Interference Dislocations

• Interference dislocations are topological singularities defined by points or lines where wave amplitude vanishes and the phase is undefined.
• They arise in various systems—including optics, quantum condensates, and electronic materials—facilitating direct measurements of wave topology and defect structures.
• Experimental methods like shift-interferometry and coherent X-ray diffraction provide practical insights into dislocation dynamics and lattice defect imaging.

Interference dislocations are topological singularities—points, lines, or manifolds at which the phase of a complex wave field is undefined because the amplitude vanishes—arising generically in any system governed by the superposition principle. In both classical and quantum wave physics, these dislocations manifest as abrupt terminations, bifurcations, or reconnections of interference fringes and encode quantized topological charges (vorticities). The phenomenon bridges singular optics, condensed matter, photonics, materials science, and nonlinear dynamics, offering real-space probes of wave topology, defect structures, and dynamical instabilities.

1. Fundamental Theory and Mathematical Characterization

A complex scalar or vector field $$\Psi(\mathbf{r}, t) = A(\mathbf{r}, t) e^{i\phi(\mathbf{r}, t)}$$ exhibits a phase dislocation (also called a wavefront dislocation, fork, vortex, or singularity) where the amplitude vanishes, $A(\mathbf{r}_0, t_0) = 0$, and the phase $\phi$ is undefined. Around such a point (or line in 3D), the phase accumulation along a closed path $\mathcal{C}$ is quantized: $$Q = \frac{1}{2\pi} \oint_{\mathcal{C}} \nabla \phi \cdot d\mathbf{r} \in \mathbb{Z}$$ The integer $Q$ is the topological charge or vortex strength. This singularity structure is universal: it appears in quantum condensates, optical fields, electron waves, plasmonic/phononic/metamaterial systems, and even macroscopic hydrodynamics (Kamchatnov et al., 2014, Leonard et al., 2019, Dutreix et al., 2020, Zhuang et al., 29 Oct 2024, Leonard et al., 3 Nov 2025).

2. Interference Dislocations in Physical Systems

Optical and Photonic Systems

In singular optics, interference of two or more coherent beams can produce dislocation patterns such as forked fringes, abrupt fringe terminations, or moiré-type dislocations near structured sources. For instance, in shift-interferometry of spatially modulated emission—such as excitons in TMDC monolayers—the combination of emission from multiple sub-sources yields adjacent interference dislocations (forks) by moiré beating: $$I(\mathbf{r}) \sim \sum_{\mathbf{r}_s} \frac{P(\mathbf{r}_s)}{|\mathbf{r}-\mathbf{r}_s|} \cos\big[\mathbf{k}(\mathbf{r}_s)\cdot\delta\mathbf{r} + q_t y\big]$$ The zeros of the total field correspond to forks straddling the emission spot, their positions and orientations controlled by the emitter profile and interferometric shift (Leonard et al., 3 Nov 2025). In microcavity polariton scattering, the superposition of incident and scattered waves at an obstacle leads to nodal (zero) points: $$\psi(\mathbf{r}) \approx e^{ikx} + \frac{f(\varphi)}{\sqrt{r}}e^{i(kr + \pi/4)}$$ with rich phenomenology: isolated vortices, edge dislocations (“forks”), vortex–antivortex pairs and resonant-driven nodal line patterns (Kamchatnov et al., 2014).

Quantum Condensates

In Bose-condensed systems, conventional topological defects (vortices, skyrmions) manifest as phase singularities with quantized circulation in the order parameter. However, interference dislocations in quantum condensate experiments can also originate from the superposition of coherent matter-waves emitted from multiple spatially separated sources with no intrinsic vorticity: $$I(\mathbf{r}) \propto \cos(q_t y)\cos[k\delta x \cos \gamma]$$ Here, the forked fringes arise not from intrinsic defects but from converging matter-wave flows, a distinction proven by absence of expected vortex-pair structures and by precise agreement of the fork locations with two-source interference geometry (Leonard et al., 2019, Leonard et al., 3 Nov 2025).

Topological Materials and Electronic Systems

In graphene and other systems with internal pseudospin textures, intervalley interference between sublattice-resolved electron propagators generates phase singularities in the local transmission or conductance: $$\phi_{BA}(\mathbf{r}) = \Delta K \cdot \mathbf{r} + 2\theta_r$$ For monolayer graphene, this produces wavefront dislocations of charge $Q=2$; for Bernal bilayer, $Q=4$. These singularities are robust against probe details and mass gap, reflecting the valley-dependent Berry phase and serving as direct local probes of topological band structure (Zhuang et al., 29 Oct 2024, Dutreix et al., 2020).

Magnetohydrodynamic and Plasma Waves

In solar coronal loops, interference between kink (m=1, transverse) and sausage (m=0, longitudinal) MHD wave modes of distinct frequency leads to singularities in observed Doppler velocity maps at loci where the projected complex amplitude vanishes: $F(z, t) = C_k e^{-i\Delta\omega t} + C_s$ The spatial and temporal location of these dislocations—confirmed by monodromy integrals of $\pm2\pi$—are determined by amplitude ratios, phase differences, and projection effects. Only coupled kink–sausage interference can explain the observed, localized crest endpoint singularities (Ariste et al., 2015).

Crystalline Dislocations Observed by Coherent Diffraction

In coherent X-ray diffraction imaging (CXRD), dislocations in the crystal lattice impart a topological phase winding in the scattered amplitude: $$A(\mathbf{q}) = \int \rho(\mathbf{r}) e^{i \mathbf{q}\cdot\mathbf{r} + i\mathbf{q}\cdot \mathbf{u}(\mathbf{r})} d\mathbf{r}$$ where $\mathbf{u}(\mathbf{r})$ encodes the Burgers vector. This results in destructive interference at the Bragg condition (intensity dip) and splitting of Bragg peaks, with the angular separation reflecting the strength and structure of the core. CXRD thus directly images lattice dislocations as phase singularities in reciprocal space (Jacques et al., 2010).

3. Classification and Topological Properties

Interference dislocations are classified by local phase structure and their effect on interference fringes:

• Edge/fork dislocations: Singularities where a fringe terminates or splits. Local topological charge $m = \pm1$.
• Vortex–antivortex pairs: Oppositely charged zeros coming together and annihilating, e.g., in the Nye–Hannay scenario (Kamchatnov et al., 2014).
• Screw/gliding dislocations: Appearing as tilted lines in space–time or parameter–frequency plots, arising from band-touching points or dynamical phase slip (Ariste et al., 2015, Dutreix et al., 2020).
• Moiré/fork pairs adjacent to extended sources: Result from superposition of locally varying wavevectors rather than intrinsic vorticity (Leonard et al., 3 Nov 2025, Leonard et al., 2019).

Topological invariance is guaranteed by winding number quantization; for example, the phase accumulation around a band-closing point in parameter space connects directly to a topological invariant (e.g., winding number in Su–Schrieffer–Heeger chains (Dutreix et al., 2020)).

4. Analytical Methods and Predictive Tools

Prediction of dislocation positions in complex fields involves:

• Direct solution of complex amplitude zeros: Solve $\Re \psi(\mathbf{r}) = 0$ and $\Im \psi(\mathbf{r}) = 0$ simultaneously.
• Partial-wave and Green’s function expansions: Accurate for obstacles or scattering problems; truncation at physically motivated $|n| \lesssim ka$ suffices for polaritons (Kamchatnov et al., 2014).
• Born approximation for simple estimates: Provides scaling for distance of singularities from obstacles or sources.
• Fourier filtering and phase mapping: For transmission/conductance in electronic systems, FFT filtering at intervalley momentum isolates relevant singularities (Zhuang et al., 29 Oct 2024).
• Phase retrieval and ptychography: In CXRD, iterative algorithms reconstruct real-space dislocation lines from measured diffraction patterns (Jacques et al., 2010).
• Numerical simulation: For structured emission or spatially inhomogeneous sources, superposition of local fringe patterns predicts fork formation (Leonard et al., 3 Nov 2025).

5. Experimental Manifestations and Diagnostics

Table: Representative Manifestations of Interference Dislocations

Physical System	Diagnostic Observable	Characteristic Dislocation Signature
Microcavity polaritons	Far-field photoluminescence	Forks, vortex pairs, π-phase jumps, nodal valleys
Exciton condensates	Shift-interference fringes	Isolated forks, geometry-determined locations
Graphene	Conductance/STM maps	Extra wavefronts at probe site, charge = Berry phase
Coronal MHD waves	Doppler velocity (CoMP)	Crest terminations, topological phase ∮dχ=±2π
Bulk crystals	Coherent X-ray Bragg diffraction	Split peaks, central dip, angular separation ∝

The position, orientation, and charge of dislocations are sensitive to experimental geometry but robustly reflect the underlying wave topology, pseudospin structure, or interference mechanism.

6. Relevance to Topology, Defect Physics, and Dynamics

Interference dislocations serve as real-space diagnostics of:

• Topological phase transitions: Fringe count changes encode topological invariants such as winding numbers, as observed in photonic SSH chains (Dutreix et al., 2020) and graphene Berry phase measurements (Zhuang et al., 29 Oct 2024).
• Plastic instabilities and mutual dislocation dynamics: Mutual (interference) interactions between many dislocations, mediated by mobile impurities, drive collective stick–slip (“serration”) phenomena in metallic alloys—the Portevin–Le Châtelier effect. A dome-shaped regime in (drive, effective temperature) emerges, governed by pinning/unpinning time scale matching, not present for single or non-interacting dislocations (Leoni et al., 2013).
• Nonlinear and many-body effects: In polariton fluids, linear dislocation patterns give way to nonlinear structures (oblique solitons) when the local healing length exceeds the dislocation width. The crossover condition is $d \ll \xi$ (Kamchatnov et al., 2014).
• Elastic deformation and defect core imaging: Coherent diffraction visualizes the phase discontinuity of lattice dislocations, resolving their structure even inside the bulk (Jacques et al., 2010).

7. Broad Implications and Future Directions

Interference dislocations unify disparate physical contexts under a framework of phase topology and singular optics:

• Provide direct, quantitative, real-space measurements of otherwise abstract topological invariants in quantum materials, metamaterials, and photonic systems.
• Serve as diagnostics of wavefunction structure, superposition effects, coherence length, and local spatial modulations, with sensitivity to Berry phase, winding, and symmetry.
• Underpin advanced imaging techniques (coherent X-ray, STM, shift-interferometry) offering nanometric resolution of core structure and strain.
• Motivate generalized applications to non-Hermitian and long-range coupled systems, as well as to the mapping of dynamic stick–slip phenomena in dislocation assemblies.
• In singular optics and photonics, inform the design of tailored emission, beam shaping, and manipulation of topological light and matter fields.

In all contexts, the interplay between interference, topology, and singularity imparts both fundamental insight and practical diagnostic power, with advances driven by the synergy between precise experimental techniques and sophisticated theoretical analysis.