Open Access
Add to BibSonomy
Abstract
Systems of controllable orbital angular momentum (OAM) require more compact, higher conversion efficiency and more tolerable wavelength or polarization. We introduce an optical vortex switch based on a multiplexed volume grating (MVG). The MVG recorded in a piece of photo-thermo-refractive (PTR) glass exhibits high diffraction efficiency (DE, also known as conversion efficiency in transporting), sensitive angular selectivity, and polarization-insensitivity. The effects of the incident divergence angle and polarization on the DE and the far-field diffraction profiles are demonstrated and investigated. It turns out that the divergence angle of the probe beam can greatly affect the DE. The fluctuation of the DE caused by polarization variation is less than 1.59%. This switch can be potentially applied in vortex tweezers, optical communication, and high power systems.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Light beams with spiral wavefront called optical vortices exhibit zero intensity in the center and spiral phase, carrying orbital angular momentum (OAM) [1]. As a result, when a vortex beam interacts with small objects varied from microscopic particles to biological cells, the OAM of the beam can be delivered to the objects which then spiral around the propagation axis. This characteristic is widely used for manipulating matters in tweezers and optical spanners [2,3]. On the other hand, theoretically infinite orthogonal OAM modes enable high information capacity for optical communication [4–6]. The entanglement of two photons carrying opposite OAM has been used to testify the Bell inequality violation [7]. Therefore, the optical vortex can also be applied in quantum teleportation [8] and quantum image memory [9]. In order to obtain optical vortices, OAM generation methods inside and outside the laser cavity have been deeply developed over decades [10–13]. Moreover, modern optical systems require more efficient, flexible, and compact components to realize the controllable mode modulation.
Up to now, there have been several methods to control the OAM mode. OAM-tunable lasers using mode converter [14], tailored fibers [15], meta-surfaces [16] and spot-defect mirror [17] are reliable techniques for generating stable Laguerre-Gaussian beams. In addition, OAM generators outside the cavity, such as the Dammann gratings [18], liquid crystal fork gratings [12], seem to be more portable comparing with the intrinsic complex structure of the OAM-tunable lasers. Most of these techniques are based on the traditional thin diffraction elements, typically as spatial light modulators (SLMs). Moreover, due to diffraction in the Raman-Nath regime, these thin elements do neither have high conversion efficiency, angular and wavelength selectivity nor have the capability of multiplexing, which restricts the applications in information modulation and high power systems.
Fortunately, there are other kinds of diffraction gratings called volume Bragg gratings (VBGs) exhibiting volume effect and characterizing only one diffraction order under the particular Bragg conditions [19]. As a result, they have excellent angular and spectral selectivity, which are thereby widely used as filters [20], beam controllers [21], waveguides in augmented reality [22], etc. In addition, VBGs can be multiplexed to store large amount of data, forming multiplexed volume gratings (MVGs), which plays an essential role in volume holographic imaging (VHI) [23], angle amplifiers [24] and high-power beam combining [25]. The most promising application is to multiplex spatial modes, where the stored data of MVG in VHI are substituted by light phase modes [26,27]. However, in Ref. [26] the incident beam was modulated and separated into several modes by the designed MVG, leading to uneven and low diffraction efficiencies (DE). Furthermore, the used material PQ/PMMA is an organic photopolymer, which is flawed by shrinking deformation and cannot be used in high power systems.
In this paper, an MVG recorded in photo-thermo-refractive (PTR) glass is designed and fabricated to realize optical vortex switching by rotating the MVG to specific Bragg angles. Effects of the divergent angle and polarization of the incident beam on the DE and far-field profiles are investigated. The experimental results are highly consistent with the theoretical ones. To our best of knowledge, this is the first time to impose the MVG on switching OAM modes with high diffraction efficiency. This kind of grating is of guiding significance to the tunable optical mode multiplexing and generation in high-power systems, quantum communication, and optical tweezers.
2. Modelling and design of the MVG
The principle of our tunable OAM method utilizing an MVG is illustrated in Fig. 1. We apply multiple times recording in a piece of volume holographic material with different interferograms.As a result, it forms refractive index variation inside the material (approximately in proportion to the intensity of the interferogram) at corresponding fringe tilt angles. As depicted in Fig. 1(b), the gray-scaled fringes $G_m$ ($m=1,2,\ldots$) are generated by the interference of two conventional plane waves, one of which is embedded in different topological charges of $l_m$ (for instance here gives $l_1=-1, l_2=0, l_3=1$). According to the principle of holography, the diffracted wave can reconstruct the recorded object phase, thus carrying OAM. Thanks to the instinctive angular selectivity of volume gratings, the output OAM modes can be tuned by controlling the incident angle, at one’s convenience, by rotating the MVG along the axis perpendicular to the plane of incidence ($y$-axis in Fig. 1(a)).
In order to realize OAM switching, we record the interfered waves in the medium with an unchanged interference angle but at different rotation angles. According to the Bragg conditions of volume gratings [19], the resultant readout angles and the periods of each channel in the MVG are approximately identical. Note that the Bragg conditions can be presented as $K$ vector circles, yielding to $2n\Lambda \textrm {sin}[(\theta '_s-\theta '_r)/2]=\lambda$, where $\theta '_r$ and $\theta '_s$ represent the incident and diffracted angles inside the medium, $\Lambda$ and $n$ represent the period and average refractive index of the grating, $\lambda$ is the wavelength of the incident beam. As shown in Fig. 2, two concentric circles colored with purple and red illustrate two different wavelengths in the recording and reconstruction process, wherein $\beta_i=2\pi n/\lambda_i$ $(i=1,2)$, $\vec {\rho }$ and $\vec {\sigma }$ are wave vectors of the recording reference and object waves, respectively. And similarly, $\vec {r}$ and $\vec {s}$ are wave vectors of the reconstruction beam and diffracted beam, respectively. $\vec {K}$ is the grating vector and values $2\pi /\Lambda$. Thus these vectors satisfy $\vec {K}=\vec {r}-\vec {s}=\vec {\rho }-\vec {\sigma }$, then giving the structure parameters of the $m$-th channel in MVG from the Bragg conditions as
Fig. 1. (a) Scheme of OAM modulation and switching by rotating the MVG. The exampled MVG is exposed three times by (b) the corresponding interferograms marked as $G_1-G_3$. The topological charge of the diffracted beam switches to $l_m$ as the MVG rotates along the $y$-axis at the Bragg angle $\theta _{r,m}$ ($m=1,2,3$ for each channel) with respect to the propagation direction of the incident Gaussian beam.
Fig. 2. Schematics of the $\boldsymbol {K}$ vector circles of the volume grating for different wavelength reconstruction. The angles are counted in the anti-clockwise direction.
For the sake of storing as many channels as possible and at the same time lessening cross-talk, 2 mm of the thickness $d$ and $6.55^\circ$ of the interval tilt angle $\phi$ between channels of the gratings are taken into consideration. The recording and reconstruction wavelengths are set to 325 nm and 1064 nm, then the grating parameters for each channel can be obtained by Eqs. (1) and shown in Table 1. CH4-6 are under the conditions that the reconstruction beams are incident at the angles of the diffracted beams of CH1-3, respectively, resulting in reversed OAMs. Incidentally, the MVG can also be operated at other wavelengths, while the incident and readout angles should be tuned to the corresponding Bragg angles calculated according to Eqs. (1).
Table 1. Designed characteristic parameters of each grating channel recorded at 325 nm and working at 1064 nm. Optimized refractive index modulation ($\Delta n$), as well as diffraction efficiencies (DE) of non-divergent (Non-div.) and divergent (Div. for $b=480$$\mu$rad) incident are shown.
Next, considering the best RIM for maximum DE, we apply the matrix-based algorithm proposed by Ingersoll et al. [28] to calculate the diffraction efficiencies for all channels in the MVG. The DE for $m$-th channel is thus characterized as
Note that the incident divergence angle $b$ has a non-negligible influence on the DE due to the super-narrow angular spectrum of the VBG, Eq. (2) should be modified by the Gaussian function integral [29]
Fig. 3. (a) Diffraction efficiency (DE) distribution of the MVG associated with the refractive index modulation $\Delta n$ and the deviation of incident angle $\Delta \theta _r$ at the divergence angle $b=480$ $\mu$rad. Two specific refractive index modulation (RIM) values $\Delta n$ of extrema of the DE are marked. (b) Angular spectra of the two $\Delta n$ values with ($b=480$ $\mu$rad) and without the incident divergence angle.
3. Fabrication and test procedure
We utilize the photo-thermo-refractive glass as the holographic volume material benefiting from its low optical losses, high damage threshold, and excellent environmental stability [31]. As depicted in Fig. 4(a), the PTR sample is set on an electric rotator and exposed by two interfered expanded and collimated waves split from a He-Cd laser at a wavelength of 325 nm. For CH1 and CH3, a spiral phase plate (SPP with topological charge -1 and 1, respectively) is inserted in one of the arms, and for CH2, the two arms are both unperturbed plane waves. The multi-exposed procedure is implemented by rotating the electric rotator after finishing the exposed channel. Note that the exposure order and dosage pose an influence on the consequent DE. We subtly adjust the exposure time of the three channels and set CH2 as the final exposure order to minimize the effect of fringe elimination caused by the adjacent exposure order. In addition, for tilt channels CH1 and CH3, there is energy intensity variation compared to the non-tilt channel CH2, resulting in decreased interferogram contrast. The maximum and minimum of the interference intensity are respectively described as $I_{max}=I_{\rho }+I_{\sigma }+2\sqrt {I_{\rho }I_{\sigma }}$ and $I_{min}=I_{\rho }+I_{\sigma }-2\sqrt {I_{\rho }I_{\sigma }}$. Under the condition of linear response, the resultant RIM $\Delta n \propto I_{max}-I_{min} \propto \sqrt {I_{\rho }I_{\sigma }}$. If the cross-section intensities of the two beams are identical, then the effective dosage is in proportion to $\sqrt {\textrm {cos}\theta _{\sigma }\textrm {cos}\theta _{\rho }}$. In this experiment, the original dosage for CH2 is set to 200 $mJ/cm^2$, then dosages for CH1 and CH3 are accordingly identical and equal to 203.15 $mJ/cm^2$, which can be finely adjusted by exposure time using an electronic shutter.
Fig. 4. Experiment setup of (a) exposure recording and (b) reconstruction of the MVG. M1-M4: mirrors; BS: beam splitter; SPP: spiral phase plate; $\frac {\lambda }{2}, 2\alpha$: half-wave plate at an angle of $2\alpha$ between the fast axis and $x$-axis; PM1, PM2: power meters
After the exposure, the PTR sample is repeatedly heat-treated until all channels reach the designed $\Delta n$ in Table 1. Method in Ref. [29] is utilized to evaluate the $\Delta n$ of each heat-treatment from the experimental angular spectrum. However, since the exposed and heat-treated PTR sample cannot be exposed again, maximizing the DE of all channels is a challenge. Even so, we fabricated the MVG with three-time heat treatment under the temperature 500 ℃ for sequentially 60, 20, 10 minutes. In order to measure the angular spectrum, $\Delta n$ and the intensities of diffracted beams for all channels, the flexible setup of reconstruction is demonstrated and shown in Fig. 4(b). We employ a TE-polarized Nd: YAG laser at the wavelength of 1064 nm with the original divergence angle of 480 $\mu$rad (retrieved by an $M^2$ measurement system) as the incident Gaussian beam. To investigate the dependence of the incident divergence angle on the DE and far-field profiles, a $5\times$ beam expander has been applied to reduce the incident divergence angle. Then, the probe beam’s waist width is expanded from 0.9894 mm to 4.947 mm, and the divergence angle is correspondingly reduced to 96 $\mu$rad. A half-wave plate is used to adjust the polarization of the reconstruction beam at $2\alpha$ angle between the fast axis and $x$-axis, yielding to the polarization direction of $\alpha$ of the linear-polarized beam. The fabricated sample is set on an electric rotator, followed by two power meters (PM1 and PM2) to measure intensities for the transmitted beam ($I_t$) and the diffracted beam ($I_d$). Then the relative DE is described by $I_{d}/(I_{d}+I_{t}) \times 100\%$. A CCD camera is set at about 60 cm away from the sample (farther than the Rayleigh length) to image the diffracted beams.
4. Results and discussions
4.1 Effects of the divergence angle
The angular spectra of the six channels probed by the original incident laser at $b=480$ $\mu$rad are shown in Fig. 5(a), in which the DE of all channels all surpass 80%. As shown in Fig. 5(b,c), after the beam expansion, the maxima of DE for all channels can reach over 92.68%, which means the conversion efficiency would be dramatically affected by the incident divergence angle. To determine the influence of the divergence angle on the angular selectivity, we have calculated the theoretical $\Delta \Theta$ (the full width at half maximum of the angular spectrum) for all separated channels, which gives that the values for all channels are the nearly identical, and equal to $1.2498^\circ ,1.2580^\circ$, and $1.4676^\circ$ at b = 0, 96, and 480 $\mu$rad, respectively. It turns out that, theoretically, the angular spectrum widens with a larger divergence angle. Note that the experimental results in Fig. 5(d) are inconsistent with the theoretical conclusion, which may mainly result from the crosstalk and the measurement error. However, for six channels characterized as the full width at half maximum (FWHM) of the angular spectrum are less than 1.5 mrad, bringing out the theoretical storage resolution of more than 10 channels/degree.
Fig. 5. Measured angular spectra for all six channels of the fabricated MVG at (a) $b=480$ $\mu$rad and (b) b=96 $\mu$rad, as well as the theoretical and experimental (c) maximum DE and (d) $\Delta \Theta$ (the full width at half maximum of the angular spectrum).
Figure 6 shows the profiles of the diffracted beam for all channels. After the beam expansion, the CCD camera is set about 450 cm away from the sample to ensure the far-field diffraction. It can be seen that the reconstructed vortex beams at $b=96$ $\mu$rad are less elliptical than those at $b=480$ $\mu$rad. An opaque black screen (not shown) is utilized behind the sample and partially truncates the diffracted beams to testify the topological charges of the diffracted beams. Resultant edge-diffraction patterns in Fig. 6(b,d) can determine topological charges of the generated OAM beam: the redundant fringes of the fork-shaped fringes imply the value, and the orientation (red arrows) reads the handedness [32]. For example, as Fig. 6(b1) shows, the redundant fringe is 1, and the fork fringe orients to the downside, so the topological charge is 1.
Fig. 6. (a)(c) Measured diffraction intensity profiles for all six channels of the fabricated MVG and (b)(d) corresponding edge-diffraction patterns at the incident divergent angles of (a)(b) 480 $\mu$rad and (c)(d) 96 $\mu$rad. Hatched area shows the position of the screen and red arrows indicate the direction of the fork in the edge-diffraction patterns. The intensity distribution is normalized.
4.2 Polarization dependence
According to Ref. [19], the diffraction efficiency of TM mode can be calculated by replacing a coupling constant $\kappa _{TM}=-\kappa _{TE}\textrm {cos}(\theta _r'-\theta _s')$ to that of TE mode, resulting in $\eta _{TM}=\eta _{TE}\textrm {cos}^2(\theta _r'-\theta _s')$ under the Bragg conditions. For arbitrary linear polarized beam, the DE can be described as $\eta (\alpha )=\eta _{TE}\textrm {cos}^2\alpha +\eta _{TM}\textrm {sin}^2\alpha =[1-\textrm {sin}^2\alpha \textrm {sin}^2(\theta _r'-\theta _s')]\eta _{TE}$, where $\alpha$ is the polarization direction. As Eqs. (1) indicate, the variation between $\eta _{TE}$ and $\eta _{TM}$ is slight for a larger grating period. Figure 7(a) shows the measured maximum of DE for CH1-6 at different polarization directions under the Bragg conditions. The DE fluctuations for all channels are less than 1.59%, while CH1 shows the maximum fluctuation and is marked in the inset in Fig. 7(a). The measured polarization–dependence of CH1/CH4 is in good agreement with the calculated one retrieved by the angular spectrum. Furthermore, the intensity distributions of CH1 under different $\alpha$ are also investigated and depicted in Fig. 7(b). There shows hardly variation in both horizontal and vertical distributions. It turns out that the MVG can be generally regarded as a polarization-independent device, which is an extraordinary advantage comparing with polarization-dependent SLM.
Fig. 7. (a) Measured polarization dependence on the maximum of the DE for six channels. The inset figure illustrates the experimental (solid circles for CH1 and dashed ones for CH4) and the retrieved values (solid lines for CH1 and dashed ones for CH4). (b) Horizontal and vertical relative intensity distributions of CH1 at different polarization direction incidence.
5. Conclusion
In conclusion, a multiplexed volume-grating-based optical vortex switch recorded in PTR glass has been designed and fabricated. The topological charge of the diffracted beam can be flexibly switched to -1, 0, and 1 by rotating the grating at the respective Bragg angle. The diffraction efficiencies of all channels are all pass over 92.68%, and the angular selectivity is less than 1.5 mrad at the incident divergence angle of 96 $\mu$rad. Note that the absolute DE can be further improved by anti-reflection coating. Although the fabricated MVG working at 1064 nm in this paper gives the best DE, it can be undoubtedly operated at other wavelengths. Since the DE still remains at the maximum, the incident angle of each channel should be also changed according to the Bragg conditions. The polarization dependence of the DE and diffracted intensity profiles has been demonstrated. It turns out that the switch can work under different kinds of polarization with slight DE variation (<1.59%) and without changing the diffracted light structure. This high diffraction-efficient and polarization-insensitive optical vortex switch can be promisingly applied in vortex tweezers, optical communication, high power beam shaping and combining.
Funding
National Natural Science Foundation of China (61875212, U1831211); Shanghai Strategic Emerging Industry Development Special Fund (31011442501217020191D3101001); National Key Research and Development Program of China (2020YFA0714500).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]
2. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]
3. T. Xu, S. Wu, Z. Jiang, X. Wu, and Q. Zhang, “Regulating trapping energy for multi-object manipulation by random phase encoding,” Opt. Lett. 45(7), 2002–2005 (2020). [CrossRef]
4. J. Wang, “Advances in communications using optical vortices,” Photonics Res. 4(5), B14–B28 (2016). [CrossRef]
5. A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014). [CrossRef]
6. L. Gong, Q. Zhao, H. Zhang, X.-Y. Hu, K. Huang, J.-M. Yang, and Y.-M. Li, “Optical orbital-angular-momentum-multiplexed data transmission under high scattering,” Light: Sci. Appl. 8(1), 27 (2019). [CrossRef]
7. P.-A. Moreau, E. Toninelli, T. Gregory, R. S. Aspden, P. A. Morris, and M. J. Padgett, “Imaging bell-type nonlocal behavior,” Sci. Adv. 5(7), eaaw2563 (2019). [CrossRef]
8. S. Liu, Y. Lou, and J. Jing, “Orbital angular momentum multiplexed deterministic all-optical quantum teleportation,” Nat. Commun. 11(1), 3875 (2020). [CrossRef]
9. D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Single-photon-level quantum image memory based on cold atomic ensembles,” Nat. Commun. 4(1), 2527 (2013). [CrossRef]
10. S. Khonina, V. Kotlyar, M. Shinkaryev, V. Soifer, and G. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39(5), 1147–1154 (1992). [CrossRef]
11. W. Ji, C.-H. Lee, P. Chen, W. Hu, Y. Ming, L. Zhang, T.-H. Lin, V. Chigrinov, and Y.-Q. Lu, “Meta-q-plate for complex beam shaping,” Sci. Rep. 6(1), 25528 (2016). [CrossRef]
12. P. Chen, B.-Y. Wei, W. Ji, S.-J. Ge, W. Hu, F. Xu, V. Chigrinov, and Y.-Q. Lu, “Arbitrary and reconfigurable optical vortex generation: a high-efficiency technique using director-varying liquid crystal fork gratings,” Photonics Res. 3(4), 133–139 (2015). [CrossRef]
13. J. Zhao, Y. Jin, D. He, F. Kong, J. Chen, H. Dai, and J. Mo, “Vortex volume gratings with high diffraction efficiency at 1064 nm,” in Second Target Recognition and Artificial Intelligence Summit Forum, vol. 11427 T. Wang, T. Chai, H. Fan, and Q. Yu, eds., International Society for Optics and Photonics (SPIE, 2020), pp. 376–382.
14. S.-C. Chu, T. Ohtomo, and K. Otsuka, “Generation of doughnutlike vortex beam with tunable orbital angular momentum from lasers with controlled hermite-gaussian modes,” Appl. Opt. 47(14), 2583–2591 (2008). [CrossRef]
15. T. Wang, A. Yang, F. Shi, Y. Huang, J. Wen, and X. Zeng, “High-order mode lasing in all-fmf laser cavities,” Photonics Res. 7(1), 42–49 (2019). [CrossRef]
16. Q. Jiang, Y. Bao, J. Li, L. Tian, T. Cui, L. Sun, B. Du, B. Li, B. Bai, J. Wang, H. Sun, B. Shen, H. Zhang, F. Lin, X. Zhu, and Z. Fang, “Bi-channel near- and far-field optical vortex generator based on a single plasmonic metasurface,” Photonics Res. 8(6), 986–994 (2020). [CrossRef]
17. Q. Liu, Y. Zhao, M. Ding, W. Yao, X. Fan, and D. Shen, “Wavelength- and oam-tunable vortex laser with a reflective volume bragg grating,” Opt. Express 25(19), 23312–23319 (2017). [CrossRef]
18. J. Yu, C. Miao, J. Wu, and C. Zhou, “Circular dammann gratings for enhanced control of the ring profile of perfect optical vortices,” Photonics Res. 8(5), 648–658 (2020). [CrossRef]
19. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” The Bell Syst. Tech. J. 48(9), 2909–2947 (1969). [CrossRef]
20. S. Wu, X. Yuan, X. Zhang, J. Feng, K. Zou, and G. Zhang, “Broadband angular filtering with a volume bragg grating and a surface grating pair,” Opt. Lett. 39(14), 4068–4071 (2014). [CrossRef]
21. I. Ciapurin, V. Smirnov, G. Venus, L. Glebova, E. Rotari, and L. Glebov, “High-power laser beam control by ptr bragg gratings,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference and Photonic Applications Systems Technologies, (Optical Society of America, 2004), p. CTuP51.
22. M. V. Shishova, S. B. Odinokov, A. Y. Zherdev, and D. S. Lushnikov, “Recording of multiplexed volume gratings via a phase mask for augmented reality waveguides,” Appl. Opt. 60(4), A140–A144 (2021). [CrossRef]
23. A. Sinha, G. Barbastathis, W. Liu, and D. Psaltis, “Imaging using volume holograms,” Opt. Eng. 43(9), 1959–1972 (2004). [CrossRef]
24. P. Chen, D. He, Y. Jin, J. Chen, J. Zhao, J. Xu, Y. Zhang, F. Kong, and H. He, “Method for precise evaluation of refractive index modulation amplitude inside the volume bragg grating recorded in photo-thermo-refractive glass,” Opt. Express 26(1), 157–164 (2018). [CrossRef]
25. D. Ott, I. Divliansky, B. Anderson, G. Venus, and L. Glebov, “Scaling the spectral beam combining channels in a multiplexed volume bragg grating,” Opt. Express 21(24), 29620–29627 (2013). [CrossRef]
26. S. Vyas, P.-H. Wang, and Y. Luo, “Spatial mode multiplexing using volume holographic gratings,” Opt. Express 25(20), 23726–23737 (2017). [CrossRef]
27. A. Bekshaev, S. Sviridova, A. Popov, and A. Tyurin, “Generation of optical vortex light beams by volume holograms with embedded phase singularity,” Opt. Commun. 285(20), 4005–4014 (2012). [CrossRef]
28. G. B. Ingersoll and J. R. Leger, “Spectral interference in multiplexed volume bragg gratings: theoretical calculations and experimental verification,” Appl. Opt. 53(24), 5477–5485 (2014). [CrossRef]
29. P. Chen, Y. Jin, D. He, J. Chen, J. Xu, J. Zhao, Y. Zhang, F. Kong, H. He, and J. Shao, “Design and fabrication of multiplexed volume bragg gratings as angle amplifiers in high power beam scanning system,” Opt. Express 26(19), 25336–25346 (2018). [CrossRef]
30. N. V. Nikonorov, E. I. Panysheva, I. V. Tunimanova, and A. V. Chukharev, “Influence of glass composition on the refractive index change upon photothermoinduced crystallization,” Glass Phys. Chem. 27(3), 241–249 (2001). [CrossRef]
31. J. Lumeau and E. D. Zanotto, “A review of the photo-thermal mechanism and crystallization of photo-thermo-refractive (ptr) glass,” Int. Mater. Rev. 62(6), 348–366 (2017). [CrossRef]
32. A. Bekshaev and K. Mohammed, “Spatial profile and singularities of the edge-diffracted beam with a multicharged optical vortex,” Opt. Commun. 341, 284–294 (2015). [CrossRef]
