Table of contents

This collection of papers arose from an Advanced Research Workshop on Singular Optics, held at the Bogolyubov Institute in Kiev, Ukraine, during 24–28 June 2003. The workshop was generously financed by NATO, with welcome additional support from Institute of Physics Publishing and the National Academy of Sciences of Ukraine. There had been two previous international meetings devoted to singular optics, in Crimea in 1997 and 2000, reflecting the strong involvement of former Soviet Union countries in this research. Awareness of singular optics is growing within the wider optics community, indicated by symposia on the subject at several general optics meetings.

As the papers demonstrate, the field of singular optics has reached maturity. Although the subject originated in an observation on ultrasound, it has been largely theory-driven until recently. Now, however, there is close contact between theory and experiment, and we speculate that this is one reason for its accelerated development.

To single out particular papers for mention here would be invidious, and since the papers speak for themselves it is not necessary to describe them all. Instead, we will confine ourselves to a brief description of the main areas included in singular optics, to illustrate the broad scope of the subject.

Optical vortices are lines of phase singularity: nodal lines where the intensity of the light, represented by a complex scalar field, vanishes. The subject has emerged from flatland, where the vortices are points characterized by topological charges, into the much richer world of vortex lines in three dimensions. By combining Laguerre–Gauss or Bessel beams, or reflecting light from plates with spiral steps, intricate arrangements can be generated, with vortices that are curved, looped, knotted, linked or braided.

With light whose state of polarization varies with position, different singularities occur, associated with the vector nature of light. These are also lines, on which the electric (or magnetic) polarization ellipse is purely circular (C lines) or purely linear (L lines). The patterns of ellipse-fields are different for purely paraxial and fully three-dimensional fields.

White-light diffraction generates richly coloured vortices—the colours of dark light. The description of these chromatic effects, and also those associated with polarization singularities, leads to new applications of coherence theory.

For non-monochromatic light, it is natural to seek singularities of the full electromagnetic field, rather than of the electric or magnetic field separately. Such electromagnetic singularities are the Riemann–Silberstein vortices; these are relativistically covariant nodal lines of a complex scalar field constructed from the electromagnetic field.

Optical fields have dynamical aspects, particularly those associated with angular momentum. Although angular momentum is not inevitably associated with optical singularities, in practice the two phenomena can occur together. Orbital angular momentum is associated with the spatial structure of light, and in beams with optical vortices it can be used to rotate particles in the field. Spin angular momentum is associated with the polarization structure of the light. There are tricky questions associated with the angular momentum of light in a refracting medium, echoing the Abraham–Minkowski controversy about linear momentum.

In optically nonlinear materials (leading to second-harmonic generation, for example), new classes of phenomena can occur, such as, for example, dynamical interaction between vortex lines, whose stability needs to be considered.

At a more fundamental level, it is important to investigate quantum effects associated with optical singularities, and a start has been made. The dark centre of an optical vortex can be regarded as a window onto the vacuum fluctuations of quantum optics, with the quantum core emerging as a distinct entity when the classical light is intense. And for light in a rapidly and inhomogeneously flowing material, horizons can develop, analogous to those surrounding black holes in general relativity, and these new optical singularities can be regarded as wave catastrophes, and new associated quantum effects anticipated.

Three decades after wave dislocations were introduced as 'a new concept in ... wave theory', these phase singularities have been extensively explored and are now familiar. New ideas—in addition to those described in this special issue—continue to emerge. For example, x-ray vortices were observed recently; there is a proposal to create lenses to form atomic beams containing vortices; and astrophysical applications have been suggested for both photon orbital angular momentum and optical vortices. We can safely assume that the science of wave singularities will develop further, and diffuse into new areas of physics.

A unity of Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) beam families is proposed by introducing an additional parameter. Continuous changing of the introduced parameter allows one to transform HG beams into LG beams in a continuous way, keeping some important properties of both families, for example, structural stability under propagation. The generalized beams (called Hermite–Laguerre–Gaussian beams) are investigated by theoretical and experimental means.

We investigate the l = 1 mode structure of a weakly guiding strongly anisotropic optical fibre, in which the anisotropy axis uniformly rotates in a transverse plane with z increasing. An analytical solution of the vector wave equation for this problem is presented, based on the effective reduction of a twisted fibre equation to a straight fibre one. It is demonstrated that at certain values of the anisotropy pitch in the rotating coordinate frame, rigidly connected with the anisotropy axes, the l = 1 modes of such fibres are presented by almost pure linear optical vortices. Polarization corrections to the propagation constant are found.

We present an elaborated holographic technique for synthesis of optical beams possessing optical vortices with fractional topological charge. The technique is based on the production of sub-harmonic diffraction orders. The diffraction orders corresponding to the fundamental period have embedded vortices with integer topological charge and sub-harmonic diffraction orders have optical vortices with fractional topological charge. An analytical approach for charge-1/2 optical-vortex beam structure description is presented.

With an example of the Laguerre–Gaussian LG₀₁ mode we show that the free-space evolution of a vortex light beam can be interpreted as the mechanical motion of a 'body' whose density is determined by the mass equivalent of the beam energy. As mechanical attributes, this 'body' possesses angular velocity and moment of inertia, in addition to the common orbital angular momentum. The beam 'body' is not rigid and its radial layers glide with respect to each other analogous to the motion occurring in fluid vortices. Absolute values of the layer velocities agree with the energy transfer rates following from the consideration of the Poynting vector distribution.

Stationary lines of phase singularity for monochromatic electromagnetic fields can be defined as the zeros of a complex scalar field Ψ(r) constructed from the real fields E_real+i B_real. In terms of the Riemann–Silberstein vector F = E_real+i B_real, Ψ(r) is the time average of . For paraxial waves, Ψ(r) can be specified entirely in terms of the transverse complex electric (or magnetic) field and its spatial derivatives. The accuracy of the paraxial theory is illustrated numerically.

The nodal line singularities (optical vortices) of classical scalar optics are smoothed in quantum optics, because of spontaneous emission into unoccupied modes. The radius of the 'quantum core' surrounding each classical singularity is proportional to . A trapped excited atom, steered into a nodal line of the classical field, is a possible detector for the effect. Analogous phenomena are anticipated for other waves, for example sound, where the silence at a nodal line is disturbed by pressure fluctuations of the fluid molecules.

A new method of introducing vortex lines of the electromagnetic field is outlined. The vortex lines arise when a complex Riemann–Silberstein vector is multiplied by a complex scalar function ϕ. Such a multiplication may lead to new solutions of the Maxwell equations only when the electromagnetic field is null, i.e. when both relativistic invariants vanish. In general, zeros of the ϕ function give rise to electromagnetic vortices. The description of these vortices benefits from the ideas of Penrose, Robinson and Trautman developed in general relativity.

In recent years, singular and nondiffracting optics have been established as independent fast developing fields of modern optics. The purpose of the paper is to show that their connection can result in new prospective effects and experimental methods useful for both theoretical and applied research.

We will present experimental and theoretical studies of optical fields with subwavelength structures, in particular phase singularities and coherent detection methods with nanometric resolution. An electromagnetic field is characterized by an amplitude, a phase and a polarization state. Therefore, experimental studies require coherent detection methods, which allow one to measure the amplitude and phase of the optical field with subwavelength resolution. We will present two instruments, a heterodyne scanning probe microscope (heterodyne SNOM) and a high resolution interference microscope (HRIM). We will review some earlier work using the heterodyne SNOM, in particular the measurement of phase singularities produced by a 1 µm pitch grating with 10 nm spatial sampling. Using the HRIM we have investigated the intensity and phase distributions (with singularities) in the focal region of microlenses. The measurements are compared with the results calculated by rigorous diffraction theory.

We investigate analytically and numerically the stability of multiply-charged two-dimensional bright vortex solitons in media with focusing cubic and defocusing quintic nonlinearities. A vortex soliton becomes robust with respect to symmetry-breaking azimuthal perturbations in the self-defocusing regime above some critical beam power when its radial profile flattens. A stable high-power vortex has nearly homogeneous energy distribution across the beam with a rather sharp boundary. The dynamics of a slightly perturbed stable vortex soliton is found to be similar to oscillations of a liquid stream having a surface tension. We propose an explanation of stabilization of vortex solitons in media with competing nonlinearities based on the idea of sustaining effective surface tension.

Generic wave dislocations (phase singularities, optical vortices) in three dimensions have anisotropic local structure, which is analysed, with emphasis on the twist of surfaces of equal phase along the singular line, and the rotation of the local anisotropy ellipse (twirl). Various measures of twist and twirl are compared in specific examples, and a theorem is found relating the (quantized) topological twist and twirl for a closed dislocation loop with the anisotropy C line index threading the loop.

We describe different types of self-trapped optical beam carrying phase dislocations, including vortex solitons and ring-like soliton clusters. We demonstrate numerically how to create such nonlinear singular beams from the interaction of several fundamental optical solitons. Mutual trapping of several solitons can be regarded as a synthesis of 'soliton molecules', and it corresponds to a transfer of an initial orbital angular momentum of a system of solitons to a spin momentum of an optical vortex.

Experimental results of self-focusing and stimulated Raman scattering beam break-up into several filaments were observed under 20 ns, 0.3 cm⁻¹ spectral linewidth, 532 nm laser irradiation of nitrobenzene. Filament development in a self-focused beam, as a result of azimuthal-angle instability, was discussed.

The conversion of singular beams in a uniaxial crystal and their fine structure were experimentally studied. It is shown that the given system is able to generate singular beams in a wide spectral range. The fine structure of the 'white' singular beam has three levels: the polarization structure, the structure of the topological charge and the structure of the colours. It is revealed that the 'white' vortex can be defined by means of three experimentally measurable values: the orbital parameter S₃^L (or the ellipticity of the vortex core Q), the probability to meet such a vortex state W and the noise of the vortex core P (or the polarization degree in the given space point). The mathematical formalism for describing the vortex parameter on the complex plane and its stereographic projection is worked out. The independent characteristic of the 'white' optical vortex is also its colour gamut near the phase singularity. The degenerate and perturbed 'white' vortices are created. It is shown that the vector singularity in the 'white' beam does not have colouring, the vector singularity having gained colouring only after passing through the polarization filter.

The polarization figures of strictly monochromatic, elliptically polarized light are true ellipses only at particular points in time; at other times, and in other fields, the figures differ from ellipses, and can often exhibit a bewildering variety of complex shapes. A modified real coherency matrix of the field, M, is shown to provide a convenient representation of the polarization figures in arbitrary monochromatic, and arbitrary polychromatic, two dimensional, and three dimensional, not necessarily stationary, electromagnetic fields that are observed at arbitrary times for arbitrary intervals. The eigenvalues of M permit a simple classification of the polarization figures, and their singularities; the eigenvectors of M define the important line of sight down which the singularities are properly viewed; the zeros of the discriminant of the characteristic equation of M track the path of the singularities through space. Instructive examples are discussed that illustrate the utility of this particular matrix approach to optical polarization.

Propagation through a distorting obstacle may significantly influence the amplitude, phase and polarization state of a light beam. This potentially has consequences for the behaviour of the optical angular momentum of light. We experimentally study how both the spin and orbital angular momentum (OAM) of light behaves upon passage through microscopic optically trapped particles. Particles trapped with Gaussian and, separately, Bessel light beams in two spatially distinct sample chambers are studied with trapped objects in the first chamber acting as distorting obstacles. The Bessel beam can reconstruct its spatial form and this shows reconstruction of both spin and OAM over extended distances.

It is well known that light fields which are partially coherent and/or polychromatic do not typically possess regions of zero intensity and hence do not possess any obvious phase singularities. It is of interest to ask whether or not such fields possess singularities in some 'hidden' form, and in this paper we discuss the singular optics of partially coherent fields and the nature of the singularities in such fields.

Riemann–Silberstein (RS) vortices have been defined as surfaces in spacetime where the complex form of a free electromagnetic field given by F = E+i B is null () and they can indeed be interpreted as the collective history swept out by moving vortex lines of the field. Formally, the nullity condition is similar to the definition of C-lines associated with a monochromatic electric or magnetic field, which are curves in space where the polarization ellipses degenerate into circles. However, it was noted that RS vortices of monochromatic fields generally oscillate at optical frequencies and are therefore unobservable, while electric and magnetic C-lines are steady. Here I show that, under the additional assumption of having definite helicity, RS vortices are not only steady but they coincide with both sets of C-lines, electric and magnetic. The two concepts therefore become one for waves of definite frequency and helicity. Since the definition of RS vortices is relativistically invariant while that of C-lines is not, it may be useful to regard the vortices as a wideband generalization of C-lines for waves of definite helicity.

Wave catastrophes are characterized by logarithmic phase singularities. Examples are light at the horizon of a black hole, sound in transsonic fluids, waves in accelerated frames, light in singular dielectrics and slow light close to a zero of the group velocity. We show that the wave amplitude grows with a half-integer power for monodirectional and symmetric wave catastrophes.

The free propagation of a paraxial light beam can be exactly mapped on the free evolution of a 2D harmonic oscillator over half an oscillation period. We apply this mapping to give an analytical description of the dynamics of vortices and their relation to the orbital angular momentum of light.

When a pair of wave dislocations is created in two dimensions, or a ring of dislocation in three dimensions, higher-order local solutions are needed to exhibit the full details of the local phase pattern. The same is true for the hyperbolic interchange (reconnection) event. The contours of equal phase may or may not show local saddle points.

The vorticity of a free propagating second harmonic (SH) beam produced in a nonlinear quadratic crystal by a combined beam composed of two coaxial Laguerre–Gaussian vortex beams is analysed. In the near field of the SH radiation an aligned vortical structure with a number of double-charge vortices is created. The diffraction of the SH beam under free propagation is also discussed.

The general case of coaxial superposition of two Bessel singular beams with different radial frequencies and apertures under free space propagation is analysed. We reveal that in some cases the total topological charge of the field is not conserved under diffraction. The vortical structure of the combined beam is also discussed.

It is well known that when a light beam containing angular momentum is rotated about the propagation axis it is shifted in frequency. The magnitude of the shift is equal to the angular momentum per photon, in units of , multiplied by the rotation frequency of the beam. Here it is shown that the associated energy transfer can be understood in terms of ray optics and the torque acting on the optical component inducing the rotation.

We investigate the electric field distribution of optical dissipative solitons having a triangular symmetry. A rich structure of phase singularities is found. We study the propagative dynamics of these vortices, and discuss the possibility of controlling their motion via proper parameters.

Phase singularities appear in the diffracted far-field of an optical micro-structure if the object is of a sufficient lateral size to induce an appropriate phase delay. We present results that determine the critical dimension of a single phase bar for the generation of dislocations in the far-field. Using scalar theory, an analytical equality is derived that must be met by the structure. Because the size of the object is comparable to the wavelength, rigorous diffraction theory is used to find this feature size for a true object. Once the dislocations appear, their position is related strongly to the geometry of the object. We show theoretically and experimentally by using an interference microscope that, for a single phase bar, the distance between pairwise generated dislocations depends, to a good approximation, linearly on the width of the structure.

We analyse the diffraction of light incident on a sub-wavelength slit in a thin plate. It is found that plates with different material properties, such as conductivity and thickness, show a fundamentally different behaviour of the field near the slit. Depending on the material properties, the light transmission can either be enhanced or frustrated. A correspondence between the handedness of optical vortices and the transmission behaviour is demonstrated.

Structural units of singular paraxial ellipse fields, and their self-organization to form complementary topological networks of azimuth and eccentricity, are considered. The recently established technique of singular Stokes polarimetry is described and its capabilities are discussed. The azimuthal and eccentricity networks of generic paraxial ellipse speckle-fields were measured for the first time. Statistics of structural units and extreme are defined and compared quantitatively with theoretical predictions. Their essential difference allows us to conclude that generated speckle-fields generally possess non-Gaussian statistics. Applications of singular Stokes polarimetry as a new tool for the comprehensive characterization of ellipse fields of arbitrary complexity are discussed and demonstrated.

We have fabricated high-quality, half-integral spiral phase plates for generating optical vortices at visible and near-infrared wavelengths. When inserted in the waist of a fundamental Gaussian beam, such a device gives rise to a rich vortex structure in the far field. The near-perfect cancellation of the effect induced by two nominally identical phase plates shows that we have excellent control of the manufacturing process.

For superpositions of electromagnetic plane waves in space whose directions span a small angular range θ, the circular polarization line singularities (of the full and transverse electric and magnetic fields) form clusters, each of four closely-spaced lines. The separations of lines in each cluster are calculated analytically in terms of the transverse electric field and are of the order of λθ—much smaller than the wavelength λ and smaller still than the transverse scale λ/θ of the intensity variations of the field. The electric and magnetic surfaces of transverse linear polarization also lie close together and the separation is calculated analytically, as are the positions on these surfaces of the lines of linear polarization of the full fields. To the two lowest orders in θ, the local wavevector (geometric phase 1-form) is the same for the electric and magnetic fields. The sub-wavelength singular structures are illustrated by numerical calculations.

A Laguerre–Gaussian (LG) beam possesses a uniform orbital angular momentum of due to its helical phase term exp(−i lϕ), where l is the topological charge of the beam. The helical phase or the topological charge of any LG beam can be observed by causing it to interfere with a Gaussian beam or its mirror image. In this paper we propose a novel method of using moiré fringes to study the changes in the helical phase and the topological charge upon interference between any two LG beams of any arbitrary helicity. Using this method we are able to observe the interference purely between the helical phases which can be readily observed using white light illumination as shown in our experimental results. Such pure helical phase interference is not readily observed in conventional interferometry.

We demonstrate the generation and frequency doubling of unit charge vortices in a linear astigmatic resonator. The topological instability of the double charge harmonic vortices leads to well separated vortex cores that are shown to rotate, and become anisotropic, as the resonator is tuned across resonance.

The paraxial propagation of a beam incident along an optic axis of a biaxial crystal slab is studied in detail. Analytical descriptions are given for the Poggendorff bright and dark rings (associated with the conical singularity of the dispersion surface), and the axial spike (associated with the toroidal ring in the dispersion surface). The rings and spike depend on distance from the crystal. In sharpest focus, the rings are close and asymmetrical, and the spike is faint. Further away, the rings separate, they develop weak diffraction oscillations, and the spike grows in intensity. Eventually the oscillations disappear and the rings become symmetrical, and the axial spike dominates. The images depend on the profile of the incident beam; explicit formulae are given for a Gaussian beam and a coherently illuminated pinhole. Geometrical optics (extended to complex rays for the Gaussian beam) can describe some aspects of the images, in particular the Poggendorf dark ring, which arises from antifocusing and for which an explicit description is given.

This article falls within the scope of the special issue on Singular Optics but was actually published in the previous issue of Journal of Optics A: Pure and Applied Optics volume 6, issue 4 (April 2004), pages 289–300. It may be accessed online atstacks.iop.org/JOptA/6/289.