Hybrid active-passive optical stabilization for field-deployable atom gravimetry

Abstract

In this paper, the authors report a compact, quasi-monolithic optical distribution module (220 mm × 170 mm × 65 mm) developed toward field-deployable ⁸⁷Rb atom gravimetry. The module combines a Zerodur substrate with a PR-20 piezoelectric rotation stage having a closed-loop angular resolution of 3.5 μrad. The stage drives one wedge in a compact double-wedge layout and provides local correction along a calibrated one-dimensional actuator trajectory. Without active correction, the Raman-path fiber-coupled power decreased to 42% of its 25°C baseline under a 10°C chamber condition; one-dimensional active correction recovered it to approximately 80%. In atom-interferometer operation under the same cold condition, 100 consecutive k-reversal scan pairs showed that the branch-averaged fitted fringe visibility increased from 15.65% to 29.28%, a 1.87-fold improvement. The controller performs local search followed by bounded setpoint maintenance on a calibrated monotonic branch. These results demonstrate optical-delivery stabilization during controlled 10°C–40°C thermal cycling and fringe-visibility recovery at 10°C, providing a compact route toward more robust transportable quantum sensors.

Graphical abstract

Keywords

Atom gravimetry / Beam-pointing stabilization / Double-wedge beam steering / Fiber-coupled power stabilization

1 Introduction

Since the seminal demonstration of atom interferometry in 1991 [1], cold atom technology has evolved from a platform for fundamental physics [2–4] into a cornerstone of precision inertial sensing [5–10]. The exceptional sensitivity of these instruments has driven a transition toward practical applications in geophysics [11–14], autonomous navigation [15], and resource exploration [16], creating an urgent demand to migrate these systems from controlled laboratory environments to mobile platforms [17–20].

However, transportable operation imposes stringent constraints on Size, Weight, Power, and Cost (SWaP-C) while requiring high robustness against environmental disturbances. The laser system constitutes a critical bottleneck: it must deliver stable optical power and high polarization purity under thermal and mechanical stress [21–26]. While all-fiber architectures offer compactness, they frequently suffer from polarization extinction ratio (PER) degradation due to stress-induced birefringence. Conversely, free-space optics offer superior polarization maintenance but are inherently sensitive to thermo-mechanical misalignment. In particular, the coupling efficiency into single-mode fibers is critically sensitive to beam pointing; μradian-level angular deviations caused by minute thermal deformations can severely compromise the instrument’s signal-to-noise ratio [27–33].

While compact atom-interferometer laser and optical subsystems have advanced substantially [34–37], and representative free-space optical distributions span liter-scale packages [23,30,32], cross-study comparison remains difficult because module boundaries, temperature tests, and stability metrics are not standardized. Long-term alignment stability in hybrid glass-ceramic/metal assemblies remains a persistent challenge.

We present a compact optical distribution module designed for transportable ⁸⁷Rb atom gravimeters. The module combines a Zerodur substrate with a passive-active double-wedge correction unit, in which one piezo-actuated wedge corrects a calibrated one-dimensional component of thermal pointing drift. This reduced-order architecture trades the arbitrary two-dimensional scan capability of a fully actuated Risley pair [38–41] for smaller volume, simpler control, and potentially improved mechanical robustness. The paper first defines the laser and optical architecture, then evaluates passive and active thermal performance, and finally interprets the measured recovery with an explicitly limited one-dimensional coupling model.

2 System architecture and optical modulation scheme

The gravimeter integrates three modules: a vacuum system with a pyramidal magneto-optical trap (MOT), a timing-control unit, and a laser optical-distribution module [42–44].

A 1560 nm fiber laser seeds the system, and frequency doubling in a PPLN crystal provides the 780.24 nm output. Frequency stabilization uses modulation-transfer spectroscopy (MTS) on the ⁸⁷Rb transition, providing an absolute reference. The main output drives the compact modulation unit shown in Fig. 1, where AOM and EOM chains synthesize the required frequencies [45,46]. The first stage, AOM1, is driven at 114 MHz in a double-pass configuration; a quarter-wave plate provides polarization rotation for beam separation (Fig. 1).

2.1 Cooling and trapping (MOT and PGC)

Trapping light is generated by AOM3 at 106 MHz. Repumping light is produced by shifting the laser by 1.5 GHz with AOM2 and modulating EOM2 at 8.296 GHz, obtained by mixing 8.2 GHz with 96 MHz. A Fabry−Perot cavity selects the +1st-order sideband. For polarization-gradient cooling (PGC), a feed-forward open-loop strategy provides the required large detuning: the laser lock is disabled to apply a −100 MHz shift while the intensity is ramped down. Simultaneously, the EOM2 drive is adjusted to 8.396 GHz (8.2 GHz + 196 MHz) to maintain the repumping resonance before the laser is relocked.

2.2 Interferometry and detection

Raman beams are synthesized using the same EOM, driven at the hyperfine splitting frequency of approximately 6.834 GHz (6.8 GHz + 34 MHz). The zeroth- and +1st-order sidebands form the Raman pair, and a linear frequency chirp compensates for the gravity-induced Doppler shift. Finally, AOM3 is tuned to 114 MHz for the state-selection blow-away pulse and to 113 MHz for detection, with the repumper restored to the MOT setting.

In the present implementation, the fiber-coupled electro-optic modulator (EOM) is placed within the 1560 nm laser-source module rather than in the 780 nm free-space optical path. Conventional hybrid arrangements can introduce approximately 3.5 dB of insertion loss when coupling a free-space beam into a fiber modulator. By performing the modulation at the fundamental telecommunication wavelength, the present all-fiber arrangement simplifies the optical-distribution module, avoids an additional free-space-to-fiber coupling stage, and improves power efficiency and mechanical stability.

This 1560 nm EOM architecture has also been used in compact and onboard atom-interferometry laser systems [35,36]. The active correction unit was installed in the Raman delivery path because this path exhibited the largest low-temperature loss and its delivered power directly affects the Raman pulse areas.

3 Optical system design

3.1 Passive thermal stabilization via Zerodur technology

The 1560 nm source architecture described above removes frequency-generation hardware from the free-space bench; the remaining task is to route, split, and couple the 780 nm beams with minimal thermomechanical drift. We therefore developed a compact, quasi-monolithic distribution module on a Zerodur glass-ceramic substrate. Zerodur was selected for its near-zero coefficient of thermal expansion, which reduces substrate-driven alignment drift relative to conventional metal breadboards [47,48].

Our assembly employs a ‘hemisphere-socket’ bonding technique that addresses the alignment constraints of conventional planar adhesion. By replacing bulky mechanical mounts susceptible to drift, optical components are permanently secured to the substrate using glass hemisphere adapters nested within precision-machined spherical sockets. Illustrated in Fig. 2a, this configuration affords three rotational degrees of freedom, facilitating precision tip-tilt adjustment. This geometry was intended to improve bond-line uniformity, facilitate tip-tilt alignment, and reduce sensitivity to adhesive shrinkage and localized thermal stress.

Within the free-space module, the 780 nm input is collimated and focused by an achromatic doublet (L1) into AOM2 (1.5 GHz), which separates the cooling and Raman functions into two paths:

Cooling path (zero order): the non-diffracted beam is recollimated by L2 and routed through double-pass AOM3 (106 MHz). The double-pass geometry supports the required cooling-frequency sweep while suppressing frequency-dependent output pointing.

Raman path (first order): the diffracted beam carries the offset required for Raman interferometry and is recollimated by L3.

The two paths are subsequently routed through the module and coupled to the output fibers serving the sensor head. This layout keeps the source-side modulation functions distinct from the module-side task of stable beam delivery.

The fully assembled optical distribution module, shown in Fig. 2b, features a compact footprint of 220 mm × 170 mm × 65 mm. The bonded Zerodur structure is designed to reduce sensitivity to mechanical shock and thermal fluctuations. The package also incorporates vibration-isolation design features suitable for transport, and the module has been transported and operated across multiple cities in China. By replacing adjustable kinematic mounts with permanent bonding, the module is designed to improve mechanical robustness for mobile gravity-survey instruments.

3.2 Active beam pointing correction system

To compensate for temperature-dependent fiber-coupling loss, a compact passive-active double-wedge correction unit was integrated into the Raman delivery path. One wedge is fixed after assembly and provides static beam-path pre-alignment, whereas the second wedge is mounted on a PR-20 piezoelectric rotation stage and provides active correction. The inline transmissive arrangement avoids the additional folded path required by a fast-steering mirror and is compatible with the compact optical-module layout.

The two matched wedges are initially set with nominally opposed deviation directions so that their baseline angular deflections approximately cancel. During assembly, the wedges are rotated together to select a useful local correction direction while preserving the nominal output axis. The static wedge is then locked, and only the PR-20-mounted active wedge is driven during operation. Near- and far-field pointing were not independently mapped over temperature; the preset direction was therefore selected operationally from beam-walk observations and fiber-coupled-power recovery rather than from a measured two-dimensional thermal-drift vector.

Rotation of the active wedge moves the focused spot along a shallow trajectory in a coupling-equivalent plane. Over the calibrated operating span, this trajectory is approximated locally as q_{\text{a}}(u) ≃ q_{\text{a}0} + J(u − u_0)e_{\parallel }, where u is the actuator command, J is the local coupling-equivalent Jacobian, and e_{\parallel } is the local scan direction. The complete fiber-coupling model and the residual component normal to this direction are discussed in Section 5. For a 2.0° N-BK7 wedge, the small-angle deviation is approximately δ ≃ (n − 1)α ≃ 17.8 mrad. The PR-20 has a closed-loop stage-rotation resolution of 3.5 μrad, corresponding to an estimated local optical increment δΔφ ≃ 17.8 mrad × 3.5 μrad ≃ 0.062 μrad. This value characterizes the local increment along the wedge trajectory and is not an experimentally demonstrated isotropic two-dimensional steering resolution.

Because only scalar fiber-coupled power is detected, equal-power points on opposite sides of a local maximum cannot be distinguished directly. The control sequence therefore consists of two stages. A bidirectional scan first locates the local maximum and selects a monotonic operating branch with positive local slope. The controller then maintains a sub-maximum setpoint on that branch using u_{k+1}\text{= sat[}{u}_k+K_{\text{e}}(R_{\text{set}}-R_k)], where R_k and R_{\text{set}} are normalized to the 25°C reference. The gain was K_e = 0.1, the command was limited to ± 10, and commands were updated at 1 Hz. The 1 Hz value is the command-update rate rather than a measured closed-loop bandwidth. Because the correction is accumulated in the command state, the algorithm is an integral-like local controller rather than a static proportional controller.

The setpoint was chosen as 75% of the 25°C baseline, below the approximately 80% maximum reached during the 10°C calibration scan. This choice preserves bidirectional command margin and avoids operating near the peak, where the local power gradient approaches zero and changes sign. The photodetector (Thorlabs PDA10CS) remained outside the thermal chamber; its local ambient temperature and reference-voltage drift were not independently recorded, so detector/reference drift could not be quantitatively separated from the measured optical signal. The local actuator relation q_{\mathrm{a}}(u)\simeq q_{\mathrm{a}0}+J(u-u_0)e_{\parallel } introduced here is incorporated into the unified coupling model of Section 5. The key specifications of the passive-active double-wedge correction unit and its control parameters are summarized in Table 1.

4 Experimental validation and performance analysis

4.1 Baseline characterization and thermal drift analysis

The system’s baseline performance was calibrated at a stabilized room temperature of 25°C. With a 780 nm fiber-laser input power of approximately 500 mW and optimized RF drive powers for the AOMs, the optical distribution module delivered approximately 100 mW in the Raman output path under the Raman operating condition; with the AOM2 RF drive switched off, it delivered approximately 150 mW in the cooling output path. These powers were used as the 100% normalization references for the subsequent environmental stress tests. The corresponding input-output losses are approximately 7.0 dB for the Raman path and 5.2 dB for the cooling path under the above operating conditions. The measured transmission of the double-wedge correction unit itself was 98.2%, corresponding to an additional insertion loss of only approximately 0.08 dB.

To evaluate the environmental robustness of the Zerodur-based quasi-monolithic design, the optical module was subjected to thermal soak tests at extreme operating temperatures of 10°C and 40°C. A sufficient settling period was used to approach thermal equilibrium, accounting for the distinct thermal time constants of the glass substrate and metal components. Table 2 summarizes the normalized fiber-coupled output power observed without active correction.

At 40°C, the passive Raman and cooling outputs remained at 98% and 86% of their 25°C baselines, respectively. These results show that the assembled module retained good optical delivery after the high-temperature soak; they do not by themselves isolate the contribution of the Zerodur substrate from that of the bonded metal components. Possible contributors include differential deformation of the hybrid Zerodur/metal assembly, AOM angle or efficiency changes, polarization changes, and path-dependent fiber-coupling sensitivity; these contributions were not independently separated.

At 10°C, the passive Raman output decreased to 42% and the cooling output to 77%. The asymmetry between the two paths indicates that the response is not governed by substrate expansion alone. Differential deformation of the hybrid Zerodur/metal assembly, AOM angular sensitivity, and downstream fiber-coupling sensitivity are plausible contributors, but their individual shares were not independently measured.

The 58-percentage-point Raman-path reduction at 10°C establishes the need for active correction, but the fiber-coupled signal alone does not uniquely identify its physical origin. The 1.5 GHz Raman-path AOM has tighter angular requirements than the cooling path, and the two paths also differ in their metal-mounted optics and coupling geometry. Accordingly, the following sections use the large recovery produced by a pointing actuator as evidence for a substantial pointing-to-coupling component, while treating the detailed thermo-mechanical mechanism as an interpretation rather than a direct measurement.

Figure 3 compares the passive thermal response of the Zerodur-based module with that of a representative aluminum reference having comparable external dimensions and similar cooling-light routing. At 40°C, the Zerodur implementation remains above 80% while the aluminum reference declines continuously; at 10°C, the corresponding settled levels are approximately 75% and 35%. The comparison supports the use of a low-expansion substrate for passive alignment retention, while Table 2 separately reports the settled Raman- and cooling-path outputs of the Zerodur module.

The aluminum reference and the Zerodur module had comparable external dimensions and similar cooling-light routing. The comparison is indicative rather than a fully identical module-to-module comparison: substrate selection is a plausible primary contributor to the different thermal responses, while assembly-specific differences, including bonding uniformity and initial alignment, remain uncontrolled factors.

4.2 Verification of active correction mechanism

To test whether active beam steering could recover the severe Raman-path loss at 10°C, we actuated the dynamic wedge of the passive-active double-wedge unit.

A bidirectional step-wise search was used to determine the local power gradient and locate the maximum along the one-dimensional wedge trajectory. As shown in Fig. 4a, the procedure restored the Raman fiber-coupled power from approximately 42% to approximately 80% of the 25°C baseline. The large reversible recovery establishes a substantial pointing-to-coupling component that is correctable along the active-wedge trajectory.

The remaining 20% deficit cannot be uniquely assigned from coupled-power data alone. Within the reduced-order model of Section 5, it is represented as an uncorrected component normal to the local correction direction together with any non-pointing losses not separated by the experiment.

Increasing the EDFA output can raise the power incident on the module but cannot restore spatial mode overlap at the single-mode fiber. A separate EDFA-current-only control experiment was not performed; therefore, the present experiment demonstrates recovery by spatial correction but does not provide a direct experimental comparison with source-power rescaling.

Nevertheless, the recovered Raman power level was sufficient to drive the interferometer sequence with improved fringe visibility under the 10°C stress condition.

Figure 4b shows the search dynamics: the bidirectional scan reached a local maximum within approximately one minute. During thermal cycling, the controller was initialized on the resulting monotonic branch and used the bounded error-accumulating law in Section 3.2 to maintain a lower 75% setpoint. Thus, Fig. 4 validates local search and recovery, whereas Fig. 5 evaluates subsequent setpoint maintenance.

4.3 Closed-loop stability under thermal cycling

Because a transportable instrument may experience slow, continuous temperature drift rather than settling at a fixed temperature, the stabilization loop must also be evaluated under dynamic thermal transitions. Figure 5 therefore presents a dynamic thermal-cycling test, complementing the settled thermal-soak values in Table 2.

The transient ~35% open-loop minimum in Fig. 5 reflects dynamic multi-time-constant relaxation during temperature ramps, distinct from the settled 42% soak value in Table 2. With local feedback engaged, the red trace remains near the 75% target. The setpoint is intentionally lower than the approximately 80% static recovery ceiling to preserve bidirectional command headroom and avoid integration toward an unattainable low-temperature target. Three non-overlapping dwell intervals were available at each temperature. After excluding the first hour of each dwell, the RMS fluctuation was 0.46% ± 0.07% at 10°C and 0.31% ± 0.05% at 40°C (mean ± SD across three dwells). These values quantify repeatability of the settled monitoring signal; they do not include between-day realignment or detector calibration uncertainty. These three settled dwell intervals characterize within-cycle repeatability and should not be interpreted as a multi-day long-term-stability test.

4.4 Impact on atom-interferometer fringe contrast

To determine whether the optical recovery propagated to the sensor signal, the module was integrated into the atom gravimeter and tested at an interrogation time of T = 50 ms.

Only the Zerodur optical distribution module was placed in the 10°C chamber; the laser source, sensor head, and electronics remained at room temperature. This boundary isolates the test to optical-delivery changes more effectively than a whole-instrument thermal test, but it is not a field-environment validation.

Figure 6 compares 100 consecutive k-reversal scan pairs before and after correction, with 32 phase samples in each negative and positive branch. Each branch was fitted separately using P=a+b\text{cos(}x+c), with visibility C=b/a. Across 200 scan-resolved branch fits per condition, the branch-averaged visibility increased from 15.65% (95% CI, 15.62%–15.69%) to 29.28% (95% CI, 29.22%–29.35%), corresponding to a 1.87-fold increase. The intervals quantify within-record fitting precision rather than cycle-to-cycle repeatability; both conditions were obtained within a single thermal cycle.

The causal interpretation is limited but direct: reduced delivered Raman power shifts the fixed π/2–π–π/2 pulse sequence away from its calibrated pulse areas, whereas active correction restores the delivered power toward that operating condition. The observed contrast increase is therefore consistent with improved Raman pulse-area matching.

The negative and positive k-reversal branches show the same trend: their fitted visibilities change from 15.66% and 15.65% before correction to 29.57% and 29.00% after correction, respectively.

The improved visibility is expected to reduce the phase-readout-noise contribution under otherwise unchanged conditions. A sensitivity claim, however, requires repeated gravity measurements and Allan-deviation analysis and is outside the present scope.

5 Reduced-order interpretation of one-dimensional fiber-coupling recovery

Section 4 establishes three experimental facts: passive Raman-path power decreased to 42% at 10°C, one-dimensional wedge correction recovered it to approximately 80%, and the fitted fringe visibility increased by 1.87-fold. This section interprets those observations in four steps. Section 5.1 defines the measurement boundary and a unified coupling model. Section 5.2 analyses the local one-dimensional correction trajectory and explains why the recovery reaches approximately 80% rather than 100%. Section 5.3 reports quantitative recovery indicators and a bounded setpoint strategy. Section 5.4 relates the recovered Raman power to improved fringe visibility without extending the result to gravimeter sensitivity.

5.1 Measurement boundary and unified coupling model

The experiment measures only scalar fiber-coupled power; near-field position, far-field angle, AOM diffraction efficiency, and polarization were not independently recorded. The normalized coupled power is therefore modeled in coupling-equivalent coordinates rather than as directly measured beam coordinates. The correction loop addresses slow, quasi-static thermal drift and associated fiber-coupling degradation. Faster disturbances, such as laser intensity noise, mechanical vibration, and air-flow-induced fluctuations, are not directly corrected by the wedge actuator and require complementary stabilization methods.

At 10°C, the Raman-path output decreased to 42% of its 25°C baseline. Because a one-dimensional wedge scan recovered it to approximately 80%, the data indicate a substantial pointing-to-coupling component correctable along the actuator trajectory. The residual deficit relative to the baseline cannot be uniquely partitioned among an orthogonal coupling mismatch, temperature-dependent AOM efficiency, polarization changes, and other transmission losses.

We therefore describe the normalized coupled power by a unified Gaussian coupling model [49] R(u,T)=L(T)\cdot \text{exp}[-\Vert q_{\text{T}}(T)-q_{\text{a}}(u){\Vert }^2/{w_{\text{eq}}}^2], where q_{\text{T}}(T) is the temperature-induced coupling-equivalent displacement, q_{\text{a}}(u) is the equivalent correction produced by the active wedge at command u, w_{\text{eq}} is an equivalent coupling tolerance, and L(T)\le 1 represents temperature-dependent losses not recoverable by beam steering (AOM efficiency, polarization, transmission). The coordinates q_{\text{T}} and q_{\text{a}} are coupling-equivalent quantities; they are not independently measured near-field or far-field beam positions. For an unconstrained representative metal component of length L_{\text{m}}, the differential free expansion relative to Zerodur scales as \text{Δ}{L}_{\text{diff}}\simeq ({\alpha }_{\text{m}}-{\alpha }_{\text{z}})L_{\text{m}}\text{Δ}T. This provides only a displacement scale; the actual optical pointing change depends on mount constraints, adhesive geometry, and optical lever arms that were not individually characterized.

Temperature-dependent acoustic velocity and Bragg matching in the 1.5 GHz AOM may change both diffraction efficiency and output angle. Published temperature coefficients for other acousto-optic materials [50] provide only an order-of-magnitude indication and cannot be transferred quantitatively to the present GaP device. Accordingly, AOM-related efficiency and angular changes are retained within L(T) and q_{\text{T}}(T) as plausible but experimentally unseparated contributions.

5.2 Local one-dimensional correction and accessible maximum

Near the operating point, the wedge correction can be locally linearized as q_{\text{a}}(u)\simeq q_{\text{a}0}+J(u-u_0)e_{\parallel }, where J is the local coupling-equivalent Jacobian and e_{\parallel } defines the scan direction. The temperature-induced displacement relative to this reference is decomposed into parallel and normal components: q_{\text{T}}-q_{\text{a}0}=q_{\parallel }{e}_{\parallel }+q_{\mathrm{\perp }}{e}_{\mathrm{\perp }}. The normalized power then satisfies R(u,T)=L(T)\text{exp}[-((q_{\parallel }-J\text{Δ}u{)}^2+{q_{\mathrm{\perp }}}^2)/{w_{\text{eq}}}^2]. Because the active wedge can only vary the q_{\parallel } component, the optimum command eliminates it, giving the trajectory-constrained maximum R_{\text{traj,}max}=L(T)\text{exp}(-{q_{\mathrm{\perp }}}^2/{w_{\text{eq}}}^2). With q_{\mathrm{\perp }}>0 and/or , the model indicates that a substantial parallel component was removed while a nonzero normal residual and the non-pointing factor jointly limit further recovery. Over the calibrated optical beam-angle span \ell \approx \text{450 μrad}, the circular trajectory has a sagitta s\simeq \ell {}^2/(8\delta )\approx \text{1.4 μrad}, which is approximately 0.3% of the scan span. The corresponding PR-20 stage rotation is approximately 25 mrad, well within the stage travel. This small curvature supports local tangent linearization over the calibrated range. The linearization supports the local search logic but does not imply that the unknown thermal-drift vector is itself one-dimensional.

Under the additional pure-pointing assumption L(T)=1, let γ denote the angle between the coupling-equivalent mismatch vector and the local actuator direction. The angle γ provides a geometric representation of the same decomposition, where q_{\parallel }=q\cos \gamma and q_{\mathrm{\perp }}=q\sin \gamma. If the initial mismatch magnitude is q, the actuator removes the parallel component q\cos \gamma, leaving the residual q\sin \gamma. Because R_0=\text{exp}(-q^2/{w_{\text{eq}}}^2), the trajectory-constrained maximum becomes R_{max}(\gamma )=\text{exp}[-q^2{\sin }^2\gamma /{w_{\text{eq}}}^2]=R_0^{{\sin }^2\gamma }. This relation is illustrated in Fig. 7b, with R_0=0.42 and the marker indicating {\gamma }_{\text{eq}}\approx 30.5{}^{\circ } corresponding to R_{\text{traj,}max}\approx 80\mathrm{\%}.

Figure 7 presents this reduced-order interpretation. The scalar detector identifies the nearest point reached on the one-dimensional trajectory but cannot recover the azimuth or magnitude of the original coupling-equivalent displacement vector.

5.3 Quantitative recovery, model bounds, and setpoint selection

The directly measured recovery from 0.42 to approximately 0.80 corresponds to a recovery of {\eta }_{\text{rec}}={\displaystyle \frac{\left(0.80-0.42\right)}{\left(1-0.42\right)}}=65.5\mathrm{\%} of the power lost relative to the 25°C reference. The logarithmic coupling loss decreased by \text{Δ}E=\text{ln}\left({\displaystyle \frac{0.80}{0.42}}\right)=0.645. Under the additional idealized assumption that the full normalized loss follows the Gaussian coupling term, this corresponds to an apparent reduction of approximately 74% in the normalized coupling-loss exponent; this model-dependent metric is reported for completeness but does not carry the same weight as the directly computed 65.5% recovery. Because L(T) and q_{\mathrm{\perp }} are not independently measured, the model cannot uniquely decompose the residual deficit. However, under the Gaussian coupling model and the assumption that L(T) remains approximately constant between the before- and after-correction states, bounds can be placed: 0.80\le L(T)\le 1, and 0\le q_{\mathrm{\perp }}/w_{\text{eq}}\le \sqrt{\left(-\text{ln}(0.80)\right)}\approx 0.472. These ranges are model-dependent but illustrate that the data are compatible with a combination of non-pointing loss and an orthogonal coupling residual, without requiring either to be the sole explanation.

The closed-loop setpoint was chosen as 75% of the 25°C baseline, deliberately below the approximately 80% local trajectory maximum to preserve bidirectional command margin and avoid the peak where \text{d}R\text{/d}u\to 0. Let e_k=R_{\text{set}}-R_k denote the setpoint error and let G\text{=d}R\text{/d}u be the local power-command slope on the selected monotonic branch. Away from command saturation, the linearized error dynamics satisfy e_{k+1}\simeq (1-K_{\text{e}}G)e_k, so local convergence requires . The calibrated gain K_e = 0.1 was selected within this locally stable operating range. The 1 Hz command-update rate is not a measured closed-loop bandwidth, and the bounded command limit of \pm 10 prevents runaway accumulation toward an unattainable command.

The one-dimensional mode is appropriate when the temperature response is repeatable, a useful monotonic branch can be calibrated, and the required correction remains within actuator limits. If future vector measurements reveal a strongly two-dimensional trajectory, the second wedge can be made adjustable or fully actuated to add steering freedom, at the cost of greater volume and control complexity.

5.4 Relation to Raman pulse-area matching

For approximately unchanged relative powers of the two Raman components and fixed pulse duration τ, the effective two-photon Rabi frequency and pulse area scale as {\text{Ω}}_{\text{eff}}\propto \sqrt{P_1{P}_2} and \text{Θ}={\text{Ω}}_{\text{eff}}\tau. The recovered Raman power therefore improves pulse-area matching, consistent with the observed visibility increase under otherwise unchanged conditions.

The visibility increase is expected to reduce phase-readout noise under otherwise unchanged conditions; repeated gravity measurements and Allan-deviation analysis are required to establish a sensitivity gain.

6 Conclusion

We have presented a compact optical distribution module (220 mm × 170 mm × 65 mm) that combines Zerodur-based passive stability with local active correction by a PR-20-driven wedge. The actuator has a 3.5 μrad closed-loop stage-rotation resolution, corresponding to an estimated 0.062 μrad local optical increment along the wedge trajectory; this is not an isotropic two-dimensional steering resolution. In controlled chamber tests, the passive Raman-path fiber-coupled output decreased to 42% of its 25°C baseline at 10°C and recovered to approximately 80% after one-dimensional active correction. Under the same cold-condition experiment, the branch-averaged fitted fringe visibility increased from 15.65% to 29.28%, or 1.87-fold. The measured 80% level characterizes the trajectory-constrained maximum under the present 10°C test condition rather than the global correction limit of the module.

The large reversible recovery establishes a substantial temperature-sensitive pointing-to-coupling component. Within the reduced-order model, the actuator recovered 65.5% of the power lost relative to the 25°C reference, while the orthogonal residual q_{\mathrm{\perp }} and non-pointing factor L(T) jointly determine the accessible maximum. The model-equivalent angle and exponent-reduction metrics provide a compact interpretation of the observed scalar recovery rather than direct measurements of the physical two-dimensional drift vector.

The module has been transported between multiple laboratory sites and subsequently used in measurement campaigns. Full field qualification will require independent pointing diagnostics, repeated thermal cycles, vibration and shock tests, humidity evaluation, and gravimeter Allan-deviation measurements.

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