Open Access
Add to BibSonomy
Abstract
Complex polarization-phase transformations that are realized using easy-to-manufacture optical elements are considered. The manufacturing technology of such elements is based on the angular discretization of the required polarization and phase distributions, which allows one to make optical elements in the form of sector sandwich structures consisting of polarized and phase plates stacked together. We analyze analytically and study numerically the main types of such sector sandwich structures for the formation of cylindrical polarizations of various orders. New effects are observed, which result in the appearance of complex polarized beams with vortices of various orders, arising after the passage through polarizing plates and their combinations with differently rotated phase plates. The results of the experimental study of the formed beams using a multichannel diffraction filter are consistent with theory.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Cylindrical vector beams (CVBs) of various orders [1] are of practical interest in such fields as optical communication using mode division multiplexing [2], amplitude and polarization modulation of focal distributions [3], laser micromanipulation [4,5], and stellar coronagraphy [6,7]. Some applications can be based on the phenomenon of the so-called inverse energy flux [8], which occurs under focusing of high-order radially polarized beams. In this case, the integral inverse energy flux increases with the topological order of the radially polarized beam. Thus, the formation of high-order cylindrical beams is an urgent task.
A significant number of research papers have been recently devoted to the study of methods for producing CVBs, including beams of high orders. The main methods involve polarization transformations of the initial beam by spatial light modulators (SLMs) [9] utilizing the superposition of vector beams [10,11] and subwavelength gratings [12–14], as well as using crystalline [15–17] and polarizing film [15,18] sector plates. In fact, an interference polarizer [19,20] acts as a continuous analog of sector plates. Different vector vortex beams, including vector vortex beams with controllable degrees of freedom in trajectory shape, coherent-state phase, orbital angular momentum, and polarization, can be generated directly from a laser cavity with an external modulation [21,22]. All methods have their advantages and disadvantages. SLMs convert just a part of the transmitted light, thereby reducing the polarization extinction ratio. For a converter based on subwavelength gratings, the efficiency and, therefore, the polarization contrast change as functions of the angle of the polarization plane rotation. Note that the lack of subwavelength polarization gratings, consisting in the unevenness of Fresnel reflections, can be compensated by combining polarization and focusing elements [23,24]. It should also be emphasized that the technology of manufacturing subwavelength gratings for the infrared range is somewhat simpler due to longer wavelengths.
The main advantage of sector polarizing film is the lowest cost per unit area and ease of fabrication. The main factor that deteriorates the quality of the formed beams and, accordingly, complicates the manufacturing technology is the joints of the sectors. In addition, for converters in which circular polarization is used as the initial one [15,18–20], the generated beam is characterized by the presence of a vortex phase. We deal in this case with the so-called vortex radial polarization, while the classical radial polarization is produced using an additional phase transformation by phase plates [7]. This problem is easily solved by using a binary multichannel optical element matched to a set of optical vortices of different orders [25]. The proposed sector sandwich structures expand the functionality of the polarization converter in a controllable (the number of sectors and their position can be varied) and inexpensive (polarizing films are inexpensive and affordable, binary phase plates are easy to fabricate) way. Technologically, this is easily feasible, since the sectors of the polarizing film and the binary phase plates are made in the form of separate plates. Furthermore, the binary multichannel vortex optical element (which is easy-to-manufacture in phase form and even simpler in amplitude form) increases the number of degrees of freedom for the formation of various high-order polarizations due to polarization-phase transformations [26]. In the present work, several variants of such transformations are proposed.
2. Theoretical basis and simulation
Various types of high-order cylindrical polarizations are considered [27–33], which can be combined with the formula [27]
Various special cases can be obtained using Eq. (1), which are illustrated in Fig. 1.
As can be seen from Fig. 1 any types of first- and second-order cylindrical polarization can be realized using eight-sector polarizing plates. For third-order cylindrical polarization, a twelve-sector polarizing plate is needed. In the case of a moderate increase in the number of sectors (16 or 32 sectors), it becomes possible to generate CVBs of higher orders. With a further increase in the number of sectors, up to infinite (when the structure of the optical element and the polarization transformation can be described by a continuous function), the theory becomes simpler. However, from a practical point of view, this approach becomes meaningless, since the sectors become too narrow.
In simulating the action of sector polarizers assembled from polarizing films, we use the expression for ideal linear polarizers:
For a field with initial circular polarization ${{\textbf{c} }_0} = {{\textbf{c} }^{circ + }} = {\left( {\sqrt 2 } \right)^{ - 1}}{({1,i} )^T}$, Eq. (2) can be rewritten in the form:
Thus, if sector films are arranged along radial lines (which corresponds to the top left image in Fig. 1), then we obtain the first-order radial polarization with the first-order phase vortex, as follows from Eq. (3).
2.1 Radial polarization plate “Pr1”
An eight-sector polarizing film “Pr1” (shown in Fig. 2(a)), illuminated by a beam of circular polarization ${{\textbf{c} }^{circ + }} = {\left( {\sqrt 2 } \right)^{ - 1}}{({1,i} )^T}$, will make the different transformations in different sectors in accordance with Eq. (2). In this case, the phase is transformed in the individual components of the vector field, namely:
Fig. 2. Eight-sector polarizing film “Pr1”. (a) Polarization transformation of a circularly polarized Gaussian beam passing through an eight-sector polarizing film “Pr1”. (b) Comparative results of the focusing of an initial circularly polarized Gaussian beam (I) transmitted through an eight-sector polarizing film “Pr1” (II), and transmitted through a polarizing film with an added vortex phase of the minus first order exp (-iθ) (III). Graph (IV) shows the normalized values of |bm|2 for the corresponding fields: black color (μ = 0) for (I), red color (μ = 1.7) for (II), and blue color (μ = 0) for (III).
Figure 2(a) shows the polarization transformation of the electric field components of a circularly polarized Gaussian beam passing through an eight-sector polarizing film “Pr1”. One can see that initially the field has a uniform distribution both in amplitude and in phase (the x-component with the zero phase, and the y-component with the phase of π/2). After passing through the eight-sector polarizing film “Pr1”, both the intensity and the phase distributions change in the each component. It should be noted that now the phase difference (by π/2) is present in the x- and y-components only in the sectors θ = 135° and θ = 315°.
Figure 2(b) compares the results of focusing an initial circularly polarized Gaussian beam transmitted through the polarizing film “Pr1”. While the distribution of the total beam intensity has remained the same as the initial one, significant changes are observed in the polarization state in the peripheral region, where polarization is radial. In this case, the central part is characterized by the initial circular polarization (this fact was also noted in [34]).
To analyze the polarization and phase states of light fields, the expansion in optical vortices, exp(imφ), is often used [35,36]:
Using coefficients in Eq. (5), we can calculate the orbital angular momentum (OAM) of an arbitrary field as follows [36]:
Figure 2(b) shows the squares of the moduli of coefficients |bm|2 (N = 3) (see Eq. (5)) for the components of the corresponding fields too. One can see that there appears a second-order optical vortex for the focused field after the polarizing film. Thus, after passing through the polarizing film “Pr1”, a circularly polarized Gaussian beam with an initially zero OAM acquires a second-order optical vortex and an OAM equal to μ = 1.7.
As follows from theoretical analysis of Eq. (3), the polarizing film “Pr1” converts circular polarization into vortex radial polarization, i.e. into first-order cylindrical polarization with a first-order phase vortex. The presence of a second-order optical vortex can be explained by the interaction of polarization and phase singularities. To show it we rewrite Eq. (3) in the form:
One can clearly see from Eq. (7) that the polarizing film “Pr1” converts an initially uniform circular polarization into a nonuniform polarization which is a superposition of the same circular polarization and the circular polarization of the opposite direction with a second-order phase vortex. This result was also obtained numerically (see Fig. 2(b)). The squared modulus of the calculated coefficient |b2|2 is larger than |b0|2, since the studied field in the annular region corresponding to the second-order vortex is characterized by higher energy than that in the central part.
The obtained effect, as a rule, occurs when the circularly polarized fields are converted into another polarization state [15,18–20,34] and often leads to the spin-to-orbit angular momentum conversion [37–43]. It should be noted that various combinations of the cylindrical polarization states with a vortex phase singularity are used for optical trapping and manipulation [44,45], for optical communication with mode division multiplexing [1,2,35,36], for processing and structuring of materials [46,47], and also for the formation of subwavelength-size spherical optical traps [48,49].
To produce the classical radial polarization, additional conversion of the obtained field defined by Eq. (3) or Eq. (7) is needed. One way to remove the vortex phase in the field defined by Eq. (3) is to apply a spiral phase plate [50] or a spiral axicon [51]. Figure 2(b) shows the result of focusing a circularly polarized Gaussian beam passing through an eight-sector polarizing film “Pr1” with an added vortex phase of the minus first order. One can see that in this case, classical radial polarization is formed. In this case, optical vortices with m = ±1 of the same weights coefficients are present, and the OAM is equal to zero.
However, the fabrication of multi-level spiral optical elements faces certain challenges [52], and a binary spiral axicon has reduced diffraction efficiency. Another way is the use of binary sector plates [53], which are easy to manufacture and have no energy losses. Such phase plates correspond to the phase distribution of the cosine or sine angular functions and act similarly to a superposition of optical vortices of opposite directions, exp(imθ)±exp(-imθ). Obviously, in this case a more complex polarization phase transformation will occur, which, however, is realized using simple optical elements. This is important in practice, especially in high-power laser applications. Note that in [53] the S-waveplate was used, which makes it possible to form only first-order CVBs. S-waveplates (and q-plates) are still relatively expensive items. At the same time, optical elements are commercially produced to form the first and second order CVBs. In this paper, we take the next step by replacing the S-waveplate with an element made of polarizing film sectors to expand the functionality of the polarization converter in a controllable and inexpensive way.
Let us supplement the polarizing film “Pr1” with a two-sector binary phase plate (see Fig. 3(a)). The action of this plate is similar to that of the cosine angular function cosθ [52]. The polarization and phase transformations performed by a sandwich of the polarizing film “Pr1” and the phase plate can be described as follows:
Fig. 3. (a) Polarization phase transformation of a circularly polarized Gaussian beam passing through a polarizing film “Pr1” with an additional binary phase plate. (b)-(c) Results of focusing a circularly polarized Gaussian beam transmitted through an eight-sector polarizing film “Pr1” and an additional two-sector binary phase plate and the corresponding expansion coefficients |bm|2 (b). Similar results for a slightly rotated additional phase plate (c).
Thus, a superposition of classical radial polarization and circular polarizations of opposite directions with a phase singularity of the first and third orders is formed.
Figure 3(a) shows the results of the polarization phase transformation of a circularly polarized Gaussian beam passing through the polarizing film “Pr1” with the additional two-sector binary phase plate, which inserts an additional phase by π radians in four sectors (θ = 135°, θ = 180°, θ = 225°, and θ = 270°). Obviously, the additional phase plate does not change the beam amplitude, but produces an additional phase incursion in the corresponding sectors. In this case, the phase of the x-component looks like a first-order discrete vortex phase (Fig. 3(a), the first row), and the phase of the y-component has a more complex shape. We also considered a case when the phase plate is slightly rotated (Fig. 3(a), the second row) and the boundaries of the sectors of the polarizing film and the phase plate do not coincide (this situation may be implemented in practice).
Figures 3(b) and 3(c) show the results of focusing the corresponding fields shown in Fig. 3(a). It also presents expansion coefficients of the transverse components of the focused fields from Eq. (5). As can be seen, an additional phase plate substantially changed the focal field structure. The field intensity now has a zero value in the central part, like cylindrically polarized beams, although the distribution of polarization is hybrid. As predicted by Eq. (8), this is a superposition of classical radial polarization and circular polarizations of opposite directions with phase singularities of various orders.
The expansion of the focused field into optical vortices (see Eq. (5)) qualitatively confirms Eq. (8). As can be seen (see Figs 3(b) and 3(c)), coefficients with orders m = ±1, 3 have significant weights (a vortex of the minus first order is present in radial polarization [34]). The remaining coefficients with lower weights arise due to the use of a phase plate rather than the amplitude-phase cosine angular function [52].
Note that the rotation of the phase plate does not affect the distribution of intensity and polarization (taking into account the rotation), but there is some redistribution of the expansion coefficients of Eq. (5).
Thus, the combination of sector polarizing films and sector phase plates allows the formation of beams with various hybrid polarization and phase distributions.
2.2 Azimuthal polarization plate “Pa2”
Next, we consider an eight-sector polarizing film “Pa2” (see Fig. 4(a)), which we used to generate second-order azimuthally polarized beams. When the polarizing film “Pa2” is illuminated by a circularly polarized beam, the x- and y-components will be transmitted in the diagonal and in the vertical and horizontal sectors, respectively (see Fig. 4(a), the first row). This transformation is much simpler than that for the film “Pr1” given in Eq. (4). For cylindrical polarization to be formed, it not sufficient to use a film only, since the direction of the polarization vector should be opposite in neighboring sectors for each of the transverse components. This can be achieved by introducing a phase incursion by π radians. In this case, the use of a four-sector binary phase plate is very convenient (see Fig. 4(b), the second row).
Fig. 4. Eight-sector polarizing film “Pa2”. (a) Polarization transformation of a Gaussian beam with circular polarization when passing through an eight-sector polarizing film “Pa2” (top row) and an additional four-sector phase plate (bottom row). Comparative results of focusing of a circularly polarized Gaussian beam transmitted through an eight-sector polarizing film “Pa2” (b) and through an additional four-sector phase plate (c); graph (d) presents the normalized values of |bm|2 for the corresponding fields: red color (μ = 0) for (b) and blue color (μ = 0) for (c).
Comparative results of the focusing of the corresponding beams and their expansion in optical vortices (see Eq. (5)) are shown in Figs 4(b)–4(d). One can see that when only a polarizing film “Pa2” is used, the required transformation occurs only in the peripheral region, where the energy is almost absent (see Fig. 4(b)). The classical second-order azimuthal polarization is formed when a four-sector phase plate is used together with the polarizing film (see Fig. 4(c)). The expansion of the focused fields in optical vortices shows (see Fig. 4(d)) that in the first case there are coefficients with orders m = 0, ±4, and in the second case only with orders m = ±2, as it should be for second-order cylindrical polarization [34].
3. Experimental formation and analysis using a multi-channel diffraction filter
The simulation results for some polarization sector plates, as well as polarization plates stacked with phase sector plates, are presented below. It is known [33,34,53,54] that polarization phase states are most easily recognized by a multichannel vortex filter (see Fig. 5(a)).
Fig. 5. Simulation of the action of a multichannel filter for a circularly polarized Gaussian beam: (a) phase of a six-channel filter matched with the vortex harmonics m = ±1, ±2, ± 3, (b) input field, (c) focal distribution, and (d) distribution in the focal plane using a filter.
Figure 5(d) shows the distribution in the focal plane for a Gaussian beam with circular polarization in the case of using a six-channel filter. It is seen that the centers of all diffraction orders have zero intensity, which corresponds to the absence of conjugate orders of optical vortices in the incident field. This is completely expected.
The distribution in the focal plane for a circularly polarized Gaussian beam passing through the eight-sector polarizing film “Pr1” and the six-channel filter is shown in Fig. 6. In the intensity distribution (see Fig. 6(c)), the diffraction order corresponding to m = +2 (marked with a white circle) has a nonzero intensity in the center, which indicates the presence of a second-order optical vortex in the incident field. This result is consistent with theory (see Eq. (7)).
Fig. 6. Simulation of the action of a multichannel filter for a circularly polarized Gaussian beam transmitted through an eight-sector polarizing film “Pr1”: (a) input field, (b) focal distribution, (c) total intensity distribution in the focal plane using a filter, and (d) different components of the vector field in the focal plane using a filter.
However, more interesting results can be observed in other diffraction orders. For better visualization, the field in the focal plane is also shown in colors (see Fig. 6(d)), which correspond to different components of the vector field (red for the x-component and green for the y-component).
As can be seen from the simulation results (see Fig. 6(d)), a multichannel filter matched with a set of optical vortices can be used not only to analyze the polarization phase state of the beam, but also for the required phase transformation. It is clearly seen that in the diffraction order corresponding to m = +1 (see Fig. 6(d), marked with a yellow circle), a field with a classical first-order radial polarization is formed. In this diffractive order, the filter is matched with the complex conjugate distribution, i.e. with exp(-iθ), which is required for compensation of the vortex component (see Eq. (3)).
Thus, this is another way to obtain a given distribution using a simple binary optical element. In other diffraction orders in the focal plane, one can also observe various types of hybrid polarization states. For example, in the orders corresponding to m = +3 and m = −1 (see Fig. 6(d), marked with blue circles), a pseudo-azimuthal polarization is formed.
The distribution in the focal plane after sandwich of the polarizing film “Pr1” and the two-sector phase plate (see Fig. 7(c)) clearly indicates the presence of optical vortices of orders m=±1, +3, (marked with white circles) which is also consistent with theory (see Eq. (8) and Fig. 3(b)). Note that in this case (in contrast to the previous ones), the individual components of the vector field contain different optical vortices and have different OAMs, which is clearly shown by the intensity of the correlation peaks in Fig. 7(d).
Fig. 7. Simulation of the action of a multichannel filter for a circularly polarized Gaussian beam transmitted through an eight-sector polarizing film “Pr1” and a two-sector phase plate: (a) input field, (b) focal distribution, (c) total intensity distribution in the focal plane using a filter, and (d) different components of the vector field in the focal plane using a filter (for the x-component it is red and for the y-component it is green).
Finally, Figs 8 and 9 show the focal distributions for a circularly polarized Gaussian beam that has passed through the polarizing film “Pa2” both without (see Fig. 8) and with (see Fig. 9) the four-sector phase plate. These results to the greatest extent satisfy the initial task of forming a second-order cylindrical vector beam, which is confirmed by the distribution of intensity, the picture of the components of the vector field (see Fig. 8(d)) and the presence of maxima in the orders corresponding to second-order conjugate vortices (see Fig. 9(c)).
Fig. 8. Simulation of the action of a multichannel filter for a circularly polarized Gaussian beam transmitted through an eight-sector polarizing film “Pa2”: (a) input field, (b) focal distribution, (c) total intensity distribution in the focal plane using a filter, and (d) different components of the vector field in the focal plane using a filter (for the x-component it is red and for the y-component it is green).
Fig. 9. Simulation of the action of a multichannel filter for a circularly polarized Gaussian beam transmitted through an eight-sector polarizing film “Pa2” and a four-sector phase plate: (a) input field, (b) focal distribution, (c) total intensity distribution in the focal plane using a filter, and (d) different components of the vector field in the focal plane using a filter (for the x-component it is red and for the y-component it is green).
4. Experiments
First, we fabricated polarizing sector plates for the generation of first-order radial polarization and second-order azimuthal polarization (see Fig. 1). The plates were composed of sectors cut on a plotter with the necessary direction of polarization axes, which were then assembled on a glass substrate. As follows from the simulation results, we used sector plates in combination with phase plates having phase-shifting regions in the form of a half-plane and in the form of quadrants. Such plates were etched on quartz substrates using photomasks made with a circular laser writing system CLWS-200S.
An experimental setup for studying the formation of high-order cylindrical vector beams using sector sandwich structures is shown in Fig. 10. The linearly polarized light of a solid-state Laser was spatially filtered and collimated using a combination of a 8× micro-objective (MO) with NA = 0.1 and a pinhole (PH) with a size of 40 µm and a lens (L1) with a focal length of 250 mm. A quarter-wave plate (QP) was used to convert the initial linear polarization into circular polarization. Then, the laser beam passed through an eight-sector polarization plate PP and a sector phase plate P. The converted beam passed through a 4-f optical system consisting of lenses L2 and L3 with focal lengths of 150 and 250 mm and was directed to a multi-channel vortex filter (DOE). The laser beam passing through the filter was focused using a lens (L4) with a focal length 750 mm on the video camera Cam. A polarizer analyzer (PA) was used to analyze the polarization structure of the generated laser beam.
Fig. 10. Schematic of an experimental setup for studying the formation of high-order CVBs using sector sandwich structures: Laser is the solid-state laser (λ = 532 nm), MO is the 8× micro-objective (NA = 0.1), PH is the pinhole (aperture size 40 µm), L1, L2, L3, L4 are the lenses with focal lengths of 250, 150, 250 and 750 mm, QP is the quarter-wave plate, PP is the eight-sector polarizing plate, P is the two or four-sector phase plate, DOE is the multi-channel vortex filter, PA is the polarizer analyzer, Cam is the video camera.
At the first stage of the experiment, we checked the orientation of the polarization axes of the sectors. To this end, the sector plate P was removed from the scheme. In this case, the video camera Cam and the polarizer analyzer PA were placed in the plane of a multi-channel vortex analyzer, which coincided with that of the eight-sector polarizing plate. The obtained intensity distributions at various positions of the polarizer axis are shown in Fig. 11. One can see that the positions of the polarization axes of the sectors are fully consistent with Figs 2 and 4.
Fig. 11. Intensity distributions obtained in the transmitted light at different positions of the axis of the polarizer-analyzer in the case of using an eight-sector polarizing film “Pr1” (a) and “Pa2” (b). White arrows show the position of the axis of the polarizer analyzer.
Then, we studied the intensity distribution at the focus of the focusing lens L4 for the obtained nonuniformly polarized first- and second-order beams. To do this, we installed again the sector phase plates to the optical system and removed the multi-channel vortex analyzer DOE. The intensity distributions obtained by using eight-sector polarizing plates “Pr1” and “Pa2” and a two-sector or four-sector phase plate are shown in Figs 12(a), 12(b), and 13(a). When use was made of an eight-sector polarizing plate “Pr1” and a two-sector phase plate, we considered two different cases of the orientation of a two-sector phase plate, similarly to the data in Fig. 3. The presented images show good agreement with the simulation results, including the correspondence between the individual components of the field. As in the simulation results, a dip is observed in the center of the focal spot.
Fig. 12. Formation of high-order cylindrical vector beams using a combination of an eight-sector polarizing plate “Pr1” and a two-sector phase plate. The intensity distribution at the focus of lens L4 with non-rotated (a) and rotated (b) phase plates (the multi-channel vortex analyzer is removed from the scheme). The total intensity distribution (leftmost pictures), as well as the distributions obtained using the polarizer analyzer are presented. White arrows show the position of the axis of the polarizer analyzer. (c) Intensity distributions obtained using a multi-channel polarizer analyzer. The total intensity distribution (left picture), as well as the distributions obtained using the polarizer analyzer are presented. The central parts of the distributions containing an unmodulated part of the initial laser beam are cut out.
Finally, we studied the interaction of a multi-channel vortex filter with the generated laser beams. The results are shown in Figs 12(c), 13(b), and 14. Figures 12 and 13 show the cases with sector phase plates, corresponding to the simulation results shown in Figs 7 and 9, and Fig. 14 shows the case without phase plates, corresponding to the simulation results shown in Figs 5, 6, and 8. One can see qualitative agreement with the simulation results, including the correspondence between the individual components of the field. Maxima arise at those points of the output plane where the order of the phase singularity corresponds to the order of cylindrical polarization.
Fig. 13. Formation of high-order cylindrical vector beams using a combination of an eight-sector polarizing plate “Pa2” and a four-sector phase plate. (a) Intensity distribution at the focus of lens L4 with a phase plate (the multi-channel vortex analyzer is removed from the scheme). The total intensity distribution (leftmost picture), as well as the distributions obtained using the polarizer analyzer are presented. White arrows show the position of the axis of the polarizer analyzer. (b) Intensity distributions obtained using the multi-channel polarizer analyzer. The total intensity distribution (left picture), as well as the distributions obtained using the polarizer analyzer are presented. The central parts of the distributions containing an unmodulated part of the initial laser beam are cut out.
Fig. 14. Intensity distributions obtained using a multi-channel polarizer analyzer in the case of a multi-channel vortex analyzer illuminated by a circularly polarized laser beam (a), a circularly polarized laser beam transmitted through a sector polarizing plate “Pr1” (b) or “Pa2” (c). In cases (b) and (c), in addition to the total intensity distribution (left pictures), the distributions obtained using the polarizer-analyzer are also presented. White arrows show the position of the axis of the polarizer analyzer. The central parts of the distributions containing an unmodulated part of the initial laser beam are cut out.
5. Conclusions
We have simulated numerically the focusing of beams formed by sector sandwich structures consisting of sector polarizing films approximating nonuniform polarizations of the first and second orders and phase sector plates. New effects have been observed in the form of complex polarized beams with vortices of various orders, arising after passing through the polarizing films and their combinations with binary phase plates. It has been shown theoretically that a multichannel filter, matched with a set of optical vortices, can be used not only to analyze the polarization phase state of the beam, but also to perform the required phase transformation. Thus, in a diffraction order corresponding to a first-order vortex, a field with a classical first-order radial polarization is formed. We have fabricated sector polarizing films have to produce first-order radial polarization and second-order azimuthal polarization as well as binary phase plates needed for various polarization conversion. The experiments conducted with the manufactured plates are consistent with the simulation results. Such laser beams with complex polarization states are required in the field of laser material processing with the help of pulse radiation for the generation of laser-induced periodic surface structures (LIPSS) with the desired profile [55,56]. The local laser beam polarization direction defines a preferential orientation of sub-wavelength LIPSS appearing on the material surface. Such structures are promising for the generation of manifold functional surfaces for various applications [57,58]. The proposed sector sandwich structure has a high-damage threshold allowing the use of these plates with high power pulse laser systems. Besides, the polarization of such complex vector beams can be used as an additional degree of freedom complicating the task of signal demultiplexing for communication systems [59,60].
Funding
Russian Foundation for Basic Research (18-29-20045-mk); Ministry of Science and Higher Education of the Russian Federation (007-GZ/Ch3363/26).
Acknowledgments
This work was financially supported by the Russian Foundation for Basic Research (18-29- 20045-mk) in part of numerical calculations and experimental results and by the Ministry of Science and Higher Education (007-GZ/Ch3363/26) in part of theoretical results.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
References
1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009). [CrossRef]
2. G. Millione, Th. A. Nguyen, J. Leach, D. A. Nolan, and R. R. Alfano, “Using the nonseparability of vector beams to encode information for optical communication,” Opt. Lett. 40(21), 4887–4890 (2015). [CrossRef]
3. Z. Zhou and L. Zhu, “Tight focusing of axially symmetric polarized beams with fractional orders,” Opt. Quantum Electron. 48(1), 44 (2016). [CrossRef]
4. J. J. Saenz, “Laser tractor beams,” Nat. Photonics 5(9), 514–515 (2011). [CrossRef]
5. C. Lan, Y. Yang, Z. Geng, B. Li, and J. Zhou, “Electrostatic field invisibility cloak,” Sci. Rep. 5(1), 16416 (2015). [CrossRef]
6. D. Mawet, P. Riaud, J. Surdej, and J. Baudrand, “Subwavelength surface-relief gratings for stellar coronagraphy,” Appl. Opt. 44(34), 7313–7321 (2005). [CrossRef]
7. D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronagraph: laboratory results and first light at Palomar observatory,” Astrophys. J. 709(1), 53–57 (2010). [CrossRef]
8. S. N. Khonina, A. V. Ustinov, and S. A. Degtyarev, “Inverse energy flux of focused radially polarized optical beams,” Phys. Rev. A 98(4), 043823 (2018). [CrossRef]
9. W.-L. Hsu, K. Balakrishnan, M. Ibn-Elhaj, and S. Pau, “Infrared liquid crystal polymer micropolarizer,” Appl. Opt. 53(23), 5252–5858 (2014). [CrossRef]
10. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007). [CrossRef]
11. Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light: Sci. Appl. 8(1), 90 (2019). [CrossRef]
12. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Propagation-invariant vectorial Bessel beams obtained by use of quantized Pancharatnam–Berry phase optical elements,” Opt. Lett. 29(3), 238–240 (2004). [CrossRef]
13. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27(5), 285–287 (2002). [CrossRef]
14. S. S. Stafeev, V. V. Kotlyar, A. G. Nalimov, M. V. Kotlyar, and L. O’Faolain, “Subwavelength gratings for polarization conversion and focusing of laser light,” Photonic. Nanostruct. 27, 32–41 (2017). [CrossRef]
15. S. V. Alferov, S. V. Karpeev, S. N. Khonina, and O. Yu. Moiseev, “Experimental study of focusing of inhomogeneously polarized beams generated using sector polarizing plates,” Comp. Opt. 38(1), 57–64 (2014). [CrossRef]
16. S. V. Karpeev, V. V. Podlipnov, S. N. Khonina, V. D. Paranin, and A. S. Reshetnikov, “A four-sector polarization converter integrated in a calcite crystal,” Comp. Opt. 42(3), 401–407 (2018). [CrossRef]
17. G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32(11), 1468–1470 (2007). [CrossRef]
18. Zh. Man, Ch. Min, Y. Zhang, Z. Shen, and X.-C. Yuan, “Arbitrary vector beams with selective polarization states patterned by tailored polarizing films,” Laser Phys. 23(10), 105001 (2013). [CrossRef]
19. S. Karpeev, V. Paranin, and S. Khonina, “Generation of a controlled double-ring-shaped radially polarized spiral laser beam using a combination of a binary axicon with an interference polarizer,” J. Opt. 19(5), 055701 (2017). [CrossRef]
20. V. D. Paranin, S. V. Karpeev, and S. N. Khonina, “Generation of radially polarized beams based on the refractive elements with interference polarizing coatings,” Comp. Opt. 39(4), 492–499 (2015). [CrossRef]
21. A. Forbes, “Structured light from lasers,” Laser Photonics Rev. 13(11), 1900140 (2019). [CrossRef]
22. Y. Shen, X. Yang, D. Naidoo, X. Fu, and A. Forbes, “Structured ray-wave vector vortex beams in multiple degrees of freedom from a laser,” Optica 7(7), 820–831 (2020). [CrossRef]
23. S. A. Degtyarev, S. G. Volotovsky, and S. N. Khonina, “Sublinearly chirped metalenses for forming abruptly autofocusing cylindrically polarized beams,” J. Opt. Soc. Am. B 35(8), 1963–1969 (2018). [CrossRef]
24. S. S. Stafeev, A. G. Nalimov, M. V. Kotlyar, D. Gibson, S. Song, L. O’Faolain, and V. V. Kotlyar, “Microlens-aided focusing of linearly and azimuthally polarized laser light,” Opt. Express 24(26), 29800–29813 (2016). [CrossRef]
25. A. P. Porfirev, A. V. Ustinov, and S. N. Khonina, “Polarization conversion when focusing cylindrically polarized vortex beams,” Sci. Rep. 6(1), 6 (2016). [CrossRef]
26. S. N. Khonina, A. P. Porfirev, and S. V. Karpeev, “Recognition of polarization and phase states of light based on the interaction of nonuniformly polarized laser beams with singular phase structures,” Opt. Express 27(13), 18484–18492 (2019). [CrossRef]
27. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21(23), 1948–1950 (1996). [CrossRef]
28. M. Rashid, O. M. Marago, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A: Pure Appl. Opt. 11(6), 065204 (2009). [CrossRef]
29. J. Pu and Z. Zhang, “Tight focusing of spirally polarized vortex beams,” Opt. Laser Technol. 42(1), 186–191 (2010). [CrossRef]
30. Z.-H. Zhou, Y.-K. Guo, and L.-Q. Zhu, “Tight focusing of axially symmetric polarized vortex beams,” Chin. Phys. B 23(4), 044201 (2014). [CrossRef]
31. Zh. Man, Ch. Min, S. Zhu, and X.-C. Yuan, “Tight focusing of quasi-cylindrically polarized beams,” J. Opt. Soc. Am. A 31(2), 373–378 (2014). [CrossRef]
32. S. N. Khonina, “Vortex beams with high-order cylindrical polarization: features of focal distributions,” Appl. Phys. B 125(6), 100 (2019). [CrossRef]
33. A. L. Sokolov, “Comparative analysis of the characteristics of polar and non-polar spiral polarization rotators,” Opt. Photonics J. 10(02), 13–27 (2020). [CrossRef]
34. S. I. Kharitonov and S. N. Khonina, “Conversion of a conical wave with circular polarization into a vortex cylindrically polarized beam in a metal waveguide,” Comp. Opt. 42(2), 197–211 (2018). [CrossRef]
35. I. Moreno, J. A. Davis, I. Ruiz, and D. M. Cottrell, “Decomposition of radially and azimuthally polarized beams using a circular-polarization and vortex-sensing diffraction grating,” Opt. Express 18(7), 7173–7183 (2010). [CrossRef]
36. S. N. Khonina, V. V. Kotlyar, and V. A. Soifer, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48(10), 1543–1557 (2001). [CrossRef]
37. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006). [CrossRef]
38. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007). [CrossRef]
39. T. A. Fadeyeva and A. V. Volyar, “Extreme spin-orbit coupling in crystal-traveling paraxial beams,” J. Opt. Soc. Am. A 27(3), 381–389 (2010). [CrossRef]
40. J. Zhu, Y. Chen, Y. Zhang, X. Cai, and S. Yu, “Spin and orbital angular momentum and their conversion in cylindrical vector vortices,” Opt. Lett. 39(15), 4435–4438 (2014). [CrossRef]
41. K. Y. Bliokh, F. Rodriguez-Fortuno, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015). [CrossRef]
42. R. C. Devlin, A. Ambrosio, N. A. Rubin, J. P. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358(6365), 896–901 (2017). [CrossRef]
43. S. N. Khonina, S. V. Karpeev, V. D. Paranin, and A. A. Morozov, “Polarization conversion under focusing of vortex laser beams along the axis of anisotropic crystals,” Phys. Lett. A 381(30), 2444–2455 (2017). [CrossRef]
44. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011). [CrossRef]
45. P. Shi, L. Du, and X. Yuan, “Structured spin angular momentum in highly focused cylindrical vector vortex beams for optical manipulation,” Opt. Express 26(18), 23449–23459 (2018). [CrossRef]
46. M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett. 98(20), 201101 (2011). [CrossRef]
47. J. Ni, C. Wang, C. Zhang, Y. Hu, L. Yang, Z. Lao, and J. Chu, “Three-dimensional chiral microstructures fabricated by structured optical vortices in isotropic material,” Light: Sci. Appl. 6(7), e17011 (2017). [CrossRef]
48. Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37(8), 1286–1288 (2012). [CrossRef]
49. S. N. Khonina, A. V. Ustinov, and S. G. Volotovsky, “Shaping of spherical light intensity based on the interference of tightly focused beams with different polarizations,” Opt. Laser Technol. 60, 99–106 (2014). [CrossRef]
50. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristiensen, and J. P. Woerdman, “Helical-wave front laser beams produced with a spiral phase plate,” Opt. Commun. 112(5-6), 321–327 (1994). [CrossRef]
51. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. V. Shinkaryev, and G. V. Uspleniev, “Trochoson,” Opt. Commun. 91(3-4), 158–162 (1992). [CrossRef]
52. S. S. R. Oemrawsingh, J. A. W. van Houwelingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosterboer, and G. W. Hooft, “Production and characterization of spiral phase plates for optical wavelengths,” Appl. Opt. 43(3), 688–694 (2004). [CrossRef]
53. S. N. Khonina, A. V. Ustinov, S. A. Fomchenkov, and A. P. Porfirev, “Formation of hybrid higher-order cylindrical vector beams using binary multi-sector phase plates,” Sci. Rep. 8(1), 14320 (2018). [CrossRef]
54. S. N. Khonina, D. A. Savelyev, and N. L. Kazanskiy, “Vortex phase elements as detectors of polarization state,” Opt. Express 23(14), 17845–17859 (2015). [CrossRef]
55. J. J. J. Nivas, F. Cardano, Z. Song, A. Rubano, R. Fittipaldi, A. Vecchione, D. Paparo, L. Marrucci, R. Bruzzese, and S. Amoruso, “Surface structuring with polarization-singular femtosecond laser beams generated by a q-plate,” Sci. Rep. 7(1), 42142 (2017). [CrossRef]
56. J. J. J. Nivas, E. Allahyari, F. Cardano, A. Rubano, R. Fittipaldi, A. Vecchione, D. Paparo, L. Marrucci, R. Bruzzese, and S. Amoruso, “Surface structures with unconventional patterns and shapes generated by femtosecond structured light fields,” Sci. Rep. 8(1), 13613 (2018). [CrossRef]
57. K. Sugioka and Y. Cheng, “Ultrafast lasers — reliable tools for advanced materials processing,” Light: Sci. Appl. 3(4), e149 (2014). [CrossRef]
58. F. A. Müller, C. Kunz, and S. Gräf, “Bio-inspired functional surfaces based on laser-induced periodic surface structures,” Materials 9(6), 476 (2016). [CrossRef]
59. W. Qiao, T. Lei, Z. Wu, S. Gao, Z. Li, and X. Yuan, “Approach to multiplexing fiber communication with cylindrical vector beams,” Opt. Lett. 42(13), 2579–2582 (2017). [CrossRef]
60. B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Lightwave Technol. 36(2), 292–301 (2018). [CrossRef]
