Accelerating optical phased array calibration with improved SPGD algorithm

As a new type of beam pointing control technology, optical phased array has broad application prospects in the fields of laser radar and beam synthesis. However, the phase error in practical application brings great difficulties to beam detection and deflection, which must be calibrated. In this paper, a SPGD algorithm combined with Adam algorithm (ASPGD) is proposed and compared with the traditional optimization algorithm. The results show that the improved SPGD algorithm can double the convergence speed and increase the cosine similarity value by about 10%.


Introduction
The optical phased array is an advanced photonic device that employs optical principles and phase control techniques to precisely manipulate and align light beams.It is made up of an optical emission unit, a phase modulator, an optical waveguide network, and drive control circuits, among other integral components.Some implementations include a feedback mechanism for real-time changes that improve light wave qualities.This technology's growing popularity in LiDAR (Light Detection and Ranging) systems is attributed to its innovative beam pointing and control approach, eliminating the need for traditional mechanical rotation in a specific scanning range.Notable features include quick and accurate beam control, low power consumption, a compact form factor, and flexible adjustment.[1][2].
The theoretical derivation stipulates that the phased array's component distance must not be greater than half a wavelength in order to avoid the formation of extra energy that may interfere with the scanning beam.Nevertheless, as the waveguide spacing diminishes, thermal crosstalk between the waveguides intensifies.This, in conjunction with inevitable process errors and the influence of environmental noise, results in phase errors within the optical phased array [3].It has an effect on the propagation of the outgoing beam in the far-field, making exact localization of the main lobe difficult.As a result, optical phased array requires precise calibration before deployment to reduce inherent errors.Various calibration algorithms, such as genetic [4], stochastic parallel gradient descent (SPGD) [5], and improved SPGD with Nesterov acceleration (NSPGD) [6], have been proposed.However, when dealing with large-scale phased array, genetic algorithm exhibits sluggish convergence towards local rather than global optimal solutions.Additionally, the SPGD and NSPGD algorithms prove highly sensitive to the learning rate, with higher rates resulting in convergence failure and lower rates causing excessively slow convergence.
In this paper, we proposed an improved SPGD algorithm combined with adaptive moment estimation algorithm (ASPGD).Adaptive moment estimation algorithm (Adam) is a widely used algorithm in deep learning.We used simulation analysis to demonstrate that based on our ASPGD algorithm, the optimized OCOIP-2023 Journal of Physics: Conference Series 2700 (2024) 012002 IOP Publishing doi:10.1088/1742-6596/2700/1/012002 2 cosine similarity value can be increased by about 10 % and the convergence speed is more than twice as fast as the traditional optimization algorithm.

Ideal far-field distribution of one-dimensional equidistant optical phased array
The amplitude distribution of the optical phased array in the far field may be represented using Fraunhofer diffraction principles and excluding irrelevant imaginary and constant elements that have no bearing on the distribution of light intensity as follows: where d represents the central spacing between adjacent array elements, a signifies the width of the array element, λ corresponds to the wavelength of the light source, and θs denotes the deflection angle.The first term in equation ( 1) is referred to as the array factor U E , which gives rise to the intensity distribution envelope in the far field and is solely dependent on the width of the array element itself.On the other hand, the second term in equation ( 1), known as the array factor U A , affects the coherent characteristics of the far-field light intensity distribution and is influenced by the center-to-center spacing d of the array elements and the deflection angle θs.When the deflection angle is set to 0°, the far-field distribution of an ideal equidistant one-dimensional optical phased array is illustrated in Figure 1.

Phase calibration algorithm.
SPGD, a commonly used method in adaptive optical systems, follows the parameter updating approach described below [7].
Generate an ensemble of random perturbation voltages δu n with a mean of zero and uniform variance.Subsequently, apply a positive δu n and a negative -δu n voltage perturbation to each channel of voltage u n , giving rise to two objective functions J n+ and J n-. Compute the variation in the objective function δJ n and proceed to update the voltage values as per the subsequent equation:

𝛿𝐽
( 2 ) where n represents the iteration number, α denotes the learning rate, and u n+1 is the freshly updated voltage value.The SPGD algorithm approximates the true gradient value using δu n δJ n .The SPGD algorithm is clearly sensitive to being bound within local optima because of the unchangeable learning rate and the approach of only considering the near gradient for updating.Furthermore, the convergence speed will noticeably slow down with the increase of the scale of optical phased array.
Based on the concept of "momentum," the Adam method is a stochastic gradient descent optimization algorithm.Every time before the iterative update, the first-order and second-order moments of the gradient are computed, and the sliding average value is determined to update the current parameters.The advantages of the Adagrad method for handling sparse data and the advantages of the RMSProp algorithm for handling erratic data are combined in this concept.It finds widespread application in neural networks through adaptive modification of the learning rate based on past gradient information.Proposed by Diederik P. Kingma and Jimmy Ba in their 2014 paper [8], the fundamental parameter update process of the Adam algorithm is as follows: where n represents the iteration number,  ^ signifies the first-order moment estimation of the biascorrected gradient,  ^ corresponds to the second-order moment estimation of the bias-corrected gradient,  denotes the gradient value, α is the learning rate, β 1 and β 2 represent the initialized hyperparameters, ε is a very small value initialized to prevent division by zero errors, and u n+1 is the updated parameter.Through the introduction of first-order and second-order moment estimations of the gradient, the Adam algorithm achieves faster convergence compared to traditional gradient descent algorithms.Additionally, it eliminates the need for manual learning rate tuning and provides excellent interpretability in the setup of hyperparameters.Therefore, we combine the Adam algorithm with the conventional SPGD algorithm to yield the ASPGD algorithm, and the combined parameter update methodology is as follows: Our selected objective function J is the cosine similarity, which is calculated as follows: where I n represents the vector of the ideal far-field optical field distribution, while U n symbolizes the vector of the actual calibrated far-field optical field distribution obtained during the calibration process.The closer the value of the objective function J is to one, the smaller the phase error, indicating a more exquisite calibration of the optical phased array's phase.
The flowchart of the ASPGD algorithm is illustrated in Figure 2. At first, some parameters, such as β1, β2, α, n, ε and the first and second-order moment estimations of the gradient (m and v), are initialized.
Leveraging the applied random perturbations δu n , an approximate estimation of the gradient δu n δJn is computed.Subsequently, using these values, the bias-corrected first and second-order moment estimations (m n and v n ) are derived and employed in equation ( 11) to determine the new voltage value and calculate the updated objective function value.Finally, the iteration number n is checked against the predefined maximum iteration count N. If it is reached, the maximum value of the objective function and the corresponding set of voltage values are output; otherwise, n is increased and the next iteration is started.The ASPGD algorithm combined with the Adam algorithm can obviously change the learning rate adaptively and take into account the historical gradient information through the sliding weighted average, allowing the algorithm to converge to the extreme point more quickly and effectively than the conventional SPGD algorithm.

Simulation results
We ran simulations at various deflection angles to validate the calibration performance of the ASPGD algorithm mentioned in the second part.The software used in the simulation is MATLAB.OPA's parameters are set as follows, taking into account the actual chip structure: the operating wavelength is 1550 nm, the number of array elements is 16, the width of an array element is 1 μm, and the spacing between adjacent array elements is 3 μm.The result is shown in Figure 3.As the far-field intensity distribution of the optical phased array is shown in Figure 3, noise has a significant impact on the distribution prior to phase calibration, blurring the distinction between the main lobe and grating lobes and creating a chaotic, disorderly field distribution that makes beam scanning extremely difficult.Following calibration using the ASPGD algorithm, the far-field intensity distribution closely approximates the ideal scenario, enabling distinct differentiation of the main lobe, grating lobes, and side lobes.In Figure 3 (b) (c) (d) and (e), there are grating lobes that exhibit higher power than the main lobe.This is due to the fact that the single slit diffraction factor (array factor) will have an impact on both the main lobe and the grating lobes, suppressing the former and enhancing the latter during the steering process.For equidistant phased arrays, the grating lobes even have more energy than the main lobe during the deflection process and this presents another significant research topic in optical phased array-the optimization of grating lobes.It is evident that a robust phase calibration algorithm serves as a fundamental prerequisite for the progress of other endeavors.Furthermore, we compared the convergence performance of ASPGD to that of classical SPGD, NSPGD, and genetic algorithm, as illustrated in Figure 4.The Figure 4 shows that the genetic algorithm has the poorest convergence performance and needs the longest convergence time.On the other hand, the conventional SPGD algorithm achieves a superior global optimal solution, although it also requires a relatively long convergence time.The improved NSPGD and ASPGD achieve remarkable convergence performance, both exhibiting faster convergence rate and superior global optima.However, the convergence time of the ASPGD algorithm is 5 minutes faster than that of the NSPGD algorithm, and the value of the cosine similarity obtained is increased by 8 %.
Moreover, the ASPGD algorithm is already getting close to convergence after 5.7 minutes of running all four algorithms, while half of the convergence process has not yet been completed by the SPGD algorithm.Additionally, even if the objective function values of the genetic algorithm and NSPGD algorithm are relatively large right now, they will continue to rise in the next twenty minutes, indicating that convergence won't happen right away.This provides strong support for the superior performance of the ASPGD algorithm.
So far, we have demonstrated the workflow of ASPGD algorithm combined with Adam algorithm, and compared the convergence speed and convergence performance of ASPGD, NSPGD, SPGD and genetic algorithm to show the promising prospect of ASPGD algorithm in the application of initial phase calibration of optical phased array.

Conclusions
A unique ASPGD algorithm that combines the Adam and SPGD algorithm for initial phase calibration of optical phased array is proposed in this paper.When contrasted with conventional optimization algorithms like SPGD and genetic algorithm, the improved ASPGD method notably doubles the convergence speed.Additionally, the ASPGD algorithm can increase the cosine similarity value by about 10%.

Figure 1 .
Figure 1.The far-field intensity distribution of an optical phased array.