Harnessing coherent-wave control for sensing applications

TL;DR

The paper develops a microscopic theory of optical sensitivity under coherent wavefront control and demonstrates how phase-conjugation and remission eigenchannels can substantially enhance sensing in diffuse optical imaging, while showing that random-input illumination recovers the diffusion-based sensitivity. It provides analytic expressions for scalar-wave sensitivity, proves equivalence with diffusion in the random-illumination limit, and reveals that phase conjugation yields the largest local enhancement whereas the maximum remission eigenchannel offers a global, geometry-independent boost compatible with DOT reconstruction. Numerical studies for scalar and vector waves validate the theory, reveal the characteristic banana-shaped sensitivity maps, and quantify enhancements via FRM and MP models, including 3D considerations. The work further extends the framework to TE polarization, preserving the core insights and highlighting practical opportunities to improve depth penetration and robustness in biomedical imaging modalities such as DOT and fNIRS, with clear implications for experimental realization.

Abstract

Imaging techniques such as functional near-infrared spectroscopy (fNIRS) and diffuse optical tomography (DOT) achieve deep, non-invasive sensing in turbid media, but they are constrained by the photon budget. Wavefront shaping (WFS) can enhance signal strength via interference at specific locations within scattering media, enhancing light-matter interactions and potentially extending the penetration depth of these techniques. Interpreting the resulting measurements rests on the knowledge of optical sensitivity - a relationship between detected signal changes and perturbations at a specific location inside the medium. However, conventional diffusion-based sensitivity models rely on assumptions that become invalid under coherent illumination. In this work, we develop a microscopic theory for optical sensitivity that captures the inherent interference effects that diffusion theory necessarily neglects. We analytically show that under random illumination, the microscopic and diffusive treatments coincide. Using our microscopic approach, we explore WFS strategies for enhancing optical sensitivity beyond the diffusive result. We demonstrate that the input state obtained through phase conjugation at a given point inside the system leads to the largest enhancement of optical sensitivity but requires an input wavefront that depends on the target position. In sharp contrast, the maximum remission eigenchannel leads to a global enhancement of the sensitivity map with a fixed input wavefront. This global enhancement equals to remission enhancement and preserves the spatial distribution of the sensitivity, making it compatible with existing DOT reconstruction algorithms. Our results establish the theoretical foundation for integrating wavefront control with diffuse optical imaging, enabling deeper tissue penetration through improved signal strength in biomedical applications.

Figures (11)

• Figure 1: Schematic of scattering approach. (a) Scattering matrix element $S_{ba}$ relates field amplitudes of the incident mode $a$ (solid line arrows) and outgoing mode $b$ (dashed arrows). (b) Experimentally, one has access to a finite number of spatial channels (blue arrows), described by matrix ${\cal R}_{ba}$, which is a subset of $S$. (c) Addition of a small perturbation in the dielectric properties inside the system results in a change $\delta{\cal R}_{ba}({\mathbf r}_{0})$, which depends on its location ${\mathbf r}_{0}$.
• Figure 2: Scattering representation of the identity \ref{['EqGreenIdentity']}. Solid and dashed lines represent the propagating field and its complex conjugate, while open circles represent scatterers. Shaded tubes represent diffusive paths where both field visit the same scatterers. A more formal representation of the lower diagram is given in Fig. \ref{['fig:diagram']} in Appendix \ref{['sec:green_identity']}.
• Figure 3: Two computational approaches for evaluating optical sensitivity. (a) Method I: Sensitivity is inferred from changes in remission coefficients ${\cal R}_{ba}$ due to the introduction of a localized absorber at position ${\mathbf r}_0$; see first part in Eq. (\ref{['EqSiBorn']}). This method requires sequential simulations for each ${\mathbf r}_0$ and thus involves raster scanning. (b) Method II: Sensitivity is computed using the product of two field distributions in the second part of Eq. \ref{['EqSiBorn']}. This approach is parallel and requires only two simulations to obtain the full spatial sensitivity map across all ${\mathbf r}_0$.
• Figure 4: Comparison of microscopic and diffusive models for optical sensitivity under random input excitation. All results are averaged over 1000 disorder realizations and normalized according to Eq. \ref{['EqInt_sensitivity']}. Transport mean free path is $\ell=~$6.4 $\mu$m. (a) Sensitivity map computed using the microscopic sensitivity (MS) expression, Eq. \ref{['EqSiBorn']}, exhibits the characteristic ‘banana’-shaped pattern. Input and output port widths are $W_1=W_2=10\,\mu$m; source-detector separation is $d=128\,\mu$m $=20\times\ell$. (b) Depth cross-section of the sensitivity map along the dashed line in (a). Circles correspond to MS calculation, Eq. \ref{['EqSiBorn']}; crosses show results from the diffusion approximation (DA), Eq. \ref{['EqDiffusiveSensitivity']}. Input and detector port widths are equal. Curves are shown for $5,\,10,\,15$ and $20\,\mu$m, with fixed separation $d=128\,\mu$m $=20\times\ell$. (c) Sensitivity at the midpoint A of the banana (peak in panel b) as a function of source-detector separation. Symbols: MS results using Eq. \ref{['EqSiBorn']} for $W_1=W_2=10\,\mu$m; solid line: DA prediction based on analytic intensity profiles substituted into Eq. \ref{['EqDiffusiveSensitivity']}, as described in Appendix \ref{['sec:diffusion_in_remission_geometry']}.
• Figure 5: Sensitivity map for excitation with the maximum remission eigenchannel (MRE) and comparison with diffusion-based predictions. (a) Sensitivity map computed using the microscopically exact expression, Eq. \ref{['EqSiBorn']}, for MRE excitation. The characteristic 'banana'-shaped pattern is preserved, but overall sensitivity is enhanced compared to the random input case. Parameters: $W_1=W_2=10\,\mu$m, separation is $d=20\times\ell$, and transport mean free path $\ell=~$6.4 $\mu$m. (b) Depth cross-section along the mid-plane (dashed line in panel a), comparing sensitivity from the microscopic sensitivity (MS) formulation (Eq. \ref{['EqSiBorn']}, filled circles) and a heuristic application of Eq. \ref{['EqDiffusiveSensitivity']} using the disorder-averaged intensity of MREs (crosses, denoted as DA$^*$), see main text for details. (c)-(f) Sensitivity at the central point A in panel (a), plotted as a function of: (c) input-output separation $d$; (d) transport mean free path $\ell$; (e) number of input channels $N_1$, and (f) number of output channels $N_2$. In all cases, results are averaged over 1000 disorder realizations and normalized according to Eq. \ref{['EqInt_sensitivity']}.