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Micromachines 2026 , 17 , 592 7 of 34 delivers strong empirical performance while maintaining rigorous conver gence guaran- tees [ 31 ]. Theor etical analyses have characterized the asymptotic behavior of local BO, conﬁr…

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2.3. Alignment Optimization Algorithms

The selection of the optical alignment search algorithm fundamentally determines

both the speed and the reliability of the per-die coupling step in photonic testing. Conven-

tional procedures have historically relied on brute-force scanning or iterative hill-climbing

techniques to locate the optimal coupling position. A typical approach involves raster or

spiral scanning, where the optical ﬁber is translated exhaustively across a multidimensional

grid until an initial optical signal is detected, followed by dense sweeps to map the cou-

pling efﬁciency landscape and identify the peak. Although robust to complex landscape

topologies, these exhaustive methods require a prohibitively large number of actuator

motions and power readings. Consequently, they can consume dozens of seconds for

a single alignment, making them impractical for high-volume manufacturing scenarios,

especially when multiple degrees of freedom are involved [

]. To mitigate this measure-

ment burden, cross-scan or dimensional-reduction approaches sequentially sweep along

individual axes to identify approximate peak coordinates before performing a focused local

reﬁnement scan.

To further accelerate the alignment process, modern systems increasingly employ ad-

vanced optimization algorithms, such as gradient-based searches. These methods estimate

local coupling gradients, often through small dither motions, and steer the optical array in

the ascent direction to rapidly approach the optimum. Analogous to numerical gradient

descent or ascent, this targeted approach can converge signiﬁcantly faster than blind scan-

ning. For instance, substituting a coarse spiral scan with an optimized gradient search has

been shown to yield an approximate 5

reduction in ﬁber coupling time [

]. Despite their

simplicity and speed, gradient methods are inherently susceptible to local maxima in the

measured optical power and require careful tuning of step-size parameters, particularly

when the coupling landscape contains secondary lobes or is distorted by actuator nonlin-

earities [

]. A central tension in these systems lies in balancing aggressive, adaptive

step sizes that yield the fastest convergence on benign landscapes against conservative,

ﬁxed step sizes that maintain reliability on more challenging topologies.

As a powerful alternative, Bayesian optimization (BO) utilizing Gaussian Process (GP)

surrogates has emerged to address the limitations of both grid-based and gradient methods.

This approach treats the coupling yield as an unknown black-box function, building a

probabilistic surrogate model that is iteratively updated with each measurement [

By intelligently maximizing an acquisition function such as Expected Improvement, the

optimizer strategically balances the exploration of uncertain regions against the exploitation

of known high-power areas. This enables the algorithm to infer the global landscape

shape and locate optima using far fewer evaluations than traditional methods. Originally

popularized in complex domains like X-ray beamline autoalignment [

] and online control

of particle accelerators [

], BO excels in environments where the objective function is

expensive to evaluate and subject to noise. These characteristics are highly analogous

to optical power measurements affected by piezoelectric nonlinearities and mechanical

vibrations. Machine-learning-driven alignment has similarly demonstrated success in

multidimensional free-space optical systems and lens assemblies, proving the efﬁcacy of

Bayesian algorithms in handling complex alignments with minimal iterations [30].

Within the BO family, two distinct paradigms are particularly relevant to the photonic

alignment problem. Global BO explores the entire search space from the outset, relying

purely on the GP posterior to select the next sampling location. While this maximizes

coverage and sample efﬁciency, it scales poorly with search-space dimensionality and can

be sensitive to landscape distortions introduced by actuator nonlinearities. Conversely,

local BO restricts the search to a neighborhood around a promising region identiﬁed by

an initial coarse scan. This restricted approach sidesteps the curse of dimensionality and

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delivers strong empirical performance while maintaining rigorous convergence guaran-

tees [

]. Theoretical analyses have characterized the asymptotic behavior of local BO,

conﬁrming conditions under which locally found stationary points closely approximate

the global optimum [

]. Furthermore, replacing standard gradient-descent exploitation

with a strategy that minimizes the upper conﬁdence bound over the GP posterior yields

tighter theoretical guarantees and improved empirical results [

]. These theoretical ad-

vances directly motivate localized Bayesian strategies, which combine a coarse cross-scan

initialization with a focused GP reﬁnement step.

Beyond the optimization of single-channel algorithms, the transition toward par-

allel, multi-channel alignment represents a critical advancement for wafer-level testing.

Because modern integrated probe architectures can simultaneously engage multiple in-

put and output ﬁber pairs across a die, applying alignment algorithms concurrently to

all channels drastically shortens the overall test time. Aligning channels sequentially is

highly inefﬁcient; instead, coordinated multi-dimensional searches that optimize all optical

ports together are essential for maximizing throughput. Ultimately, the ﬁeld is rapidly

transitioning from simplistic single-variable scans toward intelligent, multi-variable, and

machine-learning-augmented optimizations. Through the integration of these advanced

algorithmic approaches, the optical alignment step can be reduced to a minor fraction of

the total test time, making active alignment on the scale of a few seconds a viable reality for

high-volume production.

2.4. Hysteresis and Nonlinearity Compensation

Piezoelectric actuators are extensively utilized in nanometric alignment systems due

to their high resolution and rapid response times. However, a persistent challenge is their

inherent hysteresis and creep. The displacement produced by the actuator at a given

drive voltage depends not only on that speciﬁc voltage but on the entire preceding voltage

history, a phenomenon driven by domain-wall motion and polarization switching within

the piezoelectric ceramic [

]. In open-loop operation, this path-dependent nonlinearity

typically introduces positioning errors spanning 10–15% of the total travel range [

]. Such

substantial deviations far exceed the strict sub-micron tolerances required for reliable

ﬁber-to-waveguide alignment in PICs.

The traditional remedy for addressing hysteresis is closed-loop control. By integrating

an embedded feedback sensor such as a capacitive sensor, strain gauge, or linear variable

differential transformer (LVDT), and utilizing PID or iterative control, systems can dynami-

cally correct positioning errors to achieve sub-nanometer accuracy. While highly effective,

closed-loop architectures introduce signiﬁcant drawbacks: they are more expensive, impose

bandwidth limitations, and add substantial hardware complexity. For dense, multi-channel

probe heads (e.g., a 16-channel array), embedding a dedicated physical sensor and its asso-

ciated wiring on every piezoelectric axis is highly impractical. Position sensors, moreover,

require space for installation that is not available in the application hereby discussed.

To circumvent these hardware limitations, the preferred strategy is open-loop control

enhanced by software-based feedforward compensation algorithms. By characterizing the

hysteresis behavior of the piezo and applying an inverse model to the drive signals, the

response of the actuator can be effectively linearized. This software-driven approach scales

effortlessly across several channels without the prohibitive overhead of extra hardware.

Over the past two decades, various frameworks have been developed to model and com-

pensate for piezoelectric hysteresis. Classical approaches include the Preisach model, which

captures hysteresis through a superposition of elementary bistable operators, and the P-I

model, which employs a weighted sum of play operators [

]. Conceptually, a play

operator acts as a mathematical representation of mechanical backlash: its output remains

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constant upon a reversal of direction until the input change exceeds a deﬁned threshold

width, after which the output linearly tracks the input. The P-I model is particularly notable

because it admits an analytical inverse that can be cascaded with the physical actuator as a

feedforward pre-distortion signal to yield a linear response.

To handle more complex behaviors like asymmetric hysteresis, saturation, and rate de-

pendence, advanced direct-inverse and modiﬁed approaches have been developed [

For example, classical symmetric P-I play operators, which apply the exact same threshold

and slope regardless of whether the voltage is increasing or decreasing, struggle to capture

the divergent ascending and descending branches of physical piezoceramics. Utilizing

modiﬁed asymmetric play operators, which assign distinct mathematical characteristics

to the forward and return paths, allows for a more accurate description of this directional

behavior with fewer model parameters [

]. Furthermore, while the classical P-I model

possesses a straightforward analytical inverse, utilizing the aforementioned modiﬁed mod-

els often introduces complex mathematical inversions that are computationally heavy for

real-time control. To resolve this, Gu et al. demonstrated a direct-inverse compensation

method that characterizes the inverse hysteresis effect directly, bypassing these heavy

calculations and achieving tracking error reductions of up to 90% [

]. Similarly, Yu et al.

utilized a rate-dependent modiﬁed P-I model to reduce linear positioning errors from 10%

to below 1% [33].

More recently, data-driven methods have emerged as a powerful alternative. Ma-

chine learning approaches, such as the sparse system identiﬁcation method proposed by

Chandra et al., can discover hysteresis models for speciﬁc piezo materials, demonstrating

residual strain errors well below 1%, which stands as a performance that rivals closed-loop

systems [

]. Ultimately, the viability of these various feedforward P-I linearization tech-

niques has been successfully validated in multi-axis optical alignment stages [

]. In

our implemented system, we calibrate each piezo axis using a variant of the P-I model. By

applying a series of voltage trajectories and recording the displacement via an external laser

sensor during the calibration phase, we identify the exact hysteresis curve. This model is

then inverted in real time by the controller to pre-distort commanded moves. Additionally,

a drift compensation table is applied via software to administer small, time-based offset

adjustments that counteract the slow creep occurring after a step maneuver.

Crucially, the impact of hysteresis extends beyond spatial accuracy; it introduces a

signiﬁcant temporal penalty. When an alignment algorithm commands a move to a lower

voltage position, the system must perform a reset-to-zero maneuver to ensure the actuator

remains on the well-characterized forward path, rather than traversing an unpredictable

return path. This reset requirement fundamentally differentiates the movement overhead

of various search algorithms. Methods that frequently require downward voltage moves,

such as Bayesian algorithms that prioritize broad exploration before exploitation, incur a

disproportionate time penalty compared to monotone hill-climbing strategies. Understand-

ing how this temporal penalty interacts with the speciﬁc search strategy of each algorithm

across different operating speeds is a key contribution of this work. As our results in

Section 4 will demonstrate, this carefully implemented open-loop strategy reliably aligns

ﬁbers with optical loss repeatability greater than 0.1 dB, validating that an algorithmic

approach to hysteresis can successfully meet the nanometric tolerances of high-volume

wafer-level photonic testing.

2.5. Algorithm Comparison Methodology

Traditional algorithm comparison often relies on averaging performance metrics such

as number of function evaluations or convergence time. However, as noted by Beiranvand

et al. [

], this approach can be misleading when the distribution of performance metrics is

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