Composite stokes singularities in coherently and incoherently superimposed vector optical fields

The composite Stokes singularities formed in the coherent and incoherent superposition of two vector monochromatic coherent optical fields are studied and compared analytically and numerically. It is shown that the dynamic behavior of composite Stokes singularities depends on the superposition scheme (coherent or incoherent superposition), initial spatial distributions of the two superimposed fields, and propagation distance, and for the coherent superposition additionally depends on the relative phase of the two fields. There exist S₁₂ (C-points), S₃₁, S₂₃ composite singularities, and L-lines, in particular, V-points and C-lines at the transverse plane. Generally, both V-points and C-lines are unstable. For the coherent superposition the degree of polarization P=1, whereas for the incoherent superposition P may be less than 1 and is variable upon propagation. The results are illustrated analytically and numerically.     

Singularities in Weyl gravitational fields

The peculiarities of the scalarS ≡R _ijkl R ^ijkl are exhibited for two axially-symmetric static (Weyl) gravitational fields. By examiningS along curved families of trajectories to the Weyl singularities, examples are found which contradict previous claims by Gautreau and Anderson regarding ‘directional singularities’. Proper circumferences about the Bach and Weyl line-mass singularity are also examined. There is no apparent correlation between the source structure and the behaviour ofS from this analysis.     

Vector singularities of superposition of mutually incoherent orthogonally polarized beams

It is shown that, at an incoherent superposition of orthogonally polarized laser beams, a special type of singularities are formed in the cross section of a combined beam in place of the well-known singularities, such as optical vortices (for scalar fields); C points, at which the polarization is circular; and L lines, along which the polarization is linear (for coherent vector fields). These new singularities are U lines, along which the degree of polarization is zero and the state of polarization is undetermined, and P points, at which the degree of polarization is equal to unity and the state of polarization is determined by the nonzero component of the combined beam. Conditions of topological stability of U and P singularities are discussed, as well as peculiarities of the spatial distribution of the degree of polarization of the field in the vicinity of such singularities. First experimental results on the reconstruction of a vector skeleton formed by U and P singularities in combined speckle fields are presented.     

Singularities in speckled speckle: Statistics

Random optical fields with two widely different correlation lengths generate far field speckle spots that are themselves highly speckled. We call such patterns speckled speckle, and study their critical points (singularities and stationary points) using analytical theory and computer simulations. We find anomalous spatial arrangements of the critical points and orders of magnitude anomalies in their relative number densities, and in the densities of the associated zero crossings.     

Vierbein singularities in CP2

The compact open four-dimensional manifold $\text{C}$P², with Euclidean metric, has recently attracted attention as an example of a spacetime in which fields of half-integral spin cannot be defined in the absence of additional structure (such as an electromagnetic background). In this note we identify the specific topological anomaly responsible for this phenomenon: $\text{C}$P² contains a class of two-dimensional spheres—“complex lines”— in small neighborhoods of each of which the two transverse degrees of freedom are forced to “twist” in a characteristic way. It is shown in detail how the twists force vierbeins—tetrads of orthonormal vector fields that play a central role in the theory of spinors—to have singularities on every complex line. As an aid to visualization we construct an example of a vierbein with, loosely speaking, the smallest possible set of singularities: It is ill-defined at every point of some one complex line and smooth everywhere else. The behavior of such a vierbein near any one of its singular points is characterized explicitly. The structure of the minimally singular vierbein is used to illuminate the observation of Hawking and Pope that in the presence of an appropriate electromagnetic background, fields of any spin can exist on $\text{C}$P² as long as their electric charges are correctly quantized, but that the charge values available to half-integral-spin fields differ from those available to integral-spin fields.     

Polarization singularities in partially coherent combined beams

Vector singularities are predicted and discovered experimentally in partially polarized combined fields formed by incoherent superposition of orthogonally polarized beams. Such singularities are U contours with zero degree of polarization and isolated P points with unit degree of polarization centered at vortices of the orthogonally polarized component of the combined beam. Crossing a U contour switches the polarization state to the orthogonal one. The above-mentioned singularities are adequately described in terms of the complex degree of polarization in the Stokes-space representation. It is shown that the field elements corresponding to the extrema of the complex degree of polarization form the vector skeleton of a partially coherent nonuniformly polarized field.     

Polarization singularities in Young's two-slit experiment

Young's interference experiment is regarded as a two-slit diffraction phenomenon, the polarization singularities in Young's two-slit configuration illuminated with two linearly, orthogonally polarized Gaussian vortex beams are studied. It is shown that generally, there exist L-lines (linearly polarization) and polarization singularities including C-points (circular polarization), S₂₃ and S₃₁ singularities even though the parameters of two beams are the same. The pair creation-annihilation and motion of polarization singularities take place upon propagation, or by varying a control parameter, such as the amplitude ratio of two beams or obscure ratio of slits etc. For a special case of the illumination with two linearly, orthogonally polarized Gaussian vortex-free beams, polarization singularities, in particular, C-points may occur if a parameter of two beams is not equal.     

String-like topological excitations of the electromagnetic field

In order to analyze the topological properties of an arbitrary configuration of the electromagnetic field, its strength F_μν is expressed in terms of new auxiliary fields which replace the gauge potential A_μ. These new fields have only physical singularities even in the presence of monopoles (no Dirac strings) and exhibit a new local O(1, 1) symmetry which replaces the gauge invariance. Boundary conditions on these fields may induce localized string-like singularities or topological defects which act as sources of magnetic field. A typical defect which emerges is a sort of tadpole formed by a non-quantized monopole attached to one or more magnetic strings of finite length. For topological reasons the total magnetic charge is quantized.     

Radio-frequency control of spontaneous emission from a coherently driven multilevel atom

The spontaneous emission properties of radio-frequency(RF)-driven multilevel atomic systems with three different configurations are investigated in details. It is demonstrated that, due to the presence of the RF-induced coherence in the multilevel system, a few interesting phenomena such as spectral-line narrowing, spectral-line enhancement, spectral-line suppression, and fluorescence-quenching can be realized under realistic experimental conditions. By inspecting single-RF-driven, double-RF-driven, and triple-RF-driven atomic systems, we find that (i) when only one RF-driven field is applied, there are two fluorescence-quenching points and four spectral lines; (ii) in the case of applying two RF-driven fields, there are three fluorescence-quenching points and five spectral lines; (iii) when applying three RF-driven fields, there are four fluorescence-quenching points and six spectral lines; and (iv) as expected, when the atomic system coupled by N RF-driven fields, there will be N + 1 fluorescence-quenching points and N + 3 spectral lines. Interestingly enough, the spectral-line enhancement, the spectral-line suppression, and the selective cancellation of fluorescence-quenching can be well controlled just by appropriately modulating the intensities and frequencies of the applied RF fields, respectively. The proposed schemes can be achieved by use of RF-driven fields into hyperfine levels in rubidium atomic systems. These investigations may find applications in high-precision spectroscopy.     

Composite spectral Stokes singularities in coherent and incoherent superpositions of partially coherent nonparaxial vortex beams

Taking the partially coherent cosh-Gaussian (ChG) vortex beam as an illustrative example, the composite spectral Stokes singularities formed by coherent and incoherent superpositions of partially coherent vortex beams beyond the paraxial approximation are studied, where the effect of superposition scheme on composite spectral Stokes singularities is stressed. It is shown that there exist s₁₂ (C-points), s₂₃ and s₃₁ composite spectral Stokes singularities, which are variable by varying a control parameter, such as the spatial correlation length, waist width, off-axis distance, or decentered parameter, as well as the propagation distance. In particular, the number, position, degree of polarization of composite spectral Stokes singularities and the critical point at which the creation–annihilation process takes place depend on the superposition scheme. A comparison with the previous work is also made.     

Singularities in speckled speckle

Speckle patterns produced by random optical fields with two (or more) widely different correlation lengths exhibit speckle spots that are themselves highly speckled. Using computer simulations and analytic theory we present results for the point singularities of speckled speckle fields, namely, optical vortices in scalar (one polarization component) fields and C points in vector (two polarization components) fields. In single correlation length fields both types of singularities tend to be more or less uniformly distributed. In contrast, the singularity structure of speckled speckle is anomalous; for some sets of source parameters vortices and C points tend to form widely separated giant clusters, for other parameter sets these singularities tend to form chains that surround large empty regions. The critical point statistics of speckled speckle is also anomalous. In scalar (vector) single correlation length fields phase (azimuthal) extrema are always outnumbered by vortices (C points). In contrast, in speckled speckle fields, phase extrema can outnumber vortices and azimuthal extrema can outnumber C points by factors that can easily exceed 10(4) for experimentally realistic source parameters.     

Short- and long-range screening of optical phase singularities and C points

Screening of signed (charged) singularities-phase vortices in scalar fields, C points in vector fields, is discussed for paraxial optical fields with short- and long-range correlations. A circular region of radius R is assumed. Short-range screening is exemplified by a Gaussian field correlator, long-range screening by a J₀ Bessel function. The short-range screening length is obtained analytically; this is found to be in substantial agreement with recent experiments. For long-range screening, an accurate asymptotic formula suitable for quantitative comparison with data (numerical or laboratory) is derived for the variance of the net charge. A J₀ correlation function is not attainable in practice, but it is shown how to generate a pseudo-long-range optical field whose correlation function closely approximates this form; screening in such a field is well described by our theoretical results for J₀. The charge variance can be measured by three different methods: by counting positive and negative singularities inside the region of interest, by counting signed zero crossings on the perimeter of this region; or by measuring phase derivatives along the perimeter. For the first method, the charge variance is calculated by integration over the charge correlation function, for the second (third) by integration over the zero crossing (phase derivative) correlation function. It is proven explicitly that, as expected, all three calculations yield the same result. It is also shown analytically that for short-range screening the zero crossings can be counted along a straight line whose length is 2πR, but that for long-range screening this useful simplification no longer holds; for this case another formula is given that is suitable for data correction. The effects of boundary smoothing are discussed, and a class of generalized exponential smoothing functions is introduced. Analytical (numerical) results are given for the large R limit of the charge variance for the short (long) range case. Finally, it is shown that for realizable optical fields, both for the short and pseudo-long-range cases, for sufficiently small R the charge variance grows as R², whereas for sufficiently large R it grows as R.     

Continuous matter fields in Regge calculus

We find that the continuous matter fields are ill-defined in Regge calculus in the physical 4D theory since the corresponding effective action has infinite terms unremovable by the UV renormalisation procedure. These terms are connected with the singular nature of the curvature distribution in Regge calculus, namely, with the presence in d>2 dimensions of the (d−3)-dimensional simplices where the (d−2)-dimensional ones carrying different conical singularities are meeting. Possible resolution of this difficulty is discretisation of matter fields in Regge background.     

Non-local specific heat — Ordered-phase results

The energy-energy correlation function C(k) is calculated for general n-component spin systems in a resummed one-loop approximation. The Goldstone mode singularities are resummed in a manner consistent with exactly known limits.     

Remarks concerning the connection between properties of the 4-point-function and the Wilson-Zimmermann expansion

This paper concerns an investigation of the Wilson-Zimmermann (or “short distance”) expansion forA(x)A(y) withx →y whereA(x) is a real scalar field fulfilling Wightman's axioms. If one assumes that such an expansion exists, where the terms of the expansion are operators relatively local toA(x), then the singularities arising in the 4-point-function forx ₃ →x ₄ must control the singularities of then-point functions (n=4, 5, 6, ...) arising forx _j →x _x+1,j=1,2,...,n?1. A similar consequence can be drawn if the terms of the expansion are assumed to exist only as bilinear-forms (Section 2). For certain classes of fields one can show that this condition necessary for the short distance expansion is indeed fulfilled (Section 3). The result of the last section is that the above mentioned condition is also sufficient for the Wilson-Zimmermann expansion, interpreted as an expansion into bilinear forms, and also as an operator expansion in a somewhat modified sense.     

87Rb NMR lineshape study of the incommensurate phase in Rb2ZnBr4

On going from the paraelectric (P) to the incommensurate (I) phase the ⁸⁷Rb $\text{1}\text{2}$ → $\text{1}\text{2}$ NMR lines in Rb₂ZnBr₄ broaden into a continuum which is limited by two edge singularities. The observed line shape and its temperature dependence are well described by the frequency distribution function predicted by the “plane wave” modulation model of the I phase. The anomalous temperature dependence of T₁ shows the presence of the phason branch in the I phase.     

Spatial dependence of correlation functions in the decay problem for a passive scalar in a large-scale velocity field

Statistical characteristics of a passive scalar advected by a turbulent velocity field are considered in the decay problem with a low scalar diffusivity κ (large Prandtl number v/κ, where v is kinematic viscosity). A regime in which the scalar correlation length remains smaller than the velocity correlation length is analyzed. The equal-time correlation functions of the scalar field are found to vary according to power laws and have angular singularities reflecting locally layered distribution of the scalar in space.