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Abstract
The synthesis of laser coherence and the accuracy of beam scanning, which are based on an optical phased array (OPA), are severely constrained by phase noise. This limitation hampers their applications in various fields. Currently, the most widely utilized calibration method is adaptive optics, which can effectively mitigate phase noise and enhance the quality of the output beam. However, because of the multiple array elements of the OPA and the large optimization range for each element, the adaptive optimization method experiences slow convergence and a high risk of falling into local optima. We propose a narrowing search range algorithm that can quickly reduce phase noise by narrowing the search range of the optimal value. After initial optimization, the SPGD algorithm was used. This study was verified through simulations and experiments utilizing the OPA of various array elements. These findings indicate that the hybrid algorithm expedites the calibration process, requires simple experimental equipment, and can be broadly utilized.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The OPA concept was first proposed 30 years ago [1]. It is a solid-state beam-scanning technique that enables chip-level beam shaping and contains no mechanical components. The phase difference between the antennas was electrically altered to achieve beam scanning, resulting in flexible high-resolution beam pointing with a broad steering range [2–6]. Furthermore, upon integration with COMS circuits, the entire system is reduced in size and power consumption, while allowing for fast and precise beam scanning to facilitate environmental sensing and information transmission. This technology has crucial research and application value in wireless laser communications [7–9], projection displays [10–12], and LiDAR [9,13,14].
Phased-array devices are sensitive to phase and external factors, such as the operating environment, which can result in phase noise, negatively impacting the quality of the synthesized beam and reducing the beam scanning accuracy. To compensate for this, scholars have proposed several techniques and algorithms, which can be categorized into four main groups. The first category encompassed the stripe interference extraction methods [15,16]. The method optimizes to an optimal value before interfering with the adjacent channel light. This process enables the observation of interference fringes generated by light emitted from two adjacent antennas in the far field. Subsequently, the phases of the two adjacent antennas were adjusted until the interference fringes generated by the light emitted from them aligned with the optimal emitting direction of the phased-array antenna arrays. However, this method requires a high demand for experimental conditions coupled with complicated and intricate steps. The second category includes the neural network prediction method [17] and deep learning [18], which use neural networks to establish a reverse relationship between the angular distribution of the outgoing beam and the phase distribution of the N array elements. However, this approach requires a large number of samples and prolonged development periods. The third type is an AO system technology without wavefront detection. Its optimization algorithms include SPGD [19–21], genetic algorithms [22], and algorithmic-level enhancements [23–26]. These algorithms directly iterate and search for the optimal value over a large range; however, their common drawback is that they require a large number of iterations and voltage transmissions, resulting in long time consumption and difficulty in meeting practical applications.
The fourth category is the recently proposed point-by-point optimization method [27]. This method uses a two-stage search approach to achieve a global search by traversing all array elements phase-by-phase. Consequently, the evaluation function approximated the ideal upper limit and accurately corrected the phase. However, the voltage required in the second stage relies on the step size set in the first stage. Therefore, the optimization time is directly influenced by this process. The deterministic method and SPGD method proposed in 2018 [28] are similar to the point-by-point optimisation method, both can pick the optimal value step by step by applying voltages in a single channel, and when the optimisation of the last channel is completed, the global optimal value can be taken. However, experiments require mechanical structures to block the channel, which imposes significant demands on the experimental conditions and is impractical for engineering applications.
This study examines how the optimal search range affects the algorithm's convergence speed, following the principle of optics, and proposes an algorithm for narrowing the search range based on phase-change characteristics. This algorithm can rapidly minimize the noise range and optimize the light spots to a superior level. Subsequently, using the SPGD algorithm, the noise is optimized to its optimal value after reducing the search range. According to the literature [29], the initial stage of SPGD exhibits a slow convergence speed. However, employing a reduced search-range algorithm can accelerate the convergence at this stage. The hybrid algorithm required a much lower number of iterations than the traditional SPGD algorithm. Compared to the point-by-point optimization method, there is no need to set a small voltage step, which is more convenient for experimental operations and reduces errors. It is more suitable for practical applications that do not require complicated experimental facilities or other data preparation.
2. Principle of optical phased array
According to the OPA principle, a beam of light is divided into multiple optical signals using an optical beam splitter. When each optical signal was given a uniform phase difference, the isophase plane ceased to be perpendicular to the waveguide direction and was deflected to a certain degree. Beams fulfilling the isophase correlation are coherent and elongated, and their directions are consistently orthogonal to the isophase plane. Hence, introducing a phase difference into the array element can deflect the beams. The structure of the OPA is shown in Fig. 1.
The Fraunhofer diffraction principle can mathematically model the one-dimensional OPA, and the far-field intensity distribution function E(θ) of the OPA can be formulated as follows:
3. Algorithm improvement
3.1 Algorithm for narrowing search range
Based on the characteristics of light propagation, a 2π phase represents a cycle. Therefore, the maximum phase noise range due to production errors is between [-2π, 2π]. The light field is most optimal when the change in phase resulting from applied voltage can compensate for the noise or is an integer multiple of 2π. This condition maximized the evaluation function. As shown in Fig. 2, the optimization range can be categorized into two main regions: a, d, e, h in the range of [0, π/2] and b, f, c, g in the range of [π/2, π]. Overall, the algorithm searches for the optimum value in the range of [-π, π]. Traditional optimization methods such as SPGD and particle swarm also search for the optimal value in this range.
The search range of the optimal solution has a significant influence on the convergence speed of the algorithm, which was simulated using the OPA theory described in Section 2 for the 16-dimensional OPA, setting the wavelength to 1550 nm and the spacing of the array elements to half a wavelength. The evaluation function used for the SPGD algorithm was the diffraction efficiency, which is the ratio of the main flap intensity to the total far-field intensity. The trend of the change in the diffraction efficiency is known to be identical to that of the side lobe suppression ratio(SLSR) obtained through the point-by-point optimization method [27]. Set the algorithm to stop iterating when it converges to 95% of its theoretical value, the number of required iterations when the noise range is [-π, π] is 174, whereas only 51 iterations are needed when the noise range is decreased to [-π/2, π/2]. We simulated other noise ranges and obtained the trend of the number of iterations and diffraction efficiency for correcting the initial phase using the SPGD algorithm as the noise range decreased. The data are averaged from 100 simulations. From Fig. 3, it shows that there is an exponential decrease in the number of iterations, while the diffraction efficiency increases exponentially with a reduction in the noise range.
Based on simulation data, it is determined that the optimized region's evaluation function undergoes a sinusoidal-like cycle, similar to the diffraction efficiency in Fig. 4, when a voltage capable of producing a phase transition from 0 to 2π is applied to one of the phased array channels. In addition, the starting point of the image changes with the overall noise of the chip.
Fig. 4. Changes in diffraction efficiency resulting from a single-channel crossing of the [0,2π] phase transition.
As shown in Fig. 4, if you set the step size to δ when optimizing a single channel, the distance between the optimal point and the selected point falls within the range [-δ/2, δ/2]. However, if only one channel is simultaneously optimized, it is necessary to mask the other channels using a mechanical mechanism or an optical switch. To achieve practical engineering applications and universality, we propose an algorithm for narrowing the search range (ANSR). First, by setting several parameters, a voltage of 2π phase shift can be generated: V2π, and the functional relationship between single channel phase shift and diffraction efficiency: f. The optimization strategy of ANSR is to use step size for the first channel δ. Apply voltage n times, nδ =V2π, which means uniformly taking n sampling points within the voltage range of [0, V2π]. The voltage corresponding to the largest evaluation function among the sampling points is updated to the voltage of the first channel. When the second channel is optimized, the first channel takes the voltage that has been updated in the previous step, and the voltages of all channels that have not been optimized after it are still taken as zero. The optimization step was repeated to obtain the voltage of the second channel. Subsequent channels are also optimized in this manner, with all channels in front of the optimized point applying the voltage value obtained after the update, and all channels in the rear set to 0, updating the optimal voltage required at the current position until all channels have been optimized. Schematic diagrams of the individual channels and the overall optimization method are shown in Figs. 5(a) and 5(b), respectively.
Fig. 5. Flowchart of ANSR algorithm. (a) Schematic of individual channel optimization. (b) Schematic of overall optimization.
The ANSR optimization strategy does not require occlusions or other structures. However, there is a disadvantage in that optimizing a single channel is based on the presence of original noise in the channel behind it. Therefore, when the voltage of the rear channel changes, the optimal value of the front channel also changes.
According to Fig. 6, there is a correlation between the number of sampling points selected within [0, 2π] and the optimized diffraction efficiency. Even if the most suitable number of sampling points is selected, the optimization remains below the diffraction limit. However, the evaluation function can quickly reach a high level of optimization when the number of sampling points is within ten, resulting in a smaller noise range. Using this optimal result as the optimization starting point in the SPGD algorithm can significantly reduce the number of subsequent iterations and the likelihood of encountering local optima. Thus, the scientific validity of the algorithm for narrowing the search range was confirmed.
3.2 Hybrid algorithm
The SPGD algorithm is a blind optimization technique for addressing multidimensional unconstrained optimization problems. They are frequently employed in adaptive optics systems to regulate wavefront correctors and counteract beam–wavefront aberrations. Using the traditional SPGD algorithm, the procedure for compensating phase noise in an OPA involves controlling each circuit voltage through the following process. First, the performance evaluation function of the system is established as the optimization objective of the algorithm, which is a function of the control signal of the OPA phase shifter. Random perturbation voltages δu, in accordance with the Bernoulli distribution, are applied to each channel of the phase shifter via the driving circuit. The infrared camera calculates the changes in the system performance evaluation function.
A mathematical derivation results in [32]:
The gradient of each channel can be expressed as the product of the change in the performance evaluation function and perturbation of the control signal. The iterative formulation of the SPGD algorithm can be expressed as
The diffraction efficiency serves as the evaluation metric, and Fig. 7. illustrates the step-by-step process clearly and concisely. Based on the above analysis, If the noise range is initially decreased and subsequently optimized using the SPGD algorithm, there can be a considerable reduction in the number of iterations required. So design a hybrid algorithm that takes the final voltage value obtained by the ANSR algorithm as the initial voltage of the SPGD algorithm, and continues to compensate for phase noise. Figure 8 shows the process of the hybrid algorithm.
4. Numerical simulation
The optimization of a single channel requires transmitting the voltage and acquiring the image M times when the number of sampling points is set to M within [0, 2π], and M is an integer ranging from 3 to 10. This completes the evaluation function for single channel acquisition and updates the evaluation function once. The SPGD algorithm sends the voltage and collects images three times in one iteration. We interpolated the convergence curve of the evaluation function obtained by the ANSR to m times its length as the number of iterations that can be compared with the SPGD algorithm, where m is:
The sending voltage and collecting images occupy most of the algorithm's running time, setting the number of iterations of the ANSR using the above method can make the single-iteration time and complexity of the two algorithms approximately the same. Thus, we believe that by doing so, the ANSR can compare the number of iterations with the SPGD algorithm.
Simulations are conducted for each dimension of the OPA chip using the aforementioned theory. The maximum range of the simulated noise was tested, and two experiments were conducted for the same noise level. The first experiment utilized a hybrid algorithm, whereas the second used the conventional SPGD algorithm. Both groups were optimized to 95% of theoretical limits. The gain coefficients of the second part of the hybrid algorithm and the SPGD algorithm are both fixed and the most suitable. The target value (dashed line) in Fig. 9 refers to 95% of the theoretical limit, and the vertical axis represents the diffraction efficiency. For comparison, we considered the number of iterations required for the hybrid and SPGD algorithms. Owing to the instability of the SPGD algorithm, we performed the experiment 100 times and averaged the results. When M is taken as 8, the ratio of the number of iterations required by the hybrid algorithm to the number of iterations required by the SPGD algorithm was set as R, and the corresponding value of R for the N-dimensional OPA was set as RN. The simulation results show that R16 = 62.49%, R32 = 55.14%, R64 = 50.21%, and R128 = 44.77%. As the number of OPA elements increases, the advantages of the hybrid algorithm become more evident, making it more suitable for practical engineering applications. Figure 9 shows a comparative plot of the convergence curves of the two algorithms in different dimensions, with the set of data closest to the average selected from the 100 experiments.
Fig. 9. Convergence curves for the hybrid and SPGD algorithms: (a) 1*16, (b) 1*32, (c) 1*64, (d) 1*128. Target value is 95% of theoretical limits.
5. Experimentation
We constructed a system to evaluate the performance of the proposed hybrid algorithm. Figure 10(a) shows a schematic of the device connection and signal transmission, and Fig. 10(b) shows the experimental object.
The 1550 nm laser's optical signal output couples through a single-mode fiber and a polarization controller onto the grating coupler of the device. Our detector measured the far-field pattern at a distance of 8 cm from the optical system to prevent long-distance interference. During calibration, the infrared camera continuously samples the light intensity within the observation range to obtain the far-field pattern and transmits it to a computer. The computer controlled the multichannel voltage source through UDP communication, providing driving power to the electrodes on the chip via the connection line. The experiments used OPA chips of 1 × 16 and 1 × 32. The phase shifter required 20 mW of power to achieve a 2π phase transition. The infrared camera captured pictures with a size of 256 × 320 pixels, and the electronic control unit had a voltage range of [0, 14]V.
The study was conducted in two groups, and each chip was calibrated using both the hybrid algorithm and the SPGD algorithm. Each calibration was repeated ten times to ensure consistency and accuracy. The hybrid algorithm uses eight sampling points. Due to factors such as chip design and experimental environment, the chip used in this experiment cannot reach the theoretical data limit. The method adopted in the experiment is to first use the SPGD algorithm to conduct more than 300 experiments on each chip, continuously adjust the parameters of the SPGD algorithm, and obtain the maximum side lobe suppression ratio that the current chip can achieve. Both sets of experiments were optimized to 95% of the above suppression ratios before stopping the iteration.
As an example, part (I)(II)(III) of Fig. 11(a) and (d) illustrate the optimization results of the hybrid algorithm for the 16-dimensional and 32-dimensional OPA, presenting the far-field spot maps at the beginning, stage 1(ANSR), and at the completion of the optimization. The values were extracted, and from Fig. 11, it is evident that the hybrid algorithm significantly improved the spot quality in the first stage. Part (I) (IV) of Fig. 11(a) and (d) illustrate the start and optimization completion of the SPGD algorithm for the 16-dimensional and 32-dimensional OPA. The (b) and (e) parts in Fig. 11 are the light intensity values of the optimization result maps extracted for uniform normalization, from which it can be seen that the ANSR algorithm provides a superior initial state for the subsequent SPGD optimization. The (c) and (f) parts in Fig. 11 are obtained by sequentially arranging the side lobe suppression ratio calculated from each image during the algorithm iteration process. It can be clearly seen that the hybrid algorithm accelerates the convergence process. The number of iterations for the ANSR algorithm is calculated using Equ. 5. The experimental group under SPGD control exhibited slower convergence during the initial period.
Fig. 11. Effect of staged optimization and spgd algorithm in different dimensions. (a)(d): calibration process of 16-dimensional and 32-dimensional opa (i) uncalibrated, (ii) after optimization by narrowing down search range, (iii) after continuing optimization with SPGD algorithm, (IV) optimization results using only the SPGD algorithm; (b)(e): Normalized light intensity values were optimized using a hybrid algorithm in two stages and SPGD algorithm; (c)(f): Convergence of the hybrid and SPGD algorithms with the number of iterations in the experiments.
The data for the specific results are shown in Table 1. From the experimental data, we can obtain the time required for 16 dimensions of SPGD algorithm is 1.71 times that of the hybrid algorithm, and for 32 dimensions, it is 1.92 times that of the hybrid algorithm. These findings confirm that the hybrid algorithm is more efficient and suitable for large-scale OPA.
Table 1. Experimental data of hybrid algorithm and SPGD algorithm at each stage
6. Error analysis
Applying a voltage that produces a phase change from 0 to 2π to one of the OPA channels results in a sinusoidal-like curve in the evaluation function of the optimized region over one cycle. Owing to the noise of the other channels, the range of intervals in the cycle for increasing and decreasing intervals are not the same. However, because we uniformly pass through several voltages in [0, 2π], the algorithm's optimization effect remains unaffected.
The power necessary for the phase shifter to achieve the 2π phase transition is fixed. However, because of the existing process limitations, not every phase shifter shares the same resistance value. Consequently, the voltage required for each phase shifter to attain the 2π phase transition differs and fluctuates around the calibration value. If the global optimum of the evaluation function falls outside the range of our sampling points, the reduced search range algorithm narrows the range below the theoretical value. Nonetheless, the probability of fulfilling this requirement was minimal. We employed identical running times and algorithm complexities for a single iteration of both the reduced search range algorithm and the SPGD algorithm during the simulation in Section 4. However, the saved time proportion of the hybrid algorithm relative to the tested SPGD algorithm in the experiments in Section 5 differed from the saved iteration proportion. On one hand, it comes from experimental conditions such as the instability of the laser and inaccurate voltage transmission values, and on the other hand, it comes from the different resistance values of each phase shifter.
7. Conclusion
In this study, we propose a hybrid calibration algorithm that combines the ANSR and SPGD algorithms to address the issue of phase noise in OPA chips. This algorithm offers several advantages over existing optimization algorithms, including fast convergence, simple experimental requirements, and universality. Our results demonstrate that the algorithm efficiently compensates for the phase noise in OPA chips. The convergence time ratio between the hybrid and SPGD algorithms was experimentally obtained, and the iteration number ratio was obtained through simulation. For the 16-dimensional OPA, the iteration times and convergence time of the hybrid algorithm were 62.49% and 58% of those of the SPGD algorithm, respectively; for the 32-dimensional OPA, the iteration times and convergence time of the hybrid algorithm were 55.14% and 52% of those of the SPGD algorithm, respectively; For 128-dimensional OPA, the number of iterations of the hybrid algorithm was 44.77% of that of the SPGD algorithm. It can be concluded that, as the number of array elements increases, the advantages of the hybrid algorithm become more apparent. The hybrid optimization algorithm in this study provides a reference for the initial phase calibration of the OPA, which is in line with practical engineering applications.
Funding
State Key Laboratory of Laser Interaction with Matter (SKLLIM2104); Jilin Province Development and Reform Commission (2022C046-2).
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.
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