Open Access
Add to BibSonomy
Abstract
A full theoretical and experimental analysis of the chain of phase singularities generated when a Gaussian beam passes a double-phase-ramp converter is presented. The overall output beam structure includes a system of interrelated optical vortices (OVs) whose trajectories form a three-dimensional singular skeleton that can be applied for the trapping and guiding of microparticles. The internal structure of each individual phase singularity is characterized by the OV topological charge and by the morphology parameters of equal intensity ellipses in the OV-core area: ellipticity (minor-to-major axes ratio) and the inclination angle. The morphology parameters’ evolution is shown to be valuable for the metrology applications.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Shaping the phase and intensity profiles of an optical beam with formation of certain special features are required in many modern branches, which use laser technologies [1]. Embedding phase singularities, or optical vortices (OV) [2] in the beam gives additional possibilities for creating efficient optical markers that are characterized by isolated intensity minima, essential amplitude and phase gradients and by specific patterns of the near-OV intensity profiles that are relatively stable and conserve during the beam propagation. Due to these exceptional properties, the phase singularities are among the basic instruments for various applications, including manipulation of microparticles in the air [3–8] or development of high-resolution and precise metrology approaches [9,10,11].
Informative reviews [12–15] comprehensively describe the fundamental features of the phase and polarization singularities, with examples of their properties, generation and promising applications. An OV can be considered as a point wavefront defect at which the phase is indeterminate and amplitude is zero. Upon the round-trip near such a point, the phase grows by 2mπ, where the integer m is called topological charge of the OV. In the OV vicinity, the wavefront has a helical shape and the energy flow possesses a circulatory component which results in the orbital angular momentum.
Generation of the optical field with OV can be performed by different techniques. The well-known phase masks for the OV generation include the spiral phase plate [16], the double-phase-ramp (DPR) converter [17], and the diffraction grating with the groove bifurcation (“fork” structure) [18]. The phase masks enable generation of OVs with any desirable topological charge m; if |m| > 1, the generated multicharged OV is unstable, and |m| single charged OVs are formed around the beam axis [19]. At present time, the best ways of a laser beam shaping employ a spatial light modulator (SLM) [20,21] and a digital micro-mirror device [22], which are simple and convenient devices used for flexible controlling of light beams in the whole space of beam propagation. Designing a phase-only mask to generate the beam with the required parameters in diffraction field is the common, simple and convenient technique.
The beams carrying multiple OVs with prescribed structure provide additional advantages for various promising techniques of OV application. The set of the OVs “nested” in a single beam are interrelated and form the so called singular skeleton of the beam, whose parameters depend on the initial conditions of the OV generation problem. In many cases, during the beam propagation the singular skeleton undergo topological reactions [23]: the phase singularities may emerge, annihilate and even collide, including, e.g., the sign reversal of the axial OV with simultaneous emergence of OVs at the beam transverse periphery [24]. A quite specific singular skeleton behavior with different topological reactions is obtained when a Laguerre-Gaussian beam diffracts at DPR converter; its sensitivity to small misalignments between the incident beam axis and the DPR was investigated in [25].
The DPR converter principle stemmed from the mechanism of optical vortices nucleation in the process of the Gaussian beam propagation through an optical wedge, detailed analysis of which is presented in [17,23–31]. This scheme of the OV generation was further developed, and the optimal theoretical parameters for generation of highly symmetric axial OVs have been discussed in [17] for the two-wedge scheme and in [29] for the stack of wedges. Analytical solution for the field of a Gaussian beam diffracted by the DPR converter placed out of the beam waist was shown and experimentally supported in [30] using the SLM.
Formation of the chain of OVs due to the Gaussian beam transformation in an optical wedge was first reported in [31]. Here, we perform the systematic study of this effect with the special focus on the characteristics of the diffracted beam singular skeleton ‘in a whole’. We classify these as “external features” of the set of phase singularities in the diffracted field, including the general description of the 3D OV trajectories and the topological reactions accompanying the diffracted beam propagation. Simultaneously, our subject is the “internal features” of the individual OVs represented by the morphology parameters of the equal-intensity ellipse near the amplitude zero [29]: ellipticity (minor-to-major axes ratio) and the ellipse orientation in the beam cross section determined by the inclination angle of the ellipse major axis with respect to the X axis.
Each individual single-charge OV, being a “part” of the singular skeleton, shows a remarkable stability to the intensity and phase perturbations and preserves its entity until it is involved in a topological reaction [23–25,32]. Therefore, a set of multiple separate OVs that do not interact explicitly during the beam propagation is the best situation for various applications. In this work, we show the way to obtain such a system of apparently independent OVs and analyze, both theoretically and in experiment, its external and internal characteristics with the special attention to possible reasonable applications.
2. Theoretical analysis
To describe the output optical field, we consider the standard diffraction problem, in which the source of light is a Gaussian beam propagating along the Z-axis. The mathematical expression for the electric field of a linearly polarized Gaussian beam can be written in the form
The diffraction element is a DPR converter having the same absolute phase gradients in both phase-ramp components [17]. It transforms the incident beam amplitude (1) according to equation $E({x,y,{z_0}} )\to E({x,y,{z_0}} )\exp [{i\Phi ({x,y,{z_0}} )} ]$ where
The phase correction (2) induces a phase discontinuity ΔΦ at every point of the X-axis; in points where ΔΦ = Nπ (odd N), including the coordinate origin, this phase discontinuity provides the destructive interference and an isolated OV formation. The value of K can be chosen deliberately depending on the beam radius at the phase-mask plane and on the required number of optical vortices in the output field. For theoretical analysis of the considered diffraction problem the Kirchhoff-Fresnel integral has been calculated. The simplified analytical solution for the amplitude of the diffracted beam formed after the Gaussian beam passes the DPR converter can be written as
All generated OVs have the same topological charges $|m |= 1$ and the same sign. For the creation of the OVs with positive charge, $m = + 1$, the phase has to grow upon the counterclockwise circulation around the coordinate origin (the same phase growth will take place near all other points of destructive interference), as is shown by arrows in Fig. 1a. The negative sign of OVs is provided by the clockwise phase growth (Fig. 1b). The illuminating beam, whose intensity contour is outlined by the dashed circle, covers 7 points of destructive interference on the phase mask shown in Fig. 1a and 9 such points in Fig. 1b. In the observation plane at some distance z, not all the initially embedded OVs satisfy the good visibility, and Fig. 1c, d expose 5 and 7 observable OVs, correspondingly. These central well-visible OVs and their morphology are discussed in the paper. The opposite signs of the synthesized OVs in Figs. 1c and 1d can be recognized by the different directions of the intensity ellipse inclination.
Fig. 1. View of the DPR-converter mask (the phase correction (3) varies from –π (black) to +π (white)) realized by the SLM for the synthesis of (a) five optical vortices with topological charge m = +1 and (b) seven OVs with topological charge m = –1 in the high-quality visible area of the output beam cross section. The dashed circle illustrates the radius of a Gaussian beam illuminating the SLM. (c, d) The calculated intensity distributions after passing a Gaussian beam through the mask at the observation distance z = zR from the DPR converter. The equal-intensity ellipses in the near-core dark areas are depicted by white contours on (c) and (d).
Following to [34] the internal structure of the OV cores is characterized by the ellipticity γ and the inclination angle $\varphi$ (Fig. 2). The ellipticity is equal to the ratio of the minor ellipse axis to the major one. Using Eq. (3), we have obtained analytical formula for the ellipticity $\gamma$ of the central OV core:
where $B = {K^2}{b_1} + \frac{4}{{\pi {c_1}}}$, $C = \sqrt {{K^2}{b_1}^2 \cdot \left( {{K^2} - \frac{8}{\pi }} \right) + \frac{{16}}{{{\pi ^2}{c_1}^2}}} $, ${a_1} = \frac{k}{2} \cdot ({R{{({z_0})}^{ - 1}} + {z^{ - 1}}} )$, ${b_1} = \frac{1}{{w{{({z_0})}^2}}}$ and ${c_1} = \frac{1}{{\sqrt {{a_1}^2 + {b_1}^2} }}$. Similarly, the analytical expression for the inclination angle $\varphi$ of the central OV can be obtained in the form:Fig. 2. Parameters of the OV core structure in the beam cross-section (XY plane) with the intensity pattern as a background. The equal-intensity contour is an ellipse with the center at the amplitude zero position. The major ellipse’s semi-axis a, minor semi-axis b and the angle φ between the positive direction of X-axis and the major semi-axis are marked.
The ellipticity and the inclination angle carry a lot of valuable information about the OV-core transformation during the beam propagation.
3. Experimental Technique
The scheme of the experimental setup is shown in Fig. 3a. It includes a beam splitter, plane mirror, plane parallel plates for the adjustment. The phase mask is formed by the SLM
Fig. 3. (a) The experimental setup shown schematically; (b) view of the phase correction (2) profile realized the SLM. (c) The intensity pattern contains five well-resolved OVs and two OVs in the low-intensity area, with equal the topological charges m=+1; (d) the interferogram of the field obtained by interference with a Gaussian beam contains seven fringe-bifurcation regions.
(Holoeye, pluto-2, Resolution 1920×1080, pixel size 8 μm) and illuminated by the He-Ne laser beam (λ = 633 nm, w0 = 0.2 mm, which corresponds to zR ≈ 20 cm). The phase mask models the DPR converter with the phase gradient parameter K = 10 (Fig. 3b). The SLM is situated at a distance z0 = 270 cm from the beam waist and inserted into an arm of the Michelson interferometer.
The interferogram and intensity distributions are detected by a CCD camera (pixel size 3.45 μm, resolution 22 × 3) which can be placed at different distances from the SLM, $z \in [20,290]\,\mbox{cm}$. The intensity image reveals the dark spots distributed along a straight line (Fig. 3c). On the interferogram (Fig. 3d), we can see the fork-like fringes patterns characteristic for the vortex structure embedded within the beam.
We applied the Fourier technique to the detected interferograms [35] to recover the phase map of the beam reflected from the SLM. The main stages of this technique are shown in Fig. 4. At the first step, the FFT-processing is applied to the registered interferogram (Fig. 3d). Next, the first-order part of the Fourier spectrum is extracted and shifted to the zero-order position (Fig. 4a). Then, the inverse Fourier transform is calculated and the phase map recovered (Fig. 4b). On the recovered phase map we can easily find the positions of the vortices forming the array. This procedure has been applied to interferograms registered for different positions of the observation plane along the Z-axis. Using this data, the singular skeleton has been reconstructed.
Fig. 4. (a) The view of the Fourier spectrum for the interferogram presented in Fig. 3d. The rectangle marks the first-order spectrum area. (b) The phase map recovered with the help of the inverse Fourier transform enables precise localization of the OVs.
To calculate the OV morphology parameters (4) and (5), we used the intensity pictures registered by the CCD camera. The image processing routines have been applied to identify dark areas within the image as in Fig. 2 and then an algorithm of the ellipse fitting was used. A series of experiments was performed to determine the ellipse parameters at distance z.
To estimate the accuracy of determining the vortex positions in the OV array, we have considered the influence of three factors during the measurements. The resolution of the CCD camera is limited by the pixel size, thus the error of measuring of the OV position by the CCD Δxccd is equal to 0.0017 cm. An error associated with the accuracy of placing camera at the z position is determined by the method of distance measurement, in our case Δx(z) amounts to 0.001 cm. An error caused by moving the picture in time due to stochastic vibrations is Δx(t). To evaluate this factor, we recorded a series of interferograms within one minute of time and explored fluctuations of the measured OV position. This procedure gave us an estimate Δx(t) = 0.002 cm. Then, the absolute error of measurement can be estimated as
Therefore, the total absolute errors of our measurements along the X and Y directions are about Δx = Δy = 0.0047 cm.Accordingly, the absolute errors of measuring of the OV core ellipticity are determined by
4. Results and discussions
In this section, the results are presented for the field obtained from the incident Gaussian beam (1) after passing through the DPR converter described by Eq. (2). The intensity patterns and phase structures show that the generated separate OVs are similar to those predicted theoretically for the chosen value of the phase gradient parameter K. The number of generated OVs is limited by SLM resolution. We demonstrate in details the case with generation of the five-OV array in the diffracted field using the DPR converter with K = 10 and the counter-clockwise phase growth (m = +1). For this purpose, we use the analytical formulas (3) – (5) and the experimental setup presented in Fig. 3. The output field characteristics are studied theoretically for the whole range of z and experimentally for the propagation distances 18 cm to 290 cm behind the DPR converter.
The results obtained are presented in a consecutive order according to our classification of the external and internal features of the singular beam structure (see Section 1). The external features of the singular skeleton are expressed by the OV trajectories for the whole range of the propagation distances z (Fig. 5). The overall 3D view of the theoretical OV trajectories with the experimental points are presented in Fig. 5a. Note that the whole singular skeleton exhibits a central symmetry with respect to the axis Z being the common axis of the incident beam and of the DPR corrector: it is invariant upon the 2D inversion transformation x → –x, y → –y (remarkably, the axial OV labeled by “0” in Fig. 5 is always fixed at the axis Z). This follows from the symmetry of the phase correction function (2) and is also seen in Figs. 1a, b and 3c, d.
Fig. 5. (a) External features of the singular skeleton in the propagating field formed by the input Gaussian beam transformed by the DPR converter of Eq. (2) with the phase gradient K = 10, topological charge of the OVs is + 1. Images (b), (c) and (d) represent the singular skeleton projections onto the longitudinal XZ, YZ planes and the transverse XY plane, correspondingly. Solid lines indicate the theoretical results, markers show the experimental data. Ordinal numbers –2nd, –1st, 0th, 1st, and 2nd denote the individual OVs.
In the 3D space, the singular skeleton consists of five nearly straight lines localized close to the XZ-plane. A better exposition of the singular skeleton details is achieved with its projections onto the coordinate planes (Fig. 5b–d). The practically rectilinear OV trajectories can be seen in the XZ plane, which is well supported by the experimental data (Fig. 5b). The singular skeleton projection onto the YZ plane (Fig. 5c) discloses the slight deviations of the OV trajectories from the initial XZ plane, and reveals the trajectories’ curvature that is best seen just behind the SLM and disappears with the diffracted beam propagation. Also, the curvature is less articulate for the OV trajectories that are remote from the nominal beam axis. This is supported by the transverse singular skeleton projection presented in Fig. 5d. It shows that the OVs in the propagating beam shift both in the radial and azimuithal directions. The first one occurs due to the natural beam divergence (widening) with the growing distance. The azimuthal deviation appears due to the transvers energy circulation induced by helicoidal wavefronts of generated OVs. Note that the counter-clockwise evolution of the OVs in Fig. 5d changes to the clockwise one if the phase mask designed for the negative-charge OV generation is applied (see Section 2, comments to Fig. 1)
Usually, the singular skeleton evolution demonstrates additional features of the OV-carrying beam, including the topological reactions [23–25,32]. In the present case, topological reactions do not occur, and the OVs evolve separately and look apparently independent through the whole range of the beam propagation distances.
Internal features of the generated phase singularities also demonstrate an intriguing evolution in the course of the beam propagation. Each of the five OVs observed in the diffracted beam possesses the same topological charge + 1 but their morphologies noticeably differ (see Fig. 6). Due to the singular skeleton symmetry with respect to the transverse plane inversion (see the discussion below Fig. 5), we may restrict ourselves to the study of the central OV and of only one pair of the off-axial OV, numbered by 1 and 2 (see Fig. 5).
Fig. 6. Internal features of the OV represented by the OV morphology parameters. (a), (b), (c) Dependencies of the ellipticity γ and (d), (e), (f) of the inclination angle φ for the OV for 0th, (upper row), 1st (middle row) and 2nd (bottom row) versus the distance z behind the SLM modeling the DPR converter with the phase correction (2). Theoretical curves are shown by solid black lines and the experimental data are represented by the markers with the error-bars.
The behavior of the ellipticity γ (Fig. 6a, b, c) is similar for different OVs, and curves in Fig. 6a, b, c differ only quantitatively. Immediately after the DPR corrector, the ellipticity grows but after some distance it starts to monotonously decrease approaching the horizontal asymptote. Its maximum value, which corresponds to the highest OV anisotropy, occurs at the propagation distance z ≈ 60 cm ≈ 3zR (see Eq. (1)). The variations of the inclination angle φ are practically identical for all OVs (Fig. 6d, e, f). The inclination angle varies monotonously from 180° (at z = 0) to the horizontal asymptote $\mathop {\lim }\limits_{z \to \infty } \varphi (z )= 107.23^\circ $ when z infinitely grows.
According to Fig. 6, the equal-intensity ellipses rotate clockwise, i.e. oppositely to the energy circulation. This feature of the OV morphology is analogous to the OV-core behavior observed in the astigmatically deformed OV beams [36,37].
5. Conclusions
In this paper, the transformations of a Gaussian beam after passage through the double-phase-ramp (DPR) converter is discussed. Our main attention is paid to the features of a set of phase singularities in the output field that form a regular quasi-linear array (chain) of the single-charged optical vortices (OVs). The theoretical results are obtained by means of the numerical analysis based on analytical expressions derived from the Kirchhoff-Fresnel integral and are supported in experiment using the setup in which the DPR action is performed by the reflecting SLM.
Earlier studies of the DPR-induced transformations demonstrated that the similar chain of OVs appears in the diffracted field because the destructive-interference conditions are realized in a series of points along the ramp edge [27,28,38]. However the full description of the OVs properties is presented for the first time. We investigated characteristics of the chain “in a whole” (mutual disposition of the OVs and their trajectories in the propagating beam) that are treated as “external features” of the singular skeleton (SS), as well as the morphology features of individual OVs as these evolve in the course of the output beam propagation. The latter constitute the “internal features” including the morphology parameters of the individual OVs: the ellipticity and orientation of the near-core equal intensity ellipses.
The important result is that the separate OVs forming the chain are located close to the x-axis parallel to the DPR edge. With growing propagation distance, they evolve regularly without any topological reactions (such as birth, annihilation and sign reversion of the optical vortex detected in other schemes of the DPR-induced beam transformation [24,25]). The OV trajectories participate in the diffracted beam widening due to divergence, and additionally perform a sort of azimuthal deviation with preserving the central symmetry with respect to the axis of propagation. This azimuthal deviations occur in agreement with the overall transverse energy circulation in the output beam.
The interesting distinctions are found for the internal parameters of the individual phase singularities. Ellipticity and the inclination angle (the ellipse’s major axis orientation with respect to the x-axis parallel to the DPR edge) depend on the propagation distance and show a certain non-linear behavior that becomes linear in the far field (z → ∞). The ellipticity reveals the maximum at a z-distance about 60 cm (approximately 3zR where zR is the Raileigh length of the incident Gaussian beam) and is higher for the OVs located closer to the axis. The inclination angle shows the equal-intensity ellipse rotation oppositely to the sense of the energy circulation in the OV core: from 180° just after the DPR converter (the ellipse is x-elongated) to the asymptotic value (in our conditions, 107.23°) in the far field. Being qualitatively similar, the evolution of the ellipticity of different OVs, constituting the chain, differs in quantitative details; the inclination angle evolution for all the inspected OVs looks the same. Small discrepancy between the theory and experiment is visible for 2nd order vortex (Fig. 6c) is probably caused by the lower contrast around the vortex on the beam periphery in comparison with the high contrast near the central OVs. In the region of lower amplitude, our algorithm of ellipse fitting has been affected by the noise connected with the pixelated structure of the SLM. So, the detailed study of the relevant problem can be a task for the future investigations.
The results of the present work are valuable in several aspects. First, they show possible approaches to the formation of light fields with the prescribed singular skeleton characteristics. This can be useful because positions and trajectories of the phase singularities can be precisely detected and measured and thus supply important spatial labels within the optical field. On the other hand, the singular skeleton of the output field obtained after the DPR converter carries distinct fingerprints of the incident beam structure and prehistory, which can be profitable in the light field diagnostics. One of the main expected applications is associated with preparing the light fields for the precise manipulation of microparticles trapped within the low-intensity regions, i.e. “attached’ to the OV trajectories.
To sum up, this paper outlines a regular approach for the full analysis of the singular optical fields formed by a Gaussian beam passing the DPR converter. Its further development and elaboration can be used for the detailed study of the DPR-based technique with exploration of its reliability, validity and accuracy in the diagnostic and metrology applications.
Funding
Ministry of Education and Science of Ukraine (MESU) (0118U001667); Ministerstwo Nauki i Szkolnictwa Wyższego (MNiSW) (DIA 2016 0079 45).
References
1. F. M. Dickey ed., Laser Beam Shaping: Theory and Techniques, (CRC Press-P2017).
2. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336(1605), 165–190 (1974). [CrossRef]
3. A. Ashkin, Optical Trapping and Manipulation of Neutral Particles Using Lasers (World Scientific2006)
4. M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 2(1), 021875 (2008). [CrossRef]
5. V. G. Shvedov, A. S. Desyatnikov, A. V. Rode, W. Krolikowski, and Y. S. Kivshar, “Optical guiding of absorbing nanoclusters in air,” Opt. Express 17(7), 5743–5757 (2009). [CrossRef]
6. F. Liu, Z. Zhang, Y. Wei, Q. Zhang, T. Cheng, and X. Wu, “Photophoretic trapping of multiple particles in tapered-ring optical field,” Opt. Express 22(19), 23716–23723 (2014). [CrossRef]
7. A. P. Porfirev and A. S. Shipilov, “Laser trapping based on photophoretic forces using a spatial light modulator,” CEUR Workshop Proc. 1638, 111–116 (2016). [CrossRef]
8. J. Zhao, I. D. Chremmos, D. Song, D. N. Christodoulides, N. K. Efremidis, and Z. Chen, “Curved singular beams for three dimensional particle manipulation,” Sci. Rep. 5(1), 12086 (2015). [CrossRef]
9. W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. Hanson, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14(1), 120–127 (2006). [CrossRef]
10. M. R. Dennis and J. B. Gotte, “Topological Aberration of Optical Vortex Beams: Determining Dielectric Interfaces by Optical Singularity Shifts,” Phys. Rev. Lett. 109(18), 183903 (2012). [CrossRef]
11. A. Popiolek-Masajada, J. Masajada, and P. Kurzynowski, “Analytical Model of the Optical Vortex Scanning Microscope with a simple phase object,” Photonics 4(4), 38 (2017). [CrossRef]
12. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001). [CrossRef]
13. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009). [CrossRef]
14. M. R. Dennis, Y. S. Kivshar, M. S. Soskin, and G. A. Swartzlander Jr, “Singular optics: More ado about nothing,” J. Opt. A: Pure Appl. Opt. 11(9), 090201 (2009). [CrossRef]
15. P. Senthilkumaran, S. Sato, and J. Masajada, “Singular Optics,” Int. J. Opt. 2012, 1–2 (2012). [CrossRef]
16. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004). [CrossRef]
17. A. N. Khoroshun, “Optimal linear phase mask for the singular beam synthesis from a Gaussian beam and the scheme of its experimental realisation,” J. Mod. Opt. 57(16), 1542–1549 (2010). [CrossRef]
18. V. Yu. Bazhenov, M.V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations of the wavefront,” JETP Lett. , 52, 429–431 (1990).
19. F. Ricci, W. Löffler, and M. P. van Exter, “Instability of higher-order optical vortices analyzed with a multi-pinhole interferometer,” Opt. Express 20(20), 22961–22975 (2012). [CrossRef]
20. J. P. Huignard, “Spatial light modulators and their applications,” J. Opt. 18(4), 181–185 (1987). [CrossRef]
21. A. Jesacher, C. Maurer, A. Schwaighofer, S. Bernet, and M. Ritsch-Marte, “Full phase and amplitude control of holographic optical tweezers with high efficiency,” Opt. Express 16(7), 4479–4486 (2008). [CrossRef]
22. Y. X. Ren, R. De Lu, and L. Gong, “Tailoring light with digital micromirror device,” Ann. Phys. 527(7-8), 447–470 (2015). [CrossRef]
23. A. Bekshaev, A. Chernykh, A. Khoroshun, and L. Mikhaylovskaya, “Singular skeleton evolution and topological reactions in edge-diffracted circular optical-vortex beams,” Opt. Commun. 397, 72–83 (2017). [CrossRef]
24. A. Khoroshun, A. Chernykh, J. Kirichenko, O. Ryazantsev, and A. Bekshaev, “Singular skeleton of a Laguerre–Gaussian beam transformed by the double-phase-ramp converter,” Appl. Opt. 56(12), 3428–3434 (2017). [CrossRef]
25. A. N. Khoroshun, A. V. Chernykh, H. O. Tatarchenko, A. Ya. Bekshaev, and A. A. Akhmerov, “Laguerre-Gaussian beam transformations by the double-phase-ramp converter: Singular skeleton formation and its sensitivity to small misalignments,” Proc. SPIE , 10612, 1–9 (2018). [CrossRef]
26. V. N. Gorshkov, A. N. Khoroshun, and M. S. Soskin, “The theory of synthesis of optical vortices by the technique of a phase wedge,” Proc. SPIE. 4705, 65–74 (2002).
27. Ya. Izdebskaya, V. Shvedov, D. Kurabtzev, A. Alexeyev, and A. Volyar, “The optical vortex generation by optical wedge,” Proc. SPIE 4607, 78–82 (2002). [CrossRef]
28. S. Vyas and P. Senthilkumaran, “Vortices from wavefront tilts,” Opt. Lasers Eng. 48(9), 834–840 (2010). [CrossRef]
29. Y. Izdebskaya, “Optical necklaces generated by the diffraction on a stack of dielectric wedges,” Phys. Lett. A 372(21), 3909–3913 (2008). [CrossRef]
30. A. N. Khoroshun, A. V. Chernykh, A. N. Tsimbaluk, J. A. Kirichenko, P. V. Yezhov, A. V. Kuzmenko, and J. T. Kim, “Properties of an Axial Optical Vortex Generated with the use of a Gaussian Beam and Two Ramp,” J. Nanosci. Nanotechnol. 16(2), 2105–2107 (2016). [CrossRef]
31. V. G. Shvedov, Y. V. Izdebskaya, A. N. Alekseev, and A. V. Volyar, “The formation of optical vortices in the course of light diffraction on a dielectric wedge,” Tech. Phys. Lett. 28(3), 256–259 (2002). [CrossRef]
32. A. Ferrando and M. A. Garcıa-March, “Theory for the control of dark rays by means of discrete symmetry diffractive elements,” J. Opt. 15(4), 044014 (2013). [CrossRef]
33. M. Abramowitz and I. Stegun, “Error Function and Fresnel Integrals,” in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (NBS Applied Mathematics Series 55, Washington, DC, 1972), pp. 296–297.
34. F. S. Roux, “Spatial evolution of the morphology of an optical vortex dipole,” Opt. Commun. 236(4–6), 433–440 (2004). [CrossRef]
35. H. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe pattern analysis for computer based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]
36. A. Bekshaev and A. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281(23), 5687–5696 (2008). [CrossRef]
37. A. Bekshaev, O. Orlinska, and M. Vasnetsov, “Optical vortex generation with a “fork” hologram under conditions of high-angle diffraction,” Opt. Commun. 283(10), 2006–2016 (2010). [CrossRef]
38. G. H. Kim, J. H. Jeon, K. H. Ko, H. J. Moon, J. H. Lee, and J. S. Chang, “Optical vortices produced with a nonspiral phase plate,” Appl. Opt. 36(33), 8614–8621 (1997). [CrossRef]
