# On the Glasner Property of Linear Maps with Prime Entries on Tori

Andrew Rajchert I would like to thank Professor Alexander Fish for acting as a
supervisor and reviewer during the construction of this paper, without whom
this project would have not been possible.

###### Abstract

We study the quantitative Glanser property in the context of maps between tori
of differing dimension instead of as a (semi-)group action. We also only
consider matrices with entries being non-constant polynomials evaluated at
primes, extending on the work of Velani and Nair [4] and Bulinski and Fish [2]
to a more general setting.

## Introduction

We begin by defining of the Glasner property.

######  Definition 1.

Let $X$ and $Y$ be metric spaces, with $Y$ compact. Let $\mathcal{M}$ be a set
of functions $f_{i}:X\to Y$ for $i\in I$. We say the triple
$(X,Y,\mathcal{M})$ has the Glasner property if for any choice of $\epsilon>0$
and any choice of infinite set $A\subseteq X$, there exists an
$f\in\mathcal{M}$ such that $f(A)$ is $\epsilon$-dense in $Y$.

By $\epsilon$-dense, we mean that $B_{\epsilon}(y)\cap f(A)\neq\emptyset$ for
any choice of $y\in Y$. It’s natural to ask what can be attained if our subset
$A$ is only finite, which leads to a quantitative variant of the Glasner
property which imposes additional constraints

######  Definition 2.

The triple $(X,Y,\mathcal{M})$ has the quantitative Glasner property if it has
the Glasner Property and there exists some $k:\mathbb{R}\to\mathbb{N}$ such
that for any $\epsilon>0$ and $A\subseteq X$ with $|A|>k(\epsilon)$, there
exists an $f\in\mathcal{M}$ such that $f(A)$ is $\epsilon$-dense in $Y$.

This is different to the standard Glasner property and quantitative variants
studied by others as here we don’t impose the restriction that $X=Y$. We study
the quantitative Glasner property in the context of the torus,
$\mathbb{T}^{n}=\mathbb{R}^{n}/\mathbb{Z}^{n}$. That is, $X$ and $Y$ are tori
(with possibly different dimension) and $\mathcal{M}$ is some subset of the
matrices with integer entries, with the functions given by matrix
multiplication.

In the context of the Glasner property, the torus is a well-studied space. Let
$f_{n}(x)=nx\pmod{1}$. When $\mathcal{M}=\\{f_{n}(x)\mid n\geq 1\\}$, Alon and
Peres show that for any choice of $\delta>0$,
$(\mathbb{T},\mathbb{T},\mathcal{M})$ has the quantitative Glasner property
with $k(\epsilon)=1/\epsilon^{2+\delta}$ as a lower bound for all small enough
$\epsilon$. Furthermore, they show this choice of $k(\epsilon)$ is optimal up
to the $\delta$ term. Alon and Peres also extend this to
$\mathcal{M}=\\{f_{p}(x)\mid p\text{ is prime}\\}$ with the same
$k(\epsilon)$. Furthermore, for any non-constant $P(x)\in\mathbb{Z}[x]$, Alon
and Peres extend their result to the case $\mathcal{M}=\\{f_{P(n)}(x)\mid
n\geq 1\\}$, with $k(\epsilon)$ depending on the degree of $P(x)$ [1]. Their
Harmonic Analysis-based approach allows for relatively minor adjustments to
prove a variety of different triples have the quantitative Glasner property,
which has been used by many to prove other variations of this statement.

One of the more important variations of Alon and Peres’ work for our purposes
is that of Nair and Velani, whose theorem we state now:

######  Theorem 1.

Fix $f\in\mathbb{Q}[x]$ to be a non-constant polynomial of degree $L\geq 1$
with $f(\mathbb{Z})\subseteq\mathbb{Z}$, and fix $\delta>0$. Then there exists
some $\epsilon_{0}=\epsilon_{0}(f,\delta)$ such that with
$\mathcal{M}=\\{f(p)\mid p\text{ is prime}\\}$,
$(\mathbb{T},\mathbb{T},\mathcal{M})$ has the quantitative Glasner property
where $k(\epsilon)=1/\epsilon^{2L+\delta}$ for all $\epsilon<\epsilon_{0}$.
[4].

Another important theorem is that of Bulinski and Fish, who also work with
polynomials, but in the higher dimensional case.

######  Theorem 2.

Let $n\geq 1$ be an integer, $\delta>0$, and $f_{i,j}\in\mathbb{Z}[x]$ be non
constant polynomials of degree $L_{i,j}\geq 1$ for $1\leq i,j\leq n$.
Additionally assume that the $f_{i,j}$ and 1 are linearly independent over
$\mathbb{Z}$ and let

$\displaystyle\mathcal{M}=\left\\{\begin{bmatrix}f_{1,1}(x)&f_{1,2}(x)&...&f_{1,n}(x)\\\
f_{2,1}(x)&f_{2,2}(x)&...&f_{2,n}(x)\\\ &\vdots&&\\\
f_{n,1}(x)&f_{n,1}(x)&...&f_{n,n}(x)\\\ \end{bmatrix},x\in\mathbb{Z}\right\\}$

Then there exists some $\epsilon_{0}=\epsilon_{0}(\mathcal{M},\delta)>0$ such
that $(\mathbb{T}^{n},\mathbb{T}^{n},\mathcal{M})$ has the quantitative
Glasner property where $k(\epsilon)=1/\epsilon^{n(n+1)(2L+1)+\delta}$ for all
$\epsilon\in(0,\epsilon_{0})$. [2]

For our work, we merge Theorems 1 and 2 by focusing on polynomials $f_{i,j}$
that are evaluated at the primes in the higher dimensional case, in addition
to expanding to the case where our two tori are of different dimensions. This
allows us to study the impact of the input space and output spaces separately,
and how they may affect the lower bound $k(\epsilon)$. Our first theorem is
the following:

######  Theorem 3.

Fix $\delta>0$, positive integers $m,n\geq 1$, and non-constant polynomials
$f_{i,j}\in\mathbb{Q}[x]$ for $1\leq i\leq n,1\leq j\leq m$ of degrees
$L_{i,j}\geq 1$ with $f_{i,j}(\mathbb{Z})\subseteq\mathbb{Z}$. Let
$L=\max(L_{i,j})$ and let $\mathcal{M}$ be the set of matrices,

$\displaystyle\mathcal{M}=\left\\{\begin{bmatrix}f_{1,1}(p_{1,1})&f_{1,2}(p_{1,2})&...&f_{1,m}(p_{1,m})\\\
f_{2,1}(p_{2,1})&f_{2,2}(p_{2,2})&...&f_{2,m}(p_{2,m})\\\ &\vdots&&\\\
f_{n,1}(p_{n,1})&f_{n,1}(p_{n,2})&...&f_{n,m}(p_{n,m})\\\
\end{bmatrix},p_{i,j}\text{ are prime}\right\\}$

Then there exists a $\epsilon_{0}=\epsilon_{0}(\mathcal{M},\delta)>0$ such
that $(\mathbb{T}^{m},\mathbb{T}^{n},\mathcal{M})$ has the quantitative
Glasner property where for all $\epsilon\in(0,\epsilon_{0})$,

$\displaystyle k(\epsilon)=1/\epsilon^{2L(m+1)n+\delta}.$

Furthermore, in the case that $m=1$, we instead have the tighter bound
$k(\epsilon)=1/\epsilon^{2Ln+\delta}$ for all $\epsilon\in(0,\epsilon_{0})$.

This Theorem can be thought of as the extension of Theorem 1 to the higher
dimensional case, whose bound we match when $m=n=1$ as expected. Additionally
we improve on the bound by Bulinski and Fish in Theorem 2. This is due to the
independent choices of $p_{i,j}$ instead of just a single prime/integer, which
allows us to treat the chosen polynomials independently.

Of interest is how our lower bound, $k(\epsilon)$, depends on different
parameters, which in our case is $n,m$ and $L$. Indeed the dependence on $n$
is obviously essential as we can fill $\mathbb{T}^{n}$ with at least
$\left\lfloor 1/\epsilon\right\rfloor^{n}$ non-intersecting spheres of radius
$\epsilon$, hence $k(\epsilon)>\lfloor 1/\epsilon\rfloor^{n}$.

While not initially obvious, the dependence on $m$ is also required. Consider
$A=\\{0,1/2\\}^{m}\subseteq\mathbb{T}^{m}$ which contains $2^{m}$ points. For
any valid choice of $f_{i,j}$, since $f_{i,j}(\mathbb{Z})\subseteq\mathbb{Z}$,
we have $\mathcal{M}\subseteq M_{n\times m}(\mathbb{Z})$. It follows for any
$M\in\mathcal{M},MA\subseteq\\{0,1/2\\}^{n}$. For any $\epsilon<1/2$, this set
is not $\epsilon$-dense, yet we can increase $|A|$ to be arbitrarily large by
increasing $m$, demonstrating the necessity of the dependence on $m$ in our
lower bound $k(\epsilon)$.

The dependence on $L$ is not obvious from the outset, and unfortunately it is
not known if this is essential. The $L$ term originates from the use of the
Hua bound to get an asymptotic upper bound on the value of an exponential sum,
which in turn originates from the typical Harmonic Analysis approach to
proving statements about the quantitative Glasner property. As the Hua bound
is effectively tight, the $L$ dependence cannot be avoided without significant
changes to the proof used.

The separate $m=1$ case originates from being able to use a tighter bound than
usual in our proof. More specifically, when given the set $\\{x_{i}\mid 1\leq
i\leq k\\}\subseteq\mathbb{T}^{m}$ and an integer $b$, a key section of the
proof requires us to bound the number of pairs $1\leq i<j\leq k$ such that
$x_{i}-x_{j}$ is rational and can be written with a denominator of at most
$b$. In the case of $m=1$, we are able to make use of Proposition 1.3 of Alon
and Peres [1], a statement which fails to extend naturally to the higher
dimensional case. When $m>1$ we instead use a weaker bound originating from
Proposition 1 from Kelly and Lê [3]. An obvious path for future work is
delving more into extending Proposition 1.3 of Alon and Peres into the higher
dimensional case.

We also prove another theorem, which instead considers the case where we have
a single choice of prime.

######  Theorem 4.

Fix $\delta>0$, positive integers $m,n\geq 1$, and non-constant polynomials
$f_{i,j}\in\mathbb{Z}[x]$, $1\leq i\leq n$, $1\leq j\leq m$ of degrees
$L_{i,j}\geq 1$ such that the $f_{i,j}$ and $1$ are linearly independent over
$\mathbb{Z}$. Let $L=\max(L_{i,j})$ and $\mathcal{M}$ be the set of matrices,

$\displaystyle\mathcal{M}=\left\\{\begin{bmatrix}f_{1,1}(p)&f_{1,2}(p)&...&f_{1,m}(p)\\\
f_{2,1}(p)&f_{2,2}(p)&...&f_{2,m}(p)\\\ &\vdots&&\\\
f_{n,1}(p)&f_{n,1}(p)&...&f_{n,m}(p)\\\ \end{bmatrix},p\text{ is
prime}\right\\}.$

Then there exists a $\epsilon_{0}=\epsilon_{0}(\mathcal{M},\delta)>0$ such
that $(\mathbb{T}^{m},\mathbb{T}^{n},\mathcal{M})$ has the quantitative
Glasner property where for all $\epsilon\in(0,\epsilon_{0})$,

$\displaystyle k(\epsilon)=1/\epsilon^{(2L+1)(m+1)n+\delta}.$

Furthermore, in the case that $m=1$, we instead have the tighter bound
$k(\epsilon)=1/\epsilon^{(2L+1)n+\delta}$ for all
$\epsilon\in(0,\epsilon_{0})$.

Due to the single choice of prime we index $\mathcal{M}$ over, this can be
thought of as an extension of Theorem 2 into the non-square and prime case.
Additionally, these restrictions do not affect the quality of the lower bound
$k(\epsilon)$ found by Bulinski and Fish, and is only slightly weaker than
Theorem 3.

## Proof of Theorem 3

Before we prove the main theorem, there is a variety of Lemmas that will be
useful. We first recall Lemma 6.2 from Alon and Peres [1] on the existence and
behaviour of bump functions.

######  Lemma 1.

For $0<\epsilon<1$, there exist non-negative functions
$g_{\epsilon}:\mathbb{R}\to\mathbb{R}$ of period 1 such that
$g_{\epsilon}(t)=0$ for $\epsilon\leq|t|\leq 1/2$ and the Fourier coefficients
$\widehat{g_{\epsilon}}(m)$ satisfy $\widehat{g_{\epsilon}}(0)=1$ and

$\displaystyle\forall m\in\mathbb{Z},\quad|\widehat{g_{\epsilon}}(m)|\leq
C\exp(-\sqrt{\epsilon|m|})$

where $C$ is an absolute constant. Furthermore, there exists another constant
$C^{\prime}>0$ such that for any $\epsilon\in(0,1)$,

$\displaystyle\int_{-\epsilon}^{\epsilon}g_{\epsilon}(z)^{2}dz=\frac{C^{\prime}}{\epsilon}.$

We will also use a Lemma on the equidistribution of irrational polynomials
from Rhin [5]. For the remainder of this paper, let $P_{N}$ denote the first
$N$ primes.

######  Lemma 2.

Let $f$ be a polynomial such that $f(x)-f(0)$ has at least one irrational
coefficient. Then,

$\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{p\in P_{N}}\exp(2\pi
if(p))=0.$

Additionally, we summarise part of the proof of Theorem 1 of Nair and Velani
in the following Lemma [4],

######  Lemma 3.

Let $f:\mathbb{N}\to\mathbb{N}$ be a non-constant polynomial, and $a/b$ be a
non-zero rational in reduced form. Then for any choice of
$\epsilon^{\prime}>0$, there exists a constant
$c_{2}=c_{2}(\epsilon^{\prime},f)$ such that,

$\displaystyle\sum_{w=1}^{T}\left|\lim_{N\to\infty}\frac{1}{N}\sum_{p\in
P_{N}}\exp\left(2\pi iwf(p)\frac{a}{b}\right)\right|\leq
c_{2}Tb^{-1/\deg(f)+\epsilon^{\prime}}.$

Our final necessary Lemma is adapted from Proposition 2.7 from Bulinski and
Fish [2], which is in turn based on Proposition 1.3 from Alon and Peres [1],
which we alter and prove in a version suitable for our purposes.

######  Lemma 4.

Let $\\{z_{1},...,z_{k}\\}\subseteq\mathbb{T}^{m}$ be a set of $k$ distinct
numbers. Let $h_{b}$ be the number of pairs $(i,j)$ such that $1\leq i<j\leq
k$ and $b$ is the smallest positive integer such that
$b(z_{i}-z_{j})\in\mathbb{Z}^{m}$. Then for any choice of $T>0$, $c>0$ and
$r\in(0,1)$, there exists a constant $C$ independent of $T$ and $k$ such that,

$\displaystyle\sum_{b=2}^{\infty}h_{b}\left(\left(cTb^{-r}+1\right)^{n}-1\right)\leq
CT^{n}k^{2-r/(m+1)}$

Furthermore in the case of $m=1$, for any $\alpha>0$, there exists a $k_{0}$
large enough such that if $k>k_{0}$,

$\displaystyle\sum_{b=2}^{\infty}h_{b}\left(\left(cTb^{-r}+1\right)^{n}-1\right)\leq
CT^{n}k^{2+\alpha-r}$

###### Proof.

We start in a similar fashion to the proof of Proposition 2.7 by Bulinski and
Fish, who initially use a result from Kelly and Lê [3]. From Bulinski and
Fish, we know $H_{b}\leq kb^{m+1}$ where $H_{b}=\sum_{s=1}^{b}h_{s}$ [2]. Fix
$Q>k^{1/(m+1)}$ such that $h_{b}=0$ for all $b>Q$. Then, by applying summation
by parts and using $H_{1}$ = 0,

$\displaystyle\sum_{b=2}^{\infty}h_{b}\left(\left(cTb^{-r}+1\right)^{n}-1\right)$
$\displaystyle=\sum_{b=2}^{Q}\left(H_{b}-H_{b-1}\right)\left(\left(cTb^{-r}+1\right)^{n}-1\right)$
$\displaystyle=H_{Q}\left(\left(cTQ^{-r}+1)\right)^{n}-1\right)$
$\displaystyle+\sum_{2\leq
b<k^{1/(m+1)}}H_{b}\left(\left(cTb^{-r}+1\right)^{n}-\left(cT\left(b+1\right)^{-r}+1\right)^{n}\right)$
$\displaystyle+\sum_{k^{1/(m+1)}\leq
b<Q}H_{b}\left(\left(cTb^{-r}+1\right)^{n}-\left(cT\left(b+1\right)^{-r}+1\right)^{n}\right)$

We now send $Q\to\infty$, which only affects the first and third terms. For
the first term, we note that we always have $H_{Q}\leq k^{2}$, which implies
the first term converges to zero. For the third term, using the same
inequality returns a telescopic sum, which can be evaluated in a relatively
straight-forward manner. For the second inequality, we first consider the
intermediate function $f(b)=\left(cTb^{-r}+1\right)^{n}$. By a binomial
expansion and the fact that $cT,r>0$, $f(b)$ is convex for all $b>0$, and
hence by a linear approximation, $f(b+1)\geq f(b)+f^{\prime}(b)$. Rearranging
this and evaluating the derivative, we have the inequality,

$\displaystyle\left(cTb^{-r}+1\right)^{n}-\left(cT\left(b+1\right)^{-r}+1\right)^{n}$
$\displaystyle\leq rcTnb^{-r-1}\left(cTb^{-r}+1\right)^{n-1}$

Applying the inequality, the bound $H_{b}\leq kb^{m+1}$ and $m-r>0$ to find
that there exists a constant $C_{1}$ depending on $c,r,m$ and $n$ such that,

$\displaystyle\sum_{2\leq
b<k^{1/(m+1)}}H_{q}\left(\left(cTb^{-r}+1\right)^{n}-\left(cT\left(b+1\right)^{-r}+1\right)^{n}\right)$
$\displaystyle\leq\sum_{2\leq
b<k^{1/(m+1)}}kb^{m+1}rcTnb^{-r-1}\left(cTb^{-r}+1\right)^{n-1}$
$\displaystyle\leq rcTnk\left(cT\cdot 2^{-r}+1\right)^{n-1}\sum_{2\leq
b<k^{1/(m+1)}}b^{m-r}$ $\displaystyle\leq rcTnk\left(cT\cdot
2^{-r}+1\right)^{n-1}\int_{2}^{k^{1/(m+1)}+1}b^{m-r}db$ $\displaystyle\leq
rcTnk\left(cT\cdot
2^{-r}+1\right)^{n-1}\left(\left(k^{1/(m+1)}+1\right)^{m-r+1}-2^{m-r+1}\right)$
$\displaystyle\leq C_{1}T^{n}k^{2-r/(m+1)}$

Putting these parts together, we find that by sending $Q\to\infty$,

$\displaystyle\sum_{b=2}^{\infty}h_{b}\left(\left(cTb^{-r}+1\right)^{n}-1\right)$
$\displaystyle\leq C_{1}T^{n}k^{2-r/(m+1)}+k^{2}\left(\left(cT\lceil
k^{1/(m+1)}\rceil^{-r}+1\right)^{n}-1\right)$ $\displaystyle\leq
CT^{n}k^{2-r/(m+1)}.$

For the case of $m=1$, we can repeat the process but instead use the bound
$H_{b}\leq(kb)^{1+\alpha}$ for any choice of $\alpha>0$, for
$k>k_{0}(\alpha)$. This comes directly from Proposition 1.3 of Alon and Peres
[1]. By splitting the sum at the index $b=k$ instead of $b=k^{1/(m+1)}$, we
find

$\displaystyle\sum_{b=2}^{\infty}h_{b}\left(\left(cTb^{-r}+1\right)^{n}-1\right)$
$\displaystyle\leq
C_{1}T^{n}k^{2+\alpha-r}+k^{2}\left(\left(cTk^{-r}+1\right)^{n}-1\right)$
$\displaystyle\leq CT^{n}k^{2+\alpha-r}.$

∎

We are now ready to prove the Theorem. Unfortunately there are a large amount
of algebraic manipulations involved in the details of this proof, but this is
done to make full use of the independent choices of primes in minimising
$k(\epsilon)$.

###### Proof of Theorem 3.

We use the Harmonic Analysis approach of Alon and Peres [1]. Let $\epsilon>0$
and suppose $X=\\{x_{1},...,x_{k}\\}$ is a set of $k$ distinct points in
$\mathbb{T}^{m}$ such that $MX$ is not $\epsilon$-dense for any
$M\in\mathcal{M}$.

Let $h_{\epsilon}(z_{1},z_{2},...,z_{n})=\prod_{i=1}^{n}g_{\epsilon}(z_{i})$
where the $g_{\epsilon}$ are bump functions from Lemma 1. We will note the
fact that the Fourier coefficients of the product of independent functions is
just the product of each functions Fourier coefficients, thus we have bounds
on the Fourier coefficients of $h_{\epsilon}$.

For any $N>1$, we can also index some elements of $\mathcal{M}$ by writing
$M=M(\mathbf{p})\in\mathcal{M}$ where
$\mathbf{p}=(p_{1,1},p_{1,2},...,p_{n,m})\in P_{N}^{mn}$ and the $f_{i,j}$ are
evaluated at $p_{i,j}$. It follows by the definition of $h_{\epsilon}$ that
for every $M\in\mathcal{M}$, there exists a $\lambda_{M}\in\mathbb{T}^{n}$
such that,

$\displaystyle 0$ $\displaystyle=\frac{1}{N^{mn}}\sum_{\mathbf{p}\in
P_{N}^{mn}}\sum_{j=1}^{k}h_{\epsilon}(Mx_{j}+\lambda_{M})$
$\displaystyle=\frac{1}{N^{mn}}\sum_{\mathbf{p}\in
P_{N}^{mn}}\sum_{j=1}^{k}\sum_{\mathbf{m}\in\mathbb{Z}^{n}}\widehat{h}_{\epsilon}(\mathbf{m})e_{\mathbf{m}}\left(Mx_{j}+\lambda_{M}\right).$

We note we attain equality in the Fourier series due to the Fourier
coefficients converging to zero faster than any polynomial, and the
periodicity of $h_{\epsilon}$. Here we let $e_{\mathbf{m}}(v)=e(2\pi
i\mathbf{m}\cdot v)$ for $v\in\mathbb{T}^{n}$. Now we separate this into parts
depending on the value of $\mathbf{m}$. Applying the triangle inequality, we
can extract the $\mathbf{m}=\mathbf{0}$ term from the series. Let
$S_{T}=\\{\mathbf{m}\in\mathbb{Z}^{n}\mid 0<\|\mathbf{m}\|_{\infty}\leq T\\}$.
Then,

$\displaystyle k$ $\displaystyle\leq\left|\frac{1}{N^{mn}}\sum_{\mathbf{p}\in
P_{N}^{mn}}\sum_{j=1}^{k}\sum_{\mathbf{m}\in
S_{T}}\widehat{h}_{\epsilon}(\mathbf{m})e_{\mathbf{m}}\left(Mx_{j}+\lambda_{M}\right)\right|$
$\displaystyle+k\sum_{\|\mathbf{m}\|_{\infty}>T}\left|\widehat{h}_{\epsilon}(\mathbf{m})\right|$

We now bound this second sum. By Lemma 1 and the fact
$\widehat{h_{\epsilon}}(\mathbf{m})=\prod_{i=1}^{n}\widehat{g_{\epsilon}}(m^{i})$
(where the $i$ superscript denotes taking the $i$-th coordinate) is invariant
under permutation of the coordinates, we know there exist constants $C$ and
$C^{\prime}=C^{\prime}(n)$ such that,

$\displaystyle\sum_{\|\mathbf{m}\|_{\infty}>T}\left|\widehat{h}_{\epsilon}(\mathbf{m})\right|$
$\displaystyle\leq 2n\sum_{\mathbf{m}\in\mathbb{Z}^{n},m^{1}\geq
T}\left|\widehat{h}_{\epsilon}(\mathbf{m})\right|$ $\displaystyle\leq
2n\sum_{\mathbf{m}\in\mathbb{Z}^{n},m^{1}\geq
T}C\exp\left(-\sum_{i=1}^{n}\sqrt{\epsilon m^{i}}\right)$ $\displaystyle\leq
C^{\prime}\sum_{m^{1}=T}^{\infty}\exp\left(-\sqrt{\epsilon m^{1}}\right).$

Directly from the proof of Proposition 6.1 by Alon and Peres [1], if we set
$T=\lfloor 4/\epsilon\log^{2}(1/\epsilon)\rfloor$, then there exists some
$\epsilon_{1}$ depending on $n$ such that when $\epsilon\in(0,\epsilon_{1})$,
this expression is less than $1/2$ . Using such a value of $\epsilon$, we can
return to our previous upper bound on $k$,

$\displaystyle\frac{k}{2}$
$\displaystyle\leq\left|\frac{1}{N^{nm}}\sum_{\mathbf{p}\in
P_{N}^{mn}}\sum_{j=1}^{k}\sum_{\mathbf{m}\in
S_{T}}\widehat{h}_{\epsilon}(\mathbf{m})e_{\mathbf{m}}\left(Mx_{j}+\lambda_{M}\right)\right|.$

Now squaring and using the Cauchy-Schwartz inequality twice, we can separate
the Fourier coefficients from the exponential terms.

$\displaystyle\frac{k^{2}}{4}$
$\displaystyle\leq\frac{1}{N^{2mn}}\left|\sum_{\mathbf{p}\in
P_{N}^{mn}}\sum_{j=1}^{k}\sum_{\mathbf{m}\in
S_{T}}\widehat{h}_{\epsilon}(\mathbf{m})e_{\mathbf{m}}\left(Mx_{j}+\lambda_{M}\right)\right|^{2}$
$\displaystyle\leq\frac{1}{N^{mn}}\sum_{\mathbf{p}\in
P_{N}^{mn}}\left|\sum_{j=1}^{k}\sum_{\mathbf{m}\in
S_{T}}\widehat{h}_{\epsilon}(\mathbf{m})e_{\mathbf{m}}\left(Mx_{j}+\lambda_{M}\right)\right|^{2}$
$\displaystyle\leq\frac{1}{N^{mn}}\sum_{\mathbf{p}\in
P_{N}^{mn}}\left(\sum_{\mathbf{m}\in
S_{T}}|\widehat{h}_{\epsilon}(\mathbf{m})|^{2}\right)\left(\sum_{\mathbf{m}\in
S_{T}}\left|\sum_{j=1}^{k}e_{\mathbf{m}}\left(Mx_{j}+\lambda_{M}\right)\right|^{2}\right)$

By Bessel’s inequality and the properties of the bump functions provided by
Lemma 1,

$\displaystyle\sum_{\mathbf{m}\in
S_{T}}|\widehat{h}_{\epsilon}(\mathbf{m})|^{2}$
$\displaystyle\leq\int_{\mathbb{T}^{n}}|h_{\epsilon}(z)|^{2}dz$
$\displaystyle=\left(\int_{\mathbb{T}}g_{\epsilon}(z)^{2}dz\right)^{n}=\frac{c_{0}}{\epsilon^{n}}.$

Using this and replacing constants by $c_{1}$, we do some careful manipulation
of our products and sums.

$\displaystyle k^{2}$
$\displaystyle\leq\frac{c_{1}}{N^{mn}\epsilon^{n}}\sum_{\mathbf{p}\in
P_{N}^{mn}}\sum_{\mathbf{m}\in
S_{T}}\left|\sum_{j=1}^{k}e_{\mathbf{m}}\left(Mx_{j}+\lambda_{M}\right)\right|^{2}$
$\displaystyle=\frac{c_{1}}{N^{mn}\epsilon^{n}}\sum_{\mathbf{p}\in
P_{N}^{mn}}\sum_{\mathbf{m}\in
S_{T}}\sum_{j=1}^{k}\sum_{l=1}^{k}e_{\mathbf{m}}\left(Mx_{j}-Mx_{l}\right)$
$\displaystyle=\frac{c_{1}}{\epsilon^{n}}\sum_{j=1}^{k}\sum_{l=1}^{k}\sum_{\mathbf{m}\in
S_{T}}\frac{1}{N^{mn}}\sum_{\mathbf{p}\in
P_{N}^{mn}}\prod_{u=1}^{n}\prod_{v=1}^{m}\exp(2\pi
im^{u}f_{u,v}(p_{u,v})(x_{j}^{v}-x_{l}^{v}))$
$\displaystyle=\frac{c_{1}}{\epsilon^{n}}\sum_{j=1}^{k}\sum_{l=1}^{k}\left[\sum_{\|\mathbf{m}\|_{\infty}\leq
T}\left(\prod_{u=1}^{n}\prod_{v=1}^{m}\sum_{p\in P_{N}}\frac{1}{N}\exp(2\pi
im^{u}f_{u,v}(p)(x_{j}^{v}-x_{l}^{v}))\right)-1\right]$
$\displaystyle=:\frac{c_{1}}{\epsilon^{n}}\sum_{j=1}^{k}\sum_{l=1}^{k}I_{j,l}.$

Here we have set $I_{j,l}$ to be the term inside the square brackets, which we
isolate from the outer sum to simplify some of the working. Further
simplifying this term,

$\displaystyle I_{j,l}$
$\displaystyle=\prod_{u=1}^{n}\left(\sum_{w=-T}^{T}\prod_{v=1}^{m}\sum_{p\in
P_{N}}\frac{1}{N}\exp(2\pi iwf_{u,v}(p)(x_{j}^{v}-x_{l}^{v}))\right)-1$
$\displaystyle\leq\left(\sum_{w=-T}^{T}\prod_{v=1}^{m}\left|\sum_{p\in
P_{N}}\frac{1}{N}\exp(2\pi
iwf_{\cdot,v}(p)(x_{j}^{v}-x_{l}^{v}))\right|\right)^{n}-1$

The inequality comes from the fact that $I_{j,l}$ is real as, by matching the
positive and negative $w$ terms together, we have the sum of complex numbers
and their conjugates. Furthermore, we introduce $f_{\cdot,v}$ to represent the
choice of $u$ which maximises this product. We now take $N\to\infty$, and as
the outer terms have no dependence on $N$, it’s enough for us to study the
inner sum, on which we apply our Lemmas.

Suppose for some $j,l$, we have $x_{j}-x_{l}$ containing at least one
irrational entry and without loss of generality, suppose $x_{j}^{1}-x_{l}^{1}$
is irrational. Then, when $w$ is non-zero, we may apply Lemma 2 to observe as
$N\to\infty$, the $v=1$ term of the product over $v$ converges to zero, thus
the product converges to zero. If $w=0$, the inner sum is identically one. It
follows that $I_{j,l}$ can be bounded from above by 0 as $N\to\infty$, and
thus can be ignored in this case with respect to our overarching bound on
$k^{2}$.

Next, suppose $x_{j}-x_{l}$ only has rational entries, and let the first non-
zero rational entry be written as a reduced fraction $a/b$. Again without loss
of generality, suppose $x_{j}^{1}-x_{l}^{1}=a/b$. By applying Lemma 3 we
observe that for any choice of $\epsilon^{\prime}>0$, there’s a constant
$c_{2}$ such that,

$\displaystyle\lim_{N\to\infty}\sum_{w=-T}^{T}\prod_{v=1}^{m}\Big{|}\sum_{p\in
P_{N}}\frac{1}{N}\exp(2\pi iwf_{\cdot,v}(p)(x_{j}^{v}-x_{l}^{v}))\Big{|}$
$\displaystyle\leq\sum_{w=-T}^{T}\Big{|}\lim_{N\to\infty}\sum_{p\in
P_{N}}\frac{1}{N}\exp(2\pi iwf_{\cdot,1}(p)(x_{j}^{1}-x_{l}^{1}))\Big{|}\leq
c_{2}Tb^{-1/L+\epsilon^{\prime}}+1$

It naturally follows $I_{j,l}$ can be bounded by
$(c_{2}Tb^{-1/L+\epsilon^{\prime}}+1)^{n}-1$ when at least one the entries of
$x_{j}-x_{l}$ is the reduced non-zero rational $a/b$. Finally we note that
when $x_{j}-x_{l}=0$, by uniqueness of our points we must have $j=l$ which can
only occur $k$ times. It is simple to see $I_{j,j}=(2T+1)^{n}-1$ for any valid
$j$.

Grouping our summation over $j$ and $l$ by the bounds we have on $I_{j,l}$, we
can continue our bound on $k^{2}$. We let the number of times we see the
rational case with denominator $b$ be $h_{b}$, the same notation of Lemma 4.
Applying this, we observe,

$\displaystyle k^{2}$
$\displaystyle\leq\frac{c_{1}}{\epsilon^{n}}\left(k\left(\left(2T+1\right)^{n}-1\right)+2\sum_{b=2}^{\infty}h_{b}\left(\left(c_{2}Tb^{-1/L+\epsilon^{\prime}}+1\right)^{n}-1\right)\right).$

The sum over $b$ can be bounded by the use of Lemma 4, with
$r=1/L-\epsilon^{\prime}$ which is positive for a small enough choice of
$\epsilon^{\prime}$. Replacing constants by $c_{i}$ as they appear, each of
which is independent of $T$ and $k$,

$\displaystyle k^{2}$
$\displaystyle\leq\frac{c_{1}}{\epsilon^{n}}\left(k\left(\left(2T+1\right)^{n}-1\right)+c_{3}T^{n}k^{2-(1/L-\epsilon^{\prime})/(m+1)}\right)$
$\displaystyle\Longrightarrow k^{2}$ $\displaystyle\leq
c_{4}\frac{T^{n}}{\epsilon^{n}}k^{2-(1/L-\epsilon^{\prime})/(m+1)}$
$\displaystyle\Longrightarrow k$ $\displaystyle\leq
c_{5}\left(\frac{T}{\epsilon}\right)^{Ln(m+1)+\epsilon^{\prime}nL^{2}(m+1)/(1-\epsilon^{\prime}L)}$

By recalling our choice of $T$, we note that for any choice of $\delta>0$,
there exists an $\epsilon$ small enough such that $T/\epsilon\leq
1/\epsilon^{2+\delta}$. By incorporating our choice of $\epsilon^{\prime}$ and
the constant $c_{5}$ into $\delta$, it follows that for any choice of
$\delta>0$, there exists a sufficiently small $\epsilon_{0}$ such that if
$\epsilon\in(0,\epsilon_{0})$ and $k>1/\epsilon^{2Ln(m+1)+\delta}$, then there
must exist a $M\in\mathcal{M}$ such that $MX$ is $\epsilon$ dense in
$\mathbb{T}^{n}$.

If instead $m=1$, we can apply the tighter bound of Lemma 4 with the same
value of $r=1/L-\epsilon^{\prime}$. Then for any choice of $\alpha>0$, if
$k\geq k_{0}(\alpha)$,

$\displaystyle k^{2}$
$\displaystyle\leq\frac{c_{1}}{\epsilon^{n}}\left(k\left(\left(2T+1\right)^{n}-1\right)+c_{3}T^{n}k^{2+\alpha-1/L+\epsilon^{\prime}}\right)$
$\displaystyle\Longrightarrow k$ $\displaystyle\leq
c_{4}\left(\frac{T}{\epsilon}\right)^{nL+(\epsilon^{\prime}+\alpha^{\prime})nL^{2}/(1-\epsilon^{\prime}L-\alpha
L)}$

We know that for any choice of $\delta>0$, $T/\epsilon\leq
1/\epsilon^{2+\delta}$. By incorporating $\alpha$ and $\epsilon^{\prime}$ into
this choice, in addition to choosing our upper bound for $\epsilon$,
$\epsilon_{0}$ as depending on $\alpha$, we find that for any choice of
$\delta>0$, there exists a sufficiently small $\epsilon_{0}$ such that if
$\epsilon\in(0,\epsilon_{0})$ and $k>1/\epsilon^{2Ln+\delta}$, then there
exists a $M\in\mathcal{M}$ such that $MX$ is $\epsilon$ dense in
$\mathbb{T}^{n}$.

∎

## Proof of Theorem 4

We now restrict Theorem 3 to the case of each entry in our set of matrices
depending on a single choice of prime, rather than independent choices of
primes for each polynomial.

There are a number of definitions and statements from [2] which will be
helpful here. We first define multiplicative complexity, a way of bounding how
linear combinations of polynomials may introduce common factors in their
coefficients.

######  Definition 3.

The vector $P(x)=[P_{1}(x),...,P_{r}(x)]\in(\mathbb{Z}[x])^{r}$ has
multiplicative complexity $Q$ if for all
$a=(a_{1},...,a_{r})\in\mathbb{Z}^{r}$ and $q\in\mathbb{Z}$ with
$\gcd(a_{1},...,a_{r},q)=1$, we have the polynomial

$\displaystyle(P(x)-P(0))\cdot a=\sum_{i=1}^{L}b_{j}x^{j}$

satisfying $\gcd(b_{1},...,b_{L},q)\leq Q$.

We will also require Corollary 2.4 from [2], however we observe from the proof
it can easily be extended to the case of non-square matrices.

######  Lemma 5.

Let $M(x)\in M_{n\times m}(\mathbb{Z}[x])$ be a matrix with polynomial entries
and $\mathbf{m}\in\mathbb{Z}^{n}\setminus\\{0\\}$ such that the entries of
$\mathbf{m}^{t}(M(x)-M(0))$ are linearly independent over $\mathbb{Z}$. Then
$\mathbf{m}^{t}M(x)$ has multiplicative complexity $Q$, where

$\displaystyle
Q=Q(M(x),\mathbf{m})=m!(n\|M(x)-M(0)\|\|\mathbf{m}\|_{\infty})^{m}.$

Here $\|M(x)-M(0)\|$ denotes the largest polynomial coefficient in any entry
of $M(x)-M(0)$.

###### Proof.

First we make note of Proposition 2.2 of Bulinski and Fish, which states that
for any linearly independent vectors $v_{1},...,v_{m}\in\mathbb{Z}^{r}$ and
coprime integers $a_{1},...,a_{m},q$, we have
$\gcd\left(\sum_{i=1}^{m}a_{i}v_{i},q\right)\leq
m!\max_{i}\|v_{i}\|_{\infty}^{m}$, where the $\gcd$ is computed by taking the
$\gcd$ of coordinate entries of the first argument and $q$ [2].

Let $P(x)=[p_{1}(x),...,p_{m}(x)]=\mathbf{m}^{t}\left(M(x)-M(0)\right)$, and
note that $P(0)=0$. It’s given these polynomials are linearly independent over
$\mathbb{Z}$, so by treating them as vectors we can apply the Proposition. It
follows that for any coprime integers $a_{1},...,a_{m},q\in\mathbb{Z}$, we
also have $\gcd(a_{1}p_{1}(x)+...+a_{m}p_{m}(x),q)\leq
m!\max_{i}\|p_{i}\|_{\infty}^{m}$, where the $\gcd$ is computed by taking the
$\gcd$ of the coefficients of the first argument and the second argument.

This is exactly the definition of multiplicative complexity, where
$a=(a_{1},...,a_{m})$. In our treatment of polynomials as vectors,
$\|p_{i}\|_{\infty}$ refers to the largest coefficient on $p_{i}$, which must
be bounded by $n\|\mathbf{m}\|_{\infty}\|M(x)-M(0)\|_{\infty}$. It follows
$\mathbf{m}^{t}M(x)$ has the desired multiplicative complexity. ∎

Finally, we further require a version of Lemma 3 from Shparlinski on
exponential sums for polynomials [6].

######  Lemma 6.

Let $P(x)=a_{0}+a_{1}x+...+a_{L}x^{L}\in\mathbb{Z}[x]$ and let
$b\in\mathbb{Z}$ be such that $\gcd(a_{0},...,a_{L},b)=1$. Then, for any
choice of $\delta>0$, there exists a constant $C$ independent of $b$ such
that,

$\displaystyle\sum_{1\leq r\leq b,\gcd(r,b)=1}\exp\left(2\pi
i\frac{1}{b}P(r)\right)$ $\displaystyle\leq Cb^{1-1/L+\delta}.$

###### Proof of Theorem 4.

We use the same strategy as in Theorem 3 by letting $\epsilon>0$, and
$X=\\{x_{1},...,x_{k}\\}$ such that for all primes $p$, the set $M(p)X$ is not
$\epsilon$-dense in $\mathbb{T}^{n}$. Following the same steps as Theorem 3,
we continue until we have the corresponding inequality,

$\displaystyle k^{2}$
$\displaystyle\leq\frac{c_{1}}{\epsilon^{n}}\sum_{\mathbf{m}\in
S_{T}}\sum_{1\leq j,l\leq k}\frac{1}{N}\sum_{p\in
P_{N}}e_{\mathbf{m}}\big{(}M(p)(x_{j}-x_{l})\big{)}$

We now take $N\to\infty$, and consider different cases depending on the value
of $x_{j}-x_{l}$. First suppose that $x_{j}-x_{l}$ has at least one irrational
entry. Using the linear independence of the $f_{i,j}$, set
$\mathbf{m}^{t}(M(x)-M(0))=[P_{1}(x),...,P_{m}(x)]=P(x)$ to be a vector of
linearly independent polynomials over $\mathbb{Z}$ and hence also over
$\mathbb{Q}$. As in the proof of Theorem 2.8 of [2], $P(x)\cdot(x_{j}-x_{l})$
will have at least one irrational non-constant coefficient. It immediately
follows from Lemma 2 that as $N\to\infty$, the inner sum will converge to 0
and the term will not contribute to anything the overall expression.

Now suppose $x_{j}-x_{l}$ is rational and non-zero, and write
$x_{j}-x_{l}=\frac{1}{b}a$ where $a\in\mathbb{Z}^{m}$ and
$\gcd(a^{1},...,a^{m},b)=1$. We recall that by Lemma 5, the multiplicative
complexity of $\mathbf{m}^{t}(M(x)-M(0))$ is given by $Q$, where, by
considering that $\mathbf{m}\in S_{T}$,

$\displaystyle Q$ $\displaystyle\leq
m!(n\|M(x)-M(0)\|\|\mathbf{m}\|_{\infty})^{m}$ $\displaystyle\leq
m!(n\|M(x)-M(0)\|)^{m}T^{m}.$

Let $P(x)=\mathbf{m}^{t}M(x)a$. Then, due to Dirichlet equidistribution of
polynomials evaluated at the primes and Lemma 6, we observe that for any
choice of $\delta>0$, there exists a constant $C_{L,\delta}$ such that

$\displaystyle\lim_{N\to\infty}\frac{1}{N}\Bigg{|}\sum_{p\in
P_{N}}e_{\mathbf{m}}\left(\frac{1}{b}M(p)a\right)\Bigg{|}$
$\displaystyle=\lim_{N\to\infty}\frac{1}{N}\Bigg{|}\sum_{p\in
P_{N}}\exp\left(2\pi i\frac{1}{b}P(p)\right)\Bigg{|}$
$\displaystyle=\frac{1}{\phi(b)}\Bigg{|}\sum_{1\leq r\leq
b,\gcd(r,b)=1}\exp\left(2\pi i\frac{1}{b}P(r)\right)\Bigg{|}$
$\displaystyle\leq C_{L,\delta}\left(\frac{Q}{b}\right)^{1/L-\delta},$

where in the last inequality, we have used that $\phi(b)\gg b^{1-\delta}$ for
any choice of $\delta>0$. By using the same $h_{q}$ from the proof of Theorem
3, we have the bound,

$\displaystyle k^{2}\leq\frac{c_{1}}{\epsilon^{n}}\sum_{\mathbf{m}\in
S_{T}}\left(\sum_{b=2}^{\infty}h_{b}\left(\frac{Q}{b}\right)^{1/L-\delta}+k\right)$

This nearly the same bound obtained by Bulinski and Fish in Theorem 2.8 of
[2], with the only difference being the $h_{b}$ counting points in
$\mathbb{T}^{m}$ instead of $\mathbb{T}^{n}$. Applying proposition 2.7 of
Bulinski and Fish, we get the bound on the sum,

$\displaystyle\sum_{b=2}^{\infty}h_{b}b^{-1/L+\delta}\leq
c_{2}k^{2-(1/L-\delta)/(m+1)}$

Furthermore, in a similar manner to Lemma 4, it is possible to show that when
$m=1$,

$\displaystyle\sum_{b=2}^{\infty}h_{b}b^{-1/L+\delta}\leq
c_{2}k^{2-1/L+\delta}$

Continuing the proof of Bulinski and Fish nearly verbatim with these bounds in
mind gives the final result. ∎

## References

  * [1] N. Alon and Y. Peres.  Uniform dilations.  Geometric and Functional Analysis, 2(1):1–28, 1992.
  * [2] Kamil Bulinski and Alexander Fish.  Glasner property for unipotently generated group actions on tori.  Israel Journal of Mathematics, Dec 2022.
  * [3] Michael Kelly and Thái Hoàng Lê.  Uniform dilations in higher dimensions.  Journal of the London Mathematical Society, 88(3):925–940, 2013\.
  * [4] R. Nair and S. L. Velani.  Glasner sets and polynomials in primes.  Proceedings of the American Mathematical Society, 126(10):2835–2840, 1998.
  * [5] Georges Rhin.  Sur la répartition modulo i des suites f(p).  Acta Arithmetica, 23(3):217–248, 1973.
  * [6] Igor Shparlinski.  On exponential sums with sparse polynomials and rational functions.  Journal of Number Theory, 60(2):233–244, October 1996.


# Taxonomy of hybridly polarized Stokes vortex beams

Gauri Arora  [  Ankit Butola<EMAIL_ADDRESS>[  Ruchi Rajput  [  Rohit
Agarwal  [  Krishna Agarwal  [  Alexander Horsch  [  Dilip K Prasad  [
Paramasivam Senthilkumaran  [

###### Abstract

Structured beams carrying topological defects, namely phase and Stokes
singularities, have gained extensive interest in numerous areas of optics. The
non-separable spin and orbital angular momentum states of hybridly polarized
Stokes singular beams provide additional freedom for manipulating optical
fields. However, the characterization of hybridly polarized Stokes vortex
beams remains challenging owing to the degeneracy associated with the complex
polarization structures of these beams. In addition, experimental noise
factors such as relative phase, amplitude, and polarization difference
together with beam fluctuations add to the perplexity in the identification
process. Here, we present a generalized diffraction-based Stokes polarimetry
approach assisted with deep learning for efficient identification of Stokes
singular beams. A total of 15 classes of beams are considered based on the
type of Stokes singularity and their associated mode indices. The resultant
total and polarization component intensities of Stokes singular beams after
diffraction through a triangular aperture are exploited by the deep neural
network to recognize these beams. Our approach presents a classification
accuracy of 98.67% for 15 types of Stokes singular beams that comprise several
degenerate cases. The present study illustrates the potential of diffraction
of the Stokes singular beam with polarization transformation, modeling of
experimental noise factors, and a deep learning framework for characterizing
hybridly polarized beams.

###### keywords:

American Chemical Society, LaTeX

Shared first authorship Indian Institute of Technology Delhi] Department of
Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016,
India Shared first authorship UiT The Arctic University of Norway] Department
of Physics and Technology, UiT The Arctic University of Norway, Tromsø, Norway
Indian Institute of Technology Delhi] Department of Physics, Indian Institute
of Technology Delhi, Hauz Khas, New Delhi 110016, India UiT The Arctic
University of Norway] Department of Computer Science, UiT The Arctic
University of Norway, Tromsø, Norway UiT The Arctic University of Norway]
Department of Physics, UiT The Arctic University of Norway, Tromsø, Norway UiT
The Arctic University of Norway] Department of Computer Science, UiT The
Arctic University of Norway, Tromsø, Norway UiT The Arctic University of
Norway] Department of Computer Science, UiT The Arctic University of Norway,
Tromsø, Norway Indian Institute of Technology Delhi] Department of Physics,
Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India
IR,NMR,UV

##  1 Introduction

Phase singularities are associated with the orbital angular momentum (OAM) 1,
and circular polarization is related to spin angular momentum (SAM) of light
2. The superposition of two beams in orthogonal polarization states with at
least one of the beams carrying phase singularity leads to the formation of
Stokes singularities 3, 4, 5. Beams with Stokes singularities are called
Stokes singular beams (SSBs), where light’s spin and orbital angular momentum
are coupled together. Several methods such as interferometric techniques 6,
implementation of spatial light modulators (SLM) 7, 8, stress engineered
optics 9, and spatially inhomogeneous wave plates 10 have been reported to
generate these beams. The polarization distribution associated with these
beams adds an extra degree of freedom in optical communication which increases
the channel capacity 11, 12, 13. The exotic properties of these beams have
been exploited in laser beam shaping 14, super-resolution microscopy 15,
chirality measurement 16, image processing 17, robust beam engineering18,
Mobius strips generation 19 and among others.

The role of SSBs in various applications depends on the net OAM (mode indices)
present in these beams. The Stokes index used to define SSBs does not provide
complete information of the vector beams in terms of their net OAM content.
Therefore, it is necessary to identify these singularities based on their
superposition modes to deploy them for a particular application. However, the
detection techniques of these beams are limited to a few, including Stokes
polarimetry 20, interferometric 21 and diffraction techniques 22. These
detection techniques can not deal with all the degenerate states associated
with Stokes singular beams. In addition, unavoidable experimental fluctuations
in phase, amplitude, and polarization together with other beam fluctuations,
makes detection a challenging task. Recently, artificial intelligence (AI) has
emerged as a powerful tool to boost scientific research in the field of optics
and photonics. The AI-based techniques are also being adopted for the
identification of multi-singularity structured field, orbital angular momentum
of vortex beams, and among others 23,24. In vector regime, machine learning-
based detection of vector vortex beams is presented in 25. However, the
classification method adopted here is based on Stokes polarimetry which does
not consider the degeneracy present in the SSBs. In addition, the report
focuses on the classification of vector vortex beams i.e., a subset of Stokes
singularities. Nevertheless, there is no report to date that deals with
identifying all the Stokes singularities and their degenerate states.

Here, we present the detection of all Stokes singularities by exploiting the
diffraction and polarization transformation patterns of the singular beams
assisted with the deep neural network. A deep neural network provides an
excellent framework for detecting hybridly polarized beams due to its ability
to recognize indiscernible features in the intensity images, that would not be
detectable merely by intensity-sensitive measurements. The method employed
here is based on the diffraction of singular beams through an equilateral
triangular aperture in combination with polarization transformation. The
resultant total and component intensities of a particular Stokes singular beam
after diffraction are utilized for classification purposes. A total of 15
classes of beams are first simulated and further experimentally generated
based on the type of Stokes singularity and the associated mode indices.

Next, the classification of all Stokes singular beams is performed and
compared by using five different deep neural networks. These networks are
trained with mixed simulation and experimental datasets which consist of 15300
images (15000 simulated and 300 experimentally acquired). A total of 90% of
the simulated images and 50% of the experimentally acquired images are used
for training and validation purposes. Finally, the optimized trained network
is tested on 10% of the simulated image and 50% of the experimental datasets.
Separate testing accuracy for both simulated and experimental datasets is
shown to demonstrate the advantage of the current framework. We found
ResNet-18 offers the best testing accuracy i.e., 97.67% and 98.67% for
simulated and experimental data, respectively. Furthermore, the current
experimental and computational approach addresses both detection and
classification tasks thus, enabling the recognition of total and component
intensities of the Stokes singular beams which could be relevant in
applications such as super-resolution optical microscopy and optical
communication. Our findings manifest the soundness of proposed method for the
detection of all Stokes singularities and novel identification method which
paves the way to scalable microscopy applications through Stokes beams
illumination.

Figure 1: An example of polarization degeneracy in Stokes singular beams with
Stokes index $\sigma_{23}=1$. Left: Phase distribution of the two superposing
vortex beams characterized by topological charges $l$ and $m$ respectively.
Orthogonal polarization basis states are marked in the inset. Centre:
Corresponding total intensity, degenerate polarization distribution and Stokes
parameter distributions. Right: Respective diffraction patterns produced by
the two beams after diffraction through a triangular aperture.

##  2 Methods

###  2.1 Stokes degenerate states

Any state of polarization of light can be defined using four Stokes parameters
$S_{0}$, $S_{1}$, $S_{2}$, and $S_{3}$ 20

$\begin{split}S_{0}&=|E_{x}|^{2}+|E_{y}|^{2};\quad
S_{1}=|E_{x}|^{2}-|E_{y}|^{2};\\\ S_{2}&=2Re(E_{x}^{*}E_{y});\quad
S_{3}=2Im(E_{x}^{*}E_{y}).\end{split}$ (1)

For inhomogeneously polarized beams, Stokes parameters are functions of $x$
and $y$. Here, the Stokes parameter $S_{0}$ gives total intensity and other
Stokes parameters give differences in intensities of orthogonal polarization
components. Using these Stokes parameters, Stokes fields $S_{jk}=S_{j}+iS_{k}$
can be constructed which can be $S_{12}$, $S_{23}$, and $S_{31}$. The
corresponding Stokes phases are $\phi_{12}$, $\phi_{23}$, and $\phi_{31}$
where $\phi_{jk}$ is argument of Stokes field defined as:
$\phi_{jk}=\tan^{-1}(S_{k}/S_{j})$. The Stokes singularities are points of
phase singularities in these Stokes phases. Stokes singularities $\phi_{12}$,
$\phi_{23}$, and $\phi_{31}$ provide information about the phase singularities
present in orthogonal polarization basis states, namely, circular (R, L),
linear (H, V), and linear (D, A) bases respectively. Here, R, L, H, V, D, and
A represent right circular, left circular, horizontal, vertical, diagonal, and
anti-diagonal polarization states. Various types of Stokes singularities are
reported in the literature which includes point and line singularities in 2D
transverse plane of the beam, and singularities present in 3D fields which
include Mobius strips, links and knots 19. In this article, we deal with point
Stokes singularities present in 2D transverse plane of the beam.

At $\phi_{12}$ Stokes singular point, polarization azimuth is indeterminate;
hence, these singularities are also known as polarization singularities. The
other two types of Stokes singularities are called Poincaré vortices. Under
paraxial approximations, Stokes singular beams can be mathematically expressed
as the superposition of two beams in orthogonal polarization states such that
at least one of them carries phase singularity and is given by

${\left|{\psi_{l,m}(r,\theta)}\right\rangle}=\cos(\chi+\frac{\pi}{4})\left|\psi_{P}^{l}(r,\theta)\right\rangle+\sin(\chi+\frac{\pi}{4})\\\
\exp(i2\gamma)\left|\psi_{Q}^{m}(r,\theta)\right\rangle$ (2)

where

$\left|\psi_{P}^{l}(r,\theta)\right\rangle=\psi^{l}(r)\exp(il\theta)\hat{P}$
(3)
$\left|\psi_{Q}^{m}(r,\theta)\right\rangle=\psi^{m}(r)\exp(im\theta)\hat{Q}.$
(4)

Here, $2\chi$ and $2\gamma$ decide the weighting factor and phase difference
between the two superposing beams respectively. The variables $l$ and $m$
represent topological charges of the vortex beams used to construct SSBs, and
$\hat{P}$, $\hat{Q}$ represent orthogonal polarization basis states. The
${\left|{\psi_{l,m}(r,\theta)}\right\rangle}$ gives the resultant amplitude
distribution due to the superposition of two beams. Bright Stokes
singularities are formed when one of the superposing beams is a Gaussian beam.
The superposition of phase singular beams in orthogonal polarization states
result in dark Stokes singularities. These singularities can be identified
using a Stokes index $\sigma_{jk}$ which is defined as
$\sigma_{jk}=\frac{1}{2\pi}\oint\nabla\phi_{jk}\cdot dl$ where $\phi_{jk}$
represents the Stokes phase, and $dl$ is the closed path of integration around
the singular point. Stokes index can also be defined as $\sigma_{jk}=m-l$,
where $m$ and $l$ are topological charges of component phase singular beams
respectively.

Stokes singular beams in which the singularities are present at the center of
the beam are represented as points on the hybrid-order Poincaré spheres(HyOPS)
or higher-order Poincaré spheres(HOPS). These spheres are geometrical
constructions where one or both the poles of the spheres represent phase
singular beams in orthogonal polarization states. Stokes singular beams with
the same Stokes index can have different intensity, polarization, and Stokes
phases distributions, which result in degenerate Stokes index states22. Stokes
singular beams generated using the superposition of phase singularities with
topological charges $l$ and $m$ such that $m-l=constant$, and $l\neq 0,m\neq
0$ are polarization degenerate. All the beams represented on a surface of a
particular HyOPS/HOPS are Stokes index degenerate 22. Also, the SSBs on a
particular longitude of a hybrid order Poincaré sphere have similar
polarization distribution but different polarization gradients 26, 22.

The degenerate polarization singular beams are composed of either different
mode indices ($l,m$) or same mode indices but with different relative
weightage of $l$ and $m$ and therefore contain different optical properties.
For example, a partially coherent beam is shown to produce different
depolarization effects despite of having the same Stokes index and
polarization distribution 27. Further, polarization degenerate beams cannot be
differentiated using the Stokes polarimetry method because of same
polarization distribution. Hence, the identification of SSBs based on their
mode indices is important. Figure 1 shows one example where the Stokes
singularities of a given index are composed of different mode indices but are
polarization degenerate. The component phases and the corresponding
polarization states (marked on the phase maps) from which the SSBs are
composed of are shown in the left side of the figure. The corresponding
resultant intensity distributions, degenerate polarization distribution and
the respective Stokes parameter distributions ($S_{1},S_{2},S_{3}$) are shown
in the middle of the figure. Due to the same polarization distributions, both
the beams have same Stokes parameter distributions. On the right side of the
figure, the respective diffraction intensity distributions produced by two
distinct SSBs are shown to distinguish the degenerate polarization states.

Figure 2: Experimental set up for generation and detection of Stokes singular
beams. SF: Spatial filter assembly, L1, L2: Lenses, Iris: amplitude aperture
to control the size of the beam, P: Polarizer, H(Q)WP: half(quarter) wave
plate, (P)BS: (polarizing) beam splitter, M1, M2: Mirrors, SPP: spiral phase
plate, Ap: Triangular aperture, CP: Circular polarizer, SC: Stokes camera, PC:
Personal computer. Figure 3: Simulation and experimental results of total
diffraction intensity patterns (T) and corresponding polarization projections
(R, L, H, V, D, A) for various Stokes singular beams. For each mode indices
$(l=l_{1},m=l_{2})$, superposition in three different polarization basis (R,L;
H,V; D,A) is considered. Figure 4: (a) Comparison of different training
strategies: We compare the data distribution of the 3 different training
strategies. The scatter plot shown here is for illustration of data
distribution. (b) Data partition for training, validation and testing of the
neural network under mix training strategy

###  2.2 Detection of Stokes singularities

The diffraction of Stokes singular beams through an equilateral triangular
aperture lifts the degeneracy associated with mode indices (refer to Fig. 1).
However, the SSBs generated using same combination of mode indices
irrespective of their orthogonal polarization states results in diffraction
degenerate states. Thus, to completely lift the degeneracy associated with
SSBs, polarization transformation after diffraction is obligatory.

The transmittance function of an equilateral triangular aperture with side
length $a$ can be mathematically written as 22:

$\begin{split}T(x,y)&=1\quad\text{for}\quad\frac{-a}{2}\leq
x\leq\frac{a}{2}\quad\text{and}\quad 0\leq y\leq-\sqrt{3}(|x|-\frac{a}{2})\\\
&=0\quad\text{otherwise}\end{split}$ (5)

The field $\left|\psi_{l,m}(r,\theta)\right\rangle$ in cartesian coordinate
system is written as $\left|\psi_{l,m}(x,y)\right\rangle$ where
$x=r\cos\theta$ and $y=r\sin\theta$. The field just after the aperture with
aperture function $T(x,y)$ is given by:

${\left|{E_{T}}\right\rangle}=T(x,y){\left|{\psi_{l,m}(x,y)}\right\rangle}$
(6)

The far-field diffraction pattern is produced at the back focal plane of a
lens with focal length $f$ and is given by:

${\left|{\mathcal{F}}(u,v)\right\rangle}=\frac{B}{i\lambda
f}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}T(x,y){\left|{\psi_{l,m}}\right\rangle}e^{-i\frac{2\pi}{\lambda
f}(ux+vy)}dxdy$ (7)

where $(u,v)$ denotes the coordinates of the Fourier plane, $B$ is a constant
and $\lambda$ defines the wavelength of light. The diffraction of the
individual superposing beams is calculated independently and added together to
obtain the total diffraction pattern of a Stokes singular beam. Notably, the
diffraction of these beams can be carried out using other apertures too, that
will result in different diffraction intensity patterns depending on their
shape and symmetry. However, the aperture should be chosen such that it can
differentiate between positive and negative phase singularities with same
magnitude of topological charge.

Diffraction is a wave phenomenon that can be used to identify the OAM of
light. Since SAM and OAM are coupled in SSBs, diffraction and Stokes
polarimetry techniques alone cannot distinguish between distinct SSBs.
Therefore, here we report the detection of these beams by combining
diffraction based Stokes polarimetry method with the deep learning techniques.
The ability of deep learning-based models to recognize intricate features in
intensity images makes it an excellent tool to classify Stokes singular beams.
Respective total diffraction intensity distributions and all the six
polarization projections (R, L, H, V, D and A) are utilized as a set to serve
as input channels for training the deep neural network (DNN). Of the
infinitely many possible classes of SSBs, we selected a total of 15 classes of
SSBs. These classes include: $\phi_{jk}^{01}$, $\phi_{jk}^{02}$,
$\phi_{jk}^{10}$, $\phi_{jk}^{12}$, $\phi_{jk}^{23}$ where ${j,k}$ run
cyclically from $1-3$ and $j\neq k$. Here, in $\phi_{jk}^{lm}$, indices $j$
and $k$ decide the orthogonal polarization states, and, $l$ and $m$ represent
the topological charges of the component vortex states of the Stokes singular
beam where the Stokes singularity is present at the center. In each of the
superpositions, other types of Stokes vortices are also present at different
locations 6. The corresponding mode indices for these classes are given by
(0,1), (0,2), (1,0), (1,2), and (2,3). Among each mode indices, three pairs of
polarization basis (R, L; H, V; D, A) are considered, contributing to 15
classes.

##  3 Experimental setup

The experimental setup used to generate and detect SSBs is depicted in Fig. 2.
The light from the He-Ne laser (632nm) is spatially filtered and collimated
using lens L1. A $45^{\circ}$ polarized beam is generated using a combination
of polarizer (P) and half wave plate (HWP). The beam is launched into a
modified Mach Zehnder type interferometer where two orthogonally linearly
polarized beams in two arms of the interferometer acquire different
topological charges using spiral phase plate (SPP). For generating bright
Stokes singularity, one SPP was removed from one of the arms of the
interferometer. The two beams combine after the beam splitter (BS). The
quarter wave plate (QWP) after BS is used to change linear basis superposition
to circular basis superposition and the HWP after the BS is used to change
(H,V) linear basis to (D,A) linear basis superposition. Further, the beam is
allowed to diffract through an equilateral triangular aperture, and the far-
field diffraction intensity pattern is recorded using the Stokes camera. The
combination of polarizer and QWP after the triangular aperture is used to
extract the component intensities of the beam which includes horizontal (H),
vertical (V), diagonal (D), anti-diagonal (A), right circular (R) and left
circular (L) polarization intensities. The total and polarization components
intensities are recorded at three different z-planes for a particular Stokes
singular beam. Based on the generation of a particular Stokes index beam, the
charge of the spiral phase plate in the two arms of the interferometer is
varied.

##  4 Results and Discussion

###  4.1 Total diffraction pattern and corresponding polarization projections
of Stokes singular beams

Simulation and experimental results which include total (T) and component
diffracted intensities (R, L, H, V, D, A) for different SSBs are depicted in
Fig. 3. These component intensities are projections of the resultant beam onto
various polarization states. The topological charges of the superposing phase
singular beams that form the Stokes singularity are mentioned in the left side
of the figure in each case. Each set contains three rows for superposition in
three different polarization basis namely circular (R, L), linear (H, V), and
diagonal bases (D, A) from top to bottom, respectively. All the polarization
component intensities (R, L, H, V, D, and A) are extracted for each basis of
superposition (R,L; H,V; and D,A). Stokes singular beams composed of different
mode indices produce different diffraction patterns. However, SSBs composed of
the same mode indices results in the same resultant diffraction pattern
irrespective of the polarization associated with component beams, and hence
are diffraction degenerate (refer to Fig. 3). To lift the diffraction
degeneracy, polarization transformation is utilized.

In the next section, we describe the deep learning procedure that is used for
the recognition of SSBs. It includes dataset preparation, selection and
optimization of deep neural networks followed by the classification results.

###  4.2 Deep Learning Models and Strategy

#### Simulation and Experimental datasets:

The datasets used in the present study consist of both simulated and
experimentally acquired images. Possible experimental fluctuations in
amplitude, phase, polarization, beam shift, and aperture shape are taken into
account while creating simulation data for the identification of SSBs. These
fluctuations may affect the accuracy of the detection techniques. Therefore,
it is imperative to include these parameters while simulating the datasets for
different classes. Moreover, this also helps in the robust training of deep
learning networks. In addition, since deep learning models require a large
amount of data for training 28, robust simulation aids in the quick
convergence of the network. Heuristically, we simulate 1000 images for each
class, resulting in 15000 total images. The experimental data contained a
total of 300 images i.e., 20 images for each class. The experimental datasets
were acquired under the specifications mentioned in section 3.

#### Comparison Models

We considered five different convolutional neural networks (CNN) based deep
learning models, namely, SqueezeNet 29, VGG 30, AlexNet 31, DenseNet 32 and
ResNet-18 33 to apply to our datasets. All of these models are pre-trained on
ImageNet data. More information about these models is provided in
Supplementary S1. The cross-entropy loss is employed and a stochastic gradient
descent optimizer is used for back-propagation. Each of the $707\times 101$
image as shown in Fig. 3 is transformed into a square image of dimension
303$\times$303 by stacking two voids of dimension 101$\times$101 and served as
an input image to the neural network. Examples of resultant input image are
depicted in Figure 4. This peculiar dimension is chosen in order to preserve
the information by symmetry, since the deep learning models resize the input
images to 224$\times$224.

Figure 5: Loss curve and confusion matrices of ResNet-18 and DenseNet. The
loss curve is shown after each epoch on the training and validation dataset.
The confusion matrix shows the classification accuracy of the network on
experimental and simulated testing datasets. Class representation: 0:
$\phi_{12}^{01}$, 1: $\phi_{12}^{02}$, 2: $\phi_{12}^{10}$, 3:
$\phi_{12}^{12}$, 4: $\phi_{12}^{23}$, 5: $\phi_{23}^{01}$, 6:
$\phi_{23}^{02}$, 7: $\phi_{23}^{10}$, 8: $\phi_{23}^{12}$, 9:
$\phi_{23}^{23}$, 10: $\phi_{31}^{01}$, 11: $\phi_{31}^{02}$, 12:
$\phi_{31}^{10}$, 13: $\phi_{31}^{12}$, 14: $\phi_{31}^{23}$. Check the
labelling of classes here and in text.

#### Training strategy

We employed 3 different strategies to train and test our model as shown in
Fig. 4. Figure 4 depicts the different strategies used for training and
testing to achieve the best classification accuracy. A scatter plot is shown
in Fig. 4 to show the data distribution in the feature space. Next, we explain
all three training and testing strategies:

(1) Training with only simulation data: We trained and validated all five
deep-learning models using simulated data and tested on the experimental data.
However, this strategy provides very poor classification performance i.e.,
43.6%. Note that this is the best achievable performance among five deep
learning models and is obtained through Alexnet. This is attributed to the
fact that experimental data have more variability between the images in each
class which is illustrated in the scatter plot in Fig. 4. In addition, slight
mismatch between the simulated and experimentally acquired datasets also
affects the classification accuracy. This strategy is further discussed in
Supplementary S2.

(2) 10-fold strategy on experimental data: We employed a 10-fold strategy 34
to train a ResNet-18 model on experimental data as shown in Fig. 4. ResNet-18
model was first pre-trained on the simulated data followed by fine-tuning
using experimental data by a 10-fold strategy (strategy 2.1 in Fig. 4). This
strategy works well and provides an accuracy of 93%. On the other hand, fine-
tuning of the ResNet-18 model (pre-trained on ImageNet) on experimental data
with a 10-fold strategy is also performed (strategy 2.2 in Fig. 4). Fine-
tuning with only experimental datasets provides an accuracy of 92%. Strategy 2
does not exploit the benefit of simulated data as strategy 2.1 (shown in Fig.
3) improves the final performance by only 1%. We discuss strategy 2 in more
detail in Supplementary 3.

(3) Mix Training: Finally, we train the deep-learning model on mix simulated
and experimental data 35 for generalized and robust training. Mix training
provides the best accuracy of 98.67% using the ResNet-18 model. The mix
strategy is discussed in detail in the next paragraph.

#### Mix Training:

For robust training of the neural networks, we performed mixed training with
simulated and experimental datasets. The dataset is divided into three parts:
training, validation and the testing set. The testing set is further
categorized into two subparts: simulated testing and experimental testing. A
total of five different deep learning models are trained separately and
optimized for a fixed batch size of 8. The models are trained on the training
set, the best hyperparameters are chosen on the validation set and the
performance of each network is shown on the testing set in Table 1. The
division of the dataset is shown in Figure 4(b). A total of 35% (105 images),
15% (45 images) and 50% (150 images) of the experimental images are used to
training, validation and testing respectively. From simulated data, 89.7%
(13455 images), 0.3% (45 images) and 10% (1500 images) images are used for
training, validation and testing purpose respectively. We chose only 0.3% of
simulated images for validation to ensure that both experimental and simulated
datasets contained an equal number of images in the validation set. Moreover,
the ratio of simulated and experimental images in the training datasets is
optimized since strategy 1 confirms that training model with only simulated
images offer poor accuracy on experimentally acquired images. To balance the
training datasets, all experimental images of the training set (105 images)
are copied 18 times resulting 1890 experimental images. This results in 7:1
ratio of simulated and experimental images in the training set. This strategy
ensures the availability of one experimental image in each batch of 8 images
during training of the network. All the models are trained for 50 epochs. The
learning rate is optimized for all models and found to be $0.001$ on the
validation set. The accuracy is calculated as the percentage of instances
rightly classified. The accuracy of each model in all the parts of the dataset
is given in Table 1.

###  4.3 Labelling hybridly polarized beams and assessing the winning models

Out of all models used in the present study, ResNet-18 and DenseNet offer the
best training accuracy of 97.81$\%$ and 98.36$\%$ respectively (see Table 1).
DenseNet performs better in the validation set followed by ResNet-18. DenseNet
offers 100$\%$ of validation accuracy while ResNet-18 provides a validation
accuracy of 97.78$\%$. In the simulation testing dataset, ResNet-18 and
DenseNet give classification accuracy of 98.33$\%$ and 97.67$\%$ respectively.
Other architecture i.e., SqueezeNet, VGG and AlexNet offer accuracy of 85.80
$\%$, 95.73$\%$ and 95.00$\%$ respectively. Interestingly, ResNet-18
outperforms all the models in the experimental testing dataset. The
classification accuracy on the experimental images using ResNet-18 is found to
be 98.67$\%$.

Further to quantify the best-performed model i.e., ResNet-18 and DenseNet, we
show the loss curve in Fig. 5. Both training and validation loss is calculated
and shown after each epoch here. It is evident from the training loss curve
that both models are converging after 50 epochs. Further, confusion matrices
of ResNet-18 and DenseNet shown in Fig. 5 demonstrate the visual clues of the
network prediction. Confusion matrices are shown for both the simulation and
experimental testing datasets. Diagonal elements of the confusion matrix show
correct prediction while off-diagonal elements are the ones wrongly classified
by the network. For example, only two images are wrongly classified by
ResNet-18 in the experimental testing datasets i.e., one instance of the class
$\phi_{12}^{01}$ which is represented as class 0 and class $\phi_{12}^{12}$
which is shown as class 4 in Fig. 5. Wrongly classified image of
$\phi_{12}^{01}$ and $\phi_{12}^{12}$ are classified as $\phi_{23}^{12}$
(represented as class 8) by ResNet-18. On the contrary, DenseNet wrongly
classifies 6 images, and hence performs slightly poorer than ResNet-18.
Nonetheless, the experimental classification accuracy of ResNet-18 and
DenseNet are 98.33$\%$ and 97.67$\%$ respectively manifesting the benefits of
using a deep learning approach to quantify SSBs in their relevant classes. The
efficacy of ResNet-18 can be attributed to their residual connections which
alleviate the degrading performance due to the vanishing gradient problem in
deep neural networks.

A deep learning model requires a substantial amount of data for training
purposes. While acquiring a large amount of experimental data is challenging
as it requires considerable time and effort, it is imperative to simulate
images that exhibit properties similar to experimental images. However, it is
almost impossible to include all experimental variations in the simulated
images and therefore, our proposed mix training strategy of a combination of
simulation and a few experimental images offers a unique solution to overcome
the aforementioned issue and successfully leverage the performance of CNNs for
the classification of Stokes singular beams. Moreover, the superior
performance by ResNet-18 asserts that a simpler network with a skip connection
performs better in the case of Stokes singular beams.

Taken together, our work presents important steps towards understanding Stokes
singular beams by a unique experimental approach and should help to
democratize their applications in super-resolution imaging, optical
communication, beam shaping and potentially new label-free imaging technique
beyond what can be achieved in the present label-free optical microscopy
community. Our ability to identify the specific class of hybridly polarized
beam can be further upscaled to real-time identification of vortex beams which
can act as a bridge between beam shaping and microscopy community to
understand and choose the appropriate beams for their precise applications.

Table 1: Model comparison based on the accuracy is given here. The accuracy is rounded to 2 digits. The best model and performance is highlighted in bold and the second best in italics. Sim Testing and Exp Testing represent simulation testing and experimental testing datasets respectively. Models | Accuracy in (%)
---|---
Training | Validation | Sim Testing | Exp Testing
SqueezeNet 29 | 90.72 | 91.11 | 85.80 | 88.00
VGG 30 | 95.85 | 97.78 | 95.73 | 93.33
AlexNet 31 | 94.71 | 92.22 | 95.00 | 94.00
DenseNet 32 | 98.36 | 100.00 | 98.33 | 96.00
ResNet-18 33 | 97.81 | 97.78 | 97.67 | 98.67

##  5 Conclusion

We demonstrated a generalized experimental and computational framework to
study the taxonomy of hybridly polarized beams. The diffraction pattern of
Stokes singular beams through an equilateral triangular aperture and their
polarization projections are utilized for training a deep neural network.
After appropriate training, deep neural networks offer excellent accuracy to
classify and detect the structured beams carrying Stokes singularities. The
scheme discussed in the article can lift all the degeneracy associated with
SSBs. Further, the detection scheme is intensity-based, and mixed simulation
and experimental images are exploited to train the neural network, which
reduces the requirement of acquiring thousands of experimental datasets. We
have achieved 97.67$\%$ and 98.33$\%$ accuracy in unknown simulated and
experimental testing datasets, respectively. In addition to the
characterization of Stokes singular beams, the present approach can also be
applied to other applications such as optical microscopy and optical
communication. It can be used to design different label-free techniques with
improved resolution.

## Acknowledgement

KA and AB acknowledges the European Research Council Starting grant (id
804233).

## Disclosures

Authors declare no competing interest.

## Author contribution

AB, PS, KA conceived the idea and supervised the work. GA, RR, PS designed
simulation, experimental system and planned the experiments. AB, RA, KA
provide inputs with simulation study. RA, AB, DKP optimized the computational
part and classify the datasets. GA, AB, RR, RA analysed the result and
prepared the figure. GA mainly wrote the manuscript and all author contributed
revised the manuscript.

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