The Geometric Phase

Abstract

The change in the phase of a beam of light produced by a cyclic change in its state of polarization is an example of the geometric phase that can be verified by interferometric measurements. This paper presents a quantum-field theoretical analysis of the geometric phase interferometer in the limit of a small photon number, as well as some experimental results that confirm that the optical effects due to the geometric phase persist down to the single-photon level.     

Quantum phase measurements and non-classical polarization states of light

Abstract

In our paper we consider the non-classical behaviour of both the Hermitian (observable) Stokes parameters of light and the phase difference of two modes that describe the quantum polarization states of optical field. To characterize the degree of polarization of light we introduce a new quantity taking into account the quantum properties of different quantum states of two orthogonally polarized modes. The problem of determination of the phase difference in two modes of optical field for the quantum polarization states of light is discussed. To describe in general such a quantum field we introduce two pairs of the phase operators: the phase angles for the Stokes parameters of light in a three-dimensional picture of the Poincaré sphere. We also consider a special type of the eight-port polarization interferometer (polarimeter) for simultaneous homodyne detection of both the Stokes parameters of light and the polarization phase operators and their fluctuations as well. Using an anisotropic (spatioperiodic) Kerr-like nonlinear medium associated with the polarization interferometer we could generate and also observe the polarization-squeezed phase states of light. The fluctuations in the phase difference between two orthogonally polarized modes for these non-classical states are less than the fluctuations for light in coherent state.     

The Adiabatic Phase and Pancharatnam's Phase for Polarized Light

Abstract

In 1955 Pancharatnam showed that a cyclic change in the state of polarization of light is accompanied by a phase shift determined by the geometry of the cycle as represented on the Poincaré sphere. The phase owes its existence to the non-transitivity of Pancharatnam's connection between different states of polarization. Using the algebra of spinors and 2 × 2 Hermitian matrices, the precise relation is established between Pancharatnam's phase and the recently discovered phase change for slowly cycled quantum systems. The polarization phase is an optical analogue of the Aharonov-Bohm effect. For slow changes of polarization, the connection leading to the phase is derived from Maxwell's equations for a twisted dielectric. Pancharatnam's phase is contrasted with the phase change of circularly polarized light whose direction is cycled (e.g. when guided in a coiled optical fibre).     

Vacuum induced berry phase: Theory and experimental proposal

Abstract

We investigate quantum effects in geometric phases arising when a two-level system is interacting with a quantized electromagnetic field. When the system is adiabatically driven along a closed loop in the parameter space, signatures of the field quantization are observable in the geometric phase. We propose a feasible experiment to measure these effects in cavity QED and also analyse the semi-classical limit, recovering the usual Berry phase results.     

Analysis of a Fibre-optic Ring Resonator with Polarization Effects: Application to Polarization Sensing with Improved Sensitivity

Abstract

A theoretical analysis for a fibre-optic ring resonator is given by assuming a polarization modulation in the loop fibre. If the change in polarization angle θ in the loop is large, the output intensity has two resonance dips separated in phase by an angle equal to 2θ, when the loop phase is scanned from 0 to 2π. When θ is small, the resonator output produces only one resonance dip and the amplitude of this resonance dip is a measure of θ. By placing a polarizer at the resonator output, a resonance peak in the intensity is produced with an amplitude that increases with increasing θ. Such a system has potential applications, for example, in Faraday current sensing, with an increased sensitivity. The effects of birefringence in the loop and the angle of polarization of the input light are also analysed.     

Pancharatnam's Phase for Polarized Light

Abstract

This paper presents a classical theoretical analysis of the geometric phase arising in a cyclic change of the polarization state of light beams. We compare the results obtained this way with the quantum mechanical analysis.     

An online triple-frequency color-encoded fringe projection profilometry for discontinuous object

Abstract

An online triple-frequency color-encoded fringe projection profilometry is proposed to measure the complex and discontinuous object at straight-line movement. N frames of color fringe patterns are specially designed. Three grayscale sinusoidal grating patterns with geometric progression frequency growth are encoded into red (R), green (G), and blue (B) channels separately to compose a color-encoded fringe pattern. If these three grayscale sinusoidal grating patterns are phase-shifted N steps with an equivalent shift phase of 2π/N, they can compose the corresponding N frames of color-encoded fringe patterns as above respectively. In order to avoid the movement’s interference to the phase shifting, position adjustment should be done to guarantee the phase-shifting direction to be perpendicular to the moving direction. While these N frames of specially designed color-encoded fringe patterns are projected onto the moving object one by one, the corresponding deformed color patterns are captured by a CCD camera in real time. By color separating, color crosstalk compensation, pixel matching, and phase calculation, three wrapped phase at different frequencies can be extracted. The unwrapped phase can be solved by a simplified algorithm based on temporal phase unwrapping method from the relationship of the three wrapped phase at the same pixel. Thus, it is very suitable to measure the online complex and discontinuous objects at straight-line movement. The experimental results show the feasibility and the validity of the proposed method.     

Polarization-independent Beam Splitter Based on Light Reflection by a Uniaxial Crystal Surface,and an Immersion Method for Determining the Crystal's Principal Refractive Indices

Abstract

Light travelling in an isotropic medium of refractive index n and incident on a uniaxial crystal whose optic axis is parallel to the surface and to the plane of incidence is reflected without a change of polarization when n = N _g = (N _o N _e)^1/2, where N _o and N _e are the crystal's ordinary and extraordinary refractive indices. This is true for all incident polarization states and at all angles of incidence and can be used to design a new polarization-independent beam splitter. For a positive uniaxial crystal (N _e > N _o), total internal reflection occurs at and above a critical angle equal to arcsin (N _o/N _e)^1/2, so that the incident light beam is deflected without attenuation or change of polarization. When n = N _g the reflectance at normal incidence for unpolarized or circularly polarized incident light is a minimum: R _0min = (N_a - N_g)/(N_a + N_g), where N _a = ½(N _o + N _e). This suggests a liquid immersion method in which n and R _0min determine N _g and N _a, hence N _o and N _e of the crystal.     

Geometric approach to the degree of polarization for arbitrary fields

Abstract

The eigenvalue decomposition of the polarization matrix is employed to find out the geometric interpretation of the traditional degree of polarization and, in particular, of the degree of polarization for arbitrary electromagnetic fields put forward recently by Setälä et al. [2002, Phys. Rev. Lett., 88, 123902]. It is shown that both measures have similar geometric meaning as a measure for the purity of the polarization state. Possible extensions to the analysis are discussed.     

The Geometric Phase in Optical Rotation

Abstract

The geometric phase arising from rotation of the plane of polarization of an electromagnetic wave is similar to that acquired by a spin-1/2 particle in a magnetic field. We describe an interferometric arrangement using an optically active medium that can be used to observe this change.     

A scheme for the simultaneous measurement of atomic position and momentum

Abstract

We propose an optical scheme for the simultaneous measurement of the position and momentum of a single atom. The scheme involves the coupling of the atom of two light fields with different spatial and polarization characteristics. The proposed technique is closely related to the Arthurs-Kelly measurement scheme; the principal difference is that in the present case the values of the position and momentum are inferred from phase shifts in electromagnetic fields rather than from shifts in the position of a pointer.     

Fast,accurate measurement of path difference with white light

Abstract

An interferometric system is described for measuring phase distributions with white light at a speed limited by the CCD frame rate using a simple phase shifting algorithm where the phase is recovered from quadrature fringe sets. The technique used is almost immune to vibration, and highly accurate. A phase measurement system is implemented using a polarizing Michelson-type interferometer together with geometric phase modulation. The measured phase distributions enabled the recovery of path differences with an accuracy of better than about λ _m /70 where λ _m is the mean wavelength.     

Geometric phase in optics and angular momentum of light

Abstract

The Physical mechanism of the geometric phase in terms of angular momentum exchange is elucidated. It is argued that the geometric phase arising out of the cyclic changes in the transverse mode space of Gaussian light beams is a manifestation of the cycles in the momentum space of the light. The apparent non-conservation of orbital angular momentum in the spontaneous parametric down conversion for the classical light beams is proposed to be related to the geometric phase.     

An experimental test of the path dependency of Pancharatnam's geometric phase

Abstract

In crystal optics the concept of the geometric phases can be demonstrated to the naked eye with comparatively simple apparatus. Pancharatnam's phase is described by an area on the Poincaré sphere which is enclosed by geodesic lines (great circles) while the concept of the Berry phase does not restrict the nature of the contour. With a simple experiment we demonstrate that Pancharatnam's phase is accompanied by a second phase which compensates the pure area effect for areas enclosed by small circles. This compensating phase changes sign at time reversal and, thus, exhibits the features of a geometric phase.     

Macroscopic jumps of the axial angular momentum variance of a bidimensionally trapped ion

Abstract

The time evolution of the axial angular momentum [Lcirc] _z of an ion confined in a bidimensional trap is investigated. We find that, under suitable initial conditions, the interaction of the ion with two properly configured classical laser beams induces a peculiar dynamical behaviour of the axial angular momentum fluctuations. We show, in fact, that there exists an instant of time at which the variance of [Lcirc] _z undergoes variations proportional to N ² further to a change of one quantum only in the initial total number N ? 1 of vibrational quanta. The non-classical origin of these macroscopic jumps is brought to the light and carefully discussed.     

A simple demonstration of the Pancharatnam phase as a geometric phase

Abstract

To demonstrate the Pancharatnam phase as a geometric (Berry's) phase, each polarization state must be obtained by projecting the previous state on it. We describe a simple interferometric arrangement for such a demonstration which only uses a single rotation linear analyser to introduce a continuously variable phase difference between the two beams.     

Phase Properties of Elliptically Polarized Light Propagating in a Kerr Medium

Abstract

Phase properties of elliptically polarized light propagating through a nonlinear Kerr medium are considered within the framework of the Pegg-Barnett Hermitian phase formalism. The joint phase probability distribution function for the phases of two orthogonal modes describing elliptical polarization of the field is calculated and its evolution discussed and illustrated graphically. The marginal phase probability distribution for the individual phases are also calculated and discussed. Analytical formulae for phase expectation values and variances are derived for the individual phases as well as for the phase difference. It is shown that in the course of propagation the correlation between the phases of the two modes builds up. This correlation is responsible for lowering phase difference variance. The expressions for the sine and cosine functions and their variances of the individual phases as well as the phase difference are obtained and discussed. The effect of randomization of individual phases and the phase difference, which is a purely quantum effect, is shown to appear during propagation. The relation between phase randomization and degradation of the degree of polarization of the light is established.     

A note on the measurement of phase space observables with an eight-port homodyne detector

It is well known that the Husimi Q-function of the signal field can actually be measured by the eight-port homodyne detection technique, provided that the reference beam (used for homodyne detection) is a very strong coherent field so that it can be treated classically [see e.g. Leonhardt, U.; Paul, H. Phys. Rev. A 1993, 47, R2460–R2463]. Using recent rigorous results on the quantum theory of homodyne detection observables [Kiukas, J.; Lahti, P. J. Mod. Opt., in press (see arXiv:0706.4436v1 [quant-ph])], we show that any phase space observable, and not only the Q-function, can be obtained as a high amplitude limit of the signal observable actually measured by an eight-port homodyne detector. The proof of this fact does not involve any classicality assumption.