Intensity Nulls Research Articles

Beam displacement tolerances on a segmented mirror for higher-order Hermite–Gauss modes

Odd-indexed higher-order Hermite–Gauss (HG) modes are compatible with four-quadrant segmented mirrors due to their intensity nulls along the principal axes, which guarantees minimum beam intensity illuminating the bond lines between the segments thus leading to low power loss. However, a misplaced HG beam can cause extra power loss due to the bright intensity spots probing the bond lines. This paper analytically and numerically studies the beam displacement tolerances on a segmented mirror for the mode. We conclude that for ‘effective’ bond lines with 6 µm width, and the beam size chosen to guarantee 1 ppm clipping loss when centered, the beam can be rotated by roughly 1∘ or laterally displaced by 4% of its beam size while keeping the total power on the bond lines under 1 ppm. We also demonstrate that the constrained beam displacement parameter region that guarantees a given power loss limit, or the beam displacement tolerance, is inversely proportional to the bond line thickness.

Topological Charge of Propagation-Invariant Laser Beams

If a vortex propagation-invariant beam is given by all its intensity nulls, then its topological charge (TC) can be defined easily: its TC is equal to the sum of topological charges of all optical vortices in these intensity nulls. If, however, a propagation-invariant beam is given as a superposition of several light fields, then determining its TC is a complicated task. Here, we derive the topological charges of four different types of propagation-invariant beams, represented as axial superpositions of Hermite–Gaussian beams with different amplitudes and different phase delays. In particular, topological charges are obtained for such beam families as the Hermite–Laguerre–Gaussian (HLG) beams and two-parametric vortex Hermite beams. We show that the TC is a quantity resistant to changing certain beam parameters. For instance, when the parameters θ and α of the HLG beams are altered, the beam intensity also changes significantly, but the TC remains unchanged.

Dividing the Topological Charge of a Laguerre-Gaussian Beam by 2 Using an Off-Axis Gaussian Beam.

In optical computing machines, many parameters of light beams can be used as data carriers. If the data are carried by optical vortices, the information can be encoded by the vortex topological charge (TC). Thus, some optical mechanisms are needed for performing typical arithmetic operations with topological charges. Here, we investigate the superposition of a single-ringed (zero-radial-index) Laguerre–Gaussian (LG) beam with an off-axis Gaussian beam in the waist plane. Analytically, we derive at which polar angles intensity nulls can be located and define orders of the optical vortices born around these nulls. We also reveal which of the vortices contribute to the total TC of the superposition and which are compensated for by the opposite-sign vortices. If the LG beam has a TC of m, TC of the superposition is analytically shown to equal [m/2] or [m/2] + 1, where [] means an integer part of the fractional number. Thus, we show that the integer division of the TC by two can be done by superposing the LG beam with an off-axis Gaussian beam. Potential application areas are in optical computing machines and optical data transmission.

An astigmatic transform of a fractional-order edge dislocation

In this work, it is theoretically and numerically demonstrated that an astigmatic transformation of a νth-order edge dislocation (shaped as a zero-intensity straight line) of a coherent light field—where ν =n + α is a real positive number, n is integer, and 0 <α <1 is fractional—produces n optical elliptic vortices (screw dislocations) with topological charge (TC) −1, which are arranged on a straight line perpendicular to the edge dislocation and found at Tricomi function zeros. We also reveal that at a distance from the said optical vortices (OV), an extra OV with charge −1 is born on the same straight line, which departs to the periphery with α tending to zero, or gets closer to the n OVs with α tending to 1. Additionally, we find that a countable number of OVs (intensity nulls) with charge −1 are produced at the field periphery and arranged on diverging hyperbolic curves equidistant from the straight line of the n main intensity nulls. These additional OVs, which we term as ‘escort’, either approach the beam center, accompanying the extra ‘companion’ OV if 0 <α <0.5, or depart to the periphery, whereas the ‘companion’ keeps close to the main OVs if 0.5 <α <1. At α =0 or α = 1, the ‘escort’ OVs are shown to be at infinity. At fractional ν, the TC of the whole optical beam is theoretically shown to be infinite. Numerical simulation results are in agreement with the theoretical findings.

Study on the propagation characteristics of elliptical Airy vortex beam

The propagation characteristics of elliptical Airy vortex beams (EAVB) with different elliptical parameter and topological charges are demonstrated experimentally and numerically. Besides, the relationship between elliptical parameter and detection probability of the orbital angular momentum(OAM) is analyzed numerically. The results show that elliptical parameter has a significant impact on the OAM spectral distribution and the propagation characteristics of the EAVB. The EAVB possess an elliptical shaped transverse pattern in the main lobe surrounded by a series of rings and has the line of intensity nulls in the focal plane. It also has two focal planes during the propagation, and the distance between the two focal plane becomes shorter with the increase of elliptical parameter. The long axis of the intensity ellipse of the EVAB rotates 90°between the second focal plane and the first focal plane. There is a linear relationship between OAM detection probability and the elliptical parameter. The experimental results are in good agreement with theory, which will be helpful for the application of EAVB in optical trapping and optical communication.

Tightly focusing vector beams containing V-point polarization singularities

We show theoretically and numerically that an n-th order vector light field containing the central V-point singularity of indefinite linear polarization with polarization singularity index n, with a 'flower' 2(n – 1)-petal polarization pattern centered on it, produces an intensity pattern with 2(n – 1) local maxima at the tight focus. Meanwhile, a vector light field with polarization singularity index –n, leading to a 'web' of polarization singularities composed of 2(n + 1) cells, is tightly focused into an intensity pattern with 2(n + 1) intensity maxima. At the intensity nulls at the focus, either 2(n – 1) or 2(n + 1) V-points with alternating + 1 or –1 indices are produced. In addition, we study more general vector fields of the order (n, m) and analytically derive their Poincare-Hopf indices for many values of n and m. Application areas of such light fields with polarization singularities are laser information technologies, laser material processing, microscopy and optical trapping.

Young's double-slit experiment with vector vortex beams.

In this Letter, Young's double-slit experiment with vector vortex beams is investigated. We present the results for various Poincaré-Hopf index beams of this class considering all four major types. Polarization associated morphological changes in the far-field interference pattern are studied both theoretically and experimentally. The Fraunhofer pattern consists of lattices of polarization singularities of the generic type, located on a line, in a direction perpendicular to the slit. The number of linear lattices varies as a function of Poincaré-Hopf index η of the beam that is diffracted, and the number of intensity nulls occurring along the vertical line is equal to |η|.

Optical beams with an infinite number of vortices

In optical data transmission with using vortex laser beams, data can be encoded by the topo-logical charge, which is theoretically unlimited. However, the topological charge of a single sepa-rate vortex is limited by possibilities of its generating. Therefore, in this work, we analyze light beams with an unbounded (countable) set of optical vortices. The summary topological charge of such beams is infinite. Phase singularities (isolated intensity nulls) in such beams typically have a unit topological charge and reside equidistantly (or not equidistantly) on a straight line in the beam cross section. Such beams are form-invariant and, on propagation in space, change only in scale and rotate. Orbital angular momentum of such multivortex beams is finite, since only a finite number of optical vortices fall into the area, where the Gaussian beam has a notable intensity. Other phase singularities are located in the periphery (and at the infinity), where the intensity is almost zero.

Propagation-invariant laser beams with an array of phase singularities

In this work, we investigate multivortex beams with an infinite topological charge. Such optical vortices contain an array of an infinite, but countable, number of phase singularities (isolated intensity nulls), which typically have the unitary topological charge and are located either uniformly (or not uniformly) on a straight line (or on several crossing straight lines) in the beam transverse cross section. Such optical vortices are form invariant and their transverse intensity distribution is conserved on propagation, changing only in scale and rotation. The orbital angular momentum of such optical vortices is finite, since only a finite number of screw dislocations is located within the area of the notable intensity of the Gaussian beam, while the other phase singularities are in the periphery (and in the infinity), where the intensity is very small. Increasing the Gaussian beam waist radius leads to a parabolic growth of the orbital angular momentum of such beams.

Converting an array of edge dislocations into a multi-vortex beam.

We theoretically show how, using a cylindrical lens, a Gaussian beam with a finite number of parallel zero-intensity lines (edge dislocations) is transformed into a vortex beam that carries orbital angular momentum (OAM) and topological charge (TC). Remarkably, while the original beam is assumed to carry a non-zero OAM and have no TC, the latter is shown to appear during free-space propagation. Considering two parallel center-symmetric zero-intensity lines located as an example, we look into the dynamics of generating two intensity nulls at the double focal length: with increasing distance between the vertical zero-intensity lines, two optical vortices are first generated on the horizontal axis before converging at the origin and then diverging along the vertical axis. Irrespective of the between-line distance, such an optical vortex has ${\rm TC} = - {2}$ at any distance from the optical axis, except for the original plane. With changing distance between the zero-intensity lines, the OAM that the beam carries is changing, taking positive and negative values, or a zero value at a certain between-line distance. We also show that if the number of zero-intensity lines is infinite, a vortex beam with finite OAM and infinite TC is generated.

Polarization singularity index determination by using a tilted lens.

The superposition of spin and orbital angular momentum states of light generates polarization singularities. By perturbing and disintegrating their component orbital angular momentum (OAM) states, the polarization singularity indices can be determined. The spatially varying polarization distribution of these beams possesses information about the helical wavefront structures of the component OAM states, although they have plane wavefronts. The polarization singular beam (PSB) is focused using a tilted lens, and the intensity distribution at a predicted position in the direction of propagation is used to determine the component OAM content in the beam. Astigmatism introduced by the tilt of the lens modulates the vortex beam to introduce intensity nulls in the propagated beam. We demonstrate by simulations and experiments the index determination of the V points and C points using a tilted lens. This method is effective in the index determination of V points and C points formed by the superposition of component scalar vortices having opposite-sign topological charges. The degeneracy of C points with the same Stokes indices can be lifted through this technique.

Optical vortex beams with the infinite topological charge

Up to now, Gaussian optical vortices (OVs) were investigated with the finite topological charge (TC). Here, we study an OV with the infinite TC. Such OVs have a countable number of phase singularities (isolated intensity nulls), which typically have the unitary TC and are located either equidistantly or not equidistantly on a straight line in the beam transverse cross section. Such OVs are structurally stable (form-invariant) and their transverse intensity is conserved on propagation, changing only in scale and rotation. Orbital angular momentum (OAM) of such OVs is finite, since only a finite number of screw dislocations are within the Gaussian beam in the area of notable intensity, whereas the other phase singularities are in the periphery (and in the infinity), where the intensity is very small. Increasing the Gaussian beam waist radius leads to the parabolic growth of the OAM of such beams. A unique feature of these beams is that their normalized OAM can be adjusted (both increased and decreased) by simple change of the waist radius of the Gaussian beam. In addition to the two form-invariant beams, we studied a Gaussian beam with a countable number of edge dislocations (zero-intensity lines), which is not form-invariant, but, after an astigmatic transform by a cylindrical lens, also becomes an infinite-topological-charge beam.

Astigmatic transformation of a set of edge dislocations embedded in a Gaussian beam

It is theoretically shown how a Gaussian beam with a finite number of parallel lines of intensity nulls (edge dislocations) is transformed using a cylindrical lens into a vortex beam that carries orbital angular momentum (OAM) and has a topological charge (TC). In the initial plane, this beam already carries OAM, but does not have TC, which appears as the beam propagates further in free space. Using an example of two parallel lines of intensity nulls symmetrically located relative to the origin, we show the dynamics of the formation of two intensity nulls at the double focal length: as the distance between the vertical lines of intensity nulls is being increased, two optical vortices are first formed on the horizontal axis, before converging to the origin and then diverging on the vertical axis. At any distance between the zero-intensity lines, the optical vortex has the topological charge TC=–2, which conserves at any on-axis distance, except the initial plane. When the distance between the zero-intensity lines changes, the OAM that the beam carries also changes. It can be negative, positive, and at a certain distance between the lines of intensity nulls OAM can be equal to zero. It is also shown that for an unlimited number of zero-intensity lines, a beam with finite OAM and an infinite TC is formed.

Experimental investigation of the energy backflow in the tight focal spot

Using two identical microobjectives with a numerical aperture NA = 0.95, we experimentally demonstrate that the on-axis intensity near the tight focal spot of an optical vortex with a topological charge 2 is zero for right-handed circular polarization and nonzero for left-handed circular polarization. This serves to confirm that in the latter case there is a reverse energy flow on the optical axis, as testified by a very weak local maximum (the Arago spot) detected at the center of the measured energy flow distribution, caused by diffraction of the direct energy flow by a 300 nm circle (the diameter of a reverse energy flow tube). The comparison of numerical and experimental intensity distributions shows that it is possible to determine the diameter of the reverse energy flow "tube", which is equal to the distance between the adjacent intensity nulls. For NA = 0.95 and a 532 nm incident wavelength, the diameter of the on-axis reverse energy flow "tube" is measured to be 300 nm. It is also experimentally shown that when an optical beam with second-order cylindrical polarization is focused with a lens with NA = 0.95, there is a circularly symmetric energy flow in the focus with a very weak maximum in the center (the Arago spot), whose distribution is determined by diffraction of the direct energy flow by a 300 nm circular region, where the energy flow is reverse. This also confirms that in this case, there is a reverse energy flow on the optical axis.

Vortex astigmatic Fourier-invariant Gaussian beams.

We find a two-parameter family of astigmatic elliptical Gaussian (AEG) optical vortices, which are free space modes up to scale and rotation. We calculate total normalized orbital angular momentum of AEG vortices, which can be an integer, fractional and zero, and which is equal to the algebraic sum of two terms reflecting the contribution of the vortex and astigmatic components of the light field. In any transverse plane, such a beam has an isolated n-fold degenerate intensity null on the optical axis (an optical vortex) embedded into an elliptical Gaussian beam. In addition to the quadratic elliptical phase, a beam has the phase of a cylindrical lens rotated by an angle of 45 degrees with respect to the principal axes of the ellipse of the Gaussian beam intensity distribution. The degenerated central intensity null in these beams does not split it into n spatially separated intensity nulls, as is usually assumed for elliptical astigmatic beams.

Lattice of C points at intensity nulls.

The dependence of fringe contrast on the state of polarization (SOP) can be altered when more than two beams interfere, and it is possible to achieve high contrast fringes. We use a six-pinhole interferometer, in which a judicious choice (but not unique) of SOPs for the interfering beams is made to realize the lattice of C points at intensity nulls. These dark C points occur due to overlapping of singularities. These are not simply a minor variation of bright C points, in which the intensity dims to zero, but are poorly understood, unusual objects in their own right. A discussion on Stokes intensity degeneracy is presented. In the ellipticity distributions, it is shown that V points occur at nulls, whereas C points occur at extrema. It is shown that destructive interference of two right (left) handed bright C points can result in a left (right) handed dark C point.

Fresnel and Fraunhofer diffraction of a Gaussian beam with several polarization singularities

Alongside phase singularities (optical vortices), there may be light fields with polarization singularities (PS), i.e. isolated intensity nulls with radial, azimuthal, or radial-azimuthal polarization around them. Here, we study Gaussian beams with several arbitrarily located PS. An analytic expression is obtained for their complex amplitude. A partial case is studied when the PS are at the vertices of a regular polygon. If the beam has one or two PS, then these are points with radial polarization. If there are four PS, then two of the points will have azimuthal polarization. It is shown that while propagating in free space, the PS can appear only in a discrete set of planes, in contrast to the phase singularities, which exist in any transverse plane. In the case of two PS, it is shown that their polarization transforms from radial in the initial plane to azimuthal in the far field.

Astigmatic transforms of an optical vortex for measurement of its topological charge.

We obtain analytical expressions for the complex amplitudes of optical vortices deformed by astigmatic transforms, i.e., passed either through a cylindrical lens or through an inclined spherical lens. We also obtain similar analytical expressions describing propagation of an optical vortex generated when a Gaussian beam illuminates an inclined spiral phase plate (SPP) or when an elliptic Gaussian beam illuminates a SPP (not inclined). All these optical vortices with a topological charge (TC) n are described by the n-th order Hermite polynomial with a complex argument. It is shown that the argument is real only on a straight line in the transverse plane of the laser beam. There are n intensity nulls on this line. The treated here astigmatic transforms are used to determine the integer TC of optical vortices. We conduct a comparative experimental study of different astigmatic transforms and we show that the transform with a cylindrical lens is the best for determining the TC. Unlike other similar works, in this study we achieve transformation of n-degenerate intensity null of an optical vortex with the TC n=100 into n isolated first-order intensity nulls.

Elliptic Gaussian optical vortices

We analyze an elliptic optical vortex embedded into a Gaussian beam. Explicit closed expressions for the complex amplitude and normalized orbital angular momentum (OAM) of such a beam are derived. The resulting elliptic Gaussian vortex (EGV) is shown to have a fractional OAM whose maximal value equal to the topological charge $n$ of a conventional Gauss vortex is attained for a zero-ellipticity vortex. As the beam propagates, the major axis of the intensity ellipse in the beam cross section rotates, making the angle of 90\ifmmode^\circ\else\textdegree\fi{} between the initial plane and the focal plane of a spherical lens. On the major axis of the intensity ellipse, there are $n$ intensity nulls of the EGV, with the distance between them varying with propagation distance and varying ellipticity. The distance between the intensity nulls is found to be maximal in the focal plane for a given ellipticity. For zero ellipticity, all intensity nulls get merged into a single $n$-times degenerate on-axis intensity null. The experimental results are in good agreement with theory.

Orbital angular momentum of an elliptic optical vortex embedded into the Gaussian beam

We consider an elliptic optical vortex embedded into a Gaussian beam. For such a beam, explicit closed-form expressions are derived for the complex amplitude and the normalized orbital angular momentum (OAM). It is shown that the elliptic Gaussian vortex (EGV) has a fractional OAM. The maximal OAM value equal to the vortex topological charge n is achieved at zero ellipticity of the vortex. The major axis of the intensity ellipse in the beam cross section rotates in propagation, turning by a 90-degree angle from the initial plane to the focal plane of a spherical lens. There are n intensity nulls on the major axis of the EGV intensity ellipse. The distance between the nulls varies during the beam propagation and with changing degree of ellipticity. At a fixed degree of ellipticity, the distance between the intensity nulls is maximal in the focal plane. At zero ellipticity, all nulls are "gathering" into a single on-axis n-times degenerate intensity null. The experimental results are consistent with the theory.