Sequential Phase Linking

• Sequential phase linking is defined as architectures and algorithms that propagate phase information stepwise to ensure robust phase synchronization across system stages.
• In nanophotonic logic, it employs phase-locking of Kerr-nonlinear resonators to achieve low-energy, high-speed optical computation with cascadability.
• The approach also facilitates phase unwrapping in interferometric imaging and real-time phase estimation in InSAR, while ensuring safe multiphase protocol interactions.

The sequential phase linking approach encompasses a set of methodologies and architectures in which phase information—interpreted as physical phase, computational phase, or protocol-progress phase—is explicitly propagated or coordinated across aligned stages, layers, or agents in a system. This enables robust cascading, enhanced resolution, or provable safety in photonic logic, quantum measurements, communication protocols, and interferometric imaging. The following sections outline key realizations across photonics, microscopy, electron-light interaction, interferometric signal processing, and theoretical computer science.

1. Fundamental Principle of Sequential Phase Linking

Sequential phase linking refers to architectures and algorithms where the output phase (physical or logical) of one stage is used, in a precisely controlled manner, as the defining reference or constraint for the subsequent stage. In various domains, this manifests as either:

• Physical phase-lock: Cascading optical or electronic modules so that their phase coherence is maintained circuit-wide, enabling phase-sensitive operations at subsequent nodes.
• Phase unwrapping/linking: Resolving multi-valued, modulo $2\pi$ phase ambiguity in interferometric measurement by stepwise, information-rich linking across spatial/temporal slices or physical layers.
• Logical phase enforcement: Explicitly encoding progress through protocol stages as phases in session-type systems for concurrent computation, ensuring safe progression and type discipline at every acquire/release transition.

This principle allows the realization of cascadable logic, absolute phase recovery, coherent electron gating, extensible phase estimation, and statically verifiable multiphase protocols.

2. Sequential Phase Linking in Nanophotonic Logic and Memory

In the context of all-optical logic and memory, sequential phase linking exploits on-chip phase coherence to enable robust cascadability of photonic logic gates and latches at the attojoule scale (Mabuchi, 2011). Here, the essential mechanism is:

• Binary signals are encoded as coherent optical states (“HIGH” $\equiv |\alpha\rangle$, “LOW” $\equiv |0\rangle$), with amplitude or phase carrying the logic value.
• Each logic or memory element comprises Kerr-nonlinear resonators whose refractive index shift depends quadratically on intra-cavity photon number, governed by the Hamiltonian

$$\hat{H}_0 = \hbar\Delta \hat{a}^\dagger \hat{a} + \tfrac{\hbar\chi}{2} (\hat{a}^\dagger)^2 \hat{a}^2$$

• Operating near the nonlinear threshold, small input changes induce abrupt $\phi(|E|)$ switching in output phase.
• Because the monolithic photonic circuit preserves absolute optical phase, the phase-shifted output (e.g., $\phi_n$) of stage $n$ becomes the local oscillator reference for the $n{+}1$ stage, enforcing a sequential phase chain.
• Interferometric logic is implemented via beam-splitters and phase shifters to combine “signal” and “bias” ports, yielding multi-input gates such as AND and NOT with phase-dependent thresholds.

This architecture supports logic gate fan-out (by splitting the coherent post-cavity output), robust bistable latches (via cross-coupled nonlinear cavities), and cascadability—all critical for integrated photonic computing. Performance is determined by parameters such as threshold energy ($\sim$500 photons/bit $\to$ 65 aJ at 1.5 µm), switching speed (10–16 ps for $\kappa/2\pi\sim10$ GHz), and phase stability across the chip (Mabuchi, 2011).

3. Sequential Unwrapping in Quantitative Phase Imaging

In dual-comb microscopy (DCM), sequential phase linking addresses the longstanding problem of phase wrapping in quantitative phase imaging, leveraging amplitude–phase coherence for robust unwrapping over large axial ranges (Mizuno et al., 2023).

The physical observable is a complex field $E(x,y,z) = A(x,y,z)e^{i\phi(x,y,z)}$, but interferometric measurement recovers only the wrapped phase $\phi_w(x,y,z) = \phi(x,y,z) \bmod 2\pi$. Unwrapping ambiguity arises when phase exceeds $2\pi$, as in most thick or multi-layer samples.

Sequential phase linking proceeds as follows:

• The confocal amplitude $A(x,y,z)$ provides localized peaks identifying interface positions with certainty (serving as $M=0$ reference).
• For each histogram along the axial direction, the algorithm proceeds from the amplitude peak, sequentially updating the $2\pi$ unwrapping integer $M(z)$ for each step:

$\phi(x,y,z) = \phi_w(x,y,z) + 2\pi M(x,y,z)$

where $M(z)$ is incremented each time a $2\pi$ phase jump is detected, guided by the expected phase slope (from refractive index $n$ and axial step $\Delta z$).

• A robust matching criterion ensures that reconstructed optical thickness between two amplitude peaks matches the absolute unwrapped phase difference:

$$n_g d = \lambda\,[(M_{z_4}{-}M_{z_2}) + (\phi(z_4){-}\phi(z_2))/2\pi]$$

This approach enables unambiguous, noise-resistant absolute phase mapping over axial ranges $>\!100$ µm with nanometer residual noise, as demonstrated for micrometer-thick cover glasses and nm-resolved surface features (Mizuno et al., 2023).

4. Blockwise Sequential Phase Linking in InSAR Time-Series

In the processing of large-scale interferometric SAR time series for earth observation, sequential phase linking appears as the problem of updating a global phase estimate as new measurements arrive, without quadratic or cubic reprocessing costs (Hajjar et al., 13 Feb 2025).

The approach is rooted in covariance fitting for phase-only vectors $\mathbf{x}\in\mathbb{T}^n$, seeking to estimate complex atmospheric or deformation phase over time. The classical method requires fitting the full $n\times n$ covariance matrix on the complex torus, an $O(n^3)$ task. Sequential covariance fitting (S-COFI-PL) adapts this as follows:

• The pre-existing phase solution $\mathbf{x}_p$ (old images) and associated covariance blocks are retained.
• When a new block of $k$ images arrives, the joint cost function (e.g., Kullback–Leibler or Frobenius) is split via Schur techniques to isolate dependence on the new block $\mathbf{x}_n$, resulting in a quadratic cost in $\mathbf{x}_n$ only.
• Majorization–Minimization (MM) optimization is then performed over $\mathbf{x}_n$ with cost $O(k^3)$ (KL) or $O(k^2p)$ (Frobenius), warm-started by the previous solution.
• Empirically, S-COFI-PL achieves phase accuracy (MSE, RMSE, UQI, SCC, SSIM) indistinguishable from batch approaches, but at a $13$–$15\%$ reduction in wall clock time and an order of magnitude improved scaling for large $n$—enabling real-time, extensible InSAR stacks (Hajjar et al., 13 Feb 2025).

5. Phase Linking Control in Electron-Light Interferometry

Sequential phase linking in ultrafast electron–light interaction is exploited in multi-zone phase-locked optical gating of free electrons traversing plasmonic near-fields (Chahshouri et al., 2023). Here:

• A slow free-electron wavepacket interacts with multiple spatially separated regions (“zones”), each supporting a localized plasmon oscillation phase-locked relative to the others.
• The coupling at each zone is $g_n = g\,e^{i[\phi_0 + (n-1)\Delta\phi]}$. The total interaction is the coherent sum $G = \sum_n g_n$, controlling the overall phase imprinted on the electron.
• Jacobi–Anger expansion yields the final electron energy-momentum spectrum with passbands or suppression set by phase offsets $\Delta\phi$, entrance phase $\phi_0$, and light polarization $\theta$:

$$|\psi(z,t)|^2 = \sum_{\ell,m} |J_\ell(2|g_z|)J_m(2|g_x|)|^2$$

• Tuning phase offsets allows constructive ($\Delta\phi=0$) or destructive ($\Delta\phi=\pi$) interference, enabling selective enhancement or suppression of recoil sidebands.
• This Ramsey-type sequential control is applicable to quantum-coherent electron shaping, PINEM state engineering, and time-resolved electron diffraction (Chahshouri et al., 2023).

6. Sequential Phase Linking in Multiparty Protocols (Session Types)

In the theory of concurrency and message-passing programs, sequential phase linking appears as manifest phased protocols in shared session type systems, as shown by Sano et al. (Sano et al., 2021):

• Each “phase” is an acquire–release cycle on a shared communication channel, made explicit via modal connectives and linear “shifts” ($\upll, \downll$ for linear acquire/release; $\upls, \downsl$ for shared).
• Cross-modal subtyping ($\upls A \leq \upll B$, etc.) enables a shared server type to be “view-cast” by clients into single-phase linear sessions, then globally returns to the shared pool for further acquire cycles.
• A coinductive “subsynchronizing” constraint ($\dsync{A}{B}{D}$) ensures that phase progressions and releases are compatible with the shared server’s offered protocol, precluding phase mismatch or unsafe reentry.
• This approach statically enforces distinct, manifest protocol phases for each client, fostering provably deadlock-free, multiphase interaction with shared providers—a property not achievable in prior, purely linear, or non-subtyped systems (Sano et al., 2021).

7. Implementation Considerations and Limitations

Across domains, practical realization of sequential phase linking is contingent on physical, algorithmic, and architectural constraints:

• Photonic systems: Achieving sub-mrad phase stability requires monolithic integration, precise fabrication (e.g., detuning control to $<\kappa$), and low-loss waveguides; scaling beyond $10^3$ gates is challenging (Mabuchi, 2011).
• Interferometric phase unwrapping: Assumes no overlapping interfaces within the confocal envelope; noise-induced phase jumps $>\pi$ require guard conditions; step size $\Delta z$ constraints are dictated by $(4\pi n/\lambda)\Delta z \ll 2\pi$ (Mizuno et al., 2023).
• Sequential InSAR: Relies on the validity of the blockwise Schur split; as stack sizes grow, assumptions of cross-covariance stationarity or phase coherence can be strained (Hajjar et al., 13 Feb 2025).
• Electron-light gating: Requires stable phase-locking of optical near-fields over $<100$ nm separations; precision polarization control is critical for selective transverse–longitudinal coupling (Chahshouri et al., 2023).
• Session-type protocols: Relies on correct implementation of subtyping, coinductive checks, and the absence of illegal channel aliasing; sharing across unreliable network layers is not addressed at the type level (Sano et al., 2021).

Application Area	Key Paper/ArXiv ID	Core Principle
Photonic logic circuits	(Mabuchi, 2011)	Gate and memory phase linking via Kerr-nonlinear interferometry
Dual-comb phase imaging	(Mizuno et al., 2023)	Amplitude–phase linking for sequential phase unwrapping
InSAR time-series analysis	(Hajjar et al., 13 Feb 2025)	Sequential blockwise covariance-fitting for phase estimation
Ultrafast electron-light interaction	(Chahshouri et al., 2023)	Coherent phase-locking of multi-zone plasmonic gating
Multiparty communication protocols	(Sano et al., 2021)	Manifest phase statics via sequentially linked session type phases

Sequential phase linking, in all these settings, enables efficient, scalable, and information-rich coordination that would be cumbersome or impossible with stateless, non-sequential, or decoupled architectures.