Open Access
Add to BibSonomy
Abstract
Visible vortex beams have a large array of applications; however, the sources are often large or complex. Here, we present a compact vortex source with red, orange, and dual wavelength emission. This Pr:Waterproof Fluoro-Aluminate Glass fiber laser uses a standard microscope slide as an interferometric output coupler, yielding high quality first order vortex modes in a compact setup. We further demonstrate the broad (∼5 nm) emission bands in the orange (610 nm), red (637 nm) and near-infrared regions (698 nm), with the potential for green (530 nm) and cyan (485 nm) emission. This is a low-cost, compact and accessible device giving high quality modes for visible vortex applications.
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
1. Introduction
Vortex beams, which posses a phase singularity giving rise to orbital angular momentum, have been extensively researched since their discovery [1]. The wide variety of applications creates a need for an equally diverse range of sources with different capabilities and wavelengths. In the visible spectrum there are many applications spanning chiral quantum technologies [2,3], chiral spectroscopy [4], chiral micromachining [5], vortex laser induced printing [6], and underwater optical communications [7].
The proliferation of vortex beam technologies will require inexpensive and readily available sources. Current techniques to generate visible vortex beams mostly rely on the conversion of the Gaussian output of visible vortex laser sources. Due to the limited range of laser gain media in the visible spectrum, these sources are typically obtained using nonlinear frequency conversion of infrared lasers [8]. This adds complexity and expense to the overall vortex source. Additionally, the beam must be converted into a vortex, which is most commonly through spiral phase plates [9,10], Q-plates [11], or spatial light modulators [12]. Additionally there are a variety of fiber-based mode conversion devices [13]. These methods use bespoke optical components that convey additional expense and complexity to the laser system, can result in low mode purity, and are typically designed for and operated at a fixed wavelength.
A more attractive proposition for vortex applications is to have direct emission of the vortex mode from a laser in a low-cost and compact device. There have been many demonstrations of direct vortex emission in near-infrared lasers. For example, in bulk gain media there has been annular pumping [14,15], coupled cavities [16,17], and off-axis pumping [18,19]; and fiber lasers have utilised intracavity spatial light modulators [20] and Q-plates [21]. In the visible region there have been investigations using the Pr:LiYF$_4$ crystalline host with off-axis pumping [22,23] and intracavity spherical aberration [24], which directly generated visible vortex modes, but were restricted to the narrow emission bands of the crystalline host. Visible vortex modes from fiber lasers have typically used mode selective couplers to convert the mode from the laser externally to the oscillator with additional phase modifications [25].
Praseodymium is a well known active laser ion with multiple emission wavelengths in the visible spectrum. In many host materials it can be optically pumped with recently developed high power blue laser diodes. This has enabled multi-watt emission from bulk solid state [22,26–28] and fiber [29–31] based sources with compact and efficient diode-pumping schemes.
In this work we present direct emission of visible first-order vortex modes from a Pr:Waterproof Fluoro-Aluminate Glass Fiber (Pr:WPFGF) [29,32] laser. Vortex emission was obtained using a standard laboratory microscope slide to convert the internal Gaussian mode of the laser into a vortex output. The vortex laser emitted both orange (610 nm) and red (637 nm) wavelengths, which were selected by simple translation of the intracavity lens. Balanced dual wavelength emission was also achieved. It was further shown, for the first time, the broad emission bands in orange and red for this laser host, with lasing achieved in the orange (606 nm - 615 nm), red (635 nm - 640 nm), and near-infrared (695 nm - 700 nm) regions, in addition to expected lasing in green (530 nm) and cyan (485 nm). The presented device presents a route to cheap, compact, robust, and multi-wavelength visible vortex sources.
2. Laser design
The configuration of the vortex laser is shown in Fig. 1(a). The gain medium was a 3000 ppm doped Pr:Waterproof Fluoro-aluminate Glass Fiber (Pr:WPFGF), which had core/cladding diameters of 8 µm/125 µm, NA of 0.235, and length 40 mm. The pump input facet was coated for antireflection of the blue pump at 442 nm and high-reflection of the orange and red laser wavelengths. The other facet was uncoated. It was pumped by a 442 nm blue laser diode (PLPT9 450D_E), which was collimated by lenses L1-2, reshaped in the fast axis by cylindrical lenses L3-4, then coupled into the fiber core through lens L5 of focal length 4 mm.
Fig. 1. (a) The laser cavity design. The Pr:WPFGF gain medium is pumped by a blue laser diode (LD). The vortex output coupler converts the internal Gaussian mode into a vortex output. (b) The wavelength selection method. The orange lasing mode has optimal recoupling with its focal length (f$_{\mathrm {orange}}$), and the longer red wavelength is suppressed due to its longer focal length (f$_{\mathrm {red}}$) and increased loss recoupling into the fiber.
The free-space section of the laser was formed by the laser mode collimation lens L6 of focal length 8 mm, which was aspheric to minimise aberrations, and a high-reflectance mirror. The overall cavity length was 200 mm. The cavity oscillated on the fundamental Gaussian mode despite the Pr:WPFGF supporting higher order modes (V-number 9.23 at 640 nm).
The vortex mode was obtained from the cavity using an interferometric mode converting output coupler [33]. This uses the principle of interferometric mode conversion [34,35], which will be briefly summarised here. The destructive superposition of two Gaussian shaped beams with an opposite horizontal lateral displacement ($d$) and opposite vertical propagation directions ($\theta$) have the combined electric field $E$ of
Therefore, the interfering beams form a non-separable, propagation invariant LG$_{01}$ mode. The full mathematical details of this technique can be found in [36].
The interferometric mode conversion was implemented in the cavity with an uncoated, standard 1 mm thickness laboratory microscope slide that was inserted into the fundamental Gaussian mode section of the laser. The front and rear surface reflections were configured to destructively interfere with each other to form a first order Laguerre-Gaussian (LG$_{01}$) vortex mode. This was achieved with the combination of a relative lateral shift from the horizontal angle of incidence, and a relative vertical tilt due to the small wedge angle of approximately 100 µrad from the manufacturing tolerances of the slide. The conversion from incident Gaussian to LG$_{01}$ is only partial; therefore, the microscope slide acted as a vortex mode output coupler.
The horizontal angle of incidence on the microscope slide vortex output coupler was maintained at approximately 8.6°, and with the refractive index $n=1.52$ this gives a lateral displacement $d = {100}\;\mathrm{\mu}\textrm{m}$. The Gaussian mode in this section of the cavity had a beam radius $w_0 = {400}\;\mathrm{\mu}\textrm{m}$. These values correspond to a mode conversion parameter of $\epsilon =0.25$, which is appropriate for high quality vortex mode production.
2.1 Wavelength selection
The Praseodymium ion is well known for having many emission lines in the visible spectrum when used as the active ion in a laser, these are also present in the Pr:WPFGF gain medium. The red emission line around 637 nm is the strongest so it must usually be suppressed for other lines to lase. It was found that the chromatic aberration of the intracavity aspheric lens could be used for wavelength selection [37], where the focal length is longer for longer wavelengths.
The schematic for the selection method is shown in Fig. 1(b), where the orange (606 nm) mode is configured to have a lower loss than the red mode. The fiber tip to lens distance is set for re-imaging and optimal re-coupling of the orange mode back into the fiber core on a round trip after reflection from the HR mirror. The red mode is imaged with a identical distances but has a longer focal length, it therefore returns to the fiber at a larger size. This reduces the re-coupling efficiency and so that the red mode has a larger round-trip loss in the cavity. When configured correctly this will raise the threshold of the red mode above the orange, so the laser will operate on the orange mode. This was found to be a robust and repeatable wavelength selection technique. It also had sufficient sensitivity to permit dual-wavelength operation. In principle this tuning method should support continuous wavelength tunability, if this was permitted by the emission bands of the gain medium.
2.2 Vortex phase interferometry
To confirm the existance of a true vortex mode it is not sufficient to only analyse the intensity profile of the beam, but the phase must be confirmed to have the correct helical structure. A common approach to this is via interferometry, where the fringe pattern resulting from the interference of the vortex with a reference reveals the phase singularity. This is usually performed with a Mach-Zehnder interferometer and has been demonstrated with a shearing interferometer [38]. Here we present a novel implementation based on lateral shearing interferometry using the radial phase of the diverging beam under test.
The shearing interferometry configuration is shown in Fig. 2. The beam under test is diverging and incident on a parallel glass plate at a non-zero angle of incidence, causing the front and rear surface reflections to have a relative lateral shear, see Fig. 2(a). This results in a relative linear phase difference between the two beams from their laterally shifted centres of curvature. The simulated result of this technique applied to a LG$_{01}$ beam is shown in Fig. 2(b). The vertical fringe pattern is the familiar result of a relative linear phase difference. The two phase singularities are laterally shifted and manifest themselves as oppositely orientated forked fringes.
Fig. 2. Schematic of the shearing interferometry to determine phase structure. (a) Shearing interferometry using an uncoated glass plate, an incident diverging test beam has a lateral shear between the back surface reflections, with a radial phase from the divergence. (b) A computed result of this scheme on a first order Laguerre-Gaussian mode.
In the presented work the shearing plate was a 1 mm thick, uncoated UV fused silica plate, which used Fresnel reflections from the front and rear surfaces. The input beam was initially collimated to a 1 mm diameter, which was then passed through a 100 mm focal length lens to provide the divergence and radial phase. The angle of incidence was 25° on the plate, and the interference pattern was imaged with a CCD camera. Using the same formula as the vortex output coupler, the lateral displacement of each beam from the centre is $d=t\theta /n = {0.3}\;\textrm{mm}$.
3. Results
The red and orange vortex modes outcoupled from the microscope slide are shown in Fig. 3, where the top and bottom rows show the red and orange vortex diagnostic data, respectively. The intensity profiles both displayed good azimuthal symmetry. The beams were propagated through a focus by an additional lens, which revealed their propagation M$^2$ parameters to both be within 0.1 of 2, as expected for LG$_{01}$ modes [39]. Spectrometer measurements (1 nm resolution) revealed the red vortex mode to be centered at 637 nm and the orange mode to be at 606 nm.
Fig. 3. The vortex laser output for the red and orange modes, showing the intensity profile, fork interferogram confirming the phase singularity, and the measured spectrum.
The phase of the output beams was investigated using the lateral shearing method outlined in Sec. 2.2. The interference patterns have excellent contrast and match the appearance of the oppositely orientated fork structures as simulated for LG$_{01}$ modes in Fig. 2(b). These measurements confirm the correct phase structure of the vortex beams.
A final confirmation of the expected operation of this laser is to confirm that the mode internal to the laser was a fundamental Gaussian mode. The internal mode was observed through the small transmission of the HR cavity mirror. It possessed good symmetry with an equal horizontal and vertical beam propagation parameter of M$^2=1.1$. This was as expected, as to yield symmetric vortex modes via interferometric mode conversion the input must possess similar levels of spatial symmetry.
Both color vortex modes had very similar properties, and in fact all that was required to switch oscillation wavelength was a longitudinal shift of the intracavity lens. To illustrate this the lens was first positioned to select the red wavelength. The lens was then progressively moved 60 µm closer to the fiber to stimulate orange lasing. The two wavelengths exited the laser collinearly from the vortex output coupler. To visualise the changing wavelength content and spatial quality of the constituent color vortices, the colors were spatially separated via a 1600 lines/mm transmission grating and imaged onto a CCD camera, see Fig. 4(a). The separated color vortex beams are shown in Fig. 4(c) for progressive lens translations, with false color applied for visualisation purposes. The results show a clear and controllable transition from red to orange emission, with dual emission stably achieved. It was possible to fine tune the balance of wavelength powers to achieve the same spectral intensity, as measured on a spectrometer, see Fig. 4(b).
Fig. 4. Dual-wavelength operation of the vortex laser. (a) Separation of the 606 nm and 637 nm beams with a 1600 lines/mm transmission grating. (b) Progressive movement of the intracavity L6 lens towards the fiber, which changed the power in each wavelength. (c) An example spectrum showing equal spectral intensity.
It was not possible to accurately measure the beam quality parameters of the vortex modes during dual-wavelength operation due to their collinear emission. However, each mode on single wavelength operation had the same high quality mode, as previously discussed, and minor adjustments to the lens position should not significantly affect the spatial laser properties, so we can confidently conclude that the modes remained high quality in dual-wavelength operation. Full mode decomposition was not available during this work, but we can also confidently predict high mode purity as has been previously demonstrated with the interferometric mode conversion technique [33,36], where up to 98% has been obtained.
3.1 Laser emission wavelengths
It has been shown that the presented laser cavity had wavelength selectivity. To explore the possible emission wavelengths of the Pr:WPFGF gain medium the vortex output coupler was removed to fully explore the emission wavelengths available, because the 1 mm thick plate had wavelength selectivity itself. The laser cavity was operated in a variety of lens position configurations, and a selection of the measured lasing spectra are presented in Fig. 5. This investigation revealed multiple laser lines spanning the orange (608 nm–615 nm), red (635 nm–640 nm), and near-infrared (695 nm–700 nm) regions. These results show the broad tunability of the Pr ion in the WPFGF host, and is in contrast to the more widely used crystalline hosts (for example LiYF$_4$), which typically have narrow and unique emission bands in these wavelength ranges. We expect that there would be similar emission and tunability in the green ($\sim$530 nm) and cyan ($\sim$485 nm) regions, where the gain medium was measured to fluoresce. The current system was unable to lase at these wavelengths due to the rear facet coating on the WPFGF fiber having insufficiently high reflectance at these wavelengths.
Fig. 5. Selection of measured emission wavelengths of the laser with no vortex output coupler in the orange (left), red (middle), and infrared (right) regions. The wavelength measurement accuracy is $\pm {1}\;\textrm{nm}$.
4. Discussion
The presented laser system successfully directly generated red and orange vortex modes, using a commonly available microscope slide as an output coupler. The equipment used in the system was configured for demonstrating the capabilities of the design and not tailored for power scaling, with output powers of up to 2.8 mW at a pump power of 600 mW. However, there are many avenues for scaling the power output of the system.
The overall reflectance of the mode transforming output coupler could be increased from 2% to approximately 8% by applying reflective coatings to the surfaces [36], rather than relying on Fresnel reflectance, which would allow optimisation and increase of the output coupling efficiency of the laser. The internal fiber facet was uncoated, resulting in Fresnel reflection losses for the intracavity mode at this interface. Applying an antireflection coating for red and orange wavelengths would significantly reduce roundtrip cavity losses from this component from approximately 10% to 0.1%. The fiber was not mechanically mounted for efficient heat conduction out of the fiber, so the input pump power was restricted to 600 mW to prevent thermal damage. Additionally, the WPFGF host can be designed as a double clad fiber, enabling efficient pumping by high-power blue laser diodes, demonstrated at up to 7 W of input power [29].
We expect that implementing these changes would allow the output power to be scaled up to the hundreds of milliwatts range. These changes would not affect the vortex mode production processes, so provide a clear path to power scaling.
5. Conclusion
We have demonstrated, for the first time, direct visible vortex emission from the Pr:WPFGF gain medium. The laser uses an easily available microscope slide to convert the internal Gaussian mode to a vortex output, with the phase singularity confirmed by a novel shearing interferometry scheme. The laser wavelength could be selected by translation of the intracavity lens, which yielded red (637 nm), orange (606 nm), and equal power dual wavelength vortex modes. This is an unusual characteristic for a vortex generation technique, where in other methods optical components are typically designed to operate at a single wavelength. Multiple lasing wavelengths were identified, for the first time, in the Pr:WPFGF host spanning orange (606 nm–615 nm), red (635 nm–640 nm), and near-infrared (695 nm–700 nm) regions, with tunable lasing in the green ($\sim$530 nm) and cyan ($\sim$485 nm) regions expected with different intracavity dielectric coatings.
This work presents a simple, accessible and compact design for high quality visible vortex mode emission, requiring no specialist equipment. The interferometric output coupling technique provides opportunities for more complex output structures [35]. We expect this design to be beneficial for applications using visible vortex modes, for example micro-machining and printing technologies.
Funding
Imperial College London (Imperial College Research Fellowship); Royal Society (IES/R3/213129); Japan Society for the Promotion of Science (22K18981, JP16H06507, JP22H05138, P18H03884); Core Research for Evolutional Science and Technology (JPMJCR1903).
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. 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. P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature 541(7638), 473–480 (2017). [CrossRef]
3. D. Green and K. A. Forbes, “Optical chirality of vortex beams at the nanoscale,” Nanoscale 15(2), 540–552 (2023). [CrossRef]
4. F. Büscher, S. Müllner, D. Wulferding, Y. G. Pashkevich, V. Gnezdilov, A. A. Peshkov, A. Surzhykov, and P. Lemmens, “Raman scattering of plane-wave and twisted light off chiral molecular liquids,” Low Temp. Phys. 47(11), 959–965 (2021). [CrossRef]
5. K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12(7), 3645–3649 (2012). [CrossRef]
6. T. Omatsu, K. Miyamoto, K.-I. Yuyama, K. Yamane, and R. Morita, “Laser-induced forward-transfer with light possessing orbital angular momentum,” J. Photochem. Photobiol., C 52, 100535 (2022). [CrossRef]
7. Y. Zhao, J. Xu, A. Wang, W. Lv, L. Zhu, S. Li, and J. Wang, “Demonstration of data-carrying orbital angular momentum-based underwater wireless optical multicasting link,” Opt. Express 25(23), 28743 (2017). [CrossRef]
8. T. Omatsu, K. Miyamoto, and A. J. Lee, “Wavelength-versatile optical vortex lasers,” J. Opt. 19(12), 123002 (2017). [CrossRef]
9. M. Beijersbergen, R. Coerwinkel, M. Kristensen, and J. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112(5-6), 321–327 (1994). [CrossRef]
10. L. Janicijevic and S. Topuzoski, “Gaussian laser beam transformation into an optical vortex beam by helical lens,” J. Mod. Opt. 63(2), 164–176 (2016). Purity and intensity profile of propagating beam after spiral phase device. [CrossRef]
11. M. Sakakura, Y. Lei, L. Wang, Y.-H. Yu, and P. G. Kazansky, “Ultralow-loss geometric phase and polarization shaping by ultrafast laser writing in silica glass,” Light: Sci. Appl. 9(1), 15 (2020). [CrossRef]
12. J. Pinnell, V. Rodríguez-Fajardo, and A. Forbes, “Probing the limits of orbital angular momentum generation and detection with spatial light modulators,” J. Opt. 23(1), 015602 (2021). [CrossRef]
13. D. Mao, Y. Zheng, C. Zeng, C. Lu, H. D. Wang, H. Zhang, W. Zhang, T. Mei, and J. Zhao, “Generation of polarization and phase singular beams in fibers and fiber lasers,” Adv. Photonics 3(01), 014002 (2021). [CrossRef]
14. D. J. Kim, J. W. Kim, and W. A. Clarkson, “Q-switched Nd:YAG optical vortex lasers,” Opt. Express 21(24), 29449 (2013). [CrossRef]
15. S. Zhang, Z. Fu, L. Lai, F. Jia, D. Qiao, Y. Fan, K. Li, and N. Copner, “Transverse mode locking of different frequency-degenerate families based on annular beam pumping,” Opt. Lett. 46(13), 3195–3198 (2021). [CrossRef]
16. W. R. Kerridge-Johns and M. J. Damzen, “Vortex laser from anti-resonant ring coupled cavities,” Opt. Express 26(25), 32839–32846 (2018). [CrossRef]
17. E. Song, T. Dai, G. Zhu, H. Wang, K. Aleksei, and X. Zhu, “Adjustable and stable beam profile generation in a Yb:YAG thin-disk laser,” Opt. Lett. 45(23), 6550 (2020). [CrossRef]
18. X. Huang, B. Xu, S. Cui, H. Xu, Z. Cai, and L. Chen, “Direct generation of vortex laser by rotating induced off-axis pumping,” IEEE J. Sel. Top. Quantum Electron. 24(5), 1–6 (2018). [CrossRef]
19. G. Lin, Y. Cao, Z. Lu, and Y. K. Chembo, “Spontaneous generation of orbital angular momentum crystals using a monolithic Nd:YAG nonplanar ring laser,” Opt. Lett. 44(2), 203–206 (2019). [CrossRef]
20. N. Zhou, J. Liu, and J. Wang, “Reconfigurable and tunable twisted light laser,” Sci. Rep. 8(1), 11394 (2018). [CrossRef]
21. H. Hu, Z. Chen, Q. Cao, and Q. Zhan, “Wavelength-tunable and oam-switchable ultrafast fiber laser enabled by intracavity polarization control,” IEEE Photonics J. 15(1), 1–4 (2023). [CrossRef]
22. X. Lin, M. Chen, Q. Feng, S. Ji, S. Cui, Y. Zhu, B. Xiao, W. Li, H. Xu, and Z. Cai, “LD-pumped high-power CW Pr3+:YLF Laguerre-Gaussian lasers at 639 nm,” Opt. Laser Technol. 142, 107273 (2021). [CrossRef]
23. A. S. Rao, T. Miike, K. Miyamoto, and T. Omatsu, “Direct generation of 523 nm orbital Poincaré mode from a diode-pumped Pr3+:LiYF4 laser with an off-axis optical needle pumping geometry,” Opt. Express 29(19), 30409–30418 (2021). [CrossRef]
24. A. S. Rao, T. Morohashi, W. R. Kerridge-Johns, and T. Omatsu, “Generation of higher-order Laguerre–Gaussian modes from a diode-pumped Pr3+:LiYF4 laser with an intra-cavity spherical aberration,” J. Opt. Soc. Am. B 40(2), 406–411 (2023). [CrossRef]
25. H. Sun, L. Wang, J. Zou, Q. Ruan, Y. Ding, C. Dong, Z. Dong, and Z. Luo, “Visible-wavelength all-fiber mode-locked vortex laser,” J. Lightwave Technol. 40(1), 191–195 (2022). [CrossRef]
26. P. W. Metz, F. Reichert, F. Moglia, S. Müller, D.-T. Marzahl, C. Känkel, and G. Huber, “High-power red, orange, and green Pr3+:LiYF4 lasers,” Opt. Lett. 39(11), 3193 (2014). [CrossRef]
27. S. Luo, X. Yan, Q. Cui, B. Xu, H. Xu, and Z. Cai, “Power scaling of blue-diode-pumped Pr:YLF lasers at 523.0, 604.1, 606.9, 639.4, 697.8 and 720.9 nm,” Opt. Commun. 380, 357–360 (2016). [CrossRef]
28. H. Tanaka, S. Fujita, and F. Kannari, “High-power visibly emitting Pr3+:YLF laser end pumped by single-emitter or fiber-coupled GaN blue laser diodes,” Appl. Opt. 57(21), 5923–5928 (2018). [CrossRef]
29. Y. Fujimoto, M. Nakahara, P. Binun, S. Motokoshi, O. Ishii, M. Watanabe, M. Yamazaki, T. Shinozaki, T. Sato, and Y. Yanomori, “2 W single-mode visible laser oscillation in Pr-doped double-clad structured waterproof fluoro-aluminate glass fiber,” in Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (IEEE, 2019), p. 1.
30. M.-P. Lord, V. Fortin, F. Maes, L. Talbot, M. Bernier, and R. Vallée, “2.3 W monolithic fiber laser operating in the visible,” Opt. Lett. 46(10), 2392 (2021). [CrossRef]
31. C. Zhang, J. Hong, L. Zhou, J. Zou, and Z. Luo, “Direct generation of 5 W all-fiber red laser at 635 nm,” Opt. Laser Technol. 160, 109050 (2023). [CrossRef]
32. Y. Fujitomo, M. Murakami, J. Nakanishi, T. Yamada, O. Ishii, and M. Yamazaki, “Visible lasers in waterproof fluoro-aluminate glass fibers excited by GaN laser diodes,” in Advanced Solid-State Lasers Congress (OSA, 2013), paper AM2A.2.
33. A.-H. Munj and W. R. Kerridge-Johns, “Unidirectional ring vortex laser using a wedge-plate shearing interferometer,” Opt. Express 31(3), 4954 (2023). [CrossRef]
34. P. Vaity, A. Aadhi, and R. P. Singh, “Formation of optical vortices through superposition of two gaussian beams,” Appl. Opt. 52(27), 6652–6656 (2013). [CrossRef]
35. M. J. Damzen, W. R. Kerridge-Johns, and J. W. T. Geberbauer, “Vortex mode transformation interferometry,” J. Opt. 22(1), 015604 (2020). [CrossRef]
36. W. R. Kerridge-Johns, J.-B. Jaillot, and M. J. Damzen, “Sampling a vortex from a Gaussian beam using a wedge-plate shearing interferometer,” Appl. Opt. 60(12), 3510–3516 (2021). [CrossRef]
37. Y. Fujimoto, O. Ishii, and M. Yamazaki, “Design of simple and compact tunable fibre laser,” Electron. Lett. 51(12), 925–926 (2015). [CrossRef]
38. B. Khajavi and E. J. Galvez, “Determining topological charge of an optical beam using a wedged optical flat,” Opt. Lett. 42(8), 1516–1519 (2017). [CrossRef]
39. S. Saghafi and C. Sheppard, “The beam propagation factor for higher order Gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998). [CrossRef]
