Open Access
Add to BibSonomy
Abstract
We demonstrate the direct generation of visible vortex beams at 640 nm and 607 nm by employing an off-axis pumping scheme in a diode end-pumped Pr3+:YLF laser. A detailed numerical analysis, based on the coherent superposition of Hermite-Gaussian modes with different amplitudes and phases, is perfectly consistent with the experimentally observed lasing modes. The maximum vortex output powers have been measured to be 808 mW and 211 mW at a pump power of 3.16 W, for the wavelengths of 640 nm and 607 nm, respectively. We also demonstrate the handedness control of the generated vortex beam. Such a visible vortex laser can potentially be applied in super-resolution fluorescent microscopes and micro-fabrication research.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical vortex beams possess annular spatial profiles and a nonzero orbital angular momentum (OAM) of ±ℓħ per photon, where ℓ is the topological charge, owing to their on-axis phase singularity [1]. Their doughnut-like intensity profile has been widely used for optical trapping, in which intense optical beams act as tweezers, enabling us to confine, guide and manipulate atoms, metallic particles, hollow dielectric particles, and even biological cells [2–5]. In addition, optical vortex beams can also significantly improve the spatial resolution of fluorescence microscopes, far beyond the diffraction limit, due to the outer region intensity depletion of a Gaussian beam via a stimulated parametric process [6]. Furthermore, optical vortex beams also provide an azimuthal phase front, exp(±iℓϕ), enabling, for example, azimuthal (helical) motion control in optical trapping systems [7–9], monocrystalline needle formation [10], and chiral structures created in azopolymers and metals [11–13]. They also increase the bit-rate of optical communication links both in free space [14] and in optical fibers [15], thanks to the spatial multiplexing/demultiplexing that arises from modulation/demodulation of the OAM degree of freedom.
There are several techniques for generating vortex beams outside a laser cavity, such as the interferometric superposition of Hermite-Gaussian (HG) modes through the use of astigmatic optical systems [1]. Perhaps more commonly used are phase elements that modulate the wavefront of light, forming a specified spatial mode after propagation—a Laguerre-Gaussian (LG) mode in this case, to be more precise. The performance of these phase elements, such as spiral phase plates [16], computer-generated holograms [17], liquid crystal based q-plates [18], conical diffraction elements [19] or spatial light modulators [20], is constantly evolving appearing new devices with even increased versatility [21]. The complete mode description in the LGp,ℓ basis requires not only an azimuthal index ℓ but also a radial index p linked to the number of radial nodes (rings). However, the purity of the spatial modes generated with such phase elements as an eigenmode is limited, because of the additional excitation of undesired higher-order radial modes [22].
While all the methods described above are based on extra-cavity configurations, optical vortex beams can also be produced directly within the laser cavity (intra-cavity configuration) as an eigenmode [23–25]. Intra-cavity configurations allow the improvement of the achievable purity obtained by the aforementioned methods, by confining all the optical power in our desired OAM mode due to the negligible overlap with the rest of the orthogonal spatial modes.
Several techniques for intra-cavity configurations have been demonstrated through the use, for example, of a doughnut-shaped pump beam in a diode end-pumped laser cavity configuration [26], spot-defect cavity mirrors [27], intra-cavity phase-only optical elements [28, 29], or off-axis pumping configuration induced by rotating the gain medium [30]. However, all these approaches only focused on generating near-infrared vortex beams, leaving a gap in the direct generation of high-quality visible vortex beams as the eigenmode (without any additional phase elements), which have the potential to be applied in super-resolution fluorescent microscopes and micro-fabrication research.
In recent years, Pr3+ doped solid-state laser materials, such as Pr3+ doped lithium yttrium tetra-fluoride (Pr3+:YLF) [31], have attracted great interest due to their excellent emission in the visible spectral region, including green (523 nm), orange (607 nm), red (640 nm), dark-red (720 nm), and their strong absorption in the blue region (∼442 nm), enabling diode-pumping. To date, several groups have successfully demonstrated continuous-wave [32], Q-switched [33], and mode-locked lasing [34] for a diode-pumped Pr3+:YLF laser. In order to go one step further, we present the direct generation of visible first-order LG modes from a Pr3+:YLF laser, by employing the off-axis pumping configuration without the use of any additional phase elements. We also conduct a theoretical analysis of the different spatial modes generated based on the coherent supposition of the HG modes. Moreover, we address controlling the handedness of the vortex mode. A maximum output power of 808 mW and 211 mW is obtained for the wavelengths of 640 nm and 607 nm, at a pump power of 3.16 W.
2. Experimental setup
Figure 1(a) shows the experimental setup used to implement our visible vortex laser. The pump source is a 3.5 W InGaN laser diode (NDB7K75) with the peak wavelength being able to be tuned from 440 nm to 455 nm, the maximum absorption band for the Pr3+:YLF crystal [31]. The pump beam is collimated by an aspheric lens (f = 4.51 mm) and a cylindrical lens (f = 250 mm), and it is then focused onto the input facet of the gain medium by a plano-convex lens (f = 35 mm), yielding an elliptical beam spot with approximate diameters of 100 μm and 40 μm along the x and y axes, respectively. The gain medium is a 5-mm-long Pr3+-doped (0.5 at.%) YLF a-cut crystal, wrapped with indium foil and mounted inside a copper holder maintained at a temperature of 8 °C by a water-cooled chiller. The crystal is cut perpendicular to the a-axis, thus the absorption peak for the σ-polarization, parallel to the a-axis, is at 442 nm.
Fig. 1 (a) Experimental setup for direct generation of a visible vortex laser beam using an off-axis pumping configuration. LD: InGaN laser diode centered at 442 nm; Col. L: collimation lens (f = 4.51 mm); Cyl. L: cylindrical lens (f = 250 mm); FL: focusing lens (f = 35 mm); HR: high-reflection coating for the lasing wavelength (640 nm or 607 nm); Pr3+:YLF: 5-mm-long gain medium; CL: cavity length; OC: output coupler. The inset shows the OC displacement for an off-axis pumping configuration example. (b) Scheme for a self-referenced laterally sheared interferometer. TG: transmission grating (10 lines/mm); 0th and 1st: diffraction orders; IL: imaging lens (f = 300 mm); CCD: CCD camera.
The plano-concave linear cavity is formed by the input facet of the gain medium and the output coupler (OC), as depicted in Fig. 1(a). The input facet of the Pr3+:YLF crystal has high- and anti-reflection coatings for the lasing (640 nm or 607 nm) and pump (442 nm) wavelengths, respectively. Two crystals with different coatings are employed to lase red (640 nm) and orange (607 nm) vortex beams within the cavity resonator. The first crystal used exhibited an input facet with reflectivities of 99.8% and 98.3% at 640 nm and 607 nm, allowing for 640 nm to lase. The input facet of the second crystal had reflectivities of 92.6% and 99.6% at 640 nm and 607 nm, allowing for 607 nm to lase. The output facet of the gain medium in both cases is anti-reflection coated for 640 nm and 607 nm. The only OC used exhibited high reflection at 640 nm and 607 nm (R = 98.7% and R = 98.8%, respectively), and a concave curvature of 150 mm radius. Figure 2(a) shows the transmission spectra measured by a spectrometer (Jasco V-750), for the OC and the two different coated crystals described above. It is worth mentioning that the OC is mounted on a 3D micro-metric translation stage to allow displacements along the x and y axes, with steps as small as Δx = Δy = 0.5 μm, enabling the spatial mode overlap efficiency to be optimized for the off-axis pumping configuration, as shown in the inset of Fig. 1(a). Figure 2(b) shows the line spectrum of the generated red (640 nm) and orange (607 nm) vortex beams by employing two crystals with different coating. The insets also show typical spatial forms (real color photographs) of the red and orange vortex beams obtained by appropriately displacing the OC. The cavity length is approximately 7 mm for the vortex beams generation. The vortex beams can also be generated in even longer cavity configurations (up to CL = 20 mm), but with slightly lower stability. The laser output beam is then collimated with a plano-convex lens (f = 300 mm), separated from the residual pump by a long-pass filter and detected by a CCD camera (not shown for simplicity). Figure 1(b) schematically describes the self-referenced laterally sheared interferometer, in which the 0th diffracted beam (transmitted vortex beam) and the 1st order diffracted beam (its laterally-shared copy) separated from the rest by a slit after traversing a diffraction grating (10 lines/mm) are interfered. We use this technique to assign the handedness ±ℓ of the vortex beam by identifying the interference patterns as described in the results.
Fig. 2 (a) Measured transmission spectra of the OC and the two differently coated crystals used. (b) Line spectrum of the generated vortex mode by employing the two different crystals. Each line is normalized with respect its own maximum. The insets show photographs of the red (640 nm) and orange (607 nm) vortex beams that can be obtained from our setup.
3. Results and discussion
Most of the results presented in this section have been obtained for a wavelength of 640 nm. Further results using a different crystal are given in the Appendix. When the OC is on axis with the 442 nm pump, the cavity generates a nice Gaussian output profile. We consider this OC position as the center of our coordinate system, (x,y) = (0,0) μm, when performing a mode mapping of the laser beam for different OC positions. The mode map shown in Fig. 3 allows us to identify the modes supported in the cavity for different off-axis pumping configurations. The OC can then be shifted horizontally and vertically from the center of coordinates, as described in the inset of Fig. 1(a), using in this case Δ x ≈ ±12.5 μm and Δ y ≈ ±37.5 μm step displacements, up to a maximum of (x,y) = (±25,±75) μm. Even though the step size is different for each axis direction, the spatial mode evolution shown in Fig. 3 obeys the principal of symmetry. However, such symmetry is broken in the mode map plot when using a c-cut crystal, as shown in the Appendix.
As the OC is displaced further from the center of coordinates, the output beam can be gradually transformed from a Gaussian to different two-petal modes (with a mixture of both modes in between), and exhibits a doughnut-shaped beam profile at the extremes of each quadrant (x,y) = (±25,±75) μm. The output power for each of the captured modes in Fig. 3 varies owing to the different mode overlap efficiency arising from misalignment of the cavity resonator. As the off-axis displacements increase further, higher-order Ince-Gaussian modes can be generated (see Appendix for an example), as previously reported for a similar laser cavity configuration [35].
A self-referenced laterally sheared interferometer as shown schematically in Fig. 1(b), in which the test wavefront and its laterally shared copy (not a plane wavefront) are interfered, is used to examine the spiral wavefront of the output mode by capturing the fork-like interference pattern. Thus, a pair of upward/downward (downward/upward) Y-shaped fringes manifest the phase singularity of an OAM mode with ℓ = +1 (ℓ = −1). The captures in Fig. 4 clearly demonstrate the handedness control of our vortex beam, showing an upward/downward double Y-shaped interference pattern corresponding to an OAM mode ℓ = +1 in (a), and a downward/upward (ℓ = −1) in (b). The insets show the spatial intensity profiles for each OAM mode after removing the self-referenced laterally sheared interferometer section. It is worth noting that the handedness of the vortex beam can be reversed only by displacing the OC from the extreme of one quadrant to the extreme of the adjacent quadrant (a lateral displacement Δx of the OC ≈ 50 μm). Although, such displacement turns to be smaller when considering a c-cut crystal instead (Δx ≈ 10 μm), as shown in the Appendix.
Fig. 4 Self-interference fringes of a (a) ℓ = +1 vortex beam (upward/downward fork-like pattern), and (b) ℓ = −1 vortex beam (downward/upward fork-like pattern) after displacing the OC laterally by Δx ≈ +50 μm. The insets show the intensity profile for each vortex beam after removing the interferometer section.
To understand why such OAM and different petal-like modes can be generated when the cavity is misaligned, we analyze the different spatial profiles obtained using our off-axis pumping configuration as a coherent superposition of Ince-Gaussian (IG) modes [36]. To simplify the analysis, we choose to work with the analogous and more common Hermite-Gaussian basis notation, HGm,n. Thus, an OC displacement in the horizontal (vertical) direction would correspond to the appearance of higher-order modes with m (n) index greater than 0, meaning that the appearing petals, or lobes, are oriented along the x (y) axis, as can be observed in Fig. 3 and other mode maps shown in the Appendix. It is well-known that misalignment of the cavity resonator in the x or y direction allows for higher-order spatial modes to selectively lase, and that the inter-modal phase in a coherent superposition of supported lasing modes can change, as driven by the Gouy phase shift [37]. Now, if we displace the OC by a small amount along both axes, Δx and Δy, we see in the mode mapping of Fig. 3 how we can simultaneously generate both HG01 and HG10 in a coherent superposition
where θ dictates the weighting for each mode, which is related to the mode lasing threshold given by the cavity resonator geometry, and φ corresponds to the inter-modal phase. Figure 5 shows an experimental example of the different spatial modes that can be generated after displacing the OC at a particular off-axis position (upper rows), and the pertinent simulations (lower rows) obtained by evaluating Eq. (1) with the appropriate θ and φ values to find the best fit for the intensity profiles from above.When considering a symmetric superposition, i.e., same weighting or lasing threshold for both HG modes (θ = π/4), the resulting output mode ℳout can go from a two-petal intensity profile aligned along the diagonal axis for φ = 0 (see Fig. 5(a)) to a doughnut-like mode with OAM ℓ = +1 for φ = π/2 (see Fig. 5(b)). In general, any LG mode can be expressed by a coherent superposition of IG or HG modes [36]. Thus, the OAM modes generated in our laser system can also be written as a superposition of HGm,n modes considering a particular inter-modal phase shift
Now, when the inter-modal phase increases up to φ = π, the two-petal intensity profile appears again, but aligned along the anti-diagonal axis this time (see Fig. 5(c)). In the case of φ = 3π/2, the spatial mode would be doughnut-shaped again, but corresponding to an OAM mode with opposite handedness (ℓ = −1), as shown in Eq. (2). Figure 5(d) shows a particular example of how the alignment of the petals can be rotated outside the diagonal/anti-diagonal basis by choosing an asymmetric weighting in Eq. (1), corresponding in this case to a perfect fit for ℳout(0.37π, π) = 0.40 HG01 − 0.92 HG10.Fig. 5 Experimental spatial modes for different OC positions (upper rows) and corresponding theoretical simulations (lower rows) based on the HG mode superposition in Eq. (1).
We can also generate different mixtures of the two-petal-like and doughnut-like modes by choosing the inter-modal phase to be between the well-defined values φ = 0, π/2, π and 3π/2. Figures 5(e) and 5(f) show two examples of spatial modes defined by ℳout(0.28π, 0.64π) and ℳout(0.24π, 0.54π), respectively. These indicate that any spatial mode generated by our system can be numerically simulated using a coherent superposition of the different spatial modes (given by the HG, LG, or IG basis) supported by the particular resonator geometry. Hence, any spatial mode generated in our system could also be generated by modulating a Gaussian beam with a computer generated hologram. More importantly, any of our mechanically generated spatial modes could go through the reverse process, i.e., demodulation by modal decomposition [38], enabling us to form a spatial-mode alphabet for free-space optical communication links.
Figure 6 shows the Pr3+:YLF laser power scaling at wavelengths of 640 nm and 607 nm. Maximum Gaussian mode output powers of 965 mW and 337 mW are obtained at a pump power of 3.16 W, corresponding to optical-optical slope efficiency of 33.7% and 15.7%, for the 640 nm and 607 nm respectively. The vortex mode (ℓ = +1) output powers are measured to be 808 mW and 211 mW, resulting in slope efficiencies of 37.3% and 16.7%, respectively. It is noteworthy that the vortex mode with ℓ = −1 shows almost the same laser performances (slope efficiency and threshold) at those of the vortex mode with ℓ = +1. The orange laser operation exhibits less output power and low slope efficiency compared with the red laser operation owing to the relatively weak emission cross section (13.6 × 10−20 cm2) for orange radiation compared with that (21.8 × 10−20 cm2) of the red radiation [39].
Fig. 6 Output powers as a function of pump power for the Gaussian mode, black squares for 640 nm emitted light and blue triangles for 607 nm, and for the vortex mode with OAM ℓ = +1, red circles for 640 nm and orange diamonds for 607 nm. Lines are the fitted slope efficiencies: 33.7% (black), 37.3% (red), 15.7% (blue) and 16.7% (orange).
Fig. 7 Spatial mode mapping of the laser output beam using a c-cut crystal for different OC positions (x,y).
4. Conclusion
In conclusion, we demonstrated the direct generation of visible (640 nm and 607 nm) vortex beams from a diode-pumped Pr3+:YLF laser by employing an off-axis pumping configuration. A detailed numerical analysis based on a coherent superposition of Hermite-Gaussian modes with different amplitudes and phases was shown to be in perfect agreement with the results of the experimental lasing modes. We also addressed selective handedness control of the vortex beam by only laterally displacing the OC. A maximum output powers of 808 mW and 211 mW for the vortex beam at 640 nm and 607 nm was achieved, at a pumping power of 3.16 W. Such a visible vortex laser can potentially be used in super-resolution fluorescent microscopes and micro-fabrications. We need to further study the feasibility of using all different mechanically generated spatial modes in a free-space communication link.
Appendix
Experimental results from a c-cut Pr3+:YLF crystal
Similar results can be obtained using a c-cut Pr3+:YLF crystal. Although, with a lower performance due to a lower emission cross-section (∼ 1.2× 10−20 cm2), the c-cut crystal can also be used to generate vortex beams at 640 nm visible light. In the same way as the experimental mode map in the main text (see Fig. 3), the mode map for a c-cut crystal is shown in Fig. 7. It now does not obey the principal of symmetry. We expect this to be caused by the birefringence of the gain medium (due to the crystal cut with respect the optical axis) breaking the symmetry of the off-axis pumping effect.
Also, we use a self-referenced laterally sheared interferometer, as shown schematically in Fig. 1(b), to examine the spiral wavefront of the output mode by capturing the fork-like interference pattern. The captures in Fig. 8 clearly demonstrate the handedness control of our laser vortex beam using a c-cut crystal, showing an upward/downward double Y-shaped interference pattern corresponding to an OAM mode ℓ = +1 in (a), and a downward/upward (ℓ = −1) in (b), after only laterally displacing the OC by Δx ≈ +10 μm now. The sudden change of helicity within the same quadrant can be explained by the higher intermodal phase (Gouy phase) introduced by the birefringence of the gain medium. The insets show the intensity profiles for each OAM mode after removing the self-referenced laterally sheared interferometer section.
Fig. 8 Self-interference fringes of a (a) ℓ = +1 vortex beam, and (b) ℓ = −1 vortex beam using a c-cut crystal after only displacing the OC laterally by Δx ≈ +10 μm. The insets show the intensity profile for each vortex beam after removing the interferometer section.
As the off-axis displacements increase further, higher-order Ince-Gaussian modes can be generated, as shown in Fig. 9 and previously reported for a similar laser cavity configuration [35]. It is worth noting that the distance between well-defined modes along the x and y axes decreases as we displace the OC further away from the center, and that the output patterns with similar displacements in both directions are not symmetric. This might be caused by the misplacement of our selected center of coordinates (based on maximum output power), inhomogenities in the Pr3+:YLF crystal, the elliptical pumping mode, or cavity instability caused by thermal lensing for different OC displacements [40].
Fig. 9 Higher-order Ince-Gaussian mode map of the laser output beam using a c-cut crystal as the OC is displaced within the first quadrant, from the center up to the approximate position of (x,y) ∼ (200,300) μm. Note that distance between modes along the x and y axes decreases as we displace the OC away from the center, meaning that this mode map is not displayed on a linear axis scale.
Funding
Japan Society for the Promotion of Science (JSPS) KAKENHI Grants (JP 17K19070; JP 18H03884) and KAKENHI Grant-in-Aid (JP 16H06507) for Scientific Research on Innovative Areas “Nano-Material Optical-Manipulation.”
References
1. L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185 (1992). [CrossRef] [PubMed]
2. Y. Song, D. Milam, and W. Hill, “Long, narrow all-light atom guide,” Opt. Lett. 24, 1805–1807 (1999). [CrossRef]
3. N. Friedman, L. Khaykovich, R. Ozeri, and N. Davidson, “Compression of cold atoms to very high densities in a rotating-beam blue-detuned optical trap,” Phys. Rev. A 61, 031403 (2000). [CrossRef]
4. P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89, 081113 (2006). [CrossRef]
5. S. Ito, H. Yamauchi, M. Tamura, S. Hidaka, H. Hattori, T. Hamada, K. Nishida, S. Tokonami, T. Itoh, H. Miyasaka, and T. Lida, “Selective optical assembly of highly uniform nanoparticles by doughnut-shaped beams,” Sci. Rep. 3, 3047 (2013). [CrossRef]
6. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19, 780–782 (1994). [CrossRef]
7. H. He, M. Friese, N. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826 (1995). [CrossRef] [PubMed]
8. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343 (2011). [CrossRef]
9. R. Paez-Lopez, U. Ruiz, V. Arrizon, and R. Ramos-Garcia, “Optical manipulation using optimal annular vortices,” Opt. Lett. 41, 4138–4141 (2016). [CrossRef]
10. F. Takahashi, K. Miyamoto, H. Hidai, K. Yamane, R. Morita, and T. Omatsu, “Picosecond optical vortex pulse illumination forms a monocrystalline silicon needle,” Sci. Rep. 6, 21738 (2016). [CrossRef]
11. J. Hamazaki, R. Morita, K. Chujo, Y. Kobayashi, S. Tanda, and T. Omatsu, “Optical-vortex laser ablation,” Opt. Express 18, 2144–2151 (2010). [CrossRef] [PubMed]
12. 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, 3645–3649 (2012). [CrossRef] [PubMed]
13. K. Masuda, R. Shinozaki, Y. Kinezuka, J. Lee, S. Ohno, S. Hashiyada, H. Okamoto, D. Sakai, K. Harada, K. Miyamoto, and T. Omatsu, “Nanoscale chiral surface relief of azo-polymers with nearfield oam light,” Opt. Express 26, 22197–22207 (2018). [CrossRef] [PubMed]
14. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488 (2012). [CrossRef]
15. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013). [CrossRef] [PubMed]
16. K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsuka, “Laguerre-gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses,” Opt. Express 12, 3548–3553 (2004). [CrossRef] [PubMed]
17. V. Y. Bazhenov, M. Vasnetsov, and M. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).
18. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006). [CrossRef]
19. C. Fallet and G. Y. Sirat, “Achromatization of conical diffraction: application to the generation of a polychromatic optical vortex,” Opt. Lett. 41, 769–772 (2016). [CrossRef]
20. N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, “Generation of high-quality higher-order laguerre-gaussian beams using liquid-crystal-on-silicon spatial light modulators,” J. Opt. Soc. Am. A 25, 1642–1651 (2008). [CrossRef] [PubMed]
21. R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to–orbital angular momentum conversion of light,” Science 358, 896–901 (2017). [CrossRef] [PubMed]
22. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-gaussian modes,” Opt. Lett. 32, 3053–3055 (2007). [CrossRef]
23. T. Omatsu, K. Miyamoto, and A. J. Lee, “Wavelength-versatile optical vortex lasers,” J. Opt. 19, 123002 (2017). [CrossRef]
24. A. Forbes, “Controlling light’s helicity at the source: orbital angular momentum states from lasers,” Philos. Trans. R. Soc., A 375, 20150436 (2017). [CrossRef]
25. X. Wang, Z. Nie, Y. Liang, J. Wang, T. Li, and B. Jia, “Recent advances on optical vortex generation,” Nanophotonics 7, 1533–1556 (2018). [CrossRef]
26. D. Kim and J. Kim, “Direct generation of an optical vortex beam in a single-frequency nd: Yvo 4 laser,” Opt. Lett. 40, 399–402 (2015). [CrossRef]
27. A. Ito, Y. Kozawa, and S. Sato, “Generation of hollow scalar and vector beams using a spot-defect mirror,” J. Opt. Soc. Am. A 27, 2072–2077 (2010). [CrossRef] [PubMed]
28. S. Ngcobo, I. Litvin, L. Burger, and A. Forbes, “A digital laser for on-demand laser modes,” Nat. Commun. 4, 2289 (2013). [CrossRef]
29. D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order poincaré sphere beams from a laser,” Nat. Photonics 10, 327 (2016). [CrossRef]
30. 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, 1–6 (2018).
31. X. Li, X. Yu, R. Yan, R. Fan, and D. Chen, “Optical and laser properties of pr3+: Ylf crystal,” Laser Phys. Lett. 8, 791 (2011). [CrossRef]
32. T. Gün, P. Metz, and G. Huber, “Power scaling of laser diode pumped pr 3+: Liyf 4 cw lasers: efficient laser operation at 522.6 nm, 545.9 nm, 607.2 nm, and 639.5 nm,” Opt. Lett. 36, 1002–1004 (2011). [CrossRef]
33. H. Tanaka, R. Kariyama, K. Iijima, K. Hirosawa, and F. Kannari, “Saturation of 640-nm absorption in cr 4+: Yag for an ingan laser diode pumped passively q-switched pr 3+: Ylf laser,” Opt. Express 23, 19382–19395 (2015). [CrossRef] [PubMed]
34. K. Iijima, R. Kariyama, H. Tanaka, and F. Kannari, “Pr 3+: Ylf mode-locked laser at 640 nm directly pumped by ingan-diode lasers,” Appl. Opt. 55, 7782–7787 (2016). [CrossRef]
35. U. T. Schwarz, M. A. Bandres, and J. C. Gutiérrez-Vega, “Observation of ince–gaussian modes in stable resonators,” Opt. Lett. 29, 1870–1872 (2004). [CrossRef]
36. T. Ohtomo, S.-C. Chu, and K. Otsuka, “Generation of vortex beams from lasers with controlled hermite-and ince-gaussian modes,” Opt. Express 16, 5082–5094 (2008). [CrossRef] [PubMed]
37. M. A. Bandres and J. C. Gutiérrez-Vega, “Ince–gaussian modes of the paraxial wave equation and stable resonators,” J. Opt. Soc. Am. A 21, 873–880 (2004). [CrossRef]
38. A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics 8, 200–227 (2016). [CrossRef]
39. 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]
40. M. Okida, T. Omatsu, M. Itoh, and T. Yatagai, “Direct generation of high power laguerre-gaussian output from a diode-pumped nd: Yvo 4 1.3-μm bounce laser,” Opt. Express 15, 7616–7622 (2007). [CrossRef] [PubMed]
