# Parachatnam-Berry Phase in Optics: Polarization Propagation in Helical
Optical Fibers

Igor Kuzmenko1,2, Y. B. Band1,2,3 and Yshai Avishai1,4 1Department of Physics,
Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
2Department of Chemistry, Ben-Gurion University of the Negev, Beer-Sheva
84105, Israel
3The Ilse Katz Center for Nano-Science, Ben-Gurion University of the Negev,
Beer-Sheva 84105, Israel
4Yukawa Institute of Theoretical Physics, Kyoto, Japan

###### Abstract

The Parachatnam-Berry phase (PBP) is a geometric phase associated with the
polarization of light propagating in optical systems. Here, we investigate the
physical principles underlying the occurrence of PBP for a single-mode light
beam propagating in a single-mode optical fiber with no significant stress-
induced birefringence wound into a circular helix configuration. We discuss
the effects of curvature and torsion of the helical fiber on the rotation of
the polarization vector and the associated PBP. We find the analytic solution
for the polarization vector and Stokes parameters of the light for any initial
polarization state of the light entering the helical fiber, the analytic
expression for the PBP of the light for periodic transport of the light, the
intensity of a superposition of the initial and final beams which depends on
the PBP, and we discuss the effects of fluctuations of the parameters
specifying the geometry and the material characteristics of the helical fiber
on the PBP. We also discuss the relationship between the PBP and the solid
angle subtended by the tangent vector of the helix plotted on the unit sphere.

##  I Introduction

The propagation of light in optical fibers is an important topic in optics.
When an optical fiber is wound into a twisted configuration, such as a helix,
the polarization state of the light propagating through the fiber can undergo
a variety of interesting effects due to the geometric properties of the
twisted fiber. One such effect arises from the Parachatnam-Berry phase (PBP),
which emerges due to the non-trivial geometry of the fiber, as has been
described in Refs. [1, 2]. This phase is acquired by a light beam when its
polarization state undergoes cyclic evolution, i.e., the PBP is a geometric
phase that arises when a polarization vector evolves along a closed path in
parameter space. In the case of a circular helix optical fiber with no
significant stress-induced birefringence, the curvature and torsion of the
coiled fiber play a crucial role in determining the PBP. In addition, as a
consequence of the helical configuration, the polarization vector undergoes
rotation as the light propagates. The mathematical formalism for calculating
the rotation and the PBP involves the Berry connection, and for the PBP it is
the integration of the Berry connection over the closed path in parameter
space. The PBP can be experimentally quantified by splitting off a portion of
the light entering and exiting the fiber and forming a superposition of these
portions of the beam, and then measuring the resulting light intensity. Phase
has broad relevance in optics; for example, Ref. [3] deals with the formation
of geometric phases during non-adiabatic frequency-swept radio frequency (RF)
pulses with sine amplitude modulation and cosine frequency modulation
functions. The study analyzes the geometric phases using a Schrödinger
equation formalism and provides solutions for sub-geometric phase components.
The results have implications for magnetic resonance imaging (MRI) and high-
resolution nuclear magnetic resonance (NMR). Here we study the physical
principles underlying the propagation of polarized light and the PBP in
helical optical fibers.

A considerable amount of research has been devoted to the Berry phase, see
Refs. [4, 5, 8, 9, 10, 11, 12, 13] and references therein, which in optics is
often called the PBP; it has applications in various optical systems.
Reference [4] develops the theoretical framework of geometric phases with
particular emphasis on what is now often called the Berry phase, which arises
during adiabatic changes in quantum systems. Geometric phases depend on the
path traversed in parameter space and have important implications in quantum
mechanics [14], mechanics in general, and optics. Reference [5] further
explores the Berry phase, emphasizing its connection to holonomy and the
quantum adiabatic theorem; it provides mathematical insights into the
geometric phase and its importance in quantum physics.

The rotation of the polarization vector of a light beam propagating in an
optical fiber has practical applications, including [7]:

  1. 1.

Optical sensing: The PBP can be used for sensing applications. Changes in the
curvature or torsion of the fiber can be detected by monitoring the
polarization state [8].

  2. 2.

Optical communications: Helical fibers can be used to create polarization-
based devices such as polarization rotators or mode converters [9].

  3. 3.

A spin-dependent deflector based on the PBP has been realized for IR
wavelengths in metallic nanogratings working in transmission [10].

  4. 4.

Lenses based on the PBP [11].

  5. 5.

The intrinsic spin-orbit interaction associated with a gradient in the PBP can
produce a novel type of electromagnetic angular momentum transfer [10, 12].

  6. 6.

Quantum optics: The PBP is relevant in quantum optics, where geometrical
phases play a crucial role [10, 12].

The PBP in twisted optical fibers provides a rich playground for exploring
geometric effects in optics. Understanding this phase can lead to novel
applications and improve our ability to manipulate light polarization in
optical systems.

The following papers are particularly relevant to the work reported here.
Reference [6] investigates the modification of polarization in low
birefringence monomode optical fibers due to geometric effects. Specifically,
it examines how bending and curving the fiber path can lead to polarization
rotation. Reference [9] analyzes the propagation of light in arbitrarily
curved step-index optical fibers. Using a multi-scale approximation scheme,
they rigorously derive Rytov’s law (the rotation of the plane of polarization
of an electromagnetic wave in a uniform inhomogeneous medium due to the PBP)
at leading order, and explore nontrivial dynamics of the polarization of the
electromagnetic field. It describes how the curvature and torsion of the
optical fiber affect the polarization vector of the light (this is an inverse
Spin Hall effect for light [9]). Measurement of the PBP have been reported in
Refs. [6, 13, 15, 16].

The outline of this manuscript is as follows: Section II presents a brief
introduction to geometric phase in mechanics, quantum mechanics and optics.
Section III discusses the geometry of an optical helix fiber and the Frenet-
Serret system, which gives the basis vectors and the generalized curvatures
for a curve in $\mathcal{R}^{n}$ (in $\mathcal{R}^{3}$ the basis vectors are
the tangent vector T, the normal N, and the binormal vector B, which form an
orthonormal basis, as well as the curvature $\kappa$ and the torsion $\tau$).
Section IV considers the dynamics of light polarization in a helical fiber as
the light propagates along the fiber, and Sec. V introduces PBP in a helical
fiber, and Sec. V.1 discusses the relationship between the PBP $\gamma_{B}$
and the solid angle subtended by the polarization vector on a unit sphere
$\Omega$. Section VI calculates the effects of fluctuations of the parameters
in the expressions for the PBP, and finally, Sec. VII contains a summary and
conclusion.

##  II Geometric phase in mechanics, quantum mechanics and optics

In classical and quantum mechanics, the term “geometric phase” refers to a
phase acquired by a system in the course of carrying out motion along a closed
curve. Often the geometric phase is studied in systems that vary
adiabatically, i.e. slowly, and often the geometric phase follows from the
geometric properties of the parameter space of a Hamiltonian system [17]. In
optics it is known as the Pancharatnam-Berry phase, but it is also sometimes
called the Pancharatnam phase or the Berry phase. The phenomenon was
independently discovered in classical optics by S. Pancharatnam in 1956[1] and
by H. C. Longuet-Higgins in molecular physics in 1958 [18]; and is manifest in
the Aharonov–Bohm effect [19]. Two additional examples of a geometric phase
arise in the Foucault pendulum, as explained in Ref. [20], and in nuclear
magnetic resonance, see Ref. [21]. A simple way to see a geometric phase is as
follows: place your right arm at your side, then move it straight in front of
you by rotating it 90 degrees, then rotate it another 90 degrees so that your
arm is extended to your right, and finally rotate your arm another 90 degrees
so that your arm is back at your side. Notice that your hand is rotated 90
degrees from its original direction. This rotation of your hand is a geometric
phase.

##  III Geometry of an optical helix fiber

A circular helix of radius $r$ and pitch $2\pi c$ is defined as a vector-
valued function

${\bf r}(t)=r\,\cos t\,\hat{\bf x}+r\,\sin t\,\hat{\bf y}+c\,t\,\hat{\bf z},$
(1)

where $\hat{\bf x}$, $\hat{\bf y}$ and $\hat{\bf z}$ are Cartesian basis
vectors, the origin of the coordinate system is on the axis of the helix, $r$
and $c$ are assumed to be positive, and the real dimensionless curve parameter
$t$ (you can think of the parameter $t$ as being the product of the beam phase
velocity and the propagation time expressed in dimensionless units). The arc-
length $s$ of the helix is

$s=\sqrt{r^{2}+c^{2}}\,t.$ (2)

Solving Eq. (2) for $t$, we obtain $t=s/\sqrt{r^{2}+c^{2}}$, i.e., we invert
$s(t)$ to get $t(s)$. Hence, the curve of the helix in Eq. (1) can be re-
parameterized to give ${\bf r}$ as a function of $s$:

$\displaystyle{\bf r}(s)$ $\displaystyle=$ $\displaystyle
r\cos\bigg{(}\frac{s}{\sqrt{r^{2}+c^{2}}}\bigg{)}\,\hat{\bf
x}+r\sin\bigg{(}\frac{s}{\sqrt{r^{2}+c^{2}}}\bigg{)}\,\hat{\bf y}$ (3)
$\displaystyle+\frac{c\,s}{\sqrt{r^{2}+c^{2}}}\,\hat{\bf z}.$

Note that $|d{\bf r}(s)/ds|=1$, which implies that ${\bf r}(s)$ is a unit-
speed curve.

The unit tangent vector is defined as ${\bf T}(s)=d{\bf r}(s)/ds$:

$\displaystyle{\bf T}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{r^{2}+c^{2}}}\,\bigg{[}-r\sin\bigg{(}\frac{s}{\sqrt{r^{2}+c^{2}}}\bigg{)}\,\hat{\bf
x}$ (4)
$\displaystyle+r\cos\bigg{(}\frac{s}{\sqrt{r^{2}+c^{2}}}\bigg{)}\,\hat{\bf
y}+c\,\hat{\bf z}\bigg{]}.$

The unit normal vector ${\bf N}(s)$ can be obtained from the equation $d{\bf
T}(s)/ds=\kappa\,{\bf N}(s)$, where the curvature $\kappa$ is given by

$\kappa=\Big{|}\frac{d{\bf T}(s)}{ds}\Big{|}=\frac{r}{r^{2}+c^{2}},$ (5)

and using these results we obtain:

${\bf N}(s)=-\cos\bigg{(}\frac{s}{\sqrt{r^{2}+c^{2}}}\bigg{)}\,\hat{\bf
x}-\sin\bigg{(}\frac{s}{\sqrt{r^{2}+c^{2}}}\bigg{)}\,\hat{\bf y}.$ (6)

The binormal vector is given by ${\bf B}(s)={\bf T}(s)\times{\bf N}(s)$:

$\displaystyle{\bf B}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{r_{0}^{2}+c^{2}}}\,\bigg{[}c\sin\bigg{(}\frac{s}{\sqrt{r^{2}+c^{2}}}\bigg{)}\,\hat{\bf
x}$ (7)
$\displaystyle-c\cos\bigg{(}\frac{s}{\sqrt{r^{2}+c^{2}}}\bigg{)}\,\hat{\bf
y}+r\,\hat{\bf z}\bigg{]}.$

By differentiating ${\bf B}(s)$ we find $d{\bf B}(s)/ds=-\tau\,{\bf N}(s)$,
where the torsion of the helix is $\tau=-{\bf N}(s)\cdot d{\bf B}/ds$:

$\tau=\frac{c}{r^{2}+c^{2}}.$ (8)

More generally, a Frenet–Serret system [22] describes differentiable curves in
${\mathcal{R}}^{n}$. Although the curve ${\bf x}(t)$ is given as a function of
the dimensionless curve parameter $t$, the Frenet–Serret system is usually
written in terms of the arc-length $s$, so

${\bf r}(s)=x_{1}(s)\,\hat{\bf x}_{1}+x_{2}(s)\,\hat{\bf
x}_{2}+\ldots+x_{n}(s)\,\hat{\bf x}_{n},$ (9)

where $x_{i}(s)$ are smooth functions of the arc-length $s$, and the unit
tangent vector is given by ${\bf T}(s)=\frac{d{\bf r}(s)}{ds}$. In three-
dimensional Euclidean space, ${\mathcal{R}}^{3}$, the normal vector ${\bf
N}(s)$ and the binormal vector ${\bf B}(s)$ are given by

$\displaystyle{\bf N}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa(s)}\,\frac{d{\bf T}(s)}{ds},$ (10)
$\displaystyle{\bf B}(s)$ $\displaystyle=$ $\displaystyle{\bf T}(s)\times{\bf
N}(s),$ (11)

The curvature $\kappa(s)$ measures the failure of the curve ${\bf r}(s)$ to be
a straight line, while torsion $\tau(s)$ measures the failure of the curve to
be planar:

$\displaystyle\kappa(s)$ $\displaystyle=$ $\displaystyle\frac{|{\bf
r}^{\prime}(s)\times{\bf r}^{\prime\prime}(s)|}{|{\bf r}(s)|^{3}},$ (12)
$\displaystyle\tau(s)$ $\displaystyle=$ $\displaystyle\frac{[{\bf
r}^{\prime}(s)\times{\bf r}^{\prime\prime}(s)]\times{\bf
r}^{\prime\prime\prime}(s)}{|{\bf r}^{\prime}(s)\times{\bf
r}^{\prime\prime}(s)|^{2}},$ (13)

where ${\bf r}^{\prime}(s)=d{\bf r}(s)/ds$, ${\bf
r}^{\prime\prime}(s)=d^{2}{\bf r}(s)/ds^{2}$ and ${\bf
r}^{\prime\prime\prime}(s)=d^{3}{\bf r}(s)/ds^{3}$. The Frenet–Serret system
describes the curvature $\kappa(s)$ and torsion $\tau(s)$ of the curve and the
derivatives of the tangent, normal, and binormal unit vectors in terms of each
other,

$\displaystyle\frac{d{\bf T}}{ds}$ $\displaystyle=\kappa{\bf N},$ (14)
$\displaystyle\frac{d{\bf N}}{ds}$ $\displaystyle=-\kappa{\bf T}+\tau{\bf B},$
$\displaystyle\frac{d{\bf B}}{ds}$ $\displaystyle=-\tau{\bf N}.$

The unit vectors ${\bf T}$, ${\bf N}$, ${\bf B}$, and the scalars $\kappa$ and
$\tau$, are together called the Frenet–Serret apparatus. In
${\mathcal{R}}^{n}$ there are $n$ generalized curvatures, and $n$
Frenet–Serret unit basis vectors. [In the Wolfram Mathematica language, which
was used to carry out the calculations reported here, the Frenet–Serret system
is called ‘FrenetSerretSystem’; the Frenet–Serret equations are coded as a
function of the parameter $t$, not the arc-length $s$.]

Figure 1: Helix ${\bf r}(s)$ (black), and ${\bf T}(s)$ (red), ${\bf N}(s)$
(yellow), ${\bf B}(s)$ (blue) plotted with parameters $r=1$ and $c=1/4$. The
triad of unit vectors are plotted for $s=2$.  Figure 2: Tangent vector ${\bf
T}$ on the Poincaré sphere plotted with parameters $r=1$ and $c=1/4$.

Figure 1 shows a helix ${\bf r}(s)$ with parameters $r=1$ and $c=1/4$ plotted
in black, and the unit vectors ${\bf T}(s)$ (red), ${\bf N}(s)$ (yellow),
${\bf B}(s)$ (blue) evaluated at $s=2$. A triad of unit vectors can be plotted
at any value of arc-length $s$.

Figure 2 shows the tangent vector ${\bf T}(s)$ on the unit sphere as a blue
curve, where the unit vector ${\bf T}(s)$ goes from the origin to the blue
curve. The solid angle on the sphere subtended by ${\bf T}(s)$ as $s$ moves
from $s=0$ (see the black arrow from the origin of the sphere to the sphere
surface) and back to this same point is colored red, but is semitransparent.
We call this area $\Omega(r,c)$:

$\Omega(r,c)=\frac{2\pi c}{\sqrt{r^{2}+c^{2}}}.$ (15)

The figure was drawn for $r=1$ and $c=1/4$. As $r$ decreases (increases) the
area decreases (increases), and as $c$ decreases (increases) the area
increases (decreases).

##  IV Dynamics of light polarization in a helix fiber

For a fiber wound into a circular helix (e.g., by wrapping the fiber around a
cylinder), an analytic solution for the polarization state of the light as a
function of arc-length in a fiber with negligible birefringence is possible as
both $r$ and $c$ are constants independent of $s$. The propagation equations
for the polarization vector, which we call ${\bf A}(s)=A_{n}(s){\bf
N}(s)+A_{b}(s){\bf B}(s)$, where the subscripts $n$ and $b$ stand for normal
and binormal, is given by [6, 23]

$i\,\frac{d}{ds}\left(\\!\\!\begin{array}[]{c}A_{n}(s)\\\
A_{b}(s)\end{array}\\!\\!\right)=\left(\\!\\!\begin{array}[]{cc}-\frac{\alpha\,\kappa^{2}}{2}&i\tau\\\
-i\tau&\frac{\alpha\,\kappa^{2}}{2}\end{array}\\!\\!\right)\left(\\!\\!\begin{array}[]{c}A_{n}(s)\\\
A_{b}(s)\end{array}\\!\\!\right).$ (16)

Here $\alpha$ is a constant given by the product of the radius of the optical
fiber, $\rho$, and a constant depending on the wavelength of the light and the
elastic material properties of the fiber, $a$, $\alpha=\rho a$ [6, 23]. With
the most general initial conditions given by

$A_{n}(0)=p,\qquad A_{b}(0)=e^{i\phi}\sqrt{1-p^{2}},$ (17)

the solution to Eq. (16) takes the form,

$\displaystyle A_{n}(s)$ $\displaystyle=$ $\displaystyle
p\,\cos(\omega_{s}s)+\frac{2\tau
e^{i\phi}\sqrt{1-p^{2}}+i\,\alpha\kappa^{2}p}{2\omega_{s}}\,\sin(\omega_{s}s),$
$\displaystyle A_{b}(s)$ $\displaystyle=$ $\displaystyle
e^{i\phi}\sqrt{1-p^{2}}\cos(\omega_{s}s)$ (18) $\displaystyle-$
$\displaystyle\frac{2\tau
p+i\,\alpha\kappa^{2}e^{i\phi}\sqrt{1-p^{2}}}{2\omega_{s}}\,\sin(\omega_{s}s),$

where $\omega_{s}=\frac{1}{2}\sqrt{\alpha^{2}\kappa^{4}+4\tau^{2}}$.

Figure 3: Amplitudes $A_{n}(s)$ and $A_{b}(s)$ of the normal and binormal
vectors in the polarization vector ${\bf A}(s)$ for $p=1/\sqrt{2}$ and
$\phi=0$ (linearly polarized light with polarization at a 45∘ angle from the
$x$-axis). Re$[A_{n}(s)]$ is the solid blue curve, Im$[A_{n}(s)]$ is the
dashed red curve, Re$[A_{b}(s)]$ is the solid green curve, and Im$[A_{b}(s)]$
is the dashed purple curve.  Figure 4: Amplitudes $A_{n}(s)$ and $A_{b}(s)$ of
the normal and binormal vectors in the polarization vector ${\bf A}(s)$ for
left circular polarized light, $p=1/\sqrt{2}$ and $\phi=\pi/2$. The
Re$[A_{n}(s)]$ is the solid blue curve , Im$[A_{n}(s)]$ is the dashed red
curve, Re$[A_{b}(s)]$ is the solid green curve, and Im$[A_{b}(s)]$ is the
dashed purple curve which overlaps the solid blue Re$[A_{n}(s)]$ curve.
Figure 5: Amplitudes $A_{n}(s)$ and $A_{b}(s)$ of the normal and binormal
vectors in the polarization vector ${\bf A}(s)$ for right circular polarized
light, $p=1/\sqrt{2}$ and $\phi=-\pi/2$. The Re$[A_{n}(s)]$ is the solid blue
curve, Im$[A_{n}(s)]$ is the dashed red curve, Re$[A_{b}(s)]$ is the solid
green curve, and Im$[A_{b}(s)]$ is the dashed purple curve.

Figure 3 plots the amplitudes $A_{n}(s)$ and $A_{b}(s)$ versus $s$ for
linearly polarized light with polarization at a 45∘ angle from the $x$-axis,
i.e., probability amplitude $p=1/\sqrt{2}$ and phase $\phi=0$. The parameter
values used are $\alpha=1$, $r=1$ and $c=1/4$. The amplitudes have a period
$T_{s}=2\pi/\omega_{s}=4\pi/\sqrt{\alpha^{2}\kappa^{4}+4\tau^{2}}$. The
Im$[A_{n}(s)]$ and Im$[A_{b}(s)]$ curves are 180∘ out of phase; Re$[A_{n}(s)]$
and Re$[A_{b}(s)]$ are out of phase by a much smaller angle.

Figure 4 plots the amplitudes $A_{n}(s)$ and $A_{b}(s)$ versus $s$ for left
circular polarized light, i.e., probability amplitude $p=1/\sqrt{2}$ and phase
$\phi=\pi/2$. In this case, Re$[A_{n}(s)]$ equals Im$[A_{b}(s)]$ so these
curves lie one on top of the other. Figure 5 is somewhat similar to Fig. 4,
except for right circular polarized light, $p=1/\sqrt{2}$ and $\phi=-\pi/2$,
the Re$[A_{n}(s)]$ and Im$[A_{b}(s)]$ curves are distinct.

For understanding the physics of the polarization it is convenient to define
the Stokes vector ${\bf S}(s)$ [24], which is a real vector that can be
plotted on the Poincaré sphere (as can the complex polarization vector ${\bf
A}(s)$ be plotted on the complex Bloch sphere [25]):

$\displaystyle{\bf S}(s)$ $\displaystyle=$
$\displaystyle\big{(}A_{n}^{*}(s),A_{b}^{*}(s)\big{)}\,\bm{\sigma}\,\left(\\!\\!\begin{array}[]{c}A_{n}(s)\\\
A_{b}(s)\end{array}\\!\\!\right).$ (21)

Here the Pauli vector is defined as $\bm{\sigma}=\sigma_{x}\hat{\bf
x}+\sigma_{y}\hat{\bf y}+\sigma_{z}\hat{\bf z}$, and $\sigma_{x}$,
$\sigma_{y}$ and $\sigma_{z}$ are the Pauli matrices. We now plot the
Cartesian components of ${\bf S}(s)$ versus $s$ in Fig. 6 for the linearly
polarized case where the amplitudes $A_{n}(s)$ and $A_{b}(s)$ are shown in
Fig. 3. For this case the period of Stokes vector ${\bf S}(s)$ is half of the
period $T_{S}$ of the amplitudes, where $T_{S}=12.5281$, as can be seen in all
the figures for the amplitudes shown above.

Figure 6: Polarization vector components of ${\bf S}(s)$ versus arc-length $s$
for initial conditions $p=1/\sqrt{2}$ and $\phi=0$ with with parameters $r=1$
and $c=1/4$.

Figure 7 shows the trajectory of the Stokes vector ${\bf S}(s)$ on the
Poincaré sphere for $p=1/\sqrt{2}$ and $\phi=0$. The black arrow shows the
polarization vector at $s=0$. As propagation proceeds, the tip of ${\bf S}(s)$
follows the red curve shown in the figure, making one full revolution at
$s=T_{s}/2$ and two full revolutions at $s=T_{s}$. As the initial conditions
change the solid angle subtended by ${\bf S}(s)$ on the Poincaré sphere
changes; as $p$ increases from zero to one, the solid angle subtended
increases. When $p=0$ and $p=1$, there is no dependence on $\phi$, but when
$p\neq 0$, first the solid angle increases, then decreases, and it might again
increase. The solid angle subtended equals the PBP, and more will be said
about the PBP in Sec. V.

The position of the polarization vector on the Poincaré sphere can be
quantified by the polar angle $\theta(s)$ and azimuthal angle $\phi(s)$.
Figure 8 plots these angles versus $s$ for the linear polarization case with
$p=1/\sqrt{2}$ and $\phi=0$. The discontinuities in $\phi(s)$ arises because
of the way the azimuthal angle was calculated; it ranges from $-\pi$ to $\pi$.
Clearly these angles are also periodic.

Figure 7: Polarization vector ${\bf S}$ plotted on the Poincaré sphere.
Figure 8: The polar and azimuthal angles of the polarization vector ${\bf
S}(s)$, $\theta(s)$ and $\phi(s)$, versus $s$.

##  V Pancharatnam-Berry phase in a helix fiber

The geometrical phase can defined as follows [4, 17]:

$\displaystyle\gamma_{B}$ $\displaystyle=$
$\displaystyle-i\int\limits_{0}^{T_{s}}\Big{(}A_{n}^{*}(s)\,\frac{dA_{n}(s)}{ds}+A_{b}^{*}(s)\,\frac{dA_{b}(s)}{ds}\Big{)}\,ds$
(22) $\displaystyle=$
$\displaystyle-\frac{2\pi[(1-2p^{2})\alpha\kappa^{2}-4p\sqrt{1-p^{2}}\,\tau\sin\phi]}{\sqrt{\alpha^{2}\kappa^{4}+4\tau^{2}}}.$

Here $T_{s}=4\pi/\sqrt{\alpha^{2}\kappa^{4}+4\tau^{2}}$ is the period of the
amplitudes $A_{n}(s)$ and $A_{b}(s)$ in Eq. (18).

In order to demonstrate that the PBP can be represented as a solid angle on
the Poincaré sphere it is convenient to rewrite Eq. (16) as follows:

$i\,\frac{d}{ds}\left(\\!\\!\begin{array}[]{c}A_{n}(s)\\\
A_{b}(s)\end{array}\\!\\!\right)=\mathbb{B}\cdot\bm{\sigma}\,\left(\\!\\!\begin{array}[]{c}A_{n}(s)\\\
A_{b}(s)\end{array}\\!\\!\right).$ (23)

The $s$-independent vector $\mathbb{B}$ is given by

$\mathbb{B}=-\tau\,\hat{\bf y}-\frac{1}{2}\,\alpha\kappa^{2}\,\hat{\bf z},$

and its absolute value is $|\mathbb{B}|=2\pi/T_{s}$. Hence we can express
$\gamma_{B}$ as

$\gamma_{B}=-\frac{2\pi}{T_{s}}\int\limits_{0}^{T_{s}}\hat{\mathbb{B}}\cdot{\bf
S}(s)\,ds,$ (24)

where $\hat{\mathbb{B}}=\mathbb{B}/|\mathbb{B}|$ is the unit vector in the
direction of $\mathbb{B}$, and the vector ${\bf S}(s)$ is defined in Eq. (21).
It should be noted that the solution of Eq. (23) describes precession of the
vector ${\bf S}(s)$ about the axis given by the unit vector
$\hat{\mathbb{B}}$. Therefore, the projection of ${\bf S}(s)$ onto the
precession axis, $\hat{\mathbb{B}}\cdot{\bf S}(s)$, does not depend on $s$ and
is given by

$\hat{\mathbb{B}}\cdot{\bf
S}(s)=\frac{(1-2p^{2})\alpha\kappa^{2}-4p\sqrt{1-p^{2}}\,\tau\sin\phi}{\sqrt{\alpha^{2}\kappa^{4}+4\tau^{2}}}.$
(25)

Substituting this equation into Eq. (24), we obtain Eq. (22).

Figure 9 plots the PBP $\gamma_{B}$ versus the probability amplitude $p$ and
initial angle $\phi$ for parameters $r=1$ and $c=1/4$. $\gamma_{B}$ has a
maximum of $\pi$ at $p=0.970339$ and $\phi=3\pi/2$, and a minimum of $-\pi$ at
$p=0.241748$ and $\pi=\pi/2$. Note the symmetry of $\gamma_{B}$: making the
transformation $p\to-p$ and $\phi\to-\phi$ (i.e., $\phi\to\phi+\pi$) leaves
$\gamma_{B}$ unchanged, as is clear from Fig. 9.

Figure 9: The PBP $\gamma_{B}$ (divided by $\pi$) versus the probability
amplitude $p$ and initial angle $\phi$ for parameters $r=1$ and $c=1/4$.
$\gamma_{B}/\pi$ ranges from -1 and 1.  Figure 10: Intensity $I$ plotted as a
function of the initial conditions $p$ and $\phi$ that is obtained upon taking
a coherent superposition of a fraction of the incident beam at $s=0$ and the
same fraction of the beam at $s=T_{s}$.

Figure 10 plots the intensity $I$ obtained upon taking a coherent
superposition of a fraction of the incident beam at $s=0$ and the same
fraction of the beam at $s=T_{s}$. The intensity is proportional to
$\cos^{2}(\gamma_{B}/2)=(1+\cos\gamma_{B})/2$, which ranges from 0 to 1. Note
that $s=T_{s}$ is not the arc-length at which the helix tangent vector returns
to its initial value ${\bf T}(0)$. This is clear from Eq. (4); the period for
the tangent vector returning to its original position is
$2\pi\sqrt{r^{2}+c^{2}}=\tfrac{2\pi}{\sqrt{\kappa^{2}+\tau^{2}}}\neq
T_{s}=\tfrac{4\pi}{\sqrt{\alpha^{2}\kappa^{4}+4\tau^{2}}}$.

###  V.1 Relationship between $\gamma_{B}$ and $\Omega$

Here we compare the solid angle $\Omega(r,c)$ given in Eq. (15) and the PBP
$\gamma_{B}$ given in Eq. (22). Note that $\gamma_{B}$ depends on the initial
conditions parameterized by $p$ and $\phi$, see Eq. (17), and the parameter
$\alpha$ (which depends on the fiber radius and elastic properties of the
material). In contradistinction, $\Omega(r,c)$ is a function only of the helix
radius $r$ and pitch $2\pi c$ (it does not depend upon $\alpha$, nor on the
initial conditions). Solving Eqs. (5) and (8) for $r$ and $c$ in terms of
$\kappa$ and $\tau$, we get $r=\frac{\kappa}{\kappa^{2}+\tau^{2}}$,
$c=\frac{\tau}{\kappa^{2}+\tau^{2}}$, and substituting these equations into
Eq. (15), we can express $\Omega(\kappa,\tau)$ as a function of the curvature
$\kappa$ and the torsion $\tau$:

$\Omega(\kappa,\tau)=\frac{2\pi\tau}{\sqrt{\kappa^{2}+\tau^{2}}}.$ (26)

Comparing Eqs. (26) and (22) reveals that

$\gamma_{B}=\pm\,\Omega(\kappa,\tau),$ (27)

provided that

  1. 1.

$\alpha=2/\kappa$ (which is very improbable given that $\alpha$ depends on the
material characteristics of the fiber),

  2. 2.

the light is circularly polarized: $p=\tfrac{1}{\sqrt{2}}$,
$\phi=\pm\tfrac{\pi}{2}$.

Unfortunately, Eq. (27) is frequently cited in the literature without
specifying these provisions, e.g., see Refs. [2, 26, 13].

##  VI Fluctuations of the PBP due to fluctuations of the helix parameters

Let us now examine the fluctuations of $\gamma_{B}(\kappa,\tau,\alpha)$ as a
result of possible fluctuations in the fiber parameters $r$, $c$ and $\alpha$.
For example, let us assume that there is a Gaussian probability distribution
for some of the parameters $r$, $c$ and $\alpha$, with expectation values
designated as $r_{0}$, $c_{0}$ and $\alpha_{0}$, and the variances
$\sigma_{r}^{2}$, $\sigma_{c}^{2}$ and $\sigma_{\alpha}^{2}$. Assuming that
the standard deviations of the variables are smaller than the corresponding
mean values, $\sigma_{r}\ll r_{0}$, $\sigma_{c}\ll c_{0}$ and
$\sigma_{\alpha}\ll\alpha_{0}$, and neglecting terms of order $\sigma^{4}$, we
find the expectation value $\bar{\gamma}_{B}$ of
$\gamma_{B}(\kappa,\tau,\alpha)$ and its variance $\sigma_{\gamma_{B}}^{2}$:

$\displaystyle\bar{\gamma}_{B}$ $\displaystyle=$
$\displaystyle\gamma_{B}(r_{0},c_{0},\alpha_{0})+\frac{\partial^{2}\gamma_{B}(r_{0},c_{0},\alpha_{0})}{\partial
r_{0}^{2}}\sigma_{r}^{2}$ (28) $\displaystyle+$
$\displaystyle\frac{\partial^{2}\gamma_{B}(r_{0},c_{0},\alpha_{0})}{\partial
c_{0}^{2}}\sigma_{c}^{2}+\frac{\partial^{2}\gamma_{B}(r_{0},c_{0},\alpha_{0})}{\partial\alpha_{0}^{2}}\sigma_{\alpha}^{2},\quad$
$\displaystyle\sigma_{\gamma_{B}}^{2}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{\partial\gamma_{B}(r_{0},c_{0},\alpha_{0})}{\partial\ln
r_{0}}\bigg{)}^{2}\frac{\sigma_{r}^{2}}{r_{0}^{2}}+\bigg{(}\frac{\partial\gamma_{B}(r_{0},c_{0},\alpha_{0})}{\partial\ln
c_{0}}\bigg{)}^{2}\frac{\sigma_{\tau}^{2}}{\tau_{0}^{2}}$ (29)
$\displaystyle+$
$\displaystyle\bigg{(}\frac{\partial\gamma_{B}(\kappa_{0},\tau_{0},\alpha_{0})}{\partial\ln\alpha_{0}}\bigg{)}^{2}\frac{\sigma_{\alpha}^{2}}{\alpha_{0}^{2}}.$

Let us consider two special cases. (a) For linearly polarized light with
$p=1$, we find

$\displaystyle\bar{\gamma}_{B}$ $\displaystyle=$
$\displaystyle\frac{2\pi\alpha_{0}r_{0}^{2}}{\sqrt{c_{0}^{2}(r_{0}^{2}+c_{0}^{2})^{2}+\alpha_{0}^{2}r_{0}^{4}}}+\frac{2\pi\alpha_{0}r_{0}^{2}c_{0}^{2}}{(c_{0}^{2}(r_{0}^{2}+c_{0}^{2})^{2}+\alpha_{0}^{2}r_{0}^{4})^{5/2}}$
(30)
$\displaystyle\times\bigg{\\{}\frac{2\sigma_{r}^{2}}{r_{0}^{2}}\,c_{0}^{2}\,\Big{[}c_{0}^{2}\big{(}r_{0}^{2}+c_{0}^{2}\big{)}^{2}\big{(}c_{0}^{2}-3r_{0}^{2}\big{)}$
$\displaystyle-\alpha_{0}^{2}r_{0}^{4}\big{(}3r_{0}^{4}+5c_{0}^{2}\big{)}\Big{]}$
$\displaystyle+\frac{\sigma_{c}^{2}}{c_{0}^{2}}\Big{[}2c_{0}^{2}\big{(}r_{0}^{2}+c_{0}^{2}\big{)}^{2}\big{(}r_{0}^{4}+3r_{0}^{2}c_{0}^{2}+6c_{0}^{4}\big{)}$
$\displaystyle-\alpha_{0}^{2}r_{0}^{4}\big{(}r_{0}^{4}+12r_{0}^{2}c_{0}^{2}+15c_{0}^{4}\big{)}\Big{]}$
$\displaystyle-3\sigma_{\alpha}^{2}\,r_{0}^{4}\big{(}r_{0}^{2}+c_{0}^{2}\big{)}^{2}\bigg{\\}},$
$\displaystyle\sigma_{\gamma_{B}}^{2}$ $\displaystyle=$
$\displaystyle\frac{4\pi^{2}\alpha_{0}^{2}r_{0}^{4}c_{0}^{4}(r_{0}^{2}+c_{0}^{2})^{2}}{(c_{0}^{2}(r_{0}^{2}+c_{0}^{2})^{2}+\alpha_{0}^{2}r_{0}^{4})^{3}}\bigg{(}\frac{4c_{0}^{4}\sigma_{r}^{2}}{r_{0}^{2}}+\frac{(r_{0}^{2}+3bc_{0}^{2})^{2}\sigma_{c}^{2}}{c_{0}^{2}}$
(31)
$\displaystyle+\frac{(r_{0}^{2}+c_{0}^{2})^{2}\sigma_{\alpha}^{2}}{\alpha_{0}^{2}}\bigg{)}.$

(b) For circularly polarized light with $p=1/\sqrt{2}$ and $\phi=\pi/2$,

$\displaystyle\bar{\gamma}_{B}$ $\displaystyle=$ $\displaystyle\frac{4\pi
c_{0}(r_{0}^{2}+c_{0}^{2})}{\sqrt{c_{0}^{2}(r_{0}^{2}+c_{0}^{2})^{2}+\alpha_{0}^{2}r_{0}^{4}}}+\frac{4\pi\alpha_{0}^{2}r_{0}^{4}c^{3}_{0}}{(c_{0}^{2}(r_{0}^{2}+c_{0}^{2})^{2}+\alpha_{0}^{2}r_{0}^{4})^{5/2}}$
(32) $\displaystyle\times$
$\displaystyle\bigg{\\{}6\frac{\sigma_{r}^{2}}{r_{0}^{2}}\,\big{[}\alpha_{0}^{2}r_{0}^{4}+r_{0}^{4}c_{0}^{2}-c_{0}^{6}\big{]}$
$\displaystyle-3\frac{\sigma_{c}^{2}}{c_{0}^{2}}\,\big{[}r_{0}^{6}-2\alpha_{0}^{2}r_{0}^{4}+3r_{0}^{4}c_{0}^{2}+11r_{0}^{2}c_{0}^{4}+7c_{0}^{6}\big{]}$
$\displaystyle-\frac{\sigma_{\alpha}^{2}}{\alpha_{0}^{2}}\,\frac{r_{0}^{2}+c_{0}^{2}}{c_{0}^{2}}\,\big{[}c_{0}^{2}(r_{0}^{2}+c_{0}^{2})^{2}-2\alpha_{0}^{2}r_{0}^{4}\big{]}\bigg{\\}},$
$\displaystyle\sigma_{\gamma_{B}}^{2}$ $\displaystyle=$
$\displaystyle\frac{4\pi^{2}\alpha_{0}^{2}r_{0}^{4}c_{0}^{4}(r_{0}^{2}+c_{0}^{2})^{2}}{(c_{0}^{2}(r_{0}^{2}+c_{0}^{2})^{2}+\alpha_{0}^{2}r_{0}^{4})^{3}}\,\bigg{(}4c_{0}^{4}\frac{\sigma_{r}^{2}}{r_{0}^{2}}+(r_{0}^{2}+3c_{0}^{2})^{2}\frac{\sigma_{c}^{2}}{c_{0}^{2}}$
(33)
$\displaystyle+(r_{0}^{2}+c_{0}^{2})^{2}\frac{\sigma_{\alpha}^{2}}{\alpha_{0}^{2}}\bigg{)}.$

Note that the expectation value $\bar{\gamma}_{B}$ depends on the variances of
$r$, $c$, and $\alpha$ i.e.,
$\bar{\gamma}_{B}\neq\gamma_{B}(r_{0},c_{0},\alpha_{0})$. Hence, average
experimental results depend on the variances.

##  VII Summary and Conclusions

The concept of geometric phase is fundamental and establishes a deep
connection between classical mechanics, quantum mechanics and optics (where it
is referred to as the PBP $\gamma_{B}$). In this work, we have analyzed the
geometric phase $\gamma_{B}$ for light propagating in a circular helix optical
fiber. Using the Frenet-Serret system, the curvature $\kappa$, the torsion
$\tau$, and the three orthogonal vectors, the tangent vector ${\bf T}$, the
normal vector ${\bf N}$, and the binormal vector ${\bf B}$ along the helical
curve are obtained. An analytic expression for the amplitudes $A_{n}(s)$ and
$A_{b}(s)$ in the polarization vector ${\bf A}(s)=A_{n}(s){\bf
N}(s)+A_{b}(s){\bf B}(s)$ as a function of the arc-length $s$, and the PRB
$\gamma_{B}$ for arbitrary initial conditions of the polarization are derived
in Secs. IV and V, respectively. Section V.1 shows that, even for circularly
polarized light, $\gamma_{B}$ is not equal to the solid angle $\Omega(r,c)$ on
the sphere subtended by ${\bf T}(s)$ as $s$ moves from $s=0$ and back to the
same point. $\gamma_{B}$ depends on the parameter $\alpha=\rho a$, which is
the product of the radius of the optical fiber, $\rho$, and a constant $a$
which depends on the wavelength of the light and the elastic properties of the
fiber. In contrast, $\Omega(r,c)$ is not dependent on $a$. The dependence of
the mean and standard deviation of the PBP $\gamma_{B}$ on fluctuations of the
parameters $r,c,\alpha$ is discussed in Sec. VI. The expectation value of
$\gamma_{B}$ is dependent upon the variances of $r$, $c$, and $\alpha$.
Although it is well established that disorder can result in the localization
of light, as evidenced by previous research [27, 28], a comprehensive
understanding of the impact of disorder on light polarization and the PBP
remains a topic for further investigation.

## Acknowledgements

We acknowledge useful conversations with Professor Marek Trippenbach. We
dedicate this article to Professor Sergey Geredeskul and his wife Victoria who
were murdered in their home in Ofakim, Israel on October 7, 2024 by Hamas
terrorists.

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