Limits to surface-enhanced Raman scattering near arbitrary-shape scatterers

Various scatterers such as rough surfaces or nanostructures are typically used to enhance the low efficiency of Raman spectroscopy (surface-enhanced Raman scattering). In this work, we find fundamental upper bounds on the Raman enhancement for arbitrary-shaped scatterers, depending only on its material constants and the separation distance from the molecule. According to our metric, silver is optimal in visible wavelengths while aluminum is better in the near-UV region. Our general analytical bound scales as the volume of the scatterer and the inverse sixth power of the distance to the active molecule. For periodic scatterers, a second bound with surface-area scaling is presented. Simple geometries such as spheres and bowties are shown to fall short of the bounds. However, using topology optimization based inverse design, we obtain surprising structures maximizing the Raman enhancement. These optimization results shed light to what extent our bounds are achievable.


INTRODUCTION
Surface-enhanced Raman scattering (SERS) is a common method used to enhance the low efficiency of Raman spectroscopy. Active molecules are placed in the vicinity of various scatterers that both concentrate the incoming pump field at the Raman material's location as well as enhance the radiated Stokes signal emitted by the Raman material. Optimized textured surfaces and nanostructures with various shapes have been studied, but no study thus far has looked at the possibility of an upper limit to the enhancement achievable in SERS. In our work, we find fundamental upper bounds on the Raman enhancement for arbitrary-shaped scatterers, depending only on its material constants and the separation distance from the molecule. Our method is based on decomposing the problem into two linear problems: concentration of the incident field on the molecule and a dipole emission at the Raman-shifted frequency. Analytical bounds on the power radiated by a dipole near a scatterer of arbitrary shape were previously reported and are based on optimization under energy-conservation constraints. 1 We apply the same method to the field concentration problem and obtain a final bound on SERS enhancement. According to our metric, silver is optimal in visible wavelengths while aluminum is better in the near-UV region. Our general analytical bound scales as the volume of the scatterer and the inverse sixth power of the distance to the active molecule. Tighter bounds can be computed by simple numerical integration using the free-space Green's function. We also compare the bounds to the actual performance of simple structures (spheres and bowties). Numerical computations show that such geometries fall short of the bounds, suggesting further design opportunities for future improvement. While the bounds can be reached in the quasistatic limit, both the bounds and the actual performance increase with the scatterer's volume and further shape optimization is required to get closer to the limits for large scatterers. In the case of periodic structures, a second bound for the concentration problem is obtained using reciprocity. The local field enhancement of the scattering problem can in fact be related to the radiative enhancement of the emission problem as well as the finite incident power per unit cell. The use of the radiative bound for the concentration problem leads to a second SERS bound proportional to the surface area of the unit cell. This area-scaling bound highlights the fact that for a periodic structure, only a fraction of the actual volume of the scatterer is effectively used in the scattering. Comparing the two bounds leads to a tighter limit and suggests an optimal period depending on the volume of the scatterer. For large periods, the volume-scaling bound is limiting because of the reduced interactions between the scatterers of the array, while for smaller periods the performance of the array is limited by the area-scaling bound since the intensity received by each sphere is reduced compared to the non-periodic case. Motivated by the results from the bounds, we also perform freeform shape optimization ("topology optimization", TO) of the total Raman enhancement, and obtain surprising structures ∼ 60× better than optimized coupled-sphere or bowtie. Such results are a proofof-concept of Raman TO in 2d systems, and the significant enhancements suggest promising results for future improvements in practical 3d Raman sensing.

RAMAN BOUNDS
We consider the configuration represented in the inset of Fig. 1-a to describe Raman scattering. An incident "pump" planewave E inc is scattered by the nanostructure, leading to a near-field enhancement. A Raman-active molecule close to the structure then acquires a dipole moment proportional to the enhanced field, p = α R E where α R is the Hermitian (usually real-symmetric) Raman polarizability tensor. The power radiated by the dipole in the far field at the Raman frequency is our quantity of interest and is given by P = |α R E| 2 P p where P p is the power radiated by a unit-vector dipolep = α R E/|α R E|. If we note P b the power radiated by a unit-vector dipole in the background medium (in absence of the scatterer), the Raman enhancement can then be defined as: where . is the induced norm (which gives an upper bound on the magnitude ofp for any E orientation). We see that the enhancement comes from two parts: radiation enhancement (q rad = P p /P b ) and local field (or focusing) We can bound the total enhancement by bounding both contributions, as explained in detail in Ref. 2. In particular, if we assume that the tensor α R is isotropic and consider that the scatterer is enclosed in a simple structure (such as a half-space or the outside of a sphere) that is separated from the Raman-active molecule by a distance d, we obtain analytical bounds in the limit of small d (assuming a free-space background): where χ(= − 1) is the material susceptibility, V the scatterer volume, k the wavenumber and the subscripts p and r denote the pump and Raman frequencies at which the variables are evaluated. β is a geometrical factor equal to 1/(6π) for a full sphere, 1/(12π) for a half-sphere, and 1/(32π) for a half-plane. 2 The fundamental limit in Eq. 2 scales as V /d 6 . The factor 1/d 6 is related to both the radiation of the dipole (1/d 3 ) and the coupling to it (1/d 3 ), while the factor V is due to the planewave coupling. In practice, the Raman frequency shift is small enough so that the bounds do not change much when the expressions are simply evaluated at the same (pump or Raman) frequency. In this case, the bound is simply proportional to |χ| 2 /Imχ 3 . This material figure of merit can be used to compare the optimal performance of different materials and is shown in Fig. 1-a. We note that silver (Ag) has the highest bound at visible wavelengths but is outperformed by aluminum (Al) in the near-UV region.
We also simulated two of the most common nanostructures used in SERS: triangular prisms used in a bowtie configuration, and a sphere. Results are shown in Fig. 1-b where we used silver, which is the best-performing Raman material at visible frequencies and which also satisfies the resonance condition for χ, unlike e.g. gold.
The geometry of the triangles was taken from Ref. 3. We plotted both the shape-independent bound (∼ Eq.2) by considering the exterior of a spherical shell and using the largest volume of all scatterers (4-triangle bowtie), as well as a shape-dependent bound for each structure (given in Ref. 2). Results in Fig. 1-b show that all structures fall short of the shape-independent bound by several orders of magnitude. The performances of bowties also lie far from their shape-specific limits. Only the sphere approaches its bound, at frequencies greater than the plasma frequency of silver.
Using reciprocity, we also found a second bound in the case of periodic structures, as shown in Ref.
2. This new bound is now proportional to |χ| 2 /Imχ 2 (rather than |χ| 2 /Imχ 3 ) and scales as the surface area of the unit cell instead of the volume of the scatterer. This area-scaling bound highlights the fact that for a periodic structure, only a fraction of the actual volume of the scatterer (proportional to the unit-cell area) is effectively "used" in the scattering. Combining the volume-scaling and area-scaling bounds leads to a tighter bound with different behavior as a function of the period. Overall, maximum enhancement due to periodicity occurs at the smallest possible period, where increased interactions between the scatterers dominate the decrease in incident intensity received by each unit cell. 2

TOPOLOGY OPTIMIZATION
Encouraged by results from bounds, we also performed freeform shape optimization ("topology optimization", TO) to maximize Raman enhancement as discussed in details in Ref. 4. The Raman scattering process is modeled as described in the previous section using two sequentially-coupled frequency-domain electromagnetic simulations. In particular, the system is first excited by an incident planewave at the pump frequency. Afterwards, the Raman molecule is modeled as a dipole source at the Raman-shifted frequency (with dipole moment given by the polarizability tensor α R multiplied by the electric field obtained in the first simulation) which then allows to compute the total emitted power. Simulations are for two-dimensional (2d) structures with in-plane polarization (Fig 2-a).
The use of density-based topology optimization operates by introducing a continuous design field to control the physical material distribution, enabling the use of adjoint sensitivity analysis and gradient-based optimization algorithms to efficiently solve design problems with potentially billions of design degrees of freedom. Hence, the approach provides near-unlimited design freedom, with a computational complexity dominated by the solution of the Maxwell equations, utilizing mature finite-element techniques. 4 Solutions are presented in Fig 2-b, where topology optimization results are also compared to reference geometries (coupled triangles and discs). Reference structures are parameter-optimized (through the radius of the discs and through the tip-angle and the side length of the triangles) to maximize performance at the targeted wavelength (532 nm). We see that topology optimization leads to a surprising structure that fully encloses the molecule and that, in some sense, is a fusion of different features tailored to enhance either focusing or emission. 4 The TO structure has a maximum enhancement of ≈ 8 · 10 5 , which is 60× larger than bowtie-antenna enhancement (≈ 1.3 · 10 5 ) and 2.6 · 10 3 × larger than coupled-discs enhancement (≈ 3 · 10 2 ). These extreme enhancement results, serve as a proof-of-concept suggesting promising improvements for practical 3d structures.
Topology optimization has also been used to optimize focusing and radiation enhancements separately, showing the need to optimize both processes simultaneously. 4 When compared to the fundamental upper bounds presented in section 2, we find that the TO structure (optimized for radiation) comes to within a factor of 4 of the radiation bound. This is a reasonable difference that can be attributed to the length-scale imposed during optimization as well as simplifications leading to the bound. On the other hand, the TO structure optimized for focusing is ∼ 6 · 10 2 smaller than the bound. This suggests that the upper limit may not be tight, in the sense that it may be difficult to achieve optimal coupling to both far-field and near-field, as is required by the focusing bound. The discrepancy may also be due to the fact that the outside radius of the domain Ω D was fixed in the optimization. The focusing bound rather suggests the need of a large volume for a good coupling to the planewave. This can for example be seen in the area-scaling bound for periodic structures where we obtained a |χ| 2 /Imχ 2 dependence due to the finite incident power per unit cell. A similar dependence on the material metric has indeed been noticed in optimized structures. 4 In general, it remains an open problem, for future theoretical and numerical investigations, to understand practical limitations related to the near-field focusing problem.