positive fresh MDBPs by creating a bounded derivation of maximal size $m^{2}$.

To this end, instead of $\textit{In}_{p}$ we introduce relations
$\textit{In}^{i}_{p}$ for each $i$ up to $m^{2}$ to record that a fresh
instance can reach place $p$ in $i$ steps. That is,
$\textit{In}^{i}_{p}({\bar{s}})$ is true iff a fresh instance with
$\textit{In}({\bar{s}})$-record can reach place $p$ in $i$ steps.

Next we adapt the rules.

Traversal Rules For each transition $t$ from a place $q$ to a place $p$ and
for each $i$ up to $m^{2}$ we adapt a _traversal rule_ as follows:

$\textit{In}^{i+1}_{p}({\bar{s}})\leftarrow E_{t}^{i}({\bar{s}})$

Note that, as a difference from the general case, here $E_{t}^{i}({\bar{s}})$
denotes the execution condition evaluated over the initial database (rather
than on the extended database as $E^{\prime}_{t}({\bar{s}})$ would denote),
and where $\textit{In}({\bar{s}})$ condition is replaced with
$\textit{In}^{i}_{q}({\bar{s}})$. The database relations are not changed.

Generation Rules Similarly, for each transition $t$ from above we introduce
the following _generation rule_ :

$R^{\prime}(\bar{u})\leftarrow
E_{t}^{i}({\bar{s}}),B_{t}^{i}({\bar{s}}^{\prime},\bar{u}),{\bar{s}}={\bar{s}}^{\prime}$

As pointed in the observations negation in $E_{t}$ and $B_{t}$ does not make
reasoning more complex since negation is on the base relations that are not
updated by the process.

Summary Let $\Pi_{\mathcal{P}}^{\textsl{ro,fr}}$ be the non-recursive Datalog
program with stratified negation that encodes the rowo process $\mathcal{P}$
obtained from $\Pi_{\mathcal{P}}^{\textsl{po,fr}}$ substituting the traversal
and generation rules with the rules above. The rest of the program is the same
as in the positive variant.

######  Lemma C.1.

Let $R^{\prime}(\bar{u})$ be a fact defined over $\textit{adom}^{*}$, then the
following is equivalent:

  * •

there is an execution in $\mathcal{B}$ that produces $R(\bar{u})$

  * •

$\Pi_{\mathcal{P}}^{\textsl{ro,fr}}\cup\mathcal{D}\models R^{\prime}(\bar{u})$

Let $\Pi_{Q}^{\textsl{test}}$ be the test program based on a query $Q$ as
defined for the general variant.

######  Theorem C.2.

The following is equivalent:

  * •

$Q$ is instable in $\mathcal{B}$;

  * •

$\Pi_{\mathcal{P}}^{\textsl{ro,fr}}\cup\mathcal{D}\cup\Pi_{Q}^{\textsl{test}}\models\textit{Instable}$

##### Rowo Closed MDBPs

We now consider a possibly cyclic rowo
$\mathcal{B}=\langle\mathcal{P},\mathcal{I},\mathcal{D}\rangle$ under closed
executions. We adapt the encoding from the acyclic closed variant (which can
be obtained from closed positive). The main difference is that each instance
is encoded independently of the others. I.e., we encode an execution of a
single instance as a tuple $\bar{\omega}$ of the form

$\bar{\omega}=\langle o;t_{h_{1}},\dots,t_{h_{i}}\rangle.$

meaning that instance $o$ traverses first $t_{h_{1}}$ then $t_{h_{2}}$, and so
on.

Similarly we adapt $R^{i}$’s and $\textit{State}^{i}$’s from the general case
such that:

  * •

$R^{i}(o;t_{h_{1}},\dots,t_{h_{i}};{\bar{s}})$ denotes that the instance $o$
after traversing $t_{h_{1}},\dots,t_{h_{i}}$ produces $R({\bar{s}})$; and

  * •

$\textit{State}^{i}(o;t_{1},\dots,t_{i};p)$ denotes that the instance $o$
after traversing $t_{1},\dots,t_{i}$ is located at place $p$.

Similarly to the previous variant, cycles can be dealt with bounded
derivations of maximal length $m^{2}$, so $i$ ranges from $1,\dots,m^{2}$.
Then similarly to the closed variant, we use $\textit{In}^{0}(o;{\bar{s}})$ to
associate instance $o$ with the input record $\textit{In}({\bar{s}})$. In this
way, we obtain the facts that can be produced by each instance. Then we
introduce additional rules that combine facts produced by different instances.
Assume we are given a query $Q(\hbox to0.0pt{\hskip
2.0347pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$X$}})\leftarrow
R_{1}(\bar{u}_{1}),\dots,R_{n}(\bar{u}_{n})$ that we want to check for
stability. To this end, we introduce the following relations.

  * •

Path is a relation with arity $m^{2}+1$ that contains legal paths of an
instance. $\textit{Path}(o;\bar{t},\bar{\epsilon})$ is true iff $\bar{t}$ is a
legal path in $\mathcal{P}$ for instance $o$. For technical reasons we
introduce $\epsilon$ to denote an empty transition. Then, $\bar{\epsilon}$ is
vector of $\epsilon$ that we use to fill in remaining positions in Path
($|\bar{\epsilon}|=m-|\bar{t}|$).

  * •

$R^{\prime}$ is an auxiliary relation of size $1+m^{2}+\textit{arity\/}(R)$
that we introduce for each $R$ in $\mathcal{B}$ to store $R$-facts produced by
an instance. That is, $R^{\prime}(o;\bar{t},\bar{\epsilon};{\bar{s}})$ in true
iff $R({\bar{s}})$ is produced after $o$ traversed $\bar{t}$.

  * •

$\textit{Exec}^{j}$ are relations of arity $(m^{2}\times j)+j$ for every
$j=1,\dots,k$ that combines legal paths for different $n$ instances where $n$
is is the number of atoms in the query. Then,
$\textit{Exec}^{j}(o_{1},\bar{t}_{1},\dots,o_{j},\bar{t}_{j})$ is true iff
tuple $\bar{t}_{l}$ is a legal path for instance $o_{l}$ and if $o_{h}=o_{l}$
then $\bar{t}_{h}=\bar{t}_{l}$. This relation we use to record all
combinations of instances that can contribute to create a new query answer,
thus if two facts are produced by the same instance ($o_{h}=o_{l}$) then the
facts have to be produced on the same legal path ($\bar{t}_{h}=\bar{t}_{l}$).

Again $i$ ranges from $1,\dots,m^{2}$. Now we define a program that generates
those relations.

Initialization Rules First we adapt the _initialization rules_ for a single
instance:

$\displaystyle\textit{State}^{0}(o;q)\leftarrow\textit{true}\text{ iff $q$ is
the starting place of instance $o$, and }$ $\displaystyle R^{0}(O;\hbox
to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}})\leftarrow\textit{In}^{\prime}(O;{\rule{5.0pt}{0.5pt}\hskip
0.50003pt}),R(\hbox to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}}).$

Traversal Rules Similarly, we adapt traversal rules to be for a single
instance, as follows:

$\displaystyle\textit{State}^{i+1}(O;\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}},t;p)\leftarrow\textit{State}^{i}(O;\hbox
to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}};q),E_{t}(O)$

for each transition $t$ from place $q$ to place $p$ and $E_{t}(O)$ is the same
as $E_{t}$ except that each atom $\textit{In}({\bar{s}})$ is replaced with
$\textit{In}^{0}(O,{\bar{s}})$ and  $\overline{\hbox{}}$$T$ is a vector of
different variables of size $i$.

Generation Rules For a transition $t$ with writing rule $W_{t}\colon
R(\bar{u})\leftarrow B_{t}(\bar{u})$, the generation rules become:

$\displaystyle R^{i+1}(O;\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}},t;\bar{u})$
$\displaystyle\leftarrow\textit{State}^{i+1}(O;\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}},t;p),B_{t}(O;\hbox
to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}};\bar{u}).$

Here $B_{t}(O;\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}};\bar{u})$
is obtained analogously to $E_{t}(O)$.

Copy Rules Then we adapt the copy rules as follows:

$\displaystyle R^{i+1}(O;\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}},t;\hbox
to0.0pt{\hskip
1.459pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$U$}})$
$\displaystyle\leftarrow R^{i}(O;\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}};\hbox
to0.0pt{\hskip
1.459pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$U$}}),\text{
and }$ $\displaystyle R^{\prime}(O;\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}},\bar{\epsilon};\hbox
to0.0pt{\hskip
1.459pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$U$}})$
$\displaystyle\leftarrow R^{i}(O;\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}};\hbox
to0.0pt{\hskip
1.459pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$U$}})$

where the size of $\epsilon$-vector $|\bar{\epsilon}|=m^{2}-i$.

Summary We denote the above program as $\Pi_{\mathcal{P}}^{\textsl{ro,cl}}$.
Then we have that the following holds:

######  Lemma C.3.

Let $o$ be an instance in $\mathcal{B}$, a list of transitions $\bar{t}$ in
$\mathcal{B}$ of size $i$, and $R(\bar{u})$ a fact, then the following is
equivalent:

  * •

after $o$ traverses $\bar{t}$ the fact $R(\bar{u})$ is produced;

  * •

$\Pi_{\mathcal{P}}^{\textsl{ro,cl}}\cup\mathcal{D}\models
R^{i}(o;\bar{t};\bar{u})$.

Combining Rules Then we need to combine the atoms produced by the different
instances, e.g.,

$R^{i}_{1}(o_{1};\bar{t}_{1};{\bar{s}}_{1})$ $\vdots$
$R^{i}_{n}(o_{n};\bar{t}_{n};{\bar{s}}_{n}).$

and ensure that atoms produced by one instance are all produced following one
path. To do this we need to ensure that if
$o_{i}=o_{j}\Rightarrow\bar{t}_{i}=\bar{t}_{j}$. This is achieved with the
_combining rules_.

First we copy all legal paths of an instance from $\textit{State}^{i}$ into
Path relation:

$\displaystyle\textit{Path}(O;\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}},\bar{\epsilon})\leftarrow\textit{State}^{i}(O;\hbox
to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}};{\rule{5.0pt}{0.5pt}\hskip
0.50003pt})\text{ where }|\epsilon|=m^{2}-i.$

Then we initialize $\textit{Exec}^{1}$ with all legal paths of an instance.

$\textit{Exec}^{1}(O;\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}})\leftarrow\textit{Path}(O;\hbox
to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}).$

Then we combine different paths in the following way. If instance $O_{l}$
executes $\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{l}$
and the same instance executes $\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{i+1}$
then $\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{l}$
and $\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{i+1}$
must be the same. This is captured with the following rules:

$\displaystyle\textit{Exec}^{j+1}(O_{1},$ $\displaystyle\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{1},\dots,O_{l},\hbox
to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{l},\dots,O_{j},\hbox
to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{j},O_{j+1},\hbox
to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{j+1})\leftarrow$
$\displaystyle\textit{Exec}^{j}(O_{1},\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{1},\dots,O_{l},\hbox
to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{l},\dots,O_{j},\hbox
to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{j}),$
$\displaystyle\textit{Path}(O_{j+1};\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{j+1}),$
$\displaystyle O_{l}=O_{j+1},\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{l}=\hbox
to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{j+1}$

for every $l\in 1,\dots,i$.

If the instance $O_{j+1}$ is different from all the other instances
$O_{1},\dots,O_{j}$ then executing path of $O_{j+1}$ can be any legal path

$\displaystyle\textit{Exec}^{j+1}(O_{1},$ $\displaystyle\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{1},\dots,O_{l},\hbox
to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{l},\dots,O_{j},\hbox
to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{j},O_{j+1},\hbox
to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{j+1})\leftarrow$
$\displaystyle\textit{Exec}^{j}(O_{1},\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{1},\dots,O_{l},\hbox
to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{l},\dots,O_{j},\hbox
to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{j}),$
$\displaystyle\textit{Path}(O_{j+1};\hbox to0.0pt{\hskip
1.11632pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$T$}}_{j+1}),$
$\displaystyle\neg(O_{1}=O_{j+1}),\neg(O_{2}=O_{j+1}),\dots,\neg(O_{j}=O_{j+1}).$

$Q^{\prime}$-rule Then, $Q^{\prime}$ collects what has been produced for
relations $R_{1},\dots,R_{n}$ for the give query $Q(\hbox to0.0pt{\hskip
2.0347pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$X$}})\leftarrow
R_{1}(\bar{u}_{1}),\dots,R_{n}(\bar{u}_{n})$ with the rule

$\displaystyle Q^{\prime}(\hbox to0.0pt{\hskip
2.0347pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$X$}})\leftarrow$
$\displaystyle\textit{Exec}^{n}(O_{1},\hbox to0.0pt{\hskip
2.91666pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$W$}}_{1},\dots,O_{n},\hbox
to0.0pt{\hskip
2.91666pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$W$}}_{n}),$
$\displaystyle R^{\prime}_{1}(O_{1};\hbox to0.0pt{\hskip
2.91666pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$W$}}_{1};\bar{u}_{1}),$
$\displaystyle\dots$ $\displaystyle R^{\prime}_{n}(O_{n};\hbox to0.0pt{\hskip
2.91666pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$W$}}_{n};\bar{u}_{n}).$

Test Rule The _test rule_ is then as before:

$\textit{Instable}\leftarrow Q^{\prime}(\hbox to0.0pt{\hskip
2.0347pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$X$}}),\neg
Q(\hbox to0.0pt{\hskip
2.0347pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$X$}}).$

Summary Let us denote with $\Pi_{\mathcal{P},Q}^{{\textsl{test,ro}}}$ the
testing program for $Q$ defined above. The program is non-recursive Datalog
with stratified negation.

Let $\mathcal{D}_{\mathcal{I}}$ be the database that encodes the instance part
In and that contains database $\mathcal{D}$.

######  Theorem C.4.

The following are equivalent:

  * •

$Q$ is instable in $\mathcal{B}$ under closed executions;

  * •

$\Pi_{\mathcal{P}}^{\textsl{ro,cl}}\cup\mathcal{D}_{\mathcal{I}}\cup\Pi_{\mathcal{P},Q}^{{\textsl{test,ro}}}\models\textit{Instable}$.

##### Rowo MDBPs

For arbitrary rowo, similarly to the positive variants, the encoding is
obtained combining the encodings for the fresh and the closed variants.

To combine what comes from the instances in the process and the new ones it is
enough to add rules that will combine the program for closed
($\Pi_{\mathcal{P},Q}^{\textsl{ro,cl}}$) and
fresh($\Pi_{\mathcal{P},Q}^{\textsl{ro,fr}}$).

To this end, for a given query $Q(\hbox to0.0pt{\hskip
2.0347pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$X$}})\leftarrow
R_{1}(\bar{u}_{1}),\dots,R_{n}(\bar{u}_{n})$, we introduce relations:

  * •

$B^{i}_{Q}$ of arity $\textit{arity\/}(R_{1})+\dots+\textit{arity\/}(R_{n})$
that contains on the first $i$ arguments what comes from a mixture of existing
and new process instances while the others come only from existing process
instances, for $i=1,\dots,n$.

##### Encoding into Non-Recursive Datalog

Let $\mathcal{B}=\langle\mathcal{P},\mathcal{I},\mathcal{D}\rangle$ be a rowo
MDBP.

Now we define rules that compute relations introduced above.

First we consider what is produced by the running instances

$\displaystyle B^{0}_{Q}(\hbox to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}}_{1},\dots,\hbox
to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}}_{n})\leftarrow$
$\displaystyle\textit{Exec}^{n}(O_{1},\hbox to0.0pt{\hskip
2.91666pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$W$}}_{1},\dots,O_{n},\hbox
to0.0pt{\hskip
2.91666pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$W$}}_{n}),$
$\displaystyle R^{\prime}_{1}(O_{1};\hbox to0.0pt{\hskip
2.91666pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$W$}}_{1};\bar{u}_{1}),$
$\displaystyle\dots$ $\displaystyle R^{\prime}_{n}(O_{n};\hbox to0.0pt{\hskip
2.91666pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$W$}}_{n};\bar{u}_{n}).$

Then, for the $i$-th atom we both consider the case in which it was produced
by a new instance (1) and the case it was produced by the instances already in
the process (2). These cases are added to the combinations obtained for the
atoms from $1$ to $i-1$. We do this for every $i=1,\dots,n$.

$\displaystyle B^{i}_{Q}(\dots,$ $\displaystyle\hbox to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}}_{i-1},\hbox
to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}}_{i},\hbox
to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}}_{i+1},\dots)\leftarrow$
(4) $\displaystyle B^{i-1}_{Q}(\dots,\hbox to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}}_{i-1},{\rule{5.0pt}{0.5pt}\hskip
0.50003pt},\hbox to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}}_{i+1},\dots),R^{\prime}_{i}(\hbox
to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}}_{i})$
$\displaystyle B^{i}_{Q}(\dots,$ $\displaystyle\hbox to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}}_{i-1},\hbox
to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}}_{i},\hbox
to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}}_{i+1},\dots)\leftarrow$
(5) $\displaystyle B^{i-1}_{Q}(\dots,\hbox to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}}_{i-1},\hbox
to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}}_{i},\hbox
to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}}_{i+1},\dots).$

Then, we add the $Q^{\prime}$-_rule_ to collect what has been produced by the
process for relations $R_{1}(\bar{u}_{1}),\dots,R_{n}(\bar{u}_{n})$ as
follows:

$\displaystyle Q^{\prime}(\hbox to0.0pt{\hskip
2.0347pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$X$}})\leftarrow
B^{n}_{Q}(\hbox to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}}_{1},\dots,\hbox
to0.0pt{\hskip
1.51389pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$Y$}}_{n}).$

The above rules extend the testing program
$\Pi_{\mathcal{P},Q}^{{\textsl{test,ro}}}$ for normal cyclic arbitrary closed.
We denote the new testing program with
$\Pi_{\mathcal{P},Q}^{{\textsl{test,ro,op}}}$.

  * •

$Q$ is instable in $\mathcal{B}$;

  * •

$\Pi_{\mathcal{P}}^{\textsl{ro,cl}}\cup\Pi_{\mathcal{P}}^{\textsl{ro,fr}}\cup\mathcal{D}_{\mathcal{I}}\cup\Pi_{\mathcal{P},Q}^{{\textsl{test,ro,op}}}\models\textit{Instable}$.

Then we set
$\Pi_{\mathcal{P},Q}^{\textsl{ro}}=\Pi_{\mathcal{P}}^{\textsl{ro,cl}}\cup\Pi_{\mathcal{P}}^{\textsl{ro,fr}}\cup\Pi_{\mathcal{P},Q}^{{\textsl{test,ro,op}}}$,
and the claim follows from there.

####  C.2. Proof of Proposition 9.2

######  Proof C.5.

Assume the instance $o$ is at place $p$ and it has an input record
$I(\bar{s})=M_{S}(o)$. To show the claim it is sufficient to guess a closed
execution $\Upsilon$ consisting of the traversals by $o$, and then verify
whether atoms $A_{1},\ldots,A_{n}$ can be produced by such execution. In the
case of singleton rowo MDBPs under closed executions, a closed execution is
uniquely determined by a path in $\mathcal{P}$. Thus, we guess a path
$t_{1},\ldots,t_{m}$ in $\mathcal{P}$ that starts in $p$. This guess is
polynomial in the size of $\mathcal{P}$. For all transitions on the path, we
further guess assignments $\alpha_{1},\ldots,\alpha_{m}$ for the execution
conditions $E_{t_{1}},\ldots,E_{t_{m}}$. Then for each writing rule
$W_{t_{i}}$ of the transition $t_{i}$ we guess up to $n$ assignments
$\beta^{l}_{t_{i}}$ for $1\leq l\leq n$, because a rule may need to produce
more than one fact, but no more than the size of the query. In principle, only
a subset of $W_{t_{1}},$ …, $W_{t_{m}}$ may be needed to produce atoms
$A_{1},\ldots,A_{n}$. Wlog we can guess the assignments for all. Now we
verify. Firstly, we verify whether the path can be traversed by the instance.
This is, if for every execution condition $E_{t_{i}}$ the ground query
$\alpha_{i}E_{t_{i}}$ evaluates to true in
$\mathcal{D}\cup\\{\textit{In}(\bar{s})\\}$. Secondly, we verify whether
$A_{1},\ldots,A_{n}$ are produced on the path by checking if for every $A_{i}$
there exist a writing rule $W_{t_{j}}\colon A_{t_{j}}\leftarrow B_{t_{j}}$ and
assignment $\beta^{l}_{t_{j}}$ such that the ground query
$\beta^{l}_{t_{j}}B_{t_{j}}$ evaluates to true in
$\mathcal{D}\cup\\{\textit{In}(\bar{s})\\}$ and $A_{i}$ is equal with the head
of the writing rule $\beta^{l}_{t_{j}}A_{t_{j}}$. Since all guesses and checks
are polynomial in the size of $\mathcal{B}$ the claim follows directly.


Further author information: (Send correspondence to Meiji M. Nguyen: E-mail:
<EMAIL_ADDRESS>
# GPI 2.0: Optical Designs for the Upgrade of the Gemini Planet Imager
Coronagraphic system

Meiji M. Nguyen  Space Telescope Science Institute, 3700 San Martin Dr,
Baltimore, MD 21218, USA  Bryony F. Nickson  Space Telescope Science
Institute, 3700 San Martin Dr, Baltimore, MD 21218, USA  Emiel H. Por  Space
Telescope Science Institute, 3700 San Martin Dr, Baltimore, MD 21218, USA
Rémi Soummer  Space Telescope Science Institute, 3700 San Martin Dr,
Baltimore, MD 21218, USA  John G. Hagopian  NASA Goddard Space Flight Center,
8800 Greenbelt Rd, Greenbelt, MD 20771, USA  Lambda Consulting LLC/Advanced
NanoPhotonics, 4437 Windsor Farm Rd Harwood, MD 20776, USA  Bruce Macintosh
Stanford University, 450 Serra Mall, Stanford, CA 94305, USA  Jeffrey Chilcote
University of Notre Dame, 1220 E. Angela Blvd. South Bend, IN 46617, USA
Laurent Pueyo  Space Telescope Science Institute, 3700 San Martin Dr,
Baltimore, MD 21218, USA  Marshall Perrin  Space Telescope Science Institute,
3700 San Martin Dr, Baltimore, MD 21218, USA  Quinn Konopacky  University of
California, San Diego, 5998 Alcala Park, San Diego, CA 92110, USA

###### Abstract

The Gemini Planet Imager (GPI) is an integral field spectrograph (IFS) and
coronagraph that is one of the few current generation instruments optimized
for high-contrast direct imaging of substellar companions. The instrument is
in the process of being upgraded and moved from its current mount on the
Gemini South Observatory in Cerro Pachón, Chile, to its twin observatory,
Gemini North, on Mauna Kea (a process colloquially dubbed “GPI 2.0”). We
present the designs that have been developed for the part of GPI 2.0 that
pertains to upgrading various optical components of the GPI coronagraphic
system. More specifically, we present new designs for the apodizer and Lyot
stop (LS) that achieve better raw contrast at the inner working angle of the
dark zone as well as improved core throughput while retaining a similar level
of robustness to LS misalignment. To generate these upgraded designs, we use
our own publicly available software package called APLC-Optimization that
combines a commercial linear solver (Gurobi) with a high contrast imaging
simulation package (HCIPy) in order to iteratively propagate light through a
simulated model of an apodized phase lyot coronagraph (APLC), optimizing for
the best coronagraph performance metrics. The designs have recently finished
being lithographically printed by a commercial manufacturer and will be ready
for use when GPI 2.0 goes on-sky in 2023.

###### keywords:

Exoplanets, Direct imaging, Coronagraphy, Instrumentation, Linear optimization

##  1 INTRODUCTION

The Gemini Planet Imager [1] is a near-infrared high-contrast direct imager
designed to search for and characterize sub-stellar companions [2] and disks
[3] at small angular resolution relative to their host star. The primary
mission of the instrument is an 890 hr observation campaign, the Gemini Planet
Imager Exoplanet Survey (GPIES)[4]. After nearly 7 years of continuous
operation since 2013, the instrument was decommissioned in August 2020 in
order to be moved from the Gemini South telescope in Cerro Pachón, Chile to
the Gemini North telescope atop Mauna Kea in Hawaii. Since both Gemini North
and South are twin observatories, the instrument would sit in the same
position when installed on Gemini North (behind the Cassegrain focus of the
8.1-m telescope) as it was used in on Gemini South.

There are several major components of the planned upgrade of GPI including an
upgraded AO-system which uses a faster realtime computer and a pyramid
wavefront sensor, better observatory software, a more sensitive IR tip/tilt
camera, various improvements to the IFS, and updated coronagraphic masks (see
Fig. 4 of Chilcote et al. 2020[5]). In this study, we will be presenting the
results for the updated coronagraphic masks, specifically the new apodizers
and Lyot stops which will go into the apodizer and Lyot wheels of GPI
respectively (see Fig. 1). We first start by presenting the theoretical
framework and software tools we use to generate the apodizer and LS designs in
Section 2. Then we discuss the three sets of new designs (where a ”set” is an
apodizer with its corresponding LS) that have been settled upon and physically
sent out for manufacture in Section 3. Finally, we examine the theoretical
performance of the GPI 2.0 designs and the expected improvement they should
provide using various high contrast imaging metrics such as the throughput,
raw contrast, and robustness to misalignment tolerance in Section 4. Though
the new apodizers and Lyot stops have just recently finished fabrication, the
move of GPI from Gemini South was delayed due to the pandemic and other
factors, so laboratory testing of these optics was not yet available at the
time of this report.

---
Figure 1:  (Left) A 3D solid model of GPI showing the three main components of
the instrument: the AO system (black), the Calibration system (blue), and the
IFS system (red). (Right) A more detailed schematic diagram showing all the
various sub-components of GPI. In this study, we are only concerned with
upgrades to the coronagraph subsystem, specifically changes to the apodizer
wheel and Lyot wheel. Both figures are taken from De Rosa et al. (2020) [6].

##  2 THE OPTIMIZATION PROCESS

Figure 2:  A basic schematic of the planes of propagation in an APLC. This
figure is taken from Fig. 1 of N’diaye et al. (2015)[7]. Figure 3:  Example
images of the pupil/apodizer, focal plane, and Lyot stop masks in the
appropriate order to match the direction of light propagation as the diagram
above. These are the binary representations of some of the previous generation
GPI 1.0 masks. See Section 3 for the actual GPI 2.0 masks.

The infrastructure we use to generate the optical designs is all contained
within a publicly available STScI github repository called APLC-
Optimization[8, 9] 111https://github.com/spacetelescope/aplc_optimization.
There are two main components of this optimization code: (1) An APLC simulator
built using the high-contrast imaging python package HCIPy [10]; and (2) A
commercial linear solver called Gurobi
222https://www.gurobi.com/products/gurobi-optimizer/ that iteratively
propagates light through the APLC simulator, varying the shape of the apodizer
and minimizing the loss function of the output coronagraphic point spread
function (PSF) with respect to various performance metrics such as the raw
contrast or core throughput. To simulate an APLC, we need to have four optical
planes of propagation (see Fig. 2). A detailed mathematical description of the
light propagation through each plane in an APLC is beyond the scope of this
paper (see Soummer et al. 2007 [11] for a full treatment), but the two basic
operations are: (1) Fourier (or inverse Fourier) transforms to propagate
between pupil planes and focal planes (or vice versa), and (2) multiplications
(or convolutions depending on whether the multiplication is done in a
conjugate Fourier space) by masks to selectively filter light in the
appropriate planes (namely multiplication/convolution by either an apodizer
mask, a focal plane mask, or a Lyot stop mask). We can follow the light
propagation through the APLC using the following basic algorithm (represented
in Fig. 2):

Plane A) The light comes in from the entrance pupil and gets multiplied by the
apodizer mask.

Plane B) We take the Fourier transform of this light to propagate it to the
focal plane, then multiply the result by the focal plane mask (FPM).

Plane C) We take the inverse Fourier transform of this light to propagate it
to the exit pupil plane, then multiply the result by the Lyot stop mask.

Plane D) We take the Fourier transform of this light to propagate it to the
final image plane where the detector sits.

##  3 THREE NEW DESIGNS

###  3.1 Apodizers

The optimization process assumes a fixed primary pupil shape and fixed focal
plane mask since these surfaces are not mutable on GPI. This leaves the
apodizer and Lyot stop planes (in planes A and C respectively) as the two
surfaces we can design for. After various rounds of optimizations and testing,
a total of three separate apodizer designs were settled upon (with each
apodizer design paired with a corresponding Lyot stop optimized for it). For
simplicity, we will refer to these three designs as LS03Symm, DualPlane, and
DualPlaneSymm, with the reasoning behind the names to be described below. For
all three of the designs, we have overlaid an evenly spaced diffraction
grating (also known as an astrometric grid) on top. The astrometric grid
creates a diffraction pattern of regularly spaced out copies of the core PSF,
called ”satellite spots” (or ”satspots” for short), that are useful for
astrometry and calibration purposes.

Design 1: LS03Symm) The first apodizer design, LS03Symm, assumed a fixed Lyot
stop geometry during the optimization process based on a horizontally
symmetrized version of one of the previous generation GPI 1.0 Lyot stops
(specifically mask ”080m12_03”, hence the reason why we have given it the name
”LS03” along with the suffix ”-Symm”, see Table 2 in 3). The reason we used a
symmetrized version of the LS03 Lyot stop was because of memory constraints on
the server computers on which the optimizations ran. We found that there was a
memory bottleneck during the optimization process and introducing symmetry
into the LS reduced the computational overhead enough to bypass this
bottleneck. We optimized this design for the GPI 1.0 K1 FPM (m/2 = 3.476
$\lambda/D$) which is designed for H band, but we expect this mask to perform
similarly in other spectral bands as long as the ratios in wavelengths are
equal. We set the target contrast for the dark zone to be $10^{-8}$ over a
20$\%$ bandpass, corresponding to one order of magnitude higher contrast than
the specification of the GPI 1.0 coronagraphs[12, 13].

Design 2: DualPlane) The second apodizer design, DualPlane, allowed both the
apodizer and Lyot stop to vary during the optimization process (hence the name
”Dual”Plane). This design was optimized to produce a ”grayscale” solution,
meaning the surface of the apodizer can vary in its transmissivity anywhere
from 0 (fully absorptive) to 1 (fully transmissive). This is in contrast to
the other two apodizer designs which are ”binary” solutions, meaning the
surface is manufactured to be either fully absorptive or fully transmissive
with no partial transmissivity in between. For the details on how this
particular design was developed, see Por et al. (2022) [14].

Design 3: DualPlaneSymm) The third apodizer design, DualPlaneSymm, is
partially based on Design 2, assuming a fixed Lyot stop geometry during the
optimization process that is a vertically symmetrized version of Design 2’s LS
(hence the name DualPlane+”-Symm”). We symmetrized the LS for the same memory
constraint reason as Design 1’s symmetrized LS. We also used similar
optimization parameters (target contrast, bandpass, K1 FPM mask) that Design 1
used.

LS03Symm Apod | DualPlane Apod | DualPlaneSymm Apod
---|---|---
|  |
|  |
LS03Symm LS | DualPlane LS | DualPlaneSymm LS
|  |
|  |
Figure 4:  Row 1) FITS file representations of the three new apodizer designs.
Row 2) Actual photographs taken of two of the freshly manufactured apodizers.
The apodizers are inside the white square in the center of the image (the
clear plastic around the outside is simply the storage case). To give a sense
of scale, the diameter of each apodizer design is approximately 1.2 cm. Row 3)
FITS file representations of the three new LS designs (corresponding to the
three new apodizer designs). Row 4) An actual photograph taken of one of the
freshly manufactured Lyot stops. The opaque part of the LS is etched using
wire-cut electrical discharge machining (EDM). To give a sense of scale, the
diameter of each LS is just under 1 cm, and the struts holding the central
obscuration in place have a thickness of approximately 180 microns.

###  3.2 Lyot stops

GPI 1.0 had nine different Lyot stop masks available on a Lyot wheel in the
instrument, each having a name starting with the string ”080m12” (which refers
to the fact that the GPI 1.0 MEMS DM has a serial number 80 and the mask
version was modified in 2012). See Footnote
333http://docs.planetimager.org/pipeline/gpireference/coronagraph.html  for a
full description of the Lyot stop specifications. The main difference between
the different Lyot stop designs was the width of the ”spiders” masking the
four support struts of the secondary mirror on Gemini and the width of the
circular tabs masking the bad actuators (See Fig. 3 for an example of a GPI
1.0 LS mask).

For the GPI 2.0 upgrade, we reduced the number of available Lyot stops from
nine to just three, each corresponding to one of the three new apodizer
designs described above in Section 3.1. This decision was made because the new
apodizer designs are finely optimized to produce a target dark zone when their
corresponding LS is also being used, so mix-and-matching Lyot stops or
utilizing additional Lyot stops which are not optimized for the specific
apodizer in place is pointless as the resulting dark zone will produce very
poor raw contrast.

###  3.3 The Fabrication Process

The original GPI 1.0 apodizers were fabricated using microdot printing as
described in Sivaramakrishnan et al. (2009[15] and 2010 [16]). For the GPI 2.0
upgrade, we contracted a commercial manufacturer, Advanced
NanoPhotonics444https://www.advancednanophotonics.com/, to produce the
apodizers and Lyot stops using their proprietary fabrication process[17, 18].
The apodizer masks are transmissive, so to produce them we start with a fully
transmissive substrate of infrasil glass and then pattern catalyst from which
dark carbon nanotubes are grown using chemical vapor deposition at around
$\sim 600^{\circ}C$. The carbon nanotubes result in 100% blocking of incoming
light while minimizing reflected light (total hemispheric reflectance is 0.5%
in visible wavelengths, 0.2% at infrared). Some of the manufacturing
requirements of the apodizers are an excellent transmitted wavefront error (we
achieved $\sim 5$ nm RMS) and high spectral transmission (greater than 90%
from 250-500 nm and less than 92.5% from 500-2500 nm).

##  4 THEORETICAL PERFORMANCE

The primary metrics we use to assess the performance of a coronagraph are the
raw contrast, the core throughput, the inner working angle (IWA) of the dark
zone, the outer working angle (OWA) of the dark zone, and the misalignment
tolerance. The first and third designs, LS03Symm and DualPlaneSymm, were
optimized to have significantly better raw contrast (excluding aberrations) of
approximately $10^{-8}$ across the entirety of the dark zone between the inner
and outer working angle (see Figs. 6 and 6). This improved raw contrast comes
at the cost of having a significantly smaller OWA compared to the GPI 1.0
masks as well as decreased throughput compared to the second design,
DualPlane. The DualPlane design, has comparable raw contrast to the GPI 1.0
designs (with a slight improvement of approximately $7*10^{-6}$ raw contrast
right at the IWA), but does not have the issue of a reduced OWA. In fact, the
DualPlane design technically has an ”infinite” OWA because the raw contrast
does not go back up after some characteristic angular separation which is what
characterizes an OWA in other coronagraphic dark zones.

The designs were optimized without the effect of the astrometric grid
included. When the astrometric grid is overlaid, this produces an effect
similar to a diffraction grating, causing evenly spaced copies of the core PSF
to be created at four radial symmetric positions angularly separated from the
central PSF - the ”satspots” (short for ”satellite spots”). See Figs. 8 and 8
for the coronagraphic PSFs and raw contrast curves with the effect of the
satspots included.

The nominal throughput of the new GPI 2.0 designs are either comparable or
significantly improved compared to previous generation designs (see Table 1).
More specifically, the first and third designs, LS03Symm and DualPlaneSymm,
have comparable combined throughput (which is calculated by taking the nominal
apodizer throughput and nominal LS throughput and multiplying them together)
of 23.9% and 27.1% respectively compared to a GPI 1.0 H-band design’s combined
throughput of 26.0%. The stand out case in terms of throughput is the second
design, DualPlane, which has a combined throughput of 35.6%. There is
typically a trade off in coronagraph design between raw contrast and
throughput, so it is natural to expect that the first and third designs
perform worse in terms of throughput since they were optimized to perform
significantly better in raw contrast (and vice versa for the second design).

For a discussion of the misalignment tolerancing, see Appendix A.

Table 1: Theoretical Performance Metrics Design Name | Apodizer | Lyot Stop | Combined | Raw Contrast | Average Raw Contrast
---|---|---|---|---|---
| Throughput | Throughput | Throughput | at (IWA) 3 $\lambda/D$ | from 3 - 12 $\lambda/D$
LS03Symm | 0.36 | 0.66 | 0.24 | $6\times 10^{-8}$ | $3\times 10^{-8}$
DualPlane | 0.49 | 0.73 | 0.36 | $3\times 10^{-6}$ | $3\times 10^{-7}$
DualPlaneSymm | 0.40 | 0.68 | 0.27 | $6\times 10^{-8}$ | $3\times 10^{-8}$
GPI 1.0 (H-band) | 0.37 | 0.70 | 0.26 | $5\times 10^{-6}$ | $3\times 10^{-7}$

Figure 5:  Coronagraphic PSFs (with no aberrations) for the three new designs
as well as an old GPI 1.0 H-band design for reference.

Figure 6:  Raw contrast (with no aberrations) vs. angular separation for the
three new designs - LS03Symm (orange), DualPlane (red), DualPlaneSymm (green)
- as well as an old GPI 1.0 H-band design for reference (blue). The raw
contrasts are calculated directly from the coronagraphic PSFs in Fig. 6.

Figure 7:  Coronagraphic PSFs (with no aberrations) for the three new designs
with the four ”satspots” caused by the diffracting astrometric grid included.
An old GPI 1.0 H-band design is included for reference.

Figure 8:  Raw contrast (with no aberrations) vs. angular separation for the
three new designs with the four ”satspots” (radially symmetric copies of the
core PSF caused by the diffracting astrometric grid placed on top of the
apodizers) included. Similar to Fig. 6, we show the raw contrasts for the
three new designs - LS03Symm (orange), DualPlane (red), DualPlaneSymm (green)
- as well as an old GPI 1.0 H-band design for reference (blue). The raw
contrasts are calculated directly from the coronagraphic PSFs in Fig. 8.

###### Acknowledgements.

This research project was funded by the STScI Discretionary Research Fund
(D0101.90238). Advanced Nanophotonics/Lambda Consulting performed research
with funding from the University of Notre Dame and matching funds from the
NASA SBIR program to develop transmissive apodizers for the Gemini Planet
Imager upgrade. E.H.P. is supported by the NASA Hubble Fellowship grant #HST-
HF2-51467.001-A awarded by the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astronomy,
Incorporated, under NASA contract NAS5-26555. The Gemini Planet Imager 2.0
project upgrade is supported by the National Science Foundation under Grant
No. AST-1920180, and also significantly supported by the Heising-Simons
Foundation. The international Gemini Observatory is a program of NSF’s
NOIRLab, which is managed by the Association of Universities for Research in
Astronomy (AURA) under a cooperative agreement with the National Science
Foundation on behalf of the Gemini Observatory partnership: the National
Science Foundation (United States), National Research Council (Canada),
Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia,
Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia,
Inovações e Comunicações (Brazil), and Korea Astronomy and Space Science
Institute (Republic of Korea).

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##  Appendix A Lyot Stop Robustness to Misalignment

Based on engineering specifications and tolerances for GPI 2.0 and the Gemini
North Telescope, we decided that we needed to accommodate a minimum LS
misalignment tolerance of 0.5%, which corresponds to approximately 50 microns
of misalignment tolerance on the actual component (0.5% of approximately 1cm).
The LS misalignment tolerance was an important metric during the optimization
process because making the LS more robust to this comes with the trade-off of
reduced throughput or raw contrast. This trade-off was not well understood
before we performed this study. Though we were not able to perform a full
parameter space study to explore the trade-offs, we were able to run a few
rounds of optimizations to explore the trade-off relationship between two of
the parameters, specifically between the core throughput and the LS
robustness. We ran two sets of optimizations using the Design 3
(DualPlaneSymm) template, keeping all parameters (e.g. the wavelength, FPM,
OWA, target raw contrast) the same except the IWA and the LS robustness, and
found there was a very clear negative, linear relationship between the core
throughput and LS robustness (see Fig. 9 below). This made it fairly easy to
then determine what was an optimal amount of robustness to include in the LS
optimization, since we want to maximum throughput, but we need to meet the
minimum LS robustness tolerance of 0.5% which, since the designs were
optimized at a resolution of 1000x1000 pixels, corresponds to a minimum
tolerance of 5 pixels.

Figure 9:  Maximum core throughput vs. LS robustness to translational
misalignment for two different sets of designs (optimized for an IWA of 2.7
$\lambda$/D vs. 3 $\lambda$/D). The designs were optimized at a 1000x1000
pixel resolution, so a LS misalignment tolerance of 0.5% corresponds to 5
pixels of translational robustness tolerance. These designs used the LS for
Design 3 (DualPlaneSymm).