Mid-Anneal Measurement Techniques

• Mid-anneal measurement is a technique for probing intermediate states during annealing, capturing transient structural and electronic changes in materials.
• The method is applied across domains, from in-situ optical scatterometry in coatings and controlled charge collection in silicon diodes to quench-based readouts in quantum annealing.
• It delivers actionable insights into process thresholds and defect evolution, guiding improved manufacturing protocols and device designs.

Searching arXiv for the cited works to ground the article in recent and original sources. Mid-anneal measurement denotes measurement performed during an annealing process, or after an intermediate, standardized annealing interval, rather than only before and after annealing. In optical-coating studies it refers to in-situ imaging of scatter while a sample is being heated and held at temperature; in irradiated-silicon and cryogenic-CCD studies it refers to repeated characterization at controlled intermediate anneal states or at equivalent reference times such as $60^\circ\mathrm{C}$; in quantum annealing it refers to approximate probing of the evolving solution distribution at intermediate anneal fractions, typically by quench-based readout rather than by direct access to the instantaneous quantum state (1901.11400, Kałuzińska et al., 3 Mar 2025, Pelofske et al., 2019).

1. Conceptual scope and terminology

The term is used in several technically distinct ways. In optical scatterometry for amorphous coatings, the defining feature is genuinely in-situ observation: coated optics are illuminated and imaged while they are being annealed, so that scatter can be indexed by anneal time or instantaneous temperature without removing the sample from the thermal environment (Rezac et al., 2022, 1901.11400). In irradiated silicon-diode studies, the term is more operational: a “mid-anneal state” is the intermediate part of an isothermal annealing history, after the very early beneficial phase has largely unfolded but before reverse annealing dominates; for the 8-inch p-type HGCAL pad diodes studied at $60^\circ\mathrm{C}$, this corresponds to the region around the maximum of charge collection efficiency and the minimum of saturation voltage, reached after $O(10^2)$ minutes (Kałuzińska et al., 3 Mar 2025). In cryogenic CCD anneal cycling, mid-anneal measurements are the snapshots taken at $153\,\mathrm{K}$ after each intermediate warm stage during a stepwise $143$–$188\,\mathrm{K}$ anneal protocol, rather than measurements during the warm state itself (Parsons et al., 2021). In thin LGAD studies, the corresponding intermediate regime is the $80$–$1000$ min window at $60^\circ\mathrm{C}$, after the principal gain-layer changes have occurred and before very long reverse annealing dominates (Cindro et al., 2020).

Quantum annealing uses the phrase differently. On a D-Wave–type machine there is no literal measurement of the instantaneous quantum state midway through the anneal; instead, intermediate probing is approximated by following the default anneal schedule up to a chosen time and then rapidly quenching to $s=1$ for readout (Pelofske et al., 2019). A later formal analysis treats mid-anneal measurement abstractly as measurement at an intermediate time $60^\circ\mathrm{C}$0 that maximizes the probability of a target configuration, especially when desirable configurations are low-lying excited states rather than the final ground state (Takahashi et al., 27 Jul 2025). This suggests a family resemblance rather than a single universal protocol: all usages are concerned with extracting information from a partially annealed state, but the operational meaning depends strongly on the physical system.

2. Experimental realizations and measured observables

Across domains, mid-anneal measurement combines a controlled anneal schedule with observables chosen to reveal intermediate structural, electrical, or computational states.

Domain	Measurement configuration	Principal outputs
Optical coatings	Imaging scatterometer built into a vacuum annealing oven, or industrial air-annealing oven with viewports	BRDF at a fixed angle, spatial scatter images, TIS or point-scatterer statistics
Irradiated silicon diodes	IR-top TCT with IV/CV during isothermal anneal	$60^\circ\mathrm{C}$1, CCE, $60^\circ\mathrm{C}$2, effective doping trends
Cryogenic irradiated CCDs	Fe55 X-ray, dark current, and trap pumping after each anneal step	Parallel CTI, dark current, trap landscapes
Quantum annealers	Custom anneal schedule plus quench and repeated shots	Slice distributions, energies, Hamming distances, freeze-out indicators

In optical-coating work, the measurement apparatus is an imaging scatterometer integrated with the annealing environment. The vacuum implementation uses a $60^\circ\mathrm{C}$3 SLED at near-normal incidence, collection at a fixed scattering angle of $60^\circ\mathrm{C}$4, a single converging lens, an iris, a $60^\circ\mathrm{C}$5 low-noise CCD, and a $60^\circ\mathrm{C}$6 bandpass filter, with images acquired every $60^\circ\mathrm{C}$7 (1901.11400). The air-annealing implementation uses an industrial oven with front and rear optical ports, normal-incidence $60^\circ\mathrm{C}$8 SLD illumination, imaging at $60^\circ\mathrm{C}$9, and one bright plus one dark image per minute, enabling time-stamped BRDF and image sequences during ramps and dwells (Rezac et al., 2022). The corresponding scatter metric is the bidirectional reflectance distribution function,

$O(10^2)$0

with image-based calibration used to convert CCD counts into scattered power per unit solid angle (1901.11400).

In irradiated silicon-diode studies, the central observables are charge collection and effective field coverage. The collected charge is defined from the transient current waveform as

$O(10^2)$1

and the charge collection efficiency is

$O(10^2)$2

The same work uses a saturation voltage $O(10^2)$3, extracted from two linear regimes in $O(10^2)$4, as an effective marker of the onset of full-field coverage (Kałuzińska et al., 3 Mar 2025). In long-term temperature-dependent annealing studies on the same 8-inch p-type material, $O(10^2)$5, IV/CV, and charge collection are measured from $O(10^2)$6 to $O(10^2)$7, with Hamburg-model-based fits used to infer time constants and activation energies (Diehl et al., 18 Dec 2025).

Cryogenic CCD mid-anneal work uses a different observable set. A proton-irradiated CCD280 is characterized after each anneal step using Fe55 X-ray data for parallel CTI, dark-current measurements, and trap pumping to extract emission-time-constant distributions. In the LGAD studies, CV/IV measurements at $O(10^2)$8 and timing plus charge measurements at $O(10^2)$9 are repeated between anneal steps, so that full depletion voltage, gain-layer depletion voltage, leakage current, collected charge, and time resolution can be tracked over $153\,\mathrm{K}$0 min of annealing at $153\,\mathrm{K}$1 (Parsons et al., 2021, Cindro et al., 2020).

3. Temporal protocols and data reduction

Mid-anneal measurement is fundamentally a matter of synchronizing data acquisition with anneal history. The optical-coating apparatus described for vacuum operation is designed around a LIGO-like protocol with a ramp rate of roughly $153\,\mathrm{K}$2, design temperatures up to $153\,\mathrm{K}$3, and continuous acquisition every $153\,\mathrm{K}$4, so that scatter can be replotted against time or temperature during ramps or during fixed-temperature soaks (1901.11400). The air-annealing scatterometer similarly acquires one bright and one dark image every $153\,\mathrm{K}$5, together with incident power and thermocouple temperature, so that BRDF$153\,\mathrm{K}$6 and image morphology can be analyzed throughout the thermal cycle (Rezac et al., 2022).

Image reduction in these optical systems follows a common pattern. Background is removed by subtracting either SLED-off images from SLED-on images or dedicated dark frames, a region of interest is defined around the illuminated area, and counts are normalized by exposure time and incident power. The vacuum implementation explicitly uses SLED on/off subtraction, region masking, pixel-wise BRDF conversion, and scalar outputs such as average BRDF over the illuminated region, TIS over the field of view, and the count and brightness of point-like scatterers (1901.11400). The air system adds a concentric-ROI extrapolation, in which BRDF-related counts are fit as a linear function of ROI area and the intercept is taken as the localized scatter without diffuse background contamination (Rezac et al., 2022).

The silicon-diode work defines time not only by elapsed oven time but by total equivalent $153\,\mathrm{K}$7 time, combining the in-reactor offset and subsequent isothermal steps through Arrhenius/Hamburg scaling. The detailed 2023 campaign includes measurements at $153\,\mathrm{K}$8 and $153\,\mathrm{K}$9 min at $143$0, making it possible to resolve the non-monotonic CCE$143$1 and $143$2 curves across beneficial and reverse annealing (Kałuzińska et al., 3 Mar 2025). The CCD anneal-cycling experiment uses approximately five-day dwells at each temperature step and repeats the entire characterization back at $143$3 after every warm stage, with CTI fits excluding the first $143$4 rows because of slow-trap effects near the readout (Parsons et al., 2021).

Quantum-annealing slicing is the most algorithmically explicit realization of temporal indexing. For a slice at time $143$5, the schedule follows the default anneal up to $143$6, then performs the steepest allowed ramp to $143$7 by $143$8, and collects typically $143$9 shots. Repeating this over many $188\,\mathrm{K}$0 values yields time-resolved estimates of the best energy, Hamming distance between adjacent slices, and qubit-level flipping histories (Pelofske et al., 2019).

4. Characteristic regimes in thermal annealing

In material systems, mid-anneal measurement is most informative when the anneal passes through distinct regimes. For optical coatings, the relevant contrast is between benign scatter reduction and damage-induced scatter growth. Initial vacuum tests on single-layer quarter-wave Ti:Ta$188\,\mathrm{K}$1O$188\,\mathrm{K}$2 to $188\,\mathrm{K}$3 found no increase in diffuse background at $188\,\mathrm{K}$4 and no significant change in the number or brightness of point-like scatterers, consistent with the expectation that $188\,\mathrm{K}$5 is below typical crystallization onset ranges for Ta$188\,\mathrm{K}$6O$188\,\mathrm{K}$7 (1901.11400). A larger vacuum study to $188\,\mathrm{K}$8–$188\,\mathrm{K}$9 found that the scatter of three of four coated optics decreased during annealing, by $80$0–$80$1 for tantala and by up to $80$2 for titania-doped tantala, while the fourth remained constant; angle-resolved measurements before and after vacuum annealing suggested improvement in three of four samples (Capote et al., 2020). Air annealing extends the accessible regime: samples annealed to $80$3–$80$4 showed a decrease in BRDF with no signs of damage, whereas a sample ramped to $80$5 showed a sudden, essentially permanent factor-of-$80$6 BRDF increase starting near $80$7, and a $80$8, $80$9 h soak produced progressive crystallization that emerged tens of minutes into the soak and saturated after roughly $1000$0–$1000$1 h (Rezac et al., 2022).

In neutron-irradiated 8-inch p-type silicon diodes, the $1000$2 anneal history is explicitly divided into short-time beneficial annealing, intermediate or mid-anneal behavior, and late reverse annealing. The intermediate regime is roughly $1000$3–$1000$4 min at $1000$5: CCE is near its maximum, reverse annealing has started but is not yet dominant, and $1000$6 passes through its minimum (Kałuzińska et al., 3 Mar 2025). The time of that optimum depends on material: FZ $1000$7 and $1000$8 devices peak around $1000$9 min, whereas EPI $60^\circ\mathrm{C}$0 devices peak around $60^\circ\mathrm{C}$1 min. The same study reports that all thicknesses exhibit a non-monotonic CCE$60^\circ\mathrm{C}$2 at $60^\circ\mathrm{C}$3, with early increase from beneficial annealing and later gradual decline from reverse annealing, and that the mid-anneal point serves as an operationally relevant “best-case” estimate of bulk performance (Kałuzińska et al., 3 Mar 2025). The later temperature-dependent extension shows that the same minimum in $60^\circ\mathrm{C}$4 shifts from $60^\circ\mathrm{C}$5–$60^\circ\mathrm{C}$6 min at $60^\circ\mathrm{C}$7 to days or weeks at lower temperatures, enabling direct translation to HL-LHC warm-stop scenarios (Diehl et al., 18 Dec 2025).

Thin LGADs show a different but related pattern. The principal effect of annealing on timing resolution and collected charge is not very large and mainly occurs within the first few tens of minutes at $60^\circ\mathrm{C}$8, reflecting a decrease of around $60^\circ\mathrm{C}$9 in active initial acceptor concentration in the gain layer. By the $s=1$0–$s=1$1 min interval, gain-layer depletion voltage, leakage current, collected charge, and timing performance are largely stabilized; only at very long times does reverse annealing in the bulk significantly increase the field in the gain layer and hence the gain at a given voltage (Cindro et al., 2020).

The cryogenic CCD study shows that a mid-anneal regime need not correspond to a performance extremum. As the device is warmed stepwise from $s=1$2 to $s=1$3 and remeasured at $s=1$4, the trap landscape changes appreciably: continuum traps decrease, the divacancy peak increases, and the total number of traps in the probed $s=1$5 range decreases slowly. Yet the parallel CTI measured at $s=1$6 remains effectively unchanged within errors, implying that the net population of CTI-effective traps is largely preserved even as the defect landscape reorganizes (Parsons et al., 2021).

5. Quantum-annealing interpretations

In quantum annealing, mid-anneal measurement is primarily a diagnostic or algorithmic construct rather than a direct physical readout of the instantaneous state. The D-Wave 2000Q study makes this explicit: one never gets a literal mid-anneal measurement of the instantaneous quantum state, only classical bitstrings at the end of a programmed anneal. The workaround is “slicing,” in which a custom anneal schedule follows the default path up to time $s=1$7, then quenches as steeply as hardware allows to $s=1$8, after which the final bitstring distribution is interpreted as an approximate snapshot of the configuration near $s=1$9 (Pelofske et al., 2019). With $60^\circ\mathrm{C}$00 shots per slice and $60^\circ\mathrm{C}$01 temporal resolution, the method reveals a pronounced energy decrease and subsequent plateau, Hamming-distance stabilization, and qubit-dependent freeze-out. For an optimized QUBO, the global freeze-out point was inferred around slice $60^\circ\mathrm{C}$02 of a $60^\circ\mathrm{C}$03 anneal (Pelofske et al., 2019).

A later formal treatment quantifies when such intermediate measurement can be advantageous. For a target probability $60^\circ\mathrm{C}$04, the proposed effectiveness metric is

$60^\circ\mathrm{C}$05

This metric is large only when annealing improves the target probability above the initial random level and when the maximum occurs strictly before the final readout time (Takahashi et al., 27 Jul 2025). The same analysis shows that mid-anneal measurement is most effective when desired solutions are low-lying excited states close in energy to the ground state, and when the Hamming distance between the ground and desired excited states is small. In fully connected Ising models, the effectiveness remains nonzero with increasing system size, indicating that the phenomenon is not confined to very small toy instances (Takahashi et al., 27 Jul 2025).

The quantum-annealing literature therefore uses mid-anneal measurement in two complementary senses: as a hardware-limited quench-based approximation to intermediate-time probing, and as a theoretical strategy for harvesting populations that are transiently favorable even when the final ground state is not the desired solution.

6. Limitations, ambiguities, and scientific significance

A persistent source of ambiguity is that “mid-anneal measurement” is not methodologically uniform. In optical-coating work it often means truly in-situ high-temperature measurement, in silicon-detector studies it may mean characterization after an intermediate standardized anneal interval at a reference temperature, and in quantum annealing it is commonly an approximation based on quench-based readout rather than direct access to the evolving state (Rezac et al., 2022, Kałuzińska et al., 3 Mar 2025, Pelofske et al., 2019). This suggests that the term is best treated as a context-dependent descriptor of intermediate anneal interrogation rather than as the name of a single technique.

Each realization has its own systematic limitations. Optical scatterometry in hot ovens must contend with blackbody radiation, heater flashing, contamination, thermal expansion, stray light, and single-angle geometry; even the air-annealing system notes that heater-element flashing is a major source of BRDF noise and that thermocouple temperature is only a proxy for actual coating temperature (Rezac et al., 2022). The vacuum optical apparatus similarly relies on SLED on/off subtraction, bandpass filtering, and careful masking to isolate coating scatter from ceramic and metal backgrounds (1901.11400). In silicon detectors, $60^\circ\mathrm{C}$06 is not identical to a classical depletion voltage, CC-extracted and CV-extracted saturation voltages differ, and the use of only $60^\circ\mathrm{C}$07 injection precludes explicit extraction of separate trapping times (Kałuzińska et al., 3 Mar 2025). In the cryogenic CCD protocol, “mid-anneal” states are reconstructed at $60^\circ\mathrm{C}$08 after each warm stage rather than measured during the warm stage itself, and CTI fits intentionally exclude the first $60^\circ\mathrm{C}$09 rows because of slow-trap behavior (Parsons et al., 2021). Quantum slicing is limited by the anneal-schedule slope constraint, finite time resolution, and the fact that the resulting samples are classical bitstrings obtained after the quench rather than direct snapshots of superposition or coherence (Pelofske et al., 2019).

Despite this heterogeneity, the scientific role of mid-anneal measurement is consistent across fields. In optical coatings it identifies whether and when annealing begins to create irreversible scatter through crystallization, blistering, or related damage, and it provides empirical guidance for safe temperature ceilings and modified anneal recipes (1901.11400, Rezac et al., 2022). In irradiated silicon detectors it locates the window of maximum CCE and minimum $60^\circ\mathrm{C}$10, anchors Arrhenius/Hamburg extrapolations to realistic warm-stop conditions, and informs detector-thickness allocation and operating-bias strategy for HL-LHC systems (Kałuzińska et al., 3 Mar 2025, Diehl et al., 18 Dec 2025). In cryogenic CCDs it shows that defect populations can evolve without materially changing CTI up to $60^\circ\mathrm{C}$11, implying that CTI correction below that range need not depend strongly on detailed thermal history (Parsons et al., 2021). In quantum annealing it exposes freeze-out, schedule sensitivity, and the circumstances under which intermediate-time readout can outperform conventional end-of-anneal measurement for desired excited-state solutions (Pelofske et al., 2019, Takahashi et al., 27 Jul 2025).

Mid-anneal measurement is therefore best understood as an intermediate-state characterization paradigm. Its central value lies not in replacing terminal measurements, but in revealing kinetics, thresholds, and transient regimes that are invisible to purely pre- and post-anneal comparisons.