Extraction of Bend-Resolved Modal Basis in Deformed Multimode Fiber

TL;DR

This work addresses the challenge of interpreting deformation-induced modal dynamics in bent multimode fibers by introducing a two-stage SVD framework that builds a deformation-resolved modal basis from speckle-field statistics. First, per-bend SVDs extract deformation-specific orthonormal mode sets from speckle-correlation matrices, then a second-stage SVD on the concatenated sets yields a global bend-resolved basis that robustly spans the deformation-induced subspace. The approach is data-driven and model-free, enabling deformation tracking, distributed sensing, and integration with mode-division multiplexing without requiring detailed fiber geometry or waveguide solutions. Practically, the resulting basis provides physically interpretable signatures of mechanical perturbations and supports predictive fault diagnostics in deployed fiber networks and sensing systems.

Abstract

Mode mixing in optical fibers caused by mechanical bending induces perturbations that distort the spatial field profile of coherent beams as they propagate through few-mode or multimode fibers. The observed output from a bent fiber commonly appears as complex speckle, which is challenging to relate directly to the underlying deformation, particularly in continuously varying systems such as aerially deployed fibers or fiber-integrated sensors in mechanical structures. We introduce a novel method for constructing a complete deformation-resolved orthonormal modal basis that captures the optical response of a multimode fiber across a range of controlled mechanical deformations. The basis is derived via a two-stage singular value decomposition framework that initially constructs deformation-specific orthonormal mode sets from speckle pattern correlation matrices and subsequently decomposes the aggregated sets to produce a unified functional basis that comprehensively spans the deformation-induced modal subspace supported by the fiber. This hierarchical framework yields an energy-balanced representation that isolates statistically dominant field components across all deformation states, approximates superpositions of the fiber's propagation-invariant modes, systematically encodes deformation-induced perturbations, and supports robust decomposition of output fields across varying mechanical conditions. Such a basis enables tracking of mechanically induced modal evolution in deployed fibers, supporting distributed sensing, network resilience, and predictive fault diagnostics, with potential for integration into mode-division multiplexing systems.

Figures (12)

• Figure 2: Illustration of the experimentally obtained field emerging from a linearly deformed fiber under coupling conditions $\bigl\{ \mathrm{LG}_{\ell 0} \mid \ell \in \{ -1, 0, 1 \} \bigr\}$. (a) Volumetric visualization of the evolving speckle amplitude at the distal facet, resulting from fixed proximal facet coupling at $(\upsilon,\nu) = (-3, -9)$, for deformation states $\Delta p_{\text{mts}} = \bigl\{0.5\cdot(p-1) \mid p \in {N},\, p \leq N_\mathrm{B}\bigr\}\,\unit{\milli\metre}$, where $p \in \left\{1, \dots, N_{\mathrm{B}}\right\}$ indexes the deformation state. Highlighted cross-section shows the amplitude quarter-volume $\bigl|S_{\mu}^{(p)}\!(x_0, y_0)\bigr|$ for $(x_0, y_0) = \bigl\{(-r, 0)\mathbin{,}(0, r)\bigr\}$. (b) Cumulative power distribution averaged over all input displacements and deformations, $\overline{P}(r,\theta) = (N_\mathrm{B}N_\mathrm{S})^{-1}\sum_{p=1}^{N_\mathrm{B}}\sum_{\mu=1}^{N_\mathrm{S}}\bigl|{S_{\mu}^{(p)}\!(r,\theta)}\bigr|^{2}$, with corresponding cross-sections showing per-deformation contributions, $\overline{P}^{(p)}\!(x_0,y_0)\mathbin{=}{N}_\mathrm{S}^{-1}\sum_{\mu=1}^{N_\mathrm{S}}|{S_{\mu}^{(p)}\!(x_0,y_0)}|^{2}$ for $(x_0, y_0) \in \bigl\{(\pm{r}, 0), (0,\pm{r})\bigr\}$, where $N_{\mathrm{B}}=101$ is the number of bend configurations, and $N_{\mathrm{S}}=729$ is the number of speckle patterns per deformation.
• Figure 3: Intermodal power exchange within the fiber’s modal basis is shown for an optical field propagating through a U-shaped bend under linear deformations of $\Delta{p}_{\text{mts}} = \qtylist[list-open-bracket=\{,list-close-bracket=\},list-units=brackets,list-pair-separator={\text{,}\,},list-final-separator={\text{,}\,}]{25.0;37.5}{\milli\metre}$, corresponding to bend radii of $r_\mathrm{B} = \qtylist[list-open-bracket=\{,list-close-bracket=\},list-units=brackets,list-pair-separator={\text{,}\,},list-final-separator={\text{,}\,}]{79.3;50.1}{\milli\metre}$. Symmetry preservation in the field is illustrated through spatial profiles derived from an input displaced $\mathrm{LG}_{00}$ mode, defined as a superposition of fiber modes at positions $\bigl\{(x_{\upsilon\nu},y_{\upsilon\nu})\bigr\} = \bigl\{(-\frac{3}{5}r_\text{core},0),(0,-\frac{3}{5}r_\text{core})\bigr\}$, where $r_\text{core} = \qty{12.5}{\micro\metre}$. Segmental lengths $L_s \mid s\in{\{2,\dots,6\}}$ scaled by the factor of $0.01\cdot{(\uppi R_{0})}^{-1}$, reducing the effective propagation distance to $\widehat{z}=\qty{1}{\centi\metre}$ to aid visualization of intermodal coupling.
• Figure 4: Normalized modal power overlap for selected fiber modes $\{\mathrm{LP}_{\ell n}\mid(\ell,n)\in\{(1,1),(0,2),(2,2),(0,3)\}\}$, shown as a function of linear deformation values $\Delta{p}_{\text{mts}} = \qtylist[list-open-bracket=\{,list-close-bracket=\},list-units=brackets,list-separator={,},list-final-separator={,\,\dots\,,}]{0.0;5.0;10.0;50.0}{\milli\metre}$, corresponding to bend radii $r_\mathrm{B} = \qtylist[list-open-bracket=\{,list-close-bracket=\},list-units=brackets,list-separator={,},list-final-separator={,\,\dots\,,}]{137.5;125.9;114.2;21.0}{\milli\metre}$. The configuration involves a U-shaped bend and OAM-carrying inputs $\mathrm{LG}_{\ell 0}\mid\ell\in\{\pm 1,0\}$, launched at a fixed entrance position $(x_{\upsilon\nu}, y_{\upsilon\nu}) = \left( -\tfrac{3}{13}r_\text{core},\ -\tfrac{9}{13}r_\text{core} \right)$, where $r_\text{core} = \qty{12.5}{\micro\metre}$. Segmental lengths $L_{s}\mid{s}\in \{2,\dots,6\}$ are scaled by a factor $0.01\cdot(\uppi R_{0})^{-1}$, reducing the effective propagation distance to $\widehat{z} = \qty{1}{\centi\metre}$ to enhance visual clarity in the illustrated power dynamics.
• Figure 5: Measurement system for tracking mode field distortions in a controllably bent fiber. (a) Experimental optical setup enabling spatial scanning across the input facet of a SI-MMF, repeated over a series of controlled bending configurations imposed by the MTS. (b) Idealized geometric model of the U-shaped fiber deformation, illustrating the predictable mechanical profile used to generate repeatable modal distortions. Fiber parameters: SiO2 core with $\varnothing_{\text{core}}=\qty{25(3)}{\micro\metre}$, F:SiO2 cladding with $\varnothing_{\text{cladd}}=\qty{125(2)}{\micro\metre}$, $\mathrm{N\!A} = 0.1(0.015)$, $\alpha\approx\qty{0.048}{\decibel\per\metre}$ at $\lambda=\qty{850}{\nano\metre}$. Linear MTS is set to discrete positions $\Delta{p}_{\text{mts}}=\qtylist[list-open-bracket=\{,list-close-bracket=\},list-units=brackets,list-separator={,},list-final-separator={,\,\dots\,,}]{0.0;0.5;50.0}{\milli\metre}$ with a positioning accuracy of $\overline{\delta}_{\text{mts}}=\pm\qty{1.6}{\micro\metre}$. Notes: BS -- beamsplitter, CAM -- camera, HWP -- half-wave plate, SI-MMF -- step-index multimode fiber, L -- lens, LCoS SLM -- liquid crystal on silicon spatial light modulator, LED -- light-emitting diode source at $\qty{850}{\nano\metre}$, MTS -- motorized translation stage, ND -- neutral-density filter, PMF -- polarization maintaining fiber.
• Figure 6: An overview of the experimentally constructed deformation-resolved sets of singular modes, ordered by decreasing singular values. (a) A plot illustrates the variation in the spatial profiles of singular modes, where $\langle{\Psi_{\eta}^{(p)}}|{\Psi_{\eta^{\prime}}^{(p^{\prime})}}\rangle = \delta_{pp^{\prime}}\delta_{\eta\eta^{\prime}}$, under approximately equal coupling strengths as the fiber undergoes linear deformation via MTS translation. The modes are derived from speckle patterns generated through $\mathrm{LG_{00}}$ excitation. (b) Relative coupling strengths for all linear deformations $\Delta{p}_\text{mts}$, visualized through plots of singular values derived from the cumulative energy content $\zeta_{\eta}^{(p)}\!\in[0,1]$, expressed as $\zeta^{(p)}_{\eta} = \sum_{i=1}^{\eta}\sigma_{i}^{(p)}\bigl(\sum_{i=1}^{N_\mathrm{S}}\sigma^{(p)}_{i}\bigr)^{-1}$. The average of the given quantities is indicated by the black dash-dotted line for $\mathrm{LG_{00}}$ and the white dashed line for $\mathrm{LG_{\pm10}}$, respectively.