Refractive Bi-Conic Axicon (Volcone) for Polarization Conversion of Monochromatic Radiation

A new element is proposed for producing an azimuthally polarized beam with a vortex phase dependence. The element is formed by two conical surfaces in such a way that the optical element resembles a mountain with a crater on top, like a volcano (volcanic cone is volcone). The element in the form of a refractive bi-conic axicon is fabricated by diamond turning, in which an internal conical cavity is made. Polarization conversion in this optical element occurs on the inner surface due to the refraction of beams at the Brewster angle. The outer surface is used to collimate the converted beam, which significantly distinguishes the proposed element from previously proposed approaches. The paper describes a method for calculating the path of beams through a refractive bi-conic axicon, taking into account phase and polarization conversions. In the case of incident circularly polarized radiation, azimuthally polarized ring-shape beam radiation is generated at the output. The proposed element is experimentally made of polymethyl methacrylate on a CNC milling machine. The experiment demonstrates the effectiveness of the proposed element.

Structured laser beams have recently attracted much attention due to the promise of their interaction with azopolymers [31,32], which are photosensitive materials containing azofunctional groups. A distinctive feature of such materials is reversible photoisomerization and mass transfer induced by optical forces. The polarization anisotropy of azopolymer films makes it possible to use beams with radial and azimuthal polarizations both to analyze the molecular structure of an azopolymer [33] and to form a complex microrelief on its surface [34,35].

Theoretical Description of the Method for Calculating the Beam Path
We propose a new refractive optical element for the generation of an azimuthally polarized beam with a vortex phase function. The element is formed by two conical surfaces in such a way that the inner conical cavity is made from the side of the pointed part of the truncated cone, so that a figure of rotation is formed, resembling a mountain with a crater at the top like a volcano. The central section of the element is shown in Figure 1a.
From bottom to top, a circularly polarized beam (green) is incident on the inner conical surface at the Brewster angle. The reflected part of the beam has azimuthal polarization. The second (outer) conical surface reflects the generated collimated beam forward. There is a technological offset ∆ from the outer radius of the bi-conic axicon. The offset is needed to ensure high quality of the working area of the second conical surface.
Let us present the derivation of the main formulae. Obviously, the incident and output beams must be collimated.
The proposed element is made of polymethyl methacrylate on a CNC milling machine. During experimental studies, the bi-conic axicon was illuminated by a laser at a wavelength of 532 nm. The element material index for the given wavelength is:  Let us present the derivation of the main formulae. Obviously, the incident and output beams must be collimated. The element was made from a cylinder with a radius R 0 . The radius of the exit pupil of the axicon is denoted by the letter R.
The technological offset is expressed by the formula: Photonics 2022, 9, 421 4 of 13 Then the offset of the edge of the working conical surface from the top edge of the element has the form: Letter designations of angles and sizes are introduced in Figure 1. Let us write down the basic formulae for the geometrical parameters of the axicon. The input beam is incident on the inner conical surface of the element at the Brewster angle The beam reflected from the inner surface is s-polarized. Further, this beam again is incident on the inner surface of the axicon at an angle ϕ, is refracted at an angle ψ, and is incident at an angle σ on the outer conical surface of the element, where total internal reflection occurs. The reflected beam exits the element through the top face of the element parallel to the optical axis.
The angles during interaction with the axicon surfaces can be calculated using the formulae: The element profile is defined by the polygon: Let a plane-parallel beam with circular polarization is incident on the bi-conic axicon. The Jones vector of the incident beam is expressed as: When the initial beam is incident on the bi-conic axicon, a completely s-polarized beam is reflected. Thus, the volcone works as a polaroid that transmits polarization along the polar vector ϕ.
The Jones matrix of a linear polaroid is expressed as: If we take into account the spatial rotation of matrix (19) by the angle ϕ + π/2, then the Jones matrix will take the form [1,77]: After passing the incident beam through the bi-conic axicon, we obtain the following expression for the output field polarization: It can be seen from Formula (21) that the output beam has azimuthal polarization with a first-order vortex phase.

Calculation of the Path of the Beams through a Volcone
The beam path in the element was calculated using software designed to take into account polarization ray tracing. We consider polarization state of each ray in Jones notation. Jones vector is transformed with Jones matrix which is automatically calculated for interaction of each ray with each surface. We calculate conversion efficiency as a ratio of passed integrated energy to incident integrated energy.
It is quite obvious that only a small part of the energy of the incident beam will form the output beam. The path of useful beams through a bi-conic axicon is shown in Figure 2.

Calculation of the Path of the Beams through a Volcone
The beam path in the element was calculated using software designed to take into account polarization ray tracing. We consider polarization state of each ray in Jones notation. Jones vector is transformed with Jones matrix which is automatically calculated for interaction of each ray with each surface. We calculate conversion efficiency as a ratio of passed integrated energy to incident integrated energy.
It is quite obvious that only a small part of the energy of the incident beam will form the output beam. The path of useful beams through a bi-conic axicon is shown in Figure  2. In this case, the energy efficiency of the element is 6.6%. The path of all the beams with a ray tracing depth of 5 is shown in Figure 3. In this case, the energy efficiency of the element is 6.6%. The path of all the beams with a ray tracing depth of 5 is shown in Figure 3. In this case, the energy efficiency of the element is 6.6%. The path of all the beams with a ray tracing depth of 5 is shown in Figure 3.  Figure 3 shows that, at a distance from the element, side beams are eliminated and a collimated azimuthally polarized beam is formed. Part of the energy is converted into a useful beam that propagates along the optical axis and is collimated. As can be seen in Figure 3, the remaining part of the energy is dissipated to the sides and at a short distance from the element completely leaves the zone of the useful beam. This fact is confirmed with experimental results.
We also considered the case of misalignment of the incident beam and the bi-conic axicon axis. The path of the beams through the bi-conic axicon for an incident beam inclination of 5° to the optical axis of the element is shown in Figure 4.  Figure 3 shows that, at a distance from the element, side beams are eliminated and a collimated azimuthally polarized beam is formed. Part of the energy is converted into a useful beam that propagates along the optical axis and is collimated. As can be seen in Figure 3, the remaining part of the energy is dissipated to the sides and at a short distance from the element completely leaves the zone of the useful beam. This fact is confirmed with experimental results.
We also considered the case of misalignment of the incident beam and the bi-conic axicon axis. The path of the beams through the bi-conic axicon for an incident beam inclination of 5 • to the optical axis of the element is shown in Figure 4.  We conducted a study of the tolerance when the incident beam hits the icon of the conical axis at small angles. The incident beam tilt gives the resulting ray ellipticity. The dependence of ellipticity on the angle of inclination of the incident beam to the optical axis of the element is shown in Figure 5. We calculate ellipticity that is introduced when the incident beam falls into the element not exactly perpendicularly using proper accounting of the Jones vector for each ray. We conducted a study of the tolerance when the incident beam hits the icon of the conical axis at small angles. The incident beam tilt gives the resulting ray ellipticity. The dependence of ellipticity on the angle of inclination of the incident beam to the optical axis of the element is shown in Figure 5. We calculate ellipticity that is introduced when the incident beam falls into the element not exactly perpendicularly using proper accounting of the Jones vector for each ray. Figure 5 shows that the maximum ellipticity of one of the output rays depends almost linearly on the angle of displacement. This dependence shows the deviations of the polarization of the output beam with partially azimuthal polarization from an ideally polarized beam with ideal azimuthal polarization with an increase in the angle of inclination of the input beam with circular polarization. Figure 6 shows exactly how the polarization of individual vectors changes when the angle of incidence of the initial beam deviates from the axis of the optical system. As can be seen, despite the appearance of ellipticity of the vectors that make up the azimuthal po-Photonics 2022, 9, 421 7 of 13 larization, this ellipticity introduces less distortion into the overall polarization distribution and the average coefficient of ellipticity is significantly lower than the maximum. We conducted a study of the tolerance when the incident beam hits the icon of the conical axis at small angles. The incident beam tilt gives the resulting ray ellipticity. The dependence of ellipticity on the angle of inclination of the incident beam to the optical axis of the element is shown in Figure 5. We calculate ellipticity that is introduced when the incident beam falls into the element not exactly perpendicularly using proper accounting of the Jones vector for each ray.  Figure 5 shows that the maximum ellipticity of one of the output rays depends almost linearly on the angle of displacement. This dependence shows the deviations of the polarization of the output beam with partially azimuthal polarization from an ideally polarized beam with ideal azimuthal polarization with an increase in the angle of inclination of the input beam with circular polarization. Figure 6 shows exactly how the polarization of individual vectors changes when the angle of incidence of the initial beam deviates from the axis of the optical system. As can be seen, despite the appearance of ellipticity of the vectors that make up the azimuthal polarization, this ellipticity introduces less distortion into the overall polarization distribution and the average coefficient of ellipticity is significantly lower than the maximum.

Materials and Methods
The calculated axicon was made by lathe turning. The material used was PLEXI-GLAS GS polymethyl methocrylate with a refractive index of 1.4958 for the 532 nm wavelength. The radius of the cutting edge of the tool used was 1 mm. The spindle speed during processing was equal to 1000 rpm, with the feed of 0.3 mm/rev. The specified turning regimes after the polishing made it possible to obtain an axicon surface of optical quality. The cylinder was not pre-treated. The appearance of the manufactured axicon is shown in Figure 7.

Materials and Methods
The calculated axicon was made by lathe turning. The material used was PLEXIGLAS GS polymethyl methocrylate with a refractive index of 1.4958 for the 532 nm wavelength. The radius of the cutting edge of the tool used was 1 mm. The spindle speed during processing was equal to 1000 rpm, with the feed of 0.3 mm/rev. The specified turning regimes after the polishing made it possible to obtain an axicon surface of optical quality. The cylinder was not pre-treated. The appearance of the manufactured axicon is shown in Figure 7. GLAS GS polymethyl methocrylate with a refractive index of 1.4958 for the 532 nm wavelength. The radius of the cutting edge of the tool used was 1 mm. The spindl speed during processing was equal to 1000 rpm, with the feed of 0.3 mm/rev. The speci fied turning regimes after the polishing made it possible to obtain an axicon surface o optical quality. The cylinder was not pre-treated. The appearance of the manufactured axicon is shown in Figure 7. The schematic of the experimental setup for studying a laser beam formed by an axicon is shown in Figure 8. The initial linearly polarized laser beam of a solid-state lase with a wavelength of 532 nm was expanded and spatially filtered using a system con sisting of a microobjective MO1 (20×, NA = 0.4), a pinhole PH (aperture size 40 μm), and a lens L1 (focal length 150 mm). A quarter-wave plate QWP was used to form circula polarization. The schematic of the experimental setup for studying a laser beam formed by an axicon is shown in Figure 8. The initial linearly polarized laser beam of a solid-state laser with a wavelength of 532 nm was expanded and spatially filtered using a system consisting of a microobjective MO1 (20×, NA = 0.4), a pinhole PH (aperture size 40 µm), and a lens L1 (focal length 150 mm). A quarter-wave plate QWP was used to form circular polarization. The linear polarizer P was used to analyze the polarization state of the output radiation, which was directed at the translucent screen S. Then the image of the collimated azimuthally polarized beam was projected through the lens D2 onto the matrix of the video camera.

Results and Discussion
As can be seen from the images of intensity distributions (Figure 9) at different screen distances from the bi-conic axicon, as a result of the passage of monochromatic circularly polarized radiation with a wavelength of 532 nm directed at the inner conica surface of the bi-conic axicon, the main part of the radiation is localized in two annular regions. These annular light distributions retain their structure when removed. From Figure 10 we can conclude that inner beam has an azimuthal polarization [5].
In this case, the inner ring is formed by a collimated azimuthally polarized laser beam with a small divergence. The remaining untransformed energy forms a strongly divergent outer ring and leaves the working area when removed. The linear polarizer P was used to analyze the polarization state of the output radiation, which was directed at the translucent screen S. Then the image of the collimated azimuthally polarized beam was projected through the lens D2 onto the matrix of the video camera.

Results and Discussion
As can be seen from the images of intensity distributions (Figure 9) at different screen distances from the bi-conic axicon, as a result of the passage of monochromatic circularly polarized radiation with a wavelength of 532 nm directed at the inner conical surface of the bi-conic axicon, the main part of the radiation is localized in two annular regions. These annular light distributions retain their structure when removed. From Figure 10 we can conclude that inner beam has an azimuthal polarization [5].
In this case, the inner ring is formed by a collimated azimuthally polarized laser beam with a small divergence. The remaining untransformed energy forms a strongly divergent outer ring and leaves the working area when removed.
The size of the image on the screen (Figure 9) corresponds to 160 mm. As can be seen from the images, the output beam (inner ring) with azimuthal polarization is collimated and its divergence does not exceed 6 mrad. The outer ring has a divergence of about 36 degrees that correspond to theoretical divergence angle of 34.9 degree. The inner ring has the theoretical radius R = 11 mm and thickness t = 4 mm. Experimentally measured meanings are approximately the same. screen distances from the bi-conic axicon, as a result of the passage of monochromatic circularly polarized radiation with a wavelength of 532 nm directed at the inner conical surface of the bi-conic axicon, the main part of the radiation is localized in two annular regions. These annular light distributions retain their structure when removed. From Figure 10 we can conclude that inner beam has an azimuthal polarization [5].
In this case, the inner ring is formed by a collimated azimuthally polarized laser beam with a small divergence. The remaining untransformed energy forms a strongly divergent outer ring and leaves the working area when removed.  Photonics 2022, 9,0 In this case, the inner ring is formed by a collimated azimuthally beam with a small divergence. The remaining untransformed energy f divergent outer ring and leaves the working area when removed.  The size of the image on the screen (Figure 9) corresponds to 160 mm. As can from the images, the output beam (inner ring) with azimuthal polarization is co and its divergence does not exceed 6 mrad. The outer ring has a divergence of degrees that correspond to theoretical divergence angle of 34.9 degree. The inner the theoretical radius R = 11 mm and thickness t = 4 mm. Experimentally m meanings are approximately the same.
This radiation is localized in the inner ring and the local area inside it. H there is a non-zero intensity in the center caused by the presence of a vortex phas This radiation is localized in the inner ring and the local area inside it. Howev is a non-zero intensity in the center caused by the presence of a vortex phase [7 the manufacture of the inner conical surface, it is quite difficult to perform turn polishing in the area of the center of rotation without error. For the optical exp this area was closed to eliminate the effect of manufacturing inaccuracy. The ge parameters of the conical surfaces, the ratio of the size of the inner cone and th one, and the size of the overlap area in the center have an effect on the width a of the rings in the area of the distribution of laser radiation intensity after passin This radiation is localized in the inner ring and the local area inside it. However, there is a non-zero intensity in the center caused by the presence of a vortex phase [78,79]. In the manufacture of the inner conical surface, it is quite difficult to perform turning and polishing in the area of the center of rotation without error. For the optical experiment, this area was closed to eliminate the effect of manufacturing inaccuracy. The geometric parameters of the conical surfaces, the ratio of the size of the inner cone and the outer one, and the size of the overlap area in the center have an effect on the width and size of the rings in the area of the distribution of laser radiation intensity after passing the biconical axicon. In this experiment, we did not adhere to the goal of obtaining the necessary intensity distribution or the necessary dimensions of the annular intensity distribution of azimuthally polarized radiation.
Further, we investigate the transformation of the polarization of the input radiation from circular polarization to azimuthal. The images in Figure 10 show the intensity distributions that are obtained for different orientation of the analyzing polarizer. The white arrow shows the orientation of passing axis of analyzer-polarizer P.
The measured polarization extinction ratio from the images in the Figure 10 is 1:220 on average. The ratio of the energies of the outer ring with respect to the radiation concentrated in the ring with azimuthal polarization corresponds to 1:16, which corresponds to the efficiency modeling in Section 2.
However, a nonzero intensity is observed in the center, caused by the presence of a vortex phase [78,79]. The polarization state of this region corresponds to the circular polarization of the input beam. This feature is consistent with the calculation results (Formula (21)).
In the introduction, we mentioned that there are various approaches to the formation of inhomogeneously polarized beams. A feature of the proposed approach is to use both surfaces of the bi-conic axicon: polarization conversion occurs on the inner surface due to the refraction of beams at the Brewster angle and the outer surface is used to collimate the converted beam. This is the essential difference between the proposed element and the previously proposed approaches.
Note that the use of two surfaces for different transformations of the laser beam is only possible for a refractive element. The second (collimating) surface can be replaced by a diffractive element, but this will complicate the optical design and increase the cost.
Besides, the bi-conic axicon can be applied for high-power laser radiation. This makes it possible to compensate for the not very high efficiency of the polarization conversion.

Conclusions
We have presented simulation and experimental results of the proposed new refractive polarizing bi-conic axicon. The theoretical description and numerical investigation show that the suggested bi-conic axicon can produce an azimuthally-polarized ring-shape beam radiation at a given wavelength. Provided experiments verified the conceptual model and efficiency of proposed element. We have shown that the bi-axicon converts the input circularly polarized laser beam incident on the inner conical surface into an azimuthally polarized beam with a vortex phase dependence. We have demonstrated that the beam is also collimated by the outer surface of the axicon. Despite quite low efficiency (6.6%) of our element, it is sufficient for various microscopic researches [17][18][19], optical microtrapping [20][21][22], laser microablation, and microinteraction [23][24][25][26][27][28][29][30][31][32], as well as other applications where only a few percent of input laser energy or less is sufficient.

Conflicts of Interest:
The authors declare no conflict of interest.