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Micromachines 2026 , 17 , 592 24 of 34 T able 2. A verage 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 …

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(600.3 steps, 334.11 a.u.) renders the approach entirely impractical for high-throughput

manufacturing. The remaining techniques, that are Cross, Var-Grad, and Reference Coarse,

exhibit heightened sensitivity to landscape topology, peaking at a 75% success rate. No-

tably, Var-Grad operates with the highest speed (22.5 steps, 59.28 a.u.) but lacks requisite

dependability. Consequently, Local BO presents the optimal synthesis of speed, accuracy,

and reliability for physical alignment tasks. Securing a 100% success rate is paramount

for strict production environments to eliminate manual intervention and preserve critical

operational throughput. Figure 9 shows the data proﬁle for experimental results, plotting

the fraction of problems solved as a function of total alignment time. The discrete, step-like

appearance of these proﬁles is simply a consequence of the limited experimental sample

size (N = 8).

50 100 150 200 250 300 350 400

Time (a.u.)

0.0

0.2

0.4

0.6

0.8

1.0

Fraction of Problems Solved (Tolerance=5%)

Cross

Fix-Grad

Global BO

Local BO

Ref-Coarse

Ref-Coarse+Fine

Var-Grad

Figure 9. Data proﬁle for experimental results. Local BO and Fix-Grad methods show steep proﬁles

indicating rapid, reliable convergence to 100% success. Var-Grad converges fastest on easy problems

but plateaus at 75%, while Ref-Coarse/Ref-Coarse+Fine require many measurements.

Following the methodology of Beiranvand et al. [

], data proﬁle analysis elucidates

comprehensive convergence behaviors that simple central tendency metrics obscure. Specif-

ically, the Local BO proﬁle exhibits a steep initial trajectory, achieving 100% success rate

within approximately 30 steps. The Fix-Grad method similarly reaches absolute reliability

by 80 steps, demonstrating a steady, albeit slower, progression. Conversely, algorithms

such as Global BO, Var-Grad, and Cross exhibit rapid initial climbs but prematurely plateau

at success rates of 87.5%, 75%, and 75%, respectively, indicating fundamental failure cases.

Moreover, the exhaustive reference techniques Ref-Coarse and Ref-Coarse+Fine require

hundreds of measurements merely to attain sub-optimal success rates between 75% and

87.5%. Ultimately, this statistical visualization demonstrates that while Var-Grad yields

the lowest average step count (22.5), its early performance plateau and 25% failure rate

render it signiﬁcantly inferior to the robust Local BO and Fix-Grad algorithms for rigorous

production deployment.

4.2. Simulation Results

Table 2 summarizes the average number of measurement steps required by each

algorithm across the four simulation scenarios. Results are averaged over multiple mock

ﬁles generated from the experimental data. Global BO requires the fewest steps (

∼

25),

followed closely by Var-Grad (

∼

26) and Local BO (

∼

31). The intelligent sampling of

Bayesian and Gradient methods reduces the measurement burden by over 20

compared

to Ref-Coarse+Fine scanning (540 steps). The step counts are independent of the operating

scenario, as they reﬂect algorithmic logic rather than timing. However, these step counts

directly impact total time, which varies signiﬁcantly across scenarios as shown next.

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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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0.6

0.8

1.0

(a) LS-H

0.0

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0.6

0.8

1.0

(b) HS-H

0.0

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0.8

1.0

0.0

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1.0

(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.

https://doi.org/10.3390/mi17050592