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Micromachines 2026 , 17 , 592 25 of 34 The performance of the seven investigated algorithms was evaluated across the 72 sim- ulated test cases derived from experimental data. W e utilize data proﬁles to r epr esent the f…

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Table 2. Average number of measurement steps.

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

Ref-Coarse 961.0 961.0 961.0 961.0

Ref-Coarse+Fine 539.6 539.6 539.6 539.6

Cross Scan 75.6 75.6 75.6 75.6

Local BO 30.7 31.3 31.2 30.9

Global BO 24.6 24.7 23.9 24.6

Var-Grad 26.1 26.1 26.1 26.1

Fix-Grad 34.4 34.4 34.4 34.4

Table 3 shows total alignment time including movement and settling delays. Here, op-

erating speed and hysteresis effects become apparent. The experimental data highlight the

substantial throughput advantages of high-speed actuation. Transitioning from low-speed

to high-speed regimes yields a roughly 24-fold acceleration for the Global BO algorithm,

reducing alignment time from 53.06 to 2.24 a.u. under non-hysteretic conditions (HS-NH).

However, managing uncompensated hysteresis introduces severe temporal overhead due

to mandatory reset-to-zero motions, which inﬂates Global BO execution times by approx-

imately 92% (from 2.24 to 4.29 a.u. in HS-H). Among the evaluated methods, Var-Grad

achieves the absolute fastest alignment at 1.18 a.u. (HS-NH), although subsequent analysis

reveals this speed incurs a signiﬁcant reliability penalty. Conversely, Fix-Grad maintains

a highly competitive balance, completing alignments in 1.44 a.u. (HS-NH) and 3.32 a.u.

(HS-H) to offer a rapid, dependable alternative to Bayesian approaches.

Table 3. Average total alignment time (a.u.).

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

Ref-Coarse 33.39 30.64 382.60 315.01

Ref-Coarse+Fine 21.26 18.10 286.07 205.53

Cross Scan 2.88 2.44 35.42 24.88

Local BO 3.60 1.54 85.29 26.63

Global BO 4.29 2.24 103.26 53.06

Var-Grad 2.39 1.18 51.00 18.97

Fix-Grad 3.32 1.44 72.47 20.95

Finally, to evaluate overall optimization quality and robustness, Table 4 details the

achieved optical power as a percentage of the global maximum. Simulation results highlight

the superior precision of the Fix-Grad method, which consistently achieves 99.4% of the

global maximum power. Global BO and Local BO also perform excellently, achieving

around 98% accuracy. Var-Grad shows improved accuracy compared to initial tests (98%),

making it a competitive high-speed option.

Table 4. Average ﬁnal power coupling (% of global maximum).

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

Ref-Coarse 99.9 99.9 99.9 99.9

Ref-Coarse+Fine 100.0 100.0 100.0 100.0

Cross Scan 98.2 98.2 98.2 98.2

Local BO 98.2 98.3 98.3 98.3

Global BO 98.2 98.0 97.9 98.4

Var-Grad 98.0 98.0 98.0 98.0

Fix-Grad 99.4 99.4 99.4 99.4

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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.

0.0

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1.0

(b) HS-H

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(d) HS-NH

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

https://doi.org/10.3390/mi17050592