Optical power auto-alignment method with eugenics sorting for enhancing the alignment speed and robustness of fiber-grating couplers

Lian Duan; Lian Duan; Cong Xu; Cong Xu; Shunshun Zhong; Shunshun Zhong; Haibo Zhou; Haibo Zhou; Ji-an Duan; Ji-an Duan

doi:10.1364/OE.470642

1. Introduction

Recently, silicon-based optical waveguides with high refractive indices have become a key aspect of optoelectronic communication [1–3]. Owing to the mismatch in dimensions, a grating coupler is employed to connect the waveguides and exteriors (optical fiber). However, low efficiency coupling between couplers and fibers is a significant problem. In general, the coupling loss between couplers and fibers consists of three aspects: mode-matching [4], waveguide transmission [5], and alignment error losses [6]. The first two losses are determined by fabricating the precision level of the grating coupler. The alignment error loss in the entire space is determined by the relative location, including the lateral (Δx, Δy), vertical Δz, and pitching angle Δθ deviations. Several studies have focused on optimizing mode-matching and waveguide transmission losses [4,5], [7–10]. However, in practical coupling and packaging processes, the alignment error loss is often more easily overlooked but has an impact on the optical power output. Zheng (2019) investigated the effects of the alignment parameters on fiber-grating coupling [10]. The simulated results illustrated that the pitching angle deviation has a larger influence on the center wavelength offset than on the horizontal distance and vertical height. However, the best coupling parameters for enhancing the alignment error loss have not been verified experimentally.

The premise of the alignment parameter (tolerance) analysis is made up of a precise multi-axis movement system and optical power search algorithms [11]. The automated alignment process is typically divided into two stages: initial alignment and fine alignment. In the initial alignment process, a power meter is used as feedback to move the coupling devices from a weak signal position to another position where optical power is available for detection. After the initial alignment, precision alignment of the maximum optical power is performed. A simple method for optical power search is an optical power meter search of each point of the search region to find the maximum value. The number of points detected depends on the accuracy of the platform. However, this method neglects the geometric errors of the platform and consumes considerable time and calculation. More broadly, maximum optical power auto-alignment can be treated as a single-objective optimization problem. The numerical analysis methods, such as Hill-climbing method [12], Hamilton algorithm [13], Pattern search method [14] and least square method [15], have their own advantages and disadvantages. However, these methods do not effectively use the information from the searched locations to re-route ranges and paths. Long search times and lack of accuracy are the reasons why numerical analysis methods are becoming obsolete in industrial applications. Compared with numerical analysis method, swarm intelligence algorithms are available for fast and accurate alignment of the entire space [16–19]. Among them, the particle swarm optimization (PSO) algorithm has been validated for optical power search in fiber-laser coupling [18]. The search speed and accuracy are not robust because of the inevitable randomness of the particle initialization distribution. Moreover, a large number of space-detected points leads to the promotion of accumulated geometric errors. Therefore, an optical power auto-alignment method with high robustness and a small number of particles is important for optical device coupling.

In the past few decades, chaos has received extensive attention in various fields such as engineering, economics, numerical analysis and information security. This is because chaotic systems have many good dynamical properties, including infinite duration, stochasticity, ergodicity, long-term unpredictability and sensitivity to initial conditions [19,20]. In this study, an optical power auto-alignment method with a chaotic-aided PSO algorithm was proposed for fast and precise optical power auto-alignment between a single-mode fiber (SMF) and silicon-based grating coupler. Chaotic mapping with the eugenics mechanism was introduced to readjust and re-sort the initial particle distribution. A series of simulations and experiments was performed to verify the advantages of this method. The remainder of this paper is organized as follows. In Section 2, the mathematical model of coupling between grating couplers and SMFs is established, and the proposed optical power auto-alignment method is presented. Section 3 presents the simulation and experimental results of the optical power auto-alignment, and a series of chaotic maps are compared to illustrate the superiority of the proposed method. Finally, a brief conclusion is presented in Section 4.

2. Modeling and method

2.1 Modeling

In the fiber-grating coupling process, light is transmitted to the grating structure through optical fibers and couples into the optical waveguide using diffraction grating. The coupling model of the grating coupler and SMF is described by the coupling efficiency $\eta $. The coupling efficiency is mainly influenced by two factors: the size of the grating structure, including the top layer, oxide layer, and silicon substrate, and the parameters of the incident Gaussian beams, including the incident angle and relative position between the beam waist and grating coupler (as shown in Fig. 1). Herein, the research focused on the parameters of the incident Gaussian beams under the condition of consistent grating structure size. According to a previous study [21], standard Gaussian beams in the $({x,\; z} )\; $ plane are emitted from the end face of the fiber and spread in the z-direction. The coupling efficiency was described as the ratio of the optical power injected into waveguide ${\textrm{P}_{in}}$ to the incident beam power ${\textrm{P}_{inc}}$, as expressed in Eq. (1):

(1)$$\eta = {{{\textrm{P}_{in}}} / {{\textrm{P}_{inc}}}} = t \times {|{r({{z_c},{W_2}} )} |^2}$$

where t represents the factor influencing the coupling efficiency of the grating structures (a constant); ${z_c}$ represents the distance between the center of the Gaussian beam and edge of the grating; ${W_2} = {{{W_1}} / {\cos {\theta _{in}}}}$ is the radius of the Gaussian beam incident on the grating surface, where ${W_1}$ is the beam waist radius of the Gaussian beam and ${\theta _{in}}$ is the angle between the incident Gaussian beam and the direction normal to the grating surface, which follows the grating diffraction rules; and $|{r({{z_c},{W_2}} )} |$ is the influencing factor of the coupling efficiency of the Gaussian beam, revealing the connection between the optical coupling efficiency of the gratings, beam waist, and position of the beam center. The expression is as follows:

(2)$${|{r({{z_c},{W_2}} )} |^2} = \frac{{{E^{inc}}{W_2}{\pi ^{{1 / 2}}}}}{2} \cdot \exp \left\{ {{{\left( {\frac{{{W_2}}}{{4{L_c}}}} \right)}^2} - \frac{{{z_c}}}{{2{L_c}}}} \right\} \cdot \left[ {1 + erf(\frac{{{z_c}}}{{{W_2}}} - \frac{{{W_2}}}{{4{L_c}}})} \right]$$

where ${L_c}$ is the coupling length of the grating and ${E^{inc}}$ is the amplitude of the output Gaussian beam. Under the resonance condition, we assumed that the incident optical power was normalized. Thus, ${E^{inc}}{W_2}{\pi ^{1/2}} = 1.$ Moreover, Eq. (3) provides the conditions when the Gaussian beams are optimally coupled to the waveguide gratings.

(3)$$\begin{array}{l} {z_c} = {L_c}\\ {W_1} \approx 1.37{L_c}\cos {\theta _{in}} \end{array}$$

By connecting Eqs. (1), (2), and (3), the coupling efficiency between the fiber and grating couplers was obtained. To introduce the distribution of the light field into the optical power auto-alignment, the coupling efficiency $\eta $ was divided into two dimensions, expressed as $\eta = \sqrt {{\eta _x}{\eta _y}} $, where ${\eta _x}$ was the mathematical model in the x-direction and ${\eta _y}$ was the mathematical model in the y-direction, followed by a Gaussian distribution.

(4)$${\eta _x} = \frac{{{E^{inc}}{W_2}{\pi ^{{1 / 2}}}}}{2} \cdot \exp \left\{ {{{\left( {\frac{{{W_2}}}{{4{L_c}}}} \right)}^2} - \frac{{{d_x}}}{{2{L_c}}}} \right\} \cdot \left[ {1 + erf(\frac{{{d_x}}}{{{W_2}}} - \frac{{{W_2}}}{{4{L_c}}})} \right]$$

(5)$${\eta _y} = k \cdot \exp \left\{ { - k\left[ {\frac{{d_y^2m}}{2} + \frac{{{\pi^2}\theta_y^2r}}{{2{\lambda^2}}} - \frac{{{d_y}{\theta_y}z}}{{R_0^2}}} \right]} \right\}$$

In Eqs. (4) and (5), ${d_x}$ replaces the distance between the Gaussian beam and edge of grating ${z_c}$. k, r, and m are the coupling coefficients for connecting the fiber and grating couplers, as expressed in Eq. (6).

(6)$$\begin{array}{l} k = {{4R_0^2R_f^2} / {[{{{({R_0^2 + R_f^2} )}^2} + {{{\lambda^2}{z^2}} / {{\pi^2}}}} ]}}\\ r = {R^2}(z) + {R_f}^2\\ m = {1 / {R_0^2}} + {1 / {R_f^2}} \end{array}$$

where ${R_0}$ represents the mode radius of the end face of the SMF and ${R_f}$ represents half the width of the optically sensitive region. Thus, $R(z )$ is the distance function-associated fiber movement along the z-axis, as expressed in Eq. (7).

(7)$$R(z) = {R_0}\sqrt {1 + {{(\lambda z/\pi R_0^2)}^2}}$$

Previous studies have focused on the tolerance analysis of alignment parameters, showing that the vertical height has little influence on the offset of the central wavelength [10,22]. To simplify the calculation of the coupling model and collision prevention, the value of z was constant ($10\; \mathrm{\mu}\textrm{m}$). Therefore, ${d_x}$, ${d_y}$, and ${\theta _y}$ were determined as the parameters of the related dimensions and solved in the continual iteration of the algorithms.

Fig. 1. Schematic of Gaussian light beam incident on a grating coupler from the single-mode fiber.

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2.2 Proposed method

PSO is an effective algorithm for solving objective optimization problems [23–27]. A swarm of particles is randomly initialized in the search space, and each particle reveals a possible solution to the optimization problem. The fitness value of each particle is assessed using the function to be optimized. As expressed by Eq. (8), we assume that m is the population, the position of the ${i_{th}}$ particle is $\overrightarrow {{x_i}} = \{{x_{i1}^k,\; x_{i2}^k, \cdots x_{id}^k\; } \},({1 \le i \le m} ),$ and velocity $\overrightarrow {{v_i}} = \{{v_{i1}^k,v_{i2}^k, \cdots v_{id}^k} \},({1 \le i \le m} ).$ These variables are updated with iteration k, where d is the dimension of the solution vector. During the evolutionary procedure, the information exchange mechanism based on the individual best experience $\overrightarrow {{p_i}} = \{{{p_{i1}},{p_{i2}}, \cdots {p_{id}}} \},\; ({1 \le i \le m} )$ and global best experience $\overrightarrow {{p_{gb}}} $ drives particles towards the position with better fitness value (presented in Fig. 2).

(8)$$\begin{array}{l} \nu _{id}^k = \omega \cdot \nu _{id}^{k - 1} + {c_1} \cdot {r_1} \cdot ({{p_{id}} - x_{id}^{k - 1}} )+ {c_2} \cdot {r_2} \cdot ({{p_{gb}} - x_{id}^{k - 1}} )\\ x_{id}^k = x_{id}^{k - 1} + \nu _{id}^k \end{array}$$

In Eq. (8), ${c_1}$ and ${c_2}$ are acceleration coefficients, ${r_1}$ and ${r_2}$ are random numbers generated between 0 and 1, and $\omega $ is an inertia weight that balances the size of the search space. The adaptive inertia weight was adopted to establish the relationship between the fitness value and the current swarms. As expressed by Eq. (9) [28], the iteration stage of the entire search process can be determined based on the relationship between the average fitness value of swarms ${f_{avg}}$ and the current fitness value f. In the early stage, the inertia weight is taken as the maximum value ${\omega _{max}}$ to allow the particles to traverse search spaces quickly. In the later stage, if $f \ge {f_{avg}}$, then the inertia weight is adjusted based on the minimum inertia weight ${\omega _{min}}$.

(9)$$\omega = \left\{ \begin{array}{l} {\omega_{\min }} + \frac{{({\omega_{\max }} - {\omega_{\min }})({f_{\max }} - f)}}{{{f_{\max }} - {f_{avg}}}},f \ge {f_{avg}}\\ {\omega_{\max }},f < {f_{avg}} \end{array} \right.$$

Fig. 2. Optical power auto-alignment using swarm intelligence algorithm: a continuous convergence process.

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Typically, the distribution of the initial particles influences the performance of the PSO algorithm. Particles with better positions are more beneficial for the global optimum search and convergence vector acceleration than those with worse positions. The traditional PSO algorithm randomly creates the initial particles in the configured region of the optimization problem. The convergence process is volatile and exhibits a high degree of randomness. Several investigations have proposed efficient methods to improve the distribution of initial particles [29–32]. Among them, the combination of the PSO algorithm and chaos improved the optimization mechanism of the local extremum in the optimizing process. Chaos is a singular steady-state evolutionary behavior of nonlinear dynamical systems that characterizes the essence of complex phenomena in nature and human society [33]. The ergodicity of chaos drives particles to visit arbitrary points within a limited space. In addition, the pseudo-randomness inside the chaos system drives the distribution of particles to be regular with rules.

Chaotic mapping has always been used in the reconstitution of initial particles and in an interferential network of optimum fitness values. Figure 3(a) presents a histogram distribution of initial particles with different chaotic mappings, including logistic [20], tent [34], Chebyshev [35], circle [36], piecewise linear chaotic (PWLCM) [37], and Cubic mapping [38]. The parameter settings and functions of the chaotic mappings are listed in Table 1.

Fig. 3. Chaotic mapping for initialization of particle swarm optimization algorithm: (a) histogram distribution of 10⁵ initial particles with different chaotic mappings; (b) schematic of eugenics mechanism for initial particles optimization

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Table 1. Functions of chaotic mapping and corresponding parameter settings

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The different mappings studied have unique distribution characteristics. Logistic mapping has a homogeneous distribution in the range [0.1, 0.9] and intensive aggregation in the range [0,0.1] and [0.9,1]. Because many particles are focused on both ends of the search space, finding the optimal point is difficult. In contrast, circle mapping has a local aggregation in the central region. Analogously, the Chebyshev and Cubic mappings also exhibit inhomogeneous distribution characteristics. Both tent and PWLCM mapping have homogeneous distributions in the entire search space. Because the optical power distribution in an SMF is complicated and uncertain, a particle distribution with high fitness values requires simulation validation. In an optical power auto-alignment system, the total number of detected points is equal to the product of the iteration and initial particles. Although chaotic mapping helps the PSO algorithm optimize the initial distribution, a mass of initial points adds inefficient computation and geometric errors.

In this study, a filtering mechanism based on eugenics was proposed to retain particles for iteration. Initialization with chaotic mapping was considered as a pre-treatment. We determined the optical power (fitness value) of the initial particles and sorted them according to the fitness value. Higher values indicated that the detected point was closer to the global optimal point. Thus, high values were preserved, whereas low values were removed. A schematic of the proposed method is shown in Fig. 3(b). In the iteration process, the particles following the PSO algorithm learn from the population. Population dimensions that are too small are harmful to the communication of information; hence, a balance between the initial particles to be preserved and the population dimensions is critical for this optimization method. With the disturbance of eugenics, particle initialization is completed along with the initial iteration. Then, maximum optical power alignment is performed with the iteration of the adaptive PSO (APSO) algorithm. Figure 4 presents the schematic flow of the proposed method.

Fig. 4. Flow diagram of proposed chaotic-aid adaptive particle swarm optimization (APSO) algorithm

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3 Simulation and experiment results

3.1 Simulation

Prior to the simulation experiments, a series of parameters were determined. The input vectors ${d_x}$, ${d_y}$, and ${\theta _y}$ (refer to Equations (4)∼(7)) were contained in a range that is equal to the optical power alignment space. With the help of auxiliary software by FDTD Solutions (Lumerical Solutions), a coupling tolerance analysis of the optical model was carried out. The structural parameters of the SMF and grating coupler are listed in Table 2. Where ${h_{top}}$, ${h_{Si{O_2}}}$ and ${h_{bottom}}$ represented the height of Si top-layer, SiO₂ layer and Si substrate layer, respectively. Also, the etch depth $e$ was $0.1\mu m$, grating period $\mathrm{\Lambda }$ was $0.638\mu m$, duty cycle f was 0.5, and the number of period was 53.

Table 2. Parameters of coupling simulation between grating couplers and single-mode fibers

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The relationship between the coupling efficiency and position of the SMF ($x,y,{\theta _{in}}$) is shown in Fig. 5. The maximum coupling efficiency in the y-axis direction is 36%, which is less than the maximum coupling efficiency in the x-axis direction (51%). This is due to the error caused by the different modelling dimensions (2D modelling in x-axis and 3D modelling in y-axis). When the x- and y-axes deviated from the optimum location, the optical coupling efficiency consistently decreased. Over ${\pm} 10\mathrm{\;\ \mu m}$, no optical power was detected. Figures 5(c) and (d) show the relationship between the incident angle of the SMF and optical coupling efficiency. When the incident angle changed $1^\circ $, the optical wavelength shifted $15\,\textrm{nm}$. At $1500\,\textrm{nm}$ wavelength, an incident angle of 8° obtained the maximum coupling efficiency, and the tolerance range was − $[{ - 2.8^\circ , + 2^\circ } ]$. In the actual alignment process, the fiber was fixed on the fixture, moving from the negative x-axis to the positive direction. Therefore, the range of the input vectors for optical power alignment was determined to be ${d_x} \in [{ - 5,75} ],\,{d_y} \in [{ - 20,20} ]$, and ${\theta _y} \in [{6,10} ]$. Other parameters of the PSO algorithm were set as ${c_1} = {c_2} = 1.9$, ${\omega _{max}} = 0.9$, and ${\omega _{min}} = 0.4$.

Fig. 5. Effect of alignment parameters on coupling efficiency: (a) horizontal position x; (b) vertical position y; (c) incident angle ${\theta _{in}}$, and (d) wavelength offset against incident angle

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Because the theoretical maximum coupling efficiency calculated using the mathematical model was 0.78, we set the iteration termination condition as 0.772 (99%). In the first part of the simulation, tent chaotic mapping with eugenics was performed to adjust the particle number after sorting. The initial number of particles N was set to 30, and the number of screened particles $N^{\prime}$ was the adjusted variate $({N^\mathrm{^{\prime}}} < N)$. The number of simulation repetitions was set to 30, and the results are shown in Fig. 6. The value of the detected spatial points ${N_{total}} = N + N^{\prime} \times t$. With increasing screened particles, the average number of iterations required to achieve the target decreases. Too few screened particles break the associated and learning mechanisms of swarm intelligence algorithms, leading to an extreme optimum (two screened particles are shown in Fig. 6(b)). Too many screened particles reduce the effectiveness of eugenics, making little difference from traditional chaotic mapping. The promotion of the population brings little quality information because there is no increase in quality particles. The standard deviation reflects the degree of dispersion of a dataset (robustness of the algorithms). Using over five screened particles, the algorithm exhibited an excellent robust performance (standard deviation $\le 2.5$). Combining the results, we set the fine range of screened particles to 10–15, which maintains a good balance between eugenics and swarm information exchange.

Fig. 6. Simulation results of different screened particles in chaotic-aid particle swarm optimization algorithm with eugenics: (a) average iteration and detected spatial points statistics with screened particles; (b) result of 2 screened particles in 30 simulations; (c) result of 10 screened particles in 30 simulations; (d) result of 15 screened particles in 30 simulations.

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In the second part of the simulation, the different algorithms were compared by their coupling efficiency, including the proposed chaotic-aid APSO algorithm with eugenics (CAPSOE) (tent, circle, and logistic mapping) and the basic APSO algorithm without chaotic-aid. The simulation results are presented in Table 3 and Fig. 7. Table 3 shows that all the algorithms achieved a 100% success rate. The basic APSO algorithm leveraged the learning capabilities of swarm intelligence algorithms and adaptive inertia weight to escape the local optimum; whereas, the CAPSOEs achieved the target from fewer particles. Of the CAPSOE algorithms, the circle mapping presented an excellent performance in the optical coupling process: requiring few iterations to achieve the target value of coupling efficiency, driving a few points to detection, and showing strong robustness. Logistic mapping also provided a fast search speed and few detected points; however, it had weaker robustness than circle mapping. Tent, PWLCM, cubic, and Chebyshev mappings made little advance in terms of search speed and robustness, being stronger than the basic APSO algorithm only in terms of the reduction in detection points. The results illustrate that the distribution of optical power is not uniform in the detected space. In addition, the basic APSO with eugenics was carried out for comparison. A random distribution before iteration and sorting is used to help the iteration at the initial stage. The simulated results showed that eugenics can also help basic PSO algorithms to improve search speed (132 detected points down to 80). However, the stability of detection decreased from 2.74 to 3.4, because of the irregularity of particle distribution. In conjunction with Fig. 3(a), the strong performance of circle mapping revealed that optical power focuses more on the intermediate than marginal region. The initial particles with circle mapping can better match the rules of the optical power distribution because the initialization of circle mapping offers more particles in the range [0.1, 0.9] (intermediate region). Logistic mapping adversely affects particle distribution: based on the eugenics mechanism, some points with low fitness values were filtered out. The quality of the detected particles is ensured for the fast detection of a global optimal position. However, extreme distribution makes it impossible to guarantee that low-quality particles will be screened out during each initialization. Thus, the robustness of logistic mapping is weaker than that of circle mapping.

Fig. 7. Results of 30 simulations of chaotic-aid particle swarm optimization algorithm with circle mapping: (a) with eugenics; (b) without eugenics

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Table 3. Detected performance of different algorithms on coupling simulation between single mode fiber and grating couplers

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Figure 7 presents the effects of the eugenic mechanism on circle mapping. Comparing the two results, eugenics affected the initial distribution of the particles. When the iteration begins, the particles screened by eugenics occupy a higher position and the information conveyed between the particles is closer to the global optimum. Therefore, in the first iteration, the circle mapping with the eugenics mechanism had higher fitness values. From the curves of the optical coupling efficiency, circle mapping with the eugenics mechanism had significantly faster convergence. Generally, the eugenics mechanism screens out initial particles with a better quality for iteration. Although this method breaks the characteristics of population diversity in swarm intelligence optimization algorithms, it plays a unique role in small-population applications. In actual optical device coupling processes, displacement errors accumulate owing to optical power alignment. These errors tend to cause poor coupling or are trapped in a light-finding blind spot. Therefore, less space-point detection can reduce the error generation.

3.2 Experiments

The experimental platform mainly included two 4-degree-of-freedom coupling platforms, fiber fixtures, and a visual system that was set as a mirroring distribution (as shown in Fig. 8(a)). In addition, tunable lasers and three-ring polarizers were connected to the input optical fiber, and an optical power meter was connected to the output optical fiber. To maintain a smooth coupling environment, an air-floating platform was adopted underneath the entire platform. The chip carrier was equipped with a negative-pressure adsorption device, which was used to adsorb and fix the silicon optical chip. A vertical-viewing charge-coupled device (CCD) was adopted to move the SMF to the grating workspace. A horizontal-viewing CCD was used to measure the incident angle of the fiber and the distance between the fiber and chip. The moving range of the x- and y-directions was $20\; \textrm{mm}$, and the repeat positioning accuracy was $0.5\; \mathrm{\mu}\textrm{m}$, controlled by closed-loop stepper motors. Based on the simulated results of the incident angle (shown in Fig. 5(c)), we designed a slot with an inclined angle of $8^\circ $ to fix the fiber. Figures 8(b) present the morphology of the grating couplers and local characteristics obtained using an ultra-deep field microscope (VHX-5000, Keyence). The SMFs used in this study were Corning SMF-28 flat-tip fibers. The core diameter was approximately $8.2\; \mathrm{\mu}\textrm{m}$, and the mode-field diameter was approximately $10\; \mathrm{\mu}\textrm{m}$. The incident optical power from single-mode fiber to grating coupler was 1 mW. The fiber-grating coupling experiments were divided into two parts: initial and precise alignments. The specific process was as follows:

(1) The incident SMF was moved to the left region of the grating edge, avoiding the fiber being too close to the waveguide transmission and falling into the uncoupled region. In the vertical waveguide direction, the center of the SMF was close to the geometric center of the gratings. To prevent subsequent angle adjustments leading to collisions, we set the distance between the SMF and grating surface to $10\; \mathrm{\mu}\textrm{m}$. Also, the output SMF was moved to the right region of the output grating edge.
(2) A spiral scanning method was adopted to conduct the initial alignment, increasing the optical power to a precise alignment threshold of $50\; \mathrm{\mu}\textrm{W}$, both in incident SMF and output SMF. If the initial alignment reached the threshold, we recorded the initial position and proceeded to the next step. If the initial alignment did not reach the threshold, the z-axis was moved closer to the grating surface and rescanned.
(3) The swarm of search particles was initialized at the range $X\epsilon [{ - 5,75} ],$ $Y\epsilon [{ - 20,20} ],$ and ${\theta _{in}}\epsilon [{7.5^\circ ,8.5^\circ } ]$. The optical power value from the power meter was used as a fitness value and the position values were fed back from the motion control card as position information. The largest iteration was set to 100, and the maximum optical power value was recorded as the average value of 10 repeated alignments.

Fig. 8. (a) Experimental equipment of optical coupling between fibers and grating coupler; (b) the morphology of grating couplers.

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Both the incident SMF and output SMF are moved in the maximum optical power detection. The experimental groups included three algorithms: basic APSO (Sample 1), chaotic-aid APSO without eugenics (Sample 2), and CAPSOE (Sample 3). The initial number of particles number was set to 30, and the number of screened particles was set to 10. The initial and final positions of the particles with the maximum optical power were recorded for comparison. The experimental results are shown in Fig. 9 and Table 4. Figure 9(a) illustrates the detected maximum optical power value using the basic APSO algorithm. During the experiments, the maximum optical power value was $108.34\; \mathrm{\mu}\textrm{W}$. It can be clearly observed that the basic APSO algorithm was trapped in the local optimum. Although the adaptive inertia weight helped the particles jump out of the traps, the coupling efficiency decreased significantly. The experimental results were different from those of the simulation, particularly the number of iterations. A possible reason for this was the accelerated errors of 3D movement (expressed as product of the positioning errors of the displacement and rotation axes). In the iteration process, every detected point had fine geometry errors, which affected the next iteration and caused it to jump out of the traps. Figure 9(b) illustrates the detection process of the maximum optical power value by CAPSOE and chaotic-aid APSO without eugenics. The termination condition was to achieve the maximum power of the basic APSO ($108.34\; \mathrm{\mu}\textrm{W}$). The initial particles followed a circle-mapping distribution. The curves of both chaotic-aid APSO algorithms successfully achieved the target value. With eugenics helping, the optical power alignment process was less trapped in the local optimum. This was mainly due to the higher qualified positions of the initial particles. Thus, Sample 3 reached the target earlier than Sample 2 (without eugenics) or Sample 1 (basic APSO). Table 4 shows the statistics of the three algorithms in experiments with 10 repetitions. The detected points in Sample 3 decreased by 78% and 90.3% from those in Samples 2 and 1, respectively. The standard deviation in Sample 3 decreased by 31% from Sample 2 and 50% from Sample 1. A marked increase in the detected speed and robustness was observed. The differences between the algorithms in the experiments were greater than those in the simulations. The main reason for this was that the geometric errors of the platform accumulated during the iterations. Smaller numbers of detected points in the eugenics mechanism required fewer times to move. Therefore, the relative motion accuracy of the space region was enhanced, leading to a fast and robust optical power search.

Fig. 9. Experimental results of maximum optical power alignment in fiber-grating coupling: (a) alignment using basic adaptive particle swarm optimization (APSO); (b) alignment using chaotic-aid APSO; (c) the coupling efficiency versus the position of y direction; (d) the coupling efficiency versus the position of x direction; (e) the coupling efficiency versus coupling angle $\theta $.

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Table 4. statistics of an experiment repeated 10 times with three algorithms.

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After the detection of the position with maximum optical power, a step-by-step measurement was carried out to verify the simulated distribution of optical power efficiency. At the optimum coupling position, the results of displacement deviation of d_y, d_x and $\theta $ direction are shown in Fig. 9(c), (d), and (e), respectively. The effect of the deviation of the single-mode fibers in the y-direction on the coupling efficiency is essentially symmetric, because both the single-mode fiber and the grating structure have a near-Gaussian distribution in the y-direction mode field, and the best coupling power is obtained when the center of the two modes coincide. The deviation curve of the single-mode fibers in the x-direction tends to match the mathematical model, and the variation curve of the coupling efficiency due to the positive and negative displacement deviations is not symmetrical, with the tolerance for positive deviation being slightly lower than that for negative deviation. This is because the beam exiting the single-mode fibers is essentially circular, while the inclined angle of incidence makes the mode field above the grating not symmetrically circular. The maximum optical coupling efficiency was lower than simulated results. The main losses include the optical fiber to optical waveguide, the optical waveguide transmission process and the optical waveguide to fiber process. Combined with Fig. 9(c), (d) and (e), the proposed CAPSOE algorithm can accurately find the maximum optical power point.

The simulation results were obtained without errors. For the basic APSO algorithms, information acquisition relied on abundant particles passing through the entire space. The efficiency and search speed of this strategy mainly depended on the strengths and weaknesses of the initial particle distribution. More geometric errors accelerated by a large amount of detection also decreased efficiency and search speed; therefore, the experimental performance of the basic APSO algorithm was worse than that of the simulation results. Compared with the basic APSO algorithms, the CAPSOE algorithm used chaotic mapping and eugenics for optimization. From the simulation and experimental results, the advantages of CAPSOE are as follows:

(1) The reduction in the detected points enhances the search speed for optical power auto-alignment. The eugenics mechanism is used as an artificial sorter to select the high fitness value (optical power) particles for iteration. Generally, the particles in the PSO algorithm tend to be close to the position with a high fitness value in each iteration. The high fitness value of the initial particles provides the impulse for searching for the optimal extreme in the early iteration. Meanwhile, owing to the distribution obtained via chaotic mapping, most particles have the potential to traverse space. CAPSOE presents a good balance between local detection (exploitation) and global detection (exploration) in optical power alignment.
(2) Chaotic mapping and sorting reduce the randomness of the particle distribution and enhance robustness. Chaotic mapping tends to have a relatively fixed initial distribution. Taking the circle mapping for example, the initial particles were mostly distributed in the middle region of the entire space in every simulation or experiment. The distance between the initial and target positions was relatively fixed (under the condition of the same optical fiber). Therefore, the robustness of the detected points in the chaotic mapping distribution was higher than that of the random distribution. Moreover, the eugenics mechanism decreased the difference between a series of chaotic maps. The screened particles always exhibited high fitness values corresponding to positions closer to the maximum optical power point. This is another reason for the enhanced robustness of swarm intelligence algorithms.

The accuracy of the platform is very important for optical communication applications. However, geometric errors are inevitable during the coupling process because of machining errors and the mechanical vibrations of the platform. Considering industrial efficiency and measurement accuracy, a reliable auto-alignment method can be used to replace manual alignment. The proposed CAPSOE method detects the maximum optical power in the SMF-grating coupling using a few steady steps to complete detection and promote efficiency and robustness. We believe that this method can be extended to other engineering problems in finding optimal targets. This method can provide theoretical support for fast and accurate alignment and coupling, especially in optical device coupling.

4 Conclusion

This paper presented a simple but effective alignment method for fiber and grating coupling. The process involves detecting the space position with maximum optical power. A PSO algorithm was adopted to solve the optimal problems. Using chaotic mapping and the eugenics mechanism, we improved the performance of the algorithm in terms of coupling efficiency and robustness. The simulation results illustrated that this method had a 100% success rate in determining the maximum optical power point. Compared to the basic PSO algorithm, the method based on circle mapping had the best performance: it reduced the number of detected points by 48% and promoted robustness by 18%. The experimental results proved the simulation results; the detected points reduced by 90.3% and robustness increased by 50%. Because of eugenics, the high-quality initial particles were not prone to falling into local extremes. The advantages of this method are the sorting and screening of initial particles, which can optimize the distribution of the initial particles. In addition, the eugenics mechanism reduces the number of swarms required for the iterative execution. This method can be extended to other practical engineering measurements to determine optimal extremes.

Funding

National Natural Science Foundation of China (51975590).

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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Mapping	Function	Parameter
Logistic	$x_{k + 1} = μ x_{k} (1 - x_{k}), x_{k} \in (0, 1)$	$\begin{array}{l} x_{1} = 0.152 \\ μ = 4 \end{array}$
Tent	$x_{n + 1} = {\begin{cases} x_{n} / a, 0 \leq x_{n} \leq a \\ (1 - x_{n}) / (1 - a), a \leq x_{n} \leq 1 \end{cases}$	$\begin{array}{l} x_{1} = 0.152 \\ a = 0.7 \end{array}$
Chebyshev	$x_{n + 1} = \cos (k \cos^{- 1} x_{n}), x_{n} \in [- 1, 1]$	$\begin{array}{l} x_{1} = 0.152 \\ k = 5 \end{array}$
Circle	$x_{n + 1} = mod (x_{n} + b - (\frac{a}{2 π}) \sin (2 π x_{n}), 1)$	$\begin{array}{l} x_{1} = 0.152 \\ a = 2.2, b = 0.5 \end{array}$
PWLCM	$x_{n + 1} = {\begin{cases} x_{n} / p, x_{n} \in (0, p) \\ (1 - x_{n}) (1 - p), x_{n} \in [p, 1) \end{cases}$	$\begin{array}{l} x_{1} = 0.002 \\ p = 0.7 \end{array}$
Cubic	$x_{n + 1} = 4 x_{n}^{3} - 3 x_{n}, - 1 < x_{n} < 1$	$x_{1} = 0.152$

Grating coupler		SMF
Parameter	Value	Parameter	Value
$W_{1} (μ m)$	4.9	Numerical aperture (NA)	0.14
$L_{c} (μ m)$	15	$θ_{i n} (^{\circ})$	4-12
$z_{c} (μ m)$	6	$λ (n m)$	1550
$h_{t o p} (μ m)$	0.22	$d_{c o r e} (μ m)$	8.2
$h_{S i O_{2}} (μ m)$	2	$d_{c l a d d i n g} (μ m)$	125
$h_{b o t t o m} (μ m)$	5

Algorithm	Particles in first iteration	Experimental times	Success times	Average iterations	Detected points	Standard deviation
Sample 1	30	10	10	42.5	1275	6.6
Sample 2	30	10	10	18.6	558	4.8
Sample 3	10	10	10	12.4	124	3.3

Mapping	Function	Parameter
Logistic	$x_{k + 1} = μ x_{k} (1 - x_{k}), x_{k} \in (0, 1)$	$\begin{array}{l} x_{1} = 0.152 \\ μ = 4 \end{array}$
Tent	$x_{n + 1} = {\begin{cases} x_{n} / a, 0 \leq x_{n} \leq a \\ (1 - x_{n}) / (1 - a), a \leq x_{n} \leq 1 \end{cases}$	$\begin{array}{l} x_{1} = 0.152 \\ a = 0.7 \end{array}$
Chebyshev	$x_{n + 1} = \cos (k \cos^{- 1} x_{n}), x_{n} \in [- 1, 1]$	$\begin{array}{l} x_{1} = 0.152 \\ k = 5 \end{array}$
Circle	$x_{n + 1} = mod (x_{n} + b - (\frac{a}{2 π}) \sin (2 π x_{n}), 1)$	$\begin{array}{l} x_{1} = 0.152 \\ a = 2.2, b = 0.5 \end{array}$
PWLCM	$x_{n + 1} = {\begin{cases} x_{n} / p, x_{n} \in (0, p) \\ (1 - x_{n}) (1 - p), x_{n} \in [p, 1) \end{cases}$	$\begin{array}{l} x_{1} = 0.002 \\ p = 0.7 \end{array}$
Cubic	$x_{n + 1} = 4 x_{n}^{3} - 3 x_{n}, - 1 < x_{n} < 1$	$x_{1} = 0.152$

Grating coupler		SMF
Parameter	Value	Parameter	Value
$W_{1} (μ m)$	4.9	Numerical aperture (NA)	0.14
$L_{c} (μ m)$	15	$θ_{i n} (^{\circ})$	4-12
$z_{c} (μ m)$	6	$λ (n m)$	1550
$h_{t o p} (μ m)$	0.22	$d_{c o r e} (μ m)$	8.2
$h_{S i O_{2}} (μ m)$	2	$d_{c l a d d i n g} (μ m)$	125
$h_{b o t t o m} (μ m)$	5

Optical power auto-alignment method with eugenics sorting for enhancing the alignment speed and robustness of fiber-grating couplers

Abstract

1. Introduction

2. Modeling and method

2.1 Modeling

2.2 Proposed method

3 Simulation and experiment results

3.1 Simulation

3.2 Experiments

4 Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Tables (4)

Equations (9)

Optics Express

Algorithm	Particles in first iteration	Number of experiments	Number of successes	Average iterations	Detected points	Standard deviation
CAPSOE (Tent mapping)	10	30	30	3.5	82.5	2.8
CAPSOE (Logistic mapping)	10	30	30	4	70	2.74
CAPSOE (Circle mapping)	10	30	30	3.9	69	2.24
CAPSOE (Chebyshev mapping)	10	30	30	4.7	77	2.72
CAPSOE (PWLCM mapping)	10	30	30	5.6	86	2.81
CAPSOE (Cubic mapping)	10	30	30	4.65	76	2.72
Basic APSO with eugenics	10	30	30	4.95	80	3.4
Basic APSO	30	30	30	3.4	132	2.74