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Interference of optical beams with optical vortices is often encountered in singular optics. Since interferometry makes the phase observable by intensity measurement, it brings out a host of applications and helps to understand the optical vortex. In this article we present an optical vortex interferometer that can be used in optical testing and has the potential to increase the accuracy of measurements. In an optical vortex interferometer (OVI), a lattice of vortices is formed, and the movement of the cores of these vortices is tracked when one of the interfering beams is deformed. Instead of multiple vortices in an OVI, an isolated single vortex also finds applications in optical testing. Finally, singularity in scalar and vector fields is presented, and the relation between them is illustrated by the superposition of these beams.

Phase singularities in light waves appear at points or lines in a beam cross section, where the phase of the wave changes abruptly [

In this paper we present the use of OVs in interferometry. In part I, we present optical vortex interferometer (OVI), which refers to interferometric system in which OV lattice plays a crucial role in the measurement process. In part II, we present the use of isolated single vortex in optical metrology. In the last section of part II, we discuss the role of vortex interferometry in the realization of inhomogeneously polarized beams.

When two beams interfere in space, bright and dark surfaces due to interference are formed in the volume of overlap. When viewed at an observation plane, these surfaces appear as interference fringes in a conventional interferometer. When one of the interfering beams is modified, the fringe pattern undergoes a change, and this is tracked in conventional optical testing. But when three or more waves interfere, light vanishes at lines rather than on surfaces. In 2D, these lines appear as dark points instead of fringes. The central part of each such dark point is a vortex point, that is, an isolated point where phase is undetermined. The OVI focuses on the distribution and dynamics of these dark points.

The regular lattice of optical vortices generated by wave interference was a subject of interest prior to the work on OVI. The first papers focused on physical questions concerning the properties of electromagnetic field or more specifically the phase singularity itself [

Here we want to focus our attention on metrological aspects of vortex lattice in interference fields [

Six cubic beam splitters generate a set of three plane waves:

The Intensity distribution of the interference field obtained by three-plane waves (experiment). The position of vortex points is marked by plus signs or crosses to distinguish between two different topological charges.

The resultant field, due to the interference, consists of vortices arranged in a regular fashion. The net charge in the pattern is zero, which means that there is an equal number of positive and negative charges in the lattice. The vortex lattice can be decomposed into two sublattices each consisting of vortex points having the same topological charge (Figure

(a) A set of CALs (dashed lines) of two interfering waves

The vortex lattice shown in Figure

The

After presenting the specific basic properties of the vortex lattice formed by the interference of three-plane waves, let us present how OVI is useful in optical testing. The sample under measurement is introduced in one or more beams of the OVI. This in turn disturbs the vortex lattice geometry. These changes can be related to the value of the physical quantity being measured. The most basic example is wave tilt measurement [

The position of the vortex points (expressed in pixel number) as measured without the wedge (circles) and with the wedge (crosses), (a) wedge inserted horizontally, (b) wedge inserted at

Using the properties listed above, we can derive formulas for the wave tilt through

Two methods have been proposed to analyze vortex lattice dynamics. The first method is based on the analysis of vortex triplet geometry. Vortex triplet consists of three vortex points which do not lie along a single line [

The vortex lattice consists of two sublattices which are marked by plus signs and circles. The CALs of wave pairs

Both methods are compared, and proper formulas for computing wave tilt are derived in [

The angle of wave tilt (wave rotation) determined for

Promising versions of the OVI are compact setups using one [

The trajectory of vortex points measured when a quarter wave plate is inserted between the Wollaston prism and rotated (a). The same situation applies but the quarter wave plate is wedge-like. Figure on the right shows part of the whole image.

A more sophisticated setup is shown in Figure

(a) A new polarimetric system consists of polarizer P, Wollaston prism

The OVI was also applied for the wavefront reconstruction [

The accuracy and resolution of the OVI strongly depends on the accuracy of vortex point’s localization. A few localization methods have been specifically designed for the OVI. The most basic is the one which searches intensity minima [

Other ways of vortex lattice generation are also possible. We describe here the use of amplitude splitting and wavefront splitting interferometers for the same. Three pinholes on an opaque screen are illuminated by a plane wave, or a spherical wave and the diffracted waves behind the pinholes interfere to form the vortex lattice. The schematic of the experimental setup is shown in Figure

Schematic of the three pinholes arrangement for vortex lattice generation.

Diffraction through the pinhole arrangement falls under the category of interference by wavefront splitting. It is also possible to generate vortex lattices by shear interferometers in which amplitude splitting of spherical waves is employed [

(a) Lateral shear interferometer in which the incident wave is split into two and sheared. (b) Schematic showing that the phase difference between sheared plane waves is constant. (c) Schematic showing the phase difference between sheared spherical waves is not constant and varies linearly in the direction of shear.

The fringes in the interference pattern represent phase gradients of the test wavefront. If the test wavefront has a small curvature, the interference pattern exhibits straight fringes, perpendicular to the direction of shear. The number of fringes in the interference pattern is a function of both the curvature and magnitude of shear.

In the vortex lattice generation by interference of three-plane waves, the phase difference between any two waves at the observation plane is found to vary in a linear way. Linearly varying phase difference leads to straight fringes. Hence, for a given interference pattern

Experimental setup for the Mach-Zehnder interferometer configuration with shear plate inducted in one arm.

(a). Interferogram recorded with the Mach-Zehnder interferometer configuration, (b) Formation of fork fringe pattern when fourth beam is added, indicating the presence of vortex dipole arrays.

There is also another area of research, namely, the vortex metrology. In vortex metrology [

The study of optical vortices in microscopy was started with Tychynsky [

The interference fringes obtained in an interferometer represent contour lines of phase difference between a test and a reference wave. By using suitable reference waves in conventional interferometers, interference fringes that are functions of polar coordinates can be obtained [

Optical vortices play a crucial role in obtaining interferograms that are radial or spiral. The reference wave used in these interferometers consists of vortex-like phase variation. To obtain radial interferograms, reference wave consisting of an optical vortex of multiple charge is useful. The reference wave that is used to get radial interferogram is given by

The phase distributions of (a) reference wave

The phase variations of the two simulated test wavefronts are

The interferogram of Figure

Interferogram when the test wave

The interferogram of Figure

Interferogram when the test wave

Shear interferometer eliminates the need for a known reference as the interference is between the test wavefront and a sheared copy of it. The fringes obtained give the phase gradient of the wavefront if the test wavefront is appropriately sheared.

While the lateral shear interferometer reveals phase gradients in Cartesian coordinates, it is essential to modify the way shear is applied for obtaining gradient in polar coordinates. The gradient of the complex amplitude

Bryngdahl and Lee [

Grating that introduces a constant radial displacement between sheared wavefronts [

Segmented spiral grating that can used to introduce a constant azimuthal displacement between sheared wavefronts [

In the grating shown in Figure

For constant azimuthal displacement, simple rotation about the centre of the beam will not suffice as in, simple rotation, the azimuthal displacement is different at different radial locations. In the grating shown in Figure

Knowledge about the nature of interferograms formed by the interference of various types of wavefronts with vortex beams is useful in many situations. Fringes obtained by the interference of vortex beams are characterized by the birth of new fringes from the middle of the interferogram. In conventional interferograms, extrema are surrounded by closed fringes, and no fringe terminates or originates at the centre of the interferogram.

When the vortex-bearing wavefronts are plane and their vortex cores coincide, purely radial fringes occur as in Figure

Fringes obtained due to interference between a vortex of charge +3 with (a) another beam with charge −3 (b) an on-axis plane wave (c) an off-axis plane wave. Interferograms obtained when both the interfering beams have off centred vortices of (d) opposite unequal charges (e) same signed unequal charges.

The radial fringes shown in Figure

Fringes obtained due to interference between a vortex of charge 3 with (a) Spherical beam of positive curvature (b) spherical beam of negative curvature (c) conical beam. (d) Interference between an off centred vortex and a spherical wave (e) Interference between negatively charged vortex and a spherical beam with positive curvature.

In this section, we show the synthesis of a vortex beam by a superposition of beams with a polarization singularity using an interferometer. In other words, scalar beams with a phase singularity and vector beams with a polarization singularity can be mutually converted.

The optical beam generated in an optical resonator is a solution of the wave equation. When the scalar wave equation, commonly referred to as the Helmholtz equation, is solved, the solution is a linearly polarized optical beam with homogeneous spatial distribution across the beam cross section. Hermite-Gaussian (HG) beams are the typical paraxial solutions obtained in the orthogonal coordinate system. In the cylindrical coordinate system, Laguerre-Gaussian (LG) and Bessel-Gaussian (BG) beams are derived and are well known as vortex beams. The electric field of scalar LG beam of degree

If the polarization of the optical beam is inhomogeneous, optical beams must be derived by solving the vector wave equation. The solutions of this equation are different from those of the Helmholtz equation. LG [

The lowest-order mode of vector LG beams is obtained for _{01} mode because there is no electric field in the direction of beam propagation. The other beam shown in Figure _{01} mode). The small circle in the center is the dark area due to the polarization singularity on the beam axis. In what follows, we will show that these beams can also be synthesized by a superposition of scalar beams with a phase singularity.

The lowest-order vector LG beams. (a) Azimuthally (TE_{01}) and (b) radially (TM_{01}) polarized, respectively. The arrows indicate the direction of the temporal electric field.

As shown in Figure _{01} mode with _{10} mode with _{10} mode with _{01} mode with _{01} beams with inverse handedness of both spin and orbital angular momenta, that is, right- and left-hand circularly polarized LG_{01} beams superposed with beams carrying left- and right-hand orbital angular momenta, respectively. Note that there is _{01} beams, namely, subtraction and addition in Figures

The conversion between scalar and vector LG beams. (a) Azimuthally and (b) radially polarized beams.

These transformations between scalar and vector LG beams have been experimentally demonstrated by Tidwell et al. [_{00} mode). Although the manipulation of polarization is not difficult, the production of higher transverse modes such as HG_{01} and LG_{01} modes needed the use of unconventional phase elements. For the HG_{01} mode, a half part of a TEM_{00} mode beam was passed through a tilted glass plate, whose angle was adjusted to obtain a _{01} mode beams were produced by passing a TEM_{00} mode beam through a spiral phase delay plate, which had a spiral ramp made from thin-film-coated glass plate resulting in the relative phase shift of _{01} beams was sensitive to the intensity profile error of an input TEM_{00} mode beam. The second one using LG_{01} beams had a low conversion efficiency (<50%) because two circularly polarized beams were combined by a conventional polarization beam splitter for linear polarization. These problems have been solved based on the fact that a linearly polarized HG_{01} mode is a superposition of two linearly polarized LG_{01} modes carrying inverse orbital angular momenta [_{01} mode is expressed by a superposition of two LG_{01} modes with inverse orbital angular momenta and _{10} mode is expressed by an addition of two LG_{01} modes. Since the polarizations are linear and the intensity patterns are doughnut, the drawbacks in the previous methods are improved. In addition, this method implies that vector LG_{01} beams with a polarization singularity on the beam axis can be converted from scalar LG_{01} beams with a phase singularity on the beam axis similar to the conversion shown on the right-hand side of Figure

Conversion between linearly polarized HG and LG modes.

The generation of pure radially and azimuthally polarized beams directly from a laser cavity has also been demonstrated [_{01} and HG_{10} beams are generated and combined in a laser cavity using linear polarization optics and a

In the following, we show that the conversion between scalar and vector LG beams mentioned above is generally concluded. The upper row of (_{01} modes with opposite spin and orbital angular momenta in the following way:

The electric fields of scalar and vector BG beams,

In this paper we present a detailed analysis about the interferograms that form optical vortex lattices and also how these vortex lattices are used in the measurement process. An example of the displacement of the vortex cores in the lattice by the introduction of a wedge plate in the interferometer is illustrated. The various vortex lattice generation methods including the one based on multiple pinhole diffraction and another using a shear interferometer are presented. As far as beams with single optical vortex are concerned, the role played by a single vortex in realizing interferograms in polar coordinates and shear interferometry in polar coordinates are dealt with. Interferograms of vortex-infested beams with plane, spherical, and conical beams and their applications are discussed. Finally, scalar beams with phase singularity and vector beams with polarization singularity are presented. The conversion between scalar and vector LG beams and the conversion between linearly polarized HG and LG modes are also illustrated. Realization of radial and azimuthal polarization states by superposition of orthogonally polarized beams are also discussed.