## Abstract

Light beams carrying orbital angular momentum (OAM) have important implications for future classical and quantum systems. In many applications, controlled switching of the OAM state at high speed is crucial, while accelerating the switching rate presents a long-standing challenge. Here we present a method for flexibly switching the OAM state of light based on a coherent laser array system. In the system, the output structured light beam is tailored by the coherent combination of array elements. By employing an OAM mode sorting assisted phase control subsystem, which continuously performs the optimization algorithm, the dynamic wavefront distortion of the combined OAM beam could be compensated. Meanwhile, our approach allows one to achieve fast states switching of the combined OAM beam via programming the cost function of the algorithm. The results of Monte-Carlo simulations demonstrate the feasibility of the proposed method, and the mode purity and power scaling potential of the controllably generated OAM beam are discussed. This theoretical work could be beneficial to the future implementation of rapidly switchable OAM beams at practical output power.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Recently, optical vortex beams with helical phase structures and carrying orbital angular momentum (OAM) have been intensively explored owing to the unique optical and dynamic characteristics [1–3]. Harnessing the OAM of light has given rise to widespread applications such as classical and quantum communications [4–6], particle manipulation [7,8], enhanced resolution in optical imaging and microscopy [9,10], and rotation probing [11–13]. To generate OAM beams that could be utilized in these applications, various approaches have been proposed and demonstrated [14–22]. For example, the OAM beams can be formed by transferring the fundamental Gaussian beams via using extra-cavity mode converters, such as spiral phase plates and slits [15,16], spatial light modulators (SLMs) [17], q-plates [18], and metamaterials [19]. Besides, intracavity approaches, namely tailoring OAM beams at source, have attracted much attention [20–22]. By designing the structure and components of the resonator, high-purity OAM beams can be generated from lasers. With the progress in the generation of OAM beams, there is a raising desire to flexibly switch the OAM state of the vortex beam at high speed and high output power, which is especially attractive for OAM-based data transmissions [23,24].

To realize the rapid switching between OAM states at practical output power, utilizing the coherent combination of laser arrays to tailor OAM beams opens up a promising way [25–28]. For one thing, the switching rate of the generated OAM beam from the coherent laser array system relies on the electro-optic phase modulators, which can operate at high frequency [27]. For another thing, coherently combining multiple lasers to form vortex beams may scale the output power to unprecedented levels [29]. In recent years, theoretical studies and proof-of-concept experiments have been carried out to promote the technical route [30–35]. Despite these advancements, implementing efficient phase control of array elements presents a serious bottleneck, since dynamic phase noise always occurs in the system and would cause the wavefront distortion and mode purity degradation of the output OAM beam. In the past several years, enhancing the phase control capacity has been an ongoing effort [35–39]. To control the phases of laser array to desired values, there is an intuitive approach namely heterodyne or interferometric measurement by introducing reference beam. Recently, real-time (kilohertz regime) controllable generation of OAM beams in coherent beam combining (CBC) system by using interferometric measurement phase control method has been realized [35]. In a six-emitter optical phased array system that tailors OAM beams, digitally enhanced heterodyne interferometry technique has been employed for phase locking, and 12 kilohertz emitter phase actuation speeds were achieved [37]. Meanwhile, optimization algorithm based phase control approach provides another promising way. Without additional reference beam, CBC systems that employ optimization algorithm for phase control are mode-free, compact, and simple-to-implement. Particularly, in both high-power (∼7.1 kilowatts) and large-scale (107 channels) CBC systems, using optimization algorithm for phase stabilization exhibits good performance [40,41]. Most of the optimization algorithm based phase control methods presented in previous studies rely on the directly detected intensity distribution of the combined OAM beam, which brings limitation on extracting the optical field information. With the help of machine learning algorithms, more features of optical field can be extracted, thus mode switching is expected to be preliminarily achieved [39]. However, the mode switching frequency is limited by the imaging processing and network calculating time. Accurately compensating the wavefront distortion and rapidly switching the OAM state at the same time still remain a critical challenge. Here, we are motivated to efficiently extract the optical field information that feeds the high bandwidth phase control system by magnifying the difference between different OAM modes with the assistance of OAM mode sorting.

In this work, we theoretically put forward an approach to realize the rapid switching of OAM states with mode sorting assisted coherent laser array systems. As one of the most important sections of the laser array system to tailor structured light beams, the phase control section is enhanced with the assistance of OAM mode sorting based on the optical vortex field detection. When the system is in closed loop, the dynamic wavefront distortion of the generated vortex beam is compensated through optimization algorithms, whereas the rapid switching between OAM states can be realized via programming the cost function. Our results suggest that utilizing the proposed method, coherent laser array-based structured light beams tailoring system would make the controllable generation of high-power, fast switchable OAM beams a reality.

## 2. Principle and method

Figure 1(a) presents the schematic setup of the mode sorting assisted coherent laser array system for tailoring OAM beams. The seed laser output is preliminarily amplified by a pre-amplifier and split into multiple channels by a fiber splitter. The laser beam of each channel is sent through a phase modulator (PM) utilized for phase control and cascaded fiber amplifiers that scale the output power. Then, laser outputs from multiple channels are emitted by a collimator array (at the source plane) to free space. To generate the OAM beam with a topological charge (TC) of *l _{m}*, the emitted laser array that contains multiple concentrical subarrays is of circular arrangement at the source plane, as shown in Fig. 1(b).

*p*represents the serial number of subarrays, and

*N*

_{in},

*N*

_{out}are the serial numbers of the inner and outer annular subarrays, respectively. On the

*p*-th subarray,

*n*

_{0}

*p*array elements are arranged in an annular shape, and the central position of the

*j*-th beamlet of the

*p*-th radial subarray satisfies

*r*accounts for the distance between the first radial subarray (

*p*= 1) and the center of the laser array. The optical field of the laser array at the source plane is given by

*x*,

*y*) accounts for the coordinate of the source plane.

*A*

_{0},

*w*

_{0}, and

*d*denote the amplitude, waist width, and aperture diameter of each beamlet, respectively. circ(

*ρ*

_{0}) is the transmittance function of the circular aperture, namely,

In terms of [42], the range of feasible TCs is related to the number of the beamlets lying on the outer subarray, thus the specific OAM modes that can be generated by the laser array are determined. The emitted beam array passes through a high reflective mirror (HRM). The reflected part, as the high-power output of the system, would form a coherently combined OAM beam in the far-field, while the transmitted part is utilized to feed the mode sorting assisted phase control section.

In the phase control section, the transmitted beam array propagates through a 4f imaging system with a pinhole for spatially filtering. Different from ideal OAM beams, the OAM beams generated by the coherent combination of laser arrays have sidelobes that contain unwanted higher-order OAM mode components and affect the purity of the desired OAM mode [30,43]. Purification of the combined OAM beam via spatially filtering would truncate these sidelobes, thus the performance of mode sorting can be improved. The complex amplitude of the spatially filtered beam at the focal plane of L1 can be expressed as

*u*,

*v*) represents the coordinate of the Fourier plane (L1). F{·} denotes the Fourier transform operator.

*λ*,

*f*

_{1}, and

*R*are the wavelength, focal length of L1, and the radius of the pinhole, respectively. With the prior knowledge of the specific OAM modes, the OAM mode sorter can now be designed. To realize the OAM mode sorting, the phase of the purified beam is modulated by a SLM positioned at the focal plane of the second lens (L2, the focal length is

*f*

_{2}). The SLM is loaded with a specially designed complex phase mask, which can steer the vortex beam of different OAM states to different directions [44]. We suppose the steering direction of the combined OAM beam with TC =

*l*is described by the tilt angle

_{m}*α*and

_{m}*β*. Then, the complex phase mask satisfies

_{m}*ξ*,

*η*) denotes the coordinate of the Fourier plane (L2).

*k*,

*ψ*, and

*N*account for the wave number, azimuthal coordinate, and the total number of the OAM states, respectively. Subsequently, the steered beam is focused by L3 with 2f arrangement onto a PD array. The PD array has been designed and calibrated according to the steering directions of the specific OAM modes. Before each PD detector, there is a pinhole to extract the power in the circular area, which is determined by the central lobe size of the specific demodulated mode steered to the PD detector. The intensity signals detected by the PD array are sent to the controller that carries the optimization algorithm for actively phase locking and shifting. The cost function of the optimization algorithm is defined as

*J*denotes the energy encircled in a circular area of the

_{m}*m*-th PD detector.

*W*= [

*w*

_{1}

*w*

_{2}…

*w*] represents the weight vector, and

_{N}*w*∈{0,1}. When the system is in closed loop, the controller performs the optimization algorithm and applies the phase control voltages to the PMs. The cost function gradually evolves to its maximum, and correspondingly, the phase errors of the array elements that distorts the wavefront of the generated OAM beam would be compensated. To rapidly switch the OAM state of the combined vortex beam from

_{m}*l*

_{a}to

*l*

_{b}, the controller modulates the cost function of the optimization algorithm via shifting the weight vector from [0 … 0

*w*

_{a}=1 0 … 0] to [0 … 0

*w*

_{b}=1 0 … 0]. Accordingly, the relative phases of array elements would be converged to the specific values to form the vortex beam with a desired OAM mode.

## 3. Results and discussion

Without loss of generality, we investigate a 30-element coherent laser array that can generate an OAM beam with the TC ranges from -4 to +4. Figure 2(a) depicts the intensity profile of the laser array at the source plane. The waist width, wavelength, and aperture diameter of each array element are set to be *w*_{0 }= 89 μm, *λ* = 1064 nm, and *d* = 200 μm, respectively. The 30-element laser array consists of two subarrays, namely the 12-element inner subarray with the serial number *N _{in}* = 2 and the 18-element outer subarray with the serial number

*N*

_{out}= 3. The parameters

*n*

_{0}and

*r*are set to be 6 and 220 μm, respectively. According to [42], the TCs of the combined beams can range from -6 to +6. Fully consider the profile of the combined beam and the combining efficiency, the generation and control of OAM beams with the TCs range from -4 to +4 are investigated and discussed. In the phase control section, the radius of the pinhole to purify the combined beam is

*R*= 100 μm, and the focal lengths of L1, L2, and L3 are given by

*f*

_{1}= 5 cm,

*f*

_{2}= 5 cm, and

*f*

_{3}= 100 cm, respectively. The complex phase mask generated by the SLM is shown in Fig. 2(b). The complex phase mask loaded SLM can steer the eight OAM modes to eight directions. In our investigation, the PD array is of two-dimensional annular arrangement, and the radius of the PD array is set to be 9.5 mm. In practical implementation, the arrangement and size of the PD array can be adjusted by modulating the focal length of L3 and the complex phase mask.

Here we exhibit the phase distributions of the laser array at the source plane and the corresponding intensity distributions of the diffracted optical fields at the receiver plane to specifically illustrate the function of the complex phase mask. Figures 3(a1)–3(a8) display the relative phases of the array elements utilized to generate the OAM beam with, in turn, TC=+1, +2, +3, +4, -1, -2, -3, and -4. As we can see from the intensity distributions of the diffracted optical fields [e.g. Figures 3(b1)–3(b8)] correspond to the phase distributions shown in Figs. 3(a1)–3(a8), eight beamlets (seven dark-hollow beams and a Gaussian-like beam) are azimuthally arranged. As for the different OAM states, the Gaussian-like beamlets with concentrated energy occur at different positions. Therefore, the function of the complex phase mask can be concluded as an OAM mode sorter that magnifies the difference between the different test OAM modes with the envisaged OAM mode. It is worth noting that spatial filtering of the coherently combined OAM beams plays an important role in discrimination of OAM modes at the receiver plane. For comparison, Figs. 4(a)–4(h) exhibit the intensity distributions at the receiver plane for sorting the coherently combined OAM beams without spatial filtering, and the TCs are +1, +2, +3, +4, -1, -2, -3, and -4, respectively. As is mentioned above, coherently combined OAM beams contain unwanted OAM mode components in the sidelobes. When spatial filtering is not employed, these OAM sidebands of the combined beam cannot be truncated and would form the petals surround each PD of the PD array. For one thing, the formation of these petals reduces the percentage of energy in the central lobe of the demodulated beam. For another thing, the distance between the surrounding petals and the central position of the PD could approach the distance between the adjacent PDs, indicating that the petals of each PD would be detected by the adjacent PDs when the PD array is of compact design. Therefore, the purification of combined OAM beams by spatial filtering is essential for the subsequent OAM mode sorting and optical field information extraction. In our previous studies, the intensity distribution of the combined OAM beam is directly utilized for extracting the phase information, which always brings limitations [38,39]. Now with the assistance of OAM mode sorting based on the complex phase mask, the phase control capacity of the coherent laser array systems that tailor mode switchable OAM beams is expected to be enhanced.

To demonstrate the feasibility of the proposed OAM state switching approach, Monte-Carlo simulations have been conducted. The initial phase errors of the laser array system are generated randomly, and our motivation is to explore that whether these phase errors can be efficiently compensated. In the phase control section, the controller carries the optimization algorithm to realize the compensation of undesired phase errors and desired phase shifting by iterations. The cost function, which is modulated by the programmable weight vector *W*, determines the output OAM mode of the phase-locked laser array. To generate the OAM beam with TC=+1, +2, +3, +4, -1, -2, -3, and -4, the *W* is set to be *W*_{1}, *W*_{2}, *W*_{3}, *W*_{4}, *W*_{5}, *W*_{6}, *W*_{7}, and *W*_{8}, respectively. As for the elements of the weight vector *W _{q}*,

*w*= 1 and other elements equal 0.

_{q}Figure 5 shows the results of the Monte-Carlo simulations. The optimization algorithm is performed during the phase control process, and we take the stochastic parallel gradient descent (SPGD) algorithm as an example [45,46], which has been verified in both high-power and large-scale coherent laser beam combining systems [40,41,47]. For each desired OAM state, 100 simulations have been carried out. The convergence curves of the cost function are depicted in Figs. 5(a1)–5(a8), which correspond to the cases of TC=+1, +2, +3, +4, -1, -2, -3, and -4, respectively. The results indicate that the cost function can get converged within 250 steps of iterations on average, while the phase locking for generating an OAM beam with a larger |*l _{m}*| requires more steps (for several randomly generated phase errors, 300 steps are required). The inset figures show the average phase distributions, which verify that the helical phase structure with the desired TC could be finally formed when the cost function gets converged. The normalized average far-field intensity profiles of the laser array when the cost function gets converged have been investigated as well. We assume that a focus lens with a focal length of 50 mm is positioned behind the collimator array, and the combined OAM+1, OAM+2, OAM+3, OAM+4, OAM-1, OAM-2, OAM-3, and OAM-4 beams could be observed at the focal plane (observation plane), which are presented in Figs. 5(b1)–5(b8), respectively. One can see that during the phase control process, the phase errors are compensated by performing the SPGD algorithm, and undesired local optima of the cost function are avoided with the assistance of the above-mentioned OAM mode sorting. As a result, the optical field of the combined OAM beam is turning to the ideal OAM beam with the desired OAM state.

Then, the accuracy of the proposed approach has been evaluated based on the analysis of OAM spectra. Different from ideal OAM beams, sidelobes exist and surround the main ring of the coherently combined OAM beams, as shown in Figs. 5(b1)–5(b8). These sidelobes contain unwanted OAM mode components that affect the purity of the desired OAM mode, which has been discussed in previous studies [30,43]. When our proposed method is implemented in practical settings, the sidelobes should be filtered out by the receiving aperture with a finite diameter, which can be thought of beam purification. Therefore, it is helpful to investigate the OAM spectrum in a limited circular area [30,39]. In the polar coordinate system, the optical field of the combined beam at the observation plane is expressed as *E*_{combined} (*ρ*, *φ*), where (*ρ*, *φ*) denotes the coordinate of the observation plane. The optical field of the combined beam truncated by the receiving aperture with a diameter of *D _{t}* is given by

*E*(

_{t}*ρ*,

*φ*) =

*E*

_{combined}(

*ρ*,

*φ*) • circ (2

*ρ*/

*D*). The optical field of the truncated beam can be represented by the composition of angular harmonics, namely,

_{t}The purity of the *l*-order OAM mode *P _{l}* is expressed as

According to the definitions of OAM mode purity and OAM spectrum [48], the OAM spectra of the combined OAM beam truncated by the receiving aperture can be obtained by numerically calculating Eqs. (8) and (9). The diameter of the truncated aperture *D _{t}* is set to be 0.2 mm, and the infinite ranges of

*l*and

*n*in Eqs. (8) and (9) are bounded as [-80, +80] for available practical analysis and ensuring the high accuracy of calculation [49]. Figure 6 exhibits the average OAM spectra of the truncated combined beams when the cost function gets converged. For all of the cases, the purity of the desired OAM modes in the truncated aperture exceeds 0.99 after iterations, thus we manifest that to generate the OAM beams with various TCs, the proposed control approach is always of high accuracy that ensures the efficient compensation of wavefront distortion.

To further illustrate the capacity of OAM state switching, the switching process has been investigated. Here we take the switching from OAM+1 state to OAM-3 state as an example. Figure 7(a) shows the evolutions of the normalized energy collected by the first and the 7-th PD detectors. The circular areas of the PD detectors to collect the intensity signals and the initial/final phase distributions at the receiver plane are depicted in the inset figures. When the weight vector is modulated from *W*_{1} to *W*_{7}, the cost function is accordingly changed from *J* = *J*_{1} into *J* = *J*_{7}. The controller performs the optimization algorithm continuously to make the changed cost function evolve to its optimum (within 250 steps). According to the execution process of SPGD algorithm [46], the execution time of a single iteration step *τ* is determined by the formula *τ* = 6*τ _{c}* + 2

*τ*+ 3

_{PD}*τ*, where

_{PM}*τ*,

_{c}*τ*, and

_{PD}*τ*represent the corresponding operation times of the controller, PD, and PM, respectively. In terms of the current hardware capacity, we take the typical case of the controller with the main frequency of 1.2 gigahertz, the PD with the bandwidth of 1 gigahertz, and the PM with the bandwidth of 1 gigahertz as an example, thus the execution time of a single iteration step is calculated as 10 nanoseconds. Certainly, the execution time of a single iteration step can be further reduced by employing the controller of higher main frequency and the PD and PM with higher bandwidth, indicating that higher switching speed of OAM states could be realized. During the state switching process, the intensity distributions at the receiver plane, intensity profiles of the combined OAM beam, and the phase distributions of the combined beam are displayed in Figs. 7(b1)–7(b6), Figs. 7(c1)-7(c6), and Figs. 7(d1)–7(d6), respectively. From left to right, they are in turn, the cases of the first, second, 10

_{PM}^{th}, 30

^{th}, 50

^{th}, and 250

^{th}steps. With the increase of the cost function, the energy of the diffracted pattern at the receiver plane evolves from being concentrated at the first PD detector to being concentrated at the 7-th PD detector. Meanwhile, the dark-hollow shaped intensity profile and helical wavefront (with the desired TC) of the combined beam in the far-field are gradually formed. The results indicate that by modulating the cost function and performing the optimization algorithm, the switching between different OAM states could be realized.

In the above analysis, the feasibility of implementing the proposed approach to compensate the wavefront distortion of the combined vortex beam and switch the OAM state has been verified. In the coherent laser array system, the high rate of electro-optic PMs and the high execution frequency of the optimization algorithm ensure the high switching rate of the output OAM beam based on the proposed approach. The complex phase mask, as a flexible and robust OAM mode sorter, can demultiplex a large number of OAM modes simultaneously. With the number of OAM modes increases, the power of the sampled beam that inputs to the mode sorter can be adaptively adjusted to maintain the intensity detected by the PD array and correspondingly ensure the efficient discrimination of OAM modes at the receiver plane. Therefore, these features of the OAM mode sorting assisted phase control subsystem could benefit the number scalability of the controllable generation of OAM modes. Furthermore, the average output power of the single fiber laser beam in coherent laser beam combining systems has reached kilowatts level [50]. Therefore, employing our proposed approach into the coherent laser array system holds great promise for high-power and rapidly switchable OAM beams.

## 4. Conclusion

In conclusion, we theoretically propose an approach to flexibly switch the OAM state of the vortex beams generated by the coherent combination of fiber lasers. In the coherent laser array system, we enhance the phase control section by employing the mode sorting based optical vortex fields detection, which magnifies the difference between the OAM states to solve the serious phase-locking difficulty in structured light tailoring. During the closed-loop control process, the wavefront distortion compensation and OAM state switching are realized by performing the optimization algorithm (e.g. SPGD algorithm) and modulating the cost function (enabled by modulating the weight vector). Detailed Monte-Carlo simulations, analysis of OAM spectra, and illustration of the OAM state switching process have demonstrated the feasibility of the proposed approach, which is an important step towards making the high-power, rapidly switchable OAM beams become a practical reality.

## Funding

National Natural Science Foundation of China (62075242); Natural Science Foundation of Hunan Province (2019JJ10005); Hunan Provincial Innovation Construct Project (2019RS3017).

## Acknowledgment

The authors appreciate Xiaoming Xi and Wei Liu for valuable discussions. Portions of this work were presented at Asia Communications and Photonics Conference in 2020, T3A. 4.

## Disclosures

The authors declare no conflicts of interest.

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