Topology-optimized distributed 3d anisotropic Raman emission

Abstract: Topology optimization (TO) of 3D surface-enhanced Raman scattering (SERS) substrates faces challenges in managing field singularities and modeling orientation-averaged anisotropic molecules. We present 3D TO for manufacturable SERS substrates that maximize spatially averaged signals from randomly oriented, anisotropic molecules in both elastic and inelastic scattering. A new trace formulation provides a closed-form rotational average of anisotropic Raman tensors, which are not equivalent to isotropic molecules because of tensor nonlinearity. Optimized silver and Si3N4 devices show that lengthscale constraints are sufficient to suppress designs that rely on unphysical mathematical field divergences at sharp corners. Metallic designs deliver broadband enhancement and remain robust to typical Raman shifts, whereas dielectric designs yield narrower, quality-factor-limited gains that are inferior to metallic designs for quality factors below about 500. Our approach readily incorporates additional physics, such as a nonlinear damage model. Together, these results provide a practical route to improved manufacturable SERS substrates and extend naturally to other distributed-emitter design problems.

• The paper introduces a topology-optimization (TO) framework for 3D surface-enhanced Raman scattering (SERS) substrates designed to maximize Raman emission from anisotropic molecules, addressing issues related to field singularities and orientational averaging.
• The study utilizes a trace identity for the radiated Poynting flux forming an objective, which minimizes field singularities.
• The research demonstrates optimized designs for silver and silicon nitride surfaces, including manufacture feasible solutions resulting in high enhancements of order of magnitude compared to conventional designs.

Overview

This paper presents a topology-optimization (TO) framework for three-dimensional surface-enhanced Raman scattering (SERS) substrates that maximizes spatially and orientationally averaged Raman emission from randomly distributed, anisotropic molecules. The work addresses two obstacles specific to 3d inverse design of SERS: the presence of non-integrable electromagnetic field singularities at sharp conical tips, which are stronger than their 2d counterparts, and the fact that orientational averaging of an anisotropic Raman polarizability tensor does not reduce to any effective isotropic model. The authors formulate the objective via a trace identity requiring only 1–3 frequency-domain Maxwell solves per objective evaluation, derive a closed-form rotationally averaged correlation term for complex-symmetric polarizability tensors using Weingarten–Collins calculus on SO(3), and demonstrate optimized silver and Si$_3$N$_4$ metasurfaces under manufacturability constraints.

Trace formulation for anisotropic emitters

The SERS figure of merit $g$ is the ensemble average of radiated Poynting flux over molecular positions $\mathbf{r}_0$ and orientations $Q \in \mathrm{SO}(3)$, computed exactly as an overlap integral between the pump field $\mathbf{E}_\text{p}$ at $\omega_\text{p}$ and a reciprocal field $\mathbf{E}'_\text{e}$ obtained by solving Maxwell's equations at $\omega_\text{e}$ with the detector mode as source. The objective takes the trace form

$$g = \operatorname{Tr}\left[ M_\text{e}^{-\dagger} O\, M_\text{e}^{-1} B \right],$$

where $_4$0 is the ensemble-averaged source-correlation matrix. Because the output operator $_4$1 is low rank (rank 2 for dual-polarization detection into one direction; effectively rank 1 since emission is dominated by the pump polarization), only two or even one reciprocal solves are needed beyond the pump solve, plus one adjoint solve per gradient.

The central analytical contribution is the rotationally averaged dyadic term for a complex-symmetric Raman tensor $_4$2. Using the Collins–Śniady fourth-moment formula for Haar-distributed rotations, the authors show

$_4$3

with $_4$4 and $_4$5, where $_4$6 and $_4$7. This result is not equivalent to the isotropic form $_4$8: the extra phase-sensitive term vanishes if and only if $_4$9 is proportional to the identity, because $g$0 requires equality in the Hilbert–Schmidt Cauchy–Schwarz bound $g$1. In Raman-spectroscopy language, the coefficient of this extra coupling is precisely the "anisotropic invariant" $g$2, so the correction is controlled by the standard depolarization measure. Notably, for elastic single-channel detection with a linearly polarized homogeneous pump field ($g$3), the integrand coincides with the isotropic form even for anisotropic molecules—clarifying when the common $g$4 heuristic remains exact.

The appendix also derives strict bounds on the error incurred by optimizing under an isotropic assumption: for uniaxial molecules the normalized enhancement-factor error is bounded within $g$5, and for traceless tensors (e.g., the $g$6/$g$7 modes of CCl$g$8) within $g$9.

Numerical implementation

Design variables are density fields $\mathbf{r}_0$0 processed by Helmholtz/conic filtering (20 nm minimum lengthscale), explicit minimum-linewidth constraints on both solid and void phases, and subpixel-smoothed projection (SSP) allowing $\mathbf{r}_0$1 while retaining differentiability. A metal-specific material interpolation avoids artificial bulk-plasmon resonances at intermediate densities. Optimization uses CCSA (CCSAQ in NLopt) with adjoint gradients, implemented in Gridap.jl with zeroth-order Nédélec elements (~2 nm elements near metal design regions to resolve skin depth). Typical metal optimizations required roughly three days on 40 CPUs. Mirror symmetries in $\mathbf{r}_0$2 and $\mathbf{r}_0$3 reduce the computational cell fourfold; removing them produced negligible performance change, and at least one mirror symmetry emerged spontaneously even from asymmetric initializations—a useful empirical regularity.

Hotspots and lengthscale regularization

Near a 3d conical tip the field scales as $\mathbf{r}_0$4, so the $\mathbf{r}_0$5 objective scales as $\mathbf{r}_0$6 and diverges for tip exponents $\mathbf{r}_0$7—a stronger pathology than in 2d, where divergence occurs only at particular corner angles for metals. Without regularization, TO produces many artificial hotspots, including ones seeded by mesh roughness alone, and these do not converge with resolution; physically they would be cut off by nonlocal quantum effects below ~10 nm. The paper's practical finding is that filtering combined with explicit manufacturing constraints suffices to suppress such singularities and steer optimization toward fabricable structures, without the optimizer stalling at constraint-violating intermediates. For the 2d-patterned constant-cross-section designs, the residual 90° top/bottom edges fall outside the non-integrable angle range for Ag and are always integrable for dielectrics.

Metallic results

For Ag-on-Ag substrates (period 184 nm, targeting the primary plasmonic resonance at the pump wavelength, normal incidence), optimized 2d-patterned designs exceed an array of radius-82 nm spheres with 20 nm gaps by more than an order of magnitude. Emission is almost entirely co-polarized with the pump, and monopolarized designs match bipolarized designs in total enhancement, halving the number of reciprocal solves. Across ten random initializations the final objective varied with ~55% standard deviation—an honest acknowledgment of local-optima sensitivity—yet most designs converged to a shared "rounded corners in $\mathbf{r}_0$8" motif, indicating a robust underlying geometry principle. Level-set remeshing shifted resonances by ~5 nm and increased peak enhancement several-fold (surface plasmons are sensitive to interface details); a realistic ~2 nm oxide coating reduces enhancement by roughly a factor of 2 in high-resolution 2d calculations. These uncertainties are secondary to the order-of-magnitude gains relative to benchmarks.

Dielectric results and the metal–dielectric comparison

Si$\mathbf{r}_0$9N$Q \in \mathrm{SO}(3)$0 designs (period 238 nm, guided-mode resonance near the operating wavelength) were regularized by artificial damping bounding $Q \in \mathrm{SO}(3)$1, yielding final structures with $Q \in \mathrm{SO}(3)$2–300 after the loss is removed. In this regime the metallic surface outperforms both dielectric designs by at least an order of magnitude, despite the freeform 3d dielectric delivering nearly an order of magnitude over its 2d-patterned counterpart. The physical explanation is bandwidth: the metal resonance has $Q \in \mathrm{SO}(3)$3, roughly ten times the dielectric bandwidth. The implication is that for moderate-$Q \in \mathrm{SO}(3)$4 applications, plasmonic broadband enhancement dominates; dielectric advantages require accessing much higher $Q \in \mathrm{SO}(3)$5, which trades off robustness, bandwidth, and fabrication tolerance. Field localization also differs qualitatively: plasmonic designs concentrate energy at metal surfaces and gaps, whereas dielectric designs concentrate fields in voids.

Anisotropy and inelastic scattering

When isotropically optimized designs are re-evaluated with the full anisotropic figure of merit, they remain within ~20% of designs explicitly optimized for uniaxial molecules (within 10% at resonance peaks), consistent with the theoretical bounds above; geometries are nearly indistinguishable. This means practitioners can optimize under the simpler isotropic model without meaningful penalty for typical anisotropic molecules.

For inelastic scattering (532 nm pump, 549 nm emission, ~582 cm$Q \in \mathrm{SO}(3)$6 shift), metallic designs optimized elastically degrade only slightly, owing to their broadband response, and inelastically optimized metal designs differ mainly by a small spectral shift. Dielectric designs optimized elastically suffer significant degradation at shifted emission wavelengths; inelastic re-optimization retunes the single resonance but yields no better peak than the elastic design evaluated at the pump. Attempts at doubly resonant dielectric designs produced more complex geometries without superior performance in the chosen unit cell. The conclusion is that dielectric SERS is ill-suited to large Raman shifts unless larger cells supporting multiple overlapping resonances are employed.

Nonlinear damage regularization

As a demonstration of framework extensibility rather than a core mechanism, the authors embed a differentiable quenching model in which the effective polarizability is suppressed by a sigmoid when local intensity exceeds a threshold $Q \in \mathrm{SO}(3)$7. Lower thresholds drive the optimizer toward spatially spread field distributions and qualitatively different topologies. However, a design optimized at $Q \in \mathrm{SO}(3)$8 improves over the damage-free baseline by only ~1.8× at low evaluation thresholds, indicating that geometric lengthscale constraints already suppress most damaging hotspots in these metallic designs.

High-$Q \in \mathrm{SO}(3)$9 considerations and limitations

Several caveats bear directly on the reported results. First, all dielectric comparisons are made in the artificially damped $\mathbf{E}_\text{p}$0 regime; recent sum-of-squares bounds indicate that even lossy periodic systems can achieve arbitrarily large Raman response via delocalized high-$\mathbf{E}_\text{p}$1 states when the period approaches the wavelength, but such states are excluded here by the imposed bandwidth floor and period choice. Extending to genuinely high-$\mathbf{E}_\text{p}$2 targets faces known stiffness/ill-conditioning of single-frequency objectives, though eigenfrequency-shift methods have recently been proposed to mitigate this; the authors also note that many low-$\mathbf{E}_\text{p}$3 local optima exist, so high-$\mathbf{E}_\text{p}$4 optimization may require careful initialization from known bound-states-in-the-continuum. Second, the ~55% variance across random initializations for metal designs means absolute performance figures carry substantial local-optima uncertainty, even though morphology converges. Third, validation is entirely computational: no experimental fabrication or measurement is presented, and mesh discretization plus gray-interface layers introduce spectral shifts of order 5 nm. Finally, the orientation-average assumes uniformly distributed molecules in a fluid layer; other distributions (surface deposition, channels) are accommodated by the trace framework but were not optimized here.

Conclusion

This work delivers a complete computational pipeline for 3d TO of manufacturable SERS substrates: a closed-form, provably non-isotropic rotationally averaged objective for anisotropic Raman tensors; lengthscale constraints sufficient to eliminate divergent tip singularities inherent to the $\mathbf{E}_\text{p}$5 objective in 3d; and quantitative design guidance showing metallic surfaces superior for $\mathbf{E}_\text{p}$6 and robust to anisotropy and Raman shifts, while dielectric advantages await controllable high-$\mathbf{E}_\text{p}$7 multi-resonant design. The framework extends naturally to other distributed-emitter problems (LEDs, thermal emitters, scintillators), with experimental validation and very-high-$\mathbf{E}_\text{p}$8 optimization remaining the principal open questions.