
The setup of the Yb:YAG thindisk vortex laser is shown in Fig. 1. The 220μm thin Yb:YAG disk crystal has a Yb^{3+} doping concentration of 7% (Dausinger + Giesen, Stuttgart, Germany), and is placed inside a 48pass pump module^{47}. It is pumped by a fibercoupled diode laser at a wavelength of 940 nm with a maximum output power of 500 W. The pump spot diameter on the disk is adjusted to ~3.3 mm by tuning the collimating lens system. The Yb:YAG disk is highreflection coated at the lasing wavelength (1030 nm) and used as a folding mirror in a Zshaped cavity containing a focusing section formed by two concave mirrors (CM). The other mirrors are highreflection mirrors (HR) at 10101070 nm, except for the output coupler (OC), which has a transmittance of 15%. A watercooled copper aperture (HA) with a diameter of 2.9 mm is placed inside the cavity to assist the selection of the LG vortex beams. The output laser beam is separated by a wedge, with a fraction of the power injected into a homemade MachZehnder (MZ) interferometer, followed by a chargecoupled device (CCD) camera to confirm the profile of the spiral phase. The MZ interferometer is comprised of two halftohalf nonpolarizing beam splitters (NPBS), a convex lens (F1), and two HRturning mirrors. The LG beam is divided into two identical beams by the first NPBS. One beam is directly incident on the CCD. The other beam passes through F1 to enlarge its cross section, which creates an approximate reference spherical wave or plane wave depending on the horizontal distance between F1 and the CCD camera. It is coherently and collinearly combined with the first beam before reaching the CCD camera. With this MZ interferometer, the spiral interference pattern and the chirality can be characterized.
Fig. 1 Setup of the vortex thindisk oscillator and the homemade MZ interferometer. CM, concave mirrors with a radius of curvature of 150 mm; OC, output coupler with an output coupling ratio of 15%; HR, highreflection mirrors; NPBS, nonpolarization beam splitter with 45° incidence angle and splitting ratio of 50%; HA, hard aperture with a diameter of 2.9 mm; F1, convex lens with 50 mm focal length. The distance between the two concave mirrors is approximately 152 mm. The FS is antireflection coated for laser wavelength on both sides with a thickness and diameter of 2 mm and 0.5 inch, respectively.
Whether the LG or other transverse modes oscillate in the laser cavity depends on their gain integrals. Generally, the fundamental mode (LG_{00}) dominates, because it has the lowest threshold owing to the preferred mode matching between the laser and pump beam. For the LG_{0l} mode, the threshold pump power $ {P}_{pth} $ is given by Ref. 48:
$$ {P}_{pth}=\frac{h{\nu}_{p}{I}_{0}}{{\tau }_{c}{\eta }_{q}\left[1\mathit{exp}\left({\alpha }_{p}{d}_{0}\right)\right]}{\left[{\int }_{cavity}^{}{r}_{p}\left(x,y,z\right){s}_{0}\left(x,y,z\right)dV\right]}^{1} $$ (1) where $ h $ is the Planck constant; $ {\nu }_{p} $ is the pump frequency; $ {d}_{0} $ is the length of the thindisk gain medium; $ {\alpha }_{p} $ is the absorption coefficient for the pump in the disk medium; $ z $ is the axial distance perpendicular to the disk surface; $ {\tau }_{c} $ is the cavity photon lifetime; $ {I}_{0} $ is the saturation intensity, and $ {\eta }_{q} $ is the fraction of absorbed pump photons that leads to the subsequent population of the upper laser level. The normalized distributions of the pump and laser beams, $ {r}_{p}\left(x,y\right) $ and $ {LG}_{0l}\left(x,y\right) $, are given by
$$ {\int }_{disk}^{}{r}_{p}\left(x,y\right)dS\equiv 1 $$ (2) $$ {\int }_{disk}^{}{LG}_{0l}\left(x,y\right)dS\equiv 1 $$ (3) The LG_{0l} beam modes oscillating in a cavity consisting of circular plane mirrors are described by the following equation^{49}:
$$ {LG}_{0l}\left(r,\varphi \right)={E}_{r}\mathit{exp}\left(il\varphi \right),\left(l=integer\right) $$ (4) where $ {E}_{r} $ is the invariant field distribution. The beam mode has a phase term $ exp\left(il\varphi \right) $ associated with the azimuthal angle, and can be calculated by solving the scalar Helmholtz equation under the paraxial approximation^{50} and expressed as:
$$ \begin{aligned}{LG}_{p=0,\pm l}=&\sqrt{\frac{2!}{\pi \left(\leftl\right\right)!}}\frac{1}{\omega \left(z\right)}{\left(\frac{\sqrt{2}r}{\omega \left(z\right)}\right)}^{\leftl\right}\mathit{exp}\left(\frac{{r}^{2}}{{\omega \left(z\right)}^{2}}\frac{ik{r}^{2}z}{2\left({z}_{R}^{2}+{z}^{2}\right)}\right) \\\times& {L}_{0}^{\leftl\right}\left(\frac{{2r}^{2}}{{\omega \left(z\right)}^{2}}\right){exp}\left[i\left(\leftl\right+1\right)ta{n}^{1}\left(\frac{z}{{z}_{R}}\right)\right]{exp}\left(\mp il\varphi \right) \end{aligned}$$ (5) where $ r $ and $ \varphi $ are the radial and azimuthal coordinates, $ k $ is the wavenumber, $ {z}_{R} $ is the Rayleigh range, $ \omega \left(z\right) $ is the 1/e radius of the beam at position $ z $, $ p $ is the radial order, and $ l $ is the azimuthal angular order with $ p=0 $. The high order modes with $ p=0 $ and a nonzero $ l $ comprise an azimuthal phase term $ \mathit{exp}\left(\mp il\varphi \right) $, and possess an orbital angular momentum of $ \pm l\hslash $ for each photon, which corresponds to the spiral phase front. The radius of LG_{0l} is calculated by the following equation:
$$ \omega \left(z\right)={\omega }_{0,l}\left(z\right)=\sqrt{\leftl\right+1}{\omega }_{\mathrm{0,0}}\left(z\right) $$ (6) where $ {\omega }_{\mathrm{0,0}}\left(z\right) $ is the radius of the fundamental laser beam calculated using “ABCD” beam transfer matrices. As mentioned previously, the thindisk gain medium is pumped by a multimode fiber laser through a 48pass module, with the resulting pump beam having an approximately topflat profile. Hence, the intensity distribution of the pump beam can be expressed as
$$ {r}_{p}(x,y)=\frac{1}{\pi {r}_{pump}^{2}} $$ (7) In our experiment, we fixed the pump beam radius on the thindisk medium and varied the laser beam size on the thin disk by adjusting the distance between the two concave mirrors within the range of the stability zone or by using concave mirrors with different radii of curvature. In reality, there is a certain curvature of the gain crystal owing to processing tolerances, which can lead to a small difference between the actual mode size in the resonant cavity and the calculated size.
To stimulate higherorder transverse modes inside the thindisk oscillator, the ratio of the fundamental laser beam to the pump spot size should be decreased; in other words, the pump spot should be larger than the LG_{00} mode size. As can be seen from Fig. 2c, when the radius of the LG_{00} beam is 325 μm (a ratio of 19.7%), the LG_{02} beam has the lowest threshold, and can oscillate first in the cavity. By contrast, when the radius of the LG_{00} beam is increased to 625 μm (the ratio increased to 37.9%), the threshold of the LG_{01} mode decreases while that of highorder beam increases. Thus, to generate a pure LG_{01} beam, we decreased the radius of the LG_{00} beam gradually to ensure that the other beammode thresholds were always higher than that of the LG_{01} mode. This was achieved by changing the distance between the two concave mirrors and forcing the oscillator to operate in different stability regions with different intracavity laser mode sizes. A watercooled copper aperture was added to further refine the control of the oscillating modes and suppress any residual higherorder beams.
Fig. 2 a Schematic of the pump beam (blue area) and LG_{0l} beam (red area and orange area) on the thin disk gain medium (black area). b Corresponding beam crosssections of the LG_{0l} (l = 04) beam. The shaded area indicates the pumped region. c d Lasing thresholds of each order of the LG_{0l} beam at different ratios between the sizes of the LG_{00} mode and the pump mode.

Initially, the oscillator emitted in the LG_{00} mode stably (Fig. 3a) with a lasing threshold pump power of 30 W and an output power of 6 W. As the pump power increased, the beam profile changed from the standard LG_{00} mode to the tophat mode (Fig. 3b) and later to the imperfect LG_{01} mode with a lowintensity contrast between the ring and center part of the beam profile (Fig. 3c). The output mode became the standard LG_{01} at a pump power of 105 W with the output power of 18 W (Fig. 3d). This process can be explained as follows. The various transverse modes in the oscillator are influenced by both the diffraction losses and gains mentioned above. At low pump powers near the lasing threshold, the effect of diffraction loss is more pronounced. The loworder LG_{00} mode experiences a lower diffraction loss, and has advantages in terms of mode competition. Thus, it is the first to experience lasing, emitting a beam that has a bright central region. As the pump power increases, the influence of the modedependent gain dominates the mode competition dynamics. At this point, the LG_{01} mode overcomes the lasing threshold and appears simultaneously with the LG_{00} mode, gradually leading to a tophat profile for the combined beam. With a further increase in the pump power, the power growth of LG_{01} is more rapid than that of the LG_{00} and other modes, and it consumes more of the available gain in the laser medium, suppressing other modes and leading to the emission of the pure LG_{01} mode.
Fig. 3 a−d Output beam profiles at an output pump power of 6.0 W, 9.6 W, 14.5 W, 18.0 W, respectively. e−h Simulated beam profiles as a combination of the LG_{00} and LG_{01} modes.
A numerical simulation based on the model presented in Ref. 51 was performed to characterize this process by expressing the total beam profile as a combination of the LG_{00} and LG_{01} modes.
$$ {I}_{\left(\mathrm{0,0}\right)\to \left(\mathrm{0,1}\right)}\left(r\right)={I}_{\mathrm{0,0}}\left(r\right)+\sigma {I}_{\mathrm{0,1}}\left(r\right) $$ (8) where $ {I}_{\mathrm{0,0}} $ and $ {I}_{\mathrm{0,1}} $ are the normalized intensity profiles of the LG_{00} and LG_{01} modes respectively, and $ \sigma $ is an empirically deduced scaling factor determining the relative intensity of the two modes. The simulated combined beam profiles with the best matching $ \sigma $ value are shown in Fig. 3e−h, matching well with the experimentally measured profiles. Fig. 4a, b display the cross sections along the xaxis for the experimental and simulated beam profiles, respectively, indicating a good match between the two.
After the initial process, we obtained a highpower pure LG_{01} mode with the increase of the pump power. The power measurement results are shown in Fig. 5; a maximum output power of 101 W was achieved at a pump power of 310 W. The overall relationship between the output power and pump power is linear. The slope and opticaltooptical efficiencies are 40.7% and 33.3%, respectively. As shown in the graph, the output power does not saturate even at the maximum pump power, indicating a large potential for higherpower vortex beam generation. At the maximum output power of 101 W, the power was continuously recorded within one hour (Fig. 6a), showing a good stability of 0.37% (root mean square, rms). We have also measured the output spectrum of the vortex beam using a commercial optical spectrum analyzer with a resolution of 0.02 nm (Fig. 6b). The laser spectrum is centered at a wavelength of 1030.516 nm with a bandwidth of 0.14 nm at full width at half maximum. Fig. 7 shows the beam profiles of the output vortex laser at different power levels. At an output power of 49 W, a slight distortion in the beam profile began to appear (Fig. 7d). This was likely due to thermally induced stress in the gain crystal, and we did not increase the pump power beyond 310 W to prevent damage. As the next step, we investigate the optimization of water cooling to suppress thermal effects and pursue higherpower LGmode vortex beams.
Fig. 6 a Power stability measurement at the maximum output power. b Measured spectrum at the maximum output power.
To determine the chirality of the output vortex laser beam, an antireflectioncoated fusedsilica plate was inserted into the cavity in front of the output coupler. Initially, the surface of the plate was normal to the beam path. By adjusting the tilt angle of the plate to 2.5° or −2.5°, we can respectively obtain either the LG_{01} or the LG_{01} mode. Fig. 8 shows the interferograms between the 101W LG mode (beam vortex chirality of both l = 1 and l = −1) with either a plane wave (Fig. 8i−l) or spherical wave (Fig. 8e−h). Under planewave interference (Fig. 8i−l), a pair of Yshaped forks with opposite orientations can be observed. In the case of spherical wave interference (Fig. 8e−h), a spiralling pattern in either the counterclockwise or clockwise direction can be observed, with the number of spiral arms equivalent to the topology charge $ \leftl\right $. The experimental results are in very good agreement with the simulated results, as shown in the third and fourth columns of Fig. 8. We also measured the radial intensity contrast of the vortex, which is approximately 11.4 dB and 11.2 dB along the x and y axes (Fig. 9a). The generated vortex beam has excellent beam quality at an output power of 101 W with $M_x^2 $ = 2.09 and $M_y^2 $ = 2.02 (Fig. 9b), which is close to the ideal value (M^{2} = 2).
Fig. 8 Beam profiles and interference patterns of the LG beams. a−d Measured and simulated intensity distribution of the LG_{01} and LG_{01} beams. e−h Measured and simulated spiral phase structure of the LG_{01} and LG_{01} beams when interfered with a spherical wave. i−l Measured and simulated spiral phase structure of the LG_{01} and LG_{01} beams when interfered with a plane wave.
Fig. 9 a Radial distribution of the spatial intensity for the LG vortex beam at an output power of 101 W. b Beam quality measurement of the 101W LG vortex beam.
Such a laser source is beneficial for applications in optical ablation and microstructure manufacturing, because the OAM of photons can be transferred to the processed material. For highefficiency optical ablation, a highpower vortex beam enables highquality processing with less debris, a clearer outline of the ablated zone, and a smoother surface, owing to the effect of rotational motion^{12}. Additionally, a highpower vortex laser beam can twist the metal to form chiral nanoneedles during the ablation process^{14}. Besides, compared to the Gaussian beam, the vortex beam has a significant advantage in drilling submillimetersized throughholes on stainless steel without moving the optical system^{52}.
The vortex beam power in this experiment is not yet saturated, and a higher pump power is withheld to avoid damaging the disk crystal. In the future, we will optimize the crystal coolant temperature and increase the pump spot area on the disk crystal so that a higher absolute pump power can be applied. Based on this, a new resonant cavity can be designed to optimize mode matching to the largest possible pure LG_{01} mode, together with the implementation of hard apertures with more precise diameters for finer mode selection. The system can also be combined with active multipass cell^{53} technology to increase the output efficiency and power. With these measures, the direct generation of a vortex beam from a thindisk oscillator with an output power higher than 1 kW is expected in the future.
The generation of significantly higher topological orders is also feasible based on the thindisk architecture demonstrated in this study. According to the model presented in this work, the mode size ratio of the fundamental laser beam to the pump spot size should be decreased to stimulate higherorder transverse modes. This can be achieved by increasing the pump spot size or decreasing the laser spot size with a suitable cavity design. Additionally, a ringshaped pump beam can be applied to achieve mode matching between the pump beam and highorder vortex beam. Furthermore, a defectspot mirror and hard aperture of suitable size can also be utilized to suppress the oscillation of the loworder or undesired highorder modes.

According to Fox–Li iterationbased algorithms, the selfreproducing process of the beam modes in a cavity consisted of circular plane mirrors is described as^{49}
$$\begin{aligned} {u}_{q+1}=&\left(\frac{1}{\prod _{1}^{q+1}{\gamma }_{n}}\right)\upsilon \\=&\frac{i}{2\lambda }{\int }_{0}^{a}{\int }_{0}^{2\pi }{u}_{q}\left({r}_{1},{\varphi }_{1}\right)\frac{{e}^{ikR}}{{R}_{q}}\left(1+\frac{b}{{R}_{q}}\right){r}_{1}{\rm d}{\varphi }_{1}{\rm d}{r}_{1} \end{aligned}$$ (9) With
$$ {R}_{q}=\sqrt{{b}^{2}+{r}_{1}^{2}+{r}_{2}^{2}2{r}_{1}{r}_{2}\mathrm{cos}({\varphi }_{1}{\varphi }_{2})} $$ (10) where $ \upsilon $ is an invariable distribution function; $ {u}_{q+1} $ and $ {u}_{q} $ are the fields after q and q+1 transits; a is the radius of the circular mirror; and b is the distance between two adjacent mirrors; $ {\gamma }_{n} $ is a complex constant independent of position coordinates.
For $ b\gg a $ and $ \dfrac{{b}^{2}}{{a}^{2}}\gg \dfrac{{a}^{2}}{b\lambda } $, the solutions to the integral equation are given by:
$$ \upsilon \left(r,\varphi \right)={T}_{n}\left(r\right)\mathrm{exp}\left(il\varphi \right),\left(l=integer\right) $$ (11) where $ {R}_{n}\left(r\right) $ satisfies the following reduced integral equation:
$$ {T}_{n}\left({r}_{2}\right)\sqrt{{r}_{2}}={\gamma }_{n}{\int }_{0}^{a}{K}_{n}\left({r}_{2},{r}_{1}\right){T}_{n}\left({r}_{1}\right)\sqrt{{r}_{1}}d{r}_{1} $$ (12) with
$$ {K}_{n}\left({r}_{2},{r}_{1}\right)=\frac{{i}^{n+1}k}{b}{J}_{n}\left(k\frac{{r}_{1}{r}_{2}}{b}\right)\sqrt{{r}_{1}{r}_{2}}{e}^{\frac{ik\left({r}_{1}^{2}+{r}_{2}^{2}\right)}{2b}} $$ (13) where $ {J}_{n} $ is a Bessel function of the first kind and nth order.
The invariable field is calculated by solving the scalar Helmholtz equation^{50} as shown in Eq. 5.
The LG_{0l} modes with zero radial order p and a nonzero azimuthal order l are typical optical vortex laser beams comprising an azimuthal phase term $ \mathrm{e}\mathrm{x}\mathrm{p}(\mp il\varphi ) $ corresponding to the spiral phase front.

The Poynting vector $ \overrightarrow{S} $ for a linearly polarized LGmode optical vortex beam is expressed as^{56}:
$$ \overrightarrow{S}={\varepsilon }_{0}\left(\frac{{\omega }_{0l}krz}{{z}_{R}^{2}+{z}^{2}}\hat r+\frac{{\omega }_{0l}l}{r}\hat{\varphi }+{\omega }_{0l}k\hat{z}\right){\left{LG}_{0,l}\right}^{2} $$ (14) where $ {\omega }_{0l} $ is the angular frequency. The Poynting vector follows a spiral path along the propagation direction, and its rotation direction is determined by the sign of the azimuthal order l. Therefore, the direction of the Poynting vector is not parallel to the propagation direction. In the experiment, when no fusedsilica plate was inserted, the LG_{01} and LG_{01} modes experienced similar losses and could be generated simultaneously. However, environmental perturbations can cause sudden jumps between the two chiral modes. Consequently, a fusedsilica plate was inserted into the beam path. Its surface is almost normal to the optical beam, but is slightly tilted either in the clockwise or counterclockwise direction by ~2.5°. Thus, the symmetry in the chirality is destroyed, with one of the chiral modes experiencing an increased loss and suppression of its generation. By switching between the two FS angles, a stable output can be sustained in one of the LG chiral modes.

The electric field distribution of a vortex beam can be approximated by^{56}
$$ {E}_{1}={A}_{1}\mathrm{exp}\left(\frac{{\rho }^{2}}{{\omega }_{0}^{2}}\right)\mathrm{exp}\left(il\varphi \right) $$ (15) where $ {A}_{1} $ represents the amplitude and $ \rho =\sqrt{{x}^{2}+{y}^{2}} $. The distributions of the plane and spherical waves generated by our MZ interferometer (Fig. 1) are
$$ {E}_{2}={A}_{2}\mathrm{exp}\left(\frac{{\rho }^{2}}{{\omega }_{0}^{2}}\right)\mathrm{exp}\left(ikx\right) $$ (16) $$ {E}_{3}={A}_{3}\mathrm{exp}\left(\frac{{\rho }^{2}}{{\omega }_{0}^{2}}\right)\mathrm{exp}\left(ik\sqrt{{d}^{2}+{\left({xx}_{0}\right)}^{2}+{y}^{2}}\right) $$ (17) where $ {A}_{2} $ and $ {A}_{3} $ represent the amplitudes of the plane and spherical waves, respectively; d is the distance between the lens generating the spherical waves (lens F1 in Fig. 1) and the CCD; and $ {x}_{0} $ is the horizontal offset of the lens relative to the beam. The intensities on the CCD due to interference between the plane waves or spherical waves with the original LG vortex beam are given by
$$ {I}_{2}={\left{E}_{1}+{E}_{2}\right}^{2}=\left[{A}_{1}^{2}+{A}_{2}^{2}+2{A}_{1}{A}_{2}cos\left(l\varphi +kx\right)\right]\mathrm{exp}\left(\frac{2{\rho }^{2}}{{\omega }_{0}^{2}}\right) $$ (18) $$\begin{aligned} {I}_{3}=&{\left{E}_{1}+{E}_{3}\right}^{2} \\=&\left[{A}_{1}^{2}+{A}_{3}^{2}+2{A}_{1}{A}_{3}cos\left(l\varphi +k\sqrt{{d}^{2}+{\left({xx}_{0}\right)}^{2}+{y}^{2}}\right)\right]\\&\mathrm{exp}\left(\frac{2{\rho }^{2}}{{\omega }_{0}^{2}}\right)\end{aligned} $$ (19) In the experiment, the beam from the first NPBS that passed through the lens F1 served as the reference beam. By shifting the horizontal position of F1 away from the beam axis while maintaining the spatiotemporal overlap of the reference and LG beams on the CCD, the reference beam exhibits either spherical or planewave characteristics, and spiral or forkshaped interference patterns can be generated. Fig. 8e−h show the experimentally measured and simulated spiral patterns, respectively. The simulated results were obtained by overlapping and interfering the center of the spherical waves with that of the LG beam. Fig. 8i−l show the experimentally measured and simulated forkshaped interference patterns, respectively. It can be seen that in both the experimental and simulated results, the fork stripes are curved instead of being straight. This is because the reference plane waves, which are generated and simulated by shifting the center of the spherical waves horizontally, are only approximations of the true plane waves. The number and direction of forks, including the direction of curvature, agreed well between the experiments and simulations.
100W Yb:YAG thindisk vortex laser oscillator
 Light: Advanced Manufacturing 4, Article number: (2023)
 Received: 08 August 2023
 Revised: 18 November 2023
 Accepted: 20 November 2023 Published online: 29 December 2023
doi: https://doi.org/10.37188/lam.2023.040
Abstract:
Optical vortices carrying orbital angular momentum and spiral wavefront phases have garnered increasing research interest owing to their numerous applications. Here, we present a simple yet effective approach to generate powerful optical vortices directly from a thindisk laser oscillator. The demonstrated source delivered Laguerre–Gaussian beams with an output power of up to 101 W. To the best of our knowledge, this is the highest output power of all optical vortex laser oscillators. The highpower vortex output will have significant implications for laser ablation and micromachining at high throughput and for largearea applications. Additionally, it serves as a new platform for the further development of more complex highpower opticalvortex beams.
Research Summary
New powerful vortex laser source: 100W thindisk vortex laser oscillator
Optical vortices carrying orbital angular momentum and spiral wavefront phases have garnered increasing research interest owing to their numerous applications. Jinwei Zhang and Lisong Yan from Huazhong University of Science and Technology and colleagues now present a simple yet effective approach to generate powerful optical vortices directly from a thindisk laser oscillator. The demonstrated source delivered Laguerre–Gaussian beams with an output power of up to 101 W, which is the highest output power of all optical vortex laser oscillators. The highpower vortex output will have significant implications for laser ablation and micromachining at high throughput and for largearea applications. Additionally, it serves as a new platform for the further development of more complex highpower opticalvortex beams.
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