Research Article Vol. 28, No. 4 / 17 February 2020 / Optics Express 4444
Inverse design of nanoparticles for enhanced
Raman scattering
RASMUS E. CHRISTIANSEN,1,2,* JÉRÔME MICHON,3 MOHAMMED
BENZAOUIA,4 OLE SIGMUND,1 AND STEVEN G. JOHNSON2
1Department of Mechanical Engineering, Technical University of Denmark, Nils Koppels Allé, Building
404, 2800 Kongens Lyngby, Denmark
2Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
3Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA
02139, USA
4Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA
*raelch@mek.dtu.dk
Abstract: We show that topology optimization (TO) of metallic resonators can lead to ∼102 ×
improvement in surface-enhanced Raman scattering (SERS) efficiency compared to traditional
resonant structures such as bowtie antennas. TO inverse design leads to surprising structures
very different from conventional designs, which simultaneously optimize focusing of the incident
wave and emission from the Raman dipole. We consider isolated metallic particles as well as
more complicated configurations such as periodic surfaces or resonators coupled to dielectric
waveguides, and the benefits of TO are even greater in the latter case. Our results are motivated
by recent rigorous upper bounds to Raman scattering enhancement, and shed light on the extent
to which these bounds are achievable.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In this paper, we present freeform shape optimization ("topology optimization," TO [1–3]) of
metallic nanoparticles for Raman scattering [4–6], and obtain non-intuitive structures ∼ 60×
better (in terms of emitted power) than optimized coupled-sphere [7–9] or bowtie [10–14]
antennas (Sec. 3.1) for identical separation distance to the Raman molecule. Our current results
are a proof-of-concept of Raman TO in 2d systems, and the resulting dramatic enhancements
suggest exciting opportunities for future improvements in practical 3d Raman sensing. Stated
briefly, Raman scattering consists of an incident wave at a frequency ω1 interacting with a
molecule that subsequently emits electromagnetic radiation at ω2. A nanostructure can enhance
both the incident-wave absorption by a focusing effect as well as the emission by a Purcell effect
[15–20]. Figure 1(a) shows a schematic of this process, in which the molecule is surrounded
by an unknown arrangement of metal (e.g. silver); TO is used to tailor this arrangement to
maximize the Raman scattering (Sec. 2), resulting in surprising structures such as the one shown
in Fig. 1(b). In addition to optimizing isolated resonators coupled with incident planewaves
(Fig. 1(a)), we also optimize Raman scattering on periodic surfaces as well as resonators coupled
with input/output waveguides, as depicted in Fig. 2(b) and 2(c), respectively. We show that
it is important to consider the full Raman process combining both focusing and emission, as
optimizing either emission or focusing alone sacrifices a factor of ∼ 5× (Sec. 3.2). When the
Raman shift ω1 − ω2 is more than a few percent (i.e., comparable to the bandwidth of a single
plasmonic resonance), we show that this shift must be taken into account in the optimization, or
one may sacrifice a factor of ∼ 5× (Sec. 3.5). We find that the huge enhancements observed
from TO structures compared to simple geometries are in qualitative agreement with recently
discovered theoretical upper bounds to Raman scattering [21]. Quantitatively, for a material with
#382827 https://doi.org/10.1364/OE.28.004444
Journal © 2020 Received 13 Nov 2019; revised 29 Nov 2019; accepted 29 Nov 2019; published 3 Feb 2020
Research Article Vol. 28, No. 4 / 17 February 2020 / Optics Express 4445
susceptibility χ, the key figure of merit for light-matter interactions is F = |χ |2/Im(χ) [22–24]
and the Raman bounds scale as a factor of F3V where V is the volume of the scatterer: a factor
of F2 maximum focusing enhancement and a factor of F maximum emission enhancement. The
enhancement achieved by our TO designs scale roughly linearly with V in agreement with the
upper bounds (Sec. 3.3), but they scale roughly with F2 instead of F3 (Sec. 3.4) suggesting, in
part, that in practice one cannot simultaneously achieve ideal focusing and emission enhancement.
Fig. 1. (a) Sketch of the Raman scattering design problem. (b) |E2 |-field emitted through
Raman scattering from a molecule (point dipole) placed at the centre of a (gray) topology-
optimized silver structure at λ1 = λ2 = 532 nm. The color scheme is saturated due to the
diverging field at the source.
Fig. 2. Problem configurations. (a) ARaman molecule (blue) in air background, surrounded
by the design domain ΩD (gray), excited by an incident planewave (green), the total emitted
power (red) through Γout is sought maximized. (b) A Raman molecule positioned on a
periodically patterned surface (period: a), surrounded by ΩD, excited by a planewave at
normal incidence, maximizing the emitted power normal to the surface. (c) A Raman
molecule positioned inside ΩD, excited by an incident wave, coupled into the domain via a
waveguide (input), maximizing the emitted power into another waveguide (output).
In conventional Raman spectroscopy, the very small Raman cross-section of most chemicals
results in Raman radiation typically on the order of 0.001% of the power of the pump signal
[4]. This low efficiency is overcome in surface-enhanced Raman spectroscopy (SERS) by
placing the chemicals of interest in the vicinity of a scatterer, typically a surface or collection
of nanostructures, which increases the overall signal that can be collected [15–20,25]. This
technique has allowed for efficiencies up to 12 orders of magnitude larger than that of traditional
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Raman spectroscopy, yielding detection levels down to the single molecule [26,27] and opening
up applications in the fields of biochemistry, forensics, food safety, threat detection, and medical
diagnostics [19,20]. While many different materials and antenna geometries have been used for
SERS substrates [28–30], so far the optimization of these geometries has been limited to one
or two degrees of freedom in designs such as bowtie antennas [10,31–34]. Comparison with
the recently derived upper bound to Raman enhancement showed these geometries to be greatly
suboptimal, with performance several orders of magnitude below the bounds [21]. Although it is
possible that the bounds can be tightened (e.g. by incorporating additional physical constraints
[35]), the fact that current SERS geometries are so far below these new theoretical limits made
us wonder if dramatic improvements might be attainable by TO.
In this work we thus take a hitherto unexplored approach in the context of Raman scattering, in
which the problem of designing metallic nanostructures to enhance the process is formulated as a
mathematical optimization problem and solved using density-based topology optimization [1]. In
the design process, we simultaneously optimize the focusing of the incident field and the emission
from the molecule (dipole), which is shown to lead to structures with higher performance than
only optimizing for one process. In brief, density-based topology optimization operates by
introducing a continuous design field to control the physical material distribution, enabling the
use of adjoint sensitivity analysis [36] and gradient-based optimization algorithms [37,38] to
efficiently solve design problems with potentially billions of design degrees of freedom [39] .
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
[40]. A suite of well-understood tools, developed or matured over the last decades, are used
to solve the optimization problem, interpolate material parameters, control design variations
and ensure physical admissibility of the design [38,41–44]. In addition, geometric length-scale
constraints are employed to ensure that all features of the final designs are above a specified size
(which may be chosen to comport with fabrication constraints) [45]. Over the past 20 years,
TO has been applied to an increasingly wide range of problems in electromagnetic design [2,3].
Our work on Raman optimization couples multi-frequency focusing and emission problems.
Maximizing the emission alone (for a dipole source) corresponds to maximizing the local density
of states (LDOS) [46,47], a formulation that has been employed for TO of optical cavities [48,49].
The focusing problem alone is related to lens and antenna design among others, for which TO
has also been applied (both to small metallic particles such as those considered here and to
much larger structures, such as metalenses) [44,50–54]. Nonlinearly coupled electromagnetic
resonances at multiple frequencies were optimized via TO for second harmonic generation [55].
2. Model formulation
The Raman scattering phenomenon (sketched in Fig. 1(a)) is modelled as two sequentially-
coupled, time-harmonic, classical electromagnetic problems [40,56] in a domain Ω. The first
models an in-plane polarized planewave propagating through the domain, interacting with any
material distributed inside. The second models the excitation of a Raman molecule in Ω by the
electric field resulting from the first problem. The Raman molecule is modelled as a dipole
source, with its dipole moment given by a polarizability tensor multiplied by the electric field,
excited by the incident pla
formally as (newave at the p)osition of the dipole [18]. The problem may be stated
∇ × µ−1∇ × E1(r) − ω21ε(r)E1(r) = f1(r), r ∈ Ω, (1)
( f (r) = −) ω22 2αE1(r0)δ(r − r0), (2)
∇ × µ−1∇ × E2(r) − ω22ε(r)E2(r) = f2(r), r ∈ Ω. (3)
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Here Ω ∈ R2 represents the modelling domain and r and r0 denote points in Ω. The material
parameters µ and ε(r) are the magnetic permeability and the electric permittivity, respectively
(non-magnetic materials are assumed in the following, i.e. µr ≡ 1). Further, ωi = c/λi denotes
the angular frequency with c being the speed of light and λi the wavelength, Ei the electric field
and fi the excitation sources for i( ∈ 1, 2. Finally α de)notes the polarizability tensor, which in this
work is taken to be the identity αE1(r0) = IE1(r0) .
The problem of enhancing the power emitted from the Raman molecule at λ2, by tailoring
the material distribution in the design domain ΩD, is formulated as a continuous optimization
problem and solved using density-based topology optimization. In its basic form the optimization
problem may be written as ∫
maxξ(r)∈[0,1] Φ(ξ) = 〈S2(ξ)〉 · n dr, (4)
Γout
( ) s.t. 0 ≤ ξ(r) ≤ 1, r ∈ ΩD. (5)
Here 〈S 〉 = Re 1E ×H∗2 2 2 2 denotes the time averaged Poynting vector computed from E2,
which in turn is obtained by solving Eqs. (1)–(3) for a given ξ(r) and (•)∗ denotes the complex
conjugate. The vector n is the outward pointing normal vector for the integration curve Γout
(Fig. 1(a)). The permittivity distribu(tion ε(r) is determined by the value of ξ(r) through theinterpolation [44], ) ( )
ε(ξ) = n(ξ)2 − κ(ξ)2 − i 2n(ξ)κ(ξ) ,
n(ξ) = nM1 + ξ(nM2 − nM1 ), (6)
κ(ξ) = κM1 + ξ(κM2 − κM1 ).
Here Mi denotes the materials considered (metal or air) and n and κ denote the refractive index
and extinction coefficient of Mi, respectively. Finally, i denotes the imaginary unit.
Density-based topology optimization is used to solve Eqs. (4)–(5) where the physical
admissibility of the final material distributions is assured using filtering, thresholding and
continuation techniques developed in the mechanics community [42,43]. This allows the
enforcement of a minimum length scale (6 nm) on all features in the material distribution using
geometric length-scale constraints [45].
We consider the three configurations shown in Fig. 2. The first (Fig. 2(a), Sec. 3.1–Sec. 3.6)
is used to benchmark the proposed approach. It consists of a Raman molecule (blue dot) placed
in an air background with ΩD (gray region) surrounding the molecule. An incident planewave
propagates through the domain from left to right (green) and the power emitted through Γout by
Raman scattering (red) from the molecule is sought maximized.
The second problem (Fig. 2(b), Sec. 3.7) models out-of-plane Raman scattering and considers
a metallic surface with periodic patterning placed in a dielectric medium. The model problem
is x-periodic with period a. It consists of a spatial region containing the design domain with a
Raman molecule placed at its centre, truncated from below by a metallic surface. An incident
planewave propagates through the domain from top to bottom and the power emitted from the
molecule is sought maximized in the direction normal to the metallic surface. The Raman
scattering process from the different molecules on the surface is assumed to be incoherent.
Therefore a k-space integration technique, the array scanning method [57,58], is used to compute
the power emitted from each individual Raman molecule situated in the periodic background.
The third model problem (Fig. 2(c), Sec. 3.8) concerns in-plane Raman scattering into a
waveguide (output). A Raman molecule is excited by an incident field coupled into the system
from a second waveguide (input). The Raman molecule is positioned at the centre of ΩDMetal
(dark gray), in which a metallic nanostructure is designed. In the remaining part of the design
domain, ΩDDielectric (light gray), a dielectric structure, aimed at coupling the light into and out of
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the waveguides, is designed simultaneously. For this problem, the difference in the power emitted
by the Raman molecule through Γout and Γin is sought maximized.
The model domains are truncated using different boundary conditions depending on which
problem configuration is considered. For the isolated resonators (Fig. 2(a)) and the resonators
coupled with the input/output waveguides (Fig. 2(c)), a perfectly matched layer is used to truncate
Ω. For the Raman molecules placed on a periodic surface (Fig. 2(b)), Floquet-Bloch boundary
conditions are used in plane, a perfectly matched layer truncates the top boundary and a perfect
electric conduction condition is imposed on the bottom boundary.
The model problems are implemented in COMSOL Multiphysics [59] and the optimization
problem is solved using the Globally Convergent Method of Moving Asymptotes [38] utilizing a
maximum of 3 inner iterations per design iteration. Details concerning the numerical simulation,
optimization, post-processing, and evaluation processes are given in Appendix A and B.
3. Results
This section details eight studies conducted to investigate various aspects of the Raman scattering
enhancement (Raman enhancement) problem. Study-specific parameters are given in each of the
following subsections, while a general set of parameters used across all studies may be found in
Appendix B.
3.1. Comparison to known solutions
To demonstrate the proposed approach, we design a silver nanostructure that enhances Raman
scattering from an isolated Raman molecule (Fig. 2(a)) excited at λ1 = 532 nm, assuming a
negligible Raman shift (λ2 = λ1); non-negligible shifts are considered in Sec. 3.2 and Sec. 3.5.
For context, the achieved enhancement is compared to enhancements generated by two simple
parameter-optimized references.
Regarding the reference geometries, it is well known that placing a Raman molecule between
two carefully scaled circular metallic cylinders (a 2d analogue of a coupled-sphere antenna)
significantly enhances the Raman scattering process [7–9], while placing it between two
identical metallic triangular structures (a bowtie antenna) further enhances the process [10–14].
Both these references are parameter-optimized to maximize their performance at the targeted
wavelengths. For the coupled-cylinder antenna, the radius of the two discs is optimized over
Rcsa ∈ [10 nm, 50 nm], and we find the largest enhancement for Rcsa = 48 nm. The bowtie
antenna is optimized by sweeping the tip-angle over, θbta ∈ [10o, 180o], and the side length over
Lbta ∈ [20 nm, 100 nm], and we find θ obta = 15 and Lbta =(70 nm t)o yield the largest enhancementat λ1 = λ2 = 532 nm.
Figure 3 shows the enhancement of the emitted power Eq. (4) versus wavelength for each of
the three structures, relative to the power emitted from the Raman molecule in free space. The
coupled-cylinder antenna (black) is seen to achieve a nearly wavelength-independent enhancement
of ≈ 3 · 102, while the bowtie antenna (red) achieves a peak enhancement of ≈ 1.3 · 104 at 532
nm. Interestingly, the intricate geometry of the topology-optimized nanostructure (blue), fully
encapsulating the Raman molecule, achieves an enhancement of ≈ 8.0 · 105. This is an increase
by a factor of ∼ 60× relative to the bowtie antenna. The TO structure is in some sense a fusion of
different features, tailored to enhance either the focusing or emission process, as will be studied
more closely in Sec. 3.2. To our knowledge, no metallic structure for Raman enhancement
similar to the optimized structure has previously been proposed, nor is such a structure likely to
arise from applying any traditional design rules or intuition.
A finding worth noting for this study, is that starting from an initial material configuration of
ξ(r) = 0.0 ∀r ∈ ΩD results in the design process converging to a structure not encapsulating the
Raman molecule. Importantly, this non-encapsulating design achieves an enhancement which is
lower by more than a factor of ∼ 2× compared to the fully encapsulating design in Fig. 3. In
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Fig. 3. Raman enhancement as a function of wavelength, for a molecule placed at the center
of different silver nanostructures (dark blue dot) relative to a molecule placed in free space.
A topology-optimized structure (blue), a bowtie antenna (red) and coupled-cylinder antenna
(black) are considered, all designed to maximize enhancement at λ = 532 nm.
general, any structure found with gradient-based TO may only be a local optimum to the design
problem. Alternative global optimization formulations are rarely practical or even feasible in
large design spaces for non-convex problems [60] and do anyway not guaranty global minima.
Gradient-based optimization from different starting points (a "multistart" algorithm) can explore
different local optima but in this work, the main goal is to find a structure much better than what
could be easily designed by hand. Comparison to theoretical upper bounds is another route to
gauging global optimality of TO structures [22,61].
3.2. Raman vs. focusing vs. emission
A question worth asking, as it could potentially reduce the computational effort associated
with the design procedure by a factor of two, is whe(ther it is) necessary to consider the fulltwo-step Raman scattering process in the design procedure, or if it is possible to achiev(e simila)r
performance by only considering an energy-focusing Eq. (1) or a dipole-emission Eq. (3)
problem. To answer this question using the proposed approach, we consider the following three
cases. A dipole-emission case, an energy-focusing case and (a case conside)ring the full two-stepprocess. All cases assume a 50 nm Raman shift, with λ1 = 532 nm and λ2 = 582 nm.
For the dipole-emission case, the optimization problem Eqs. (4)–(5) remains unchanged,
while the dipole considered in Eq. (3) has its orientation fixed along the y axis and its magnitude
kept constant through(out the o)ptimization, i.e. f = −ω22 2I〈1, 0〉δ(r− r0), effectively removing the
first model problem Eq. (1) as f2 no longer depends on E1. This corresponds to maximizing
the local density of states for the dipole [48]. For the energy-focusing case, the optimization
problem is changed to one of solely maximizing |E1 |2 for an incident (planewa)ve at the position
of the dipole [44,50] effectively removing the second model problem Eq. (3) , as the emission
process is no longer considered.
Figure 4 reports the enhancement of the three objectives attained by introducing the nanos-
tructures optimized for the emission (red), focusing (black) and two-step Raman scattering
(blue) process. The leftmost bar group reports the emission enhancement for a dipole with unit
magnitude oriented along the y axis, when placed at the center of each of the three designs. The
middle bar group reports the enhancement of |E |21 at the dipole position for each design, while
the rightmost bar group reports the enhancement achieved for the full two-step Raman scattering
process. From the figure, it is seen that optimizing for a given process yields the best performance
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for that process. In particular, by explicitly optimizing for the two-step Raman scattering process,
an increase in the enhancement by a factor of more than 4.5× is achieved. It is interesting to
observe that the design optimized for the full Raman scattering process qualitatively consists of a
fusion of geometric features found in the focusing and the emission designs, e.g the closed cavity
(emission) and the protruding features on the right side of the design (focusing). This suggests
that a combination—and, by extension, a trade-off— between the features is required to achieve
the largest Raman enhancement. By evaluating the emission and focusing components of the
enhancement bound presented in Ref. [21] for the red and black structures in Fig. 4, we discover
the following. The emission enhancement achieved by the red structure (≈ 103) comes within a
factor of four of the bound (≈ 3.5 · 103), where difference may be explained by the length-scale
imposed on the design. The focusing enhancement achieved by the black design (≈ 103) is, on
the other hand, nearly three orders of magnitude from the focusing bound (≈ 6 · 105). Hence, it
has not been possible to reach the focusing bound, the magnitude of the discrepancy suggesting
that the bound may not be tight.
Fig. 4. Objective enhancement achieved at λ1 = 532, λ2 = 582 nm by introducing optimized
silver structures for pure dipole emission [Leftmost bar group]; pure energy focusing [Middle
bar group]; full Raman scattering process [Rightmost bar group]. Structures optimized for
dipole emission (red); energy focusing (black); full Raman scattering process (blue). The
focal point (dipole position) is indicated using a dark blue dot.
3.3. Size scaling
It was recently shown that an upper bound for enhancing the Raman scattering process using
metallic nanostructures scales linearly with the volume (area in 2d) of the design region [21].
To investigate if this scaling is found using the proposed design approach, we perform a study
considering three different outer radii ofΩD. These are Rout = 50 nm (black), Rout = 75 nm (red)
and Rout = 100 nm (blue), respectively. All other parameters are kept constant across the three
cases and an assumption of a negligible Raman shift is made, i.e. λ1 = λ2 = 532 nm.
Figure 5 shows the Raman enhancement relative to a molecule in free space versus the area
of ΩD with the three designs and their performance plotted in black, red and blue and a trend
line of slope 1 included to guide the eye. The data shows an approximately linear scaling of the
emission enhancement with volume (area) in agreement with the upper bounds derived in Ref.
[21], demonstrating that the proposed approach finds the expected volume scaling.
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Fig. 5. Raman enhancement versus ΩD area, for a molecule placed at the center of the
sliver structures (dark blue dot) relative to free space. A reference line (gray) with slope 1 is
included to guide the eye. Nanostructures optimized for Rout = 50 nm (black), Rout = 75
nm (red) and Rout = 100 nm (blue) are shown. Approximately linear scaling is observed.
3.4. Material scaling
The upper bound for enhancing the Raman scattering process, derived in Ref. [21], scales
cubically with the material-related quantity F = |χ |2/Im(χ). Here a factor of F2 stems from
energy-focusing enhancement while the remaining factor of F stems from dipole-emission
enhancement. To investigate this behaviour, we conduct a materials study considering four metals,
silver (Ag), gold (Au), copper (Cu) [62], and platinum (Pt) [63]. All parameters, other than the
material parameter ε(r), were kept constant across the four cases and a negligible Raman shift
was assumed with λ1 = λ2 = 532 nm.
Figure 6 shows the Raman enhancement obtained by placing the Raman molecule at the center
of the optimized structures relative to free space versus |χ |2/Im(χ) for the Cu- (magenta), Au-
(black), Pt- (red) and Ag-design (blue). To guide the eye, two trend lines (gray) with slope
2 and slope 3 are plotted with the data. From a linear fit of the data, an approximate scaling
with F2 is observed. Trusting that TO indeed provides highly optimized results, as confirmed
by the previous examples, this data suggests that the theoretical bounds may not be tight for
two reasons. First, it may not be possible to create a design which achieves ideal focusing- and
ideal emission-enhancement simultaneously, but that a trade-off must be made. The observation
is supported by the data in Fig. 4, where significantly different geometries are observed when
targeting the maximization of either focusing or emission, suggesting that different geometries
are required to achieve either the highest attainable focusing or the highest attainable emission.
Second, combining the observed F2 scaling in Fig. 6 with the finding in Sec. 3.2 that the focusing
bound is not reached when optimizing purely for focusing, leads to the suggestion that the bound
might not be tight. In conclusion, we find that the optimized structures (while vastly superior
to bowtie antennas) still fall far short of the upper bound [21], especially for large values of F.
Future analytical work including additional constraints, e.g. coupling the two processes, may
lead to a tighter bound.
3.5. Accounting for the Raman shift
Depending on the molecule(s) and energy transitions of interest for a given Raman scattering
problem, the Raman shift between the wavelengths λ1 and λ2 will be different. In this section,
we investigate the benefits of accounting for the Raman shift in the design process. To this end,
three cases are considered (Fig. 7) with Raman shifts of 0 nm (blue), 50 nm (red), and 100 nm
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Fig. 6. Raman enhancement for a Raman molecule (dark blue dot) versus |χ |2/Im(χ)
relative to free space. Reference lines with slopes 2 and 3 (gray) are included. Optimized
structures using Cu- (magenta), Au- (black), Pt- (red) and Ag-parameters (blue).
(black), respectively. The wavelength of the incident field is kept constant at λ1 = 532 nm and
the wavelength of the light emitted by the Raman molecule is adjusted accordingly.
Fig. 7. Raman enhancement for different Raman shifts for a Raman molecule (dark blue
dot) placed at the centre of different silver nanostructures optimized for λ1 = 532 nm and
λ2 = 632 nm (black) λ2 = 582 nm (red) and λ2 = 532 nm (blue). The enhancement is
reported for 0 nm [Leftmost], 50 nm [Middle] and 100 nm [Rightmost] bar groups.
Figure 7 presents the three designs along with the power-emission enhancement attained for
each design when evaluated at 0 nm shift (leftmost bar group), 50 nm shift (middle bar group),
and 100 nm shift (rightmost bar group). A first observation is that all three structures share
the same overall geometrical features and thus in that sense look similar. However, looking
more closely at each structure it is clear, that both the size and shape of each feature change
across the designs. Furthermore, looking at the differences in enhancement, it is clear that these
delicate feature changes are reflected in the performances of the structures. Unsurprisingly, each
design exhibits the largest enhancement at the Raman shift it was optimized for. Specifically,
an enhancement difference of a factor of ∼ 1.6× is observed for the 50 nm Raman shift while a
factor of ∼ 4.5× is observed for the 100 nm Raman shift, clearly illustrating the performance
benefit of accounting for the Raman shift as part of the design process.
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3.6. Wavelength dependence
The importance of adjusting an electromagnetic structure to the operating wavelength, such as
changing the length of an antenna, is well known. In this study we demonstrate that significantly
larger design adjustments than simple scaling are required to achieve the best Raman enhancement
for a given operating wavelength. It is shown how the optimized nanostructure geometry, as
well as the emission enhancement, changes significantly with wavelength over the interval
λ1 = λ2 ∈ [250 nm, 490 nm]. The study considers smaller designs with Rout = 30 nm and
assumes a negligible Raman shift. In a sense, it probes a similar quantity as the material-
dependence study in Sec. 3.4, seeing as |χ |2/Im(χ) for silver varies by orders of magnitude
across the wavelength interval. However, this study considers the added complexity that the
wavelength changes along with ε(r).
The top panel of Fig. 8 shows nine silver nanostructures (rainbow-colored) optimized and
evaluated for the reported wavelengths (colored dots). The quantity [|χ |2/Im(χ)]2 for silver
is overlaid for easy reference (black). It is clearly observed that the enhancement factor of
the optimized structures is strongly dependent on [|χ |2/Im(χ)]2. Furthermore, it is seen that
the optimized nanostructure geometries change significantly from left to right, starting with a
reflector-like geometry for λ = 250 nm and ending with a geometry resembling the red structure
in Fig. 4 optimized to maximize dipole emission. Interestingly, around λ = 320 nm (near
the minimum of [|χ |2/Im(χ)]2) the optimized structure is seen to experience a topological
change which fundamentally changes the geometry from the reflector-like type to structures fully
encapsulating the emitter. The bottom panel of Fig. 8 shows the per-wavelength max-normalized
enhancement attained for each of the nine structures at each wavelength. From the panel it is seen
that each of the optimized nanostructures outperformed the other structures at the wavelengths
for which they were optimized. Furthermore, it is seen that as the wavelength increases the
performance sensitivity increases. The data clearly shows the need for tailoring the nanostructure
geometry to the particular operating conditions if the highest performance is sought.
3.7. Out-of-plane Raman scattering
A common configuration for Raman-sensing applications is the distribution of Raman molecules
over a surface, illuminated by some out-of-plane source with the scattered light collected out-of-
plane as well. In the following study we demonstrate the strength of the proposed approach for
such problem configurations (Fig. 2(b)).
We consider a periodically-patterned platinum surface (period a = 150 nm) submerged in
a water background (εH2O ≈ 1.77 around 500 nm). For computational simplicity, the study
assumes a single Raman molecule per unit cell, situated at a fixed position at the center of the 150
nm x 200 nm design domain, which is again placed immediately on top of the platinum surface.
A circular region of radius Rin = 10 nm, centered at the Raman molecule, is kept free of platinum
throughout the optimization. The model problem is excited at λ1 = 532 nm with the Raman
molecules emitting at λ2 = 549 nm. The molecules in the periodic array are assumed to scatter
incoherently, as this most accurately models the physical scattering process. To account for this
incoherent scattering, the array scanning method is used in the design process to compute the
Raman scattering from a single molecule in the periodic model using 50 k-points in [−π/a, π/a]
[57,58].
As a reference, we consider a periodic array (a = 150 nm) of circular platinum discs resting on
top of the platinum surface, with the Raman molecules placed at the center of the gap between
the discs (blue dot in Fig. 9(d)). The separation distance from the molecule to the discs is taken
to be 10 nm, identical to the radius of the platinum-free circular region in the optimization case.
Figure 9 shows the optimized (left) and reference (right) surface structures and their behaviour
under illumination at λ1 = 532 nm with a single Raman molecule placed in the center unit
cell. The top row shows the periodic platinum surface structures (black) in water background
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Fig. 8. Raman enhancement relative to a molecule emitting in free space as a function of
pump and emission wavelength (λ1 = λ2). [Top] Nine optimized silver nanostructures with
outer design domain radius Rout = 30 nm (rainbow) and the achieved enhancement (colored
dots). The quantity [|χ |2/Im(χ)]2 is overlaid as a reference (black) and the position of the
Raman molecule relative to the nanostructures is indicted using dark blue dots. [Bottom] The
per-wavelength best design max-normalized enhancement for each of the nine wavelengths
and each of the nine designs.
(gray) with the blue dot indicating the position of the Raman molecule. The middle row shows
|E1 | at λ1 = 532 nm for identical incident field strength using identical color scales. From the
panels it is clear that the optimized structure provides a stronger focus of the incident field at
the Raman molecule position. The bottom row presents |E2 | at λ2 = 549 nm using identical
color scales, clearly illustrating the significantly enhanced emission for the optimized surface
structure. Computing the power emitted by the Raman molecule for both cases reveals a Raman
enhancement by a factor of ∼ 64× for the optimized structure relative to the reference. While this
example assumes a 2d model, it demonstrates the potential of applying the proposed approach to
the design of nanostructures for full 3d out-of-plane Raman scattering problems.
3.8. In-plane Raman scattering
In the final study we consider an in-plane Raman scattering problem with input and output
waveguides (Fig. 2(c)). This study demonstrates the vast design freedom inherent to the proposed
approach, which allows for the design of extreme enhancement structures in configurations
where it would otherwise be difficult, if not impossible, to achieve good enhancement using
conventional design techniques or intuition.
The objective of the design problem is to maximize the difference between the power emitted
through Γout and through Γin. This is achieved by designing a Pt nanostructure inΩDMetal together
with a Si3N4 dielectric structure in ΩDDielectric . Figure 10(a) shows the optimized nanostructure
obtained by solving the design problem at λ1 = 532 nm, λ2 = 549 nm.
Research Article Vol. 28, No. 4 / 17 February 2020 / Optics Express 4455
Fig. 9. [left] Optimized and [right] reference periodically-structured platinum surfaces
for the out-of-plane emission problem (Fig. 2(b)). (a) Optimized surface (black) in water
background (gray), with single Raman molecule (blue dot). (b) |E1 |-field at λ1 = 532 nm
resulting from the interaction between the incident field and the topology-optimized surface.
(c) |E2 |-field emitted at λ2 = 549 nm from a Raman molecule positioned at the center of
the middle unit cell. (d) Reference surface (black) in water background (gray), with single
Raman molecule (blue dot). (e) |E1 |-field at λ1 = 532 nm resulting from the interaction
between the incident field and the reference surface. (f) Emission |E2 |-field emitted at
λ2 = 549 nm from the Raman molecule.
As a reference, a bowtie antenna is parameter-optimized over its tip angle θ o obta ∈ [10 , 90 ],
side length Lbta ∈ [20 nm, 100 nm], and rotation angle relative to horizontal θrot ∈ [−90o, 90o].
The parameters leading to the largest objective value are found to be θ obta = 10 , Lbta = 62.5 nm,
and θ orot = 25 . For the dielectric material distribution in the reference a simple extension of the
two waveguides towards the bowtie antenna is used. The reference nanostructure is shown in
Fig. 10(b).
Figure 10(c)–10(d) shows the incident field at λ1 = 532 nm for the optimized and reference
structure, respectively. Identical color scales are used for both plots, clearly showing the enhanced
focusing of E1 at the position of the Raman molecule. The rotated orientation of the reference
structure, introduced to maximize the objective, means that the bowtie antenna is not capable of
creating a highly localized focus between its tips. In contrast, the significantly more intricate
geometry of the optimized design has no problem providing such enhancement.
Figure 10(e)–10(f) shows the field emitted by the Raman molecule at λ2 = 549 nm for the
optimized structure and the reference structure, respectively. Again, identical color scales are used
for the two plots. From these it is obvious that the optimized structure generates a significantly
enhanced emission relative to the reference. In fact, the difference between the two structures
in terms of emitted power through Γout is a factor of ∼ 345×, i.e. more than two orders of
magnitude.
Research Article Vol. 28, No. 4 / 17 February 2020 / Optics Express 4456
Fig. 10. [Top row] Optimized and [bottom row] reference structures and fields for the
in-plane emission problem (Fig. 2(c)). (a) Optimized platinum (black) and Si3N4 structure
(white) with Si3N4 input/output waveguides (white) and water background (gray). (b))
Reference platinum bowtie antenna (black) input/output waveguides (white) and water
background (gray). (c)-(d)) |E1 |-field at λ1 = 532 nm resulting from the interaction between
the incident field through the lower left waveguide and the (c) optimized and (d) reference
structure. (e)-(f) |E2 |-field emitted at λ2 = 549 nm from a Raman molecule positioned at
the center of (e) the optimized and (f) reference platinum nanostructure.
4. Conclusion
In this paper we proposed a TO-based approach for designing Raman scattering enhancing
metallic nanostructures and presented a structure that increases the enhancement by a factor of
∼ 60× compared to a parameter-optimized bowtie antenna (Sec. 3.1). Through a number of
studies we documented: the importance of considering the full Raman scattering process in the
design procedure (Sec. 3.2); that the expected linear scaling [21] of the Raman enhancement
with design volume (area) is achieved (Sec. 3.3); that the Raman enhancement scales with
[|χ |2/Im(χ)]2 rather then the predicted scaling of [|χ |2/Im(χ)]3 [21], possibly due to a trade
off between the focusing and emission enhancement processes (Sec. 3.4); the importance of
accounting for the Raman shift in the design procedure (Sec. 3.5); the importance of tailoring
the nanostructure geometry to the operating wavelength as large geometric and associated
performance changes occur as the wavelength is varied (Sec. 3.6). Finally, we demonstrated
that the TO-based approach may be used to achieve ∼ 64× enhancement for out-of-plane Raman
scattering (Sec. 3.7) and ∼ 345× enhancement for in-plane Raman scattering in a two-waveguide
setup (Sec. 3.8), relative to simple parameter optimized reference structures.
While all studies in this work consider 2d model problems, the demonstration of extreme
enhancements compared to well known reference geometries is promising. Hence, an important
next step is the extension of the proposed approach to three dimensions. Applying the approach
for Raman scattering problems modelling realistic operating conditions may help reveal hitherto
undiscovered nanostructures for extreme Raman enhancement. If such structures are found, they
Research Article Vol. 28, No. 4 / 17 February 2020 / Optics Express 4457
may serve to improve existing Raman scattering based tools significantly as well as enable the
development of novel tools.
An important topic for future work is to incorporate variability in the location of the Raman
molecule. In some experimental situations, the Raman molecules are suspended in a fluid and
the objective is to maximize the average emission over all possible molecule locations [32,64,65].
Naively, this would require solving the model problem for a vast number of different dipole
locations and averaging the emission, making such a task computationally infeasible for all but
the smallest of problems. However, efficient "trace" formulations have been developed for related
problems involving a distribution of random emitters—thermal emission [66,67] and spontaneous
emission [67]—and we believe that similar techniques are applicable to the Raman problem.
Another future step is the investigation of the sensitivity of the optimized structures towards
perturbations of their geometry, such as those encountered in fabrication. Results in previous
studies on designing plasmonic nanostructures using TO, e.g. [52], as well as several results in
this work, suggest that the optimized nanostructures are sensitive to geometric variation, as small
variations lead to large performance changes. By employing well known robust optimization
techniques [43,68] it may be possible to reduce the geometric sensitivity of the nanostructures,
hereby bridging a gap between simulations and experiment.
Appendix A. Optimization, post processing and evaluation procedure
The physics is modelled in COMSOL Multiphysics [59] and the optimization problem is solved
using the Globally Convergent Method of Moving Asymptotes (GCMMA) [38].
The following stopping criterion is used to terminate the optimization:
if i ≥ imin then
if β < βmax then
β = 2β
i = 1
else
if |Φi − Φi−n |/|Φi | ≤ 0.001 ∀ n {1, 2, . . . , 10} then
Terminate optimization.
end if
end if
end if
Here i denotes the current optimization iteration, imin = 70 denotes the minimum number of
iterations taken. β denotes the thresholding strength used in the filtering procedure and βmax = 32.
Φi denotes the objective function value at the i’th iteration and n ∈ N+.
After the design process is completed a post-processing procedure is performed to extract the
final nanostructure from the optimized material distribution. In this procedure the final (r)-field
is sampled in a Cartesian (x,y)-grid with 0.1 nm resolution. The sampled distribution is smoothed
using a simple cone-shaped kernel [41] with 1.5 nm filter radius to remove all sub-nanometer
kinks left by the 1 nm topology optimization mesh. The final geometry is then extracted as the
0.5-contour of the smoothed field. Finally, the geometry is rescaled to have an inner radius of
rinner = 10 nm for consistency across designs.
The reported performances and fields are evaluated using the final post processed design,
discretized using an unstructured triangular body fitted mesh with 1 nm side length for the
structure. Quadratic continuous Lagrange basis functions are used to resolve the physics during
the evaluation step.
Note that the exact position and magnitude of the Raman enhancement peak depend strongly
on the final geometry. This means that a size scaling of a structure by a few percent may result in
a shift of several nm in the emission peak position as well as a change in the overall enhancement.
Research Article Vol. 28, No. 4 / 17 February 2020 / Optics Express 4458
Appendix B. Study parameters
This appendix lists the parameters used in setting up the model and associated optimization
problems. Unless specified otherwise below or in the individual sections detailing each study,
the parameters listed in the following are used.
B. 1. Model domain
For the model domain sketched in Fig. 2(a) the following values are used: The model domain Ω
is a square with side length, LΩ = 600 nm. Ω is surrounded by a perfectly matched layer with a
depth of DPML = 300 nm. The design domain ΩD is centered in Ω and taken to be a perfect 2d
torus with inner radius, Rin = 10 nm and outer radius, Rout = 100 nm. The Raman molecule is
modelled as a point dipole source placed at the center of ΩD. The power emitted by the dipole
at λ2 is computed by evaluating Eq. (4) over ΓRE, a circular curve centered on the dipole with
radius: RΓRE] = 250 nm.
For the model domain sketched in Fig. 2(b) the following values are used: The model domain
Ω is a rectangle of width, a = WΩ = 150 nm and height HΩ = 600 nm. Ω is truncated from
above by perfectly matched layer with a depth of DPML = 300 nm. Ω is truncated from below
using a perfect electric conductor boundary condition: n × E = 0. Ω is truncated by a Floquet
Bloch boundary condition on the left and right boundaries to model the x-periodicity. The top
500 nm of Ω is taken to contain water (εr ≈ 1.77). The bottom 100 nm of Ω is taken to contain
platinum [63]. The design domain ΩD is placed immediately on top of the platinum region. The
design domain ΩD has a width of WΩD = 150 nm and a height HΩD = 200 nm. The design
domain ΩD has a region fixed to contain water at its center with radius Rin = 10 nm. The Raman
molecule is modelled as a point dipole source placed at the center of ΩD. The power emitted by
the dipole at λ2 is computed by evaluating Eq. (4) over ΓRE, a straight horizontal line 50 nm
below the top boundary of Ω.
For the model domain sketched in Fig. 2(c) the following values are used: The model domain
Ω is a square with side length, LΩ = 1200 nm centered at (0 nm,0 nm). Ω is surrounded by a
perfectly matched layer with a depth of DPML = 300 nm. The design domain ΩDMetal is centered
at (-300 nm,-300 nm). ΩDMetal is a square of side length LΩD = 200 nm with a circular holeMetal
of radius Rin = 10 nm at its center. The Raman molecule is modelled as a point dipole source
placed at the center of ΩD. The design domain ΩDDielectric is centered at (-200 nm,-200 nm).
ΩDDielectric is the difference between a square of side length LΩD = 400 nm and Ω . TheDielectric DMetal
input waveguide runs horizontally and is centered at y = −300 nm. It starts at the left edge of
Ω and runs until ΩDDielectric . The output waveguide runs vertically and is centered at x = 300
nm. It starts at the top edge of Ω and runs until ΩDDielectric . The width of both waveguides is 150
nm. The power emitted by the dipole at λ2 into the input and output waveguides is computed
by evaluating Eq. (4) over Γin and Γout respectively. Γin is a vertical line through the input
waveguide at x = −300 nm. Γout is a horizontal line through the output waveguide at t = 300 nm.
B. 2. Numerical solution
For the numerical solution of all model problems the geometry is discretized using an unstructured
first order finite element mesh [40]. The design domain is discretized using triangular elements
with a uniform side length of h = 1 nm, dictating the resolution of the design. The remaining
model domain is discretized using a non-uniform mesh of triangular elements with a smallest side
lengths of h = 1 nm near the design domain and a maximum side length of h = 1/16 effective
wavelength (λ/n), to ensure the accuracy of the model.
The design problems considered in this work were solved on a laptop with 16 GB RAM and
an Intel(R) Core(TM) i5-7200 CPU @ 2.50 GHz processor. The approximate wall time spent per
design iteration for the design problem in Sec. 3.1 was 90 seconds, resulting in approximately 15
Research Article Vol. 28, No. 4 / 17 February 2020 / Optics Express 4459
hours spent executing the full design process. It is noted that these numbers are naturally strongly
dependent on the number of design iterations, number of elements in the mesh, and by extension
on the size of the model and design domain, and finally that computational speed was not the
focus of this work. ( )
For the model problems sketched in Fig. 2(a)–2(b) the incident-field problem Eq. (1) , is
solved using the scattered-field approach [40], where the background field is taken to be a perfect
planewave.
For the model problems sketched in Fig. 2(c) the incident-field problem is solved using
the total-field approach [40]. First the lowest-(order m)ode confined to the input waveguide iscomputed. This mode is then used to excite the system and a total-field problem is solved.
For all model problems the emitter problem Eq. (3) is solved for the total field.
B. 3. Physics related
The physics related parameters are chosen as follows: The wavelength of the incident field is
taken to be λ1 = 532 nm. The wavelength of the emitted field is taken to be λ2 = 532 nm. The
relative permittivity and relative permeability of air are taken to be εair = µair = 1.0. The relative
permittivity and relative permeability of water are taken to be εwater = 1.77, µwater = 1.0 at the
operating wavelengths. The relative permittivity and relative permeability of silver (Ag), gold
(Au) and copper (Cu) are taken from [62]. The relative permittivity and relative permeability of
platinum (Pt) are taken from [63]. The relative permittivity and relative permeability of Si3N4
are taken to be εSi3N4 = 2.05, µwater = 1.0 at the operating wavelengths. The speed of light is
taken to be c = 3 · 108 m/s.
B. 4. Design related
For the designs considering the model problem sketched in Fig. 2(a) mirror symmetry is imposed
on the designs normal to the horizontal axis due to the nature of the model problem, where the
planewave in Eq. (1) propagates through the domain along the horizontal axis.
For the designs considering the model problem sketched in Fig. 2(b) mirror symmetry is
imposed on the design normal to the vertical axis due to the nature of the model problem, where
the planewave in Eq. (1) propagates through the domain along the vertical axis.
B. 5. Optimization related
For the model problem sketched in Fig. 2(a) the following values are used: As an initial guess for
all optimizations a full metal disc is used, i.e. ξinitial(r) = 1 ∀ r ∈ ΩD. The filter radius in the
smoothing operation is, rf = 10 nm. (rf = 5 nm was used for the study detailed in Sec. 3.6.) The
thresholding level in the thresholding operation is, η = 0.5. The initial thresholding strength is,
βinitial = 8. The thresholding strength is increased gradually during the optimization through the
values, β = 8, 16, 32. A minimum length scale of all features in the designs is ensured using
geometric length-scale constraints with cLS = 625, ηe = 0.75, ηd = 0.25. The length-scale
constraints are only enforced for β = 32 to allow the design to form freely without limiting
topology changes in the earlier stages of the optimization.
For the model problem sketched in Fig. 2(b) the following values are used: As an initial
guess a full metal region is used, i.e. ξinitial(r) = 1 ∀ r ∈ ΩD. The filtering radius used in the
smoothing operation is, rf = 10 nm. The thresholding level used is in the thresholding operation
is, η = 0.5. The initial thresholding strength used in the thresholding operation is, βinitial = 8.
The thresholding strength is increased gradually during the optimization through the values,
β = 8, 16, 32. A minimum length scale of all features in the designs is ensured using geometric
length-scale constraints using cLS = 625, ηe = 0.75, ηd = 0.25. The length-scale constraints are
only enforced for β = 32 to allow the design to form freely without limiting topology changes in
the earlier stages of the optimization.
Research Article Vol. 28, No. 4 / 17 February 2020 / Optics Express 4460
For the model problem sketched in Fig. 2(c) the following values are used: As an initial guess
for ΩDMetal a full metal region is used, i.e. ξinitial(r) = 1 ∀ r ∈ ΩDMetal . As an initial guess for
ΩDDielectric a full dielectric region is used, i.e. ξinitial(r) = 1 ∀ r ∈ ΩDDielectric . The filtering radius in
the smoothing operation is: rf = 10 nm. The thresholding level in the thresholding operation is:
η = 0.5. The initial thresholding strength used is in the thresholding operation is: βinitial = 8.
The thresholding strength is increased gradually during the optimization through the values:
β = 8, 16, 32. A minimum length scale of all features in the designs is ensured using geometric
length-scale constraints using cLS = 625, ηe = 0.75, ηd = 0.25. The length-scale constraints is
only imposed for β = 32 to allow the design to form freely without limiting topology changes in
the earlier stages of the optimization.
Funding
Villum Fonden (8692); U.S. Army Research Office (W911NF-13-D-0001); National Science
Foundation (1453218, 1709212).
Acknowledgments
The authors thank Juejun Hu for insightful discussions.
Disclosures
The authors declare no conflicts of interest.
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