11institutetext:  School of Physics and Astronomy, University of Glasgow,
Glasgow G12 8QQ, UK
E-mail<EMAIL_ADDRESS>(corresponding author)

# On lines of constant polarisation in structured light beams

S.M. Barnett 11 F.C. Speirits 11** J.B. Götte 1111**

###### Abstract

We show that Skyrmion field lines, constructed from the local Stokes
parameters, trace out lines of constant optical polarisation.

Structured light beams are characterised by an engineered spatial variation of
amplitude, phase and polarisation [1, 2, 3, 4]. Important examples of these
include beams carrying orbital angular momentum [5, 6, 7, 8, 9], helicity
lattices [10, 11, 12, 13, 14, 15], and the vector vortex beams [16, 17, 18,
19]. Some of these beams, in particular those with spatially varying
polarisation, have been shown to exhibit Skyrmionic structure [20, 21].
Typically, these have a polarisation pattern in the transverse plane that, at
its centre, has one polarisation but at the outer reaches of the plane has the
orthogonal polarisation. In general, all possible polarisations appear at some
point in this transverse plane in a winding pattern, and Skyrmions are
characterised by a corresponding winding number, the Skyrmion number [20, 21,
22, 23, 24, 25, 26]. There exist numerous variations on this theme [22, 23,
24, 25, 26]. What has not yet been identified, however, is the physical
significance of the Skyrmion field itself: we rectify that in this letter.

We present an unexpected property of Skyrmion field lines that has application
whether or not a structured light beam has a non-zero Skyrmion number, as long
as the polarisation pattern covers the whole Poincaré sphere continuously
[27]. Put simply, it is that Skyrmion field lines trace out contours of
constant polarisation. Moreover, all such lines of constant polarisation are
Skyrmion field lines. Several important properties of structured light beams
then follow from the mathematical properties of the Skyrmion field. The
central theme of our paper is the application of these ideas to paraxial light
beams, but we conclude with a brief discussion of these ideas in other fields
of physics, including electron [28, 29, 30] and neutron [31] optics and also
gravitational waves [32].

Skyrmion field lines for paraxial light beams are defined in terms of the
normalised Stokes parameters, $S_{1}$, $S_{2}$ and $S_{3}$ [33]. The $i^{th}$
component of the Skyrmion field is [20, 21]

$\Sigma_{i}=\frac{1}{2}\varepsilon_{ijk}\varepsilon_{pqr}S_{p}\frac{\partial
S_{q}}{\partial x_{j}}\frac{\partial S_{r}}{\partial x_{k}}\,,$ (1)

where $\varepsilon_{ijk}$ is the alternating or Levi-Civita symbol and we
employ the summation convention in which a summation is implied over repeated
indices. The specific form of $\Sigma_{i}$ is crucial to an appreciation of
the link with lines of constant polarisation. For this reason it is worth
writing explicitly one of the Cartesian components of $\bm{\Sigma}$

$\displaystyle\Sigma_{z}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varepsilon_{pqr}S_{p}\left(\frac{\partial
S_{q}}{\partial x}\frac{\partial S_{r}}{\partial y}-\frac{\partial
S_{r}}{\partial x}\frac{\partial S_{q}}{\partial y}\right)$ (2)
$\displaystyle=$ $\displaystyle S_{1}\left(\frac{\partial S_{2}}{\partial
x}\frac{\partial S_{3}}{\partial y}-\frac{\partial S_{3}}{\partial
x}\frac{\partial S_{2}}{\partial y}\right)$
$\displaystyle+S_{2}\left(\frac{\partial S_{3}}{\partial x}\frac{\partial
S_{1}}{\partial y}-\frac{\partial S_{1}}{\partial x}\frac{\partial
S_{3}}{\partial y}\right)$ $\displaystyle+S_{3}\left(\frac{\partial
S_{1}}{\partial x}\frac{\partial S_{2}}{\partial y}-\frac{\partial
S_{2}}{\partial x}\frac{\partial S_{1}}{\partial y}\right)\,.$

Note that this $z$-component depends on the variation of the Stokes
parameters, and therefore of the polarisations, only in the $x$ \- and $y$ \-
directions. Each term, moreover, depends on all three Stokes parameters.

The Skyrmion number associated with our structured beam is readily obtained by
integration over the plane transverse to the direction of propagation. If we
take this direction to define our $z$ \- axis, then the Skyrmion number is

$n=\frac{1}{4\pi}\int\Sigma_{z}\,dxdy\,,$ (3)

where the integral runs over the whole transverse plane. This value is
typically an integer, although structures with non-integer Skyrmion number can
be constructed [20]. Our present concerns do not involve Skyrmions or the
Skyrmion number explicitly, but rather focus on the Skyrmion field. This
exists wherever there is a continuously spatially varying polarisation apart
from a few special cases, such as where there is a polarisation variation only
in one direction. We note that the Skyrmion field is a transverse field in
that

$\bm{\nabla}\cdot\bm{\Sigma}=0\,,$ (4)

and therefore the integral over any closed surface is zero
$\oint\bm{\Sigma}\cdot d\,\mathbf{S}=0$. The only exception to this condition
will occur if there are lines along which the polarisation is undefined.

Figure 1:  Plot of a line of constant elliptical polarisation and the local
coordinate system $\mathbf{u},\mathbf{v},\mathbf{w}$ at $p$.

Let us turn to the properties of lines of constant polarisation. As is well
known, structured light beams are threaded by lines of constant polarisation.
The most studied example is the C-lines along which the polarisation is purely
left- or right-handed circularly polarised [1, 34]. There is nothing in this
context, however, that is specific to circular polarisation, and we can trace
such contours of constant polarisation for any chosen polarisation. Consider a
point $p$ in a structured paraxial light beam, as depicted in Fig. 1. From
this point there extends a line (in two directions) along which the
polarisation is the same as at $p$. Note that the amplitude and phase will
not, in general, remain the same along this line. Let us introduce a local
right-handed Cartesian coordinate system $(\mathbf{u,v,w})$ at $p$, in which
the line of constant polarisation extends in the direction $\mathbf{u}$. As
the polarisation in the direction $\mathbf{u}$ (and $-\mathbf{u}$) is
unchanged, it follows that the direction of the Stokes vector $\mathbf{S}$ is
also unchanged:

$(\mathbf{u}\cdot\bm{\nabla})\mathbf{S}=\frac{\partial}{\partial
u}\mathbf{S}=0\,,$ (5)

where ${\bf u}$ is a unit vector in the direction of the coordinate $u$. We
can write the components of the Skyrmion field at $p$ in the $u,v,w$ basis and
find

$\displaystyle\Sigma_{u}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varepsilon_{pqr}S_{p}\left(\frac{\partial
S_{q}}{\partial v}\frac{\partial S_{r}}{\partial w}-\frac{\partial
S_{r}}{\partial v}\frac{\partial S_{q}}{\partial w}\right)$
$\displaystyle\Sigma_{v}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varepsilon_{pqr}S_{p}\left(\frac{\partial
S_{q}}{\partial w}\frac{\partial S_{r}}{\partial u}-\frac{\partial
S_{r}}{\partial w}\frac{\partial S_{q}}{\partial u}\right)$ (6)
$\displaystyle\Sigma_{w}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varepsilon_{pqr}S_{p}\left(\frac{\partial
S_{q}}{\partial u}\frac{\partial S_{r}}{\partial v}-\frac{\partial
S_{r}}{\partial u}\frac{\partial S_{q}}{\partial v}\right)\,.$

The derivatives of the Stokes parameters with respect to $u$ are zero and it
follows that $\Sigma_{v}=0=\Sigma_{w}$, and therefore that the Skyrmion field
line points in the direction of constant polarisation. This is our principal
result.

It is straightforward to confirm that the Skyrmion field is independent of the
basis used to denote the Stokes vectors and hence the identification of the
Skyrmion field lines with lines of constant polarisation holds for every
possible polarisation. Such a global transformation changes the polarisation
at every point in the field but does not alter the Skyrmion field, which is
associated with lines of constant polarisation, but not the specific
polarisation along these lines. Identifying a Skyrmion field line does not
determine the polarisation along the field line, merely the line itself. More
formally, the Skyrmion field is invariant under any unitary transformation of
the Poincaré sphere and so is not dependent on the basis used to express
$S_{1}$, $S_{2}$ and $S_{3}$. In this way, the Skyrmion field extends the
characterization of lines of constant circular or linear polarisation [35] to
every polarisation.

The mathematical properties of the Skyrmion field allow us to make general
statements about lines of constant polarisation. The simplest and most
important among these follows from the transverse nature of the Skyrmion
field, $\bm{\nabla}\cdot\bm{\Sigma}=0$. Like other transverse fields, such as
the magnetic induction $\mathbf{B}$ in electromagnetism, Skyrmion field lines
cannot start or end (no monopoles) and nor can they branch or coalesce. The
identification of Skyrmion field lines with lines of constant polarisation
means that the same properties must hold for lines of constant polarisation.
The only exception to this rule occurs along lines at which the polarisation
is undefined, where several lines of different polarisation can meet. At such
lines, however, the transverse condition on $\Sigma$ will fail.

One remaining subtlety needs to be addressed. This is the fact that lines of
constant polarisation do not have a preferred sense of direction: such a line
is independent of whichever direction we choose to move along it. The Skyrmion
field line, however, has a specific direction; if we change the sign of
$\Sigma$, then it reverses its direction along the line of constant
polarisation. In this sense, at least, there seems to be a significant
difference between lines of constant polarisation and Skyrmion field lines and
we should explain the origin of this difference.

We have seen that the Skyrmion field lines do not determine the local
polarisation, merely the local direction along which the polarisation does not
vary. For the structured light beam, however, there is a further class of
symmetries that leaves the pattern of lines of constant polarisation
unchanged. This is to apply the operation of complex conjugation to the
polarisations. To be specific, let $\mathbf{e}_{\text{H}}$ and
$\mathbf{e}_{\text{V}}$ be the real unit vectors corresponding to horizontal
and vertical polarisation, so that left- and right-circular polarisations are
$\mathbf{e}_{\text{L}}=(\mathbf{e}_{\text{H}}+i\mathbf{e}_{\text{V}})/\sqrt{2}$
and
$\mathbf{e}_{\text{R}}=(\mathbf{e}_{\text{H}}-i\mathbf{e}_{\text{V}})/\sqrt{2}$.
If we apply the complex conjugation operation then the left- and right-
circular polarisations switch, but the horizontal and vertical are unchanged,
as are all the other possible linear polarisations, and we arrive at an
alternative (but physically allowed) polarisation pattern with the same lines
of constant polarisation. This transformation is antiunitary in nature [36].
Such transformations are familiar from the study of time-reversal and CP
symmetries in particle physics [37]. The complex conjugate transformation
coupled with rotations provides a second set of symmetries under which the
polarisation changes but the lines of constant polarisation do not. The
Skyrmion field lines, however, switch direction under the antiunitary
transformation as their value is based on a right-handed coordinate system for
the Poincaré sphere. The complex conjugation operation applied to
polarisation, however, changes a right-handed arrangement of the Stokes
parameters into a left-handed one and, in doing so, flips the sign of the
Skyrmion field.

It is interesting to ask if there is always a Skyrmion field in a structured
beam. The answer follows from the discussion in the preceding paragraph; if,
for example, the field is unchanged by performing complex conjugation on the
polarisation then the Skyrmion field is everywhere equal to its negative and
therefore is equal to zero. Simple examples are the radially and azimuthally
polarised beams [16] for which the polarisation is everywhere linear. For such
beams we have _planes_ of constant polarisation rather than lines.

The connection between the Skyrmion field and lines of constant polarisation
has been established here for paraxial structured light beams, but we may
expect it to have wider applications. For electrons and neutrons, a similar
association will hold for the Skyrmion field lines and lines along which the
particle spin does not change. For non-paraxial optical fields there exists a
variety of features that can be associated with Skyrmions and, by extension,
with a Skyrmion field. It will be interesting to see how these are related to
the spatial arrangement of spin-related properties of the electromagnetic
field. Finally, gravitational waves have two orthogonal polarisations and so
we would expect Skyrmion field lines to be associated, also, with spatial
variations of the polarisation of these fields.

In summary, we have shown that lines of constant polarisation in any
structured paraxial light beam are identified with Skyrmion field lines. It
follows that there is a more intimate relationship between structured light
beams and Skyrmion fields than simply whether or not a particular beam has an
associated Skyrmion number or Skyrmionic structures.

###### Acknowledgements.

This work was supported by a Royal Society Research Professorship, grant
number RP150122, and the UK Engineering and Physical Sciences Research
Council, grant numbers EP/R008264/1 and EP/V048449/1.
_Data availability statement_ : No new data were created or analysed in this
study.

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The function $\left\|x\right\|^{\alpha}$ is $\frac{\alpha-1}{n}$-locally
smooth for $1\leq n-1<\alpha-2\leq n$.

###### Proof

Without loss of generality, assume $\left\|y\right\|\leq\left\|x\right\|$
which implies
$\left\|y\right\|-\left\|x\right\|\leq\left\|x-y\right\|\leq\left\|x\right\|+\left\|y\right\|\leq
2\left\|x\right\|$. Therefore,

$\displaystyle{\displaystyle\left\|\nabla^{2}f(x)-\nabla^{2}f(y)\right\|_{\mathrm{op}}}$
$\displaystyle\leq\left(\alpha-1\right)\left|\left\|y\right\|^{\alpha-2}-\left\|x\right\|^{\alpha-2}\right|+4\left(\alpha-2\right)\left\|x\right\|^{\alpha-3}\left\|x-y\right\|+2\left(\alpha-2\right)\left\|x\right\|^{\alpha-4}\left\|x-y\right\|^{2}$
$\displaystyle\leq\left(\alpha-1\right)\left\|x-y\right\|^{\frac{\alpha-2}{n}}\left(\left\|x\right\|^{\frac{\left(n-1\right)\left(\alpha-2\right)}{n}}+\ldots+\left\|y\right\|^{\frac{\left(n-1\right)\left(\alpha-2\right)}{n}}\right)$
$\displaystyle+4\left(\alpha-2\right)\left\|x\right\|^{\alpha-3}\left\|x-y\right\|+2\left(\alpha-2\right)\left\|x\right\|^{\alpha-4}\left\|x-y\right\|^{2}$
$\displaystyle\leq\left(n\left(\alpha-1\right)+6\left(\alpha-2\right)\right)\left\|x-y\right\|^{\frac{\alpha-2}{n}}\left(1+\left\|x\right\|^{\frac{\left(n-1\right)\left(\alpha-2\right)}{n}}+\left\|y\right\|^{\frac{\left(n-1\right)\left(\alpha-2\right)}{n}}\right).$

This is the desired result.

###### Lemma 24

Suppose $\pi=e^{-U}$ satisfies $\alpha$-mixture weakly smooth. Let
$p_{\mu,0}=N(0,\frac{1}{L}I)$. Then $H(p_{\mu,0}|\pi_{\mu})\leq
U(0)+\frac{NL\mu^{1+\alpha}}{(1+\alpha)}d^{\frac{2}{2\wedge
p}}-\frac{d}{2}\log\frac{2\Pi e}{L}+\frac{Nd}{1+\alpha}=O(d).$

###### Proof

Since $U$ is mixture weakly smooth, for all $x\in\mathbb{R}^{d}$ we have

$\displaystyle U_{\mu}(x)$ $\displaystyle\leq U(0)+\langle\nabla
U(0),x\rangle+\frac{L}{1+\alpha}\sum_{i}\left\|x\right\|^{1+\alpha_{i}}+\frac{NL\mu^{1+\alpha}}{(1+\alpha)}d^{\frac{2}{2\wedge
p}}$ $\displaystyle\leq
U(0)+\frac{L}{1+\alpha}\sum_{i}\left\|x\right\|^{1+\alpha_{i}}+\frac{NL\mu^{1+\alpha}}{(1+\alpha)}d^{\frac{2}{2\wedge
p}}.$

Let $X\sim\rho=N(0,\frac{1}{L}I)$. Then

$\displaystyle\mathbb{E}_{\rho}[U(X)]$ $\displaystyle\leq
U(0)+\frac{NL\mu^{1+\alpha}}{(1+\alpha)}d^{\frac{2}{2\wedge
p}}+\frac{L}{1+\alpha}\sum_{i}\mathbb{E}_{\rho}\left(\left\|x\right\|^{1+\alpha_{i}}\right)$
$\displaystyle\leq U(0)+\frac{NL\mu^{1+\alpha}}{(1+\alpha)}d^{\frac{2}{2\wedge
p}}+\frac{L}{1+\alpha}\sum_{i}\mathbb{E}_{\rho}\left(\left\|x\right\|^{2}\right)^{\frac{1+\alpha_{i}}{2}}$
$\displaystyle\leq U(0)+\frac{NL\mu^{1+\alpha}}{(1+\alpha)}d^{\frac{2}{2\wedge
p}}+\frac{L}{1+\alpha}\sum_{i}\left(\frac{d}{L}\right)^{\frac{1+\alpha_{i}}{2}}$
$\displaystyle\leq U(0)+\frac{NL\mu^{1+\alpha}}{(1+\alpha)}d^{\frac{2}{2\wedge
p}}+\frac{Nd}{1+\alpha}.$

Recall the entropy of $\rho$ is
$H(\rho)=-\mathbb{E}_{\rho}[\log\rho(X)]=\frac{d}{2}\log\frac{2\Pi e}{L}$.
Therefore, the KL divergence is

$\displaystyle\mathbb{E}(\rho|\pi)$
$\displaystyle=\int\rho\left(\log\rho+U\right)dx$
$\displaystyle=-H(\rho)+\mathbb{E}_{\rho}[U]$ $\displaystyle\leq
U(0)+\frac{NL\mu^{1+\alpha}}{(1+\alpha)}d^{\frac{2}{2\wedge
p}}-\frac{d}{2}\log\frac{2\Pi e}{L}+\frac{Nd}{1+\alpha}$ $\displaystyle=O(d).$

This is the desired result.

###  F.2 Proof of Lemma 25

###### Lemma 25

Assume $\pi_{\mu}=e^{-U_{\mu}(x)}$ then

$\mathbb{E}_{\pi_{\mu}}\left[\left\|\nabla
U(x)\right\|^{2}\right]\leq\frac{2NL}{\mu^{1-\alpha}}d^{\frac{2}{p}}d^{\frac{2}{p}}+2\left(\frac{NL\mu^{1+\alpha}}{(1+\alpha)}d^{\frac{2}{2\wedge
p}}\right)^{2},$

for $d$ sufficiently large.

###### Proof

Since $\pi_{\mu}$ is stationary distribution, we have

$\displaystyle\mathbb{E}_{\pi_{\mu}}\left[\left\|\nabla
U_{\mu}(x)\right\|^{2}\right]$
$\displaystyle=\mathbb{E}_{\pi_{\mu}}\left(\triangle
U_{\mu}\left(x\right)\right)$
$\displaystyle\stackrel{{\scriptstyle}}{{\leq}}\frac{NL}{\mu^{1-\alpha}}d^{\frac{2}{p}},$

where the last step comes from Lemma 10that $\nabla U_{\mu}\left(x\right)$ is
$\frac{NL}{\mu^{1-\alpha}}d^{\frac{2}{p}}$-Lipschitz,
$\nabla^{2}U_{\mu}\left(x\right)\preceq\frac{NL}{\mu^{1-\alpha}}d^{\frac{2}{p}}I$.
In addition,

$\displaystyle\mathbb{E}_{\pi_{\mu}}\left[\left\|\nabla
U(x)\right\|^{2}\right]$ $\displaystyle\leq
2\mathbb{E}_{\pi_{\mu}}\left[\left\|\nabla
U_{\mu}(x)\right\|^{2}+\left\|\nabla U_{\mu}(x)-\nabla
U(x)\right\|^{2}\right]$
$\displaystyle\leq\frac{2NL}{\mu^{1-\alpha}}d^{\frac{2}{p}}d^{\frac{2}{p}}+2\left(\frac{NL\mu^{1+\alpha}}{(1+\alpha)}d^{\frac{2}{2\wedge
p}}\right)^{2},$

where the last step follows from Lemma 10. This gives the desired result.

###  F.3 Proof of lemma 35

###### Lemma 26

If $U$ satisfies Assumptions 1and 3, then

$U(x)\geq\frac{a}{2\beta}\|x\|^{\beta}+U(0)-\frac{L}{\alpha+1}\sum_{i}R^{\alpha_{i}+1}-\frac{b}{\beta}.$
(35)

###  F.4 Proof of Lemma 27

###### Lemma 27

Assume that $U$ satisfies Assumptions 1 and 3, then for $\pi=e^{-U}$ and any
distribution $p$, we have for $\beta>0$,

$W_{\beta}^{\beta}(p,\
\pi)\leq\frac{4a}{\beta}\left(1.5+\tilde{d}+\tilde{c}_{\mu}\right)H(p_{\mu,k}|\pi_{\mu})+\frac{4a}{\beta}\left(1.5+\tilde{d}+\tilde{c}_{\mu}\right),$

where

$\displaystyle\tilde{c}_{\mu}$
$\displaystyle=\frac{1}{2}\log(\frac{2}{\beta})+\frac{L}{\alpha+1}\sum_{i}\left(\frac{2b}{a}\right)^{\frac{\alpha_{i}+1}{\beta}}+\frac{b}{\beta}+|U(0)|+\frac{NL\mu^{1+\alpha}}{(1+\alpha)}d^{\frac{2}{2\wedge
p}},$ (36) $\displaystyle\tilde{d}$
$\displaystyle=\frac{d}{\beta}\left[\frac{\beta}{2}log\left(\Pi\right)+\log\left(\frac{4\beta}{a}\right)+(1-\frac{\beta}{2})\log(\frac{d}{2e})\right].$
(37)

###  F.5 Proof of Lemma 28

###### Lemma 28

If the potential $U$ satisfies $\beta$-dissipative and $\alpha$-mixture weakly
smooth with $2\alpha_{N}\leq\beta$, then let $p_{k}$ be the distribution of
$x_{k}$ of ULA with a step size satisfying
${\displaystyle\eta\leq\frac{1}{2}\left(1\wedge\frac{a}{2N^{2}L^{2}}\right)}$,
we have for any even integer $s\geq 2$,

$\mathrm{M}_{s}(p_{k}+\pi)\leq\mathrm{M}_{s}(p_{0}+\pi)+C_{s}k\eta,$

where

$C_{s}\stackrel{{\scriptstyle\triangle}}{{=}}{\displaystyle\left(\frac{3a+2b+3}{1\wedge
a}\right)^{\frac{s-2}{\beta}+1}s^{s}d^{\frac{s-2}{\beta}+1}},$

$\mathrm{M}_{s}(p_{0}+\pi)\leq
2(\frac{3a+b+3}{a})^{s/\beta}s^{s/\beta}d^{s/\beta}.$

###### Proof

Since $U$ satisfies $\alpha$-mixture weakly smooth, we have

$\displaystyle\left\|\nabla U(x)\right\|$
$\displaystyle\leq\sum_{i}L_{i}\left\|x\right\|^{\alpha_{i}}$
$\displaystyle\leq\sum_{i}L\left\|x\right\|^{\alpha_{i}}$
$\displaystyle\leq\sum_{i}L\left(\left\|x\right\|^{\alpha_{N}}+1\right)$
$\displaystyle\leq NL\left(\left\|x\right\|^{\alpha_{N}}+1\right)$

where $2\alpha_{N}\leq\beta$ by our assumption. Moreover, $U$ also satisfies
$\beta$-dissipative. From (Erdogdu and Hosseinzadeh,, 2020) Proposition 2 and
Lemma 22, we obtain the desired result.

###### Lemma 29

[(Nguyen et al.,, 2021) Lemma F.16] If $\xi\sim N_{p}\left(0,I_{d}\right)$
then $d^{\left\lfloor\frac{n}{p}\right\rfloor}\leq
E(\left\|\xi\right\|_{p}^{n})\leq\left[d+\frac{n}{2}\right]^{\frac{n}{p}}$where$\left\lfloor
x\right\rfloor$ denotes the largest integer less than or equal to $x.$ If
$n=kp,$ then $E(\left\|\xi\right\|_{p}^{n})=d..(d+k-1)$.

###### Lemma 30

[(Nguyen et al.,, 2021) Lemma C.2] Assume $\pi=e^{-U(x)}$ is $\alpha$-mixture
weakly smooth. Then

$\mathbb{E}_{\pi}\left[\left\|\nabla U(x)\right\|^{2}\right]\leq
2NL^{2}d^{\frac{3}{p}},$

for $d$ sufficiently large.

###### Lemma 31

[(Nguyen et al.,, 2021) Lemma 2.1] If potential
$U:\mathbb{R}^{d}\rightarrow\mathbb{R}$ satisfies an $\alpha$-mixture weakly
smooth for some $0<\alpha=\alpha_{1}<...<\alpha_{N}\leq 1$, $i=1,..,N$
$0<L_{i}<\infty$, then:

$U(y)\leq U(x)+\left\langle\nabla U(x),\
y-x\right\rangle+\sum_{i}\frac{L_{i}}{1+\alpha_{i}}\|y-x\|^{1+\alpha_{i}}.$
(38)

## Acknowledgements

This research was funded in part by the University of Mississippi summer
grant.

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# Bridging Dictionary: AI-Generated Dictionary of Partisan Language Use

Hang Jiang<EMAIL_ADDRESS>0009-0000-9523-136X MIT Center for Constructive
Communication & MIT Media LabCambridgeMAUSA ,  Doug Beeferman<EMAIL_ADDRESS>0009-0005-5879-5744 MIT Center for Constructive Communication & MIT Media
LabCambridgeMAUSA ,  William Brannon<EMAIL_ADDRESS>0000-0002-1435-8535 MIT
Center for Constructive Communication & MIT Media LabCambridgeMAUSA ,  Andrew
Heyward<EMAIL_ADDRESS>MIT Center for Constructive Communication &
MIT Media LabCambridgeMAUSA and  Deb Roy<EMAIL_ADDRESS>0000-0002-2780-4768
MIT Center for Constructive Communication & MIT Media LabCambridgeMAUSA

###### Abstract.

Words often carry different meanings for people from diverse backgrounds.
Today’s era of social polarization demands that we choose words carefully to
prevent miscommunication, especially in political communication and
journalism. To address this issue, we introduce the Bridging Dictionary, an
interactive tool designed to illuminate how words are perceived by people with
different political views. The Bridging Dictionary includes a static,
printable document featuring 796 terms with summaries generated by a large
language model. These summaries highlight how the terms are used distinctively
by Republicans and Democrats. Additionally, the Bridging Dictionary offers an
interactive interface that lets users explore selected words, visualizing
their frequency, sentiment, summaries, and examples across political divides.
We present a use case for journalists and emphasize the importance of human
agency and trust in further enhancing this tool111HJ and DB are both co-first
authors. HJ led the evaluation and writing and DB led the tool development..
The deployed version of Bridging Dictionary is available at
https://dictionary.ccc-mit.org/.

natural language processing, text analysis, political science, journalism

††conference: The 27th ACM SIGCHI Conference on Computer-Supported Cooperative
Work & Social Computing (CSCW); November 9-13, 2024; San Jose, Costa
Rica††ccs: Human-centered computing Empirical studies in HCI††ccs: Human-
centered computing Web-based interaction††ccs: Human-centered computing
Visualization toolkits††ccs: Computing methodologies Natural language
processing

##  1\. Introduction

Polarization is a significant feature of the political landscape in the United
States (Poole and Rosenthal, 1984; McCarty et al., 2016; Heltzel and Laurin,
2020). Previous research has shown that Republicans and Democrats often use
and interpret words differently, even when speaking the same language (Li et
al., 2017; R. KhudaBukhsh et al., 2021). This linguistic divide poses
considerable challenges to public discourse, particularly in journalism, where
the use of words without an understanding of their varying connotations across
political communities can have serious consequences. Journalists face the
added difficulty of manually reading and editing news content, a process that
is not only time-consuming but also prone to errors due to the constantly
evolving connotations of words online. To address this problem, we introduce
the Bridging Dictionary (BD), a tool designed to automatically identify
controversial terms across the political divides and to summarize their
different usages. We provide not only a useful resource for the academic
community, journalists, and wider audiences but also highlight the importance
of considering human agency and trust in developing human-AI systems.

##  2\. Related Work

#### Polarized language use in NLP

Researchers in political science and Natural Language Processing (NLP) have
found that there is a partisan difference in language understanding (Li et
al., 2017). R. KhudaBukhsh et al. (2021) used modern machine-translation
methods to show that the Republican and Democratic communities use English
words differently. For instance, there are different connotations when
partisans use “undocumented workers” or “illegal aliens” to discuss the same
group of people. To address this issue, Webson et al. (2020) developed an NLP
method to mitigate the political bias of text representations, showing that it
improves the viewpoint diversity of document rankings. Recently, NLP
researchers have also proposed novel methods to quantify and debias the
political bias of language models (Liu et al., 2021, 2022). However, previous
work tends to focus on measuring and debiasing NLP models instead of
facilitating humans in writing less biased content. Our work fills the gap by
leveraging NLP to help humans understand the language bias and facilitate them
in writing and editing through an interactive tool.

#### NLP for qualitative analysis and sensemaking

NLP has been used to develop computational tools for qualitative analysis and
sensemaking (Crowston et al., 2012; Guetterman et al., 2018). Among these
tools, topic models are particularly prevalent for text analysis across
various domains (Yan et al., 2013; Hu et al., 2019; Isoaho et al., 2021).
However, traditional NLP models often lack world knowledge, resulting in
limited insights and interpretability (Barde and Bainwad, 2017). The modern
large language models (LLMs) have enabled users to interact with unstructured
data through simple queries to extract more nuanced and interpretable
insights. Recent studies have explored the application of LLMs to
automatically extract and summarize valuable information from texts (Chew et
al., 2023; Xiao et al., 2023; Gero et al., 2024; Overney et al., 2024).
Despite these advancements, there is a lack of research focusing on how LLMs
can assist journalists by providing qualitative insights for writing and
editing. This study aims to bridge this gap by introducing an interactive tool
to summarize the varying usage of terms across political divides, thereby
guiding journalists in their word choices in news writing.

##  3\. System Overview

(a) The front page of the interactive Bridging Dictionary demo. The user can
type any term they are interested in.

(b) The front page of “Bridging Dictionary: Paper Edition” with GPT-generated
summaries for representative terms.

Figure 1. Selected views of the Bridging Dictionary: an interactive front page
with user text input, and a static paper edition page.

The Bridging Dictionary (BD) comprises two main components: (1) a paper
dictionary and (2) an interactive demo. As illustrated in Figure 1, the paper
dictionary presents 796 representative terms in a print-ready format,
supplemented with summaries generated by an LLM. The interactive demo, on the
other hand, enables users to explore the usage of a given term within
Republican and Democratic communities in greater detail. BD leverages
gpt-3.5-turbo, a widely-recognized LLM, via OpenAI’s API to generate these
summaries. The term usages are sampled from a Twitter dataset (Jiang et al.,
2022), which includes 4.7 million partisan-generated tweets (amounting to 100
million tokens) from each side during the 2020 American election. Users can
customize both the dataset and the available LLMs from OpenAI.

###  3.1. Paper Dictionary

We generate a static printable document called “Bridging Dictionary: Paper
Edition” that comprises 796 terms with LLM-generated summaries. These terms
are curated algorithmically: we identify words and multi-word phrases that (1)
occur sufficiently often within both partisan communities and (2) have
significant differences between the two communities, either in sentiment score
or in usage frequency. The parameters for these operations (i.e., the
thresholds for “sufficient” and “significant”) are adjusted manually by an
editor.

###  3.2. Interactive Demo

The interactive demo is a web-based generative dictionary providing greater
detail and flexibility than the print version. It is implemented with the
Streamlit framework (Streamlit, 2021). Whenever a user types a phrase, the
system provides a few functions to explore how two communities (Republicans
and Democrats) use this term, with alternative suggestions and visualizations
of the input data.

(a) Statistics

(b) Summary

(c) Definition

(d) Topic scatterplot

(e) Sample list

Figure 2. Six core sections in the interactive demo to help users explore how
two communities use a term differently.

Statistics. This section offers an overview of tweets containing the specified
term from each community. The column “Matches per Thousand Tweets” indicates
the frequency of tweets that include the term (e.g., “climate change”) per
1,000 tweets within a community. The accompanying pie chart illustrates the
proportion of term usage between two partisan communities. Sentiment scores
represent the average sentiment for tweets from a community that matched the
term, with higher scores indicating a more positive sentiment. Colored text
highlights the comparative scores between the two communities, helping users
to interpret the data easily.

Summary. This section features LLM-generated summaries that explain the usage
of the term across divides and propose alternative terms. The generation
follows a standard retrieval-augmented generation (RAG) procedure, as outlined
by Gao et al. (2023). Initially, the system randomly samples up to 50 tweets
containing the term from a specific community, creates a prompt using these
tweets, and prompts the LLM to produce summaries with a simple query.
Importantly, the LLM is unaware of the community identity and only uses the
sampled tweets for its summarization.

Definition. This section generates a dictionary-style definition of a term
from each community’s perspective. This follows the same RAG process as the
Summary section but asks the model for a definition instead of a summary.

Topic scatterplot. This section presents a two-dimensional interactive
scatterplot that organizes individual tweets by topic, grouping similar topics
closely together. Users can explore specific tweets by hovering over the
corresponding points, enabling them to review the sampled tweets used for
summary and definition generation. The process follows a well-established
pipeline (e.g., BERTopic (Grootendorst, 2022)) transforming raw text into a
scatterplot. It involves computing the embedding for each tweet using a
sentence embedding model, then projecting these embeddings into two
dimensions. By default, the points are clustered and color-coded based on the
outcome of a clustering algorithm applied to these embeddings. For projection,
we employ UMAP (McInnes et al., 2018), and for clustering, we use HDBSCAN
(McInnes et al., 2017).

Sample list. This section lists the sampled tweets from each group, allowing
users to read the information source.

##  4\. Discussion and Evaluation

After deploying the Bridging Dictionary, we interviewed a professional
journalist from Frontline at PBS and received positive feedback. Based on the
interview, our statistics and summary features significantly aid in
understanding the political divide in language use. Additionally, the topic
scatterplot and sample list features are relied upon for supporting LLM-
generated content and assisting journalists in making informed word choices.
During our interview, two promising directions emerged: (1) enhancing the
connection between LLM-generated content and information sources by
considering human agency and trust in human-AI interaction, and (2) broadening
the range of information sources beyond Twitter and regularly updating the
dataset to reflect the evolving nature of language across different platforms
and over time. We intend to further develop the tool based on these
suggestions, conduct a more comprehensive field study involving more
professional journalists and other users, and assess the tool’s impact on
writing and editing.

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