1. INTRODUCTION

Electromagnetic scattering at a single frequency is constrained by two loss mechanisms: material dissipation (absorption) and radiative coupling (scattering). There has been substantial research probing the limits of light–matter interactions subject to constraint of either mechanism [1–21], yet no general theory simultaneously accounting for both. In this paper, we develop a framework for upper bounds to electromagnetic response, across near- and far-field regimes, for any materials, naturally incorporating the tandem effects of material- and radiation-induced losses. Our framework relies on a power-conservation law for the polarization currents induced in any medium via a volume-integral version of the optical theorem [22–25]. An illustrative example is that of plane-wave scattering, where our bounds unify two previously separate approaches: radiative-coupling constraints leading to maximum cross-sections proportional to the square wavelength [1–6], $\max \sigma \sim {\lambda ^2}$, and material-dissipation constraints leading to cross-section bounds inversely proportional to material loss [7–9], $\max \sigma \sim |\chi {|^2}/{\rm Im}\chi$. Our framework contains more than a dozen previous results [1–5,7–9,11,12,14–17] as asymptotic limits, it regularizes unphysical divergences in these results, and it reveals new insight for optimal nanophotonic design, with applications including far-field scattering, near-field local-density-of-states (LDOS) engineering, and the design of perfect absorbers. The ramifications of our bounds for perfect absorbers are striking: we prove that independent of the geometric patterning, the minimum thickness of perfect or near-perfect absorbers comprising conventional materials is typically on the order of 50–100 nm at visible wavelengths, and closer to 1 µm at infrared wavelengths where polar-dielectric materials are resonant. These values are larger than the material skin depths, and roughly $100 \times$ larger than those suggested by previous material-loss bounds [7]. We use inverse design to discover ultrathin absorber designs closely approaching the bounds. We show that these bounds can further be utilized for the “reverse” problem of identifying optimal illumination fields, a critical element of the burgeoning field of wavefront shaping [26–29]. The framework developed here has immediate applicability to any linear or quadratic response function in electromagnetic scattering problems, including those that arise in near-field radiative heat transfer (NFRHT) [30–32], optical force/torque [14,15,33–35], high-NA metalenses [36–38], and more general nanophotonic mode coupling [39].

For many years, there was a single “channel bound” approach underlying the understanding of bounds to single-frequency electromagnetic response [1–6,10–12,14,15,20]. The approach identifies “channels” (typically infinite in number) that carry power towards and away from the scattering body [40–43], use intuition or asymptotic arguments to restrict the scattering process to a finite number of channels, and then apply energy conservation within those channels to arrive at maximal power-exchange quantities. The canonical example is in bounds for scattering cross-sections, i.e., the total scattered power divided by the intensity of an incoming plane wave. It has long been known that the maximal cross-section of a subwavelength electric-dipole antenna [44], or even a single two-level atomic transition [45], is proportional to the square wavelength; for scattering cross-sections, the bound is ${\sigma _{{\rm scat}}} \le 3{\lambda ^2}/2\pi$. These bounds are consequences of properties of the incident waves (not the scatterers): though plane waves carry infinite total power, they carry a finite amount of power in each vector-spherical-wave (VSW) basis function, and $3{\lambda ^2}/2\pi$ scattering corresponds simply to scattering all of the power in the electric-dipole channel. Related arguments can be used to bound NFRHT rates, which are constrained by restricting near-field coupling to only finite-wavenumber evanescent waves [10], absorption rates in ultrathin films, which are constrained by symmetry to have nonzero coupling to up/down plane-wave channels [11], and maximal antenna directivity [6]. All such channel bounds are consequences of radiative-coupling constraints, with optimal power-flow dynamics corresponding to ideal coupling to every channel that interacts with the scattering system. The drawbacks of channel bounds are two-fold: (1) they do not account for absorptive losses in the scatterers, and (2) except in the simplest (e.g., dipolar) systems, it is typically impossible to predict a priori how many channels may actually contribute in optimal scattering processes. Without any such restrictions, the bounds diverge.

In recent years, an alternative approach has been developed: material-absorption bounds [7–9,13,15–21] that rectify the two drawbacks of the channel approaches. These bounds identify upper limits to responses, including cross-sections [7], LDOS [19], NFRHT [13], and 2D-material response [8], that are determined by the lossiness of the material comprising the scattering body. The independence from channels provides generality and convenience, but with the key drawback that they do not account for necessary radiative damping. Very recently, for the special case of incoherent thermal or zero-point-field excitations, radiative and absorptive losses are separately identified using the ${\mathbb T}$ operator, yielding upper bounds for incoherent response functions [46–48].

In this work, we identify a single constraint that incorporates the cooperative effects of absorptive and radiative losses at any level of coherence. The constraint is the volume-integral formulation of the optical theorem (Section 2), which is an energy-conservation constraint that imposes the condition that absorption plus scattered power equals extinction, for any incident field. Channel bounds distill in essence to loosening this constraint to an inequality that scattered power is bounded above by extinction. Material-absorption bounds distill to loosening the optical-theorem constraint to an inequality that absorbed power is bounded above by extinction. Our key innovation is the recognition that one can retain the entire constraint, and enforce the requirement that the sum of absorption and scattered power equals extinction. We describe the use of Lagrangian duality to solve the resulting optimization problems, ultimately yielding very general bounds to arbitrary response functions. For the important case of plane-wave scattering (Section 3), we derive explicit bound expressions and also identify an important application: perfect absorbers. We show that our framework enables predictions of the minimal scatterer thicknesses at which perfect or near-perfect absorption may be possible, thicknesses much larger than any previous framework predicted. Our bounds explicitly account for the precise form of incident waves; for a given material and designable region, then, we can treat the illumination-field degrees of freedom as the variables and identify the optimal incoming-wave excitation (Section 4). As one example, we show that in certain parameter regimes, the extinction of an unpatterned sphere under the optimal illumination field exceeds the upper bound under plane-wave excitation, which means that as long as the incident field is a plane wave, there is no patterning of any kind that can reach the same power-response level of the optimal illumination. In the final section (Section 5), we discuss the simplicity with which our framework can be applied to numerous other scenarios, and discuss remaining open problems.

Given the variety of bounds in Refs. [1–21], as well as those contained here, a natural question is whether the bounds we present here are the “best possible” bounds, or whether they will be “superseded” later. We argue that ultimately there will be no “best” single bound, but rather a general theory comprising different bounds at different levels of a priori information that is known about a given problem. Useful analogies can be made to information theory, where Shannon’s bounds [49,50] were not a final conclusion but instead initiated an entire field of inquiry [51], as well as the theory of composite materials, where early studies into properties of simple isotropic composites [52] blossomed into a broad theoretical framework with bounds that vary with the amount of information known about the problem of interest [53–58]. In electromagnetism and optics, previous bounds [1–21] utilized information about either the number of available scattering channels or the material loss rate; in this work, we present the first bounds that combine the two, unifying the previous disconnected threads. A useful indicator of whether future bounds, with possibly more known information, will significantly alter these results is to test whether physical designs can approach these bounds, as it can almost never be guaranteed (in any field) whether given bounds are precisely achievable by real physical implementations. As we show in Section 3, in the quest for ultrathin perfect absorbers, physical designs can approach the new bounds within a factor of two, suggesting minimal opportunity for later revision.

2. GENERAL FORMALISM

Our central finding is a set of upper bounds to maximal single-frequency response. The problem of interest is to optimize any electromagnetic response function $f$ subject only to Maxwell’s equations, while allowing for arbitrary patterning within a prescribed region of space. However, Maxwell’s equations represent a nonconvex and highly complex constraint for which global bounds are not known. Instead, we use the optical theorem, and in particular a volume-integral formulation of the optical theorem, as a simple quadratic constraint for which global bounds can be derived. We start with the volume-integral version of Maxwell’s equations, which provide a simple and direct starting point to derive the optical theorem (Section 2.A). The optical-theorem constraint is quadratic, and we discuss how many previous results can be derived from weaker forms of the constraint. Then in Section 2.B, we use the formalism of Lagrangian duality to derive a single general bound expression, Eq. (6), from which many specialized results follow. In Section 2.C we consider canonical electromagnetic response functions: absorption, scattering, extinction, and LDOS. Throughout, for compact general expressions, we use six-vector notation with Greek letters denoting vectors and tensors: $\psi$ for fields, $\phi$ for polarization currents, and $\chi$ for the susceptibility tensor (which in its most general form can be a nonlocal, inhomogeneous, bianisotropic, $6 \times 6$ tensor operator [59]), and we use dimensionless units for which the vacuum permittivity and permeability equal one, ${\varepsilon _0} = {\mu _0} = 1$. The six-vector fields and polarization currents are given by

A. Optical Theorem Constraint

The optical theorem manifests energy conservation: the total power taken from an incident field must equal the sum of the powers absorbed and scattered. As discussed below, the key version of the optical theorem that enables a meaningful constraint is the version that arises from the volume equivalence principle. This principle enables the transformation of the differential Maxwell equations to a volume-integral form. It states that any scattering problem can be separated into a background material distribution (not necessarily homogeneous), and an additional distributed “scatterer” susceptibility. The total fields $\psi$ are given by the fields incident within the background, ${\psi _{{\rm inc}}}$, plus scattered fields ${\Gamma _0}\phi$ that arise from polarization currents $\phi$ induced in the volume of the scatterer, where ${\Gamma _0}$ is the background-Green’s-function convolution operator. For simplicity in the optical theorem below, we define a variable $\xi$ that is the negative inverse of the susceptibility operator, $\xi = - {\chi ^{- 1}}$. With this notation, the statement that the total field equals the sum of the incident and scattered fields can be written: ${-}\xi \phi = {\psi _{{\rm inc}}} + {\Gamma _0}\phi$. Rearranging to have the unknown variables on the left-hand side and the known variables on the right-hand side yields the volume-integral equation (VIE)

The VIE optical theorem can be derived from Eq. (2) by taking the inner product of Eq. (2) with $\phi$ (denoted ${\phi ^\dagger}$), multiplying by $\omega /2$, and taking the imaginary part of both sides of the equation, yielding

 

Fig. 1. Illustration of the two loss mechanisms in electromagnetic scattering. An incident field ${\psi _{{\rm inc}}}$ induces polarization currents $\phi$ in the scatterer. Energy dissipated inside the material corresponds to material loss, determined by the operator ${\rm Im}\xi$, which equals ${\rm Im}\chi /|\chi {|^2}$ for a linear isotropic susceptibility $\chi$. Energy coupled to the background, into far-field or near-field power exchange, corresponds to radiative loss, determined by the operator ${\rm Im}{\Gamma _0}$, where ${\Gamma _0}$ represents the background (e.g., free-space) Green’s function. Total extinction is the sum of the two and is linear in $\phi$, as dictated by the optical theorem.

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While no simplification of Maxwell’s equations will contain every possible constraint, the optical theorem of Eq. (3) has four key features: (1) it contains both the powers radiated (${P_{{\rm scat}}}$) and absorbed (${P_{{\rm abs}}}$) by the polarization currents in a single expression, (2) it is a quadratic constraint that is known to have “hidden” convexity for any quadratic objective function [67], (3) it enforces power conservation in the scattering body, and (4) it incorporates information about the material composition of the scatterer, and possibly a bounding volume containing it, while being independent of any other patterning details.

The optical-theorem constraint of Eq. (3) constrains the polarization-current vector $\phi$ to lie on the surface of a high-dimensional ellipsoid whose principal axes are the eigenvectors of ${\rm Im}{\Gamma _0} + {\rm Im}\xi$ and whose radii are constrained by the norm of ${\psi _{{\rm inc}}}$. In Supplement 1, we show that all previous channel or material-loss bounds discussed in the Introduction can be derived by applying weaker versions of Eq. (3). Channel bounds can be derived by loosening Eq. (3) to the inequality ${P_{{\rm scat}}} \le {P_{{\rm ext}}}$, without the absorption term (but implicitly using the fact that absorbed power is nonnegative). Material-loss bounds can be derived by loosening Eq. (3) to the inequality ${P_{{\rm abs}}} \le {P_{{\rm ext}}}$, without the scattered-power term (but using the fact that scattered power is nonnegative). Of course, including both constraints simultaneously can only result in equal or tighter bounds.

B. Optimization Formalism

Any electromagnetic power-flow objective function $f$ is either linear or quadratic in the polarization currents $\phi$. Under a given basis, it can be generically written as $f(\phi) = {\phi ^\dagger}{\mathbb A}\phi + {\rm Im}({{\beta ^\dagger}\phi})$, where ${\mathbb A}$ is a Hermitian matrix, and $\beta$ is any six-vector field on the scatterer domain. The same basis is used to discretize ${\psi _{{\rm inc}}}$, ${\rm Im}\xi$, and ${\rm Im}{\Gamma _0}$, where the last two are now positive semi-definite matrices. Then the maximal $f$ that is possible for any scatterer is given by the optimization problem

C. Power Quantities and LDOS

If one wants to maximize one of the terms already present in the constraint, i.e., absorption, scattered power, or extinction, then the ${\mathbb A}$ and $\beta$ terms take particularly simple forms (cf. Supplement 1), leading to the bounds

Bounds on LDOS represent maximal spontaneous-emission enhancements [69–73]. Total (electric) LDOS, ${\rho _{{\rm tot}}}$, is proportional to the averaged power emitted by three orthogonally polarized and uncorrelated unit electric dipoles [74–77]. It can be separated into a radiative part, ${\rho _{{\rm rad}}}$, for far-field radiation, and a non-radiative part, ${\rho _{{\rm nr}}}$, that is absorbed by the scatterer [22]. Exact but somewhat cumbersome LDOS bounds for arbitrary materials are derived from Eq. (6) in Supplement 1; for nonmagnetic materials, the bounds simplify to expressions related to the maximum power quantities given in Eqs. (7)–(9):

The bounds of Eqs. (6)–(12) are sufficiently general to allow for arbitrary material composition (inhomogeneous, nonlocal, etc.), in which case the bounds require computations involving the ${\rm Im}{\Gamma _0}$ and ${\rm Im}\xi$ matrices. In Supplement 1, we provide a sequence of simplifications, showing step by step the increasingly simplified bounds that arise under restrictions of the incident field, material, or bounding volumes involved. In the next section, we consider the important case in which a plane wave is incident upon an isotropic nonmagnetic medium.

3. PLANE-WAVE SCATTERING

A prototypical scattering problem is that of a plane wave in free space incident upon an isotropic (scalar susceptibility), nonmagnetic scatterer. The assumption of a scalar susceptibility introduces important simplifications into the bounds. The matrix ${\rm Im}\xi$ is then a scalar multiple of the identity matrix ${\mathbb I}$,

 

Fig. 2. Plane wave of wavelength $\lambda = 360$ nm scattering from a finite Ag [79] scatterer, enclosed by a spherical bounding volume with radius $R$. The channel bound is heuristically regularized by ignoring small-scattering high-order channels. All cross-sections are normalized by geometric cross-section $A$. (a). Bound of extinction cross-section for different $R$. The general bound regularizes divergence in previous bounds and are tighter for wavelength-scale sizes. (b) Similar behavior is observed in the bounds for scattering and absorption cross-sections. (c) Per-channel extinction cross-section ${\sigma _{{\rm ext},n}}$ (defined in Supplement 1) for $R = \lambda /2$. Low-order scattering channels are dominated by radiative loss, while high-order scattering channels are dominated by material loss.

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An incident propagating plane wave (or any wave incident from the far field, cf. Supplement 1) can be decomposed in the basis ${\mathbb V}$. We write the expansion as ${\textbf{E}_{{\rm inc}}} = \frac{1}{{{k^{3/2}}}}\sum\nolimits_i {e_i}{\textbf{v}_i}$, where the ${e_i}$ are the expansion coefficients, and we factor out the free-space wavenumber $k$ to simplify the expressions below. Inserting the eigendecomposition of ${\rm Im}{\mathbb G}_0^{{\rm EE}}$ and the plane-wave expansion in this basis into Eqs. (7)–(9) gives general power bounds for plane-wave scattering:

The bounds of Eqs. (14)–(16) require knowledge of the eigenvalues of ${\rm Im}{\mathbb G}_0^{{\rm EE}}$, and thus the exact shape of the scattering body, to compute the values of ${\rho _i}$. However, analytical expressions for ${\rho _i}$ are known for high-symmetry geometries, and a useful property of the optimization problem of Eq. (4) is that its value is bounded above by the same problem embedded in a larger bounding domain. (It is always possible for the currents in the “excess” region to be zero.) In the following two sub-sections we consider the two possible scenarios one can encounter: (a) scattering by finite-sized objects, which can be enclosed in spherical bounding surfaces, and (b) scattering by extended (e.g., periodic) objects, which can be enclosed in planar bounding surfaces.

A. Finite-Sized Scatterers

Finite-sized scatterers can be enclosed by a minimal bounding sphere with radius $R$, as in the inset of Fig. 2(a). The basis functions ${\textbf{v}_i}$ are VSWs, representing orthogonal scattering channels, with exact expressions given in Supplement 1. The state labels $i$ can be indexed by the triplet $i = \{n,m,j\}$, where $n = 1,2,\ldots$ is the total angular momentum, $m = - n,\ldots,n$ is the $z$-directed angular momentum, and $j = 1,2$ labels two polarizations. In this basis, the expansion coefficients of a plane wave are given by $|{e_i}{|^2} = \pi (2n + 1){\delta _{m, \pm 1}}|{E_0}{|^2}$, where ${E_0}$ is the plane-wave amplitude. We show in Supplement 1 that values ${\rho _i}$ are given by integrals of spherical Bessel functions. With these expressions, bounds for extinction, scattering, and absorption cross-sections are easily determined from Eqs. (14)–(16) after normalization by plane-wave intensity $|{E_0}{|^2}/2$.

In Fig. 2, we compare cross-section bounds derived from Eqs. (14)–(16) to the actual scattering properties of a silver sphere (permittivity data from Ref. [79]) at wavelength $\lambda = 360\;{\rm nm}$. We choose 360 nm wavelength because it is close to the surface-plasmon resonance of a silver sphere, simplifying comparisons (instead of requiring inverse design for every data point). We also include the previously derived channel [4] and material-absorption [7] bounds for comparison, and in each case, one can see that our general bounds are significantly “tighter” (smaller) than the previous bounds, except in the expected small- and large-sized asymptotic limits. At a particular radius, the scattering response even reaches the general bound. In Fig. 2(c), we fix the radius at a half-wavelength and depict the per-channel contributions to the extinction bounds in the radiation-loss-only, material-loss-only, and tandem-loss constraint cases. Higher-order channels have increasingly smaller radiative losses (causing unphysical divergences, discussed below), such that material loss is the dominant dissipation channel. Conversely, material-loss-only constraints are inefficient for lower-order channels where radiative losses dominate. Incorporating both loss mechanisms removes the unphysical divergence, accounts for radiative losses, and sets the tightest bound among the three across all channels.

 

Fig. 3. Arbitrarily patterned SiC scatterer with maximum thickness $h$ excited by a plane wave at normal incidence and $\lambda = 11\;{\unicode{x00B5}{\rm m}}$ wavelength, where SiC is polaritonic. (a) Bounds for extinction, scattering, and absorption, compared to their values for a planar SiC [97] film. (b) Inverse-designed SiC metasurfaces (blue markers), at varying thicknesses, achieve absorption levels at 64%–95% of the global bounds (red), suggesting the bounds are “tight” or nearly so. (c) Absorption spectrum of ultrathin absorber from (b) with thickness $h = 0.4\;{\unicode{x00B5}{\rm m}}$. (Inset: inverse-design structure; blue represents SiC, white represents air.) At the target wavelength, the absorption of the inverse-designed structure is more than 10 times that of the thin film, and reaches 72% of the bound.

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For structures smaller than roughly 10 nm, instead of bulk permittivity data, one must employ a nonlocal model of the permittivity [60], which can still be subjected to bounds but requires modified techniques for modeling the polarization currents [8]. We retain small ratios of size to wavelength throughout the paper, such as in Fig. 2, to observe the relevant scalings of the classical model, and because for mid-infrared plasmonic materials, the lineshapes are quite similar while all sizes are scaled beyond 10 nm.

Technically, the channel bound diverges for any finite-sized scatterer, and the blue solid line in Fig. 2(a) should be infinitely high. To obtain a reasonable finite value, we incorporate only channels for which the sphere scattering contributions are greater than 1% of the maximal response. Yet requiring knowledge of the specific scattering structure to compute the upper limit highlights a key drawback of the channel bounds. This empirical threshold is responsible for two artifacts in the presented channel bounds. First, it results in a step-like behavior that is most prominent at small radii, where only a handful channels contribute. At each radius where a new channel is introduced for consideration (based on this threshold), there is an unphysical increase in the bound due to the larger power available for scattering, absorption, etc. Such behavior is somewhat smoothed at large radii, where the contribution from each new channel is subsumed by the large number of existing channels. Second, as we show in Supplement 1, there can potentially be large contributions from channels beyond this threshold. The arbitrary cutoff results in inaccurate and unphysical underestimates of the cross-sections, which is noticeable mostly in the large size limit in Figs. 2(a) and 2(b), where the channel bound appears to be slightly smaller than the general bound. The only way to avoid such artifacts would be to include all channels, in which case the channel bounds trivialize to infinite value for any radius.

B. Extended Scatterers

The second possible scenario is scattering from an infinitely extended (e.g.,  periodic) scatterer. Such scatterers can always be enclosed by a minimal planar “film” bounding volume with thickness $h$, as in the inset of Fig. 3(a). Then the basis functions ${\textbf{v}_i}$ of ${\rm Im}{\mathbb G}_0^{{\rm EE}}$ are known to be propagating plane waves with wave vector ${\boldsymbol k} = {k_x}{\hat {\boldsymbol x}} + {k_y}{\hat{\boldsymbol y}} + {k_z}{\hat {\boldsymbol z}}$. Now the index $i$ maps to the triplet $i = \{s,p,{{\boldsymbol k}_\Vert}\}$, where $s = \pm$ denotes even and odd modes, $p = M,N$ denotes TE and TM polarizations, and ${{\boldsymbol k}_\Vert} = {k_x}{\hat {\boldsymbol x}} + {k_y}{\hat {\boldsymbol y}}$ denotes the surface-parallel wave vector. In Supplement 1, we provide the expressions for ${\textbf{v}_i}$, and show that the eigenvalues ${\rho _i}$ are given by

Figure 3(a) compares the upper bounds for the normalized cross-sections with the cross-sections of SiC thin films at normal incidence and wavelength $\lambda = 11\;{\unicode{x00B5}{\rm m}}$, where SiC supports phonon–polariton modes. One can see that the bounds indicate that scattering, absorption, and extinction must all be small at sufficiently small thicknesses, and crossover to near-maximal possible values at roughly one-tenth of the wavelength.

 

Fig. 4. Minimum thickness required for a perfect absorber to reach 70% and 100% absorption rate under normal incidence for typical materials that are polaritonic at (a) visible [96] and (b) infrared wavelengths [97–99]. (c) Universal curve showing minimum possible thicknesses for 100% absorption as a function of perfect-absorber material figures of merit (FOM), given by $1/{\rm Im}\xi = |\chi {|^2}/{\rm Im}\chi$. The same curve is shown in the inset for 70% absorption where inverse-designed structures (triangular markers) demonstrate thicknesses within ${1.5 {-} 2.7X}$ of the bound.

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A key question for any bound is whether it is achievable with physical design. To test the feasibility of our bounds, we utilize inverse design [80–87], a large-scale computational optimization technique for discovering optimal configurations of many design parameters, to design patterned SiC films that approach their bounds. We use a standard “topology-optimization” approach [81,84] in which the material is represented by a grayscale density function ranging from zero (air) to one (SiC) at every point, and derivatives of the objective function (absorption, in this case) are computed using adjoint sensitivities. We prioritize feasibility tests (are the bounds achievable, in theory?) over the design of easy-to-fabricate structures. To this end, we utilize grayscale permittivity distributions, which in theory can be mimicked by highly subwavelength patterns of holes, but in practice would be difficult to fabricate. Recently developed techniques [88] are able to identify binary polaritonic structures that come quite close to their grayscale counterparts for many applications, and give confidence that binary structures with performance levels similar to those presented here can be discovered. We give algorithmic details for our inverse-design procedure in Supplement 1.

Figure 3(b) depicts the bounds (red solid line) and the performance of thin films (black solid line) as a function of thickness, as well as six different inverse-design structures that bridge most of the gap from the thin films to the bounds. The incident wavelength is $11\;{\unicode{x00B5}{\rm m}}$, and the period is $1.1\;{\unicode{x00B5}{\rm m}}$, with minimum feature size $0.1\;{\unicode{x00B5}{\rm m}}$. For an ultrathin absorber with thickness $0.4\;{\unicode{x00B5}{\rm m}}$, the inverse-designed metasurface can reach 72% of the global bound. In Fig. 3(c), we isolate the design at this smallest thickness and show its spectral absorption percentage, as well as its geometrical design (inset). Details of the inverse design are given in Supplement 1. Since the objective is to compare against the global, we do not impose binarization, lithography, or other fabrication constraints. It is apparent that inverse design can come rather close to the bounds, suggesting they may be “tight” or nearly so.

An important ramification of the bounds of Eqs. (18)–(20) is that they can be used to find the minimum thickness of any patterned “perfect absorber” [89–91], achieving 100% absorption or close to it. Such absorbers are particularly useful for sensing applications [90,92] and the design of ultrathin solar cells [93–95]. Absorption cross-section per area, ${\sigma _{{\rm abs}}}/A$, is the percentage absorption, while the bound on the right-hand side of Eq. (19) is a function only of the incident angle, the absorber thickness (defined as the thickness of its minimum bounding film), and its material susceptibility $\chi\! (\omega)$. For normally incident waves, we show in Supplement 1 that the minimum thickness ${h_{{\rm min}}}$ to achieve 100% absorption is given by the self-consistent equation

4. OPTIMAL ILLUMINATION FIELDS

In this section, we identify the incident waves that maximize the response bounds of Eqs. (7)–(12). There is significant interest in such wavefront shaping [26–29], in particular, for the question of identifying optimal illumination fields [15,33,34,100–103], and yet every current approach identifies optimal fields for a given scatterer. Using the framework developed above, we can instead specify only a designable region, and identify the optimal illumination field that maximizes the bound over all possible scatterers.

 

Fig. 5. Maximum extinction ${P_{{\rm ext}}}$ for arbitrary patterning and illumination, normalized by average field intensity ${I_{{\rm avg}}}$ and geometric cross-section $A$ of the bounding sphere of radius $R$. The solid red line in (a) shows the maximal extinction that can possibly be obtained by the optimal incident field, as compared to the simple plane-wave incidence shown by the solid blue line. The triangular markers give the attained extinction from an unpatterned silver sphere of radius $R$ under either optimal incidence (red triangles) or plane-wave illumination (blue triangles). (b) Three possible design regions (sphere, cube, and pyramid) and the corresponding optimal illumination fields (${\rm Im}{E_x}$) in the $x {-} z$ plane and $x {-} y$ plane (inset).

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To start, we assume that there is a basis $\Phi$ comprising accessible far-field illumination channels, such as plane waves, VSWs, Bessel beams, excitations from a spatial light modulator, or any other basis [104]. Then the incident field can be written as

Figure 5(a) demonstrates the utility of optimizing over incident fields. We consider incident fields impinging upon a finite silver scatterer within a bounding sphere of radius ${R}$ at wavelength $\lambda = 360\;{\rm nm}$ (as in Fig. 2, near the surface-plasmon resonance). We consider incident fields originating from one half-space, as might be typical in an experimental setup, and use as our basis 441 plane waves with wave vectors ${\boldsymbol k}$ whose evenly spaced transverse components range from ${-}0.8k$ to $0.8k$, where $k = 2\pi /\lambda$ is the total wave number. The 0.8 wave-vector cutoff corresponds to incident-field control over a solid angle of approximately 2.5 sr, and can be matched to the specifics of any experimental setup. We impose the constraint that the average intensity over a region that has twice the radius of the sphere must be equal to that of a unit-amplitude plane wave. Figure 5(a) shows the extinction bound evaluated for a plane wave (blue solid), as well as that for the optimal incident field (red solid). As the radius increases, incident-field shaping can have a substantial effect and yield bounds that are almost twice as large as those for plane waves ($1.94 \times$ exactly). (Each quantity is normalized by average field intensity ${I_{{\rm avg}}}$ and the geometric cross-section $A = \pi\! {R^2}$, which is why the extinction bounds may decrease with increasing radius.) Intriguingly, we show that even an unpatterned sphere (red triangles) shows performance trending with that of the bound, and for the larger radii, the unpatterned sphere under the optimal illumination field exhibits extinction values larger than the plane-wave bounds. This illustrates a key benefit of bounds: one can now conclude that an unpatterned sphere with optimal illumination fields can achieve extinction values that cannot possibly be achieved by any structure under plane-wave illumination.

Figure 5(b) further extends the optimal-illumination results, considering three designable regions: a sphere, a cube, and a pyramid. The optimal illumination patterns are shown in 2D cross-sections outside and within the designable regions. The sphere has a radius of one free-space wavelength, while the cube and pyramid have side lengths equal to twice the free-space wavelength. Within each domain, the optimal illumination fields exhibit interesting patterns that seem to put field nodes (zeros) in the interior, with the largest field amplitudes around the walls of the domains. This can be explained physically: the optimal incident fields will be those that couple most strongly to the polarization currents that exhibit the smallest radiative losses. The polarization currents that have the smallest radiative losses will tend to have oscillations with far-field radiation patterns that cancel each other, as occurs for oscillating currents along structural boundaries, such as whispering-gallery modes [107,108]. This procedure can be implemented for a beam generated by almost any means, e.g., and incident wave passing through a scatterer with a complex structural profile [109–111], precisely controlled spatial light modulators [112–115], or a light source with a complex spatial emission profile [116–118].

5. DISCUSSION AND EXTENSIONS

In this paper, we have shown that an energy-conservation law, arising as a generalized optical theorem, enables identification of maximal electromagnetic response at a single frequency. We considered: arbitrary linear and quadratic response functions, Eq. (6), power-flow quantities such as absorption and scattering, Eqs. (8) and (9), and LDOS, Eqs. (10)–(12), more specific scenarios such as plane-wave scattering and perfect absorbers, Eqs. (14)–(21), and optimal illumination fields, Eq. (24). In this section, we briefly touch on numerous other applications where this formalism can be seamlessly applied.

One important application is to understand the largest thermal absorption and emission of structured material. A direct consequence of the incoherent nature of the thermal source is that an upper bound to the average absorptivity/emissivity is given by the average of the bounds for each independent incident field in an orthogonal basis, such as VSWs for a finite scatterer. As detailed in Supplement 1, a straightforward implementation of our formalism leads to an even tighter bound than the recently published ${\mathbb T}$-operator bound of Ref. [47].

A natural extension of this work is to the emergent field of 2D materials [119–123]. From a theoretical perspective, the only difference with a 2D material is that the induced polarization currents exist on a 2D surface instead of within a 3D volume, which would change the interpretation of $\phi$ in Eq. (4), and would change the domain of the Green’s function ${\Gamma _0}$, but otherwise, the remainder of the derivation is identical. Instead of rederiving the bounds in a 2D domain, however, a simpler approach is to substitute the bulk susceptibility $\chi$ by the expression $\chi \to i{\sigma _{{2D}}}/\omega h$, where ${\sigma _{{2D}}}$ is the 2D-material conductivity, and $h$ is an infinitesimal thickness going to zero. (The bounds do not diverge because the geometric or bounding volume is also proportional to $h$, canceling the $1/h$ divergence in the material parameter.) Then, all of the bounds derived herein apply to 2D materials as well.

Another important extension is to problems of field concentration away from the scatterer itself. In surface-enhanced Raman scattering [124–126], for example, where recently material-loss bounds have been derived [17], it is important to maximize average field enhancement over a plane close to but not overlapping the scatterer itself. In this case, the objective might be the integral of the scattered-field intensity over a plane $P$, i.e., $\int_P \psi _{{\rm scat}}^\dagger {\psi _{{\rm scat}}}$. The scattering field is the convolution of the background Green’s function with the polarization fields $\phi$, such that this objective is a quadratic function of the polarization fields: ${\phi ^\dagger}[{\int_P \Gamma _0^\dagger {\Gamma _0}}]\phi$, which is exactly of the form required by Eq. (4) and thus is bounded above by Eq. (6).

Similarly, cross-density of states [127] measures the coupling strength between dipoles at two spatial locations, typically coupled via near-field interactions, for applications including Förster energy transfer [128] and quantum entanglement [129,130]. Such coupling effectively reduces to optimizing the field strength at one location from a point source at another location, mapping identically to the field concentration problem.

Maximizing optical forces and torques has been a topic of substantial interest [14,15,33–35], and is one that our framework applies to very naturally. One can compute force and torque via surface integrals of quantities related to the Maxwell stress tensor, which is a quadratic function of the electric and magnetic fields. By the same connection of the scattered fields to the induced polarization fields, it is possible to write any force/torque optimization function as a sum of quadratic- and linear-in-polarization terms, thereby equivalent to Eq. (4) and subject to the bounds of Eq. (6).

During the preparation of this paper, two preprints appeared [131,132] that contain ideas similar to those here. It is recognized in Refs. [131,132] that one can utilize the equality of absorption plus scattering and extinction, i.e., Eq. (3), as a quadratic electromagnetic constraint. They further show that an additional constraint can be identified—essentially, the real-part analog of Eq. (3). In this context, they provide bounds very similar to ours for power-flow quantities and LDOS, Ref. [131] considers the problem of directional scattering, and they both show a two-parameter dual formulation for incorporating the second constraint. Conversely, they do not have bounds for arbitrary linear and quadratic functions, i.e., our Eq. (6), or for non-scalar or nonlocal susceptibility operators, nor do they consider the possibility of bounds over all incoming wavefronts. And they do not identify the optimal value of the dual variable ${\nu ^*}$, which is important, for example, in determining the analytical bound of Eq. (21). Without an analytical value for ${\nu ^*}$, it is not possible to identify the minimum thickness of a perfect absorber.

More recent preprints have shown that one can generate an infinite set of (mostly nonconvex) constraints from spatially localized versions of the optical theorem [133,134]. There are advantages and drawbacks to such an approach relative to the one we presented here. With more constraints, one can potentially identify tighter bounds. But since most of the constraints are nonconvex, global optima are identifiable only through convex relaxations [135], which introduce two disadvantages to the computational approach. First, the bounds are numerical in nature and do not offer the intuition of semi-analytical bounds (as presented here). Second, they are computationally expensive and thus currently limited to wavelength-scale device sizes. Moreover, the non-analytical nature of the bounds precludes explicit identification of the dependence of the bounds on the incident fields, which enabled the wavefront-shaping results in Section 4, and which appears to not be possible in the approaches of Refs. [133,134]. Thus, the framework in this paper is complementary to that of Refs. [133,134], with each offering unique comparative advantages.

Looking forward, the energy-conservation approach developed here provides a framework for further generalizations and unifications. The incorporation of multiple constraints naturally leads to connections to the optimization field of semidefinite programming [135], as utilized in Ref. [136], where rapid global-optimization computational techniques are well established [105]. Away from single-frequency problems, the question of how to incorporate nonzero bandwidth in a bound framework would have important ramifications. As shown in Ref. [19], it may be possible to do so through generalized quadratic constraints based on causality. Finally, a key variable missing from semi-analytical, conservation-law-based bounds is the refractive index of a transparent medium, which does appear in bounds pertaining to the broadband absorption of sunlight [94,137–139]. Accounting for refractive index may require a unification of conservation-law approaches with, perhaps, those based on Lagrangian duality [140], or on sophisticated approaches developed in the theory of composite materials [53,58,141,142]. With such generalizations and unifications, it may be possible to understand the extreme limits of electromagnetic response in any scenario.