Can photonic heterostructures provably outperform single-material geometries?

Recent advances in photonic optimization have enabled calculation of performance bounds for a wide range of electromagnetic objectives, albeit restricted to single-material systems. Motivated by growing theoretical interest and fabrication advances, we present a framework to bound the performance of photonic heterostructures and apply it to investigate maximum absorption characteristics of multilayer films and compact, free-form multi-material scatterers. Limits predict trends seen in topology-optimized geometries -- often coming within factors of two of specific designs -- and may be exploited in conjunction with inverse designs to predict when heterostructures are expected to outperform their optimal single-material counterparts.

is the corresponding vacuum propagator acting on sources to yield their corresponding fields in vacuum-namely, via convolution of the vacuum Green's function 0 (r, r , ω k ) = δ(r − r ).I and P j represent spatial projections onto either the full or a subset V j ∈ V of the design region V , respectively.Lastly, f 0 is a quadratic function of the polarization currents |T k,m .
For the first constraint, we will take the real and imaginary parts and write the Lagrange multiplier corresponding to a given j, k, k as λ j,k,k R/I (symmetric and asymmetric constraints respectively).For the second, we will use Lagrange multipliers λ k,k ,m,m SO/AO (symmetric/asymmetric orthogonal constraint).Now we can write where L is the Lagrangian, Writing out the constraints we find ) where O lin is the linear part of the objective. with with present in every off-diagonal element.Furthermore, O quad is the quadratic part of the objective and all R, I, S, A are m × m block matrices of N × N matrices.We can compute the dual function G: We find the stationary point |T * of L by solving the relation which leads to the linear system In order for the dual to be finite, Z T T must be positive definite, so this linear system is invertible, leading to Finally, In the specific case of maximizing absorption, O lin = 0. Absorbed power is with Z the vacuum impedance, giving us O quad (being careful of the negative sign in Eq. ( S2)).Each term can be normalized as desired by the incident or total power.

SOLVING THE DUAL PROBLEM
In order to solve the convex dual problem using an interior point method, we must first find an initial feasible point.This requires choosing λ such that Z T T is positive definite, thereby ensuring that Z T T is invertible and therefore that the dual is well defined.This can be done reliably by leveraging the fact that the imaginary part of the Maxwell Green's function, Im G (k) 0 , is positive semidefinite [4].Take P j=0 = I to be the projector corresponding to global constraints, which is always enforced in our implementation.By setting all Lagrange multipliers λ = 0 except λ j=0,k,k I for all k, we set R j k,k = S j k,k = A j k,k = 0 for all k, k , j.This also sets I j k,k = 0 for all j, k = k and, although not necessary, for all j = 0, k = k .The quadratic part of L becomes with I 0 k,k taking the following block diagonal form: which is positive definite for Im χ k,m > 0. Therefore, as long as all materials have some loss (as they do in the main text), we can simply increase λ 0,k,k I for all k to make Z T T (λ) positive definite for any finite O quad .We initialize all bounds calculations in the main text (where n s = 1) by setting λ 0,k,k I = 1 ∀k.When Im χ k,m = 0, we may be able to generalize the technique utilized in Ref [1] (Supporting Information, Section 11).

DEPENDENCE ON NUMBER OF SUB-REGION CONSTRAINTS
In this section, we calculate the dependence of presented bounds on the number of sub-region constraints.For a fixed design size, we can enforce these constraints at finer regions in space through the use of additional projections P j .Any additional Lagrange multipliers can be set to 0, so enforcing additional constraints exclusively decreases the calculated bound.As a result, increasing the number of sub-region constraints tightens the bound on maximum photonic performance.
Results are shown in Fig. S1.We note that many sub-region constraints are not necessary to capture the majority of the physics when L/λ is small.Single material bounds that show imperfect absorbance exhibit a critical number of constraints around which the bound rapidly decreases and saturates.In the case of compact regions (bottom), single material bounds may not have converged for all L/λ, indicating that bounds may be possibly made tighter by increasing the number of sub-region constraints.Neither the multilayer film nor the compact region heterostructure bounds significantly change over the studied range of number of constraints.This may mean that these bounds cannot be tightened by enforcing local physics, or that the critical point of sub-region constraints has not been reached and bounds could be significantly improved by greater computational efficiency.This remains an open question and will be further studied.
We note an anomalous point: the compact structure dielectric bound (L/λ = 0.7) exhibits a greater bound with 100 than with 25 sub-region constraints.This is a numerical artifact due to the difficulty of optimizing the dual surface, even if convex, over so many parameters and again highlights the need for more efficient numerical methods.

INVERSE DESIGN DETAILS Multiple Material Topology Optimization
All calculations were run at increasing resolutions until converged.Multiple material topology optimization was done by writing = 2 + ( 1 + ( background − 1 )ρ 2 − 2 )ρ 1 for ρ 1 , ρ 2 ∈ [0, 1] and optimizing over the continuous variables ρ 1 , ρ 2 .The derivatives of the objective with respect to modifications in ρ 1 , ρ 2 were computed with Ceviche [2].The resulting optimization problem was solved with NLopt [3].Inverse designs are often binarized to reflect realistic devices.In the multi-material case, we define a binarized design as one where each pixel is exclusively one of 1 , 2 , or background .To better compare with bounds (which at high enough resolutions mimic the behavior of non-binarized devices), inverse designs were not deliberately binarized when calculating device performance.Some designs were binarized for presentation as described in the next section.

Inverse Designs in the Main Text
All topology optimized inverse designs compared to bounds are shown in Fig. S2 for multilayer films and Fig. S3 for compact structures.Multilayer films: Designs were binarized for the purposes of presentation if a binarized device could be found with performance within 1% (metals) and 2% (multi-material) of the non-binarized device.Devices solely composed of dielectric and vacuum were not deliberately binarized.Compact structures: Designs were not binarized.
The heterostructures showcase other examples where vacuum was not utilized, and therefore where these results could have been replicated in a single-material framework with a design region comprised by dielectric or metallic background and surrounded by vacuum.This phenomenon occurred when the single material metallic design vastly outperformed dielectric structures while still exhibiting non-perfect absorbance.These designs are simply metallic cavities filled by an absorbing dielectric.Interestingly, as described in the main text, the dielectric structure outperformed the metallic structure and the corresponding heterostructure utilized vacuum when L/λ = 1.
|T k ,m = 0 ∀j, k, k , T k,m | P j |T k ,m = 0 ∀j, k, k , m = m .(S1) This notation differs from the main text via ψ k,m → T k,m .As in the main text, |S k is a source k, the polarization current due to source k is |T k = m |T k,m with |T k,m the polarization current due to source k and material m and is defined in the design region V .χ k,m is the susceptibility of material m at ω k , and G (k) 0

FIG. S1 :
FIG.S1: Dependence of calculated absorbance bound on the number of sub-region constraints (i.e.number of projector Pj constraints enforced in Eq (S1)) for multilayer films (top) and compact structures (bottom).We note that multi-material multilayer films exhibit a perfect absorbance bound for L/λ ≥ 0.2.