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Micromachines 2026 , 17 , 592 6 of 34 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 ste…

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a systematic optimization of optical alignment routines speciﬁc for high-throughput PIC

testing, conﬁrming that algorithm selection has a direct and quantiﬁable impact on test

throughput and yield in production environments [

]. Fully automated edge-coupling

probers have further shown that optimized optical search routines and move sequences

can reduce per-die alignment time to the multi-second range, yielding throughputs on the

order of several thousand optical alignments per hour in single-ﬁber scenarios [10].

A complementary challenge, intimately linked to both alignment speed and precision,

concerns the motion stage itself. In automated PIC assembly and test, the demands for high

positioning precision and high throughput are fundamentally opposed: operating a motion

stage faster imposes higher control bandwidth requirements and degrades repeatability [

Mandelli et al. developed machine-learning models to predict motion stage precision as

a function of operating speed for PIC assembly systems, demonstrating that data-driven

approaches can characterize and compensate for these trade-offs without ofﬂine calibration

runs [

]. These ﬁndings directly motivate the evaluation of high-speed operating regimes

in the present work, and the need to understand how alignment algorithms degrade under

reduced repeatability conditions.

2.2. Optical Coupling Methods

Two primary optical coupling architectures are relevant to wafer-level PIC testing,

as illustrated in Figure 1. As a foundational concept, edge couplers offer high coupling

efﬁciency and broad optical bandwidth by laterally aligning an optical ﬁber to the cleaved

or polished chip edge, requiring tight positional tolerances (typically below 1

m) [

However, as illustrated conceptually in Figure 1a, pure lateral access is often physically

impossible in densely packed wafer-scale environments, necessitating the use of specialized

vertical-to-horizontal beam delivery tools. Grating couplers provide a more convenient

vertical-incidence interface compatible with full-wafer probing without edge preparation,

but at the expense of higher insertion loss, narrower wavelength bandwidth, and moderate

polarization sensitivity [23,24].

Most integrated probe card solutions for grating couplers rely either on external

positioning stages or on beam-expansion optics to reduce sensitivity to lateral misalignment,

the latter incurring an additional insertion-loss penalty [

]. The Eclipse Dynamic system

investigated here targets grating coupler interfaces, where the piezoelectric FAU provides

the ﬁne positioning needed to maximize coupling efﬁciency across the full 15

µm ×

µm

scan range.

Figure 1. Schematic diagrams of the two primary optical coupling approaches: (a) traditional

horizontal edge coupling (which, for high-density wafer-level testing, is modiﬁed using micro-lenses

to allow vertical ﬁber access) and (b) Grating coupling [26].

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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

https://doi.org/10.3390/mi17050592