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Micromachines 2026 , 17 , 592 26 of 34 In the Low-Speed regimes, the absolute time requir ed for alignment incr eases signiﬁ- cantly due to the lower slew rates and settling times. In the LS-H scenario, see Figure 10 a, …

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The performance of the seven investigated algorithms was evaluated across the 72 sim-

ulated test cases derived from experimental data. We utilize data proﬁles to represent the

fraction of problems solved as a function of the total alignment time. A solved problem

is deﬁned as achieving a coupling power within 5% of the global maximum. Figure 10

illustrates these proﬁles across the four operating regimes. Figure 10b illustrates algorithm

performance under realistic high-speed conditions with active hysteresis effects (HS-H).

The Fix-Grad algorithm demonstrates the highest reliability (95.8%) across all 72 simulated

test cases in the HS-H regime. Local BO and Global BO are competitive but exhibit lower ﬁ-

nal success rates (94.4% and 88.9%, respectively). Furthermore, while the Var-Grad method

shows the fastest initial convergence for simple scenarios, solving a signiﬁcant fraction of

cases in under 2.6 a.u., it fails to achieve absolute reliability, plateauing at a 90.3% success

rate due to its inherent sensitivity to hysteresis-induced local maxima. This highlights

the robust advantage of the deterministic Fix-Grad approach, which maintains superior

accuracy and dependability even when the mandatory reset-to-zero strategy introduces

movement overhead to compensate for piezoelectric nonlinearities.

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Time (a.u.) [Log Scale]

Fraction of Problems Solved (Tolerance=5%)

Cross Fix-Grad Global BO Local BO Ref-Coarse Ref-Coarse+Fine Var-Grad

Figure 10. Data proﬁles for all four simulation scenarios showing the fraction of problems solved

as a function of alignment time: (a) LS-H (Low-Speed with Hysteresis), (b) HS-H (High-Speed

with Hysteresis), (c) LS-NH (Low-Speed without Hysteresis), and (d) HS-NH (High-Speed without

Hysteresis). Note that the time axis is logarithmic to clearly visualize the signiﬁcant speed advantage

of active alignment methods (<5.1 a.u.) over exhaustive reference scans (>257.1 a.u.). Fix-Grad

achieves the highest and most consistent reliability (95.8%) across all scenarios, while Global BO

shows scenario-dependent reliability (88.9–93.1%) and Var-Grad shows the fastest initial convergence

but lower ﬁnal reliability (90.3%).

Comparing Figure 10b–d (HS-NH), the removal of hysteresis effects results in a left-

ward shift of the performance curves, reducing absolute alignment time by approximately

50–57% for gradient and Bayesian methods, consistent with the ﬁgures reported in Table 3.

However, the relative performance ranking remains unchanged. This suggests that while

hysteresis imposes a time penalty, the probabilistic nature of the Bayesian methods inher-

ently manages these non-linearities more effectively than pure hill-climbing approaches.

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In the Low-Speed regimes, the absolute time required for alignment increases signiﬁ-

cantly due to the lower slew rates and settling times. In the LS-H scenario, see Figure 10a,

the Ref-Coarse and Ref-Coarse+Fine scans become impractical for volume production, re-

quiring over 257.1 a.u. per die. The Cross provides a middle-ground baseline, consistently

solving approximately 90% of the problems within a predictable time budget. However, like

the gradient methods, it struggles with the highest-precision requirements. The Bayesian

methods maintain their dominance in reliability, achieving the highest ﬁnal success rates

even as the absolute time scales upward.

Data proﬁle analysis reveals distinct convergence behaviors across all evaluated sce-

narios. Local BO exhibits steep, consistent trajectories that secure high success rates within

moderate step budgets, whereas Global BO demonstrates a similar but marginally slower

initial ascent due to the absence of cross-scan initialization. Among gradient techniques, Var-

Grad displays the steepest initial rise but suffers an early performance plateau, indicating

fundamental failure cases on challenging coupling landscapes. Conversely, Fix-Grad main-

tains a steady progression toward maximum reliability, conﬁrming its status as the most

dependable gradient approach with a 100% experimental success rate. Furthermore, the

one-dimensional Cross method prematurely plateaus near 75–80% success, while exhaus-

tive reference techniques require an impractical hundreds of measurements to converge.

Analyzed according to the statistical methodology of Beiranvand et al. [

], these proﬁles

conﬁrm that Local BO and Fix-Grad offer the best combination of speed and reliability

across the operating conditions tested.

4.3. Algorithm Robustness

Table 5 quantiﬁes the convergence reliability across all simulation scenarios, measuring

the percentage of trials achieving

≥

95% of global maximum power. The reliability analysis

establishes Fix-Grad as the most robust optimization algorithm, achieving a consistent

95.8% success rate across all simulation scenarios. Global BO remains a strong contender,

achieving 93.1% success in idealized (LS-NH) conditions and maintaining around 89%

reliability in the presence of hysteresis. This represents a signiﬁcant improvement over

earlier baselines, although it does not yet match the consistency of the deterministic Fix-

Grad method.

Table 5. Convergence reliability of the alignment algorithms (deﬁned as the percentage of trials

reaching ≥95% of optimal power).

Algorithm HS-H HS-NH LS-H LS-NH

Ref-Coarse 100.0 100.0 100.0 100.0

Ref-Coarse+Fine 100.0 100.0 100.0 100.0

Cross Scan 94.4 94.4 94.4 94.4

Local BO 94.4 94.4 94.4 94.4

Global BO 88.9 90.3 88.9 93.1

Var-Grad 90.3 90.3 90.3 90.3

Fix-Grad 95.8 95.8 95.8 95.8

5. Discussion

5.1. Experimental Versus Simulation Performance

The experimental results in Table 1 provide a validation of algorithm performance

on real PIC alignment tasks. The 100% success rates achieved by Local BO and Fix-Grad

methods in experiments are particularly signiﬁcant, as they demonstrate robustness across

eight diverse coupling scenarios. This diversity stems from characterizations performed

across seven different dies distributed to cover the full wafer surface as comprehensively

as possible, spanning multiple reticle positions, supplemented by repeated measurements

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on the same die to account for environmental stability and system repeatability. By encom-

passing these variations in die location and characterization cycles, the results conﬁrm that

the proposed algorithms can generalize effectively across the typical process and placement

variations encountered in high-volume wafer-level testing.

In terms of the interpretation of experimental rankings, results reﬂect the performance

of algorithms with baseline parameters on a limited but highly realistic dataset of 8 coupling

scenarios. Local BO achieved rank 1 due to its perfect 100% success rate combined with

the fastest alignment time (99.87 a.u.) among the successful methods. Fix-Grad achieved

rank 2, also with 100% success but requiring more time (154.01 a.u.). Global BO, despite

strong performance metrics (97.60% accuracy), achieved only 87.5% success rate, placing

it at rank 3. This lower success rate in the experimental phase reﬂects the conservative

baseline parameters used at that stage, which had not yet been optimized for the speciﬁc

characteristics of the Eclipse Dynamic system.

Following the experimental phase, an iterative improvement of the methods was

conducted to reﬁne the algorithm. The simulation study, which evaluated all the seven

algorithms across 72 test scenarios (8 experiments

9 spatial offsets), employed these im-

provements. This larger statistical sample and improved parameterization led to different

performance rankings compared to the experimental results, revealing the true performance

characteristics across diverse initial conditions.

The simulation framework reliably mimics real-world constraints. The RBF inter-

polation accurately captures landscape topologies, evidenced by the strong quantitative

agreement between experimental data (e.g., Local BO experimental accuracy of 98.24%) and

simulated outcomes (98.2–98.3%), indicating minimal uncertainty from the data processing

and validating the ﬁdelity of framework.. The difference in the success rates primarily

reﬂects the evolution from baseline to changes of the algorithms and parameters, and the

statistical power of the larger simulation dataset.

Given that the simulation results employ improved methods and parameters and a

statistically rigorous evaluation across 72 scenarios, they provide the most reliable basis

for production deployment recommendations. The experimental results validate the fun-

damental algorithmic approaches and conﬁrm real-world viability, while the simulation

results guide the selection of parameters and operating regimes.

5.2. Algorithm Selection Guidelines Across Operating Scenarios

The choice of an optimal alignment algorithm for the Eclipse Dynamic system de-

pends on the operating speed capability and the effectiveness of hysteresis compensation.

Comprehensive recommendations for each of the four evaluated scenarios are provided

next, progressing from worst-case to ideal-case conditions.

5.2.1. Scenario 1: LS-H (Low-Speed with Hysteresis)

This conﬁguration establishes the baseline worst-case scenario, deﬁned by conser-

vative hardware actuation speeds coupled with uncompensated piezoelectric hysteresis.

Under these conditions, the mandatory reset-to-zero maneuvers impose substantial tem-

poral overhead, while the restricted slew rates further exacerbate the latency associated

with each trajectory adjustment. Based on these stringent operating parameters, Fix-Grad

emerges as the optimal choice for maximizing process yield (see Tables 3 and 5). While

the Var-Grad approach operates faster, its approximate 10% failure rate requires manual

intervention, preventing any speed advantage in a production setting. Furthermore, the

substantial temporal penalties imposed by mandatory reset-to-zero maneuvers dispropor-

tionately degrade the performance of Bayesian approaches in this slow-actuation regime,

leaving Fix-Grad as the most robust and practical solution.

https://doi.org/10.3390/mi17050592