$B_{0}$ that are bumped out of the first row during this process. The tableau
$T_{1}$ remaining after bumping these $m_{2}$ entries from the tableau will
have shape $\mu^{-}$ in the $spo$-bumping sequence described below. If
$m_{2}=k-\ell$, then $\mu=\mu^{-}$ and $m_{1}=0$. Otherwise, if the rightmost
box in $T_{1}$ that does not belong to $\mu$ is in column $j$, then the entry
in this box belongs to $B_{1}$ and, since bumping this entry through the
tableau causes an entry from $B_{1}$ to bump out of the first column, for each
column $i$ with $1\leq i\leq j$ there is an entry $a_{i}\in B_{1}$ in column
$i$ and an entry $a_{i-1}\in B_{1}$ in column $i-1$, with $a_{i-1}<a_{i}$,
that belongs to a row below $a_{i}$.

We can now find the image of $((\lambda,\mu,\nu),m_{1})$, using Lemma 4.16,
which is an $spo$-bumping sequence $(\lambda,\mu^{\circ},\mu^{-},\nu)$. We now
verify that the $spo$-insertion process can be reversed, as described in the
paragraphs following Theorem 4.14, using this particular sequence and these
values of $m_{1}$ and $m_{2}$. Reversing the $spo$-insertion process first
bumps the $m_{2}$ entries from $B_{0}$ as described above, leaving $T_{1}$,
which has shape $\mu^{-}$.

Continuing the reversed $spo$-insertion algorithm then adds $\ell$ empty boxes
to $T_{1}$, which are each moved through the tableau through the jeu de taquin
process, resulting in a tableau $T_{2}$ of shape $\mu^{\circ}$. Provided the
empty boxes each start in rows below the first, the tableau $T_{2}$ still
contains an entry from $B_{1}$ in each of columns $1$ through $j$ since the
jeu de taquin process cannot move an entry $a_{i-1}$ below an entry $a_{i}$,
where $a_{i-1}<a_{i}$. If an empty box starts in the first row of $T_{1}$,
then $\mu^{-}$ has fewer boxes in the first row than $\mu^{\circ}$ so, by
Lemma 4.10, $\lambda$ and $\mu^{\circ}$ have the same number of boxes in the
first row, which means that no entries are bumped out of the first row in the
next stage of the reverse $spo$-algorithm.

Since $\mu^{\circ}/\lambda$ is a horizontal strip with $m_{2}$ boxes
corresponding to columns left of column $j+1$ in $\mu^{\circ}$, any entry in
$T_{2}$ that belongs to a box of $\mu^{\circ}/\lambda$ must be at the bottom
of column $j$ or at the bottom of a column left of column $j$, so contains an
entry from $B_{1}$. Thus the final stage of the $spo$-insertion algorithm can
be completed to bump $m_{2}$ entries from $B_{1}$ from $T_{2}$, producing a
tableau $T$ of shape $\lambda$ and a $(k)$-tableau, whose product is $S$. ∎

######  Example 4.21.

Consider the $spo$-tableau  $2$ $1^{\circ}$ $3^{\circ}$ $\overline{2}$
$2^{\circ}$ and the $sp$-bumping sequence
$\left(\hskip 2.0pt\vbox{ \vbox{\vskip 3.0pt plus 1.0pt minus
1.0pt\offinterlineskip\halign{&\Vsquare{#}\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox
to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule
height=0.4pt} \vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr}}}\hskip 2.0pt,\hskip 2.0pt\vbox{
\vbox{\vskip 3.0pt plus 1.0pt minus
1.0pt\offinterlineskip\halign{&\Vsquare{#}\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule width=0.4pt\vbox
to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule
height=0.4pt} \vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr}}}\hskip
2.0pt,\hskip 2.0pt\vbox{ \vbox{\vskip 3.0pt plus 1.0pt minus
1.0pt\offinterlineskip\halign{&\Vsquare{#}\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox
to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule
height=0.4pt} \vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr}}}\hskip 2.0pt\right)$. Reversing the
$spo$-insertion process for the entry in the $(1,3)$-box bumps a $2$ out of
the first row, and then repeating the process for the entry in the $(1,2)$-box
in the resulting tableau bumps a $2^{\circ}$ out of the first column. It
follows that $m_{2}=1$ and, since $\ell=2$, $m_{1}=1$. The corresponding
$spo$-bumping sequence, via Lemma 4.16, is then $\left(\hskip 2.0pt\vbox{
\vbox{\vskip 3.0pt plus 1.0pt minus
1.0pt\offinterlineskip\halign{&\Vsquare{#}\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox
to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule
height=0.4pt} \vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr}}}\hskip 2.0pt,\hskip 2.0pt\vbox{
\vbox{\vskip 3.0pt plus 1.0pt minus
1.0pt\offinterlineskip\halign{&\Vsquare{#}\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox
to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule
height=0.4pt} \vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule width=0.4pt\vbox
to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule
height=0.4pt} \vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr}}}\hskip
2.0pt,\hskip 2.0pt\vbox{ \vbox{\vskip 3.0pt plus 1.0pt minus
1.0pt\offinterlineskip\halign{&\Vsquare{#}\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule width=0.4pt\vbox
to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule
height=0.4pt} \vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt\cr}}}\hskip 2.0pt,\hskip 2.0pt\vbox{ \vbox{\vskip
3.0pt plus 1.0pt minus
1.0pt\offinterlineskip\halign{&\Vsquare{#}\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox
to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule
height=0.4pt} \vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr}}}\hskip 2.0pt\right)$. We can use
this $spo$-bumping sequence and the given tableau to reverse the
$spo$-insertion process and decompose the tableau as a product:

$\hskip 2.0pt\vbox{ \vbox{\vskip 3.0pt plus 1.0pt minus
1.0pt\offinterlineskip\halign{&\Vsquare{#}\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$2$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$1^{\circ}$\hss} \vss\hrule height=0.4pt}
\vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox
to14.0pt{\hss$3^{\circ}$\hss} \vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$\overline{2}$\hss} \vss\hrule
height=0.4pt} \vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox
to14.0pt{\hss$2^{\circ}$\hss} \vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt\cr}}}\hskip 2.0pt=\hskip 2.0pt\vbox{ \vbox{\vskip
3.0pt plus 1.0pt minus
1.0pt\offinterlineskip\halign{&\Vsquare{#}\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox
to14.0pt{\hss$\overline{1}$\hss} \vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$\overline{1}$\hss} \vss\hrule
height=0.4pt} \vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox
to14.0pt{\hss$\overline{2}$\hss} \vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$1^{\circ}$\hss} \vss\hrule height=0.4pt}
\vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox
to14.0pt{\hss$3^{\circ}$\hss} \vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt\cr}}}\hskip 2.0pt\cdot\hskip 2.0pt\vbox{ \vbox{\vskip
3.0pt plus 1.0pt minus
1.0pt\offinterlineskip\halign{&\Vsquare{#}\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$1$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$1$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox
to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$2$\hss} \vss\hrule
height=0.4pt} \vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox
to14.0pt{\hss$2^{\circ}$\hss} \vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt\cr}}}\hskip 2.0pt.$

Note that if we consider the same $spo$-tableau but $sp$-bumping sequence
$\left(\hskip 2.0pt\vbox{ \vbox{\vskip 3.0pt plus 1.0pt minus
1.0pt\offinterlineskip\halign{&\Vsquare{#}\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox
to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule
height=0.4pt} \vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr}}}\hskip 2.0pt,\hskip 2.0pt\vbox{
\vbox{\vskip 3.0pt plus 1.0pt minus
1.0pt\offinterlineskip\halign{&\Vsquare{#}\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule width=0.4pt\vbox
to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule
height=0.4pt} \vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr}}}\hskip
2.0pt,\hskip 2.0pt\vbox{ \vbox{\vskip 3.0pt plus 1.0pt minus
1.0pt\offinterlineskip\halign{&\Vsquare{#}\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox
to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule
height=0.4pt} \vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr}}}\hskip 2.0pt\right)$, then the
corresponding $spo$-bumping sequence is
$\left(\hskip 2.0pt\vbox{ \vbox{\vskip 3.0pt plus 1.0pt minus
1.0pt\offinterlineskip\halign{&\Vsquare{#}\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox
to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule
height=0.4pt} \vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr}}}\hskip 2.0pt,\hskip 2.0pt\vbox{
\vbox{\vskip 3.0pt plus 1.0pt minus
1.0pt\offinterlineskip\halign{&\Vsquare{#}\cr\vbox{\hbox{\vrule
width=0.4pt\vbox to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss}
\vss\hrule height=0.4pt} \vrule width=0.4pt}
\kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox to14.0pt{\hrule
height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule height=0.4pt} \vrule
width=0.4pt} \kern-0.4pt}\kern-0.4pt&\vbox{\hbox{\vrule width=0.4pt\vbox
to14.0pt{\hrule height=0.4pt\vss\hbox to14.0pt{\hss$$\hss} \vss\hrule
height=0.4pt} \vrule width=0.4pt} \kern-0.4pt}\kern-0.4pt\cr\vbox{\hbox{\vrule
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Theorem 4.1 now follows. Our final example illustrates the orthosymplectic
Pieri rule.

######  Example 4.22.

Let $\lambda=(3,1)$ and $k=3$. Then $\ell=0,1,2$ or $3$. We have

$spo_{\lambda}spo_{(k)}=spo_{(6,1)}+spo_{(5,2)}+spo_{(5,1,1)}+spo_{(4,3)}+spo_{(4,2,1)}+2spo_{(4,1)}+2spo_{(3,2)}$
$+spo_{(3,1,1)}+spo_{(2,2,1)}+spo_{(5)}+2spo_{(2,1)}+spo_{(1,1,1)}+spo_{(3)}+spo_{(1)}.$

### Acknowledgement

The author wishes to thank an anonymous referee, who carefully read the paper
and provided useful suggestions.

## References


# Passive Aperiodic Optical Phased Array based on Uniform Random Shuffle

Bowen Yu
Dept. of Electrical and Computer Engineering
University of Michigan
Dearborn, MI 48128
<EMAIL_ADDRESS>
Dachuan Wu
Dept. of Electrical and Computer Engineering
University of Michigan
Dearborn, MI 48128
<EMAIL_ADDRESS>
Yasha Yi
Dept. of Electrical and Computer Engineering
University of Michigan
Dearborn, MI 48128
Energy Institute,
University of Michigan
Ann Arbor, MI 48109
<EMAIL_ADDRESS>

###### Abstract

Grating lobes arise from the periodic nature of element spacing in the optical
phased array. Essentially, the phased array performs the Spatial Fourier
Transform on light; the steering capability of the main lobe is governed by
phase shift variations among waveguides, and the Sidelobe Suppression Ratio
(SLSR) correlates with the uniformity of emitter positions. Leveraging this
understanding, we have optimized a 1×64 channel passive aperiodic OPAs with
the uniform random shuffle in the emitter’s position. Our conceptual
simulations highlight a robust steering capability (18.60° / 10nm) and SLSR
(-13.46 dB @ 0° / -8.27 dB @ ±45°), and initial measurements demonstrate the
steering capability (9.8 ° / 10nm, with smaller phase shifts design) and SLSR
(-6.1dB @ -33.4°) from the preliminary fabrication.

##  1 Introduction

Most research on the optical phased array systems (OPAs) starts from the
integrated approach from silicon on insulators (SOI) with periodically
arranged emitters [1]. Over the years, numerous review papers have been
published, offering various insights and feasibilities of OPAs and LiDAR
technologies based on photonic integrated circuits [2, 3, 4, 5, 6, 7]. Some
research efforts have been dedicated to system-level integration, employing
simple periodic emitting patterns. A notable issue with this approach is the
emergence of prominent grating lobes in the far field, attributed to the
uniform spacing of the emitter arrays. In LiDAR applications, these lobes may
be incorrectly identified as the main lobe, leading to potential inaccuracies.
To address this challenge, other research in the field of OPAs has shifted
focus to utilizing aperiodic emitter pitches, aiming to attain a broader
steering range and higher resolution [8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18]. This is done by optimizing the spacing between waveguides and expanding
the emission aperture, thus preventing aliasing across the field of view and
enabling wide-angle emissions from individual antennas. These advancements
often employ mathematical and machine learning techniques like particle swarm
optimization, genetic algorithms, NSGA-II, and pattern search methods for
optimal emitter placement. These methods have proven highly effective; for
instance, one study achieved -13.39dB @ 0° in a 1×64-channel OPAs [18]. From
the optimization perspective, it is crucial to manage the trade-off between
sidelobe level and beam width, with computational tools and simulations being
essential due to the complexity of these designs. Despite these advancements,
the ultimate spatial resolution limits of non-uniform OPAs, particularly the
theoretical limits of the Sidelobe Suppression Ratio (SLSR), remain
unresolved. Essentially, phased arrays can be viewed as the diffraction model,
performing a Spatial Fourier Transform on the light. Our previous work
demonstrated a high-efficiency 3D passive wavelength-tuning 4×16 OPAs device
on a SiN/SiO2 platform, but it showed noticeable grating lobes due to uniform
emitter spacing [7].

Leveraging the principles from the diffraction model, we optimized our
$\Omega$-shape OPAs for high SLSR and steering capability in this study. We
introduced a uniform random shuffle in emitters’ positions to break the
array’s periodicity, and integrated large passive phase delay lines for
increased phase delay variation. To optimize our simulation process, we
developed the Rapid Farfield Approximation, a simplified structural model that
significantly reduced computational efforts. The simulation results validated
our method for achieving high SLSR and steering rates. Additionally, our
preliminary fabrication and characterization, despite some fabrication errors
and equipment limits, indicate the potential viability of these strategies for
future OPAs designs.

##  2 Design and simulation

###  2.1 Correlating diffraction model to 1D OPAs design

Similar to the diffraction model, grating lobes in any optical phased array
system are caused by the intrinsic periodicity among emitting elements. The
Fourier transform effectively quantifies this periodicity within these arrays,
offering a measure of its extent. The farfield E-field equation (Eq. 1) [10]
for any arbitrary phased arrays and the Discrete Fourier Transform (DFT)
equation (Eq. 2) are the keys to the analysis:

$E(\theta)=\sum_{i=1}^{N}A_{i}e^{j(2\pi\frac{y_{i}}{\lambda}sin\theta)}e^{-j\phi_{i}}$
(1) $F(k)=\sum_{n=0}^{N-1}f(n)e^{j(2\pi\frac{n}{N}k)}$ (2)

In Eq. 1, $A_{i}$ and $\phi_{i}$ are the amplitude and phase of each emitter,
$y_{i}$ is the emitter position and $\theta$ is the viewing angle. For $f(n)$
in the DFT equation (Eq. 2), its ordinal value $n$ signifies real space
position, and its cardinal value $f(n)$ indicates the presence (1) or absence
(0) of an emitter at $n$. $k$ is the wave vector, equivalent to $1/\Lambda$,
where $\Lambda$ is the periodicity of the array. For an aperiodic arrangement,
$\Lambda$ is estimated as the arrays’ size divided by the number of emitters.

The first exponential term in Eq. 1, close to the exponential in Eq. 2,
performs a Spatial Fourier Transform on light from emitters. To achieve a high
SLSR, we first employ sets of emitters’ positional shifts via random uniform
distribution to maximize disorderness in sample variations, as it possesses
the highest entropy among different discrete distributions. Subsequently, we
apply a uniform random shuffle, specifically the Fisher-Yates Shuffle, to the
second set of positional shifts (Fig. 1). To ensure statistical consistency
(mean and variance) throughout the random generation process, we generate
shifting arrays with even distribution over a specified range, thereby
minimizing crosstalk.

Figure 1: Spatial Fourier Transform of a) periodic and b) aperiodic array
arrangement (with uniform random shuffle) sets.

The second exponential term in Eq. 1 represents the shift property within the
Spatial Fourier Transform, derived from the Huygens model. In the simplicity
of a single-mode waveguide-based design, we achieve phase delays through
varying path lengths. The phase delay for each waveguide is mathematically
expressed as:

$\phi_{i}=\frac{2\pi L_{i}n_{eff}(\lambda_{0})}{\lambda_{0}}$ (3)

###  2.2 Rapid Farfield Approximation

In this study, expanding on our prior work [7], we evaluate the impact of
aperiodic emitter layouts on steering and SLSR enhancement. Preferring
simplicity, we chose a single-layer design over a multi-layer one, easing
manufacturing while verifying concepts for future complex integrations (Fig.
2). The design uses a frequency-tunable laser source, coupled to a waveguide
via SMF-28 and an on-chip edge coupler-taper, supporting only TE0 and TM0
modes in 0.6 µm × 0.705 µm waveguides. We opt for the TM0 mode due to its
slightly better bend loss and minimal crosstalk spacing. The beam is then
split into 1×64 channels, each with an $\Omega$-shaped delay line.

Figure 2: Schematic of 1×64 passive aperiodic OPAs.

While the Spatial Fourier Transform offers a rough approximation of the
farfield pattern, employing the Finite-Difference time domain (FDTD) yields a
more accurate representation at the cost of the computational workload. To
simulate the farfield pattern efficiently using Lumerical’s FDTD, we implement
the Rapid Farfield Approximation (RFA) technique. This approach accounts for
the linearly increasing delay line length in each waveguide as we move upwards
from the bottom of the $\Omega$-shaped configuration. The passive phase delays
are thus calculated as:

$\Delta\phi=\frac{2\pi\Delta L\ n_{eff}(\lambda_{0})}{\lambda_{0}}$ (4)

$\Delta L=\alpha Y+\beta$ (5)

$\beta=\frac{[\alpha Yn_{eff}(1.55\mu m)/1.55\mu
m]\%\pi}{\pi}\times\frac{1.55\mu m}{n_{eff}(1.55\mu m)}$ (6)

In these equations, $\Delta\phi$ is the phase delay in each waveguide,
$\lambda_{0}$ is the source wavelength and $n_{eff}$ is the effective index
for TM0 at different $\lambda_{0}$. $\Delta L$ represents the delay line
length, with $\alpha$ being the path difference multiplier and $Y$ the pitch
position of the emitters. The modulus operator $\%$ is used to calculate
$\beta$, which serves as a length correction term to center the main lobe at
0° for $\lambda$0=1.55µm. The simplicity of our structure allows the FDTD
solver to focus solely on simulating the electromagnetic field around the
dielectric-air interface at the emitter’s end. This targeted approach
significantly reduces the time required for farfield formation simulations,
making large-scale and quantitative analysis feasible on our computer.

Figure 3: a) RFA simulated farfield pattern at different $\lambda_{0}$. b) The
corresponding SLSR and c) the mainlobe angles derived from these simulations.

In 2D simulations, we applied 64 positional shifts to each waveguide from 6 µm
periodic arrangement using a random uniform process, limiting maximum shifts
to ± 2.4 µm to avoid crosstalk. The path difference multiplier ($\alpha$) was
set to 16. We generated and analyzed 3000 sets, observing that SLSR decreased
correspondingly with increased spatial frequency amplitude from 0.18 to 0.5,
at $\xi$ = 0.1667 µm-1 or $\Lambda$ = 6 µm. To suppress the amplitude at
$\Lambda$ = 6 µm, a second batch was processed using uniform random shuffle,
effectively maintaining the amplitude to 0.22 (Fig. 1b) and achieving an
average SLSR of approximately -11.5 dB at 0°. We then proceeded to evaluate
the performance of the best set from the simulations at different wavelengths,
finding the peak SLSR to be -13.46 dB at 0° and -8.27 dB at ±45° (Fig. 3). All
simulations consistently showed a steering capability of 18.6°/10nm,
influenced by the fixed path difference multiplier rather than the array
arrangement. Additionally, the Full Width at Half Maximum (FWHM) of the main
lobes was consistently about 0.3° in all simulations, mainly determined by the
number of emitters. 3D simulations with a reduced path difference multiplier
($\alpha$ = 10) were conducted to anticipate fabrication variances. We tested
periodic and aperiodic OPAs, which achieved -10.0 dB SLSR (aperiodic set)
using 3D RFA. Utilizing Lumerical’s Eigenmode Expansion, we optimized the
coupler-taper for -7.30 dB coupling efficiency from the SMF-28 fiber, and the
y-branch splitter showed minimal loss (-0.534 dB per split), enhancing OPAs
performance.

##  3 Fabrication and experimentation

In the preliminary fabrication phase, we produced the two samples discussed in
the previous section with the conventional SOI fabrication techniques [19]
from the Lurie Nanofabrication Facility (LNF). Overall, the structures showed
all the necessary features of the design. However, we observed a discrepancy
in waveguide width: measurements showed the waveguide has a width of 0.7 – 0.8
µm in trapezoid shape, instead of 0.6 µm width in rectangle shape which is
used in our simulations, as illustrated in Fig. 4a – c.

Figure 4: a & b) A tilted top view of a) periodic and b) aperiodic OPAs from
the SEM. c) The cross-sectional view of a single waveguide from the aperiodic
OPAs, offering detailed structural insights. d) The simulated effective index
across various wavelengths for waveguides with differing widths. Figure 5:
Schematic of characterization setup.

To assess the steering capability, we implemented a simple 4f optical system
comprising an objective and a camera lens (Fig. 5). This setup enabled us to
calculate farfield angles through trigonometric analysis based on the stage’s
position. We conducted farfield measurements across wavelengths from 1538 to
1550 nm, constrained by the sensitivity range of our camera (Point Grey,
CMLN-13S2M). The results indicated a steering capability of 9.8°/10nm for the
aperiodic OPAs and 11.3°/10nm for the periodic OPAs (Fig. 6). These observed
values slightly differ from the simulated rate of 12.1°/10nm. On examining the
grating lobes of periodic OPAs at various angles and wavelengths, we observed
steering rates fluctuating between 11.3°/10nm near 0° and 9.7°/10nm around
40°. This variation is attributed to potential imaging distortions inherent in
our basic 4f optical system, coupled with the resolution limitations of our
3-axis stages, which have a 50 µm resolution. This resolution limitation could
introduce significant errors in angle estimation due to the typical lens
positional shifts being within 200 µm. Alternatively, we employed a method of
correlating simulated grating angles with actual grating angles and shifts in
pixels from the images, leading to an estimated steering rate of 11.90°/10 nm
for the periodic OPAs.

Figure 6: Experimental measurement result for a) Aperodic OPAs and b) Periodic
OPAs, each evaluated at different $\lambda$.

To determine the SLSR, we tuned our laser (Thorlabs, TLX-1) to 1547.7 nm, a
wavelength where our camera exhibits peak sensitivity. The farfield pattern
analysis (Fig. 7a) revealed that the SLSR of the aperiodic OPAs was -6.1 dB,
markedly lower than expected value (-10.0 dB). The mainlobe is at -33.4° from
the experiment and 41.2° from the simulation. A significant contributing
factor is the shift in the effective index due to variations in waveguide,
where an increase of 3% changes (Fig. 4d). Subsequent simulations, accounting
for this modified waveguide width (0.75 µm, rectangular), corroborated this
effect, showing a decrease in SLSR to -6.6 dB (Fig. 7b). This change also
induced wavelength range shifts, with the 0° main lobe’s wavelength moving
from 1.55 µm to 1.58 µm (SLSR = -10.1dB @ $\lambda_{0}$=1.58 µm, Fig. 7c-d).
Additionally, we noticed an uneven intensity distribution in the nearfield
pattern, likely resulting from variable losses across different waveguides,
further contributing to the diminished SLSR from farfield converging.

Figure 7: a) The measured farfield pattern for aperiodic OPAs at $\lambda$ =
1.58 µm. b) The corresponding simulated farfield for waveguides at width =
0.75 µm. c) Illustrates the simulated SLSR for waveguides with a narrower
width of 0.60 µm, while d) displays the simulated SLSR for the 0.75 µm
waveguide width.

##  4 Conclusion & Discussion

Based on the understanding from the Spatial Fourier Transform, we developed a
simple yet effective method to reduce grating lobes and increase the steering
capability in a 1×64 channel passive aperiodic OPAs. We also implement RFA, a
simplified model in farfield forming from the FDTD, to effectively conduct
large-scale and quantity simulations. From this study, the effectiveness of
the SLSR is primarily attributed to reducing the amplitude at specific spatial
frequencies, especially at $\Lambda$. This is accomplished by incorporating
uniform random permutated positional shifts to the emitter’s position. On the
other hand, the steering capability is directly tied to the passive path
differences in the $\Omega$-shaped waveguides. Greater passive path
differences between waveguides lead to larger steering capabilities, though
this comes with a trade-off in terms of lower engineering tolerance.

From the simulations, we demonstrate this aperiodic OPAs exhibiting the SLSR
of -13.46 dB at 0° and -8.27 dB at ±45°, while the steering rate is
18.60°/10nm ($\alpha$ = 16). However, due to fabrication errors and
characterization tools’ limits, the experimental SLSR is -6.1 dB at -33.4°,
and the steering rate is 9.8°/10nm ($\alpha$ = 10). With the experience gained
from the LNF, we believe that SLSR will be further improved by reducing the
fabrication error.

We acknowledge the potential for further optimization in our study. Our
approach, which introduces and visualizes the disorder into the emitter
arrangement to reduce sidelobes, draws upon principles from diffraction theory
and related research. Future improvements might include exploring different
emitter numbers and pitch sizes, as well as other possibilities like
hyperuniform distributions, known for reducing density fluctuations at larger
scales. Moreover, the concepts used in this study for 2D array arrangements
are validated, suggesting the potential for optimizing 3D OPAs structures
using similar strategies.

## References

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