Dual-Axis Evaluation Protocol

• Dual-Axis Evaluation Protocol is a systematic framework that measures two orthogonal dimensions, such as acceleration and rotation, using coordinated sensor arrays and tailored pulse sequences.
• It employs paired or orthogonally arranged sensors, along with decoupling strategies, to achieve high data rates and precision in applications like inertial sensing, radar, and calibration.
• The approach enhances measurement accuracy and real-time analysis by integrating advanced methodologies, including atom interferometry and interferometric signal processing.

A dual-axis evaluation protocol is a systematic framework for simultaneously measuring two orthogonal operational dimensions—commonly acceleration and rotation, or azimuthal and elevation velocities—within a physical or algorithmic system. Such protocols are increasingly central in inertial sensing, radar systems, and calibration methodologies, enabling enhanced multidimensional characterization in laboratory and applied environments. Dual-axis architectures typically leverage paired or orthogonally arranged sensors, algorithmic decoupling, and differential measurement strategies to achieve robust, high-throughput evaluation across the targeted axes.

1. Dual-Axis Evaluation Mechanisms

In atom interferometry-based inertial sensing (Rakholia et al., 2014, Yan et al., 2023), dual-axis protocols are realized by launching two cold atomic ensembles from spatially separated magneto-optical traps (MOTs), or by using counter-propagating atomic beams with transverse cooling. These ensembles are interrogated simultaneously using stimulated Raman pulse sequences:

• The pulse geometry is arranged so that the effective Raman wavevectors $+\vec{k}_e$ and $-\vec{k}_e$ interrogate the ensembles in opposite directions, producing Doppler-resolved transitions.
• The resulting differential phase shifts encode acceleration (as the common-mode sum) and rotation (as the difference).

In millimeter-wave radar, a dual-axis protocol is implemented by correlating received signals along two orthogonal baselines (Merlo et al., 2020). The baseline orientation enables independent measurement of azimuthal and elevation angular velocities:

• The orthogonal arrays (X and Y axis) provide separate interferometric channels.
• Combined with Doppler measurements, this yields full three-dimensional velocity evaluation: radial velocity (via Doppler), azimuthal and elevation velocities (via interferometric phase differences).

In automated accelerometer calibration (Kokuyama et al., 2022), a two-axis positioning stage enables multipoint probing over spatially distributed locations. This scans calibration artifacts (e.g., laser reflection adapters) to decouple the effects of nonrectilinear motion and local deformation.

2. Data Acquisition Rates and Sensitivity

Protocols targeting dynamic environments achieve high data rates by exploiting rapid recapture and continuous ensemble exchange (in atom interferometers) or parallel signal channels (in radars).

Atom interferometer dual-axis systems (Rakholia et al., 2014) report measurement repetition rates of 50–100 Hz, constrained by atomic recapture cycles and ensemble loading times. Sensitivities approach:

• Acceleration: $0.92\, \mu g/\sqrt{\text{Hz}}$
• Rotation: $1.07\, \mu\text{rad}/s/\sqrt{\text{Hz}}$

Continuous interferometric sensors with transverse collimation can further suppress dead time, yielding sensitivities such as:

• Rotation: $0.25\,(\mu\text{rad}/s)/\sqrt{\text{Hz}}$
• Acceleration: $0.12\, \text{mg}/\sqrt{\text{Hz}}$ (Yan et al., 2023)

Radar dual-axis protocols achieve root-mean-square velocity errors of $41.01\, \text{mm}\cdot\text{s}^{-1}$ to $45.07\, \text{mm}\cdot\text{s}^{-1}$ in three-dimensional tracking, with trajectory angle RMSEs as low as $5.11^\circ$ (Merlo et al., 2020).

Automated calibration stages obtain sub-micron spatial resolution and $\leq$0.1% dispersion by rapid, multipoint probing.

3. Experimental Architectures and Methodologies

Atom Interferometers

• Compact quartz vacuum cells house dual MOTs, with trap zones spaced $\sim$36 mm apart.
• Ensembles are launched via detuning-based optical molasses, reaching velocities $\sim$2.5 m/s.
• Stimulated Raman beams (angled $\sim$10° from normal) ensure both directionality—and Doppler resolution for dual-axis phase encoding.
• Ensembles are recaptured at the opposing trap, preserving high atom numbers for subsequent cycles and improving SNR.

Radar Interferometry

• Arrays consist of three receive antennas (square configuration, baseline $L = 7.26\,\lambda$) and a single transmitter at $41.8$ GHz.
• Signal processing involves correlation of downconverted signals from orthogonal baselines ($X$, $Y$, and $-45^\circ$) and Doppler extraction from a dedicated receiver.
• Experiments include motion control (linear guide or adjustable rails), baselined precision, and angular sweeps.

Calibration Platforms

• Two-axis linear stages automate spatial coverage over accelerometer mounting surfaces.
• Heterodyne laser interferometers deliver nanometric displacement measurements at over 450 grid points per run.
• Data reduction uses sine-fitting, spatial averaging, and deflection modeling to correct uncertainty components.

4. Mathematical Formalism and Decoupling Strategies

Dual-axis protocols rely on explicit mathematical formulations for phase extraction and uncertainty quantification.

Atom Interferometry Phase Separation:

$$\Delta \phi = k_e \cdot (a - 2 \vec{v} \times \vec{\Omega}) T^2$$

where $a$ is acceleration, $\vec{\Omega}$ rotation, $T$ interrogation time, and $k_e$ effective wavevector.

Acceleration and rotation components are decoupled: $$\phi_+ = \frac{\phi_a + \phi_b}{2}, \quad \phi_- = \frac{\phi_a - \phi_b}{2}$$

Radar Interferometric Velocity:

$f_\omega = \omega D_\lambda \cos(\alpha)$

$v_\alpha = \frac{f_\omega R}{D_\lambda}$

with $D_\lambda$ baseline in wavelengths, $R$ range, $f_\omega$ interferometric frequency.

Combined velocity vectors: $$v_\theta = \sqrt{v_{\alpha,x}^2 + v_{\alpha,y}^2}, \quad \beta = \tan^{-1}(-v_R/v_\theta)$$

Calibration Stage Deflection Ratio:

$$\frac{\delta}{d} = \frac{32 WH^3}{8L^3-4Lb^2+b^3} \cdot \frac{m(2\pi f_{\text{vib}})^2}{E}$$

with $W,H,L,b$ geometry factors, $m$ mass, $E$ Young’s modulus.

5. Applications, Impact, and Limitations

Dual-axis evaluation protocols underpin:

• Quantum inertial measurement units (IMUs) for navigation, gravimetry, and seismology (Rakholia et al., 2014, Yan et al., 2023).
• Automotive radar, robotics, and HCI, leveraging direct multidimensional velocity measurement, improved discrimination, and efficiency (Merlo et al., 2020).
• High-frequency accelerometer calibration, enabling automated, uncertainty-reduced metrology (Kokuyama et al., 2022).

Challenges include:

• Trade-off between data rate and phase sensitivity; short interrogation times diminish accumulated phase ($\propto T^2$).
• Spatial inhomogeneity, nonideal beam profiles, and ensemble temperature cause loss in interferometric contrast.
• Calibration can be limited by surface deformation, positional error, and vibrational artifacts.

Remedies are proposed, including enlarged Raman beam waists, composite pulse sequences, advanced cooling, and mechanical stabilization.

6. Comparative Advantages and Future Directions

Dual-axis protocols provide increased information content per measurement cycle, enabling simultaneous extraction of orthogonally decoupled signals and improved bandwidth. Their adoption facilitates:

• Zero dead-time operation and dynamic environment applicability (continuous atomic beams, real-time recapture).
• Reduction in system complexity and improved real-time computational efficiency (single transmitter with multi-channel radar).
• Enhanced metrological traceability and uncertainty quantification (automated calibration platforms).

Future work is directed toward increased sensitivity via advanced cooling and phase control techniques, fine-grained uncertainty modeling in calibration, and wider integration into navigation and sensing platforms operating outside laboratory settings.

7. Summary Table: Dual-Axis Evaluation Protocol Implementations

Domain	Mechanism (Axes)	Data Rate/Sensitivity
Atom Interferometer (Rakholia et al., 2014, Yan et al., 2023)	Cold ensemble exchange; counter-propagating beams (acceleration/rotation)	50–100 Hz; $\sim\mu g/\sqrt{\text{Hz}}$, $\sim\mu\text{rad}/s/\sqrt{\text{Hz}}$
Radar Interferometry (Merlo et al., 2020)	Orthogonal baselines; correlation (azimuth/elevation + Doppler)	RMSE $\sim$40–45 mm/s; RMSE $\sim$5–10°
Calibration Stage (Kokuyama et al., 2022)	Two-axis positioning; spatial multiplexing (displacement/deformation)	$<$0.1% repeatability; $\leq$1 µm error

This survey demonstrates that dual-axis evaluation protocols—whether in atomic, electromagnetic, or calibration domains—serve as foundational methodologies for multidimensional measurement, decoupling, and calibration across advanced sensing modalities.