Generation of high-efficiency, high-purity, and broadband Laguerre-Gaussian modes from a Janus optical parametric oscillator

Abstract. Laguerre-Gaussian (LG) modes, carrying the orbital angular momentum of light, are critical for important applications, such as high-capacity optical communications, superresolution imaging, and multidimensional quantum entanglement. Advanced developments in these applications demand reliable and tunable LG mode laser sources, which, however, do not yet exist. Here, we experimentally demonstrate highly efficient, highly pure, broadly tunable, and topological-charge-controllable LG modes from a Janus optical parametric oscillator (OPO). The Janus OPO featuring a two-faced cavity mode is designed to guarantee an efficient evolution from a Gaussian-shaped fundamental pump mode to a desired LG parametric mode. The output LG mode has a tunable wavelength between 1.5 and 1.6  μm with a conversion efficiency >15  %  , a controllable topological charge up to 4, and a mode purity as high as 97%, which provides a high-performance solid-state light source for high-end demands in multidimensional multiplexing/demultiplexing, control of spin-orbital coupling between light and atoms, and so on.


Introduction
Laguerre-Gaussian (LG) modes with unique spiral wavefronts are the paraxial solutions of the scalar Helmholtz equation in cylindrical coordinates, which can be distinguished by an azimuthal index l and a radial index p, i.e., LGðl; pÞ. In 1992, Allen et al. demonstrated that an LG mode carries an orbital angular momentum (OAM) of lℏ per photon, 1 where l is called the topological charge (TC). Their pioneering work has significantly boosted the applications of LG modes from optical trapping and optical tweezer to optical communications, superresolution imaging, precision measurement, quantum information processing, and so on. [2][3][4][5][6][7][8][9] In turn, these high-end demands have triggered the developments of LG-mode laser sources in recent years. [10][11][12][13][14][15][16][17] Almost all the applications benefit from the high purity of an LG laser source, such as improved signal-to-noise ratio in rotation measurement, enhanced resolution in fluorescence imaging, and optimized coupling with an OAM photonic chip. [18][19][20] High-power laser output of LG mode could provide an effective way to decrease thermal noises in gravitational-wave detection. 21 In particular, LG laser sources are expected to be wavelength-tunable for wavelength division multiplexing in OAM-based high-capacity optical communication, investigation of spin-orbital coupling with various atoms in quantum storage and isolation, and excitations of versatile fluorescence in superresolution imaging. 19,[22][23][24][25] However, these advanced applications are severely hampered by the limited wavelength bandwidth and mode purity in previous LG mode lasers. A reliable and broadband-tunable LG mode laser source does not yet exist.
The optical parametric oscillator (OPO) has been recognized as one of the most popular tunable sources. [26][27][28][29] A pump wave with frequency of ω p generates two parametric waves, 30 i.e., signal and idler waves at the frequencies of ω s and ω i , respectively, through the second-order nonlinear downconversion process. It satisfies the energy conservation of ω p ¼ ω i þ ω s . By controlling the phase-matching condition for momentum conservation, one can obtain wavelength-tunable output of the generated parametric waves. 31 An OPO system is capable of outputting broad wavelengths covering ultraviolet (UV), visible, and infrared bands, which makes it an excellent candidate for broadband output of high-quality LG modes. There are two reported configurations. One is to build a traditional OPO outputting a Gaussian mode, and then convert it to an LG mode using a spiral phase plate, a fork grating, a Q-plate, a vector vortex waveplate (VVW), or a spatial light modulator [ Fig. 1(a)]. 6,[32][33][34][35] Because these devices only introduce a spatial phase modulation, the generated beam is actually a superposition of various higherorder LG modes with the same azimuthal index l but a different radial index p, i.e., P p LGðl; pÞ. Although the conversion efficiency of a commercial device reaches 95%, it suffers from poor mode purity, 36,37 and generally, the higher the TC is, the lower mode purity becomes (typically 80% and 60% for LG(1, 0) and LG(2, 0) modes, respectively). See Note 1 in the Supplementary Material for details. The other approach is to oscillate an LG mode inside the OPO cavity [ Fig. 1(b)] by utilizing the fact that LG-pumped OPO with an LG cavity mode and an LG output mode. (c) A specially designed Janus OPO that is pumped by a Gaussian mode but outputs an LG mode. (d) A one-round-trip mode conversion without an imaging system. An LG mode passing through a VVW produces a hollow Gaussian beam, which evolves into a Gaussianlike mode after a certain propagation. However, the hollow Gaussian beam cannot recover itself without the equivalent lens as in panel (e) and neither can the LG mode. (e) A one-round-trip mode conversion inside a Janus OPO. The input coupler with a radius curvature of R1 can be seen as an equivalent lens with a focusing length of 2f ¼ R1. Therefore, the light field after the VVW, which is set at the curvature center, will recover itself at the same position after being reflected by the input coupler.
LG modes are Eigen cavity modes, 38,39 which could also extend the output wavelength. 40 In comparison to a Gaussian-mode OPO system, the frequency conversion of an LG mode is less efficient because its donut-shaped profile has a much lower power density. In addition, the output mode quality is not as good as had been hoped. The fiber laser is another potential platform for broadband output of LG mode, but it suffers from low mode purity as well. 41,42 Here, we propose and experimentally demonstrate a Janus OPO based on quasi-phase-matching (QPM) configuration [43][44][45][46][47] for highly efficient output of highly pure, broadly tunable, and TC-controllable LG modes [ Fig. 1(c)]. The Janus cavity that we have previously studied features two-faced transverse-mode structures (like the god Janus in ancient Roman mythology), which combines the advantages of both Gaussian and LG cavity modes. 48 The nonlinear crystal, i.e., a periodically poled lithium niobate (PPLN) crystal, is set next to the front mirror to fully utilize the Gaussian-like front face of Janus cavity mode for high conversion efficiency. Most importantly, due to the introduction of an intracavity imaging system, the generated parametric light is naturally converted into a designed LG mode at the output port of the cavity. The cavity loss is significantly reduced because no additional spatial mode filter is used inside the cavity. The experimental results present a high-performance LG mode source beyond the existing methods. For the generated signal LG beam, the wavelength is tunable between 1.5 and 1.6 μm, the conversion efficiency is >15%, the TC is controllable up to 4, and most importantly, the mode purity can reach 97%.

Janus Cavity Theory
As shown in Fig. 1(c), the Janus cavity has a two-faced cavity mode, distinguishing itself from the traditional cavity mode configuration [Figs. 1(a) and 1(b)]. The front face at the input mirror has a Gaussian profile to achieve a better conversion efficiency because of its higher power density relative to the LG mode. The back face at the end mirror is a donut-shaped LG profile, which guarantees the direct output of a high-purity LG mode. The key question is how to smoothly evolve the cavity mode from a Gaussian profile to an LG profile, and vice versa, without breaking the cavity mode reversibility. The general idea is to directly put a spatial phase modulator, such as a VVW, into the cavity to complete the mode conversion. 14,49 However, phase modulation alone is not sufficient to perform a perfect spatial mode conversion due to lack of necessary amplitude modulation. Let us consider an ideal LG mode at the output mirror. As shown in Fig. 1(d), it propagates through the VVW, which produces a beam superimposed by multiple modes in Part I of the Janus OPO rather than a single mode, as in a traditional cavity. 50 This superimposed beam of multiple spatial modes hardly keeps its profile during free propagation. Therefore, the VVW alone cannot convert it back into the same LG mode as the initial one [ Fig. 1(d)], which breaks the spatial mode reversibility inside the cavity. Under this situation, previous reports used an iris to filter out the unwanted high-order mode, which introduces a substantial cavity loss and severely limits laser performance. 14,49 To realize an ideal Janus OPO [ Fig. 1(c)], the mode reversibility has to be simultaneously satisfied for multiple modes in Part I of the cavity. 48 The key is to introduce an imaging system into the cavity. In our experiment, we use a concave front (input) mirror as an equivalent imaging lens for the compact Janus OPO design [ Fig. 1(c)]. Figure 1(e) shows the transformation of Janus cavity mode in a round trip. When the imaging system works properly, the multiple spatial modes will repeat themselves after passing through the equivalent lens (i.e., being reflected back at the concave front mirror). Then, the VVW can convert them back into an ideal LG mode in Part II of the Janus OPO, and the reversibility condition inside the cavity can therefore be perfectly fulfilled in principle. In addition, the cavity mode profile near the front mirror is required to match the pump Gaussian mode. In our Janus cavity design, the multiple modes after an LG mode passing through the VVW compose a so-called hollow-Gaussian beam, 50 which naturally evolves into a spatial profile very close to a Gaussian mode after a certain propagation distance [ Fig. 1(d)]. See Note 3 in the Supplementary Material for the detailed mathematics in designing a Janus cavity. In comparison to previous designs, all the spatial modes during mode conversion are fully utilized in such a Janus cavity. Therefore, the cavity loss greatly decreased and the output performance significantly improved.

Experimental Setup of the Janus OPO
Figure 2(a) shows the experimental setup of a Janus OPO for generation of an LG-mode signal beam. Its output wavelength is designed to be tunable within the optical communication band. Two concave mirrors form the input and output couplers, which are coated for high reflectivity at the signal wavelength. A PPLN crystal serves as the nonlinear medium, which has multiple channels to extend the QPM bandwidth. The pump beam is generated by a 1064-nm pulsed nanosecond laser and focused into the crystal with a spot size of 200 μm in diameter, which matches the size of signal Gaussian-like face inside the crystal (see Note 4 in the Supplementary Material). Besides the Janus cavity mode as discussed above, the polarization of the field in the cavity is also precisely controlled to facilitate the parametric downconversion and mode conversion. In the PPLN crystal, both the pump and signal waves polarize vertically to utilize the biggest nonlinear coefficient d 33 of the PPLN crystal for high conversion efficiency. By changing the temperature and selecting a channel of the PPLN crystal, the output wavelength of signal wave can range from 1480 to 1650 nm (see Note 2 in the Supplementary Material for the details). The next mode conversion subsystem includes a Faraday rotator (FR), a quarterwave plate (QWP), and a VVW (the system has a work bandwidth of 1550 nm AE 50 nm). The VVW has a distinct q factor, with its value being a positive multiple of 1/2. In the forward propagation direction, a TC of l ¼ AE2q is loaded onto the signal wave, which will be canceled when it passes through the VVW again on its way back. To generate a high-purity LG mode at the output, the cavity mode in Part I should be reconfigured to form a superposition of multiple spatial modes, which is automatically achieved by the Janus cavity design. Here, we use a concave input coupler and set the VVW at its curvature center, which composes a symmetric imaging system to satisfy the condition of multimode reversibility in Part I of the Janus OPO [ Fig. 1(e)]. In principle, an ideal LG mode propagates in Part II of the cavity. Such a stable Janus cavity mode is confirmed by numerical calculations based on the Fox-Li simulating process [ Fig. 2(b)]. The cross sections of the Janus modes for different TCs show how a Gaussian-like mode is naturally transformed to a desired LG mode (See Appendix A and Note 4 in the Supplementary Material for the details). It should be noted that the mode conversion process requires a circularly polarized signal wave on the VVW. To fulfill the polarization reversibility in the cavity, we add an FR and a QWP to accomplish the polarization control (see Note 5 in the Supplementary Material for the details).

Performance of the Janus OPO
First, we demonstrate the generation of high-purity LG(1, 0) and LGð−1,0Þ modes. A VVW with q ¼ 1∕2 is used to introduce a TC of l ¼ AE1. The sign is controlled by the orientation of the QWP. Under QPM configuration, the vertically polarized pump beam produces a vertically polarized signal beam in the PPLN crystal. After passing through the FR, the signal beam has a 45 deg linear polarization. When the fast axis of the QWP orients vertically (or horizontally), the signal polarization is further changed to a left-(or right-) circularly polarized one, resulting in l ¼ 1 (or −1) after the VVW (Fig. 2) (see Note 5 in the Supplementary Material). The intensity patterns of the output LG(1, 0) and LGð−1,0Þ modes at 1550 nm showed in Figs. 3(a) and 3(b) exhibit high-quality donut intensity distribution without observable sidelobes. Clearly, the undesired higher-order LG modes (p > 0) are significantly suppressed by the Janus OPO cavity. Further modal analyses in Figs. 3(a) and 3(b) show that the generated LG(1, 0) and LGð−1,0Þ modes have mode purities of 96.6% and 97.0%, respectively (see Note 6 in the Supplementary Material for the modal analysis process). The mode purity is greatly enhanced in comparison to the typical value of ∼80% using a VVW outside the cavity. 37 Our specially designed Janus OPO is also suitable for generating high-order, high-purity LG modes [ Fig. 2(b)]. As a demonstration, VVWs of q ¼ 1 and q ¼ 2 are used to generate LGðAE2,0Þ and LGðAE4,0Þ modes. Since LGðAEl; 0Þ modes experience the similar transverse-mode evolution in the OPO, we only show the results of LG(2, 0) and LG(4, 0) modes [ Figs. 3(c) and 3(d)]. The mode purities of 95.2% for the output LG(2, 0) mode and 93.7% for the LG(4, 0) mode are much superior to the 60% and 50% values The QWP alters the vertical polarization of the signal beam to circular polarization so that the spin-OAM conversion can happen on the VVW to achieve the desired Gaussian-to-LG mode conversion. The output LG mode can be changed by rotating the QWP or replacing the VVW. FR is used to keep the signal wave to be vertically polarized inside the PPLN crystal. (b) Janus cavity modes for l ¼ 1, 2, and 4 and their cross sections at different distances of 0, 25, 50, 75, and 130 mm away from the input coupler.
using VVWs outside the cavity. 37 It should be noted that the VVWs should be precisely collimated with the optical axis of the cavity to realize a high-purity output of LG modes. See Note 7 in the Supplementary Material for details.
In our experiment, the output wavelength of the Janus OPO can be tuned by changing the QPM channel and the temperature of the PPLN crystal. The Janus OPO shows excellent performance within the designed wavelengths ranging from 1500 to 1600 nm. As shown in Fig. 4(a), the conversion efficiency of the signal LG mode surpasses 10% in most of the working wavelengths. Under a pump power of 4.2 W, the conversion efficiencies for LG(1,0), LG(2, 0), and LG(4, 0) modes at 1550 nm reach 15.3%, 15.8%, and 15.6%, respectively. Notably, the Janus OPO maintains a high conversion efficiency for high-order LG modes. In comparison to the output performance of the signal Gaussian mode in a traditional OPO system [ Fig. 4(a)], the slightly decreasing conversion efficiencies for the outputs of LG modes can be mainly attributed to the limited mode conversion efficiency of VVWs shown in Table S1 in the Supplementary Material and the reflection losses from the FR and QWP. Figure 4(b) compares the power dependence of the output LG(1,0) mode on the pump power at 1525, 1550, 1575, and 1600 nm, respectively, whose thresholds are 1.6, 1.1, 1.2, and 1.5 W, respectively. The differences in the threshold and conversion efficiency for different wavelengths can be attributed to the fact that the intracavity optical components are not uniformly optimized at all the wavelengths. Figure 4(c) depicts the modal analysis results of the output LG(1, 0) mode at the wavelengths of 1525 and 1575 nm, which show high mode purities of 97.1% and 95.9%, respectively. The bandwidth of this Janus OPO can be further extended using ultrawideband optical components as intracavity elements.

Discussion
We have proposed and experimentally demonstrated a Janus OPO system for generating highly efficient, highly pure, broadly tunable, and TC-controllable LG modes. Such a Janus OPO distinguishes itself by possessing a two-faced cavity mode, which makes use of the distinct advantages of both the Gaussian and LG cavity modes. The front (input) face has a Gaussian profile to achieve the high-efficiency nonlinear frequency conversion, while its back (output) face is a donutshaped LG profile that guarantees the direct output of a desired high-purity LG mode from the cavity. The key to realizing such a Janus OPO is the introduction of an imaging system to facilitate the perfect intracavity mode conversion. In this work, the Janus OPO is designed for the Gaussian-to-LG mode conversion of the signal light, which can be easily adjusted to output an LG mode at the idler wavelength. The conversion efficiency of the Janus OPO could be further enhanced by use of a doublepass pump configuration. 47 In addition, by selecting proper optical components, our experimental configuration can be readily extended to visible and UV wavelength bands, as well as to generate tunable vector beams and multidimensional quantum entangled sources. The excellent features of the LG modes from our Janus OPO (e.g., wavelength tunable between 1.5 and 1.6 μm, conversion efficiency >15%, and mode purity >97%) can meet the critical requirements of high-level applications such as high-capacity optical communications, highprecision sensing and measurements, and superresolution imaging. In addition, the linewidth of LG modes could be narrowed using a continuous wave (CW) pump laser, enabling potential investigation of spin-orbital coupling with various atoms in quantum applications.

Appendix A: Experimental Setup
As shown in Fig. 2(a), a PPLN crystal with dimensions of 25ðxÞ × 12.3ðyÞ × 1ðzÞ mm 3 serves as the nonlinear medium. Both of its end faces have a transmittance >99% in the 1380-to 1800-nm wavelength range. It is mounted inside an oven with the temperature tunability up to 150°C. The input coupler (with a radius of curvature of 75 mm) is coated with a high transmittance (>99%) at 1064 nm and a high reflectivity (>99%) at 1450 to 1650 nm, while the output coupler (with a radius of curvature of 125 mm) is coated with a transmittance of 30% at 1450 to 1650 nm. The cavity length is 140 mm, satisfying the stability condition of a resonator. A pump beam (wavelength of 1064 nm, repetition rate of 22 kHz, pulse width of 45 ns, linewidth of 5 nm) is generated by a nanosecond-pulsed fiber laser (YDFLP-M7-3-PM, JPT Co.). It is focused by a lens into a 200-μm-in-diameter spot inside the crystal. The PPLN crystal has 10 channels. In the experiment, we use four channels with periods of 31.02, 30.49, 29.98, and 29.52 μm, respectively. Under the pumping wavelength of 1064 nm, the output signal wave with a wavelength bandwidth of 1.4 nm (see Notes 2 and 8 in the Supplementary Material for details) can be tuned from 1480 to 1650 nm in the temperature range from 25°C to 138°C. An FR, a QWP, and a VVW are inserted into the cavity to achieve the reversible mode conversion inside the cavity. VVW is a spatially variant half-wave plate whose optical axis rotates continuously around a singularity point. All their working wavelength bandwidths are from 1500 to 1600 nm. The VVW is placed at a distance of 90 mm away from the input coupler, where the curvature center of the input coupler is, considering the effective length due to the high refractive index of the PPLN crystal. VVWs of q ¼ 0.5, 1, 2 have been used to generate LGðl; 0Þ modes with different TCs. The output intensity patterns are recorded by a laser beam profiler (LBP, Newport Corp.).

Appendix B: Cavity Mode Simulations
The numerical simulations have been carried out based on the Fox-Li method. A one-round-trip transition of the cavity mode can be described in what follows. A parametric wave starting from the input coupler travels a distance of L A and passes through the VVW with a TC of l (or −l). After traveling a distance of L B , the parametric wave is reflected by the output coupler and propagates backward. The TC is canceled when the parametric wave passes through the VVW along the opposite direction. Finally, the parametric wave reaches the input mirror to finish its one-round-trip transition. The parametric wave repeats the cycle until a stable cavity mode is formed. Base on angular spectrum theory, the iterative procedure described above is calculated step by step using MATLAB programming.