The optical vortex generally refers to the Laguerre–Gaussian (${\mathrm{LG}}_{0l}$) mode with a spiral phase term of $\exp (il\phi )$ in its complex amplitude expressed in cylindrical coordinates, where $l$ is the topological charge. In addition to the conventional spin angular momentum (SAM), the attractive property of the optical vortex is that it can carry a well-defined orbital angular momentum (OAM) of $l\mathit{ħ}$ per photon^[1]. Due to the existence of OAM, the optical vortex exhibits wide applications in fields such as optical communication to increase the transmission capacity^[2], optical tweezers to manipulate the particles^[3], and detection of a spinning object^[4]. Up to now, various methods have been exploited for the production of optical vortices. Basically, they can be divided into two categories: intracavity mode selection and extra-cavity mode conversion. The implementation of extra-cavity mode conversion is mainly based on transformation of other transverse laser modes into a vortex beam by using optical phase elements, like cylindrical lens^[5], spiral phase plate^[6], forking grating^[7], and spatial light modulators^[8]. Such technique has advantages of simple operation, enabling generation of different orders of vortex beams; however, the other side suffers from drawbacks of low power level and wavelength sensitivity due to the dependence on material properties such as the refractive index and laser damage threshold. Alternatively, the intracavity mode-selection technique, i.e., direct generation of vortex beams from a laser cavity, can overcome these drawbacks and enable the production of optical vortices with a high power level, high beam quality, and, in particular, the potential for wavelength tuning. One notable thing in such a laser cavity is the synchronous oscillation of the vortex beams with opposite handedness due to their same spatial intensity distribution along the gain media^[9–12], which will result in a coherent superposition of these two beams to form a “petal-like” mode with zero total OAM. Fortunately, a few methods, such as intracavity nanoscale aluminum stripes^[9], propagation symmetry broken in combination with Fresnel reflection loss^[10], polarization states separation^[11], and asymmetric resonator loss induced frequency degeneracy^[12,13], have been exploited to control the handedness, thus making it possible for such vortex lasers for practical applications outside specialized laboratories.

Nowadays, generation and modulation of OAM in the laser cavities have been intensively studied with different techniques, such as spot-defect mirror^[14], annular-shape beam pumping^[15,16], and intracavity spiral phase plate^[17]. However, wavelength tuning of vortex beams with well-maintained spatial amplitude and phase structure is scarcely reported from a laser cavity. To date, the reflective volume Bragg grating (VBG) has been theoretically and experimentally proved to be an effective wavelength selector for optical vortices with well-conserved OAM^[18]. Most recently, a birefringent filter (BF) was experimentally used in a Yb:CaGdAlO₄ (Yb:CALGO) vortex laser and exhibited similar wavelength tuning ability for vortex beams^[19]. Unfortunately, all of the previous reports have been focused on the near-infrared (IR) spectral region, and the wavelength tuning range was less than 50 nm limited by the gain spectrum of laser materials or cavity design^[18,19]. So, wide tunability of the vortex laser, in particular, in the mid-IR region is desired from the viewpoint of their potential applications.

The thulium-doped yttrium aluminum perovskite (Tm:YAP) crystal has a high quantum efficiency for laser operation at 2 µm due to the existence of a self-quenching mechanism between the ${{}^3\mathrm{H}}_4$ and ${{}^3\mathrm{F}}_4$ energy levels. Laser emission from the ${{}^3\mathrm{F}}_4\to {{}^3\mathrm{H}}_6$ transition has been observed in the spectral range of 1.86–2.03 µm^[20,21]. In addition, the absorption band related to a transition from the ${{}^3\mathrm{H}}_6$ to ${{}^3\mathrm{H}}_4$ level can be easily pumped by a high-power commercial AlGaAs laser diode (LD) at 795 nm. This makes it a perfect gain media for the wavelength tuning laser in the 2 µm spectral range. In this paper, we have experimentally demonstrated a broad wavelength tuning range of the vortex laser in the 2 µm spectral range from a Tm:YAP laser by employing a BF as the wavelength selector. The produced optical vortices with a broad wavelength tuning range will have potential applications in optical communication with a combination between wavelength-division multiplexing (WDM) and OAM mode-division multiplexing (OAM-MDM).

Generation of a vortex laser in this work was realized by using an annular pumping technique, which is, in principle, based on a selection of the transverse mode in the cavity. The BF is a 2-mm-thick octagonal quartz plate (a uniaxial crystal with refractive index of $n_o=1.5209$ and $n_e=1.5291$ at 2 µm^[22]) with its optical axis parallel to the surface that has been optically polished. Since the BF was placed in the cavity at a Brewster’s angle^[23,24], the oscillated laser in the cavity has specific linear polarization. The laser with the lowest threshold power will preferentially oscillate. According to the relationship $P_{\text{th}}\propto A_{\mathrm{eff}}$^[25], where $P_{\text{th}}$ and $A_{\mathrm{eff}}$ represent threshold power and effective pump area, respectively, the mode that matches the smallest $A_{\mathrm{eff}}$ will be oscillated first. Figure 1(a) shows the calculated $P_{\text{th}}$ of the three lowest order LG modes [transverse electromagnetic (${\mathrm{TEM}}_{00}$), ${\mathrm{LG}}_{0,\pm 1}$, and ${\mathrm{LG}}_{0,\pm 2}$] versus the cavity length $L$. Here, the reshaped pump beam has a near-field annular intensity profile with a 300 µm inner diameter and an outer diameter of 500 µm. Note that ${\mathrm{LG}}_{0,+1}$ and ${\mathrm{LG}}_{0,-1}$ (or ${\mathrm{LG}}_{0,+2}$ and ${\mathrm{LG}}_{0,-2}$) are a pair of degenerate modes thus exhibiting the same threshold pump power. To obtain a single transverse mode laser operation at the ${\mathrm{TEM}}_{00}$ or ${\mathrm{LG}}_{0,\pm 1}$ mode, the cavity length $L$ should be in the range of 152–180 mm (region III) or 50–152 mm (II), respectively.

Figure 1(b) shows the schematic of the wavelength tuning vortex laser. The pump source (LD) was a fiber (100 µm core diameter and 0.22 NA) coupled LD with a center wavelength of 793 nm. The pump light was at first collimated by a plano-convex lens (${\mathrm{L}}_1$) with a focal length of 25.4 mm, then was reflected by a hollow mirror (${\mathrm{M}}_1$, 2-mm central hole), and reshaped to an annular beam. Finally, the reshaped pump beam was focused into the center of the laser crystal by a lens (${\mathrm{L}}_2$) with $f=150\mathrm{mm}$. The gain medium is an $a$-cut YAP crystal (orthorhombic, $Pnma$ space group) doped with 3% (atomic fraction) thulium ion (${\mathrm{Tm}}^{3+}$). The crystal has the dimensions of 6 mm (a) × 4 mm (b) × 4 mm (c), and its two end faces were antireflective coated at 1900–2150 nm. To mitigate the thermal load, the crystal was wrapped with indium foil and tightly mounted in a copper holder which was water cooled to 15.0°C. The resonator was a standard hemispherical cavity including a plane input mirror (IM) and a plane-concave output coupler (OC) with radius of curvature of $R_{\mathrm{OC}}=200\mathrm{mm}$. In this work, the experimentally optimized transmission of the OC was 10%. Beam profiles and the corresponding self-interference patterns were recorded by using a mid-IR CCD camera (Xeva-1.7) together with a homemade Mach–Zehnder interferometer.

For comparison, we at first studied the tuning performance of the fundamental mode by adjusting the cavity to a length of $L=160\mathrm{mm}$, i.e., in region III, as shown in Fig. 1(a). Without the BF, the laser was linearly polarized (extinction ratio of 28 dB) along the $b$ axis due to the natural birefringence of the crystal. As can be seen in Fig. 2(a), the laser slope efficiency was 27.6% with respect to the absorbed pump power, and the peak wavelength of the optical spectrum at the highest power was located at 1953.8 nm. Wavelength tuning was realized by inserting the BF in the cavity at a Brewster’s angle. Since the BF mandated that the laser has horizontal polarization, wavelength tuning of the laser with polarization of $\mathrm{E}\parallel b$ or $\mathrm{E}\parallel c$ can be obtained just by rotating the laser crystal 90 deg along its $a$ axis. For the $\mathrm{E}\parallel b$ laser, the laser slope efficiency was 35.4%. As shown in Fig. 2(b), the corresponding wavelength tuning range was 104 nm from 1897.1 to 2001.5 nm at an output power of 0.69 W. For the case of the $\mathrm{E}\parallel c$ laser, the threshold absorbed pump power increased to 4.6 W due to the smaller emission cross section of Tm:YAP crystal in this direction^[26], and the laser slope efficiency was decreased to 30.1%. In this case, the tuning range was only 70 nm from 1893 to 1963.3 nm, because, in this case, a larger population inversion ratio was required for lasing, thus resulting in a narrower effective gain bandwidth. With a Glan-type polarizer, the polarization states of both laser beams were measured. Figure 2(c) shows the transmitted laser power as a function of rotation angle in polar coordinates, and the extinction ratio of $\mathrm{E}\parallel b$ and $\mathrm{E}\parallel c$ laser was measured to be 39.3 and 35.0 dB, respectively, higher than that without the BF.

Following the theoretical analysis, wavelength tuning of the optical vortices (${\mathrm{LG}}_{0,\pm 1}$) was thereafter studied by shortening the laser cavity to a length of 110 mm, i.e., locating in region II, as shown in Fig. 1(a). To select the pure ${\mathrm{LG}}_{0,+1}$ (or ${\mathrm{LG}}_{0,-1}$) mode, i.e., handedness control, additional asymmetric loss that was introduced by slightly misaligning the laser cavity, is necessary to reduce the frequency difference of the intracavity eigen modes, thus forming an LG mode with well-defined handedness^{[12,13,27,28]}. For the $\mathrm{E}\parallel c$ laser, which has a lower gain as in the case of the fundamental beam, the threshold absorbed pump power for the ${\mathrm{LG}}_{0,+1}$ (or ${\mathrm{LG}}_{0,-1}$) mode was 4.8 W. A slight change of the pump power or rotation of the BF will destroy the spatial structure of the optical vortex, leading to mode instability or mixing. We attributed this to the thermal lens effects at a high level of absorbed pump power; this was confirmed from the slightly improved stability by decreasing the temperature of the cooling system. Figure 3 shows the simulated beam radius in the crystal as a function of the absorbed pump power, i.e., different thermal lens effect^[29]. For example, when the absorbed pump power of $\mathrm{E}\parallel c$ was 5.5 W, the calculated focal length of the thermal lens was about $f=181\mathrm{mm}$, corresponding to a beam radius of 318 µm. This means the laser was operated at the edge of region III, as seen in Fig. 3, thus leading to an emission of a mixed mode, as shown by the inset. Therefore, it seems unsuitable for wavelength tuning of the vortex beams for such $\mathrm{E}\parallel c$ polarization due to the high threshold pump power and serious thermal lens effects.

Next, we studied the wavelength tuning performance of the vortex laser for $\mathrm{E}\parallel b$ polarization by rotating the laser crystal. Figure 4(a) shows the output power with respect to the absorbed pump power. Without the BF, the threshold absorbed pump powers were 2.73 W and 2.57 W for the ${\mathrm{LG}}_{0,+1}$ and ${\mathrm{LG}}_{0,-1}$ modes, corresponding to the slope efficiencies of 25% and 27.1%, respectively. The inset shows their optical spectra with the peak wavelength located at 1950.9 nm and 1953.8 nm, respectively. Similarly, the vortex laser was linearly polarized along the $b$-axis direction owing to the natural birefringence. By inserting the BF into the laser cavity, wavelength tuning of ${\mathrm{LG}}_{0,+1}$ and ${\mathrm{LG}}_{0,-1}$ modes was, respectively, realized. To confirm the OAM of the produced optical vortices, we measured the beam profiles and the corresponding self-interference fringe patterns by using a CCD camera and a homemade Mach–Zehnder interferometer. As shown in Fig. 4(b), a clean doughnut-beam profile without other obvious transverse modes and the clear interference fringes indicate the high purity of the generated ${\mathrm{LG}}_{0,+1}$ (or ${\mathrm{LG}}_{0,-1}$) mode.

For the ${\mathrm{LG}}_{0,+1}$ mode, the maximum output power was 0.64 W with a slope efficiency of 29.2%, and, in the whole pump power range, the optical vortex can be well maintained. As shown in Fig. 5(a), the wavelength was tunable from 1934.9 to 1993.3 nm at an output power of 0.3 W, corresponding to a range of 58 nm. Figure 5(b) shows the optical spectra at different emission peaks, the spectral full width at half-maximum (FWHM) in the full range was less than 50 pm, which should be helpful for the high-purity vortex laser operation^[28]. In comparison, the ${\mathrm{LG}}_{0,-1}$ mode exhibited a better performance, which may be caused by the lower introduced losses for generation of such a mode. A maximum output power of 0.76 W was achieved with a slope efficiency of 35.5%. The wavelength tuning range was broadened to more than 76 nm, from 1920.4 to 1996.6 nm. It should be emphasized that a wider tuning range from 1913.4 to 2008.4 nm could be obtained in this case; however, the spatial phase front has been destroyed, leading to beam distortion at the extended spectral region. Nevertheless, the range achieved in the present work is much broader than the previous reports based on the VBG or BF operating in the near-IR region^[20,30–34].

In summary, we reported on the first, to the best of our knowledge, wavelength-tunable vortex laser in the 2 µm spectral range by employing a BF. A continuous tuning range of 76 nm from 1920.4 to 1996.6 nm has been demonstrated for the ${\mathrm{LG}}_{0,-1}$ mode, with a typical linewidth of less than 50 pm. In the present work, further power scaling of the vortex laser or a broader wavelength running range was limited by thermally induced beam distortion and mode mixing. Therefore, a cavity design with a large beam waist and optimization of the cooling system will be the next step to improve the laser performances. Nevertheless, this work, as a proof of principle study, shows that the BF can be employed as an effective wavelength selector for the ${\mathrm{LG}}_{0,\pm 1}$ mode in the 2 µm spectral range with a well-maintained spiral phase front. Such a tunable vortex laser providing a specific wavelength selection and reliable topological charge conversion should have potential application in the field of high-capacity optical communications with wavelength and OAM-MDM.