4\chi^{2})}|\sin(\omega
t+\tilde{\varphi}+\Psi/2)|,\quad\eta_{RL}\approx
2E_{0}g\frac{e^{-\frac{\xi^{2}}{1+4\chi^{2}}}}{(1+4\chi^{2})}|\cos(\omega
t+\tilde{\varphi}+\Psi/2)|.\end{split}$ (18)

In contrast to the RR case here their oscillations are purely temporal and
depend on both $\Psi$ and $\tilde{\varphi}$, see Fig. 3. As it shows,
variation of the phases, like in previous case, results in a shift of the main
maximum from the center. Namely, it is located at the origin $t=0$ for
$\tilde{\varphi}=0$ at $\Psi=\pi$, whereas for $\tilde{\varphi}=\pi/2$ at
$\Psi=0$. As we will see in the next section, off-centering of the main
maximum results in passing from unimodal to bimodal profile of created pairs.

Figure 3: Time evolution of the invariant electric field $\epsilon$ at
$\xi=\phi=\chi=0$ for counterpropagating circularly e-polarized focused
Gaussian laser pulses in RL configuration with CEP
$\tilde{\varphi}=0,\leavevmode\nobreak\ \pi/2$ and with relative phase
$\Psi=\>0,\>\pi/4,\>\pi/2,$ and $\pi$. Laser parameters are the same as in
Fig. 1.

##  III Pair production: results and discussion

According to Eq. (1), the pair production rate depends exponentially and
monotonously on the spatiotemporal distribution of the invariant electric
field in the focal region, which in turn is controlled by the phase shifts
$\Psi$ and $\tilde{\varphi}$. Hence, we present and discuss the results of
calculation of differential particle production rate for various polarizations
of the collided pulses and in dependence on the values of $\Psi$ and
$\tilde{\varphi}$. The main goal is to demonstrate how its features can be
natively understood in terms of the underlying invariant EM field
spatiotemporal structure.

###  III.1 Differential pair production rate (linear polarization)

Figure 4: Spatial distributions in longitudinal coordinate $\chi$ of particles
created by colliding linearly polarized laser pulses with
$\Psi=0,\;\pi/2,\;\pi$ and $\tilde{\varphi}=0$, $\pi/4$, $\pi/2$. Laser
parameters are the same as in Fig. 1.

Figure 5: Temporal distributions of particles created by colliding linearly
polarized laser pulses with $\Psi=0,\;\pi/2,\;\pi$ and $\tilde{\varphi}=0$,
$\pi/4$, $\pi/2$. Laser parameters are the same as in Fig. 1.

Let us first discuss pair production in the case of linearly polarized
colliding pulses. The differential particle distributions in longitudinal
$z$-coordinate, calculated by means of Eqs. (2) – (6) for the values $0$,
$\pi/2$, $\pi$ of relative phase $\Psi$ and for CEP
$\tilde{\varphi}=0,\pi/4,\pi/2$, are shown in Fig. 4. The distributions
possess a spiky structure, with the peaks positions sensitive to $\Psi$ but
independent of $\tilde{\varphi}$. This feature is obvious from the form of the
simplified expression (9) of the invariant $\epsilon$ in electric regime.
Furthermore, the production rate is maximal for $\Psi=0$ and
$\tilde{\varphi}=\pi/2$. As $\Psi$ varies, the peaks are shifted from the
focus center (as is the case e.g., for $\Psi=\pi/2$ or $\pi$), hence the
distribution becomes asymmetric (changes from unimodal to bimodal) and the
production rate is reduced. For $\Psi=\pi$ the bimodal distribution becomes
symmetric. In contrast to the position, the separation of the peaks is merely
independent of phase shifts (remains about $\pi\Delta^{2}=0.0314$ in
dimensionless units used at the figure). The same is approximately true also
for the peak widths. It is seen that for the adopted values of parameters pair
production results the generation of narrow (FWHM = $0.0636\mu m$) particle-
antiparticle bunches localized in longitudinal direction.

Similar features are observed also in a temporal distribution (see Fig. 5),
where, however, the distribution profiles (peak heights as well as their
locations) are sensitive to both phase shifts. In all the cases shown, in
agreement with Eq. (9), the temporal distribution is unimodal and maximal for
$\tilde{\varphi}+\Psi/2\approx\pi/2$ and bimodal symmetric for
$\tilde{\varphi}+\Psi/2\approx 0$ or $\pi$. The peaks FWHM width is here as
narrow as $200$as. As expected from Fig. 1, the most prolific production rate
is observed in Fig. 5(c) for $\tilde{\varphi}\leavevmode\nobreak\
=\leavevmode\nobreak\ \pi/2$ and $\Psi=0$, i.e., for an in-phase configuration
of the counterpropagating beams.

###  III.2 Differential pair production rate (circular polarization)

For colliding circularly e-polarized laser pulses it has been observed earlier
Banerjee and Singh (2018) that the structure of the invariant electric and
magnetic fields and the pair production rate depend on their relative
handedness. In particular, in RR configuration the invariant fields and the
differential particle production rates do not reveal any CEP dependence. When
the counterpropagating pulses are in-phase, a broad unimodal temporal particle
distribution is produced. On the other hand, the colliding pulses in RL
configuration produce particle bunches localized in time and with notable CEP
dependence. It is, therefore, natural to consider these two cases separately.

Figure 6: Spatial longitudinal distributions of $e^{+}e^{-}$ pairs created in
RR configuration for $\Psi=0,\,\pi/4,\,\pi/2,\,3\pi/4,\,\pi$. The laser
parameters are same as in Fig. 1.

For RR configuration, since the invariant fields trivially depend on time and
are independent on $\tilde{\varphi}$ [see Eq. (16) and note that the same is
true exactly], it is enough to present the spatial distribution of created
pairs only in dependence on $\Psi$, see Fig. 6. The maximal production rate is
achieved with $\Psi=0$, in this case the distribution looks unimodal and
symmetric. The minimal production rate corresponds to a symmetric bimodal
distribution at $\Psi=\pi$, while for the intermediate values of $\Psi$ the
distribution is bimodal but asymmetric. The FWHM width (about $0.0764\mu m$)
of the peaks, as well as separation between them ($\lambda/2=0.5\mu m$), are
both insensitive to the phase $\Psi$. All these results are in obvious
agreement with our above discussion of the invariant field structure [see Eq.
(16) and Fig. 2].

For RL configuration, in contrast, the invariant fields trivially depend on
position, but are sensitive to both phase shifts, hence it is enough to
present only temporal evolution of the production rate, but in dependence on
both phase shifts, see Fig. 7.

Figure 7: Evolution of the $e^{+}e^{-}$ pair production rate in RL
configuration for $\tilde{\varphi}=0,\leavevmode\nobreak\
\pi/4,\leavevmode\nobreak\ \pi/2,\leavevmode\nobreak\ 3\pi/4$ and for
$\Psi=\leavevmode\nobreak\ 0,\leavevmode\nobreak\ \pi/4,\leavevmode\nobreak\
\pi/2,\leavevmode\nobreak\ 3\pi/4,\leavevmode\nobreak\ \pi$. Laser parameters
are the same as in Fig. 1.

As in previous cases, at variation of phases the distribution profile changes
from unimodal through asymmetric bimodal to symmetric bimodal. In agreement
with Eq. (18), the particular form of the distribution depends solely on the
combination $\tilde{\varphi}+\Psi/2$, with maximal and minimal production
rates when it is close to $\pi/2$ and to zero or $\pi$, respectively. The FWHM
width of the generated particle bunches is about $220as$, much shorter than
laser pulse duration $10fs$. As before, it is insensitive to both phases
$\tilde{\varphi}$ and $\Psi$, as well as to separation between the peaks when
the profiles are bimodal, which is about $1.6fs$.

By comparing the figures of this section, we conclude that for the same values
of parameters the circularly polarized RR-configuration with $\Psi=0$
maximizes the total number of created pairs. Namely this configuration was
discussed in greater details (in particular, in dependence of amplitude and
focusing degree) in Ref. Bulanov et al. (2006).

##  IV conclusion

It was proposed Bulanov et al. (2006); Bulanov et al. (2010a); Hebenstreit et
al. (2010); Bulanov et al. (2005); Bulanov (2004); Wöllert et al. (2015);
Gonoskov et al. (2012) that coherent superposition of focused optical laser
pulses is favorable for future observations of spontaneous pair production
below the Schwinger limit because of constructive interference, which notably
increases the peak field strength. In this context, it is natural to analyze
phase dependence of the arising interference pattern, as well as the
possibilities for phase control of the corresponding pair production rate.
Here we have done it with respect to both phase shifts inherent to the
problem, the carrier envelope phase $\tilde{\varphi}$ of individual pulses,
and their relative phase shift $\Psi$. As shown, their variation shifts the
spiky interference pattern of the invariant electric field with respect to the
carrier envelope, leaving the widths of the peaks unaltered.

The same conclusion broadly refers to the pair production rate, which depends
on the invariant fields monotonously. Indeed, in all the cases considered here
we could relate the peculiarities of particle distribution to the underlying
invariant field structure. However, since due to exponentiation in Eqs. (1) or
(2), at the level of production rate only those peaks of the invariant
electric field that are closest to the center of the spatiotemporal envelope
remain significant, the resulting spatial and temporal distributions of
created pairs can look either nearly unimodal or bimodal. For the same reason,
their longitudinal and temporal spike widths are much smaller than of the
original invariant field structure, meaning time-localized (during hundreds
attoseconds) formation of extremely short (tens nanometers long) electron-
positron bunches. The post production dynamics of such dense bunches may be
highly non-trivial Bulanov (2004) and may need a separate study.

We predict that the total pair production is maximal when one of the spikes is
located near the center of the spatiotemporal envelope (and distribution of
created pairs looks approximately unimodal), and minimal when the neighboring
spikes are off-centered but located symmetrically. The particular phase shifts
required for each case depend on polarization of the pulses. Among the
considered cases, for the parameters adopted here the global maximum is
achieved with a circularly polarized RR configuration considered in Ref.
Bulanov et al. (2006).

Possibility of phase control of Schwinger pair production under discussion may
be useful, e.g., to increase the attainable intensity of tightly focused
colliding laser pulses by reducing pair production and hence preventing field
depletion at their crossing, or, conversely, to measure the typically unknown
field structure and phase relations of extremely strong laser pulses in a way
similar to proposed in Ref. Mackenroth et al. (2010) by using the multiphoton
Compton scattering.

###### Acknowledgements.

A. M. F. acknowledges support from the MEPhI Academic Excellence Project
(Contract No. 02.a03.21.0005), Tomsk State University Competitiveness
Improvement Program, and the Russian Fund for Basic Research (Grant
16-02-00963a).

##  Appendix A Simplified temporal envelope for counterpropagating pulses in a
focal region

As shown in Ref. Narozhny and Fofanov (2000), finite pulse duration can be
incorporated into a focused pulse model roughly by introducing an individual
envelope factor $g(\varphi)$ for each pulse [such that $g(0)=1$ and
$g(\varphi)$ vanishes for $|\varphi|\gtrsim\omega\tau$, where $\tau$ is pulse
duration]. Let us show that inside a focal region of counterpropagating pulses
one can with high accuracy rather use a single common envelope instead.
Consider their total (for definiteness, electric) field

$\textbf{E}_{tot}=g_{1}\textbf{E}_{f}+g_{2}\textbf{E}_{b},$ (19)

where $\textbf{E}_{f,b}$ are the fields of forward and backward propagating
pulses [see Eqs. (3), (4)],
$g_{1}=g_{1}(\varphi/{\omega\tau})=g_{1}((t-z/c)/\tau)$ and
$g_{2}=g_{2}(\varphi^{\prime}/{\omega\tau})=g_{2}((t+z/c)/\tau)$ – their
individual envelopes. For the sake of simplicity, we assume
$g_{1}(\varphi)=g_{2}(\varphi)=\exp(-4\varphi^{2}/\tau^{2})$, as used
throughout the paper. By rewriting

$\textbf{E}_{tot}=\sqrt{g_{1}g_{2}}\Big{(}\sqrt{g_{1}/g_{2}}\textbf{E}_{f}+\sqrt{g_{2}/g_{1}}\textbf{E}_{b}\Big{)},$
(20)

we define $g=\sqrt{g_{1}g_{2}}=\exp(-4t^{2}/\tau^{2}-4z^{2}/{c^{2}\tau^{2}})$,
then the weight factors $g_{f}=\sqrt{g_{1}/g_{2}}$ and
$g_{b}=\sqrt{g_{2}/g_{1}}$ read $g_{f,b}=\exp(\pm 8zt/{c\tau^{2}})=\exp(\pm
8\chi t^{\prime}L/{c\tau})$, where $\chi=z/L$ and $t^{\prime}=t/\tau$ are the
dimensionless longitudinal coordinate and time normalized by laser duration.
Even though for the parameters used here ($\tau=10fs,\lambda=1\mu
m,\Delta=0.1$) we have $L/{c\tau}=5.3$, pair creation is localized in a tiny
region $|\chi|\lesssim 0.02$ and $|t^{\prime}|\lesssim 0.15$ (see Figs. 4-7),
where both weight factors $g_{f,b}$ are close to unity. Hence one can
optionally use for the total field a modified common envelope function
$g=\sqrt{g_{1}g_{2}}$, which is extremely useful to explain the results in a
qualitative way. The numerical results are in good agreement in both cases of
using either $g_{1}$ and $g_{2}$ or the approximate common envelope $g$.

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