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Micromachines 2026 , 17 , 592 8 of 34 constant upon a reversal of dir ection until the input change exceeds a deﬁned thr eshold width, after which the output linearly tracks the input. The P-I model is particularly notab…

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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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non-normal, particularly as the number of optimization problems increases. They advocate

for more rigorous statistical approaches including performance proﬁles and data proﬁles.

Performance proﬁles [

] measure the fraction of problems solved within a given

performance ratio relative to the best solver. Data proﬁles [

] extend this concept by

plotting the fraction of problems solved as a function of the number of function evaluations,

providing insight into convergence speed and robustness. These proﬁle-based methods are

particularly appropriate for optimization algorithm comparison, as they account for the

full distribution of performance rather than relying on potentially misleading averages.

3. Methodology

3.1. Eclipse Dynamic System Architecture

To address the severe spatial constraints associated with conventional lateral edge

coupling in high-density wafer-level environments, the Eclipse Dynamic system employs

lensed ﬁbers and 3D-printed micro-lenses [

] to redirect the optical signal from horizontal

propagation to vertical access. This conﬁguration enables dense wafer-level probing while

remaining compatible with standard prober geometries and integrated electro-optical

test operation.

Figure 2 presents the system-level architecture of the Eclipse Dynamic probe card

during wafer-level testing. The tester provides both the electrical test program and the

optical test program. The optical path comprises a source/modulation stage and a detection

stage connected to the integrated FAU in the probe head, while the mechanical alignment

loop is closed through the controller and the piezoelectric drive electronics. The FAU

is integrated into the probe head together with the electrical probes, such that optical

and electrical contact are established on the same wafer site during probing. Rather than

actuating individual ﬁbers independently, the system displaces the entire FAU as a single

rigid body in the X–Y plane, with differential actuation used for tilt compensation. This

architecture eliminates the need for external optical positioning stages and enables compact,

high-throughput wafer-level electro-optical testing.

Electrical Test

Testing program

Optical Test

Testing program

Optical

Modulator

Photoelectric

Converter

Optical Alignment Algorithm

Controller

Piezo Power Supplier

Piezo Voltage Boost

Probe Head

Align loop

Fiber Array Unit

PCB

Tester

Piezo

Wafer

Probes

Figure 2. System-level block diagram of the automated wafer-level photonic testing platform, show-

ing the tester, controller, optical source/detection chain, piezoelectric drive electronics, integrated

probe head, and wafer under test.

https://doi.org/10.3390/mi17050592