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Generating controllable Laguerre-Gaussian laser modes through intracavity spin-orbital angular momentum conversion of light

The rapid developments in orbital-angular-momentum-carrying Laguerre-Gaussian (LG0 l) modes in recent years have facilitated progresses in optical communication, micromanipulation and quantum information. However, it is still challenging to efficiently generate bright, pure and selectable LG0 l laser modes in compact devices. Here, we demonstrate a low-threshold solid-state laser that can directly output selected high-purity LG0 l modes with high efficiency and controllability. Spin-orbital angular momentum conversion of light is used to reversibly convert the transverse modes inside cavity and determine the output mode index. The generated LG0 1 and LG0 2 laser modes have purities of ~97% and ~93% and slope efficiencies of ~11% and ~5.1%, respectively. Moreover, our cavity design can be easily extended to produce higher-order Laguerre-Gaussian modes and cylindrical vector beams. Such compact laser configuration features flexible control, low threshold, and robustness, making it a practical tool for applications in super-resolution imaging, high-precision interferometer and quantum correlations.


Introduction
Laguerre-Gaussian (LG p l ) modes, characterized by the azimuthal index l (any integers) and the radial index p (zero or positive integers), are the eigen solutions of the paraxial wave equation in cylindrical coordinates 1  Doppler effect, and so on [4][5][6][7][8][9][10][11][12][13][14][15] . The evolving practical applications inevitably require high-quality LG p l laser modes. For example, a pure LG 0 l mode has a more uniform donut-intensity distribution along the propagation direction, which can increase the depth resolution in super-resolution imaging 16 . In the field of precision measurement, it has been shown that higher-order LG p l modes could reduce thermal noise in the LIGO system for gravitational-wave detection because of their more homogeneous power distributions 17,18 . The high-precision optical interferometry in the LIGO system requires a high-purity LG p l mode. In quantum information science, high-purity LG p l modes could increase hybrid azimuthal-radial quantum correlations [19][20][21] . However, it is still a challenge to efficiently generate high-purity LG p l modes in a simple optical setting. In this Letter, we experimentally demonstrate an intracavity spin-orbital angular momentum conversion method to produce selected high-purity LG 0 l laser modes, which can be feasibly extended to generate a general LG p l laser mode.
The common way to generate a LG 0 l beam is to add the azimuthal phase term of il e  to a Gaussian mode using a fork-grating, a q-plate, a spiral phase plate, and so on 22 . However, directly adding a spiral phase to the Gaussian mode will generate a hypergeometric-Gaussian (HyGG) mode, which has a definite l index but an expansion of the p index 23,24 , which decreases the mode purity and conversion efficiency of the desired LG 0 l beam. Furthermore, the power weightings of these undesired LG p l (with p > 0) modes have positive correlations to the modulus of the l index. These undesired radial modes hidden in the LG 0 l beam will decrease the imaging resolution in the superresolution microscope, reduce the accuracy of the high precision interferometer 25 , contaminate the quantum information processing 19,20 , and induce uncertainties in the nonlinear interactions 26 .
Researchers have turned to intracavity modulation for generating high-purity LG 0 l modes, since an optical cavity can perform mode selection and directly output laser modes 27,28 .
Unfortunately, because LG 0 l modes with opposite handedness (for example, l = 1 and l = -1 modes) have the same loss and are degenerate in an active optical cavity, their coherent superpositions usually form petal-intensity patterns 29,30 . The superposition of two petal-intensity patterns may form a donut-intensity profile but not a pure LG 0 l mode 31 . Special components, like nanoscale stripes and oblique etalons, are proposed to discriminate the degenerate LG 0 l modes inside active cavities. However, the additional components with circular asymmetries spoil the mode purities of 3 the generated LG 0 l modes, which has impeded their further development 32,33 . Recently, q-plates, a kind of planar phase-modulation devices, is utilized inside the laser cavity for generating highpurity LG 0 l modes 34 . However, the extremely high pump power needed for operation severely limits its practical applications 34 . To generate useful LG 0 l laser modes with high mode purities, low lasing thresholds and controllable mode indices, the laser cavity needs to be carefully designed with proper control and selection of intracavity components.
In our experiment, we put a vortex half-wave plate (VWP, Thorlabs, Inc.), a Faraday rotator (FR) and a quarter-wave plate (QWP) inside a Nd: YVO4 laser cavity, which can achieve cavity mode reversibility through intracavity spin-orbital angular momentum conversion. The process is self-producing after each round trip. The QWP generates a circularly-polarized state carrying a spin angular momentum (SAM) of ±ћ per photon depending on its handedness 35 . The interaction between a photon and an optically anisotropic medium will change the value of the photon's SAM and induce a geometric phase shift 36 . The VWP, featuring the artificial helical anisotropic parameter space, can add a spiral geometric phase to an incident mode by flipping the handedness of its circular polarization, i.e., spin-orbital angular momentum conversion. Therefore, the VWP couples the SAM or polarization of the intracavity field into the OAM, and flexibly controls the TC 37 . The modulus and sign of the TC can be changed by using different VWPs and rotating the QWP, respectively. By adding a pinhole to suppress the undesired radial modes, the laser outputs a selected high-purity LG 0 l laser mode for example, a LG 0 1 (LG 0 2 ) mode with a mode purity of ~97% (~93%), at the wavelength of 1064 nm in our experiment. The specially-designed compact cavity requires only a few intracavity components, which benefits the system stability, loss control and practical operation.
The lasing thresholds for lower order LG 0 l modes are comparable to that of the Gaussian mode and the slope efficiencies of the LG 0 1 and LG 0 2 lasing modes are ~11% and ~5.1%, respectively. The beam quality factor and mode stability at high pump power also show the excellent laser performance. The flexibility of the cavity design is also reflected in its ability to generate laser outputs of vector beams with cylindrical symmetry in polarization by slightly modifying the intracavity elements 38 .

Principle of intracavity spin-orbital angular momentum conversion.
In a stable laser cavity, a light beam should reproduce itself after each round trip when the cavity resonates. Since each point on the VWP is a half-wave retarder, a linearly-polarized light beam, propagating forward and then reflecting backward through the same VWP, is equivalent to experience a full-wave plate, in which the initial polarization and wavefront are recovered (Fig. 1a).
To couple the spin and orbital angular momenta, a QWP is needed to turn the linearly-polarized 4 state to the circularly-polarized state (see Supplementary Note 1 for the working principle of the VWP). The combination of the FR and QWP makes the light beam reversible when its initial polarization direction is parallel to the optical axis of the QWP (Fig. 1b). Therefore, combining the FR, QWP and VWP, we achieve a complete reversible cycle inside a laser cavity for the output of a selected OAM-carrying mode (Fig. 1c). Here, the FR enables mode reversible propagation. The system is greatly simplified in comparison to the previous intracavity geometric-phase-control configuration 34 . Figure 1d We choose to work under the condition of LA < f and LB < f. Considering the refractive indices of the gain medium, FR and QWP in the first part of the cavity, the actual LA can be a bit larger than f 5 without breaking the cavity stability condition. To optimize mode matching, LA is set to approach f so that the cavity mode has a diameter large enough at the position of the lens and a diameter comparable to that of the pump beam at the position of the gain medium. The pump beam is focused on the crystal with a 200 um-in-diameter Gaussian spot. The VWP is placed in Part 2 of the cavity, adjacent to the lens to take advantage of the large mode size, because the large mode size can effectively overcome the imperfect singularity of the VWP. A polarization beam splitter (PBS) is inserted between the gain medium and the FR to eliminate the unwanted vertically-polarized light caused by the imperfect polarization modulation from the FR, QWP and VWP. A 1 mm-in-diameter pinhole is placed at the front of the output coupler to filter out the higher-order transverse modes so that the cavity can output the desired LG 0 l beam. Therefore, in an ideal case with a highly-efficient intracavity mode conversion, a LG 0 0 mode is expected to oscillate in Part 1 of the cavity, while a LG 0 l mode oscillates in the regime of Part 2. Under such experimental configuration, higher-p-order LG modes are greatly suppressed and the laser will output a selected high-purity circularly-polarized LG 0 l mode. Figure 2b shows an output LG 0 1 laser mode generated by the VWP of m = 1. The LG 0 -1 mode can be generated (Fig. 2c) by rotating the QWP with an angle of 90°. Replacing the VWP with m = 2, we have generated the LG 0 2 (or LG 0 -2 ) mode by orienting the fast axis of the QWP horizontally (or vertically) as shown in Fig. 2d (or Fig. 2e). All the intensity profiles present high-quality donut shape. Simulations based on Fox-Li method have been carried out to find the steady cavity mode and the corresponding output mode for each cavity configuration 40 , which validate our experimental setup (see Supplementary Note 6 for the numerical simulation). Since the experimental setup is similar for generating RCP LG 0 |l| and LCP LG 0 -|l| modes, we will only present the laser properties of RCP LG 0 1 and LG 0 2 laser modes in the following.

Detailed parameters of the experimental setup.
As shown in Fig. 2a, an a-cut 0.5 at% Nd-doped YVO4 crystal with its dimensions of 3 mm × 3 mm × 8 mm serves as the gain medium. One of its ends, as the input coupler, is coated with the reflectivity >99.9% at 1064 nm and transmittance >97.7% at 808 nm; the other end has a high transmittance of >99.8% at 1064 nm. The crystal is mounted on a water-cooled copper holder maintained at the room temperature and is oriented to generate horizontally-polarized light. The output coupler is a plane mirror with a transmittance of 10% at 1064 nm. The lens with a focal length f is inserted into the cavity to keep the cavity satisfying the stability condition. The pump source is a fiber-coupled laser diode outputting a continuous-wave at 808 nm wavelength, which is

Results and Discussion
Lasing threshold and slope efficiency. The result shows the main power weighting of LG 0 2 mode to be 92.7% (Fig. 4b). The active cavity plays an important role in the transverse-mode selection, which increases the mode purities appreciably, as compared to the achieved mode purities through extracavity modulation using VWPs (typically ~80% for LG 0 1 mode and ~60% for LG 0 2 mode) 24 . 7

Beam quality factors.
We further measure the beam quality factor to characterize the output LG 0 l laser mode. The output power is set to be ~40 mW. We record output intensity patterns of the LG 0 1 and LG 0 2 modes (Figs. 5a and 5b) at different propagation distances separately. The corresponding distances are 20 cm, 30 cm, 40 cm, 50 cm and 60 cm from left to right in Fig. 5, respectively. The intensity patterns preserve the well-defined donut shape with negligible higher-order radial modes during the propagation. The LG 0 1 and LG 0 2 modes, respectively. Our experimental results show the good beam quality of the generated LG 0 l modes.

Laser output of cylindrical vector beams.
Laser beams with different axial symmetries in polarization, the so-called cylindrical vector beams, can also be generated by different combinations of VWPs and linearly-polarized directions of incident beams 38,42 . As shown in Fig. 1(a), a single VWP can achieve a reversible propagation cycle inside a laser cavity, which can be used to directly produce laser outputs of cylindrical vector beams.
The new setup satisfies the self-reproducing and stability conditions for a cavity (see Supplementary   Fig. 4). The light beam incident on the VWP is fixed into a horizontal polarization in our experiment, so the output polarization distribution can be directly controlled by rotating the VWP (See Supplementary Equation (3)). A linear polarizer is used to analyze the polarization distribution of the output beam. Figures 6a and 6b show the output cylindrical vector beams with a well-defined donut-intensity distribution by orienting the VWP of m = 1 with different initial angles of φ0 = 45° and φ0 = 0°, respectively. By orienting the linear polarizer at the angles of 0, 45°, 90°, and 135°, the measured intensity distributions are shown in Figs. 6d and 6e, which indicate that the output beams shown in Figs. 6a and 6b are azimuthal and radial vector beams, respectively 43 . Using the VWP of m = 2, we have generated a higher-order cylindrical vector beam with negligible radial modes (Fig.   6c). The measured intensity distributions after the linear polarizer, as shown in Fig. 6f, reveal the cylindrical polarization distribution of Fig. 6c. The cylindrical vector beams with a well-define donut-intensity profile and controllable polarization distribution can be applied in optical communication, material processing, superfocusing, etc [43][44][45][46] . 8 In summary, we have demonstrated a Nd: YVO4 laser generating high-purity LG 0 l laser modes.

Conclusions
The cavity design satisfies the light reversibility and stability conditions for a cavity. The degeneracy of opposite-handedness LG modes is lifted through coupling SAM into OAM for the field mode in the laser cavity. The output power dependences on the pump power show low lasing thresholds (0.7 W for LG 0 1 and 1.8 W for LG 0 2 ) and high slope efficiencies (11.06% for LG 0 1 and 5.11% for LG 0 2 ) for the generated LG 0 l modes. The mode purities are up to ~97% and ~93% for the output LG 0 1 and LG 0 2 modes, respectively. Changing the fast-axis orientation of the QWP, the generated LCP LG 0

Supplementary Information
Supplementary Note 1 | Working principle of the VWP.
The VWP can be seen as a spatially variant half-wave plate, whose fast axis rotates continuously over the area of the optic around a singularity point. Its transmission efficiency is up to 96%. The orientation of its fast axis can be expressed as: where φ is the azimuthal angle, φ0 is the orientation of the fast axis at φ = 0 and m is a positive integer determined by the VWP. Its Jones Matrix can be written as 1 : ( If we apply it to a horizontally-polarized light beam, the output result can be expressed as: (3) l | 2 yields the power weighting of the corresponding LG p l mode. Supplementary Equation (6) shows that the HyGG mode is the superposition of LG p l modes with different p indices. In our laser cavity, the undesired higher-order LG p l modes (with p > 0) are suppressed by the transverse mode selection of the laser cavity.
As shown in Figs. 1c and 1d has an angle of 135° with the initial polarization direction. In this process, the LCP OAM mode turns into a linearly-polarized OAM mode with a TC of -l0. Twisted by the FR again, the polarization state rotates 45° clockwise and recovers to the initial horizontal direction. The reflection of the input coupler inverses the TC to its initial value of l0, while keeps horizontal polarization unchanged.
Such cycle repeats in the cavity. Similarly, orienting the fast axis of the QWP along the vertical axis, a LCP OAM mode with TC of l0-m will be generated.

Supplementary Note 3 | Highly stable intensity profile at different pump powers.
To demonstrate the excellent mode stability of the output LG 0 l beams, we record the intensity patterns at a propagation distance of 50 cm at different pump powers. Supplementary Figure 1a shows the intensity patterns of the LG 0 1 laser mode at pump powers of 0.68 W, 1.10 W, 1.33 W, and 1.50 W from left to right, respectively. The stable intensity patterns illustrate that increasing pump power has little influence on the mode purity. The output LG 0 2 mode is robust as well, which can be seen from Supplementary Fig. 1b  The experimental setup is shown in Supplementary Fig. 4. An a-cut 0.5 at% Nd-doped YVO4 crystal with its dimensions of 3 mm × 3 mm × 8 mm serves as the gain medium. One of its ends, as the input coupler, is coated with reflectivity of >99.9% at 1064 nm and transmittance of >97.7% at 808 nm; the other end has a high transmittance of >99.8% at 1064 nm. The crystal is mounted on a water-cooled copper holder maintained at the room temperature and oriented to generate horizontally-polarized light. The output coupler is a plane mirror with a transmittance of 10% at An end-pumped solid-state laser comprises of a Nd:YVO4 crystal, a lens and an output coupler working at the wavelength of 1064 nm. The Nd:YVO4 crystal is pumped by an 808 nm-wavelength fiber-coupled diode laser. Following the Nd: YVO4 crystal, a polarization beam splitter (PBS) and a vortex half-wave plate (VWP) are placed in the cavity successively. All of them are antireflection coated and oriented correctly, so that the cavity satisfies light reversibility condition.

Supplementary Note 6 | Numerical simulations.
Based on the Fox-Li method, an optical cavity can be expanded into a transmission line by replacing cavity mirrors with equivalent lenses 8 . For simplifying the calculation process, we only calculate the scalar cavity mode in a passive optical cavity, in which the polarization transformation, gain and loss are ignored. A pinhole is needed to filter out undesired higher-order radial modes. We set it on the output coupler because the distance between them is very small in the experimental setup. The distance between the lens and the VWP is neglected as well. Besides, the simplified model of scalar field allows us to replace the VWP with two spiral phase plates (SPPs) of opposite TCs. Therefore, one round-trip path of our laser cavity is equivalent to a complete cycle shown in Supplementary   Fig. 5. A scalar light field starting from the front mirror travels a distance of LA. It passes through the lens and the SPP with TC of l, and then reaches the pinhole after propagating a distance of LB.
A fraction of light is blocked by the pinhole while the majority propagates a distance of LB. After passing through the other SPP with a TC of -l and another lens, the spiral phase front is cancelled out. The light beam travels a distance LA to the front mirror and finishes one round trip. The light beam repeats the cycle until a stable cavity mode is formed. The iterative procedure described above is calculated step by step by Matlab programming. Supplementary Figure 6a shows the simulated intensity pattern of the output LG 0 1 beam propagating a distance of 30 cm, which is nearly the same as the experimental result shown in Supplementary Fig. 6b. Furthermore, the intensity profiles of 21 the stable intracavity mode at different locations have been also calculated. Supplementary Figure   7 presents intensity profiles at the distances of 200 mm, 150 mm, 100 mm, 50 mm, and 0 mm away from the front mirror, respectively, which show that the mode in LA evolves from a donut-intensity pattern on the lens into a Gaussian pattern on the front mirror.