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.     

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.     

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.     

Berry's Phase in the Coherent Excitation of Atoms

Abstract

The adiabatic variation of a Hamiltonian can cause the wavefunction, governed by the Hamiltonian, to acquire an unexpected phase. The existence of this phase (Berry's phase) is an additional element in the well-known quantum adiabatic theorem. Berry's phase is observable in the interference between two identically prepared systems only one of which is adiabatically varied. We show that a single quantum system prepared in a superposition of the eigenstates of its Hamiltonian leads to observable Berry phase effects at all times. We examine this principle within the context of two-level optical resonance.     

Fiber Optics: Research and Development

Abstract

With twisted stacks of N polarizers P or retarders R, the polarization of a light beam can be cycled around the Poincaré sphere on N similar arcs of great P or small R circles. We calculate the phase changes around these cycles (geometric for P; geometric + dynamical for R). In the continuum limit N → ∞ of a smoothly twisting medium, a P stack forces the light to follow its changing polarization, and the phase is the solid angle of the associated loop on the sphere; for an R stack, on the other hand, it is only in the adiabatic limit of slow twist (where the dynamical phase is large) that the geometric phase corresponds to that of the loop specified by the changing eigenpolarization of the medium. The predicted phase shifts are observed as fringe shifts in an interferometer for N=2, 3 and 4.     

Polarizationally Non-sensitive Pseudo-holographic Computer Reconstruction of Random Rough Surface

Abstract

The function, describing a profile of a random rough surface (RRS) is expanded in a Fourier series, i.e. the surface is considered as a composition of sinusoidal gratings. The total diffracted optical field from this RRS is a sum of the fields due to all harmonic gratings, since Kirchhoff's condition for ‘locally flat surface’ is realized for each harmonic grating at a given light wavelength and at an appropriate choice of the basic grating period. The registered s and p components of the diffracted (+1 diffraction orders of each harmonic gratings), incident and mixed optic fields are separated with an optical analyser. These fields are experimentally measured and from these values the phase and the amplitude of each grating are determined. The profile of the surface is reconstructed for s and p polarization of the light scattered field, when the electric vector of the incident light concludes an arbitrary angle with the incidence plane. The mean roughness is determined in both cases. It is shown, that both reconstructions of the profile and the determination of the mean roughness are not dependent on the polarization of the incident light. The separation of the s and p components is of great importance at the two-dimensional reconstruction, when independent of incident light polarization (s or p), the scattered optical field is always depolarized. In this case the profile of the two-dimensional surface can be easily reconstructed with s or p component of the mixing and diffracted fields.     

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 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.     

Polarization Aberration Effects Propagated in Optical Systems

Abstract

The propagation and imaging of polarized light through optical systems described by a polarization aberration expansion is treated by combining a scalar operator calculus with the Jones calculus. The martrix-operator framework provides a means for handling diffraction and propagation in optical systems containing polarization aberrations. An expansion for the polarization aberration function of an optical system, similar to the expansion of the wavefront aberration function into defocus, tilt, piston, and higher-order terms, is analysed. These polarization aberration terms introduce phase changes in the diffraction image proportional to the first and second derivatives of the non-polarization aberrated image structure.     

Polarimetry of scattered light using heterodyne detection

Abstract

Heterodyne detection has been used to measure the polarization state of light back-scattered from various targets (including flame-sprayed aluminium, sandpaper and painted surfaces). The samples are illuminated with a linearly polarized single-frequency continuous-wave CO₂ laser operating at a wavelength of 10.6 μm. The back-scattered co-polarized and cross-polarized components are both coherently detected by beating with an optical local oscillator. This process allows the relative amplitudes and phases of the two components to be measured and hence the light's polarization state can be evaluated. When the target undergoes movement, the scattered light demonstrates the usual properties of dynamic speckle, and the technique allows observation of the time evolution of the polarization ellipse.     

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.     

Intensity-dependent Phase Shifts in Nonlinear Coupling Devices

Abstract

For nonlinear coupling devices, general solutions using Stokes parameters fail to give complete information of phase shifts, as they only provide the phase-shift difference between the two outputs of coupling waveguides (or two cores for fibre coupling devices). Thus the standard Stokes parameter formulation is not sufficient for some applications in which the nonlinear phase shifts through the devices are of great concern. The analysis given here presents complete and exact solutions for nonlinear phase shifts in optical coupling devices for the first time. This reveals unusual behaviour of the nonlinear phase shifts because, in a certain input power range, a small change of input power can bring about a large change in the nonlinear phase shift. Some basic characteristics of the nonlinear phase shifts and their potential influence on application of coupling devices to all-optic signal processing are discussed.     

Necessary and Sufficient Conditions for Thin Phase Imagery

A thin phase structure is rendered visible when the optical image-forming system produces interference between the direct light and the diffracted light. The condition required to relate explicitly some uniquely identifiable terms of the coherent transfer function of the optical system with the image irradiance variations associated with the interference terms is derived. This treatment is employed firstly to establish the necessary and sufficient conditions for rendering visible a thin phase structure, and secondly to define a unique effective transfer function to map the object's thin phase structure into image irradiance terms. Thus, thin phase imagery is represented by a linear filtering process as previously suggested by Menzel. Finally, the equivalence between Menzel's phase contrast function and the effective transfer function is indicated, as well as the advantages of employing the present symmetry-based approach.     

Achromatic phase shifting for polarization interferometers

Abstract

The use of an achromatic phase shifter with a polarization interferometer to make very accurate measurements of small optical path differences with white light is described.     

Polarimetric Study of the Phase Conjugate Wave in an Isotropic Medium

Abstract

We derive expressions for the third-order nonlinearities responsible for the optical phase conjugation via backward degenerate four-wave mixing using various combinations of left and right circularly polarized beams in isotropic media. The effects of polarization conjugation and the nulling effect of phase conjugate waves are also discussed in an isotropic medium.     

Solitary Waves of Maxwell's Equations in Nonlinear Anisotropic Media

Abstract

The (1 + 1)-D solitary wave solutions of Maxwell's equations in nonlinearity induced anisotropic media (in liquids such as carbon disulphide, and in crystals, etc.) are investigated. We find that there is no arbitrarily linearly polarized (in the x-y plane perpendicular to the propagation direction z) soliton solution from Maxwell's equations except that with linear polarization either in alignment with or orthogonal to the geometric axis of the light induced refractive index change. This contradicts the prediction of the vector nonlinear Schroedinger equation (an approximation of Maxwell's equations) which yields soliton solutions with an arbitrary linear polarization. However, Maxwell's equations are found to admit stable elliptically polarized solitary wave solutions which reduce to the stable circularly polarized solitary wave solutions of the vector nonlinear Schroedinger equation when the induced refractive index change approaches zero.     

The Phase Rotor Filter

Abstract

We report creation by photolithography techniques of the phase rotor filter, an optical element whose complex transmittance depends in a linear fashion on the azimuth angle. Relationships are given that describe the scalar diffraction of coherent light by the rotor filter. The results of the numerical simulation and experiments are discussed.     

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.     

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.