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Abstract
We report the direct generation of optical vortex arrays in a diode-pumped bulk Yb:CALGO laser. By adjusting the angles of the input mirror, output coupler and laser crystal to achieve off-axis-pumped condition, the ring-shaped LG0,±1, two to six order vortex array have been obtained. The phase singularities of optical vortex arrays were tunable from 1 to 6. When the pump power was 13.41 W, the maximum output power of LG0,±1, double-vortex array, three-vortex array and four-vortex array were 1.93 W, 1.5 W, 1.02 W and 0.79 W respectively. The chirality could be adjusted by rotating the output coupler. The topological charges of each phase singularity were determined by interference pattern. Theoretical simulation including intensity distribution, interference fringes and phase distribution have been analyzed. The simulations are in good accordance with experimental results.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Vortex beams (VBs) carrying OAM have been widely used in materials processing [1,2], optical imaging [3,4], underwater optical communication [5], and particle manipulation [6]. Laguerre-Gaussian (LG) beams, especially the LG0,n mode, as the most common and convenient VB obtained in the laboratory, have been extensively studied in recent years [7–10]. Commonly, LG0,n has only one phase singularity. Optical vortex arrays (OVA) have multiple phase singularities, thus, they are considered to have broader application prospects in many fields, such as particle manipulation [11], optical communication [12,13] and Bose–Einstein condensate [14]. Various methods have been proposed for developing OVA, including extra-cavity and inner-cavity methods. The typical extra-cavity methods are based on multiple-beam interference, phase diffractive optical elements, mode conversion, spatial light modulators, and metamaterial [13,15–18]. S. Vyas et al. presented OVA generation using Michelson interferometer and Mach–Zehnder interferometer to realize three-wave interference [15]. G. X. Wei et al. used a phase-only diffractive optical element for generating OVA according to the fractional Talbot effect [16]. Y. F. Chen’s group generate OVA by transforming a standing-wave LG mode into the crisscrossed Hermite-Gaussian (HG) modes via a π/2 cylindrical lens mode converter [17]. J. A. Anguita demonstrated high-quality and coherently superimposed optical vortices using an optical arrangement based on spatial light modulators [13]. J. J. Jin et al. demonstrated a nano-slit metasurface that could generate multi-channel vortex light with equal energy [18]. However, these OVA generators often suffer from several drawbacks. For instance, optical elements may suffer from low damage threshold which makes it difficult to generate higher power OVA. Also, there is a contradiction between the complexity of fabrication procedures and the overall performance. In recent two decades, the technology of generating various vortex array beams in inner-cavity such as semiconductor microcavity or thin-slice lasers has been developed [19–29]. The superposition of different LG or HG modes are usually utilized to describe the inner-cavity OVA. The OVAs in a vertical-cavity surface-emitting laser was discovered by J. Scheuer and M. Orenstein [19]. K. Otsuka demonstrated the OVAs in a thin-slice solid-state LiNdP4O12 laser pumped by a wide aperture laser diode [23]. Dong et al. obtained structured optical vortices in a diode-pumped continuous-wave Yb:YAG/YVO4 Raman microchip laser, the maximum output power was 1.16 W [27]. By adjusting the pump aperture and lateral displacement of the laser diode to control the lateral mode-locking effect, Shen et al. obtained an OVA beam in thin-slice solid-state laser [28]. Chen et al. used a microchip laser pumped with a tilted annular beam to generate an array of vortices, the maximum output power was 2.01 W [29]. For semiconductor microcavity or thin-slice lasers, the optical elements of resonator are integrated together. The OVAs generated by titling the cavity strongly relied on the pump power and tilting angle since saturated population inversion inside the gain medium depended on these two factors [29]. Therefor the OVA could only be obtained in a narrow range of pump power. For example, in Ref. [27], double-vortex array was observed under pump power from 2 to 4 W. However, in all-solid-state laser based on bulk crystal, all elements are separated from each other, so the degree of freedom is added for off-axis-pumped condition and each type of OVA could be generated in a wide range of pump power.
In this article, we have achieved the direct generation of vortex arrays in an all-solid-state bulk Yb:CALGO laser. By adjusting the angles of the input mirror (IM), crystal and output coupler (OC), OVAs with tunable phase singularities from 1 to 6 are obtained. The distribution of phase singularity of 1 to 4 order OVA was analyzed by interferometry. The output power and slope efficiencies were measured. Theoretical simulations were analyzed, and the results matched the experimental data quite well.
2. Experimental setup
A schematic experimental setup of the diode-pumped Yb:CALGO OVA laser is shown in Fig. 1. The pump source is a fiber-coupled laser diode with an output wavelength of 976 nm. The fiber is a multimode fiber with core diameter of 100 µm and numerical aperture of 0.22. The fiber is connected to a 1:2 collimating coupler via an adaptor. The pump light is incident into the Yb:CALGO crystal through the fiber and collimating coupler. The laser gain medium is an a-cut Yb:CALGO (4 mm*4 mm*8 mm, 3 at.%) crystal wrapped in a piece of indium foil and installed in a copper radiator cooled by water at 14 °C. The Yb:CALGO crystal has a very high thermal conductivity (6.9 Wm−1K−1) and extremely broad emission bandwidth (0.75×10−20 cm2 for σ-polarization and 0.25×10−20 cm2 for π-polarization at 1040 nm, the emission bandwidth: 80 nm), thus allowing high-power diode pumping [30]. Both ends of the input mirror (IM) are coated with high transmission at 976 nm and high reflection at 1030∼1080 nm. The output coupler (OC) of the cavity is a plane-concave mirror with a curvature radius of 100 mm (transmissivity of 5% at 1030∼1080 nm). The output laser beam profile and beam interference fringe are recorded by a charge-coupled device (CCD) camera (Dataray, S-WCDLCM-C-UV).
3. Results
The OVAs were realized by rotating the laser crystal, IM and OC. The rotation angles of the crystal, IM and OC were defined as αx,y, βx,y and θx,y, respectively. For example, θx are the angle between x-axis and OC surface. Firstly, the cavity was adjusted to obtain TEM00 mode. Then, rotating the laser crystal (βx=−0.34°, βy=−0.41°), when OC was on axis (θx=θy=0), the doughnut-shaped beam was realized, as shown in Fig. 2(c). To determine the topological charges of LG modes, a simple interferometer with reflective plane-concave mirror [10] was used to observe the interference pattern. Figure 2(i) is the interference pattern of the beam in Fig. 2(c). The clockwise spiral with counterclockwise fork-shaped pattern in the center indicates that there are two topological charges l=1 and l=−1, so the generated beam is LG0,±1 vortex beam with two chirality coexisting. The OC was then rotated in positive or negative direction (θy=0.04° or −0.04°), doughnut-shaped beam could be maintained quite well (as shown in Fig. 2(d) and (b)). Figure 2(j) is the interference pattern of LG beam in Fig. 2(d), and the clockwise pattern indicated that the topological charge was l=1 and the generated beam was LG0,1 vortex beam. In Fig. 2(h) the topological charge was l=−1 since the interference pattern was counterclockwise, therefore, the generated beam in Fig. 2(b) was LG0,−1 vortex beam. With increasing rotating angle of OC (θy=0.08° or −0.08°), the pattern changed from LG modes to double-vortex array (DVA) which had two phase-singularities, as shown in Fig. 2(a) and 2(e). Figure 2(f) and 2(g) are interference pattern of DVA in Fig. 2(a) when the center of spherical wave was located at upper and lower dark hole respectively. Spiral structures with two opposite handedness in Fig. 2(f) and 2(g) give clear evidences for the two-vortex array because it possesses two phase singularities with topological charge l=−1 (upper phase singularity) and l=+1 (lower phase singularity). Figure 2(k) and 2(l) are interference fringes for Fig. 2(e). The counterclockwise and clockwise spiral fringes demonstrate helical phase distribution of the transverse pattern with l=+1 (upper phase singularity) and l=−1 (lower phase singularity). From the contrast of Fig. 2(f) -(l), when the OC was on axis, both chirality of the LG0,1 vortex beam could exist in the cavity, while OC was off axis by tilting the angle of OC, the topological charge of the generated vortex beam could be tuned from −1 to +1, which indicates the chirality of vortex beam could be adjusted by rotating the output coupler. The reason is the symmetry of the cavity was detuned so the helical phase distribution of the vortex beam would be changed, which was manifested by topological charge varying to the contrary sign.
By further adjusting αx,y, βx,y and θx,y (Table 1), the cavity could generate three vortex array (TVA) and four vortex array (FVA). Figure 3(a1), (b1), (c1), (c3) show the transverse pattern of TVA and FVA recorded by the CCD. Interference diagrams of TVA and FVA are shown in Fig. 3(a2)-(a4), 3(b2)-(b4), 3(c2) and 3(c4). In Fig. 3(a2)-(a4), the clockwise, counterclockwise, clockwise spiral fringe in the upper, middle, lower phase singularity represented the topological charges of TVA were [1, −1, 1]. By slightly adjusting θx of OC, the chirality of the phase singularity of the TVA could change to the opposite (Fig. 3(b1)). From Fig. 3(b2)-(b4), the interference patterns demonstrated the topological charges of TVA were [−1, 1, −1]. It’s worth noting that the black line on the images was due to the destroyed filter. Similarly, the chirality distribution of the FVA in Fig. 3(c1) is [1, −1; −1, 1], as shown in Fig. 3(c2). Figure 3(c3) was the intensity image with opposite chirality distribution. The chirality distribution [−1, 1; 1, −1] was demonstrated in Fig. 3(c4).
Fig. 3. The light intensity map of three to six vortex array and interference pattern of three and four vortex array obtained from the experiment.
Table 1. The rotating angles of the optical elements generating OVAs
The rotating angles of optical elements are shown in Table 1. We define κ to represent the overall amplitude of the laser resonator adjustment, $\kappa \textrm{ = }\alpha _\textrm{x}^2 + \alpha _\textrm{y}^2 + \beta _\textrm{x}^2 + \beta _\textrm{y}^2 + \theta _\textrm{x}^2 + \theta _\textrm{y}^2$. It can be seen from Table 1, with the increasing of rotating angles of IM, laser crystal, and OC, the cavity tends to generate higher-order vortex arrays. The reason is mainly that, the symmetry of the cavity is detuned with the increase of rotating angle, leading to the increase of cavity diffraction loss, high-order transverse modes are easier to oscillate in the resonator, and the superposition the different laser modes constitute higher-order vortex arrays.
We measured the power curves of LG01, DVA, TVA and FVA, as shown in Fig. 4. When the pump power was 13.41 W, the maximum output power of LG01, DVA, TVA and FVA were 1.93 W, 1.5 W, 1.02 W and 0.79 W respectively. The slope efficiencies were η1=18.58%, η2=16.56%, η3=11.40% and η4=9.28%, respectively. The output slope efficiency of the laser was reduced when the phase singularities increased due to the increased diffraction loss. For TVA, the pattern of vortex array could not be formed until the pump power was higher than 9.6 W, because under 9.6 W, the lasing mode was single HG mode. And for FVA, the point was 7.48 W. TVA had a higher pump value than FVA when the vortex array was formed, because the oscillation modes constituting TVA were more complicated than FVA. In next simulation part, we demonstrate the TVA is the superposition of four LG modes LG1,1, LG1,−1, LG0,1 and LG0,−1, FVA is the linear combination of a frequency-degenerated family, which are LG1,0, LG0,2 and LG0,−2. Therefore, a higher pump power is required to achieve multiple modes lasing in TVA. Besides 1 to 4 phase singularities, by increasing the angle of OC, we also observed optical vortex array with 5 and 6 phase singularities under pump power of 19W (Fig. 3(d1) and (d2)). The limitation on the number of phase singularities is mainly due to the mode matching between pump beam and the resonator mode. Mode overlap will reduce by increasing the elements’ rotating angle, but diffraction loss will grow, correspondingly higher order modes oscillate in the resonator. But another result is the increase of lasing threshold, which may restrain several modes’ oscillating. One can increase the pump power to make sure all needed mode oscillating, but another risk is laser crystal’s damage. For example, we just obtained the output power of six vortex array (0.54 W), the power of five vortex array and the interference pattern were not measured, because there was damage in the crystal.
Fig. 4. Output power dependence of the OVAs on the pump power. The solid lines are the linear fit of the experimental data. The red points (9.6W, 7.48W) are the pump power when the output arrays are stable.
4. Simulation
The vortex arrays are formed with stationary multimode oscillation which can be described as linear combination of LGp,l or HGm,n modes. Here we choose LGp,l modes as the bases. The multi-mode oscillation leads to transverse mode-locking effect, and forms a stationary transverse pattern. The electrical field of vortex array can be described as follows [31]
Table 2 shows the detailed parameters of LG modes participating in forming vortex arrays with phase singularity from 1 to 6. Vortex array with 1 phase singularity is the doughnut-shaped beam LG0,1 or LG0,−1, the angular index l=1 or −1 depends on the helical phases. Vortex array with 2 phase singularities is formed by two doughnut-shaped beams LG0,1, LG-1 and LG10. Vortex array with 3 phase singularities is the superposition of four LG modes LG1,1, LG1,−1, LG0,1 and LG0,−1, Vortex array with 4 phase singularities is the linear combination of a frequency-degenerated family $2p + |l |= 2$[31], which are LG1,0, LG0,2 and LG0,−2. Vortex array with 5 phase singularities is composed of LG1,1, LG1,−1 and LG0,3. Similarly, for six vortex array, the lasing modes are LG1,0, LG0,3 and LG0,−3.
Table 2. Participating transverse modes and their weights for forming vortex arrays
Figure 5(a1)-(j1) shows the transverse pattern of vortex arrays. Figure 5(a1) and 5(b1) are the doughnut-shaped beam LG0,1 and LG0,−1. Figure 5(c1) and 5(d1) are vortex array with 2 phase singularities. Figure 5(c1) is the array with helical phase with l=1 at lower singularity and l=−1 at upper singularity. Figure 5(d1) is the pattern whose helical phases are opposite to Fig. 5(c1). Figure 5(e1) and 5(f1) are the array with 3 singularities. Figure 5(g1) and 5(h1) are the array with 4 singularities. Figure 5(i1) and 5(j1) are the arrays with 5 and 6 singularities respectively. Figure 5(a2)-(j2) are the interference pattern of vortex arrays. Figure 5(a3)-(j3) are the phase distribution of the vortex arrays. The counterclockwise phase variation of 2π in Fig. 5(a3) indicates doughnut-shaped beam is LG mode with topological charge l=1. The simulation in Fig. 5(b3) is opposite to Fig. 5(a3). In Fig. 5(c3), phase variation of 2π (counterclockwise) around lower singularity and −2π (clockwise) around upper singularity represent l=1 and l=−1 respectively. The simulation in Fig. 5(d3) is opposite to Fig. 5(c3). In Fig. 5(e3), phase variations around upper, middle and lower singularity are (2π, −2π, 2π). Therefore, the topological charges are [+1, −1, +1]. The simulation in Fig. 5(f3) is opposite to Fig. 5(e3). Similarly, the phase variations in Fig. 5(g3) indicated the topological charges are [+1, −1; −1, +1]. And the opposite one in Fig. 5(h3) is [−1, +1; +1, −1]. Figure 5(i3) and Fig. 5(j3) are phase distributions of five and six vortex array, respectively. The calculated phase distribution further confirms that the vortex arrays have phase singularities. From Fig. 5 we can see the simulations are in good agreement with the experimental results.
Fig. 5. Theoretically calculated transverse patterns (a1)–(j1), interference patterns (a2)–(j2) and phase (a3)–(j3) of vortex arrays.
5. Conclusion
Compared with several OVA reports published in recent years [27,29], we obtained the highest power of DVA and TVA. The maximum output power of DVA and TVA in our experiment were 1.5 and 1.02 W, and those in other groups were 1.1 [27] and 0.6 W [29] respectively. The maximum power of FVA was 0.79 W, a little lower than Dong’s result (1.16 W) [27]. It’s worth noting that unlike thin disk or microchip laser, we obtained vortex arrays in a bulk all-solid-state laser. The vortex arrays were achieved by tilting each cavity elements, such as laser crystal, input mirror, and output coupler, thus it provides more degrees of freedom for off-axis-pumped condition to generate multiple LG modes which form vortex arrays. The topological charges of each phase singularity were determined by interference pattern. By adjusting rotating angle of output coupler, the topological charges of one to four vortex arrays could be tuned to the opposite values, leading to adjustable chirality of vortex array. Finally, vortex arrays with tunable phase singularities from 1 to 6 were achieved.
In conclusion, we have obtained optical vortex arrays with tunable phase singularity from 1 to 6 in a diode-pumped bulk Yb:CALGO laser. By adjusting the angle of the IM, OC and laser crystal, the ring-shaped LG0,±1, two to six vortex array have been obtained. When the pump power was 13.41 W, the maximum output power of LG0,1, DVA, TVA and FVA were 1.93 W, 1.5 W, 1.02 W and 0.79 W respectively. The topological charges of each phase singularity were determined by interference pattern. The chirality was adjustable by tilting output coupler. Theoretical simulation including intensity distribution, interference fringes and phase distribution have been analyzed. The simulations are in good accordance with experimental results.
Funding
Local science and technology development project of the central government (YDZX20203700001766); Innovation Group of Jinan (2018GXRC010); National Natural Science Foundation of China (11404196, 11525418, 11974218, 91750201, 91950106); National Key Research and Development Program of China (2019YFA0705000).
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.
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