rus
DOI: 10.22184/1993-7296.FRos.2023.17.4.308.317
We have considered the diffraction of the first-order Bessel beam by zone plates with two odd open Fresnel zone to generate chain-like beams with an embedded phase singularity. We have shown that the capsule size depends on the number of the second odd open Fresnel zone and the zone plate focal length. The change of the zones relative illumination leads to the change of the contrast between dark and light regions. The best contrast corresponds to the equal illumination of the zones. We have experimentally generated a chain-like beam with an embedded vortex by the first-order Bessel beam diffraction by zone plates with the first and the ninth open Fresnel zones. We experimentally proved the dislocation presence and investigated the main beam features. We demonstrated sufficiently good agreement between experimental and numerically calculated results.
We have considered the diffraction of the first-order Bessel beam by zone plates with two odd open Fresnel zone to generate chain-like beams with an embedded phase singularity. We have shown that the capsule size depends on the number of the second odd open Fresnel zone and the zone plate focal length. The change of the zones relative illumination leads to the change of the contrast between dark and light regions. The best contrast corresponds to the equal illumination of the zones. We have experimentally generated a chain-like beam with an embedded vortex by the first-order Bessel beam diffraction by zone plates with the first and the ninth open Fresnel zones. We experimentally proved the dislocation presence and investigated the main beam features. We demonstrated sufficiently good agreement between experimental and numerically calculated results.
Теги: bessel beam binary amplitude diffraction mask chain-like beam fresnel zones phase singularity бинарная амплитудная дифракционная маска зоны френеля пучок бесселя фазовая сингулярность цепочноподобный пучок
Hollow
Chain-Like Beams
D. Yu. Cherepko 1, N. D. Kundikova 1, 2, I. I. Popkov 2
South Ural State University, Chelyabinsk, Russia
Institute of Electrophysics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia
We have considered the diffraction of the first-order Bessel beam by zone plates with two odd open Fresnel zone to generate chain-like beams with an embedded phase singularity. We have shown that the capsule size depends on the number of the second odd open Fresnel zone and the zone plate focal length. The change of the zones relative illumination leads to the change of the contrast between dark and light regions. The best contrast corresponds to the equal illumination of the zones. We have experimentally generated a chain-like beam with an embedded vortex by the first-order Bessel beam diffraction by zone plates with the first and the ninth open Fresnel zones. We experimentally proved the dislocation presence and investigated the main beam features. We demonstrated sufficiently good agreement between experimental and numerically calculated results.
Keywords: Chain-like beam, Fresnel zones, binary amplitude diffraction mask, phase singularity, Bessel beam
Article received on: 13.05.2023
Article accepted on: 02.06.2023
INTRODUCTION
Nowadays, structured light beams have gained much interest due to its unique amplitude and phase structures. The attention is also motivated by modern applications for advanced laser material processing [1–3], optical manipulation [4–6], communication [7–9] and for data storage [10].
The beams with the self-similar structure of intensity distribution along a direction of propagation form the class of beams useful for optical microparticle manipulation. The chain-like beam is a result of Gaussian beam diffraction by a zone plate with several open odd zones [11,12]. A diffraction pattern of a Gaussian beam passing through a fractal zone plate is a light beam with a self-like intensity structure along the beam propagation direction [13]. The usage of generalized zone plates increases the number of focal points, saving the property of self-similarity [14]. The main difference between the beams described in [13, 14] and chain-like beams [11, 12] is the intensity distribution between principal and secondary focal points and the number of focal points [11–14].
The combination of properties of the Bessel beams of the first order and chain-like beams generates a new class of beams with the property of self-similarity. Such kind of beams increase the fields of application and can help to observe new effects of the spin-orbit interaction of light [15–17]. One can generate beams with inhomogeneous intensity distribution along the propagation direction and the sequence of focused optical vortices along the propagation direction using a spiral fractal zone plate [18] and a helical vortex phase mask [19].
We propose to embed dislocation into a beam with inhomogeneous intensity distribution along the propagation direction by the zone plates illumination with a beam carrying a topological charge. To prove that, we investigate the properties of chain-like beams with phase singularity generated by the first-order Bessel beam diffraction by an amplitude binary zone plate with two open odd Fresnel zones numerically and experimentally.
NUMERICAL SIMULATION
OF THE HOLLOW CHAIN LIKE BEAM PROPERTIES
We have considered the diffraction of the first-order Bessel beam by an amplitude zone plate with two open odd Fresnel zones. The paraxial approximation was used for numerical simulation. The parabolic equation was solved by the spectral method, based on a two-dimensional Fourier transform. The wavelength of radiation diffracting on the mask was λ = 632.8 nm, and two odd Fresnel zones were open on a zone plate. The radius Rm of the Fresnel zone with the number m is
Rm = √—mFλ, (1)
where F is the zone plate focal length. A binary amplitude mask has the first Fresnel zone radius R1 = 0.96 mm. The main focus is on the distance z1 = R12 / λ = F = 145 cm. Additional focal spots were at the distances z2 = F / 3 = 48 cm and z3 = F / 5 = 29 cm. Figure 1 shows the mask with the first and ninth open Fresnel zones. The same binary amplitude mask was used in [11, 12, 20].
Diffraction of a Gauss beam by the zone plate, depicted in Fig. 1, results in a chain of light capsules forming a chain-like beam. Diffraction of the first-order Bessel by the same zone plate leads to a chain-like beam with a wavefront dislocation. Figure 2 shows diffraction trees for the first and the second cases. One can see from Fig. 2 that the main difference between the two beams is the presence of a dark channel at the beam axis along the propagation direction. The dark channel proves the phase singularity presence.
We have studied the first-order Bessel beam diffraction by the zone plates with the different odd open Fresnel zones by numerical simulations. Figure 3 shows the first-order Bessel beam diffraction trees for the zone plate with the first and the fifth open Fresnel zones, for the zone plate with the first and the ninth open Fresnel zones, and for the zone plate with the first and the thirteenth open Fresnel zones. We altered the Bessel beam parameters to allow an equal energy amount to pass through each zone.
One can see from Fig. 3 that the number of chosen zones influences the capsule’s longitudinal and transversal size. The increase in the difference between the zones numbers results in a decrease in the capsule size. Considering our experimental setup, we have carried out all the following numerical simulations with the zone plates with the first and the ninth open Fresnel zones.
The relative amount of energy passed through two open zones of a binary amplitude mask depends on the beam width. It has been demonstrated numerically and experimentally that a decrease in the Gaussian beam width increases the capsule sizes and the focus depth, decreases the intensity in the focus’s region, and does not affect the capsule position [12]. The relative portions of energy passed through two open Fresnel zone of the zone plate are different for a Gaussian beam and the first-order Bessel beam.
Let us consider how the first-order Bessel beam width influences the beam transverse and longitudinal intensity distribution. One can see from Fig. 4 that the alteration of the first-order Bessel beam width results in the change of the relative portions of energy passed through two open Fresnel zone. If the radius of the first ring of the first-order Bessel beam is equal to 2.3 mm (Fig. 4a), the value of the energy transmitted through the ninth zone relative to the energy transmitted through the first zone J9 / J1 = 0 will be equal to zero. If the radius of the first ring is equal to 3.45 mm (Fig. 4b) and 4.6 mm (Fig. 4c), the relative energy will be J9 / J1 = 0.5 and J9 / J1 = 2.5, respectively.
Figure 5 shows the first-order Bessel beam diffraction trees for the beam with the radius of the first ring equal to 2.3 mm (Fig.5a), 3.45 mm (Fig. 5b), and 4.6 mm (Fig. 5c).
One can see from Fig. 5 that the alteration of the first ring radius of a first-order Bessel beam, diffracted by the zone plate, results in the change of the contrast between the dark and the light areas of the chain-like beam with the phase singularity. If the Bessel beam illuminates only the first Fresnel zone, the diffraction tree will be similar to the diffraction by a circular aperture. Light capsules appear if the first ring radius of the Bessel beam increase. If equal energy passes through the first and the ninth Fresnel zones, the contrast between the dark regions and the light capsules will be maximal.
The beam profiles at the distance of 1.26 m for two beams with diffraction trees depicted in Fig.5b and Fig. 5c are presented in Fig. 6. The dashed lines in Fig. 5 show that the center of the light capsule is located at a distance of 1.26 m. One can see from Fig. 6 that if the value of J9 / J1 increases, the contrast decreases.
We have researched the influence of the zone plate focal length on the beam diffraction tree. We have changed the radii of the first and the ninth zone according to Eq. (1) and altered the Bessel beam parameters in such a way as to keep the value J9 / J1 unchanged and equal to 1. Figure 7 shows the results of the computer simulation. The dashed line marks the distance of 1 meter from the zone plate. One can see from Fig. 7 that the increase in the zone plate focal length leads to a change in the sizes and positions of capsules. However, the main beam features remain unchanged. So, we can control the capsule’s size and position by changing the zone plate focal length.
The computer simulation has allowed us to choose the parameter of the zone plate and the Bessel beam to generate the hollow chain-like beams.
EXPERIMENTAL GENERATION OF THE HOLLOW CHAIN-LIKE BEAM
Figure 8 shows the experimental set-up designed to generate and study the properties of the chain-like beams with a phase singularity.
Two amplitude masks are the key elements of the setup. The first mask is an amplitude mask (the Bessel mask) obtained in the following way. We interfered two beams, namely, a Gaussian beam and a first-order Bessel beam in a computer experiment and then printed the negative image of the interference pattern in an enlarged size. The printed negative image was photographed using a film with high resolution. To generate the first-order Bessel beam, we used that film 3.5 × 2.5 cm2 in size. The second mask is a binary amplitude mask 3.5 × 2.5 cm2 in size with the first and ninth open Fresnel zones and the first Fresnel zone radius R1 = 0.96 mm, generated on the computer (256 × 256 pixels) and further printed on the transparency with resolution 600 dpi (Fig. 2). We used the output of a He-Ne laser with a wavelength of 632.8 nm and a power of 1.5 mW.
We expanded the laser beam by an optical system to a width of 3 cm. The expanded collimated coherent monochromatic light interacted with the Bessel mask. The diffracted beam was focused to select the proper order from the diffracted beam in the focal plane by diaphragm. The selected first-order Bessel beam was collimated and interacted with the second amplitude mask. The value J9 / J1 was equal to 7. We registered the cross section of the resulting chain-like beam with the phase singularity by a CCD camera at different propagation distances. The CCD camera (VEC‑545) with an image area of 5.81 × 4.29 mm2 and pixel elements of 2 592 × 1 944, each pixel of about 2,2 μm.
Figure 9 shows the beam transverse intensity distribution at a distance of 1.6 m. The left half of the image is the experimental image, and the right half is the calculated image at the same distance from the mask and J9 / J1 = 7. Figure 9 demonstrates a reasonably good agreement between the experimental and calculated intensity distribution.
We have installed additional optical elements forming an interferometer into the experimental setup to prove the dislocation presence in the beam under investigation. We have split the laser beam into two ones. The first beam has been used to obtain the beam under investigation, and the second beam expanded and collimated, has interfered with the beam under investigation. Figure 10 shows the interference pattern of the Gaussian beam with the chain-like beam with the phase singularity. The interference patterns are the same, but a dashed line in Fig. 10 b shows a “fork” position. A fork-like dislocation (Fig. 10) with a difference of one arm corresponds to phase winding by 2π around the vortex core and proves the vortex presence in the beam.
Using a CCD matrix, the intensity distribution of the hollow chain-like beam has been recorded at distances ranging from 100 to 201 cm with a step of 1 cm. We have processed the experimentally recorded diffraction patterns to obtain the diffraction tree in the following way. A section fitting the length of the beam diameter has been selected from every diffraction pattern so that the thickness of each section corresponds to one pixel of the CCD camera (in the actual case). Then, placing them consecutively in the growing order of z, the image representing a part of the diffraction tree of the beam under investigation has been built up. The left part of Fig. 11 presents the processed experimental data. The right half of Fig. 11 shows the diffraction tree calculated numerically under the same parameters. One can easily see from Fig. 11 that the diffraction tree, constructed from the experimental data, depicts the main features of the diffraction tree calculated numerically, the light beam resembles a chain-like structure, and the dark channel on the beam axis demonstrates the vortex presence.
CONCLUSIONS
In summary, we have studied the diffraction of the first-order Bessel beam by zone plates with two odd open Fresnel zone. The result of the diffraction is a chain-like beam with embedded wavefront dislocation or a hollow chain-like beam. We have shown numerically that the number of the second odd open Fresnel zone influences the capsule’s longitudinal and transversal size; namely, the increase of the difference between the zone numbers leads to a decrease in the capsule size. The change in the focal length of the zone plate results in the capsule size changing; the variation of the relative illumination leads to the alteration of the contrast between dark and light regions. The equal energy passed through each two zone results in the best contrast. We have experimentally generated a hollow chain-like beam using the first-order Bessel beam diffraction by zone plates with the first and the ninth open Fresnel zones. We experimentally have proved the dislocation presence and studied the main features of the beam; a comparison of numerically calculated results and experimentally obtained results demonstrate sufficiently good agreement. So, we proposed a new approach to hollow chain-like beam generation and used this approach to generate the beam experimentally.
AUTHORS
Cherepko Dmitry Yurievich, Graduate Student, South Ural State University, Chelyabinsk, Russia.
Popkov Ivan Igorevich, Candidate of Sciences (Phys.&Math.), Researcher, Institute of Electrophysics, Ural Branch of The Russian Academy of Sciences, Yekaterinburg, Russia.
ORCID:0009-0008-4259-4376
Kundikova Natalia Dmitrievna, Dr. of Sciences (Phys.&Math.), professor, Head of laboratories, Institute of Electrophysics, Ural Branch of the Russian Academy of Sciences, Yekaerinburg; Head of department, South Ural State University,
Chain-Like Beams
D. Yu. Cherepko 1, N. D. Kundikova 1, 2, I. I. Popkov 2
South Ural State University, Chelyabinsk, Russia
Institute of Electrophysics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia
We have considered the diffraction of the first-order Bessel beam by zone plates with two odd open Fresnel zone to generate chain-like beams with an embedded phase singularity. We have shown that the capsule size depends on the number of the second odd open Fresnel zone and the zone plate focal length. The change of the zones relative illumination leads to the change of the contrast between dark and light regions. The best contrast corresponds to the equal illumination of the zones. We have experimentally generated a chain-like beam with an embedded vortex by the first-order Bessel beam diffraction by zone plates with the first and the ninth open Fresnel zones. We experimentally proved the dislocation presence and investigated the main beam features. We demonstrated sufficiently good agreement between experimental and numerically calculated results.
Keywords: Chain-like beam, Fresnel zones, binary amplitude diffraction mask, phase singularity, Bessel beam
Article received on: 13.05.2023
Article accepted on: 02.06.2023
INTRODUCTION
Nowadays, structured light beams have gained much interest due to its unique amplitude and phase structures. The attention is also motivated by modern applications for advanced laser material processing [1–3], optical manipulation [4–6], communication [7–9] and for data storage [10].
The beams with the self-similar structure of intensity distribution along a direction of propagation form the class of beams useful for optical microparticle manipulation. The chain-like beam is a result of Gaussian beam diffraction by a zone plate with several open odd zones [11,12]. A diffraction pattern of a Gaussian beam passing through a fractal zone plate is a light beam with a self-like intensity structure along the beam propagation direction [13]. The usage of generalized zone plates increases the number of focal points, saving the property of self-similarity [14]. The main difference between the beams described in [13, 14] and chain-like beams [11, 12] is the intensity distribution between principal and secondary focal points and the number of focal points [11–14].
The combination of properties of the Bessel beams of the first order and chain-like beams generates a new class of beams with the property of self-similarity. Such kind of beams increase the fields of application and can help to observe new effects of the spin-orbit interaction of light [15–17]. One can generate beams with inhomogeneous intensity distribution along the propagation direction and the sequence of focused optical vortices along the propagation direction using a spiral fractal zone plate [18] and a helical vortex phase mask [19].
We propose to embed dislocation into a beam with inhomogeneous intensity distribution along the propagation direction by the zone plates illumination with a beam carrying a topological charge. To prove that, we investigate the properties of chain-like beams with phase singularity generated by the first-order Bessel beam diffraction by an amplitude binary zone plate with two open odd Fresnel zones numerically and experimentally.
NUMERICAL SIMULATION
OF THE HOLLOW CHAIN LIKE BEAM PROPERTIES
We have considered the diffraction of the first-order Bessel beam by an amplitude zone plate with two open odd Fresnel zones. The paraxial approximation was used for numerical simulation. The parabolic equation was solved by the spectral method, based on a two-dimensional Fourier transform. The wavelength of radiation diffracting on the mask was λ = 632.8 nm, and two odd Fresnel zones were open on a zone plate. The radius Rm of the Fresnel zone with the number m is
Rm = √—mFλ, (1)
where F is the zone plate focal length. A binary amplitude mask has the first Fresnel zone radius R1 = 0.96 mm. The main focus is on the distance z1 = R12 / λ = F = 145 cm. Additional focal spots were at the distances z2 = F / 3 = 48 cm and z3 = F / 5 = 29 cm. Figure 1 shows the mask with the first and ninth open Fresnel zones. The same binary amplitude mask was used in [11, 12, 20].
Diffraction of a Gauss beam by the zone plate, depicted in Fig. 1, results in a chain of light capsules forming a chain-like beam. Diffraction of the first-order Bessel by the same zone plate leads to a chain-like beam with a wavefront dislocation. Figure 2 shows diffraction trees for the first and the second cases. One can see from Fig. 2 that the main difference between the two beams is the presence of a dark channel at the beam axis along the propagation direction. The dark channel proves the phase singularity presence.
We have studied the first-order Bessel beam diffraction by the zone plates with the different odd open Fresnel zones by numerical simulations. Figure 3 shows the first-order Bessel beam diffraction trees for the zone plate with the first and the fifth open Fresnel zones, for the zone plate with the first and the ninth open Fresnel zones, and for the zone plate with the first and the thirteenth open Fresnel zones. We altered the Bessel beam parameters to allow an equal energy amount to pass through each zone.
One can see from Fig. 3 that the number of chosen zones influences the capsule’s longitudinal and transversal size. The increase in the difference between the zones numbers results in a decrease in the capsule size. Considering our experimental setup, we have carried out all the following numerical simulations with the zone plates with the first and the ninth open Fresnel zones.
The relative amount of energy passed through two open zones of a binary amplitude mask depends on the beam width. It has been demonstrated numerically and experimentally that a decrease in the Gaussian beam width increases the capsule sizes and the focus depth, decreases the intensity in the focus’s region, and does not affect the capsule position [12]. The relative portions of energy passed through two open Fresnel zone of the zone plate are different for a Gaussian beam and the first-order Bessel beam.
Let us consider how the first-order Bessel beam width influences the beam transverse and longitudinal intensity distribution. One can see from Fig. 4 that the alteration of the first-order Bessel beam width results in the change of the relative portions of energy passed through two open Fresnel zone. If the radius of the first ring of the first-order Bessel beam is equal to 2.3 mm (Fig. 4a), the value of the energy transmitted through the ninth zone relative to the energy transmitted through the first zone J9 / J1 = 0 will be equal to zero. If the radius of the first ring is equal to 3.45 mm (Fig. 4b) and 4.6 mm (Fig. 4c), the relative energy will be J9 / J1 = 0.5 and J9 / J1 = 2.5, respectively.
Figure 5 shows the first-order Bessel beam diffraction trees for the beam with the radius of the first ring equal to 2.3 mm (Fig.5a), 3.45 mm (Fig. 5b), and 4.6 mm (Fig. 5c).
One can see from Fig. 5 that the alteration of the first ring radius of a first-order Bessel beam, diffracted by the zone plate, results in the change of the contrast between the dark and the light areas of the chain-like beam with the phase singularity. If the Bessel beam illuminates only the first Fresnel zone, the diffraction tree will be similar to the diffraction by a circular aperture. Light capsules appear if the first ring radius of the Bessel beam increase. If equal energy passes through the first and the ninth Fresnel zones, the contrast between the dark regions and the light capsules will be maximal.
The beam profiles at the distance of 1.26 m for two beams with diffraction trees depicted in Fig.5b and Fig. 5c are presented in Fig. 6. The dashed lines in Fig. 5 show that the center of the light capsule is located at a distance of 1.26 m. One can see from Fig. 6 that if the value of J9 / J1 increases, the contrast decreases.
We have researched the influence of the zone plate focal length on the beam diffraction tree. We have changed the radii of the first and the ninth zone according to Eq. (1) and altered the Bessel beam parameters in such a way as to keep the value J9 / J1 unchanged and equal to 1. Figure 7 shows the results of the computer simulation. The dashed line marks the distance of 1 meter from the zone plate. One can see from Fig. 7 that the increase in the zone plate focal length leads to a change in the sizes and positions of capsules. However, the main beam features remain unchanged. So, we can control the capsule’s size and position by changing the zone plate focal length.
The computer simulation has allowed us to choose the parameter of the zone plate and the Bessel beam to generate the hollow chain-like beams.
EXPERIMENTAL GENERATION OF THE HOLLOW CHAIN-LIKE BEAM
Figure 8 shows the experimental set-up designed to generate and study the properties of the chain-like beams with a phase singularity.
Two amplitude masks are the key elements of the setup. The first mask is an amplitude mask (the Bessel mask) obtained in the following way. We interfered two beams, namely, a Gaussian beam and a first-order Bessel beam in a computer experiment and then printed the negative image of the interference pattern in an enlarged size. The printed negative image was photographed using a film with high resolution. To generate the first-order Bessel beam, we used that film 3.5 × 2.5 cm2 in size. The second mask is a binary amplitude mask 3.5 × 2.5 cm2 in size with the first and ninth open Fresnel zones and the first Fresnel zone radius R1 = 0.96 mm, generated on the computer (256 × 256 pixels) and further printed on the transparency with resolution 600 dpi (Fig. 2). We used the output of a He-Ne laser with a wavelength of 632.8 nm and a power of 1.5 mW.
We expanded the laser beam by an optical system to a width of 3 cm. The expanded collimated coherent monochromatic light interacted with the Bessel mask. The diffracted beam was focused to select the proper order from the diffracted beam in the focal plane by diaphragm. The selected first-order Bessel beam was collimated and interacted with the second amplitude mask. The value J9 / J1 was equal to 7. We registered the cross section of the resulting chain-like beam with the phase singularity by a CCD camera at different propagation distances. The CCD camera (VEC‑545) with an image area of 5.81 × 4.29 mm2 and pixel elements of 2 592 × 1 944, each pixel of about 2,2 μm.
Figure 9 shows the beam transverse intensity distribution at a distance of 1.6 m. The left half of the image is the experimental image, and the right half is the calculated image at the same distance from the mask and J9 / J1 = 7. Figure 9 demonstrates a reasonably good agreement between the experimental and calculated intensity distribution.
We have installed additional optical elements forming an interferometer into the experimental setup to prove the dislocation presence in the beam under investigation. We have split the laser beam into two ones. The first beam has been used to obtain the beam under investigation, and the second beam expanded and collimated, has interfered with the beam under investigation. Figure 10 shows the interference pattern of the Gaussian beam with the chain-like beam with the phase singularity. The interference patterns are the same, but a dashed line in Fig. 10 b shows a “fork” position. A fork-like dislocation (Fig. 10) with a difference of one arm corresponds to phase winding by 2π around the vortex core and proves the vortex presence in the beam.
Using a CCD matrix, the intensity distribution of the hollow chain-like beam has been recorded at distances ranging from 100 to 201 cm with a step of 1 cm. We have processed the experimentally recorded diffraction patterns to obtain the diffraction tree in the following way. A section fitting the length of the beam diameter has been selected from every diffraction pattern so that the thickness of each section corresponds to one pixel of the CCD camera (in the actual case). Then, placing them consecutively in the growing order of z, the image representing a part of the diffraction tree of the beam under investigation has been built up. The left part of Fig. 11 presents the processed experimental data. The right half of Fig. 11 shows the diffraction tree calculated numerically under the same parameters. One can easily see from Fig. 11 that the diffraction tree, constructed from the experimental data, depicts the main features of the diffraction tree calculated numerically, the light beam resembles a chain-like structure, and the dark channel on the beam axis demonstrates the vortex presence.
CONCLUSIONS
In summary, we have studied the diffraction of the first-order Bessel beam by zone plates with two odd open Fresnel zone. The result of the diffraction is a chain-like beam with embedded wavefront dislocation or a hollow chain-like beam. We have shown numerically that the number of the second odd open Fresnel zone influences the capsule’s longitudinal and transversal size; namely, the increase of the difference between the zone numbers leads to a decrease in the capsule size. The change in the focal length of the zone plate results in the capsule size changing; the variation of the relative illumination leads to the alteration of the contrast between dark and light regions. The equal energy passed through each two zone results in the best contrast. We have experimentally generated a hollow chain-like beam using the first-order Bessel beam diffraction by zone plates with the first and the ninth open Fresnel zones. We experimentally have proved the dislocation presence and studied the main features of the beam; a comparison of numerically calculated results and experimentally obtained results demonstrate sufficiently good agreement. So, we proposed a new approach to hollow chain-like beam generation and used this approach to generate the beam experimentally.
AUTHORS
Cherepko Dmitry Yurievich, Graduate Student, South Ural State University, Chelyabinsk, Russia.
Popkov Ivan Igorevich, Candidate of Sciences (Phys.&Math.), Researcher, Institute of Electrophysics, Ural Branch of The Russian Academy of Sciences, Yekaterinburg, Russia.
ORCID:0009-0008-4259-4376
Kundikova Natalia Dmitrievna, Dr. of Sciences (Phys.&Math.), professor, Head of laboratories, Institute of Electrophysics, Ural Branch of the Russian Academy of Sciences, Yekaerinburg; Head of department, South Ural State University,
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