# Failure of quasi-isometric rigidity for infinite-ended groups

Nir Lazarovich  and  Emily Stark

###### Abstract.

We prove that an infinite-ended group whose one-ended factors have finite-
index subgroups and are in a family of groups with a nonzero multiplicative
invariant is not quasi-isometrically rigid. Combining this result with work of
the first author proves that a residually-finite multi-ended hyperbolic group
is quasi-isometrically rigid if and only if it is virtually free. The proof
adapts an argument of Whyte for commensurability of free products of closed
hyperbolic surface groups.

##  1\. Introduction

Infinite-ended groups display a strong form of quasi-isometric flexibility:
Papasoglu–Whyte [PW02] proved the quasi-isometry type of an infinite-ended
group depends only on the quasi-isometry types of its one-ended factors.
Commensurability classes, on the other hand, are often finer. For example,
while there is one abstract commensurability class of closed hyperbolic
surface groups, there are infinitely many abstract commensurability classes of
their free products. Indeed, Whyte [Why99, Theorem 1.6] proved that free
products of two hyperbolic surface groups are abstractly commensurable if and
only if they have equal Euler characteristic.

We adapt Whyte’s argument to free products of groups with a nonzero
multiplicative invariant, like Euler characteristic. Consequently, we show
quasi-isometric rigidity fails for many infinite-ended groups, where a
finitely generated group $G$ is quasi-isometrically rigid if any group quasi-
isometric to $G$ is abstractly commensurable to $G$, meaning the groups
contain isomorphic finite-index subgroups. In particular, by [Laz], our main
result gives the following:

######  Theorem 1.1.

A residually-finite multi-ended hyperbolic group is quasi-isometrically rigid
if and only if it is virtually free.

Whether all hyperbolic groups are residually finite is an open question of
considerable interest in the field. We remark that in the proof of Theorem 1.1
residual-finiteness is only used to ensure that the one-ended factors of the
multi-ended hyperbolic group have some proper finite-index subgroup.

Quasi-isometric rigidity is a driving problem in geometric group theory. Many
well-studied groups have been shown to be quasi-isometrically rigid, including
non-abelian free groups [Sta68, Dun85], fundamental groups of closed
hyperbolic surfaces [Tuk88, Gab92, CJ94], non-uniform lattices in rank-one
symmetric spaces [Sch95], mapping class groups [BKMM12], and uniform lattices
in certain hyperbolic buildings [BP00, Hag06, Xie06]. Fundamental groups of
closed hyperbolic $3$-manifolds and free products of closed hyperbolic surface
groups are non-examples, and this paper gives many more. We refer the reader
to the book of Drutu–Kapovich [DK18] for additional background.

We show quasi-isometric rigidity for infinite-ended groups fails in the
presense of a suitable _multiplicative invariant_ (also known as _volume_ cf.
[Rez96]).

######  Definition 1.2.

Let $\mathcal{C}$ be a family of groups closed under taking finite-index
subgroups. A multiplicative invariant on $\mathcal{C}$ is a function
$\rho:\mathcal{C}\rightarrow\mathbb{R}$ so that if $G\in\mathcal{C}$ and $H$
is a finite-index subgroup of $G$, then $\rho(H)=[G:H]\rho(G)$.

######  Example 1.3.

The following are examples of multiplicative invariants.

  1. (1)

Euler characteristic for the family groups with a finite classifying space;
see discussion on generalizations in work of Chiswell [Chi04];

  2. (2)

$\ell^{2}$-Betti numbers for the family groups with a classifying space with
finite skeleta in each dimension; for background, see work of Lück [L0̈2];

  3. (3)

Volume for the family of fundamental groups of closed hyperbolic manifolds of
dimension $n\geq 3$ by Mostow Rigidity [Mos73]

  4. (4)

Volume from an upper or lower volume in the sense of Reznikov [Rez96, Section
1]. Examples include deficiency volume and rank volume [Rez96] and a volume
from topological complexity due to Lazarovich [Laz].

Let $\mathcal{C}$ be a family of one-ended groups and suppose that
$\mathcal{C}$ is closed under taking finite-index subgroups. Suppose
$\mathcal{C}$ has a multiplicative invariant that is nonzero for every element
of $\mathcal{C}$.

######  Theorem 1.4.

Let $G_{i}\in\mathcal{C}$, and suppose $H_{i}\leq G_{i}$ is a proper finite-
index subgroup for some $n>1$ and $1\leq i\leq n$. Then, the groups
$G_{1}*\ldots*G_{n}$ and $H_{1}*\ldots*H_{n}$ are quasi-isometric and not
abstractly commensurable.

######  Corollary 1.5.

If $G$ is a finitely presented infinite-ended group that is not virtually free
and whose one-ended factors are in $\mathcal{C}$ and contain non-trivial
finite-index subgroups, then $G$ is not quasi-isometrically rigid.

###### Proof of Theorem 1.1.

A multi-ended hyperbolic group which is not virtually free has finitely many
one-ended factors. If the group is residually finite, then each factor is a
residually-finite hyperbolic group and, in particular, has a proper finite-
index subgroup. Every hyperbolic group has finite asymptotic dimension
[Roe05], and so there exists $m$ such that the asymptotic dimension of all the
factors is bounded by $m$. Let $\mathcal{C}$ be the family of one-ended
hyperbolic groups whose asymptotic dimension is bounded by $m$. By [Laz, Proof
of Theorem A], the family $\mathcal{C}$ has a multiplicative invariant, namely
$\underline{C}_{2,m}$, that is nonzero for every element of $\mathcal{C}$. The
conclusion now follows from Corollary 1.5. ∎

The free products $G_{1}*\ldots*G_{n}$ and $H_{1}*\ldots*H_{n}$ are quasi-
isometric by work of Papasoglu–Whyte [PW02] since $G_{i}$ and $H_{i}$ are
quasi-isometric for each $i$. It remains to prove the groups are not
abstractly commensurable. To do this, given a multiplicative invariant $\rho$
for a family of groups $\mathcal{C}$, we construct a multiplicative invarant
$\chi_{\rho}$ on the family of free products whose non-free free factors are
elements of $\mathcal{C}$, and we adapt an argument of Whyte [Why99, Theorem
1.6]. We note that Sykiotis [Syk18] defines a related multiplicative invariant
for infinite-ended groups through a limiting process; our construction is
simpler and more direct.

To prove that two groups are quasi-isometric, often one finds geometric
actions of the groups on the same proper metric space - in this case, the
groups are said to have a common model geometry. Theorem 1.4 can be used to
exhibit quasi-isometric groups with no common model geometry. Indeed, such
examples arise via finitely generated groups that are not quasi-isometrically
rigid, but are action rigid, meaning that whenever the group and another
finitely generated group act geometrically on the same proper metric space,
the groups are abstractly commensurable. Many examples of action rigid free
products are given in [SW, MSSW], including free products of closed hyperbolic
manifold groups.

### Acknowledgements

NL was supported by Israel Science Foundation (grant no. 1562/19). ES was
supported by NSF Grant No. DMS-2204339.

##  2\. Preliminaries

We will use the notion of a graph of spaces and a graph of groups; we refer
the reader to [Ser77, SW79] for additional background.

######  Definition 2.1.

(Graph notation.) A graph $\Gamma$ is a set of vertices $V\Gamma$ together
with a set of edges $E\Gamma$. An edge $e$ is oriented with an initial vertex
$\iota(e)$ and a terminal vertex $\tau(e)$. For every edge $e$ there is an
edge $\bar{e}$ with reversed orientation so that $\iota(e)=\tau(\bar{e})$.
Further, $e\neq\bar{e}$ and $e=\bar{\bar{e}}$.

We often suppress the orientation of an edge if it is not relevant to a
construction.

######  Definition 2.2.

A _graph of spaces_ $(X,\Gamma)$ consists of the following data:

  1. (1)

a graph $\Gamma$ called the underlying graph,

  2. (2)

a connected space $X_{v}$ for each $v\in V\Gamma$ called a _vertex space_ ,

  3. (3)

a connected space $X_{e}$ for each $e\in E\Gamma$ called an _edge space_ such
that $X_{\bar{e}}=X_{e}$, and

  4. (4)

a $\pi_{1}$-injective map $\phi_{e}:X_{e}\rightarrow X_{\tau(e)}$ for each
$e\in E\Gamma$.

The _total space_ $X$ is the following quotient space:

$X=\left(\bigsqcup_{v\in V\Gamma}X_{v}\bigsqcup_{e\in
E\Gamma}\bigl{(}X_{e}\times[0,1]\bigr{)}\right)\;\Big{/}\;\sim,$

where $\sim$ is the relation that identifies $X_{e}\times\\{0\\}$ with
$X_{\bar{e}}\times\\{0\\}$ via the identification of $X_{e}$ with
$X_{\bar{e}}$, and the point $(x,1)\in X_{e}\times[0,1]$ with $\phi_{e}(x)\in
X_{\tau(e)}$.

######  Definition 2.3.

A _graph of groups_ $\mathcal{G}$ consists of the following data:

  1. (1)

a graph $\Gamma$ called the _underlying graph_ ,

  2. (2)

a group $G_{v}$ for each $v\in V\Gamma$ called a _vertex group_ ,

  3. (3)

a group $G_{e}$ for each $e\in E\Gamma$ called an _edge group_ , such that
$G_{e}=G_{\bar{e}}$,

  4. (4)

and injective homomorphisms $\Theta_{e}:G_{e}\rightarrow G_{\tau(e)}$ for each
$e\in E\Gamma$ (called _edge maps_).

If $\mathcal{G}$ is a graph of groups with underlying graph $\Gamma$, a _graph
of spaces associated to $\mathcal{G}$_ consists of a graph of spaces
$(X,\Gamma)$ with points $x_{v}\in X_{v}$ and $x_{e}\in X_{e}$ so that
$\pi_{1}(X_{v},x_{v})\cong G_{v}$ and $\pi_{1}(X_{e},x_{e})\cong G_{e}$ for
all $v\in V\Gamma$ and $e\in E\Gamma$, and the maps $\phi_{e}:X_{e}\rightarrow
X_{\tau(e)}$ are such that $(\phi_{e})_{*}=\Theta_{e}$. The _fundamental
group_ of the graph of groups $\mathcal{G}$ is $\pi_{1}(X)$. The fundamental
group is independent of the choice of $X$ and is denoted by
$\pi_{1}(\mathcal{G})$.

If $G\cong\pi_{1}(\mathcal{G})$, where $\mathcal{G}$ is a graph of groups,
then we may view each vertex group of $\mathcal{G}$ as a subgroup of $G$ (up
to conjugacy). Suppose that $G=\pi_{1}(\mathcal{G})=\pi_{1}(X)$, where
$\mathcal{G}$ is a graph of groups and $X$ is the corresponding total space.
If $G^{\prime}\leq G$, then $G^{\prime}=\pi_{1}(X^{\prime})$, where
$X^{\prime}$ covers $X$. This cover $q:X^{\prime}\rightarrow X$ naturally
yields a graph of spaces decomposition of $X^{\prime}$ with underlying graph
$\Gamma^{\prime}$, where the edge and vertex spaces are components of the
preimages of the edge and vertex spaces in $X$. Hence, $G^{\prime}$ is the
fundamental group of a corresponding graph of groups $\mathcal{G}^{\prime}$.
Each vertex group of $\mathcal{G}^{\prime}$ is then conjugate to a subgroup of
a vertex group of $\mathcal{G}$.

######  Definition 2.4.

With the notation from the previous paragraph, if $G_{v}$ is a vertex group of
$\mathcal{G}$, we say a vertex group $G_{u}^{\prime}$ of
$\mathcal{G}^{\prime}$ covers $G_{v}$ if $G_{u}^{\prime}$ is conjugate to a
subgroup of $G_{v}$ via the covering of the corresponding graphs of spaces.

We make use of the following observation.

######  Observation 2.5.

Suppose $\mathcal{C}$ is a family of groups closed with respect to taking
finite-index subgroups and with a multiplicative invariant $\rho$. Suppose
$\mathcal{G}$ is a graph of groups with each vertex group in $\mathcal{C}$ and
$G=\pi_{1}(\mathcal{G})$. Suppose $G^{\prime}$ is an index-$k$ subgroup of $G$
and $\mathcal{G}^{\prime}$ is its corresponding graph of groups decomposition.
Then for each vertex group $G_{v}\in\mathcal{G}$, if
$K=\\{G^{\prime}_{u_{1}},\ldots,G^{\prime}_{u_{n}}\\}$ is the collection of
vertex groups of $\mathcal{G}^{\prime}$ covering $G_{v}$, then

$\sum_{i=1}^{n}\rho(G^{\prime}_{u_{i}})=k\rho(G_{v}).$

##  3\. Multiplicative invariants and commensurability

######  Notation 3.1.

For $r\geq 0$, let $F_{r}$ denote the free group of rank $r$. Throughout this
section let $\mathcal{C}$ be a family of one-ended groups that is closed with
respect to taking finite-index subgroups and has a multiplicative invariant
$\rho$. Let $\mathcal{C}^{\prime}$ denote the family of groups of the form
$G=G_{1}*\ldots*G_{n}*F_{r}$ for some $r\geq 0$, $n\geq 0$, and
$G_{i}\in\mathcal{C}$. Equivalently, $G\in\mathcal{C}^{\prime}$ if $G$ is the
fundamental group of a graph of groups $\mathcal{G}$ with trivial edge groups
and each vertex group either trivial or in $\mathcal{C}$.

We will work in the setting of free products, rather than the general
infinite-ended case, as the arguments are cleaner. See Remark
LABEL:rem:infendedgen regarding the more general setting.

######  Definition 3.2.

Let $\rho$ be a multiplicative invariant on the family of groups
$\mathcal{C}$. Define a function
$\chi_{\rho}:\mathcal{C}^{\prime}\rightarrow\mathbb{R}$ by

$\chi_{\rho}(G_{1}*\ldots*G_{n}*F_{r})=1-r-n+\sum_{i=1}^{n}\rho(G_{i}).$

######  Remark 3.3.

The function $\chi_{\rho}$ is a variant of the Euler characteristic. Viewing
$G\in\mathcal{C}^{\prime}$ as the fundamental group of a graph of groups, each
vertex in the graph of groups decomposition labeled $G_{i}\in\mathcal{C}$ is
assigned $\rho(G_{i})$ rather than $1$. In particular, if the multiplicative
invariant $\rho$ is Euler characteristic, then $\chi_{\rho}$ is as well.

To prove that $\chi_{\rho}$ is a multiplicative invariant, we will use the
following.

######  Definition 3.4.

Let $\mathcal{G}$ be a graph of groups with underlying graph $\Gamma$, trivial
edge groups, and with nontrivial vertex groups
$G_{1},\ldots,G_{n}\in\mathcal{C}$. Define

$\chi_{\rho}(\mathcal{G})=\chi(\Gamma)-n+\sum_{i=1}^{n}\rho(G_{i}).$

######  Lemma 3.5.

Let $G=G_{1}*\ldots G_{n}*F_{r}\in\mathcal{C}^{\prime}$, and suppose
$G\cong\pi_{1}(\mathcal{G})$, where $\mathcal{G}$ is a graph of groups with
underlying graph $\Gamma$, trivial edge groups, and with nontrivial vertex
groups $G_{1},\ldots,G_{n}\in\mathcal{C}$. Then,
$\chi_{\rho}(G)=\chi_{\rho}(\mathcal{G})$.

###### Proof.

By definition,

$\displaystyle\chi_{\rho}(G)$ $\displaystyle=$ $\displaystyle
1-r-n+\sum_{i=1}^{n}\rho(G_{i}),\textrm{ and }$
$\displaystyle\chi_{\rho}(\mathcal{G})$ $\displaystyle=$
$\displaystyle\chi(\Gamma)-n+\sum_{i=1}^{n}\rho(G_{i}).$

Thus, we must show $\chi(\Gamma)=1-r$. Indeed, it follows from the definition
of the fundamental group of a graph of groups that
$F_{r}\cong\pi_{1}(\Gamma)$. So, $\chi(\Gamma)=\chi(F_{r})=1-r$. ∎

######  Proposition 3.6.

The function $\chi_{\rho}:\mathcal{C}^{\prime}\rightarrow\mathbb{R}$ is a
multiplicative invariant.

###### Proof.

Let $G=G_{1}*\ldots*G_{n}*F_{r}\in\mathcal{C}^{\prime}$. Realize $G$ as the
fundamental group of a graph of groups $\mathcal{G}$ whose underlying graph
$\Gamma$ has vertex set $V\Gamma=\\{v_{0},\ldots,v_{n}\\}$ and (unoriented)
edge set $E\Gamma=\\{e_{1},\ldots,e_{n},a_{1},\ldots,a_{r}\\}$, where
$e_{i}=\\{v_{0},v_{i}\\}$ for $1\leq i\leq n$, and $a_{i}$ is a loop based at
$v_{0}$ for $1\leq i\leq r$. Suppose each edge group is trivial, the vertex
group $G_{v_{0}}$ is trivial, and the vertex group $G_{v_{i}}$ is $G_{i}$.

Let $G^{\prime}\leq G$ be an index-$k$ subgroup. As discussed in Section 2,
$G^{\prime}\cong\pi_{1}(\mathcal{G}^{\prime})$, where $\mathcal{G}^{\prime}$
is a graph of groups with:

  * •

$k$ vertices with trivial vertex group that each cover $G_{v_{0}}=\left\langle
1\right\rangle$,

  * •

$k$ edges with trivial edge group that each cover $G_{a_{i}}=\left\langle
1\right\rangle$ for $1\leq i\leq r$,

  * •

$k$ edges with trivial edge group that each cover $G_{e_{i}}=\left\langle
1\right\rangle$ for $1\leq i\leq n$, and

  * •

for $i\in\\{1,\ldots n\\}$, there exists $n_{i}\in\mathbb{N}_{\geq 1}$ and
$n_{i}$ vertices $v_{i}^{1},\ldots,v_{i}^{n_{i}}$ with vertex group
$G_{v_{i}^{j}}$ that each cover $G_{v_{i}}$. Further,

$\sum_{j=1}^{n_{i}}\rho(G_{v_{i}^{j}})=k\rho(G_{i})$

by Observation 2.5 since $\rho$ is a multiplicative invariant.

Thus, $\chi_{\rho}(\mathcal{G}^{\prime})=k\chi_{\rho}(\mathcal{G})$. So, by
Lemma 3.5,
$\chi_{\rho}(G^{\prime})=\chi_{\rho}(\mathcal{G}^{\prime})=k\chi_{\rho}(\mathcal{G})=k\chi_{\rho}(G)$.
∎

As explained in the introduction, Theorem 1.4 follows from the next theorem.
The proof is analogous to Whyte’s argument in [Why99, Theorem 1.6].

######  Theorem 3.7.

Let $\mathcal{C}$ be a family of groups closed with respect to taking finite-
index subgroups and with a multiplicative invariant $\rho$ that is nonzero for
every element of $\mathcal{C}$. For $m>1$ and $1\leq i\leq m$, let
$G_{i}\in\mathcal{C}$, and suppose $H_{i}\leq G_{i}$ is a proper finite-index
subgroup. Then, the groups $G=G_{1}*\ldots*G_{m}$ and $H=H_{1}*\ldots*H_{m}$
are not abstractly commensurable.

###### Proof.

Suppose towards a contradiction that there exist finite-index subgroups
$G^{\prime}\leq G$ and $H^{\prime}\leq H$ of index $k$ and $k^{\prime}$,
respectively, so that $G^{\prime}\cong H^{\prime}$. Suppose that
$H^{\prime}\cong G^{\prime}\cong G_{1}^{\prime}*\ldots*G_{n}^{\prime}*F_{r}$
where $r\geq 0$ and for each $i\in\\{1,\ldots,n\\}$, the group
$G_{i}^{\prime}$ covers $G_{j}$ and $H_{\ell}$ for some
$j,\ell\in\\{1,\ldots,m\\}$. By Proposition 3.6,
$\chi_{\rho}(G^{\prime})=k\chi_{\rho}(G)=k^{\prime}\chi_{\rho}(H)$. We will
show that $k=k^{\prime}$ by relating these numbers to $r,n$, and $m$. This
will complete the proof since $\chi_{\rho}(H)<\chi_{\rho}(G)$ because $\rho$
is a multiplicative invariant.

######  Claim 3.8.

$k=k^{\prime}=\frac{1-r-n}{1-m}$.

###### Proof of Claim..

We prove the claim for $k$; the proof for $k^{\prime}$ is analogous. First, by
Proposition 3.6,

(1) $\displaystyle\chi_{\rho}(G^{\prime})$ $\displaystyle=$ $\displaystyle
k\chi_{\rho}(G)$ (2) $\displaystyle=$ $\displaystyle
k\left(1-m+\sum_{i=1}^{m}\rho(G_{i})\right).$

Next, by the definition of $\chi_{\rho}$,

(3) $\displaystyle\chi_{\rho}(G^{\prime})$ $\displaystyle=$ $\displaystyle
1-r-n+\sum_{i=1}^{n}\rho(G_{i}^{\prime}).$

For $1\leq i\leq m$, let $K_{i}=\\{G_{i_{j}}^{\prime}\\}_{j=1}^{n_{i}}$ and
$i_{j}\in\\{1,\ldots,n\\}$ be the collection of free factors of $G^{\prime}$
so that $G_{i_{j}}^{\prime}$ covers $G_{i}$. Then, for all $1\leq i\leq m$,

$\sum_{j=1}^{n_{i}}\rho(G_{i_{j}}^{\prime})=k\rho(G_{i})$

by Observation 2.5. Since the $K_{i}$ partition the set
$\\{G_{i}^{\prime}\\}_{i=1}^{n}$, Equation (3) can be rewritten as

(4) $\displaystyle\chi_{\rho}(G^{\prime})$ $\displaystyle=$ $\displaystyle
1-r-n+\left(\sum_{j=1}^{n_{1}}\rho(G_{i_{j}}^{\prime})+\ldots+\sum_{j=1}^{n_{m}}\rho(G_{i_{j}}^{\prime})\right)$
(5) $\displaystyle=$ $\displaystyle 1-r-n+k\sum_{i=1}^{m}\rho(G_{i}).$

Combining Equation (2) and Equation (5) yields

(6) $\displaystyle k\left(1-m+\sum_{i=1}^{m}\rho(G_{i})\right)$
$\displaystyle=$ $\displaystyle 1-r-n+k\sum_{i=1}^{m}\rho(G_{i}).$

Thus, $k=\frac{1-r-n}{1-m}$. ∎

Therefore, $k=k^{\prime}$, completing the proof of the theorem. ∎

######  Remark 3.9.

Note that in the proof of Claim 3.8 we have not used that the multiplicative
invariant $\rho$ is non-zero. In fact, the same proof works for the
multiplicative invariant $\rho(G)=1/|G|$ (where $1/\infty=0$).

######  Remark 3.10.

(Weakening the hypotheses.) If $G=G_{1}*\ldots*G_{m}$ and
$H=H_{1}*\ldots*H_{m}$ are in the family $\mathcal{C}^{\prime}$ and
$\chi_{\rho}(G)>\chi_{\rho}(H)$, then the proof of the theorem above shows
that $G$ and $H$ are not abstractly commensurable. So, for example, one can
weaken the hypothesis of the theorem to asking that $H_{i}\leq G_{i}$ is a
proper finite-index subgroup for only one $i\in\\{1,\ldots,n\\}$.

The general infinite-ended case now follows from Theorem 3.7.

###### Proof of Corollary 1.5.

Suppose $G$ is a finitely presented infinite-ended group. Work of Stallings
and Dunwoody [Sta68, Dun85] proves that $G$ is the fundamental group of a
finite graph of groups with finite edge groups and vertex groups that have at
most one end. If $G$ is not virtually free then there is at least one one-
ended factor in this decomposition. Suppose the one-ended vertex groups in
this decomposition are $G_{1},\ldots,G_{n}$ for some $n\geq 1$ and belong to a
family of groups $\mathcal{C}$ that is closed under taking finite-index
subgroups. Suppose $\mathcal{C}$ has a multiplicative invariant that is
nonzero for every element of $\mathcal{C}$. Suppose further that each one-
ended factor of $G$ contains a non-trivial finite-index subgroup. The group
$G$ is quasi-isometric to the free product $G_{1}*G_{1}*\ldots*G_{n}$ by work
of Papasoglu–Whyte [PW02]. Thus, $G$ is not quasi-isometrically rigid by
Theorem 3.7. ∎

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#

S. V. Gurevich  Institute for Theoretical Physics, University of Münster
Wilhelm-Klemm-Str. 9, D-48149 Münster, Germany  Departament de Física and
IAC3, Universitat de les Illes Balears, C/ Valldemossa km 7.5, 07122 Mallorca,
Spain  F. Maucher  Departament de Física and IAC3, Universitat de les Illes
Balears, C/ Valldemossa km 7.5, 07122 Mallorca, Spain  Faculty of Mechanical,
Maritime and Materials Engineering; Department of Precision and Microsystems
Engineering, Delft University of Technology, 2628 CD, Delft, The Netherlands
J. Javaloyes  Departament de Física and IAC3, Universitat de les Illes
Balears, C/ Valldemossa km 7.5, 07122 Mallorca, Spain

# Aberration-driven tilted emission in degenerate cavities

S. V. Gurevich  Institute for Theoretical Physics, University of Münster
Wilhelm-Klemm-Str. 9, D-48149 Münster, Germany  Departament de Física and
IAC3, Universitat de les Illes Balears, C/ Valldemossa km 7.5, 07122 Mallorca,
Spain  F. Maucher  Departament de Física and IAC3, Universitat de les Illes
Balears, C/ Valldemossa km 7.5, 07122 Mallorca, Spain  Faculty of Mechanical,
Maritime and Materials Engineering; Department of Precision and Microsystems
Engineering, Delft University of Technology, 2628 CD, Delft, The Netherlands
J. Javaloyes  Departament de Física and IAC3, Universitat de les Illes
Balears, C/ Valldemossa km 7.5, 07122 Mallorca, Spain

###### Abstract

The compensation of chromatic dispersion opened new avenues and extended the
level of control upon pattern formation in the temporal domain. In this
manuscript, we propose the use of a nearly-degenerate laser cavity as a
general framework allowing for the exploration of higher contributions to
diffraction in the spatial domain. Our approach leverages the interplay
between optical aberrations and the proximity to the self-imaging condition
which allows to cancel or reverse paraxial diffraction. As an example, we show
how spherical aberrations materialize into a transverse bilaplacian operator
and, thereby, explain the stabilization of temporal solitons travelling off-
axis in an unstable mode-locked broad-area surface-emitting laser. We disclose
an analogy between these regimes and the dynamics of a quantum particle in a
double well potential.

The understanding of self-organized spatiotemporal patterns is key in
photonics and the formation of shocks, vortices, tilted waves, cross-roll
patterns, weak optical turbulence, and localized structures was observed
experimentally and studied theoretically in large-aspect-ratio lasers, see
e.g., (L-CSF-94, ; Lega_1994, ; HMT-PRL-95, ; SSW-PRL-97, ; TSW-PRL-98, ; ABR-
PR-99, ; BTB-NAT-02, ; AFO-AAM-09, ; GBG-PRL-10, ; J-PRL-16, ; GPTT-PRL-21, ).
Another recent example of multidimensional spatio-temporal self-organization
is the experimental observation of spatiotemporal mode-locking in multimode
optical fibers (WSP-Nature-20, ; DXL-PRL-22, ; WRC-Optica-22, ).

Dispersion compensation consists in combining elements with opposed chromatic
properties to achieve an overall partial or total cancellation of the second
order dispersion. This simple yet powerful idea permitted exploring the
influence of higher order contributions. While optical temporal localized
structures often result from the balance between self-phase modulation and
anomalous second-order dispersion (LCK_NP_10, ; HBJ-NAP-14, ; MJB-PRL-14, ),
it was recently proven that third and fourth-order dispersion lead to
unforeseen effects such as the stabilization of solitons and frequency combs
(TG-OL-10, ; PGL-OL-14, ; TABB-PRA-20, ), symmetry breaking (LMK-PRL-13, ),
the control of modulational instabilities (TML-OL-07, ; DKB-OL-13, ) or the
realization of purely quartic solitons (BSS-NAC-16, ; DRH-APLP-21, ) as
predicted in (KH-OC-94, ).

The paraxial diffraction emerging as a beam propagate is mathematically
equivalent to that of second-order chromatic dispersion. Optical cavities in
which the path of light is folded onto itself may contain a transverse plane
that is its own image after a round-trip (A-APO-69, ; SIEGMAN-BOOK, ). This
so-called stigmatic condition is equivalent to an effective cancellation of
paraxial diffraction. Imposing additionally the nullification of the round-
trip wavefront curvature does not only achieve the self-imaging condition
(SIC) for the field intensity, but also for its _amplitude_. Self-imaging
cavities have attracted a great deal of attentions for their rich spatio-
temporal dynamics (TSW-PRL-98, ; HBE-OE-08, ; LKF-OE-10, ; J-PRL-16, ; BVM-
Optica-22, ; NathanPHD, ) and for their application in speckle-free imaging
(KLR-Optica-16, ; MEY-NNP-21, ), frequency comb multiplexing (PWC-Optica-22,
), controllable and reconfigurable multimode fields (EMF-PRL-22, ; BVM-PR-23,
), see (CCB-NRP-19, ) for a review. The proximity of the SIC was used to
manipulate the spatial coherence of the field (TVF-JOSAA-91, ; MEY-NNP-21, ;
EMF-PRL-22, ; KBJ-APLP-22, ; POT-NAP-22, ), create a perfect coherent absorber
(SWH-SCI-22, ), realising topological band structures (YZH-NATCOM-22, ), or to
form propagation invariant beams (CBT-OE-18, ). Aberrations become crucial as
the SIC is approached (Clark2020, ; PhysRevA.104.013524, ; BVM-Optica-22, ),
yet, their effect on spatio-temporal laser dynamics received comparatively
less attention.

Figure 1: Schematic of the fundamental Hermite-Gauss (HG) mode $A_{s}(x)$
(blue) and the corresponding potential $V(x)$ (orange) of Eq. (1) in the real
(left) and Fourier space (right) for (a) $b<0$, $c>0$ and $s=1$ (b) $b>0$,
$c>0$ and $s=1$. The resulting double-well potential in the Fourier space with
the minima located at $k_{0}$ shown in (b) correspond to a tilted HG mode in
the real space.

Photonic crystals also allow for dispersion control (VBH-Nature-05, ; FLFP-
OE-06, ) and lead to the zero-diffraction regime (SSH-PRE-06, ). Higher order
spatial operators occur in fiber lasers (KZC-PhotRes-21, ) and in optical
cavities either at the onset of bistability (TML-PRL-94, ) or containing a
photorefractive or semiconductor medium (Lega_1994, ; HMT-PRL-95, ; SSW-
PRL-97, ; MM-PRE-02, ). Diffraction control was also proposed using
metamaterials (GCdST-PRA-07, ; KTV-JOSAB-09, ; GGVdS-PRA-08, ; TKG-PRA-11, )
or atomic resonances (FLS-NAP-09, ). In this paper, we study how aberrations
appear as leading effects in nearly-degenerate broad-area surface-emitting
lasers. We show that these effects can crucially modify the spatio-temporal
mode-locking dynamics and give rise to either spatially tilted pulses or
spatio-temporal solitons. Our results represent a step further towards the
understanding of spatio-temporal mode-locking (WSP-Nature-20, ; DXL-PRL-22, ;
WRC-Optica-22, ), the realization of fully confined three-dimensional light
bullets (J-PRL-16, ; GRM-PRL-17, ; GJ-PRA-17, ) and may lead to new ideas and
applications for beam steering, tweezing (ADB-OL-86, ) and tailored optical
energy potential landscapes (WAE-LPR-13, ; BCL-NAR-21, ).

The dynamics of the transverse profile of the field close to the SIC and in
presence of spherical aberrations is equivalent to the dynamics of a quantum
harmonic oscillator featuring a fourth-order derivative

$\displaystyle-i\partial_{\theta}A$
$\displaystyle=\left(cx^{2}+b\partial_{x}^{2}+s\partial_{x}^{4}\right)A\,,$
(1)

where $c$, $b$ and $s$ are real coefficients that correspond to the residual
wavefront curvature, diffraction, and spherical aberration, respectively. We
will focus our attention on spherical aberration as it is often the dominant
form of aberration. A detailed derivation is provided in the Appendix I.
Notice that Eq. (1) was studied already albeit in a different context in (SHV-
PRE-06, ).

It is well known that the potential $V\left(x\right)=c\,x^{2}$ in Eq. (1)
supports localized solutions for $bc<0$ (SIEGMAN-BOOK, ). Setting
$A(x,\theta)=A_{{\rm s}}(x)e^{-i\omega\theta}$ with $\omega$ denoting the
frequency of the eigenmode and imposing boundedness of the the solution
defines a singular Sturm-Liouville problem (SLP) that allows determining
$\omega$:

$\left(\omega+cx^{2}+b\partial_{x}^{2}+s\partial_{x}^{4}\right)A_{s}=0\,,\lim_{x\rightarrow\pm\infty}A_{s}=0$
(2)

For $bc<0$ and $s=0$ the solutions of Eq. (2) are the so-called Hermite-Gauss
(HG) modes $H_{n}(\frac{x}{\sigma})$ with $\sigma=\sqrt{-b/c}$ and frequencies
$\omega_{n}=\dfrac{b}{\sigma^{2}}(2n+1)$ while $n\in\mathbb{N}$ is the modal
index. We will see later that this situation corresponds to the case of a
stable cavity devoid of aberration.

The HG modes are invariant under Fourier transform since the multiplicative
and differential terms in Eq. (1) are exchanged upon performing the latter.
However, the role played by the fourth order derivative in Eq. (1) becomes
more clear in Fourier space, where the SLP becomes

$\displaystyle\left(\omega+c\partial_{k}^{2}-bk^{2}+sk^{4}\right)\hat{A}_{s}$
$\displaystyle=0\,,$ (3)

Here, we defined $\hat{A}_{s}$ as the Fourier transform of $A_{s}$. Hence, the
presence of a fourth order derivative in Eq. Eq. (1) converts the situation to
that of a particle in single or a double-well potential. In case of $b<0$ and
$c>0$ the potentials $V\left(x\right)$ or
$\hat{V}\left(k\right)=-b\,k^{2}+sk^{4}$ both feature a minimum at the origin
as shown in Fig. 1 (a).

Figure 2: Numerical solution of Eq. (1) for different values of $b$ and fixed
values of $s=1$ and $c=4.97\times 10^{-4}$. White line indicates the second
moment.

The bounded solutions of Eq. (1) remain practically unaffected by small values
of $s$, since the solutions remain concentrated at low values of $k$ for which
$bk^{2}\gg sk^{4}$. Let us know consider the situation where $b$ changes its
sign, which usually signals the transition from a stable to an unstable
cavity. In that case, the potential $\hat{V}(k)$ develops a negative curvature
around $k=0$ for $b>0$. If there were no aberrations present in the system,
i.e. $s=0$, the laser would simply turn off as there is no transverse mode to
support emission. Mathematically speaking, this means that the bounded
solutions of the SLP cease to exists.

However, the situation can be remedied by the fourth order derivative as
depicted in Fig. 1(b): For $s>0$, two new minima emerge symmetrically at
$k_{0}=\pm\sqrt{b/2s}$ which allows bounded solutions to continue to exist in
the otherwise unstable domain. The ground state solution $\hat{A}_{s}(k)$ in
Fourier space can be approximated by a superposition of two bell-shaped
functions localized around $\pm k_{0}$ which, in real space, amounts to a
strongly modulated eigenmode with wavelength $\lambda_{\perp}=2\pi/k_{0}$,
akin to the interference between Bose-Einstein condensates in a double-well
potential (Ketterle:Science:1997, ; Ketterle:PRL:2004, ). This situation is
depicted in Fig. 2 where we solved for the ground state of Eq. (1) numerically
using complex time evolution. Note, that the waist of the mode decreases
monotonically upon approaching $b=0$, but remains finite at $b=0$. This is due
to the fact that the higher order derivative start to play a more dominant
role. In the unstable range $b>0$ we notice that the second order moment
increases, albeit at at a faster rate.

Figure 3: Evolution of $\left|\Gamma_{0}\left(x,0\right)\right|^{2}$and
$\left|\Psi_{0}\left(x,0\right)\right|^{2}$ as a function of $s$. Traces are
shifted for clarity. From top to bottom, $s=\left(5,\,1,\,0.5,\,0.1\right)$.
Other parameters are $\left(b,\,c\right)=\left(0.7873,\,0.0004973\right)$.

Figure 4: Evolution of $\Gamma_{n}\left(x,0\right)$and
$\Psi_{n}\left(x,0\right)$ as a function of the modal index $n$. We note the
even/odd alternating sequence. Traces are shifted for clarity. From top to
bottom, $n=\left(3,\,2,\,1,\,0\right)$. Other parameters are
$\left(b,\,c,\,s\right)=\left(0.7873,\,0.0004973,1\right)$.

While we could not obtain closed form analytical solutions of Eq. (2) for
$s\neq 0$, the following change of variable

$\displaystyle A\left(x\right)$ $\displaystyle=$ $\displaystyle
A_{s}\left(x\right)\exp\left[ik_{0}x-\omega\theta\right]\,,\;k_{0}=\sqrt{\frac{b}{2s}}$
(4)

permits cancelling the first-order derivative and leads to

$\left(\omega-\frac{b^{2}}{4s}+cx^{2}-2b\partial_{x}^{2}+i\sqrt{8bs}\partial_{x}^{3}+s\partial_{x}^{4}\right)A_{s}=0\,.$
(5)

Slowly varying envelope solutions, where the characteristic length scale for
the envelope $A_{s}$ is much larger than $2\pi/k_{0}$ correspond to the
inequality $\left|\partial_{x}A_{s}\right|\ll\left|k_{0}A_{s}\right|$. If this
condition is fulfilled, one may neglect the third and fourth order derivative
in Eq. (5), leading to

$\displaystyle\left(\omega-\frac{b^{2}}{4s}+cx^{2}-2b\partial_{x}^{2}\right)A_{s}$
$\displaystyle=$ $\displaystyle 0\,.$ (6)

By comparing Eq. (1) with Eq. (2) we note that the latter is again a Hermite-
Gauss equation where the parameter $b$ has been replaced by $-2b$, which fully
explains the increased diverging rate of the second moment in Fig. 2. We note
that the frequencies $\omega$ are shifted due to the additional term
$b^{2}/\left(4s\right)$.

In summary, we disclose the surprising result that an unstable cavity does not
necessarily turn off upon crossing the SIC, and lasing eigenmodes can still be
supported by spherical aberrations. The modes can be approximated by

$\displaystyle A\left(x,\theta\right)$ $\displaystyle=$ $\displaystyle
H_{n}\left(\frac{x}{\sigma}\right)\exp\left[i\left(k_{0}x-\omega_{n}\theta\right)\right]\,,$
(7) $\displaystyle\omega_{n}$ $\displaystyle=$
$\displaystyle\frac{b^{2}}{4s}-\frac{2b}{\sigma^{2}}\left(2n+1\right)\,,$ (8)
$\displaystyle\sigma^{2}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2b}{c}}\,,\,k_{0}=\sqrt{\frac{b}{2s}}\,.$ (9)

Figure 5: Birth of two symmetrically located minima for the Fourier potential
$\hat{V}$ for a stable, marginal and unstable cavity with (from left to right)
$b_{x}=\left(-0.2,0,0.05\right)$ Other parameters are
$\left(b_{y},s\right)=\left(-0.8,1\right)$.

Since $k_{0}\in\mathbb{R}$, it is convenient to define two families of
eigenfunctions $\Gamma_{n}\left(x,\theta\right)$ and
$\Psi_{n}\left(x,\theta\right)$ as:

$\displaystyle\Gamma_{n}\left(x,\theta\right)$ $\displaystyle=$ $\displaystyle
H_{n}\left(\frac{x}{\sigma}\right)\cos\left(k_{0}x\right)e^{-i\omega_{n}\theta}\,,$
(10) $\displaystyle\Psi_{n}\left(x,\theta\right)$ $\displaystyle=$
$\displaystyle
H_{n}\left(\frac{x}{\sigma}\right)\sin\left(k_{0}x\right)e^{-i\omega_{n}\theta}.$
(11)

The latter are depicted in Fig. 3 and Fig. 4, respectively. We observe the
effect of the parameter $s$ on the ground states $\Gamma_{0}$ and $\Psi_{0}$
in Fig. 3. In order to stabilize a bounded eigenmode, the decrease of $s$ must
be compensated by additional oscillations and, therefore, an increase in
$k_{0}\sim 1/\sqrt{s}$. In Fig. 4 we also depict how the intensity of the
modes $\Gamma_{n}\left(x,\theta\right)$ and $\Psi_{n}\left(x,\theta\right)$
alternate between even and odd when varying the modal index $n$.

We now turn our attention to the two-dimensional (2D) transverse profile whose
evolution is governed by

$\displaystyle\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace-i\frac{\partial
A}{\partial\theta}$ $\displaystyle=$
$\displaystyle\left(c_{x}x^{2}+c_{y}y^{2}+b_{x}\partial_{x}^{2}+b_{y}\partial_{y}^{2}+s\nabla_{\perp}^{4}\right)A.$
(12)

Here, we assume that the cavity possesses two orthogonal axes denoted
$\boldsymbol{r}_{\perp}=\left(x,y\right)$ and, for the sake of generality, we
introduced dichroism in Eq. 12, i.e. $b_{x}\neq b_{y}$ and $c_{x}\neq c_{y}$.
The effect of spherical aberration in 2D translates into a bilaplacian
operator
$\nabla_{\perp}^{4}=\partial_{x}^{4}+2\partial_{x}^{2}\partial_{y}^{2}+\partial_{y}^{4}$.
We note that $\nabla_{\perp}^{4}$ is the only fourth order operator that
preserves the rotational symmetry both in real and in Fourier space, which is
consistent with the idea of spherical. The equivalent SLP for
$A\left(\boldsymbol{r}_{\perp},\theta\right)=A_{s}\left(\boldsymbol{r}_{\perp}\right)\exp\left(-i\omega\theta\right)$
reads

$\displaystyle\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace
0$ $\displaystyle=$
$\displaystyle\left(\omega+c_{x}x^{2}+c_{y}y^{2}+b_{x}\partial_{x}^{2}+b_{y}\partial_{y}^{2}+s\nabla_{\perp}^{4}\right)A_{s}.$
(13)

When $s=0$ and the two conditions $b_{x}c_{x}<0$ and $b_{y}c_{y}<0$ are
simultaneously verified, the SLP admits separable solutions that are simply
the product of the HG modes in the $x$ and $y$ directions discussed previously
(SIEGMAN-BOOK, ). We now consider the effect of aberration close to the SIC
when the cavity becomes unstable. Due to the inherent dichroism present in any
realistic experimental system, one can assume that the cavity becomes unstable
first in one direction, say the $x$-direction. Applying the Fourier transform
to Eq. 13 leads to

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left[\omega+c_{x}\partial_{q_{x}}^{2}+c_{y}\partial_{q_{y}}^{2}+\hat{V}\left(\boldsymbol{k}_{\perp}\right)\right]A_{s},$
(14)

where we defined the potential in Fourier space

$\displaystyle\hat{V}\left(\boldsymbol{k}_{\perp}\right)$ $\displaystyle=$
$\displaystyle-
b_{x}k_{x}^{2}-b_{y}k_{y}^{2}+s\left(k_{x}^{2}+k_{y}^{2}\right)^{2}\,.$ (15)

A representation of $\hat{V}\left(\boldsymbol{k}_{\perp}\right)$ for three
different values of $b_{x}$ is given in Fig. 5. We see that the transition
from a stable to an unstable cavity is obtained when $b_{x}$ changes its sign
from negative to positive which gives rise to the appearance of two new minima
as observed in the 1D case in Fig. 1. Their position is obtained by simply
setting $\partial_{k_{x}}\hat{V}=\partial_{k_{y}}\hat{V}=0$ which leads to
$k_{\perp}^{\pm}=\pm\left(\sqrt{\frac{b_{x}}{2s}},0\right)$. As already
discussed, this corresponds to off-axis emission on the $x$-axis and stable
on-axis emission on the $y$-axis. The curvature of
$\hat{V}\left(k_{\perp}\right)$ around the two minima in $k_{\perp}^{\pm}$
upon back-transforming to direct space corresponds to the effective
diffraction experienced by a wave-packet centered around this tilted
wavevector $k_{\perp}^{\pm}$. We obtain

$\displaystyle\frac{1}{2}\frac{\partial^{2}\hat{V}}{\partial k_{x}^{2}}$
$\displaystyle=$ $\displaystyle
2b_{x}\;,\;\frac{1}{2}\frac{\partial^{2}\hat{V}}{\partial
k_{y}^{2}}=b_{x}-b_{y}\;,\;\frac{\partial^{2}\hat{V}}{\partial k_{x}\partial
k_{y}}=0\,.$ (16)

The curvature in the $x$-direction changed from $-b_{x}\rightarrow 2b_{x}$,
similarly to the 1D case. However, the perpendicular direction is also
affected by the off-axis emission, leading to the substitution
$b_{y}\rightarrow b_{y}-b_{x}$. In the case of a cavity that changes behaviour
from stable $b_{x}<0$ to unstable $b_{x}>0$, the off-axis emission along the
$x$-direction that re-stabilizes the emission renders the perpendicular
direction “more diffractive” since $b_{y}-b_{x}<b_{y}$ if $b_{x}>0$. The
results of Fig. 5 entice us to seek again a modulated profile

$\displaystyle A\left(\boldsymbol{r}_{\perp}\right)$ $\displaystyle=$
$\displaystyle
A_{s}\left(\boldsymbol{r}_{\perp}\right)\exp\left[i\left(k_{0}x-\omega\theta\right)\right]$
(17)

with $k_{0}=\sqrt{b_{x}/\left(2s\right)}$. Truncating to second order we get
the approximate SLP problem

$\displaystyle\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace\frac{b_{x}^{2}}{4s}-\omega$
$\displaystyle=$
$\displaystyle\left[c_{x}x^{2}+c_{y}y^{2}-2b_{x}\partial_{x}^{2}+\left(b_{y}-b_{x}\right)\partial_{y}^{2}\right]A_{s},$
(18)

where the modification of the diffraction coefficients in Eq. 18 is fully
consistent with the discussion of the curvature of $\hat{V}$ given in Eq. 16.
The eigenmodes of such an unstable cavity are the product of a modulated HG
mode in the unstable direction and of a regular HG mode in the y-direction. As
such, we define the eigenmode family with two modal indices $\left(n,m\right)$

$\displaystyle\Gamma_{n,m}\left(\boldsymbol{r}_{\perp},\theta\right)$
$\displaystyle=$ $\displaystyle
H_{n}\left(\frac{x}{\sigma_{x}}\right)H_{m}\left(\frac{y}{\sigma_{y}}\right)\cos\left(k_{0}x\right)e^{-i\omega_{n,m}\theta}\,,$
$\displaystyle\Psi_{n,m}\left(\boldsymbol{r}_{\perp},\theta\right)$
$\displaystyle=$ $\displaystyle
H_{n}\left(\frac{x}{\sigma_{x}}\right)H_{m}\left(\frac{y}{\sigma_{y}}\right)\sin\left(k_{0}x\right)e^{-i\omega_{n,m}\theta}\,,$
$\displaystyle\omega_{n,m}$ $\displaystyle=$
$\displaystyle\frac{b_{x}^{2}}{4s}-\frac{2b_{x}}{\sigma_{x}^{2}}\left(2n+1\right)+\dfrac{b_{y}}{\sigma_{y}^{2}}(2m+1)\,,$
(19) $\displaystyle\sigma_{x}$ $\displaystyle=$
$\displaystyle\sqrt{2b_{x}/c_{x}}\,,$ $\displaystyle\sigma_{y}$
$\displaystyle=$ $\displaystyle\sqrt{\left(b_{x}-b_{y}\right)/c_{y}}\,.$

The situation considered in Eq. (1) materializes for instance in the study of
passively mode-locked integrated external-cavity surface-emitting laser
(MIXSELs) in a near-degenerate cavity (HBE-OE-08, ) as depicted in Fig. 6.
Here, the gain (G) and saturable absorber (SA) media are enclosed in a single
micro-cavity and the external mirror is assumed to be ideal while the
collimating lens is not. Our modeling approach is based on a Haus master
equation model for passive mode-locking (PML) adapted to the experimentally
relevant long cavity regime regime (CJM-PRA-16, ; GJ-PRA-17, ; SHJ-PRAp-20, ;
Hausen2020a, ), where the PML pulses become individually addressable temporal
localized states (TLSs). The Haus equation relates the slow evolution of the
three-dimensional intra-cavity field
$E\left(\boldsymbol{r}_{\perp},t,\theta\right)$ to the dynamics of the
population inversion in the gain $N_{1}\left(\boldsymbol{r}_{\perp},t\right)$
and the saturable absorber $N_{2}\left(\boldsymbol{r}_{\perp},t\right)$ as

$\displaystyle\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace\frac{\partial
E}{\partial\theta}$ $\displaystyle\negthickspace=\negthickspace$
$\displaystyle\left[\left(1-i\alpha_{1}\right)N_{1}+\left(1-i\alpha_{2}\right)N_{2}-\kappa+\mathcal{L}\right]E,$
(20)
$\displaystyle\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace\frac{\partial
N_{1}}{\partial t}$ $\displaystyle\negthickspace=\negthickspace$
$\displaystyle\gamma_{1}\left(J_{1}-N_{1}\right)-N_{1}\left|E\right|^{2},\,N_{1}(\boldsymbol{r_{\perp}},0)=J_{1},$
(21)
$\displaystyle\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace\frac{\partial
N_{2}}{\partial t}$ $\displaystyle\negthickspace=\negthickspace$
$\displaystyle\gamma_{2}\left(J_{2}-N_{2}\right)-\hat{s}N_{2}\left|E\right|^{2}\,,N_{2}(\boldsymbol{r}_{\perp},0)=J_{2}.$
(22)

Figure 6: A schematic of the MIXSEL, where both the gain (green) and the
saturable absorption (pink) are contained in the same micro-cavity. It is
coupled face-to-face to a distant external mirror by an imperfect self-imaging
system.

In this formalism, the variable $t\in\left[0,\tau\right]$ represents the
round-trip time in the external cavity and $\theta$ is a second dimensionless
time scale normalized by $\tau$. The latter corresponds to the slow evolution
of the pulse under the combined effect of gain, absorption and spatio-temporal
filtering. The two transverse dimensions are denoted
$\boldsymbol{r}_{\perp}=\left(x,y\right)$ and allow for pattern formation in
the plane perpendicular to the propagation direction of the pulse, cf. Fig. 6.
Note that in the long cavity regime, the spatio-temporal distributions of the
carriers $N_{j}\left(\boldsymbol{r}_{\perp},t\right)$ are slaved to the
evolution of the optical field (CJM-PRA-16, ; Hausen2020a, ) and do not depend
explicitly on the slow timescale $\theta$. In this regime, one can safely
assume a full recovery of the carrier between pulses and a total loss of
memory from one round-trip toward the next one. We define in Eqs. (20)-(22)
$\kappa$ as the round-trip cavity losses, and $\alpha_{j}$ are the linewidth
enhancement factors of the two active media that relax with timescales
$\gamma_{j}^{-1}$ toward the equilibrium values $J_{j}$. The ratio of the
saturation fluence of the gain and of the SA is denoted by $\hat{s}$. A
standard speudo-spectral split-step method was used for the numerical
simulations (GJ-PRA-17, ). The effective cavity _spatio-temporal linear
operator_ $\mathcal{L}$, accounts for the finite gain bandwidth and chromatic
dispersion, but also for non-perfect imaging conditions, diffraction,
parabolic wavefront curvature and mirror losses due to finite aperture.
$\mathcal{L}$ is given by

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle
d_{g}\partial_{t}^{2}+\mathcal{L}_{\perp}\,,$ (23)

where $d_{g}$ is the temporal diffusion coefficient representing the gain
bandwidth.

The Fresnel transform (PB-JOSAA-97, ) permits calculating the transverse
effects occurring at each round-trip analytically from the round-trip (ABCD)
matrix in the paraxial approximation (SIEGMAN-BOOK, ). In presence of
aberrations, its calculation is achieved by expanding the exact (non-paraxial)
operator corresponding to the lens as a sum of an ideal element plus a
deviation. The latter potentially contains all the wavefront curvature
contributions beyond the parabolic approximation. Assuming a large ratio
between the focal length of the mirror and that of the lens permits expressing
the spherical aberrations as a bilaplacian operator, see the Appendix I for
details. The transverse round-trip operator $\mathcal{L}_{\perp}$ is given by

$\displaystyle\mathcal{L}_{\perp}$ $\displaystyle=$ $\displaystyle
icx^{2}+\left(d_{f}+ib\right)\partial_{x}^{2}+is\partial_{x}^{4}\,.$ (24)

The finite size of lenses and the numerical aperture of the whole optical
system is modeled by as a soft aperture and a real transverse diffusion
parameter $d_{f}$. Finally, $b$ is the normalized paraxial diffraction
parameter, $c$ is the normalized parabolic wavefront curvature that are the
off-diagonal elements of the round-trip ABCD matrix (SIEGMAN-BOOK, ) and $s$
is the spherical aberration parameter. We note that moving the lens in Fig. 6
modifies $c$ while moving the mirror modifies both $b$ and $c$.

A qualitative model for the transverse profile of the TLSs such as the one
derived in (J-PRL-16, ; GJ-PRA-17, ) can be obtained by essentially adapting
New’s method for PML (N-JQE-74, ) to the situation at hand. This method
exploits the scale separation occurring between the pulse evolution, the so-
called fast stage in which stimulated emission is dominant, and the slow stage
that is controlled by the gain recovery processes. Assuming, as in (J-PRL-16,
; GJ-PRA-17, ), that the four-dimensional spatio-temporal profile
$E\left(x,t,\theta\right)$ can be factored as
$E\left(x,t,\theta\right)=A\left(x,\theta\right)p\left(t\right)$ with
$p\left(t\right)$ a normalized TLS profile, one obtains a Rosanov equation
(RK-OS-88, ) for the slow evolution of the TLS transverse profile
$A\left(x,\theta\right)$ as

$\displaystyle\negthickspace\negthickspace\partial_{\theta}A$
$\displaystyle\negthickspace=$
$\displaystyle\left[f\left(\left|A\right|^{2}\right)+icx^{2}+\left(d_{f}+ib\right)\partial_{x}^{2}+is\partial_{x}^{4}\right]A.$
(25)

The normalized coefficients $b$, $c$ and $s$ represent the residual paraxial
diffraction, parabolic wavefront curvature and aberration coefficients close
to SIC. Their expression and their derivation are given in the Appendix I. We
define the effective nonlinearity as

$f\left(P\right)=\left(1-i\alpha_{1}\right)J_{1}\,g\left(P\right)+\left(1-i\alpha_{2}\right)J_{2}\,g\left(\hat{s}P\right)-\kappa\,,$
(26)

with $J_{1,2}$ and $\alpha_{1,2}$ denoting the bias and the line-width
enhancement factors of the gain and absorber, $\hat{s}$ the ratio of their
saturation energies and $\kappa$ the round-trip losses. The nonlinear response
of the active material to a pulse is given by
$g\left(P\right)=\left(1-e^{-P}\right)/P$ (N-JQE-74, ; J-PRL-16, ; GJ-PRA-17,
).

Figure 7: (a) Branches of one-dimensional tilted HG modes $\Gamma_{0}$ (red)
and $\Psi_{0}$ (blue) of Eq. (25) as a function of the normalized gain bias
$J_{1}/J_{\mathrm{th}}$. The mode $\Gamma_{0}$ is stable (solid line) between
the fold $F_{2}$ (cyan circle) and the AH point $H_{22}$ as well as between AH
points $H_{23}$ and $H_{24}$ (red squares), respectively. The mode $\Psi_{0}$
gains the stability at the fold $F_{1}$ and remains stable up to the AH
bifurcation point $H_{11}$. (b,c) displays exemplary profiles of $\Gamma_{0}$
and $\Psi_{0}$ for $J_{1}=0.65J_{\mathrm{th}}$ (gray dashed line). Real
(turkis, dashed), imaginary (black, dotted) and intensity fields (blue, red,
solid), respectively, are presented. Parameters are
$\alpha_{1}=1.5,\alpha_{2}=0.5,J_{2}=-0.06,\hat{s}=15,\kappa=0.035,d_{f}=10^{-4},b=0.78,c=4.97\cdot
10^{-4},s=1.0$.

The equations (25,26) provide a unified framework which allow bridging our
results for spatio-temporal dynamics with the former results of (RK-OS-88, ;
VFK-JOB-99, ) for the case of static auto-solitons in bistable
interferometers. There, the function $g\left(P\right)$ should be replaced by
the saturated lineshape transition $\sim\left(1+P\right)^{-1}$. As such, our
discussion of the effect of aberrations close to SIC is equally valid for
temporal solitons, regular mode-locking and CW beams.

In one-dimension, Eq. (1) is recovered for the empty cavity, i.e., $f=0$.
Hence, we expect the emergence of a _stable_ family of $\Gamma_{n}$ and
$\Psi_{n}$ tilted HG modes (10,11) in the _unstable cavity_ , where the
condition $bc<0$ is violated. We performed a bifurcation analysis of Eq. (25)
in one transverse dimension using path-continuation framework pde2path
(uecker2014, ). The results are summarized in Fig. 7 in the unstable cavity
regime, where the peak powers for the fundamental tilted modes $\Gamma_{0}$
(red) and $\Psi_{0}$ (blue) are shown in the panel (a) as a function of the
gain bias $J_{1}$ normalized to the threshold value $J_{th}$. Furthermore, two
exemplary profiles of $\Gamma_{0}$ and $\Psi_{0}$ at the same fixed gain value
(gray dashed line) are depicted in Fig. 7 (b,c), respectively. For both modes,
we observe the typical subcritical transition that leads to bistable TLSs as
detailed in (MJB-PRL-14, ; GJ-PRA-17, ); The high intensity branch is stable
while the lower branch is unstable and creates a separatrix with the stable
off solution. For the parameters chosen, the leftmost limiting fold
bifurcations $F_{1}$ and $F_{2}$ are almost identical for $\Gamma_{0}$ and
$\Psi_{0}$. A small region of the Andronov-Hopf (AH) instability for
$\Gamma_{0}$ exists between points $H_{22}$ and $H_{23}$. In this region, a
small amplitude oscillation is visible in time simulations. Both modes are
limited by the AH bifurcations $H_{11}$ and $H_{24}$, respectively, for high
gain values. We stress that these nonlinear modulated HG solutions that are
solutions of Eqs. (25,26), correspond to a train of TLSs whose profile is a
tilted beam supported by spherical aberrations in an unstable cavity.

Figure 8: (a-c): Intensity profiles of the 2D pattern obtained by the
numerical simulations for three values of $b_{x}=(0.78,\,1.02,\,1.25)$,
respectively at the fixed value of the current $J_{1}=0.65J_{\mathrm{th}}$.
(d-f): Corresponding power spectra in the $(k_{x},k_{y})$ plane. Parameters
are: $b_{y}=-0.39,c_{x}=c_{y}=4.97\cdot 10^{-4}$. Other parameters as in Fig.
7.

In two spatial dimensions, we assume the system to be weakly astigmatic and,
as discussed previously, that the self-imaging condition is not reached
simultaneously for both transverse dimensions. For a cavity that is stable in
the vertical and unstable in the horizontal direction we observed the tilted
localized patterns shown in Fig. 8 (a-c) by solving Eqs. (25,26) numerically.
The typical evolution for large values of $b_{x}$ (i.e., entering more deeply
into the unstable region) is presented together with the corresponding far-
field power spectrum $|\hat{A}|^{2}$. The panels (d-f) shows the two peaks in
the far field related to the value of the transverse horizontal wave-number
$\pm k_{0,x}$. Clearly, an increase of $b_{x}$ leads to an increase of
$k_{0}$, which corresponds to a higher frequency of the mode oscillations in
the near-field. These results would correspond closely to the experimental
situation described either in (BVM-XXX-24, ) for a mode-locked vertical
surface-emitting external-cavity semiconductor laser using a saturable
absorber or in (NathanPHD, ) for a CW broad area laser.

In conclusion, we have investigated the effect of wavefront aberrations in a
degenerated cavity. We found that the interplay between spherical aberrations
and the proximity to the SIC may lead to modulated beams that can support
either CW or temporal solitons in a mode-locked broad-area VECSELs. These
modulated beams are analogous to the eigenmodes of a quantum particle in a
double well potential. They can be analytically approximated by spatially
modulated Hermite-Gauss modes. We linked the wavelength of their modulation to
the parameters of the cavity. We note that the transition from a stable
towards an unstable cavity around SIC can be obtained in two ways: Either for
$c>0$ and changing the sign of $b$ from negative to positive, or for $c<0$ and
changing the sign of $b$ correspondingly. These two situations are not
identical with respect to the effect of spherical aberrations since the off-
axis wavevector $k_{0}=\sqrt{b/2s}$ requires $b$ and $s$ to have the same
sign. As such, the sign of $s$ dictates the situation in which one can observe
these modulated Hermite-Gauss beams. While we focused on the influence of
spherical aberrations, we proposed a general framework that, in principle,
permits calculating the effect of the other Seidel aberrations such as coma,
distortion or field curvature in a cavity close to SIC. We believe that
exhibiting the link between these wavefront curvature defects and their
equivalent representations as spatial operators provides a new framework that
could lead to a range of interesting new research avenues in photonics.
Furthermore, the condition of a large ratio between the focal distances of the
aberrated lens and the non-aberrated other elements could be relaxed. In this
situation, spherical aberrations translate into a non-local spatial operators
that may lead to rich pattern formation scenarios, as observed in other fields
(FKK-PRL-03, ; MPS-PRL-16, ; JMG-PRL-17, ).

###### Acknowledgements.

We thank A. Garnache M. Guidici and M. Marconi for useful discussions
regarding aberrations and self-imaging conditions and for their comments on
this manuscript. F.M and J.J acknowledge funding from the Ministerio de
Economía y Competitividad (PID2021-128910NB-100 AEI/FEDER UE,
PGC2018-099637-B-100 AEI/FEDER UE).

## Appendix I: Spherical aberration close to the SIC

In this appendix we derive how the effect of spherical aberration can be
recast into the form of a bilaplacian differential operator in the paraxial
equation governing the field evolution after each round-trip. We will use the
formalism of wave optics provided by generalized Huygens–Fresnel transform
(HFT) for ABCD systems (SIEGMAN-BOOK, ). We note that the HFT allows for the
composition of parabolic operators, i.e. quadratic wavefront profiles induced
by parabolic lenses as well as paraxial diffraction, since the latter also
corresponds to a parabolic operator in Fourier space. We characterize our
system by an $\mathrm{ABCD}$ round-trip matrix

$\displaystyle W$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}A&B\\\ C&D\end{array}\right)$ (29)

with $\mathrm{det}\left(W\right)=AD-BC=1$. The generalized HFT for the passage
of light through a first-order optical system (PB-JOSAA-97, ) composed of
parabolic elements is given in one transverse dimension by

$\displaystyle O\left(x,l\right)=$ $\displaystyle\sqrt{\frac{-i}{\lambda
B}}e^{ikl}\int_{-\infty}^{\infty}O\left(\xi,0\right)\times$ (30)
$\displaystyle\exp\left[i\frac{\pi}{\lambda
B}\left(A\xi^{2}-2x\xi+Dx^{2}\right)\right]d\xi\,,$

where $O\left(x,0\right)$ is the incoming field passing through the system
characterized by the $\mathrm{ABCD}$ matrix $W$ and $k=2\pi/\lambda$ is the
wave-vector of light. We note that $B$ plays the role of a propagation
distance while $C$ is an inverse of a distance and represents wavefront
curvature. Further, $A,D$ are dimensionless quantities that correspond to
spatial and angular magnification, respectively.

Close to the self-imaging condition we consider the limit $B\rightarrow 0$
which renders Eq. (30) singular. This difficulty can be avoided by factoring
the quadratic form as follows:

$A\xi^{2}-2x\xi+Dx^{2}=A\left(\xi-\frac{x}{A}\right)^{2}+\left(D-\frac{1}{A}\right)x^{2}.$
(31)

Using that $\left(D-\frac{1}{A}\right)/B=C/A$, the Huygens-Fresnel integral
given in Eq. (30) becomes

$\displaystyle O\left(x,l\right)=$ $\displaystyle
e^{ikl}e^{i\frac{\pi}{\lambda}\frac{C}{A}x^{2}}\int_{-\infty}^{\infty}O\left(\xi,0\right)\times$
(32) $\displaystyle\sqrt{\frac{-i}{\lambda B}}\exp\left[i\frac{\pi A}{\lambda
B}\left(\xi-\frac{x}{A}\right)^{2}\right]d\xi\,.$

The equation (32) will be the form of the Huygens–Fresnel integral used in the
rest of the Appendix section.

As mentioned in the main text, we consider the simplest case of the self-
imaging cavity (HBE-OE-08, ) consisting of one lens of focal length $f$ and
one mirror or radius of curvature $R$, separated by distances $d_{1}$ and
$d_{2}$, see Fig. 6. The resulting round-trip propagation matrix reads

$W=D_{1}L_{0}D_{2}MD_{2}L_{0}D_{1}$ (33)

with

$\displaystyle D_{j}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1&d_{j}\\\ 0&1\end{array}\right)\,,$
(36) $\displaystyle L_{0}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1&0\\\
-\frac{1}{f_{0}}&1\end{array}\right)\,,$ (39) $\displaystyle M$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}1&0\\\
-\frac{2}{R}&1\end{array}\right)\,.$ (42)

The SIC for which one finds $W=\mathrm{Id}$ is encountered for the following
values of of $d_{1,2}$

$\displaystyle d_{1}^{*}$ $\displaystyle=$ $\displaystyle
f+\frac{f^{2}}{R}\,,$ (43) $\displaystyle d_{2}^{*}$ $\displaystyle=$
$\displaystyle f+R\,.$ (44)

Expanding to first order in $\delta d_{j}=d_{j}-d_{j}^{\text{*}}$ one obtains

$\displaystyle T$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1&2\left(\delta d_{1}+\delta
d_{2}\dfrac{f^{2}}{R^{2}}\right)\\\ -\dfrac{2}{f^{2}}\delta
d_{2}&1\end{array}\right)\,.$ (47)

We conclude that, close to the SIC, small displacements from the lens induce a
$B$ coefficient, whereas moving the mirror modifies both the values of $B$ and
$C$. Finally, we can safely assume that $A=D=1$ at first order in
$\mathcal{O}\left(\delta d_{j}\right)$. In addition to small deviations from
the SIC, we consider that the focal length of the lens depends on its radial
position and we model the spherical aberration as
$f\left(x\right)=f_{0}+\sigma x^{2}$. Denoting the field profile before and
after the lens as $E_{i}$ and $E_{o}$, respectively, we have

$\displaystyle E_{o}\left(x\right)$ $\displaystyle=$ $\displaystyle
e^{-i\frac{\pi}{\lambda}\frac{x^{2}}{f}}E_{i}\left(x\right).$ (48)

Denoting the operator corresponding to the effect of the lens by $L$, we
separate the contribution from the unperturbed lens with focal length $f_{0}$
and matrix $L_{0}$ as

$L=L_{0}+\delta L\,.$ (49)

Using that

$\frac{1}{f\left(x\right)}-\frac{1}{f_{0}}\simeq-\left(\frac{\sigma}{f_{0}^{2}}\right)x^{2}$
(50)

we write Eq. 48 as

$E_{o}\left(x\right)=e^{-i\frac{\pi}{\lambda}\frac{x^{2}}{f_{0}}}\left[1+\left(e^{\frac{i\pi\sigma}{\lambda
f_{0}^{2}}x^{4}}-1\right)\right]E_{i}\left(x\right),$ (51)

which allows to express the action of the aberration operator $\delta
L:E_{i}\rightarrow E_{o}$ as

$E_{o}\simeq\left[\exp\left(i\frac{\pi}{\lambda}\frac{\sigma}{f_{0}^{2}}x^{4}\right)-1\right]\exp\left(-i\frac{\pi}{\lambda}\frac{x^{2}}{f_{0}}\right)E_{i}\,.$
(52)

Finally, we will employ the proximity of the SIC to simplify the
representation of the round-trip operator in the presence of the aberrations.
The full operator $W$ (cf. Eq. (33)) is given by

$W=D_{1}LD_{2}MMD_{2}LD_{1}\,.$

Using Eq. (49) we can expand $W$ as

$\displaystyle W$ $\displaystyle=$ $\displaystyle W_{0}+\delta W\,,$ (53)

where we defined the unperturbed round-trip operator as
$W_{0}=D_{1}L_{0}D_{2}MD_{2}L_{0}D_{1}$. The expression of $\delta W$ reads
after simplification and to the first order in $\mathcal{O}\left(\delta
L\right)$

$\delta W=D_{1}\left(\delta
L\right)L_{0}^{-1}D_{1}^{-1}W_{0}+W_{0}D_{1}^{-1}L_{0}^{-1}\left(\delta
L\right)D_{1}\,.$ (54)

Further, we can simplify the dependence on $\delta L$ in Eq. (54) using the
exact self-imaging condition which amounts to setting $W_{0}=\mathrm{Id}$ in
Eq. (54). Indeed, since $\delta L$ is already a small quantity, the error
incurred in setting $W_{0}=\mathrm{Id}$ will be second order. We are left
calculating the two contributions of aberrations to the round-trip field
evolution. We have $W=F_{1}+F_{2}$ with

$\displaystyle F_{1}$ $\displaystyle=$ $\displaystyle D_{1}\left(\delta
L\right)L_{0}^{-1}D_{1}^{-1}\,,\,F_{2}=D_{1}^{-1}L_{0}^{-1}\left(\delta
L\right)D_{1}.$ (55)

In both cases, we can express $F_{j}$ as a double integral involving the HFT
that involve the spherical aberration of the lens. We combine the two steps
corresponding to $\gamma_{1\rightarrow 2}=L_{0}^{-1}D_{1}^{-1}$ and
$\gamma_{2\rightarrow 1}=D_{1}^{-1}L_{0}^{-1}$ into a single Fresnel
transform. Further, we employ the SIC to find

$\displaystyle\gamma_{1\rightarrow 2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1&-f\left(1+\frac{f}{R}\right)\\\
\frac{1}{f}&-\frac{f}{R}\end{array}\right)\,,$ (58)
$\displaystyle\gamma_{2\rightarrow 1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}-\frac{f}{R}&-f\left(\frac{f}{R}+1\right)\\\
\frac{1}{f}&1\end{array}\right)\,.$ (61)

As a last approximation, we consider the limit of a long cavity for which the
focal length of the mirror is large in comparison with the focal length of the
collimator lens. As such, we can define $\varepsilon=f/R\ll 1$. This allows to
approximate the operators $\gamma_{j}$ as

$\displaystyle\gamma_{1\rightarrow 2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1&-f_{0}\\\
\frac{1}{f_{0}}&0\end{array}\right)+\mathcal{O}\left(\varepsilon\right)\,,$
(64) $\displaystyle\gamma_{2\rightarrow 1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&-f_{0}\\\
\frac{1}{f_{0}}&1\end{array}\right)+\mathcal{O}\left(\varepsilon\right)\,.$
(67)

Writing the action of $F_{1}$ using three integral transforms
$F_{1}:E_{0}\rightarrow E_{3}$ leads to

$\displaystyle E_{1}\left(x\right)$ $\displaystyle=$ $\displaystyle
e^{i\frac{\pi}{\lambda
f_{0}}x^{2}}\int_{-\infty}^{\infty}E_{0}\left(\xi\right)\sqrt{\frac{i}{\lambda
f_{0}}}\exp\left[-i\frac{\pi}{\lambda
f_{0}}\left(\xi-x\right)^{2}\right]d\xi,$ $\displaystyle E_{2}\left(x\right)$
$\displaystyle=$ $\displaystyle\left(e^{i\frac{\pi}{\lambda}\frac{\sigma
x^{4}}{f_{0}^{2}}}-1\right)e^{-i\frac{\pi}{\lambda
f_{0}}x^{2}}E_{1}\left(x\right),$ (68) $\displaystyle E_{3}\left(x\right)$
$\displaystyle=$ $\displaystyle\sqrt{\frac{-i}{\lambda
f_{0}}}\int_{-\infty}^{\infty}E_{2}\left(\xi\right)\exp\left[i\frac{\pi}{\lambda
f_{0}}\left(\xi-x\right)^{2}\right]d\xi\,.$

The action of $F_{1}$ can be expressed by the following kernel

$\displaystyle E_{3}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}E_{0}\left(z\right)K_{1}\left(x,z\right)dz\,,$
(69)

where the kernel $K_{1}$ is defined as

$\displaystyle K_{1}\left(x,z\right)=$ $\displaystyle\frac{1}{\lambda
f_{0}}e^{-i\frac{\pi}{\lambda f_{0}}\left(z-x\right)\left(z+x\right)}$ (70)
$\displaystyle\times\int_{-\infty}^{\infty}\left(e^{i\frac{\pi}{\lambda}\frac{\sigma
y^{4}}{f_{0}^{2}}}-1\right)e^{i\frac{2\pi}{\lambda
f_{0}}\left(z-x\right)y}dy\,.$

For small aberration, we can expand $K_{1}$ at first order in $\sigma$ which
leads to

$K_{1}\left(x,z\right)\simeq\frac{i\pi\sigma}{\lambda^{2}f_{0}^{3}}e^{-i\frac{\pi}{\lambda
f_{0}}\left(z-x\right)\left(z+x\right)}\int_{-\infty}^{\infty}y^{4}e^{i\frac{2\pi}{\lambda
f_{0}}\left(z-x\right)y}dy\,.$

Using $t=2\pi y/\left(\lambda f_{0}\right)$ we find

$\displaystyle K_{1}\left(x,z\right)$ $\displaystyle=$ $\displaystyle
i\frac{\sigma f_{0}^{2}}{2k^{3}}e^{-i\frac{\pi}{\lambda
f_{0}}\left(z-x\right)\left(z+x\right)}\delta^{\left(4\right)}\left(z-x\right)\,,$
(71)

where we used the fact that

$\int_{-\infty}^{\infty}t^{n}e^{i\omega
t}dt=2\pi\left(-i\right)^{n}\delta^{\left(n\right)}\left(\omega\right)\,.$
(72)

Here, we defined $\delta^{\left(n\right)}$ as the $n^{\mathrm{th}}$ derivative
of the Dirac delta which associates with the local value of the
$n^{\mathrm{th}}$ derivative of a function. Since $\delta^{\left(n\right)}$ is
a well-localized function, we can set $z=x$ and obtain

$K_{1}\left(x,z\right)=i\frac{\sigma
f_{0}^{2}}{2k^{3}}\delta^{\left(4\right)}\left(z-x\right)\,.$ (73)

Performing the convolution with $\delta^{\left(n\right)}$ is identical to
taking the fourth derivative in direct space which yields the fourth
derivative contribution. The action of $F_{1}$ to the first order is

$\displaystyle F_{1}$ $\displaystyle:$ $\displaystyle E_{0}\rightarrow
E_{3}\,,$ (74) $\displaystyle E_{3}\left(x\right)$ $\displaystyle=$
$\displaystyle i\frac{\sigma
f_{0}^{2}}{2k^{3}}\partial_{x}^{4}E_{0}\left(x\right)\,.$

The action of $F_{2}$ can be found in a similar fashion using three integral
transforms $F_{2}:E_{0}\rightarrow E_{3}$ and reads

$\displaystyle E_{1}\left(x\right)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}E_{0}\left(\xi,0\right)\sqrt{\frac{-i}{\lambda
f_{0}}}\exp\left[i\frac{\pi}{\lambda f_{0}}\left(\xi-x\right)^{2}\right]d\xi,$
$\displaystyle E_{2}\left(x\right)$ $\displaystyle=$
$\displaystyle\left(e^{i\frac{\pi}{\lambda}\frac{\sigma
x^{4}}{f_{0}^{2}}}-1\right)e^{-i\frac{\pi}{\lambda
f_{0}}x^{2}}E_{1}\left(x\right),$ (75) $\displaystyle E_{3}\left(x\right)$
$\displaystyle=$ $\displaystyle\sqrt{\frac{i}{\lambda
f_{0}}}\int_{-\infty}^{\infty}E_{2}\left(\xi\right)\exp\left[-i\frac{\pi}{\lambda
f_{0}}\left(-2x\xi+x^{2}\right)\right]d\xi\,.$

Similarly, the action of $F_{2}$ can be expressed as a integral

$E_{3}=\int_{-\infty}^{\infty}E_{0}\left(z,0\right)K_{2}\left(x,z\right)dz\,,$
(76)

where we defined the kernel $K_{2}$ as

$\displaystyle K_{2}\left(x,z\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda f_{0}}e^{i\frac{\pi}{\lambda
f_{0}}\left(z-x\right)\left(z+x\right)}$ $\displaystyle\times$
$\displaystyle\int_{-\infty}^{\infty}\left(e^{i\frac{\pi}{\lambda}\frac{\sigma
y^{4}}{f_{0}^{2}}}-1\right)\exp\left[i\frac{2\pi}{\lambda
f_{0}}y\left(x-z\right)\right]dy\,.$

Expanding analogously the aberration contribution to first order in $\sigma$
and using Eq. (72), we obtain the desired kernel

$\displaystyle K_{2}\left(x,z\right)$ $\displaystyle=$ $\displaystyle
i\frac{\sigma f_{0}^{2}}{2k^{3}}\delta^{\left(4\right)}\left(z-x\right)\,.$
(78)

We conclude that, to the first order in $\sigma$, the action of $F_{2}$ is
identical to that of $F_{1}$

$\displaystyle F_{2}:E_{0}$ $\displaystyle\rightarrow$ $\displaystyle
E_{3}\,,$ (79) $\displaystyle E_{3}\left(x\right)$ $\displaystyle=$
$\displaystyle i\frac{\sigma
f_{0}^{2}}{2k^{3}}\partial_{x}^{4}E_{0}\left(x\right)\,.$

In summary, the total effect due to spherical aberration close to self-imaging
in the one-dimensional case reads

$E_{3}\left(x\right)=i\frac{\sigma
f_{0}^{2}}{k^{3}}\partial_{x}^{4}E_{0}\left(x\right)\,.$ (80)

The generalization of these calculations to two-dimensions proceeds without
difficulties and yields

$E_{3}\left(r_{\perp}\right)=i\frac{\sigma
f_{0}^{2}}{k^{3}}\nabla_{\perp}^{4}E_{0}(r_{\perp})\,.$ (81)

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distribution $T\in\mathcal{D}^{\prime}(\Omega)$, defined as
$T(\phi)=\sum_{|\alpha|=\rho+1}x^{\alpha}T(\psi_{\alpha})$ satisfies
$T|_{\Omega-O}=T_{0}$. It can be shown that
$\operatorname{deg}(T)=\operatorname{deg}(T_{0})$ [1]. Let
$T_{1}\in\mathcal{D}^{\prime}(\Omega)$ be an extension of $T_{0}$ such that
$\operatorname{deg}(T_{1})=\operatorname{deg}(T_{0})$. Then
$(T_{1}-T)\in\mathcal{E}(\Omega)$ with
$\operatorname{deg}(T_{1}-T)\leq\operatorname{deg}(T_{0})$, and we can write
$T_{1}-T=\Sigma_{|\alpha|\leq\operatorname{deg}({T_{0}})}q^{\alpha}\partial^{\alpha}\delta_{O},$
where $q^{\alpha}\in\mathbb{R}$ (Lemma 2.4).

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