1. Introduction

Optical vortex (OV) has an isolated point singularity in the wave front with a screw-type phase distribution. Since the phase is indeterminate at the singularity point, both the real and the imaginary parts of the field intensity are zero [1]. So far, the most commonly studied OV is the so-called Laguerre-Gaussian (LG) mode vortex beam with a single-ring intensity profile [2–6]. In recent years, optical vortex crystals (OVC) with more complex spatial structure, such as those with multiple separated individual singularity points [7], have aroused great interest. They are highly desirable in various applications, including in the trapping and guiding of micro-particles [8–10], quantum physics [11,12], and optical communication [13,14].

Femtosecond OV lasers, by combining orbital angular momentums (OAMs) with high peak-power and short pulse durations, play an important role in numerous applications due to their exotic structure in both time and space. These include special-structure micro-processing [15–18], subwavelength nonlinear microscopy [19] and the manipulation of micro-particles [20]. The generation methods for ultrafast vortex can generally be divided into active vortex generators and passive vortex generators [2]. Many existing passive vortex generators transform TEM_0,0 modes emitted from an ultrafast laser into LG_m,n modes using phase modulation elements, including computer-generated holograms (CGHs) [21,22],spatial light modulations (SLMs) [23] and spiral phase plates (SPPs) [24,25]. However, CGHs require additional 4f compressors to compensate their dispersion and distortion, which reduce conversion efficiency and mode purity [14]. Meanwhile, SLMs are limited by their costs, losses and laser damage threshold [16], and SPPs are limited by the wavelength of the incident laser and can only work with narrow spectral bandwidths [18]. Thus, obtaining vortices with shorter femtosecond durations with their wider laser spectrum remains a challenge. Alternatively, an astigmatic mode converter (AMC) such as those consisting of single or double cylindrical lenses [26–29], can be used to transform high-order Hermite-Gaussian (HG) modes into high-order LG modes by rephasing their decomposed terms. These passive elements can sustain wider bandwidths and higher power at lower costs. The required input high-order HG modes can be directly generated by a laser.

Active vortex laser generators – those that generate either the HG or the LG modes directly – commonly work by selectively enhancing the gain of the desired modes relative to other modes by, for example, annular pumping [30], structured mirrors [31], or by exploiting the thermal lensing effect in the laser medium [32]. Direct generation of OVs from a laser cavity is expected to provide higher efficiencies and improve the beam quality due to the inherent feedback and filtering effect in the cavity. However, state-of-the-art demonstrations of directly generated OV predominantly focus on continuous-wave output, with only limited demonstrations for ultrafast OVs so far.

Ultrafast HG or LG modes can be generated from solid-state lasers that achieve mode locking either using semiconductor saturable absorber mirrors (SESAMs) or Kerr-lens mode locking (KLM), with a wide range of pulse durations reported so far [33–36]. Beams with multiple vortices and singularities have also been generated directly from lasers, albeit only at picosecond ranges [37,38]. While OVs with the shortest duration to date have been demonstrated using SESAMs [21], SESAMs are susceptible to damage and are costly to fabricate. Furthermore, KLM has the potential to generate shorter pulses due to its effectively instantaneous response and higher nonlinear modulation depth [39–41].

In this letter, we experimentally demonstrate a KLM laser that produces femtosecond beams with either two or five optical vortices respectively. The conversion between the two states can be realized by adjusting the pitch angle of the end mirror. By accurately adjusting the location of the gain medium to control the Kerr effect in the crystal, we obtained the LG_0,0-1.5*LG_0,2 and LG_0,1-1.5*LG_1,1 modes with, respectively, 184 fs and 247 fs pulses, and 427 mW and 220 mW of average output power after mode conversion outside of the cavity. These ultrafast multiple OVs could provide additional control parameters for various applications such as vortex-based optical communication and high-resolution imaging.

2. Experimental setup

The laser oscillator uses a 3 × 3 × 3 mm³, 5-at% doped Yb:KGW crystal (cut along the Ng-axis) with antireflection (AR) coatings (980-1080 nm) in a Z-fold cavity (Fig. 1). To protect the Yb:KGW laser crystal, it is wrapped in indium foil and mounted tightly on a water-cooled copper holder, where the temperature is maintained at 18 °C. A fiber-coupled diode laser (100 µm core diameter, NA 0.22) delivers a maximum pump power of 30 W at 981 nm. The pump beam is focused to a spot size of about 80 µm using a 1:0.8 imaging system, which is slightly smaller to the designed laser mode (108 µm) in the Yb:KGW crystal. The dichroic mirror (DM) is coated for high transmission in the 800–1000 nm range and high reflection from 1000 nm to 1100 nm. The concave mirrors (C1, C2) have a radius of curvature of 300 mm and a reflectivity of more than 99.9% in the range 1000–1100 nm. The plane chirped mirror (CM) provides a group delay dispersion (GDD) of −2000 fs² per bounce to compensate the dispersion. To achieve short pulse durations while maintaining a good output power level, an output coupler (OC) with 3% transmittance is chosen for the cavity. The total length of cavity is 1.96 m, corresponding to the repetition rate of 76.55 MHz.

 

Fig. 1. Experimental setup of the KLM Yb:KGW laser. DM: dichroic mirror; HR: high reflection mirror; C1, C2: concave mirrors; CM: chirped mirrors, OC: output coupler.

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3. Results and discussion

The oscillator was first operated at the fundamental transverse mode. With KLM technique, 255-fs pulses were generated from the oscillator with an average power of 1.1 W and an optical-to-optical efficiency of 17%. Figure 2 shows the characteristics of the pulses at the fundamental transverse mode. Then the off-axis angle between the pump and laser beams in the gain medium was changed by adjusting the end mirrors, such that high-order modes could be excited in the laser. By increasing the pump power and tuning slightly the position of the gain medium, soft-aperture KLM was initiated by small mechanical perturbation by softly pushing the end mirror mounted on a translation stage. We first generated the HG_0,0-1.5*HG_2,0 and later the HG_0,1-1.5*HG_2,1 states with different off-axis angles. The average output powers were 515 mW and 263 mW in the mode-locked regime at a pump power of 5.2 W, corresponding to an optical-to-optical efficiency of 9.9% and 5%, respectively. Figure 3(a) and 3(b) show the measured time-frequency characteristic. Both pulse trains (Fig. 3(a)), when plotted on the 50 ns time scales, show intensity modulation at similar beat frequencies. Furthermore, the adjacent pulse-to-pulse time interval is about 13 ns, corresponding to the cavity length of 1.96 m. The radio frequency (RF) spectrum (Fig. 3(b)) presents the longitudinal mode frequency with a signal-to-noise ratio of more than 60 dB at 76.55 MHz and 76.44 MHz, matching the 13 ns time interval while reflecting a slight difference in the true optical path length between the two modes of operation. It is noteworthy that the beat peaks of HG_0,0-1.5*HG_2,0 are located at 27.8 MHz and 48.7 MHz, and that of HG_0,1-1.5*HG_2,1 at 27.9 MHz and 48.55 MHz. According to Ref. [42], since for each state, the sum of the two beat frequencies equals the longitudinal mode frequency, their presences can be attributed to the two transverse modes having a different frequency offset for their optical frequency combs, even though they share the same longitudinal mode frequency (see Fig. 3(f)). This frequency difference $\Delta \upsilon$ corresponds to the beat frequency as shown in Fig. 3(f) and is given by

 

Fig. 2. Mode-locked performance of the fundamental transverse mode. (a) Pulse trains; (b) RF spectrum; (c) Optical spectrum; (d) Autocorrelation (AC) trace.

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Fig. 3. Mode-locked performance of the HG_0,0-1.5*HG_2,0 and HG_0,1-1.5*HG_2,1 states. (a) Pulse trains; (b) RF spectra; (c) Optical spectra; (d) Autocorrelation (AC) traces; (e) Beam shapes; (f) Diagram indicating some of the possible beat frequencies between different transverse mode sets.

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Here $f = {c} / {{2L}}$ is the longitudinal-mode spacing, $L$ is the total cavity length, and c is the speed of light in vacuum. ${G_1} = A$ and ${G_2} = D$ with $\left[ {\begin{array}{cc} A&B\\ C&D \end{array}} \right]$ being the one-way transfer matrix between two flat end mirrors. Because the order difference $\Delta ({m - n} )$ between HG_0,0 and HG_2,0 or between HG_0,1 and HG_2,1 is identical, the beat frequencies of the two modes should be the same if they share the same optical path lengths. The distance from the end mirror HR to C2, between C2 and C1, and C1 to OC are approximated to be 590 mm, 315 mm and 1045 mm respectively, with some uncertainty of ± 10 mm arising from measurement errors in the exact beam incidence angle and position on the mirrors when tuning between the different states. We calculate the one-way transfer matrix between two flat end mirrors to be $\left[ {\begin{array}{cc} { - 0.4}&{ - 1.076}\\ { - 0.667}&{ - 0.707} \end{array}} \right]$, so the value of $\Delta \upsilon$ is 49.15 MHz, closed to the measured result of 48.7 MHz and 48.55 MHz.

The spectra (Fig. 3(c)) were centered at 1045.5 nm and had a full-width at half-maximum (FWHM) bandwidth of 6.3 nm and 5 nm. The intensity autocorrelation traces (Fig. 3(d)) indicate pulse durations of 184 fs and 247 fs respectively, assuming a sech²-shape fit. The time-bandwidth products of the generated pulses are calculated to be 0.32 and 0.34, indicating they are close to transform limited (0.315 assuming a sech² pulse shape), with a small amount of residual chirp. Since Yb:KGW supports pulses as short as 67 fs [26], there is room for further shortening of the pulses by fine-tuning the intracavity dispersion if desired.

We subsequently converted, using an AMC (Fig. 1), the femtosecond HG beams into the corresponding LG vortex beams. The AMC includes a spherical lens (f₁= 150 mm) and a matched pair of cylindrical lenses (f₂= 50 mm). The distance between the two cylindrical lenses is precisely adjusted to $\sqrt 2 {f_2}$ (70.7 mm) for the operation as a ${\pi } / {2}$ mode converter. The resulting LG beams exhibited multiple hollow structures as shown in Fig. 4(a) and Fig. 4(d). The profile deviated slightly from the ideal circular form. This could be optimized in the future by fine tuning the relative position of the spherical lens and the orientation of the cylindrical lenses. Nevertheless, as discussed below, the vortex property is still clearly present.

 

Fig. 4. (a) Beam profile, (b) measured interference pattern and (c) simulated interference pattern for the LG_0,0-1.5*LG_0,2 state. (d), (e) and (f) show the corresponding beam profile and interference patterns for the LG_0,1-1.5*LG_1,1 state.

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To verify the vortex property, we constructed a Mach-Zehnder (MZ) interferometer outside the cavity (Fig. 1). One arm of the delayed optical path allows to adjust the interval between the two femtosecond pulse trains in the time domain. The other arm contains a lens (f₃= 50 mm) that magnifies the vortex beam, which acts as an approximated spherical wave. The interference pattern between the vortex beam and the spherical wave is then characterized. The coherence length L, calculated according to formula $L = {{{\lambda ^2}}} / {{\Delta \lambda }}$, is 210 µm. Thus, the delay in the MZ needs to be adjusted to within an accuracy of 10s of µm in order to observe an interferogram of the femtosecond LG mode (Fig. 4(b) and Fig. 4(e)). The center of the spherical wave depends on where the laser beam incident on the lens. Therefore, the center of interference fringes is located at the bottom left. The other minor circular fringes are caused by dust particles covering the beam attenuators in front of the spot analyzer. In Fig. 4(b), we clearly noticed the same clockwise-oriented fork-shaped stripes with one fork on both sides of the transverse mode, indicating that the double vortices have the same spiral phase. Similarly, we observe five forked-stripes at different transverse locations as shown in Fig. 4(e), all of which consist of one fork, indicating that this LG mode has five separate phase singularities. According to the theoretical analysis for AMC [20], a HG_m,n mode will be converted into the corresponding LG_l,p, where $l = \min ({m,n} )$ and $p = |{m - n} |$. Thus, the LG modes in Fig. 4(b) and Fig. 4(e) are respectively LG_0,0-1.5*LG_0,2 and LG_0,1-1.5*LG_1,1. Based on the above assumptions, we simulated the coherent superposition of these LG modes with a spherical wave by summing their electric fields. The resulting simulated interference pattern of the LG modes are shown in Fig. 4(c) and Fig. 4(f), where both the fringe density and the fork-stripes structure agree well with the experimental results. Therefore, we believe femtosecond KLM LG_0,0-1.5*LG_0,2 and LG_0,1-1.5*LG_1,1 modes were generated from the well-regulated optical resonator. Figure 5 shows the beam profiles of the vortex beam with multiple singularities at different distances behind the cylindrical lens mode converter, indicating a good propagation stability.

 

Fig. 5. Beam propagation of multiple optical vortices behind the mode converter. (a) Beam propagation of LG_0,0-1.5*LG_0,2 state. (b) Beam propagation of LG_0,1-1.5*LG_1,1 state.

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The current experimental results have shown a reduced efficiency at higher-order transverse modes since higher-order modes have larger beam size and introduce more loss to the oscillator. In order to obtain more singularities in the future, higher pump power would be required to increase the gain of the higher-order modes, which can ensure the necessary power density for the realization of KLM. The pulse duration can be shortened further through finer dispersion compensation or using a gain medium with broader emission spectrum bandwidth. Besides, specially designed hard aperture and spot-defect mirror will be added into the oscillator to assist the KLM, higher order modes with more singularities and better beam quality will be expected for the next step.

4. Conclusion

In conclusion, we have generated multiple OVs with femtosecond duration consisting of two or five separate phase singularities using a Kerr-lens mode-locked Yb:KGW laser oscillator. To the best of our knowledge, this is the first time femtosecond optical vortices with multiple singularities have been directly generated from a laser oscillator. These pulse beams have average powers of several hundred milliwatts and pulse durations of less than 300 fs. Due to the different offset of the optical frequency combs for the different transverse modes, both OVs exhibit a beat note in their radio frequency spectrum. This beating could be eliminated in future work by controlling the one-way transfer matrix between the two end mirrors to achieve pure femtosecond multiple OVs via KLM. We believe that the present research represents an important innovation for generating femtosecond multiple optical vortices, paving the way for further studies of optical vortex crystals and their applications.