⇦Chinese Physics Letters, 2019, Vol. 36, No. 12, Article code 124203 A V-Folded Digital Laser for On-Demand Vortex Beams by Astigmatic Transformation of Hermite–Gaussian Modes * Sen-Sen Liu (刘森森), Xu-Dong Chen (陈旭东)**, Ji-Xiong Pu (蒲继雄), Zhi-Li Lin (林志立), Zi-Yang Chen (陈子阳) Affiliations Fujian Key Laboratory of Light Propagation and Transformation, College of Information Science and Engineering, Huaqiao University, Xiamen 361021 Received 5 September 2019, online 25 November 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 61605049 and 61575070, and the Natural Science Foundation of Fujian Province of China under Grant No 2018J01003, the Fundamental Research Funds for the Central Universities under Grant No ZQN-707, and the Subsidized Project for Postgraduates' Innovative Fund in Scientific Research of Huaqiao University.
**Corresponding author. Email: chenxd@hqu.edu.cn Citation Text: Liu S S, Chen X D, Pu J X, Lin Z L and Chen Z Y et al 2019 Chin. Phys. Lett. 36 124203    Abstract A V-folded digital laser using a spatial light modulator (SLM) for intra-cavity loss shaping is exploited to generate Hermite–Gaussian modes with on-demand mode order. With a $\pi$/2 astigmatic mode converter, vortex beams carrying on-demand orbital angular momentum (OAM) with a tunable range from $-11\hbar$ to $12\hbar$ are obtained. The mode order of the HG mode, hence the OAM of the vortex beam, is digitally switched by changing the phase pattern imposed on the SLM without requiring any mechanic alignment of the cavity. This work has great potential applications in various OAM-tunable vortex beams. DOI:10.1088/0256-307X/36/12/124203 PACS:42.50.Tx, 42.60.By, 42.60.Jf © 2019 Chinese Physics Society Article Text Due to their unique phase and amplitude structures, Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) beams have gained considerable attention and found a lot of applications in frontier technologies,[1] such as optical trapping and manipulation of particles,[2–4] quantum information and optical telecommunication,[5–7] and so on. Several extra-cavity generation methods for these modes have been reported by utilizing diffraction optical elements,[8] spatial light modulators (SLMs)[9] or digital micromirror devices (DMDs).[10,11] However, generally speaking, the vortex beam quality generated by most subsequent mode conversion methods is limited by poor mode purity or low conversion efficiency.[12] Meanwhile, pure HG or LG modes can be directly generated in a laser cavity. Annular pumping method is a well-known method to directly generate vortex beams in laser resonators.[13,14] However, it is difficult to obtain high order LG modes from a laser cavity with an annular pumping method. Fortunately, an astigmatic mode converter can efficiently convert a diagonal HG mode into an LG mode in a simple way.[15–17] In recent years, off-axis pumping has been widely employed to produce high-order HG modes in end-pumped lasers, which enables the generation of vortex beams with large OAM via mode transformation.[18–23] The mode order is generally limited by the laser crystal size. However, to continuously tune the HG mode order, not only the off-axis displacement of pump beam should be increased but also the position of output coupler needs to be precisely adjusted. Consequently, this scheme is less practical for generation of a vortex beam with on-demand OAM. Gain-shaping based methods have also been proposed for selective HG mode excitation, by extra-cavity shaping the pump beam of an end-pumped laser.[24–28] With these methods, no adjustment of the cavity itself was required for mode switching between different HG modes. However, the excitable HG$_{m,n}$ modes were limited to relatively low mode numbers. Recently, the application of a DMD in pump shaping has exhibited a computer-controlled selective excitation of high-order HG modes.[29] A digital laser, which utilizes an SLM as the back cavity mirror, has shown its simplicity in intra-cavity mode shaping.[30–33] By employing an SLM as an amplitude modulator for intracavity loss shaping, low order HG modes have been generated.[31,33] The output mode of digital laser is customized and switched by controlling the SLM. During the on-demand mode selection in the digital laser, neither new specially designed optical elements nor additional alignment of the laser cavity is required. However, a digital laser is limited to the damage threshold of SLM, which is generally several to dozens of watts per square centimeter. To avoid the damage of the SLM, the transverse mode size on the SLM should be as large as possible, Meanwhile, to obtain high-order modes the limitation of the crystal aperture should also be overcome; that is, the transverse mode size on the crystal should be as small as possible. To balance this contradiction, a V-folded cavity can be employed. Compared to linear geometry, V-folded laser cavity configuration allows easy control of the transverse mode sizes at different intracavity positions.[34,35] In this Letter, we demonstrated a V-folded digital laser for on-demand vortex beam generation. A computer controlled SLM serves as a folding mirror, while two spherical mirrors enclose the resonant cavity. This specially designed cavity makes a small transverse mode size in laser crystal and a large one on SLM, to avoid the limitation of the cavity aperture size on the mode order and the damage of SLM. Given the flexibility of digital laser in intra-cavity mode selection, the proposed digital laser successfully generates HG modes with on-demand order numbers. The mode order of HG modes was simply tuned with digitally controllable phase pattern imposed on the SLM, without any mechanic adjustment of the cavity nor extra-cavity shaping of the pump beam. With a $\pi$/2 astigmatic mode converter composed with two cylindrical lenses, stable vortex beams with an OAM-tunable range from $-11\hbar$ to $12\hbar$ have been obtained.
cpl-36-12-124203-fig1.png
Fig. 1. (a) Schematic of the high-order HG mode digital laser setup. (b) Experimental setup of $\pi$/2 mode converter and schematic diagram of interference setup. BS1, BS2: beam splitters, L1, L2: lenses, CL1, CL2: cylindrical lenses, M1, M2: mirrors.
The setup for the SLM-controlled high-order HG mode excitation is schematically presented in Fig. 1(a). A V-folded three-mirror cavity was employed, with a reflective SLM served as the folding mirror and two spherical mirrors on both ends. The angle between the two arms was about 4.2$^{\circ}$. These two spherical mirrors were a concave end mirror (M: HR at 1064 nm, the radius of curvature is 1000 mm) and a concave output coupler (OC: transmittance is 5% at 1064 nm, the radius of curvature is 300 mm). The gain medium used was a Nd:YAG crystal rod with a diameter of 3 mm and a length of 67 mm. A reflective SLM (Pluto-NIR-011, Holoeye) with $1920\times 1080$ pixels, a pixel pitch of 8 µm and a 60 Hz input image frame rate was used to implement intra-cavity loss shaping. The LC director of the SLM was arranged to be horizontal in our setup. A polarization beam splitter (PBS) was placed before the SLM to force the laser to oscillate in the desired polarization for the SLM (horizontal in our setup). The distance between the SLM and the end mirror was 382 mm, and that between the SLM and the OC was 718 mm. The distance between the center of gain medium and the OC was 123 mm. It should be noted that, as outlined in our previous report,[33] in a digital laser with this kind of SLM the amplitude modulation effects would play a dominant effect rather than phase modulation effects. This means that by imposing different gray levels on SLM, one can obtain different SLM reflectivity and consequently different intra-cavity loss. As we know, there are $m$ and $n$ nodal lines, respectively, in the horizontal and vertical directions for the HG$_{m,n}$ mode. Therefore, a specific phase pattern serving as localized regions of loss along the nodal lines of a particular HG$_{m,n}$ mode may be employed to select the desired mode.
cpl-36-12-124203-fig2.png
Fig. 2. Examples of (a) phase patterns imposed on the intra-cavity SLM and (b) the resultant outputs, identified as HG$_{1, 0}$, HG$_{6, 0}$ and HG$_{12, 0}$, respectively.
For simplicity, some examples of phase patterns imposed on the intra-cavity SLM are presented on the left of Fig. 2. Each phase pattern is composed of $m$ vertical strips with a gray level corresponding to lower reflectivity, in a uniform background with a gray level corresponding to higher reflectivity. The separation distances between the vertical strips are calculated according to those between the nodal lines of HG modes. The background grayscale of each phase pattern may be carefully adjusted to vary the SLM reflectivity and change the output power under the same pump power. In Fig. 2(a), the grey level of each vertical strip is 255, corresponding to a low reflectivity of SLM, while the background grayscale of each phase pattern is respectively 90, 110 and 110, corresponding to a higher reflectivity of SLM.[33] During the experiment, different phase patterns were switched on the computer that controlled the SLM. Consequently, the resultant laser output were obtained without any mechanical alignment of the cavity. However, due to the large differences among the thresholds for HG modes in a large range of mode order, an increase of pump power is necessary for high order modes. The corresponding output beam profiles are shown on the right of Fig. 2, identified as HG$_{1, 0}$, HG$_{6, 0} $ and HG$_{12, 0} $ modes, respectively. As shown in Fig. 2, the highest mode order obtained in this experiment is up to 12, which is mainly limited by the crystal aperture size in our experimental setup. To verify the purities of these HG beams, the beam quality ($M^{2}$) of each mode was measured by a commercial beam quality analyzer with a CCD camera (Spiricon, M2-200s). Figure 3 shows the near-field intensity distributions of HG$_{m, 0}$ modes ($m=0, 1, 2,\ldots, 12$), as well as the beam qualities, respectively. The experimental results show good agreement with the theoretical values of $\left( {M_{x}^{2},M_{y}^{2} } \right)=\left( {2m+1,2n+1} \right)$ for low order modes. For high order modes, the purities of HG modes slightly deteriorated, due to the increased thermal and aperture effects. The generation of HG modes with good beam qualities leads to the generation of the vortex beam with good purities.
cpl-36-12-124203-fig3.png
Fig. 3. Measured near-field intensity distribution of HG$_{m, 0}$ ($m=0, 1, 2,\ldots, 12$) modes from the digital laser with measured and theoretical $M^{2}$ values.
cpl-36-12-124203-fig4.png
Fig. 4. Simulations (first row) and experimental results (second row) of interference fringes of the transformed vortex beams with different OAMs.
A $\pi$/2 astigmatic mode converter was used to convert the HG$_{\rm m, n}$ modes into the corresponding LG$_{\rm p, l}$ vortex beam outside the cavity, with $p=\min \left( {m,n} \right)$ and $l=m-n$.[16] The experimental setup of the $\pi$/2 mode converter and the schematic diagram of interference setup are shown in Fig. 1(b). Lens L1 is a mode-matching lens. The $\pi$/2 mode converter includes two $45^{\circ}$ inclined convex-plane cylindrical lenses CL1 and CL2 with an identical focal length $f$ of 200 mm, with a separation distance of 282.8 mm ($\sqrt 2 f$). L2 is a collimating lens. After passing through the beam splitter (BS1) and the mode converter, an HG beam was transformed into a doughnut-shaped beam. To measure the topological charge (TC) $l$ of the vortex beams, an interference experiment with the transformed donut beam and a flat-wavefront reference beam was performed. As shown in Fig. 1(b), after reflection from BS1, one petal of the HG beam was filtered out with an iris diaphragm and used as a reference plane wave, which then interfered with the LG vortex beam converted. The experimental results are presented in Fig. 4. The deviation of the interference patterns from the simulation results may be caused by manufacturing limitations of the cylindrical lenses and by incorrect mode matching, which resulted in a decrease in the mode purity of the transformed vortex modes.[36] An integrated $\pi/2$ converter may be employed to improve the results.[17] However, the characteristic fork patterns clearly manifest the corresponding TCs of the principal modes. Detailed measurement for the mode purity of the transformed vortex modes can be achieved by measuring the OAM spectrum.[37]
cpl-36-12-124203-fig5.png
Fig. 5. (a) Phase pattern and (b) the resultant intensity distribution of HG$_{0, 11}$ mode. (c) Simulation and (d) experimental results of interference fringes of the transformed LG$_{0, -11}$ mode.
In addition, the excitation of HG$_{0, n }$ modes with a range of mode number $n$ from 0 to 11 has also been generated from this digital laser. The obtained HG$_{0, n}$ modes are not only limited by the crystal size but also the vertical size of the SLM, which is (8.64 mm) much smaller than the horizontal size (15.36 mm). As an example, Figs. 5(a) and 5(b) show the phase pattern and the corresponding laser output for HG$_{0, 11}$ mode, respectively. Accordingly, the experimentally measured interference fringes of transformed LG$_{0, -11}$ mode with plane reference wave is presented in Fig. 5(d), in agreement well with the simulation result as shown in Fig. 5(c). These results suggest that LG$_{0, l}$ modes with a TC $l$ range from $-11$ to 12 are obtainable by astigmatic transformation of HG modes generated from the V-folded digital laser, without any alignment of the cavity or the mode converter system outside of cavity. Here, only some examples of them are presented for simplicity. Further measurement of the spectrum of the OAM will be carried out in our future research.[37,38] However, thanks to the generation of HG modes with high beam quality, the purities of transformed LG vortex beams are only limited by the precision of the astigmatic mode converter system. In conclusion, a V-folded digital laser has been employed to generate vortex beams with on-demand OAM. Hermite–Gaussian modes have been selectively generated from the digital laser by intra-cavity loss shaping. Accordingly, the vortex LG modes have been obtained by $\pi$/2 mode converter. By proper design of the V-folded cavity, the limitation of damage threshold of the SLM and that of the crystal size are avoided to the most extent, consequently a large OAM-tunable range from $-11\hbar$ to $12\hbar$ has been obtained. The mode order of the HG mode, hence the OAM of vortex beam is simply controlled by the phase pattern imposed on the SLM, without any mechanic alignment of the cavity. The topological charges are verified by interference experiments, with the results in good agreement with the simulations. The biggest limitation of this approach is on the output power, which is imposed by the damage threshold of the SLM. Nowadays, the damage threshold of commercial SLMs with water-cooling, such as Hamamatsu X13138-03, is up to 255 W/cm$^{2}$. Further, power amplification of the vortex beams may be exploited for high power application.[39] Moreover, the damage threshold of SLM has tended to increase in recent years. In a word, given its characteristics of on-demand and flexible mode selection, this work has great potential to be further applied in various OAM-tunable vortex beams and motivate novel research. 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