1. Introduction

Optical vortex beam, referring to the light beam carrying orbital angular momentum (OAM) has attracted great interest for its phase singularity with a certain topological charge [1,2]. The expression of its optical field contains the phase factor term $exp({ \pm il\emptyset } )$, where $\emptyset $ is the azimuth angle and l is the topological charge. Such a beam with helical phase fronts and orbital angular momentum reveals a subtle connection between macroscopic physical optics and microscopic quantum optics. In recent years, varies of studies on optical vortex have been reported and the developments of such beams have led to many advanced applications such as optical handling of microscopic particles [3], atoms manipulation [4], sub-diffraction limit microscopy [5], laser material processing [6], quantum information processing and telecommunication [7].

Ultrafast vortex beams combine the advantages of traditional ultrafast lasers and vortex beams and have shown tremendous potential in a series of application fields, such as ultrafast micro-nano manipulation [8], ultrafast microprocessing [9], ultrafast filament optics [10], generation of extreme UV (XUV) OAM beams by high-order harmonics [11]. Driven by the many applications these beams have spurred, much attention has been focused on their efficient creation. Various methods have been used to generate OAM beams outside the laser cavity, such as inserting special intracavity elements in the forms of spatial light modulators (SLM) [12], q-plates [13], and other tailored metasurfaces [14,15] for generating tunable OAM beams. However, for all these methods, extra components are required and have certain disadvantages, such as low conversion efficiency, low damage threshold, and wavelength limitation. Direct generation of vortex lasers from a resonator called ‘active methods’, which is essentially accomplished by the active selection of the oscillating laser mode could be more compact and robust. These methods include using a doughnut-shaped pump beam in a diode end-pumped laser cavity configuration [16], spot-defect cavity mirrors [17], intra-cavity phase-only optical elements [18], or off-axis pumping configuration induced by rotating the gain medium [19]. Up to now, only a few explorations have been attempted on the direct generation of vortex lasers from an oscillator. The reason for this is that a laser beam with transverse higher order Gaussian mode is often unstable [20]. In 2015, a LG mode optical vortex beam was directly generated in a single-frequency solid state laser end-pumped by a ring-shaped pump beam [16]. Recently, Tang et al. reported a direct generation of continuous wave mode-locked (CWML) LG₀₁-mode vortex beam in a Ti:sapphire laser by using a spot defect spatial filter (SDSF) for intra-cavity mode selection [21]. In 2017, Qiao et al. adopted a noncollinear-pump configuration to adjust and control the high-order transverse mode and then converted into the ultrafast vortex by a cylindrical lens mode converter outside the cavity [19]. In 2021, Zhao et al. reported structured laser beams including the HG₁₀ through off-axis pumping and LG₀₁ mode generated also by employing a single-cylindrical-lens (SCL) converter in a 2 µm mode-locked (ML) solid-state laser [22]. As we know, these ever-reported methods mentioned the utilization of requiring additional or specially designed components in the laser system by employing specially designed cavity mirrors or pumping with a specially designed pump system. Otherwise, these methods mentioned above require transverse mode conversion using “π/2 converters” outside the cavity to generate vortices or vector beams by rephasing their decomposed terms.

In this paper, for the first time, we experimentally demonstrate a direct generation of ultrafast vortex beam from a Tm:CaYAlO₄ oscillator without transverse mode conversion outside the cavity. By accurately adjusted the angle of the end mirror and the distance L between the M4 (flat concave mirror) and the SESAMs to control the beam diameter of laser incidence on the gain medium in the sagittal and tangential planes, the laser emission of annular LG mode was successfully obtained from a folded-cavity resonator. With SESAMs as mode-lockers, we realized ultrafast ML vortex generation operating at 1965 nm. The generated LG₀₁ mode has a maximum output power of 327 mW, which generates a clean, topological charge-tunable ultrafast vortex beam with quite high output power. The chirality of the LG₀₁ mode can be controlled by a simple YAG crystal plate of the laser cavity. This directly emitted vortex laser generated in the ∼2 µm region from a compact and robust Tm:CYA oscillator has shown many potential applications in the areas of molecular spectroscopy and organic material processing amongst others.

2. Experiment setup and results

Figure 1 shows the experimental setup of the diode-pumped ML Tm:CYA laser. The laser crystal was pumped by a fiber coupled laser diode (LD) with 105 µm core diameter and 0.22 numerical aperture (NA) at 793 nm of the laser resonator. A telescope system with magnification of about 1:1 constituted by two identical coupling convex lenses (focal length f = 100 mm) were used to focus the pump beam into the laser crystal. The gain medium was a dopant of 4 at% with dimensions of 3 mm ×3 mm × 6.1 mm and a-cut slice-like Tm:CYA crystal for the linear polarization output. In order to protect the laser crystal from thermal fracture, it was wrapped in a copper heat sink and conductively water cooled at 18°C during the laser experiment. Two end faces of the crystal were anti-reflectively (AR) coated from 1900 to 2100 nm. The radii of the focused laser beam waist inside the gain crystal were 88 µm and 60 µm (the beam waist radii calculated by ABCD matrix under the assumption of fundamental mode) in tangential and sagittal planes respectively, which is to match the size of the pump light spot. The three plano-concave mirrors M1, M2, and M3 have the same radius of curvature (ROC) of −100 mm. The additional plano–concave mirror M4 has a ROC of −50 mm and the above four cavity mirrors were all highly reflectively coated for laser wavelength (reflectivity >99.7% from 1850nm to 2100 nm). The employed plano–concave mirrors of M3 and M4 with different ROCs were furtherly decreased to match the beam diameter on the mode-lockers. And an output ratio mirror (OC1) of 5% from 1850 to 2100 nm was used. The SESAMs (BATOP Inc) is designed to operate at 1950-2020nm with a modulation depth of 3%, a relaxation time of 10 ps, and a saturation fluence of 30 µJ/cm². The cavity mode diameter on the SESAMs was calculated to be 78 µm (under the assumption of fundamental mode). The total optical length is estimated to be ∼124.28 cm, corresponding to the round-trip time of ∼8.3 ns and fundamental repetition rate of ∼120.6 MHz. In order to measure the topological charge of the LG mode, we have added the homemade MZ interferometer outside the laser cavity, as shown in the bottom of the dashed box of Fig. 1. The interferometer consists of a half-wave-plate (HWP) and polarization beam splitter (PBS), two convex lenses F1 of 150 mm focal length and F2 of 40 mm focal length, two high reflectivity mirrors (HR) and OC2 (45°incidence, transmittance is 10% at 1900-2100 nm). The output LG mode beam is separated by using the HWP and PBS. One arm of LG mode beam was focused by a focusing lens (F1 = 150 mm) to control the beam diameter in CCD. The other arm of LG mode beam was shaped to a near-plane wave coherent with the original laser after passing through a convex short focal length-lens (F2 = 40 mm), which the distance between F2 and OC2 is much larger than the 40mm. After combination, the laser beam profile was recorded by the CCD camera.

 

Fig. 1. Schematic of the mode-locked Tm:CYA vortex laser. L1, L2: convex lens with the same focal length of 100 mm; M1, M2, and M3: plano–concave mirrors with radius of curvature (ROC) of −100 mm;M4: plano–concave mirrors with radius of curvature (ROC) of −50 mm; OC: output coupler; HWP: Half-Wave-Plate; A: Aperture diaphragm; PBS: Polarization beam splitter; F:Focus lens (F1 = 150 mm, F2 = 40 mm); HR: high reflectivity mirror; YAG: Yttrium aluminum garnet; The dashed box: (a) The homemade MZ interferometer; (b) Schematic of the spatial overlap of the pump beam, the fundamental transverse mode (TEM₀₀), and the nth-order HG transverse mode (TEM_0n) in the gain medium; (c) Trace of the end mirror moving along z-axis; (d) The coordinate diagram of the end mirror with an angle deflection estimated value of $\mathrm{\theta } = 4^\circ $ in the y-o-z plane.

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An InAs photo detector with bandwidth of 12.5 GHz and a 1 GHz digital oscilloscope (Keysight, DSO-S 104A, Santa Rosa, CA, USA) were employed to record pulse waveforms. An Optical Spectrum Analyzer (Yokogawa AQ6375 1.2∼2.4 µm) with a high resolution of 0.05 nm was used to monitor the output optical spectrum. The radio frequency (RF) spectrum was measured by a 3 GHz spectrum analyzer (Agilent, N9320B, Santa Clara, CA, USA) with a resolution bandwidth (RBW) of 10 Hz. The pulse width was recorded by a commercial autocorrelator (Pulsecheck, APE). A mid-infrared CCD (WinCamD-IR-BB, pixel size of 17 µm), together with a homemade MZ interferometer, was used to record the interference patterns of the generated scalar vortex beams.

3. Results and discussion

Figure 2 shows the dependence of the laser average output power on the incident pump power. With OC of 5%, a plano-plano high- reflectivity mirror replaced the SESAM in the cavity for generating CW lasers with the doughnut-shaped beam mode (see the inset of Fig. 2 (a)). The single- pass pump absorption ratio at 790 nm was measured to be 78%. The CW laser output power increased linearly as the pump power, with a slope efficiency of 8.74% and the maximum output average power is up to 685 mW at the incident pump power of 9 W. By using the SESAM as a mode-locker and finely aligning the distance at L=48.05 mm and deflection angle of the end cavity mirror at estimated value of $\mathrm{\theta } = 4^\circ $, stable mode-locked (SML) pulse train with doughnut-shaped beam mode of Tm:CYA laser can be observed with an oscilloscope and CCD. Increasing the pump power to 10 W, we obtained a doughnut-shaped ultrashort pulse at central wavelength of 1965nm with maximum average output power of 327 mW. As shown in the inset of Fig. 2(a), the doughnut-shaped beam mode could be maintained within a large range of the pump power between 4 W and 10 W. The intensity profile of the output beams clearly shows a donut beam shape with an intensity node at the beam center. Figure 2(b) and Fig. 2(c) demonstrates the temporal characteristics of the obtained doughnut-shaped beam mode ML pulses in the 200 µs and 200 ns time scales. The pulse-to-pulse time interval was measured to be 8.29 ns, corresponding to the repetition rate of 120.6 MHz. In the 200 µs time scale (Fig. 2 (b)), the pulses were uniform of same height without Q-switching or Q-switching ML interruption.

 

Fig. 2. (a): Input–output characteristics in CW (blue) and ML (black) of the Tm:CYA laser. Inset: output beam profiles at different pump powers; (b), (c): Doughnut-shaped ML pulse trains at the average output power of 327 mW.

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The measured CWML optical spectrum and autocorrelation trace at the maximum average output power of 327 mW were shown in Fig. 3(a) and Fig. 3(b). At incident pump power of 10 W, SML optical spectrum with broad bandwidth of ∼2.7 nm was shown in Fig. 3(a). Assuming the Gaussian-shaped hypothesis of the measured interference autocorrelation trace, the pulse duration was measured to be 2.1 ps (Fig. 3(b)). The corresponding time-bandwidth product (TBP) was calculated to be 0.442. This is very close to the Fourier transform limit value of 0.441, indicating the obtained ML pulses have a very high temporal conference. The inset shown a doughnut-shaped beam profile with maximum average output power of 327 mW. Moreover, we measured the radio-frequency (RF) spectrum of the ML pulses, and Fig. 3(d) shows a comb-like spectrum at a large scan range (3 GHz) and the RBW of 100 kHz. The corresponding pulse repetition rate was measured to be 120.6 MHz, corresponding to the laser cavity length of about 1.24 m. At a short scan range (10 MHz) and the RBW of 100 Hz, the RFs of ML pulses have a measured signal-to-noise (SNR) of 74 dBc in Fig. 3(c).

 

Fig. 3. Typical CWML pulses with average output power of 327 mW. (a) ML optical spectrums; (b) Interference autocorrelation traces of the pulses. Inset: output beam profile; (c) RF spectrum with span of 10 MHz; (d) RF spectrum of the ML pulses with span of 3 GHz.

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As shown in Fig. 4, for further verify the vortex property, we constructed a MZ interferometer outside the laser cavity and performed an interference experiment with the vortex beam and a plane wave to characterize the spatial phase of the vortex beams. Figures 4(b) shows the interference patterns of the output laser beam without chirality selected operation [23]. The chirality of the LG₀₁ mode could be controlled by a simple uncoated YAG crystal plate with dimensions of Φ 25.4 mm × 1 mm, which was inserted close to the OC as handedness selector of the laser cavity. Theoretically, the directions of Poynting vector for the linearly polarized two LG modes are different but spiral along $z$-axis synchronously, which implies that different transmission loss can be induced by Fresnel reflection [24]. Due to the output LG beams are linear polarization (Degree of Polarization (DOP) = 19 dB), a YAG crystal plate could be expected to distinguish two modes and produce remarkably pure and stable LG₀₁ mode with well determined handedness [25]. We translated the distance between the OC and the M2 mirror to find the optimal option and rotated the YAG crystal plate of deflection angle with +3°and −3°. Figures 4(a) and4(c) show the experiment interference patterns for the LG_0,+1 and LG_0,−1 mode when we tilted the YAG plate to 3° (or −3°), respectively. As we can see, typical fork-shape stripes with one fork are clearly shown, indicating that the vortex beams have a spiral phase with topological charge of +1 and −1 [26]. For further explorations about the chirality of vortex beam, we conduct numerical simulations and set $ l ={\pm} 1$ in this case. The simulated interference pattern of the LG_0,+1 and LG_0,−1 modes were also shown in Figures 4(d) and 4(f), which agree well with the experimental results.

 

Fig. 4. Measurement of phase of vortex beams. (a) and (c) are the interference patterns of the LG_0,+1 and LG_0,−1 vortices with the plane wave beam, respectively; (b) The interference patterns of plane waves without inserting YAG crystal plate; (d) and (f) are the theoretical results of the experiment interference patterns for the LG_0,+1, LG_0,−1 modes, respectively.(e) The theoretical results of incoherent superposition of LG_0,+1 mode and LG_0,−1 mode;

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By using a CCD working in the mid-infrared region, the M² factors of the CMML vortex laser in the tangential and sagittal planes were measured to be 2.37 and 2.42 (see Fig. 5(a)), which is close to the theoretical value 2 of the LG₀₁ mode [27]. The difference in beam waist position in two directions was caused by different folding angles on sagittal and tangential planes in our folded-cavity. The spatial intensity of the beam at the inset of Fig. 5(a) is slightly higher than that of the other directions, which is due to the superposition of the diffracted spot on the attenuation plate of the CCD [28]. In addition, we measured the mode intensity profile at these waists to confirm OAM in the beam, which is still an annular profile, as shown in the inset of Fig. 5(c), Fig. 5(d) and Fig. 5(e). Three points on the measured data line represent that the focus at the transverse x-axis, y-axis and the coincident point of x-axis and y- axis, respectively, which was collimated by a lens with 200 mm focal length. The far-field beam pattern was collimated by a lens with 200 mm focal length, which shows that the annular beam is completely hollow to the bottom and the ring-to-center intensity contrast is measured to be around 13 dB [see Fig. 5(b)], indicating the obtained ultrafast vortex beam has a high spatial SNR and mode contrast [29].

 

Fig. 5. (a): The measurement beam quality factor M² of the CWML Tm:CYA laser, blue and red show the transverse x and y-axis beam profiles, respectively. The inset shows the far-field beam pattern, which was collimated by a lens with 200 mm focal length at the transverse x (c), y-axis (e) and the coincident point of x-axis and y- axis(d); (b): The ring-to-center intensity contrast profile of the annular beam.

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Direct generation of vortex beams from a laser cavity is normally difficult, the main reason is that a laser cavity usually has rectangular symmetry due to the tilted surface of optical elements, resulting in HG modes being excited [30]. In our experimental setup, the end mirror has an angle (estimated value of $\mathrm{\theta } = 4^\circ $) deflection in the y-o-z planes (see Fig. 1(d)). So that an off-axis angle between the pump beam and the cavity mode axis in the gain medium was set by adjusting the angle in the y-o-z planes of the end mirror, which realizes non-colinear pumping. Here we adopt a noncollinear-pump configuration to finely adjust and control the high-order transverse mode the cavity (see Fig. 1(b)). The cavity loses the symmetry of cylindrical of the resonator due to the astigmatism caused by multiple folds of the concave mirror in the x-o-z plane. The $\textrm{Gouy}$ phase difference for the $\textrm{H}{\textrm{G}_{nm}}{\; }({n + m = 1} )$ mode in the sagittal and tangential planes can be calculated by $\Delta \varphi = |{{\varphi_T} - {\varphi_s}} | $ and the $\textrm{Gouy}$ phase difference dependent on the arm length L between M4 and SESAM [26] (see Fig. 1(c)). According to the ABCD law, accurately adjusting the angle of the end mirror and the distance L between the M4 and the SESAMs can change the beam diameter of laser incidence on the gain medium in the sagittal and tangential planes (Table 1).

Table 1. Simulation beam diameters of laser in the sagittal and tangential planes according to the ABCD law under the assumption of fundamental mode.

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Thus, by setting the off-axis angle at fixed value (θ=4°) and just easily controlling the distance L, various high-order HG transverse modes, such as HG₁₀ (Fig. 6(d), Fig. 6(l)), HG₀₁ (Fig. 6(g), Fig. 6(j), Fig. 6(n)) and HG₀₂ (Fig. 6(i)) modes, were selected to oscillate in the laser, as shown in Fig. 6. After the two kinds of astigmatism are compensated accurately (L=48.05 mm), the cylindrical symmetric cavity is formed without symmetry breaking and a mixture of LG handedness modes (see Fig. 6(c)) can be obtained in the optical resonator. By fine tuning the arm length of L = 49 mm furtherly, doughnut-shaped beams can be generated as shown in Fig. 6(m). The LG_0,+1 mode can be decomposed into a set of HG₁₀ and HG₀₁ modes with $\mathrm{\pi }/2$ phase delay. Further interference measurements showed that the single handedness vortex beam for the LG_0,+1 mode, as shown in Fig. 7.

 

Fig. 6. Different spatial modes of laser output beam by moving the SESAM along the Z axis.

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Fig. 7. Measurement of vortex beams without using YAG-plate handed-selection at L=49 mm. (a) shows the experimentally measured intensity distribution for the LG_0,+1; (b) shows the interference pattern of the LG_0,+1 vortices;

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From Fig. 6, we further found that the mode conversion is sensitive to the variation of the SESAM position. We believe that it is rooted from the two factors. The first is the change of Gouy phase difference caused by the different SESAM position (without adjusting the SESAM angle) as shown in Fig. 1 above, which is very similar to the results in Ref. [26]. The second is the changes of the distribution of laser mode incidence on the gain medium in the sagittal and tangential planes caused by the different distances between the M4 and the SESAMs. As can be seen in the Table 1, as the changing of the L, the beam size on the gain medium and the output mirror changed accordingly, which lead to the different mode conversion based on the noncollinear-pump configuration. In experiment, with careful adjustment of the SESAM position, it was easy to lock the required LG mode without adjusting the SESAM angle, as shown in Fig. 6(c) and Fig. 6(m).

Additionally, we have replaced the SESAM with a plano-plano high- reflectivity mirror (HR) and keep the other mirrors in the optical resonator unchanged at L=48.05 mm, we also observed that the clear doughnut-shaped vortex beams by YAG handedness selection. The experimentally recorded transverse intensity profile of two vortex modes is shown in Figs. 8(a) and 8(d), respectively. When passing the beam through a home-made Mach–Zehnder interferometer, the interference patterns with clear spiral structure were observed and as shown by Figs. 8(b) and 8(e), respectively. Figures 8(c) and 8(f) are the theoretical results of the experiment interference patterns for the LG_0,+1, LG_0,−1 modes, respectively, which agree well with the experimental results. Therefore, we believe that the generation of vortex beam was due to the unique structure of our folded-cavity optical resonator.

 

Fig. 8. Measurement of phase of vortex beams with HR mirror as end-cavity- mirror by YAG handedness selection . (a) and (d) are the experimentally measured intensity distribution for the LG_0,+1,LG_0,−1 modes, respectively; (b) and (e) are the interference patterns of the LG_0,+1 and LG_0,−1 vortices with the spherical wave beam, respectively; (c) and (f) are the theoretical results of the experiment interference patterns for the LG_0,+1,LG_0,−1 modes, respectively.

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4. Conclusions

In conclusion, without using any transverse mode conversion device outside the cavity, we have successfully realized direct generation of a chirality controllable LG₀₁ mode optical vortex beam from a LD pumped Tm³⁺-doped CaYAlO₄ ML laser. Ultrafast vortex beam can be achieved just by featuring pattern matching of a folded-cavity resonator and proper cavity alignment. During the experiment, we accurately adjusted the angle of the end mirror and the distance L between the M4 and the SESAMs to control the beam diameter of laser incidence on the gain medium in the sagittal and tangential planes. After the two kinds of astigmatism are compensated accurately, the cylindrical symmetric cavity is formed without symmetry breaking and a mixture of LG handedness modes co-exist can be obtained in the optical resonator. The chirality of the LG₀₁ mode can be well controlled by a simple YAG crystal plate in the cavity. Other transverse-mode operations were realized by carefully controlling the pattern matching of the folded-cavity resonator. In future work, we will try to achieve a vortex beam of higher than one unit of orbital angular momentum by increasing the pump power. The generated ultrafast vortex laser paves the way for further development of high-resolution nonlinear imaging, optical vortex infrared supercontinuum and high-intensity vortex generation and applications.