I wrote this article as part of extending my undergraduate thesis. Its certainly not original but should be of benefit to someone.

**1. Introduction **

A considerable understanding of the formal description of quantum mechanics has been achieved after Berrys discovery{BerryPhase} of a geometric feature related to the motion of a quantum system. He showed that the wave function of a quantum object retains a memory of its evolution in its complex phase argument, which, apart from the usual dynamical contribution, only depends on the geometry of the path traversed by the system. Known as the geometric phase factor, this contribution originates from the very heart of the structure of quantum mechanics. It has been over 25 Years since Berry discovered his phase. An entire industry of problems have arisen since then. It has been shown in a various reviews that a better understanding of these problems comes from the usage of topological and geometric techniques. A unified approach to geometric phase problems comes from the usage of basic topological techniques The evolution of the electric field along the curved path of a light ray is described by the Fermi-Walker parallel transport law. For the polarized light the analog of magnetic flux is the solid angle subtended on the sphere of directions k, where k represents the direction of propagation of light, which changes as light passes through the twisted fiber. This spin redirection phase is known as the Rytov-Vlasimirski-Berry{rytov,Rytov_Review,Vladimirski} phase, as opposed to the Pancharatnam-Berry Phase. Since Geometric Phases are intrinsically related to fibre-bundle topological theories, a short reviews of Fiber bundle theories is provided. Deep understanding of geometric phases phases related to time development by the Schrodinger equation and interference effects in optics needs the use of analytical methods. We suspect that in Quantum Computation research, these topological techniques will become increasingly important. The main original work in this paper will be a model of the Manual Fiber Polarization controller(MPC) ~{MPC}.

**2. Topological Phases in Optics **

To elaborate further, there are two types of topological phase in optics. The first one concerns the propagation of light, in which the polarization state changes. As has been demonstrated, in this case the light acquires an addtional phase shift, whose value is determined trajectory of the polarization state on the Poincare Sphere. This phase shift is proportional to the solid angle subtended by the polarization trajectory and is proved to be a manifestation of the Berry Phase. A well know example is the twisted monomode fiber.

This can be explained as a manifestation of the Pancharatnam-Berry Phase. A second example of Berry’s phase has been provided by the study of light propagation in coiled monomode fibers. As has been both theoretically and experimentally demonstrated, coiling the fibre induces in it so-called geometric circular birefringence, which results in the rotation of the polarization plane of a light beam transmitted through the fibre. The present work examines how a type of polarization controller can be analyzed in terms of this spin-redirection Berry Phase, and especially examines how much geometric circular birefringence contributes to the working of such controllers. In the original papers it was suspected to be stress induced birefringence. however we believe that the geometric phase explanation is more powerful. \paragraph{} There is an analogy between the change in direction of the photon obtained by using externally slowly varying parameters and by adiabatic change in the direction of the magnetic field , where the geometrical phase for the latter case is has been treated by Berry. If there is nothing to change the sign of the helicity of the photon, the helicity quantum number is an adiabatic invariant. When the photon propagates smoothly down a helical waveguide (or more pratically through a fibre) k is constrained to remain parallel to the local axis of the waveguide, since the momentum of the photon is in this direction. The geometry of a helical path of a waveguide (or fibre with a unity winding number) contrains k to trace out loop on the surface of a sphere in parameter space where the origin of this space k = 0 is singular. As was noted in previous studies the simple geometric of a circular helix gives the solid angle equal to

where is the pitch angle of the helix, or the angle between the axis of the helix and the local axis of the optical fiber.

** 2.1. Geometric Phases **

Quantum states are represented as vectors in a complex vector space. However these vectors are only defined upto a unit modulus complex number, or a phase in other words. So the two states and are indistinguishable as far as quantum mechanics is concerned – no set of measurements can discriminate them. It is interesting that although this statement looks fairly innocent at first sight, it can lead to some very profound conclusions. We shall take the well know route of considering the extra phase, geometrically in the complex plane. \newline Even though we tend to think of the amplitudes between two quantum states as being fundamental quantum entities, experimentally we can only measure probabilities. We denote the gap between two quantum states and as the mod squared overlap The rotations of a vector **r**(t) in space are described by the SO(3) group. For a one-parameter group with angular velocity

Starting with a vector in

where

\paragraph{} As is well known, in the isotropic linear medium, the Maxwells equations can be written in the form~{Geo_Schrodinger}:

where

where

On account of the Frenet Formula~{GeometryPhysics, Struik} with Eqs. (1) and (2) we have

where

In the case of a Manual Polarization Controller (MPC) the model indicates that

** 2.2. Propogation of a linearly polarized EM wave **

Let us consider a linearly polarized electromagnetic wave propagating in the direction of a wave vector **k(t)** such that **t**(0)=**t**(T) for some T > 0, i.e., the initial and final directions of the fibre coincide. If the shape of the fiber is represented by a curve C, then under the Gauss map C is mapped onto a closed curve **t**(0)=**t**(T) let us introduce an orthonormal basis

Introducing circular polarization vectors

one has

that is, the initial linear polarization is a superposition of right (s=1) and left (s=-1) polarized waves. Now, the helicity eigenstates (with eigenvalues s =

Thus, the geometric phase that appears for circularly polarized photons corresponds to rotation of the linear polarization vector

which is the basic geometric phase relation.

\paragraph{} In accordance with the popular terminology, we use the Jones vector **J**to express the electromagnetic vector in this case. In the local co-ordinate system it can be written as

satisfying

\newline N(s)=\left({cc} ik & \tau

-\tau & ik \right). \paragraph{}

**3. General theories of topological phases **

** 3.1. Berry’s Phases for the Schrodinger equation under the adiabatic approximation **

In the present section we summarize the main results obtained by Berry in his paper of 1984. Although there is no experimental evidence for the existence of magnetic charges or monopoles, the interest in such monopoles arose in the scientific area of Berry’s phases due to the fact that *fomally* certain Berry’s phase systems have the mathematical structure as the Dirac magnetic monopole. The main purpose is not to discuss mathematical superstructure, for mathematics sake, but to provide the computational and topological tools for further analysis of Topological and Geometric Phases. An excellent elaboration and review of the literature is provided in

** 3.2. Berry’s phases on a Poincare sphere **

Let us give a simple explanation for the description of a

where

where

The ellipse will be reduced to *linearly polarized light* when *circularly polarized*wave for

The Jones vector is described as a complex spinor vector

\newline The parameter S is proportional to the intensity of the wave and this parameter is normalized to 1 for unitary transformations. Then the Stokes parameters are equivalent *mathematically*to the Bloch vector components discussed above. For nonunitary transformations the magnitude of the Bloch vector, or equivalently the radius of the Poincar

These two circular polarizations are mutually orthogonal in the sense that

The above result was first given by Ross~{Ross}, it has since become part of the literature. Let us consider that there is linear birefringence induced in the helically coiled monomode fiber and other curves. In fact it is known that birefringence appears whenever the circular symmetry of an ideal fiber is broken, hence this produces an anisotropic refractive-index distribution in the core region. In general in some cases the optical axis and the normal vector (i.e.

i.e.

where *half of the solid angle* subtended by the area of the circuit on the *difference in topological phase* for two circuits which have the same dynamical phase. Since the MPC has a similar effect to quarter wave plates, we expect that similar phenomenoa will be observed. The next section will contain the proposed model for the MPC.

**4. Manual Fiber Polarization Controller **

In this section, a very simple model is proposed for modelling the Manual Fiber Polarization Controller. The modification of Polarization in Optical Fibers is increasingly important in communication research. The modification that suffers the state of polarization in an optical fiber as a result of the action of thermal and mechanical stresses or irregularities in the circular shape of the core was largely studied by several authors, a notable example is ~{Ulrich}. Ross’ concluded that the rotation of the polarization plane was due to geometric effects ~{Ross} and the first expression in the literature that extended this to noncoplanar curves was that of ~{RotPolarizationFrame} in this note we propose on the basis of a suggestion in {Mobius_Strip} an alternative model in this note. Let us first describe an elementary and somewhat primitive model of the MPC. \mathbf{r_{1}(t)}= (a\cos t,a\sin t,0),\; 0 < t < \pi \mathbf{r_{2}(t)}= (a\cos t,0, a\sin t),\; \pi < t < 2\pi Let us consider what happens when these two curves meet up. Let us call the point where they meet

Lets examine what happens around the point

Above we have outlined a piecewise description of the curve **r** = **once** differentiable on the **whole** domain t **r** has a discontinuity at the point

**r **”\left(\pi \right) = $latex \lim_{t \rightarrow \pi} \frac{**r**‘\left(t \right)- **r**‘\left( \pi \right)}{t-\pi} = \lim_{t \rightarrow \pi} \frac{**r**_{2}’\left(t \right)- **r**_{1}’\left( \pi \right)}{t-\pi}$

Hence we have

**r **

Numerator is non zero since the two terms do not approach each other at the point t=*infinite* at this point

\tau = \frac{\left(**r**‘ \times \mathbf{r}”\right).**r**”’}{\left|\mathbf{r}’ \times \mathbf{r}”\right|^{2}}

Then we can see this will be infinite at t =

This delta function models the change in polarization between left and right circularly polarized light. Further work could examine in more detail what happens at non right angles.

**5. Discussion and Conclusion **

We’ve discussed some of the background on the Optical Berry Phase, and provided a simple (perhaps even crude) model for a specific class of Polarization Controller. Further work could examine in more detail the limits of this model, and even examine the suggestion made in in regards whether twisted Mobius strips could be used in the modelling of Optical Control devices. We hope that this article inspires others to research the Geometric Phase. And highlights that the Optical Berry Phase isn’t a dead research area.