## Concept of programmable LP-mode Synthesizer

The conversion relations between LP modes and CVB/OAM modes are the theoretical basis for the proposed LP-mode Synthesizer. As we know, the CVB modes and OAM modes are two eigen-mode sets in optical fibers under the Cartesian coordinate system [38] and the circular polarization coordinate system [39], respectively. The polarization vector mode corresponds to the CVB mode whereas the phase vortex mode conveys the OAM mode. Mathematically, the LP modes are the solutions of scalar Helmholtz equations under the weakly-guiding approximation, and the superposition of near-degenerate CVB modes can form LP states. The relation of the LP modes and CVB modes could be described by [40],

$$\left[\begin{array}{c}{HE}_{m+1,n}^{e}\\ {HE}_{m+1,n}^{o}\\ {EH}_{m-1,n}^{e}\\ {EH}_{m-1,n}^{o}\end{array}\right]={F}_{m,n}\left(r\right)\left[\begin{array}{cccc}1& 0& 0& -1\\ 0& 1& 1& 0\\ 1& 0& 0& 1\\ 0& -1& 1& 0\end{array}\right] \left[\begin{array}{c}{\overrightarrow{e}}_{x}{cos}\left(m\phi \right)\\ {\overrightarrow{e}}_{y}{cos}\left(m\phi \right)\\ {\overrightarrow{e}}_{x}{sin}\left(m\phi \right)\\ {\overrightarrow{e}}_{y}{sin}\left(m\phi \right)\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& -1\\ 0& 1& 1& 0\\ 1& 0& 0& 1\\ 0& -1& 1& 0\end{array}\right]\left[\begin{array}{c}{\overrightarrow{e}}_{x}{LP}_{m,n}^{e}\\ {\overrightarrow{e}}_{y}{LP}_{m,n}^{e}\\ {\overrightarrow{e}}_{x}{LP}_{m,n}^{o}\\ {\overrightarrow{e}}_{y}{LP}_{m,n}^{o}\end{array}\right]$$

1

where “*e*” and “*o*” represent the even and odd polarization states, \({F}_{m,n}\left(r\right)\) is the radial field distribution of the corresponding scalar mode (LP) solution, *m* is the radial order of CVB modes, *n* is the azimuthal order of CVB modes, *r* is the radial coordinate, and \(\phi\) is the angular coordinate. Note that \({EH}_{m-1,n}^{e}\) is substituted by\({TM}_{0,n}\)and \({EH}_{m-1,n}^{o}\) is substituted by \({TE}_{0,n}\) when *m* = 1.

Similarly, the OAM modes having a helical wave-front phase can be formed by the superposition of CVB modes [40]. So, the OAM modes in the fiber can be expressed as linear superposition of the odd and even CVB modes with ±\(\pi /2\) phase difference. Generally, in the weakly-guiding fiber, the superposition of near-degenerate CVB modes forms LP states.

So, OAM modes can be expressed by,

$$\left[\begin{array}{c}{\overrightarrow{\sigma }}^{+}{OAM}_{+m}\\ {\overrightarrow{\sigma }}^{-}{OAM}_{-m}\\ {\overrightarrow{\sigma }}^{-}{OAM}_{+m}\\ {\overrightarrow{\sigma }}^{+}{OAM}_{-m}\end{array}\right]={F}_{m,n}\left(r\right)\left[\begin{array}{c}{\overrightarrow{\sigma }}^{+}{exp}\left(+jm\phi \right)\\ {\overrightarrow{\sigma }}^{-}{exp}\left(-jm\phi \right)\\ {\overrightarrow{\sigma }}^{-}{exp}\left(+jm\phi \right)\\ {\overrightarrow{\sigma }}^{+}{exp}\left(-jm\phi \right)\end{array}\right]=\left[\begin{array}{c}1 j j -1\\ 1-j -j -1\\ 1-j j 1\\ 1 j -j 1\end{array}\right]\left[\begin{array}{c}{\overrightarrow{e}}_{x}{LP}_{m,n}^{e}\\ {\overrightarrow{e}}_{y}{LP}_{m,n}^{e}\\ {\overrightarrow{e}}_{x}{LP}_{m,n}^{o}\\ {\overrightarrow{e}}_{y}{LP}_{m,n}^{o}\end{array}\right]$$

2

where \({OAM}_{\pm m}\) denote the field distributions of helical wave-front phase exp(± *jmφ*); the subscripts “±” of OAM separately correspond to the left-hand and right-hand helical wave-front phase. Especially, the combination of LP modes with a ±\(\pi /2\) phase difference in the weak-guiding fiber can form linearly-polarized OAM modes [41],

$$\left[\begin{array}{c}{\overrightarrow{e}}_{x}{OAM}_{-m}\\ {\overrightarrow{e}}_{y}{OAM}_{-m}\\ {\overrightarrow{e}}_{x}{OAM}_{+m}\\ {\overrightarrow{e}}_{y}{OAM}_{+m}\end{array}\right]={F}_{m,n}\left(r\right)\left[\begin{array}{c}{\overrightarrow{e}}_{x}{exp}\left(-jm\phi \right)\\ {\overrightarrow{e}}_{y}{exp}\left(-jm\phi \right)\\ {\overrightarrow{e}}_{x}{exp}\left(+jm\phi \right)\\ {\overrightarrow{e}}_{y}{exp}\left(+jm\phi \right)\end{array}\right]=\left[\begin{array}{cccc}1& 0& -j& 0\\ 0& 1& 0& -j\\ 1& 0& j& 0\\ 0& 1& 0& j\end{array}\right]\left[\begin{array}{c}{\overrightarrow{e}}_{x}{LP}_{m,n}^{e}\\ {\overrightarrow{e}}_{y}{LP}_{m,n}^{e}\\ {\overrightarrow{e}}_{x}{LP}_{m,n}^{o}\\ {\overrightarrow{e}}_{y}{LP}_{m,n}^{o}\end{array}\right]$$

3

The schematic diagram of a programmable LP-mode Synthesizer for CVB/OAM generation is illustrated in Fig. 1, which interprets literally the mode conversion relations shown in Eqs. (1) to (3). First, the LP-mode Pool should be established to stock arbitrary LP modes and output them independently according to the need. The flexible selection of active LP mode in the LP-mode Pool is obtained by the LP-mode Selection Unit. The Polarization Component Extraction Unit is used to detach the elements of the selected LP mode. After that, the phase differences among the elements are adjusted by the Phase Control Unit. The Programmable Control Unit is responsible for the configuration of the three units according to the conversion relations.

The physical implementations of the programmable LP-mode Synthesizer may take flexible forms. The LP-mode Pool accompanied with the LP-mode Selection Unit should output switchable LP mode in a general way, which could be realized by various simple and effective approaches based on commercial optical components such as LP-mode converters or MMUXes. There are also different kinds of approaches for polarization component extraction and phase delay utilizing fiber-based or free-space micro-structured optical elements such as polarization beam splitters and optical delay lines. It should be noted that the optical circuit in the programmable LP-mode Synthesizer should be built up based on weakly-coupled FMFs and matched low-modal-crosstalk mode control components to sustain the independence among the LP modes. As a principle-of-concept example, we demonstrate the programmable LP-mode Synthesizer utilizing a compact fiber ring laser and a PC in the next section.

## Experimental setup

Figure 2 shows the experimental configuration of a fiber ring laser to verify the feasibility of the programmable LP-mode Synthesizer, the cavity of which is composed of both single-mode fiber (SMF) and FMF sections. In the FMF section, the weakly-coupled MRC-FMF supports independent light propagation through 5 LP modes (LP01, LP11, LP21, LP02, LP31) with very low modal crosstalk, whose structure is shown in Fig. 6a in the Methods. The matched MMUX/MDEMUX consists of 5 cascaded MSCs with the structure shown in Fig. 6e in the Methods. It should be noted that each MSC could only perform mode conversion between the fundamental mode in the SMF and one of a pair of degenerate LP modes in the FMF. An 80:20 few-mode optical coupler (FM-OC) is utilized as the output coupler of the laser, which is a free-space wave-plate beam splitter and achieves better mode insensitivity than fused-type FM-OC [41]. A conventional single-mode pump scheme is adopted in the SMF section, consisting of 980-nm laser diode (LD), 980/1550-nm wavelength multiplexer (WMUX) for pump/signal light combination, and a piece of single-mode Erbium-doped fiber (SM-EDF) with the length of 5-m as the gain medium. The optical isolator (ISO) enables unidirectional light propagation. The single-mode optical coupler-1 (SM-OC-1) is used to split the light into 5 branches. The higher-order modes are converted back to LP01 mode by the MDEMUX and then combined to the SMF through another SM-OC-2 for the next round of signal circulation.

Compared with the schematic diagram of the programmable LP-mode Synthesizer shown in Fig. 1, we can see that the whole fiber ring laser could act as the LP-mode Pool with selectable output light from one of the 5 LP modes, while the SM-OC-1 and the Optical Switching Array could bear the function of LP-mode Selection Unit. The PC is realized by wrapping multiple turns of MRC-FMF around a 3-paddle adjusting device. For the operation to non-degenerate LP01 and LP02 modes, the PC could modify the polarization components and apply phase difference between them by twisting the 3 paddles to reallocate and select electric fields. While for the case of operation to degenerate LP11, LP21 or LP31 modes, the PC could reallocate the optical power and apply phase difference among all the 4-fold eigenmodes even if only even or odd mode is excited by the MMUX consisting of spatial-orientation-selective MSCs. So, the PC could perform the functions of both polarization and phase control. The output of the 3-dB SM-OC-3 as a reference Gaussian beam is utilized to verify the generation of OAM lasing. The spectrum properties of the laser are measured by an optical spectrum analyzer (YOKOGAWA, AQ6370C) with a resolution of 0.02 nm. The output mode profiles are observed by a charge-coupled device (CCD) camera (Xenics, Bobcat-5316). The output power is measured by an optical power-meter (EXFO, FPM-302X-FOA-22).

## Experimental results for LP generation

The independent lasing for each LP mode of the proposed fiber ring laser is first investigated. The Optical Switching Array is adjusted to excite each LP mode one by one, and the output power characteristics of output light at the 20% output port of the FM-OC are measured by the optical power-meter (EXFO, FPM-302X-FOA-22), while the intensity profile is measured by the CCD camera. The output signal power versus the pump power for five lasing modes is plotted in Fig. 3a. We can see that the lasing output power for different LP modes increases linearly with the pump power when working above the lasing threshold. The slope efficiencies of 1.17%, 0.82%, 0.56%, 0.35%, and 0.13%, and the pump power thresholds of 35, 42, 46, 48, and 53 mW are obtained for LP01, LP11, LP21, LP02, and LP31 lasing modes, respectively. The different values of slope efficiencies, and pump power thresholds among the five LP lasing modes are mainly influenced by different insertion losses for the five LP modes in the partial weakly-coupled FMF cavity.

The spectral features and the stabilities of the lasing outputs are also investigated. Figure 3c shows the optical spectra of the lasing output of five LP modes measured by the optical spectrum analyzer when the pump power is fixed to be about 400 mW. The central wavelengths of 1561.23, 1561.41, 1561.18, 1561.25 and 1561.62 nm, the 3-dB linewidths of 0.028, 0.02, 0.028, 0.024 and 0.024 nm, the side-mode suppression ratios (SMSRs) of 59, 53, 53, 46 and 37 dB are achieved for the five LP modes from low to high orders, respectively. The wavelength differences come from the different effective lengths of lasing cavities for different LP modes. The output stabilities for the five lasing LP modes are measured during the 60-min period at room temperature, and the output optical spectra are shown in Fig. 3d. The data are recorded at a 5-min interval over a 60-min period. The fluctuations of peak wavelength and average output power for each lasing LP mode are less than 0.032 nm and 0.055 mW, respectively.

## Experimental results for CVB generation

Selectable generation of CVBs is demonstrated utilizing the proposed LP-mode Synthesizer. According to the conversion relationship between LP modes and CVB modes by Eq. (1), the LP11 mode is formed by the superposition of near-degenerate CVB modes including radially-polarized TM01, azimuthally-polarized TE01, and \({\text{H}\text{E}}_{21}^{\text{e}}/{\text{H}\text{E}}_{21}^{\text{o}}\) modes. So, adjusting the Optical Switching Array to only enable the lasing of LP11 modes, and then adjusting the PC to induce the rotation and extrusion, we could obtain the TM01 or TE01 mode. The intensity distributions of the output light are recorded by the CCD camera, and the results are shown in Fig. 4a & 4b. The Fig. 4a.i & 4b.i show the doughnut-shaped intensity profiles of both TE01 mode and TM01 mode, respectively. In order to distinguish the polarization distributions of both CVB modes, a rotatable linear polarizer is placed in the light path between the lasing output and the CCD camera. Figure 4a.ii ~ v & 4b.ii ~ v show the intensity distributions with different polarization orientations (represented with white arrows) for the polarizer. The intensity patterns with two-lobe shape perpendicular to the orientation of the linear polarizer in Fig. 4a.ii ~ v show that the output lasing beam is azimuthally-polarized TE01 mode, while the intensity patterns with two-lobe shape parallel to the orientation of the linear polarizer in Fig. 4b.ii ~ v indicate that the light beam is radially-polarized TM01 mode.

Figure 4c presents the optical spectra of CVB outputs at the fixed pump power of about 400 mW. The central wavelengths of 1561.38 and 1561.45 nm, the 3-dB linewidths of 0.024 and 0.024 nm, and the SMSRs of 55 and 54 dB are measured for the TE01 and TM01 lasing modes, respectively. Figure 4d shows the output stabilities for TE01 and TM01 lasing modes at a 5-min interval during the 60-min period. The peak wavelength fluctuation and the average output power fluctuation for TE01 lasing mode are less than 0.044 nm and 0.042 mW, respectively. The fluctuations of peak wavelength and average output power for TM01 lasing mode are less than 0.040 nm and 0.030 mW, respectively.

## Experimental results for OAM generation

Selectable generation of OAMs is also verified utilizing the proposed LP-mode Synthesizer. According to the conversion relationship by Eq. (3), the OAMs can be obtained by the superposition of a specific pair of degenerate LP modes in the weakly-guiding FMF under proper polarization and phase control. For examples, the \({OAM}_{\pm 1}\)could be obtained utilizing the \({LP}_{m,1}^{e}\) and \({LP}_{m,1}^{o}\), while the \({OAM}_{\pm 2}\) could be obtained utilizing the \({LP}_{\text{2,1}}^{e}\) and \({LP}_{\text{2,1}}^{o}\). So, the Optical Switching Array should be adjusted to obtain the LP11 or LP21 lasing output for the generation of the \({OAM}_{\pm 1}\)or \({OAM}_{\pm 2}\), respectively. The PC should be carefully adjusted to rotate the lobe orientation in addition to polarization and phase control. Figure 5a shows the interference setup used to determine the charge number (± 1, ± 2) of the generated OAM beams by observing their characteristic fork patterns with a CCD camera. The reference Gaussian beam at the output of SM-OC-3 is used for the interference setup to analyze the generated OAM beams. A free-space beam combiner combines both light to form the interference pattern and a variable optical attenuator balances the power of the two optical paths. A half-wave plate is used to adjust the polarization state of the reference Gaussian beam. By properly adjusting the PC, annular intensity profiles and their corresponding patterns could be observed. Figure 5b presents the measured intensity distributions for LP modes and OAM beams, and fork intensity patterns at the laser output. The number of forks represents the topological charge of the OAM beams, which are |m|=1,2.

Figure 5c shows the optical spectra of the OAM+ 1, OAM-1, OAM+ 2, and OAM-2 outputs at the fixed pump power of about 400 mW. The central wavelengths of 1561.39, 1561.43, 1561.18, and 1560.44 nm, the 3-dB linewidths of 0.020, 0.028, 0.032, and 0.028 nm, the SMSRs of 51 dB, 53 dB, 53 dB, and 55dB, are measured for the OAM+ 1, OAM-1, OAM+ 2, and OAM-2 lasing modes, respectively. Figure 5d shows the output stabilities for all the four OAM lasing modes at a 5-min interval during the 60-min period. The fluctuation of peak wavelength for the OAM+ 1, OAM-1, OAM+ 2, and OAM-2 lasing modes are less than 0.048, 0.056, 0.072 and 0.060 nm, respectively, while the fluctuation of average output power are 0.072, 0.060, 0.041 and 0.058 mW, respectively.