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Micromachines 2026 , 17 , 592 19 of 34 Algorithm 5 Global BO 1: Define search space: x ∈ [ 0, 120 ] , y ∈ [ 0, 120 ] 2: Sample 8 Latin-Hypercube sampled points 3: Fit Gaussian Process to observations 4: for iteration = 1 …

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Algorithm 3 Cross (Two-Stage)
1: // Coarse Stage
2: for V
x
= 0 to 120 step 4 do
3: Sample power at (V
x
, 60)
4: end for
5: Find x
max
with highest power
6: for V
y
= 0 to 120 step 4 do
7: Sample power at (x
max
, V
y
)
8: end for
9: Find y
max
with highest power
10: // Fine Refinement Stage
11: for V
x
= x
max
−4 to x
max
+ 4 step 1 do
12: Sample power at (V
x
, y
max
)
13: end for
14: Find x
f ine
with highest power
15: for V
y
= y
max
−4 to y
max
+ 4 step 1 do
16: Sample power at (x
f ine
, V
y
)
17: end for
18: Move to observed maximum
• Local Bayesian Optimization (Local BO):
The Local BO algorithm first performs a cross-scan to identify the region of interest,
and then applies BO using a GP surrogate model in that local neighborhood (
±
20 V
or
±
2.5
µ
m). This hybrid approach combines the efficiency of initial cross-scan
guidance with the intelligent exploration-exploitation balance of Bayesian methods.
The complete routine is described in Algorithm 4.
Algorithm 4 (Local BO)
1: // Initial cross-scan (7 points per axis)
2: for V
x
∈ {0, 20, 40, 60, 80, 100, 120} do
3: Sample power at (V
x
, 60)
4: end for
5: Find x
max
with highest power
6: for V
y
∈ {0, 20, 40, 60, 80, 100, 120}\{60} do
7: Sample power at (x
max
, V
y
)
8: end for
9: (x
init
, y
init
) ← best point from cross-scan (13 total points)
10: Define search space: x ∈ [x
init
−20, x
init
+ 20], y ∈ [y
init
−20, y
init
+ 20]
11: Sample additional random points to reach 10 initial points in local region
12: Fit Gaussian Process to observations
13: for iteration = 1 to 100 or until convergence do
14: Select next point via Expected Improvement
15: Measure power at selected point
16: Update Gaussian Process
17: if no improvement for 8 iterations then
18: Break
19: end if
20: end for
21: Move to observed maximum
The Expected Improvement (EI) acquisition function balances exploration and
exploitation:
EI(x) = E[max(P(x) − P
max
, 0)] (4)
where P
max
is the current best observed power.
• Global Bayesian Optimization (Global BO):
The Global BO method searches the entire 120 V
×
120 V space (15
µ
m
×
15
µ
m) di-
rectly, without an initial cross-scan phase. This strategy provides the best exploration-
exploitation balance for unknown landscapes through purely probabilistic sampling.
Algorithm 5 details this approach.
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Algorithm 5 Global BO
1: Define search space: x ∈ [0, 120], y ∈ [0, 120]
2: Sample 8 Latin-Hypercube sampled points
3: Fit Gaussian Process to observations
4: for iteration = 1 to 70 or until convergence do
5: Select next point via Expected Improvement
6: Measure power at selected point
7: Update Gaussian Process
8: if no improvement for 6 iterations then
9: Break
10: end if
11: end for
12: Move to observed maximum
• Variable Gradient Ascent (Var-Grad):
Var-Grad utilizes an adaptive gradient search logic to maximize alignment speed. The
process begins with a cross-scan initialization (7 points per axis) to identify the initial
signal vicinity, followed by iterative hill-climbing using finite-difference gradient
estimates with a step size of h = 2 V (∼0.25 µm).
The efficiency of this method is driven by an adaptive learning rate (
α
) that scales
inversely with the measured power (P), governed by a gain factor k = 250:
α =
k
max(P, 10
−12
)
(5)
This mechanism allows the algorithm to take aggressive steps in low-power regions
and gradually reduce the step size as it approaches the peak, enabling high-precision
coupling that exploits the system 1 nm hardware resolution. While this approach
achieved the fastest alignment time of 1.18 a.u. in HS-NH simulations, its dependency
on local gradients makes it more susceptible to entrapment in local maxima compared
to probabilistic methods. The pseudo-code for this adaptive approach is provided in
Algorithm 6.
Algorithm 6 Var-Grad
1: // Initial cross-scan (7 points per axis)
2: for V
x
∈ {0, 20, 40, 60, 80, 100, 120} do
3: Sample power at (V
x
, 60)
4: end for
5: Find x
max
with highest power
6: for V
y
∈ {0, 20, 40, 60, 80, 100, 120} \ {60} do
7: Sample power at (x
max
, V
y
)
8: end for
9: (x
0
, y
0
) ← best point from cross-scan
10: Measure P
0
at (x
0
, y
0
)
11: α ← 250/ max(P
0
, 10
−12
) (adaptive learning rate)
12: h ← 2 (finite difference step)
13: for iteration = 1 to 10 do
14: Measure P
1
at (x + h, y), P
2
at (x −h, y)
15: Measure P
3
at (x, y + h), P
4
at (x, y −h)
16: ∇
x
← (P
1
− P
2
)/(2h)
17: ∇
y
← (P
3
− P
4
)/(2h)
18: ∆x ← α ∇
x
, ∆y ← α∇
y
19: x ← x + ∆x, y ← y + ∆ y
20: if |∆x| < 0.5 and |∆y| < 0.5 then
21: Break (convergence)
22: end if
23: end for
24: Move to best observed position
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• Fixed Gradient Ascent (Fix-Grad):
The Fix-Grad method is a traditional gradient search technique with fixed step sizes
that halve upon observing no improvement. The procedure begins with a cross-scan
(7 points per axis), and then iteratively explores neighbors in the cardinal directions
(initial step 4 V or
∼
0.5
µ
m, minimum 1 V or
∼
0.125
µ
m) to identify the optimal ascent
direction, as shown in Algorithm 7:
Algorithm 7 Fix-Grad
1: // Initial cross-scan (7 points per axis)
2: for V
x
∈ {0, 20, 40, 60, 80, 100, 120} do
3: Sample power at (V
x
, 60)
4: end for
5: Find x
max
with highest power
6: for V
y
∈ {0, 20, 40, 60, 80, 100, 120}\ {60} do
7: Sample power at (x
max
, V
y
)
8: end for
9: (x
0
, y
0
) ← best point from cross-scan
10: Measure P
current
at (x
0
, y
0
)
11: δ ← 4 (initial step size)
12: while δ ≥ 1 do
13: Measure power at (x + δ, y), (x −δ, y), (x, y + δ), (x, y −δ)
14: (x
′
, y
′
, P
′
) ← best neighbor
15: if P
′
> P
current
then
16: (x, y, P
current
) ← (x
′
, y
′
, P
′
)
17: else
18: δ ← δ/2 (reduce step size)
19: end if
20: end while
21: Move to (x, y)
Figure 8 illustrates the application of all seven alignment algorithms to a representative
mock scan scenario centered at (60 V, 60 V), providing a direct visual comparison of
their respective search strategies. Each algorithm navigates the 120 V
×
120 V operating
range using a distinct optimization philosophy: Ref-Coarse and Ref-Coarse+Fine employ
systematic raster scans; Cross scan utilizes dimensional reduction via orthogonal axis
scanning; Local and Global BO implement intelligent sampling guided by Gaussian Process
uncertainty; and Variable and Fix-Grad utilize hill-climbing techniques to track the local
power gradient.
(a) Ref-Coarse (b) Ref-Coarse+Fine
Figure 8. Cont.
https://doi.org/10.3390/mi17050592