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Abstract
Based on Dammann vortex grating and adaptive gain stochastic parallel gradient descent algorithm, we theoretically proposed a phase control technology scheme of the coherent beam combining system for generating perfect vectorial vortex beams (VVBs). The simulated results demonstrate that the discrete phase locking for different types of VVBs (including vortex beams, vector beams, and generalized VVBs) can be successfully realized. The intensity distributions, polarization orientation, Pancharatnam phases, and beam widths of different |Hm,n〉 states with the obtained discrete phase distribution further prove that the generated beams are perfect VVBs. Subsequently, the phase aberration residual for different VVBs is evaluated using the normalized phase cosine distance function, and their values range from 0.01 to 0.08, which indicates the obtained discrete phase distribution is close to the ideal phase distribution. In addition, benefitting from the high bandwidth of involved devices in the proposed scheme, the influence of dynamic phase noise can be negligible. The proposed method could be beneficial to realize and switch flexible perfect VVBs in further applications.
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1. Introduction
Vectorial vortex beams (VVBs) are a special type of structured beams with both helical phase and spatially non-uniform polarization distributions [1–3], which possess simultaneously the spin angular momentum and orbit angular momentum (OAM), and support more encode channels. Perfect VVBs independent of the polarization orders and topological Pancharatnam charges are perfect candidates for spin-orbit interaction [4], free space communication [5], quantum cryptography [6], and optical storage [7]. Therefore, various methods for generating perfect VVBs by employing the spatial light modulators (SLMs) or metasurfaces have been proposed successively [8–14]. Unfortunately, lower damage thresholds and modulation speed of these devices have difficulty in achieving high power and switchable perfect VVBs.
As a widely studied technology, coherent beam combining (CBC) provides a promising solution for greatly scaling up the output power and maintaining good beam quality [15–17]. More importantly, taking the advantage of flexible and independent regulation of amplitude, phase, and polarization of each beamlet in the laser array, the theoretical or experimental studies for vortex beams [18–22], vector beams [23,24], and Bessel-Gaussian beam [25], have been carried out. In our previous theoretical work, we proposed an approach to generate perfect VVBs based on CBC with a specially designed radial phase-locked Gaussian laser array [26].
To achieve the desired structured beams based on the CBC, the prerequisite is that the discrete phases of laser array should be arranged and locked in a specific distribution. In addressing this challenge, several approaches have been proposed [27–29]. In 2009, Hou et al. proposed that phase information can be extracted for the intensity distributions at the non-focal plane, avoiding the problem of non-uniqueness of phase information at the focal plane. With an optimization algorithm, the desired vortex beams can be generated based on the CBC [27]. Based on the heterodyne method, the phase of each beamlet can be measured and modulated, and the generation of vortex beams with topological charges ranging from 0 to 5 has been realized experimentally [28]. Recently, utilizing the unique advantages of deep learning (DL) in big data and image processing, the phase control method based on DL has been extensively investigated. In 2021, Tünnermann proposed a deep reinforcement learning model to achieve the discrete vortex phase locking in a CBC system by choosing an appropriate reward function [29]. Hou et al. developed a DL-assisted, two-stage phase control method for generating the mode-programmable vortex beams. First, a DL network is trained by a large number of intensity images on the non-focal plane or diffracted by a designed Dammann vortex grating (DVG) [30,31], the phase errors are preliminarily compensated by the trained DL network. Subsequently, the phase errors are further eliminated by an optimization algorithm, thereby achieving precise phase control of the vortex beams. However, these reported methods mostly are focused on the generation of vortex beams, the phase control methods for other types of structured beams however have received less attention.
In this work, based on the DVG and adaptive gain stochastic parallel gradient descent (SPGD) algorithm, a phase control technology scheme for generating perfect VVBs in the CBC system is proposed. For different types of VVBs (including vortex beams, vector beams, and generalized VVBs), the corresponding discrete phase distributions are optimized. With the obtained discrete phase distributions, we verify whether the generated beams are perfect VVBs or not by analyzing different parameters, such as intensity distributions, polarization orientation, Pancharatnam phases, and beam widths. In addition, the phase aberration residuals for different perfect VVBs are evaluated. Finally, based on the proposed method and devices involved, the influence of dynamic phase noise on the optimization results is discussed preliminarily. The developed phase control scheme provides a potential solution for generating high power perfect VVBs in the CBC system, making the application of high power perfect VVBs possible.
2. Principle and method
DVG is a special type of Dammann grating that combines Dammann grating with a helical phase structure, which can be used to generate vortex arrays or detect vortex beams. Its mathematical expression can be written as [32,33]:
Fig. 1. (a) The phase distribution, (b) phase distribution at the central region, (c) far field diffraction distribution, and (d) corresponding positions of vortex beams with different topological charges of designed DVG of a 5 × 5 array
Figure 2 depicts the diffraction distributions of single vortex beams or composite vortex beams with different topological charges passing through a DVG, and the insets show the shape of beam spots at the corresponding diffraction order. Obviously, it can be seen that for a single vortex beam, different topological charges correspond to different positions. While for a composite vortex beam, it is considered that two vortex beams with different topological charges after interacting with a DVG, respectively, producing two Gaussian shaped beam spots at their corresponding positions [34].
Fig. 2. The diffraction distributions of single vortex or composite vortex beams with different topological charges passing through a DVG. (a) l = −2, (b) l = 2, (c) l1= −2, l2= 2, (d) l1= −1, l2= 2. The insets depict the shape of beam spots at the corresponding diffraction order.
In previous research [26], by employing a specially designed radial phase-locked Gaussian laser array composed of two discrete vortex arrays with right-handed (RH) and left-handed (LH) circularly polarized states in turn, perfect VVBs based on CBC can be obtained. The principle underlying this research is that VVBs can be considered the superposition of RH and LH circularly polarized vortices, written as a Jones matrix in the form of [35]
The RH and LH circularly polarized vortex beams are converted into linearly polarized vortex beams with polarization directions of x and y, respectively.
Based on the above discussion and in combination with the characteristics of DVG, we proposed a phase control method of the CBC system for generating perfect VVBs, as schematically shown in Fig. 3. The laser is split into multiple channels by a splitter. In each channel, the power is amplified by a main amplifier, and the output phase is modulated by an electro-optic phase modulator. The output laser is emitted to free space from a special collimation array, which is composed of two Gaussian laser arrays with RH and LH circularly polarized states in turn. After passing through the spectroscope, the most of emitted laser array is reflected into a dynamometer, and a small portion of the transmitted beam, below the damage threshold of the SLM or DVG mask, will be used for phase control. Then, a quarter wave plate is utilized to convert RH and LH circularly polarized beams into linearly polarized beams in the x and y polarization directions, subsequently which are separated after passing through a polarization beam splitter. By using a half wave plate, the polarization direction of the reflected beam is adjusted to coincide with the polarization direction of the transmitted beam. Then two beams can coaxially transmit after passing through a beam splitter. A 4f imaging system with an adjustable diameter circular aperture pinhole is used to filter unwanted sidelobes. The two beams are then incident on an SLM or a DVG mask loaded with the phase of the designed DVG, which is on the front focal plane of the lens. Finally, a photodetector (PD) array in the back focal plane of the lens is used to receive the diffraction beam intensity. The size of the PD array is designed to correspond to the diffraction order of the DVG. In addition, a pinhole is placed before each PD to receive only the beam intensity in the central region. The detailed description of the simulation is given in Supplement 1, S1.
Fig. 3. Schematic illustration of phase control of CBC system to tailor perfect VVBs with a designed DVG. QWP: quarter wave plate, M: mirror, HWP: half wave plate, PBS: polarization beam splitter, BS: beam splitter, SLM: spatial light modulator.
According to Eq. (2), vortex beams and vector beams are two special types of VVBs. The vortex beam is composed of RH and LH circularly polarized vortex beams with the same topological charges, the vector beam is composed of RH and LH circularly polarized vortex beams with the opposite topological charges, and the generalized VVBs are decomposed into RH and LH circularly polarized vortex beams with the different topological charges. Therefore, using our proposed approach and based on Fig. 2, after diffraction of the designed DVG, the vortex beams will form a Gaussian shaped spot at the corresponding position, while the vector beams will generate double positional symmetric Gaussian shaped spots, and the VVBs will produce two Gauss shaped spots at the asymmetric position. Therefore, on the one hand, classification of different types of structured light fields can be achieved based on the proposed method. On the other hand, based on the intensity signals detected by the PD array, the phase controller continuously and independently adjusts the phase modulator in each laser module using an optimization algorithm, thereby regulating the output phase of each laser module. Through continuous feedback iterations, phase locking of the desired VVBs can be achieved.
3. Results and discussions
We choose the adaptive gain SPGD algorithm as the optimization algorithm [36], and the cost function is defined as the total intensity in the target area
In the numerical simulation, the wavelength, waist width, and aperture diameter of each laser array element are set to be λ = 1064 nm, ω0 = ρ0= 0.35 mm, the radius of collimation array, the number of beamlets, and the focal length of lens are set as R1 = 3.5 mm, M = 48, and f = 500 mm. To obtain sufficient statistics for evaluating optimization effects, 20 realizations for each type of VVBs have been independently simulated. Meanwhile, all initial discrete phase and the phase perturbation amplitude are set as 0 and 0.2, respectively. We choose the maximum detected intensity by PD and use it to normalize the cost function.
For the convenience of expression, the unified representation of the VVBs is |Hm,n〉 state (where m and n are non-zero integers). Clearly, the case of m = n represents the vortex beams with the topological charge of m, m = -n accounts for the vector beams with polarization order of |m|, and |m| ≠ | n | stands for the generalized VVBs.
To quantitatively characterize the changing trend of the purity of vortex mode during the iteration process, we adopt the concept of OAM spectrum, which is formed by the normalized energy weight of OAM mode Pl. Normalized energy weight Pl can be represented as [37]
Firstly, taking |H2,2〉 (namely, the vortex beam with the topological charge of 2) as the optimized target, the distribution of discrete phases using the adaptive gain SPGD algorithm is optimized, and the results are shown in Fig. 4. Due to the randomness of phase perturbations, each iteration is different, but its overall change trend is consistent. That is, at the beginning of each iteration, the cost function changes slowly; when the number of iterations reaches a certain number, the cost function increases rapidly; when the number of iterations exceeds 500, the cost function hardly changes significantly. Figure 4(b)-(e) and (f)-(i) show the diffraction distributions and normalized OAM spectrum for iterations of 1, 150, 300, and 600, respectively. By comparison, it is found that at the beginning of the iteration, due to the phase distribution being almost zero, the Gaussian spot is located at the 0-order diffraction position, which results in the OAM spectrum being almost concentrated at the 0-order diffraction; When the number of iterations increases to 150, part of the energy has been transferred to the 2-order diffraction, and the corresponding OAM spectral intensity is 0.4611; When the iterations number increases to 300, Gaussian spot also appears at the 2-order diffraction position, and the corresponding OAM spectral intensity increases to 0.8529; When the iterations number is 600, the diffraction distribution is basically consistent with that in Fig. 2(a), and the OAM spectral intensity increases to 0.9898, which means that the discrete phase locking for vortex beam with the topological charge of 2 has been achieved through algorithm iteration.
Fig. 4. Optimized results for desired |H2,2〉 state: (a) The convergence curves of the cost function with the number of iterations, (b)–(e) and (f)–(i) are the diffraction distributions and normalized OAM spectrum with iterations of 1, 150, 300 and 600, respectively. The insets show the shape of beam spots at the corresponding diffraction order.
Similarly, we choose the |H−2,−2〉 (namely, the vortex beam with the topological charge of −2) state as the optimization target, the optimized results are shown in Fig. 5. The convergence curves change trend of the cost function are consistent with that of |H2,2〉 state. With the increase of iterations number, the intensity of the -2-order diffraction position gradually increases from zero to a Gaussian spot, and the OAM spectral intensity also increases from 0 to 0.9914, which has also been proven that the discrete phase locking for vortex beam with the topological charge of −2 has been successfully achieved.
Fig. 5. Optimized results for desired |H−2,−2〉 state: (a) The convergence curves of the cost function with the number of iterations, (b)–(e) and (f)–(i) are the diffraction distributions and normalized OAM spectrum with iterations of 1, 150, 300 and 600, respectively. The insets show the shape of beam spots at the corresponding diffraction order.
According to the principle of beam splitting and combing depicted in Fig. 3, if the incident beam is a vector beam, two Gaussian spots will be generated at the symmetrical positions about 0-order diffraction (as shown in Fig. 2(c)). Here, taking |H2,−2〉 (corresponding to the vector beam with polarization order of 2) state as the optimization target, the cost function is defined as the summation of the intensity at the -2-order and 2-order diffraction. Figure 6 gives the optimized results for |H2,−2〉 state. Clearly, the convergence speed is similar to that of a vortex beam. At the beginning of the iteration, there is almost no intensity in the central region of the - 2 and 2-order diffraction, and the corresponding OAM spectral intensity is also about 0; When the iterations number increases to 150, a small amount of energy appears in the central region of the - 2 and 2-order diffraction, but the distribution is relatively dispersed, with corresponding OAM spectral intensities of 0.0331 and 0.0351, respectively; When the iterations number is 300, the energy at the −2 and 2-order diffraction gradually concentrates towards the central region, and the OAM spectral intensity also increases to 0.3520 and 0.3894, respectively; When the iterations number reaches 600, the energy at the −2 and 2-order diffraction points is almost concentrated in the central region, and the OAM spectral intensities also reache 0.4996 and 0.4868, which is very close to the ideal case of 0.5 and 0.5. This indicates we have also successfully realized phase locking for the desired vector beams with the proposed method.
Fig. 6. Optimized results for desired |H2,−2〉 state: (a) The convergence curves of the cost function with the number of iterations, (b)–(e) and (f)–(i) are the diffraction distributions and normalized OAM spectrum with iterations of 1, 150, 300 and 600, respectively. The insets show the shape of beam spots at the corresponding diffraction order.
Similarly, for generalized VVBs, after DVG diffraction, there are also two Gaussian spots, which only appear at the two asymmetric positions with respect to 0-order diffraction, as shown in Fig. 2(d). Figure 7 displays the optimized results for the |H−1,2〉 state, and the cost function is defined as the summation of the intensity at the -1-order and 2-order diffraction. The convergence curves of the cost function, diffraction distribution, and normalized OAM spectrum are consistent with those of vector beams. Finally, when the iterations number is 600, the OAM spectral intensities of the −1 and 2-order are 0.5214 and 0.4714, respectively, which are close to ideal cases of 0.5 and 0.5. Therefore, using the proposed method, the discrete phase locking for desired generalized VVBs can also be achieved.
Fig. 7. Optimized results for desired |H−1,2〉 state: (a) The convergence curves of the cost function with the number of iterations, (b)–(e) and (f)–(i) are the diffraction distributions and normalized OAM spectrum with iterations of 1, 150, 300 and 600, respectively. The insets show the shape of beam spots at the corresponding diffraction order.
In addition, the discrete phase locking for desired higher order generalized VVBs (such as |H−2,4〉 state) is also numerically investigated, as shown in Fig. 8. For higher order vortices, to obtain more phase information, the circular aperture pinhole of the 4f imaging system is appropriately expanded (ω0 is set as 0.7 mm), so that it can be seen that the intensity distribution area on each diffraction order is also appropriately expanded, but this has no impact on the convergence curve. After 600 iterations, the OAM spectral intensities of the −2 and 4-order are 0.4345 and 0.4379, respectively, which are slightly lower than the |H−1,2〉 state, but the convergence effect is still acceptable. Furthermore, in order to verify the universality of the proposed method, more examples of higher order VVBs are given in Supplement 1, S2.
Fig. 8. Optimized results for desired |H−2,4〉 state: (a) The convergence curves of the cost function with the number of iterations, (b)–(e) and (f)–(i) are the diffraction distributions and normalized OAM spectrum with iterations of 1, 150, 300 and 600, respectively. The insets show the shape of beam spots at the corresponding diffraction order.
Based on the discrete phase information obtained from the optimization algorithm, the intensity distribution, polarization orientation, and Pancharatnam phases of |H2,2〉, |H−2,−2〉, |H−2,2〉, |H−1,2〉 and |H−2,4〉 states are numerically simulated [35,38], as shown in Fig. 9. According to the definition of VVBs, the polarization order and topological Pancharatnam charge are given as P = (m - n)/2 and l0 = (m + n)/2. For different |Hm,n〉 states, the polarization orientations separately rotate 0, 0, −4π, −3π, and −6π in a full circle, which correspond to the polarization order P of 0, 0, 1, −2, −1.5, and −3. While the Pancharatnam phases spirally vary 4π, −4π, 0, π, and 2π, which means that the topological Pancharatnam charges l0 are exactly equal to 2, −2, 0, 0.5, and 1. In addition, according to Eq. (12) of Ref. [39], the beam width ω of different |Hm,n〉 states are calculated and labeled in Fig. 9. Obviously, the beam width for different |Hm,n〉 states is basically the same, with the maximum variation not exceeding 1.1%, which indicates the beam width is independent of polarization order and topological Pancharatnam charge of VVBs, namely, the generated VVBs is perfect.
Fig. 9. Intensity distribution, polarization orientation, and Pancharatnam phase of different |Hm,n〉 states according to discrete phase distribution obtained from the proposed method. (a) m = n = 2; (b) m = n = −2; (c) -m = n = 2; (d) m = −1, n = 2; (e) m = −2, n = 4.
However, compared to the Ref. [26], the intensity distribution, polarization distribution, and phase distribution of the generated VVBs are not so perfect, because there is still a certain deviation between the discrete phase distribution obtained from the proposed method and the ideal phase distribution. Here, we adopted the Normalized Phase Cosine Distance (NPCD) proposed by Zuo et al. to evaluate the phase aberration residual, and the NPCD is defined as [40]
Table 1. NPCD for different |Hm,n〉 states at initial iteration and 1000 iterations
In practical application, the changes of refractive index induced by thermal effects in the optical fiber amplifiers, mechanical vibration of the system, and the surrounding environment both lead to the dynamical phase noise, which result in affecting the convergence speed and effectiveness. Benefiting from the high rate of electro-optic phase modulators, phase controller and high bandwidth of PD, the frequency of a single iteration approximates 100 megahertz [31,41]. For the proposed optical system, it takes 500 iterations to achieve a good convergence effect, so the bandwidth of the SPGD algorithm is about 200 kHz. According to the current experimental measurement results of optical fiber amplifier systems, the frequency of dynamical phase noise of fiber amplifiers is approximately between 10 Hz and 10 kHz [42–44]. When the bandwidth of the SPGD algorithm is far greater than the frequency of the dynamic phase noise, that is, the optimization of the SPGD algorithm has been completed, and the dynamic phase noise has not yet changed. This means that the dynamic phase noise is only equivalent to an initial phase noise and does not affect the optimization results of the SPGD algorithm.
4. Conclusions
In conclusion, based on the DVG and adaptive gain SPGD algorithm, the phase control technology scheme of the CBC system for generating perfect VVBs is proposed. By analyzing the convergence curves, diffraction distributions, and normalized OAM spectrum, the discrete phase locking of different types of VVBs (including vortex beams, vector beams, and generalized VVBs) is successfully realized. Based on obtained discrete phase distribution, the intensity distribution, polarization orientation, Pancharatnam phases, and beam widths of different |Hm,n〉 states are numerically simulated, and the results further confirm that the generated beams are perfect VVBs. In addition, the difference between the obtained discrete phase distribution and ideal phase distribution for different |Hm,n〉 states is evaluated using the NPCD. The NPCDs will decrease from 0.5 to between 0.01 and 0.08, which means the obtained discrete phase distribution is close to the ideal phase distribution. Finally, the influence of dynamic phase noise on the SPGD algorithm is analyzed. The results show that due to the high bandwidth of the devices used in the proposed phase control scheme, the bandwidth of the SPGD algorithm can approximate 200 kHz, so the impact of dynamic phase noise is negligible to the CBC system. Our work enriches the phase control scheme for generating structured beams in the CBC system and enables further applications of a high power perfect VVBs.
Funding
Youth Innovation Promotion Association XIOPM-CAS (XIOPMQCH2021003); National Natural Science Foundation of China (62005310, 62171443); Key Research and Development Projects of Shaanxi Province (2021GY-298, 2023-ZDLGY-37).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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