# Physical limits on electromagnetic response

Pengning Chao  Benjamin Strekha  Rodrick Kuate Defo  Department of Electrical
and Computer Engineering, Princeton University, Princeton, New Jersey 08544,
USA  Sean Molesky  Department of Engineering Physics, Polytechnique Montréal,
Montréal, Québec H3T 1J4, CAN  Alejandro W. Rodriguez1<EMAIL_ADDRESS>
###### Abstract

Photonic devices play an increasingly important role in advancing physics and
engineering, and while improvements in nanofabrication and computational
methods have driven dramatic progress in expanding the range of achievable
optical characteristics, they have also greatly increased design complexity.
These developments have led to heightened relevance for the study of
fundamental limits on optical response. Here, we review recent progress in our
understanding of these limits with special focus on an emerging theoretical
framework that combines computational optimization with conservation laws to
yield physical limits capturing all relevant wave effects. Results pertaining
to canonical electromagnetic problems such as thermal emission, scattering
cross sections, Purcell enhancement, and power routing are presented. Finally,
we identify areas for additional research, including conceptual extensions and
efficient numerical schemes for handling large-scale problems.

Photonics has become an indispensable tool of scientific discovery, enabling
key advances in communications [1], sensing [2, 3], photovoltaics [4],
computing [5, 6], quantum engineering [7, 8], and many other fields. At the
center of the broad applicability of optical methods is a small but powerful
set of design schemes for confining and transferring energy in time and
space—notions such as index guiding, wave interference, polaritonics, and
effective medium engineering—that provides physical intuition for extracting
concrete functionality from the abstract mathematical richness within
Maxwell’s equations. Each offers a mixture of distinctive capabilities and
limitations, and determining the best approach or combination of approaches
for any particular application remains a subject of continuing challenge for
photonics design [9].

As a concrete example, consider the problem of enhancing light-matter
interactions via the photonic local density of states (LDOS)—the Purcell
effect—reducing to the familiar Purcell factor
$F_{P}=\frac{6}{\pi^{2}}\left(\frac{\lambda}{2n}\right)^{3}\frac{Q}{V}$ in the
case where a single resonance dominates [10]. Integrated micro-resonators [11]
based on index guiding can achieve extremely long lifetimes (high quality
factors $Q$) at the expense of reduced spatial localization (large mode
volumes $V$). Electronic plasmon- and phonon-polariton resonances allow for
tight subwavelength confinement (small $V$) but suffer from high material
absorption (small $Q$) [12]. Photonic crystals [10] and bandgap engineering
provide a flexible low-loss platform for manipulating light at the wavelength
scale but are limited in practice by achievable bandwidths and a lack of forms
exhibiting omnidirectional bandgaps. Metamaterials offer conceptual simplicity
in engineering exotic dispersions and large LDOS [13] but in practice are
constrained by fabrication limitations, the breakdown of effective-medium
approximations at large wavenumbers, and challenges related to light coupling
[14, 15].

Growing out of these general design principles, continued increases in
computational power have paved the way for the development of inverse methods
that, given a set of desired electromagnetic objectives and constraints,
exploit global [16], gradient-based [17, 18, 19], and data-driven optimization
algorithms [20, 21] to search through potentially millions of structural
degrees of freedom in pursuit of ideal response characteristics. This capacity
has greatly expanded the accessible design space, and has led to vast
improvements in device performance. However, for typical problems, the vast
range of design possibilities and complicated interplay between standard
objectives and constraints also makes it practically impossible to determine
truly optimal structures, and one can at most hope for a well performing local
optimum 111Inverse methods may converge to structures that appear to reflect
intuitive design principles, such as bowtie antenna and slot waveguide motifs
for enhancing light-matter interaction [18, 204, 34].. While our arsenal of
design techniques and algorithms gives us enormous capability to tackle a wide
range of engineering applications, it cannot rigorously answer a natural
question of increasing relevance: what are the fundamental limits to optical
control and how close are existing devices to reaching them?

The notion of fundamental limits, encoded in bedrock principles like the
finite speed of light and second law of thermodynamics, is ubiquitous in
physics. Beyond added theoretical understanding, limits have and continue to
play an important role as signposts for further technological improvement.
Attempts to overcome the Abbe diffraction limit led in large part to the
development of the field of super-resolution microscopy, with diverse
techniques exploiting evanescent fields [23, 24, 25], nonlinear effects [24,
26], and active temporal control [27]. Knowledge of the physical origins of
the factors forming the Shockley-Queisser limit [28] for solar cell efficiency
pointed the way to diverse developments in concentrators [29], tandem [30],
and intermediate band photovoltaics [31]. The breakdown of familiar blackbody
limits to nanoscale separations sparked interest in super-Planckian thermal
devices [32, 33].

Figure 1: Applications of photonics and computational design. The top part of
the figure shows a collection of representative photonic functionalities. In
each instance, performance is critically effected by the possibility of using
(sub)wavelength-scale structuring to confine or transform optical fields and,
as such, is intrinsically linked to physical limits on these phenomena. The
bottom part of the figure depicts several (recent) examples of computationally
synthesized devices for achieving improved performance in the operations shown
above. Moving from left to right, these examples are a a dielectric cavity
achieving nanometer scale confinement, Ref. [34]; b a “photon extractor”
enhancing the emission collection rate of a nitrogen vacancy center, Ref. [8];
c a frequency selective emitter for thermo-photovoltaic applications, Ref.
[35]; d a photon-extractor (i.e. cavity and coupler) for diamond photonics,
Ref. [7]; e and f large area, high-efficiency, metalenses, Ref. [36] and Ref.
[37] respectively.

In this review, we first present a historical overview on the development of
electromagnetic limits, following representative examples that illustrate a
general thematic evolution essentially mirroring the history of optics itself:
from simplifying and restrictive assumptions (homogeneity, quasistatics, and
ray optics, etc.) pertinent to low-dimensional, deeply subwavelength, and
large-etalon systems, toward increasingly sophisticated wave arguments
applicable to any length scale. We then focus our discussion on an emerging
general methodology for evaluating photonic design bounds based on
optimization theory and conservation principles that follow either directly
from Maxwell’s equations or the identities of scattering theory. Originally
developed as an instrument for investigating maximal scattering cross section
limits [38, 39, 40], the framework is applicable to a broad range of design
problems where the objective can be expressed as a quadratic function of the
fields [41, 42, 43]. The constraints have clear physical meaning, limiting
both the amplitude of the polarization response, important to power transfer,
and the extent that the phase of the polarization response can be modified,
with consequences on the engineering of resonances. To better handle problems
involving near field effects and rapidly varying length scales, spatially
localized constraints can be introduced, with a denser distribution of local
constraints giving tighter bounds at the expense of higher computational
complexity [41, 42]. In this sense the framework emphasizes the complementary
role of limits and structural optimization: structural optimization enforces
Maxwell’s equations exactly (up to computational discretization) and produces
specific devices corresponding to local optima; the limits framework instead
produces bounds that apply to all possible structures via conservation-law
based constraints over spatial regions that can be viewed as a relaxation of
Maxwell’s equations.

Through instructive results concerning thermal emission, scattering/absorption
cross sections, LDOS enhancement, and power splitting, we describe physical
implications behind various components of the framework, drawing connections
to well-known prior results and demonstrating its broad applicability. For
readers interested in the mathematical details of the underlying optimization
theory, we also recommend the excellent review by Angeris et al. Ref. [44].
Finally, we detail remaining challenges and opportunities, including the need
for numerical methods that can handle larger systems, generalizations to other
physical settings beyond photonics, and potential improvements to structural
optimizations that may arise from knowledge of optimal fields.

## Historical overview

Since the first measurement of the speed of light in vacuum by Rømer, and the
subsequent postulates of special relativity demanding that information cannot
travel faster than $c$, rigorous proofs of subluminal energy velocity
$v_{e}\leq c$ have been deduced from increasingly general assumptions, moving
from homogeneous non-absorbing media, through the inclusion of dispersion,
anisotropy, and nonlocality [45, 46, 47, 48], to the simple unifying
requirement of _passivity_ : materials which do no net work on electromagnetic
fields [49, 48, 50]. In complement, there has also been great interest in
establishing limits on minimal energy velocity or “slow light”, i.e.,
engineered devices such as optical delay lines and buffers [51, 52, 53, 54,
55]. Under the approximation of zero bandwidth, the delay experienced by a
light pulse can essentially be made indefinitely long, e.g., near the band
edge of a photonic crystal where the group velocity vanishes [56]. For finite
bandwidths, however, delay-bandwidth product limits (e.g., the uncertainty
principle) must set a fundamental lower bound. For a slow light waveguide with
an idealized linear dispersion relation across the bandwidth of interest, the
delay bandwidth product limit takes the form $\Delta T\Delta
f\leq\frac{L}{\lambda_{c}}(n_{avg}-n_{min})$, where L is the length of the
device, $\lambda_{c}$ is the free-space bandwidth at the band center, and
$n_{avg}$, $n_{min}$ are the average and minimum effective index of refraction
across the bandwidth, respectively [51]. For the simple case of explicitly 1D
wave propagation and material structuring, more general bounds can be derived
without the notion of an effective index of refraction and the assumption of
idealized linear dispersion: $\Delta T\Delta
f\leq\frac{1}{2\sqrt{3}}\frac{L}{\lambda_{c}}\frac{\max\lVert\epsilon(z,f)-\epsilon_{ro}\rVert}{\epsilon_{ro}}$,
where $\epsilon(z,f)$ is the (possibly complex) relative dielectric constant
as a function of the propagation coordinate $z$ and frequency $f$,
$\epsilon_{ro}$ is the relative dielectric constant of the background medium,
and the maximum is taken across space and bandwidth [57, 58]. Further work is
needed to extend these results to accommodate rigorous full-wave descriptions
of three-dimensional structures.

Related bandwidth arguments have also been used to derive performance bounds
on optical cloaks—devices that eliminate or greatly reduce scattering of
incident light [59]. Most commonly, metamaterial cloaks are designed via
transformation optics to carry out prescribed phase and amplitude
manipulations by mapping coordinate transformations,
$\mathbf{x}\to\mathcal{J}\mathbf{x}^{\prime}$, onto effective homogenized
susceptibility parameters,
$\\{\epsilon,\mu\\}\to\\{\epsilon^{\prime},\mu^{\prime}\\}=\frac{\mathcal{J}^{\dagger}\\{\epsilon,\mu\\}\mathcal{J}}{\mathrm{det}\mathcal{J}}$,
in a design volume surrounding the object. Because light must go around the
cloaked object and maintain an unperturbed wavefront, the phase velocity in
any cloak must be superluminal, necessitating the existence of dispersion in
order to respect causality, and thereby precluding perfect cloaking in vacuum
over any finite bandwidth [60]. One way around this limitation is to consider
“ground-plane” cloaking, wherein the object to be concealed is positioned
adjacent to a reflecting boundary. With this shift in perspective, the
original causality constraint does not apply, as the reflected waves from the
cloak travel a shorter distance than reflected waves from the ground plane
itself. Yet, because the wavefront within the cloak must now be delayed, the
basic challenge posed by the need to respect delay-bandwidth limits remains,
and, in the simplifying case of 1D waves, leads to a bound on the minimum
thickness of the cloak $d\gtrsim\frac{h}{n(n-1)}\frac{\Delta\omega}{\omega}$
as a function of the object size $h$ and the bandwidth $\Delta\omega$.
Generalizing to 3D [61], any ground plane cloak must obey the inequality
$V_{c}\geq V_{c}^{\prime}/B$ where $V_{c}$ is the volume of the cloak,
$V_{c}^{\prime}$ is combined volume of cloak and object, $B$ represents the
maximum achievable index contrast (eigenvalues of
$\mathcal{J}^{T}\mathcal{J}$). While instructive, determining attainable
$\mathcal{J}$, and hence $B$, requires specific geometric analyses.

Figure 2: Overview of representative electromagnetic limits. The figure
illustrates a collection of representative photonic bounds, arranged from left
to right in roughly chronological order. a Equivalent circuit of an
omnidirectional antenna, adapted from [62]. For electrically small antennas,
taking the quasistatic limit yields the celebrated Chu limit. b Yablonovitch
limits to light trapping at a rough interface, combining geometric (ray)
optics with a statistical description of the scattering angle [63]. c and d
Limits based on passivity: for a passive device both the absorbed and
scattered power must be non-negative. c shows the feasible region for cloaking
as a function of absorption efficiency $\sigma_{sca}/\sigma_{abs}$ and net
absorption $\sigma_{abs}$ [64]. d illustrates upper bounds on scattering cross
sections dependent on the material susceptibility $\chi$. e Schematic showing
a scattering matrix description of an optical device connecting input waves to
output waves. Tools of linear algebra such as singular value decompositions
can be used to analyze limits on communication and power transfer using
lightwaves [65, 57, 66]. f Suggested initial design for a multimode 2D
Helmholtz resonator, based on the solution of the Lagrange dual problem of the
design optimization [67]. g Schematic (adapted from [68]) representing a
Lagrange dual framework for evaluating general photonic bounds using only
knowledge of the design region $\Omega$ and material susceptibility $\chi$,
through constraints based on conservation laws derived from Maxwell’s
equations. Overall we see a trend from early results that are problem or
regime specific towards more recent limits with broader applicability.

The intertwined concepts of optical delay and waveguide propagation naturally
lead to questions concerning optical confinement [11]. Two main techniques are
commonly used to confine light, without relying on coupling to material
resonances [69]: index guiding, a generalization of total internal reflection
to wavelength-scale cavities (e.g. whispering gallery or ring resonators), and
bandgap confinement, a generalization of Bragg scattering with embodiments in
photonic crystal waveguides and holey fibers [10]. Except for the simplest
cases, determining the propagation characteristics of a specific design
requires numerical computation, making sufficient conditions for the existence
of guided modes, in analogy with related variational conditions for bounds
states in quantum mechanics, conceptually and practically useful. For
instance, the displacement field $D_{c}$ of the fundamental mode of a
waveguide with permittivity profile $\epsilon$ and cladding profile
$\epsilon_{c}$ was recently shown to necessarily satisfy $\int
D_{c}^{*}(\epsilon^{-1}-\epsilon_{c}^{-1})D_{c}<0$ within the cladding [70].
Relatedly, the degree of localization achievable by a photonic crystal defect
mode is intuitively proportional to bandgap size [10], leading to recent
variational bounds on the minimum index contrast required to engineer 2D
bandgaps [71]. Generalizations to incorporate conditions for dual-polarization
and 3D localization, along with considerations of quasicrystalline [72] and
disordered media [73] remain open problems.

Light confinement is also an integral tool for enhancing light–matter
interactions in optical modulators, lasers, and quantum devices. As detailed
in the Introduction, a resonant mode enhances the power radiated by a nearby
dipolar emitter in accordance with the Purcell factor, which grows
proportionally with the quality factor $Q$ (longer lifetimes) and inversely
with mode volume $V$ (higher field intensities). Beyond modal descriptions
that do not readily generalize to multi-resonant systems [74], or that refer
to specific geometric designs, the electromagnetic local density of states
(LDOS) stands as a fundamental figure of merit quantifying optical response in
arbitrary settings. By enforcing passivity for scattered power [75], recent
quadratic optimization arguments have made it possible to constrain the
magnitude of achievable polarization response independent of geometric or
modal considerations. For a dipole emitter a small distance $d$ away from a
device enclosed within a half space, such conservation arguments yield a
material bound on the dominant contribution of evanescent fields to LDOS
enhancement at a single wavelength,
$\frac{\rho(\omega)}{\rho_{0}(\omega)}\leq\frac{1}{8\pi^{3}}(\frac{\lambda}{d})^{3}\frac{\lVert\chi(\omega)\rVert^{2}}{\imaginary\chi(\omega)}$,
where $\rho_{0}(\omega)$ is the free-space LDOS. As detailed later in this
Review, the positivity of scattered power alone cannot fully account for wave
effects and resonance conditions, and in fact is only one of several other
important constraints that limit optical response beyond quasi-static settings
[38]. Passivity arguments based on Kramers–Kronig conditions similarly place
limits on the frequency-integrated material response of a medium, yielding sum
rules of the form,
$\int_{0}^{\infty}d\omega\,\frac{\rho(\mathbf{x},\omega)-\rho_{0}(\omega)}{\rho_{0}(\omega)}=0$
[76, 77]. Both these arguments and related Thomas–Reich–Kuhn sum rules have in
turn been used to derive upper bounds on nonlinear response, e.g., molecular
hyperpolarizabilities [78, 79].

Another important aspect of localization relates to the focusing of optical
power from a source to a receiver. When restricted to systems described by
geometric optics, the conservation of etendue [80] places a lower bound on how
tight the light rays from a source can be focused down [81]; accounting for
wave effects, scattering concentration bounds over input-output modal channels
of the form $\langle\lVert
c_{out,\hat{u}}\rVert^{2}\rangle\leq\text{maxeig}(\rho_{in})$ yield the
maximum concentration of power achievable for any linear combination of output
channels represented by the unit vector $\hat{u}$ in terms of the largest
eigenvalue of $\rho_{in}$, the density matrix describing the power flow and
coherence across input channels [82]. Achromatic metalenses [83, 84, 85, 86,
87, 88] that focus several incident beams onto the same focal spot are further
restricted by causality and, thus, delay-bandwidth limitations [89].

More broadly, limits on focusing are connected to the general theme of using
light as a conduit for information and energy transfer. An early constraint
related to energy transfer is Kirchhoff’s law, equating the emissivity and
absorptivity of an object, often associated with the second law of
thermodynamics (detailed balance) and originally derived under assumptions of
geometrical optics and reciprocity [90, 91, 92]. Generalizations of this
concept via the formalism of a linear “mode-converter” have been used as
models of communication capacity, and are in principle capable of accounting
for wave effects and non-reciprocal media [66]. In particular, information
encoded in waves transferred between a source $S$ and receiver region $R$, in
free space, can be quantified via a Frobenius norm,
$\int_{V_{S}}\int_{V_{R}}\lVert\mathbb{G}_{0}(\mathbf{x},\mathbf{x}^{\prime})\rVert^{2}$,
of the vacuum Green’s function connecting the two enclosing volumes, $V_{S}$
and $V_{R}$. Adaptations to describe communication mediated by devices (e.g.,
lenses and multiplexers), which can strongly modify electromagnetic fields and
thus “channel capacity”, remain an active area of investigation [93, 43].

A related perspective on communication can be gleaned by considering heat as a
stochastic source of energy transfer. The blackbody limit as applied to
radiative heat transfer constrains the flux emitted by a macroscopic body of
area $A$ and temperature $T$ to be $H\leq\sigma T^{4}A$, where $\sigma$ is the
famous Stefan–Boltzmann constant [91]. However, this result is only applicable
to objects where all characteristic lengths are substantially larger than the
thermal wavelength $\lambda_{T}=\frac{2\pi c\hslash}{k_{B}T}$, and hence does
not explain, for instance, the power exchanged between two bodies held at
different temperatures separated by a subwavelength vacuum gap $d$; nor does
it account for the material constraints subsumed in the assumption of perfect
absorption, i.e. the difficulty of engineering absorption over a wide
bandwidth in a device of a limited size [94, 95, 96]. Just as with LDOS, the
positivity of scattered power sets material constraints on the achievable
polarization response that waves originating in one body may excite in another
[75], yielding an upper bound on the mutual absorption of light
$\propto\frac{\lVert\chi_{S}\rVert^{2}}{\mathrm{Im}\chi_{S}}\frac{\lVert\chi_{R}\rVert^{2}}{\mathrm{Im}\chi_{R}}\int_{V_{S}}\int_{V_{R}}\lVert\mathbb{G}_{0}(\mathbf{x},\mathbf{x}^{\prime})\rVert^{2}$
(generalizing the aforementioned communication bounds to incorporate material
considerations in the source/receivers), and a corresponding upper bound on
the spectrally integrated heat transfer of $H\leq\sigma
T^{4}A\frac{2}{7(kd)^{2}}\frac{\lVert\chi\rVert^{3}}{\imaginary\chi}$. Going
further, accounting for radiative losses due to mutual scattering between
bodies reveals the much tighter bound of $H\leq 2\sigma
T^{4}A\big{(}\frac{\lambda_{T}}{d}\big{)}^{2}\frac{\imaginary\chi}{\lVert\chi\rVert^{2}}\ln\left[\frac{\lVert\chi\rVert^{2}}{4\imaginary\chi}\right]$.
The origin of this reduced material scaling lies in the infeasibility of
achieving resonant optical response for all waves and is elaborated on further
below [97].

The difficulty in engineering blackbody response is directly related to limits
on the absorption of incident radiation, also known as light trapping in the
context of photovoltaic applications. The celebrated Yablonovitch limit [63],
originally derived via a statistical description of rays scattering off rough
surfaces, posits a maximum absorption enhancement factor $F\leq 4n^{2}$,
compared to the expected single pass absorption $\alpha d$ of a weakly
absorbing bulk film of thickness $d$ and absorption coefficient $\alpha$. The
dependence on the refractive index $n$ enters via the total internal
reflection angle $\theta_{c}=\arcsin(1/n)$, which sets the emission cone from
which light can escape. Analyses of maximum absorptivities for films of
thickness $d\lesssim\lambda$ have been carried out through modal decomposition
techniques [98], allowing the associated limit to be expressed as a
$\frac{1}{\Delta\omega}\sum_{m}\sigma_{m,max}$, with the maximum spectral
absorption cross section for each mode $\sigma_{m,max}$ determined by specific
material and geometric considerations. In the simplifying regime of a thin
film supporting a single guided band for each polarization, this approach
gives a limit absorption enhancement of
$F\leq\big{(}\frac{\lambda}{d}\frac{\alpha_{wg}}{\alpha}\frac{1}{2n_{wg}}\big{)}4n_{wg}^{2}$,
where $n_{wg}$ is the group index of the mode(s) and
$\frac{\alpha_{wg}}{\alpha}$ characterizes the spatial overlap between the
mode profile and absorption layer. As examined below, aside from their
practical utility in predicting performance for specific geometries, such
modal summations can be employed to gain a qualitative understanding of
achievable absorption characteristics. In contrast, limits based on maximal
material response of the form
$\frac{\sigma_{\mathrm{abs}}}{V}\leq\frac{\lVert\chi\rVert^{2}}{\imaginary\chi}$
[75] remedy the need of geometric specificity (beyond a linear volumetric $V$
dependence), but can be shown to be loose beyond quasistatic settings, or in
cases where it is not possible to achieve resonant response.

Besides transferring energy, light also imparts a force [99, 100]: the elastic
scattering of impinging photons on a body of size much larger than $\lambda$
transfers a momentum of $\Delta p=2hf/c$. For bodies with wavelength-scale
features, the impact of wave effects and nanostructuring on the scattering
cross section becomes pronounced. For quantum and thermal waves originating
within bodies—often associated with Van der Waals and Casimir forces—the
situation is even more complicated due to the broadband and incoherent nature
of thermodynamic fluctuations. Despite these challenges, a no-go theorem
establishing the impossibility of repulsive interactions between mirror-
symmetric bodies separated through vacuum [101] exists, as do recent bounds on
Casimir–Polder forces on nanoparticles [102].

Finally, as can be seen from the preceding surveys of cloaking, heat
radiation, light trapping, and optical force limits, the concept of a
scattering cross section is central to a great range of electromagnetic
problems (others include optical tweezers, laser heating, etc.); and it is for
this reason that it will occupy much of our ensuing discussion. For bodies of
dimensions $a$ much greater than $\lambda$, the scattering cross section
$\sigma_{sca}\propto a^{2}$ is known to scale like the geometric area. For
electrically small dielectric particles with $a\ll\lambda$, the well-known
Rayleigh scattering result is $\sigma_{sca}\propto a^{6}$. Resonant metallic
particles in the quasistatic regime provide larger _relative_ optical response
$\propto V$, captured in the aforementioned absorption bounds. As discussed
below, recent limit techniques make it now possible to interpolate between
these asymptotic regimes.

General scattering bounds—A common feature across the panoply of
electromagnetic limits mentioned so far is the search for simplifying
assumptions that “relax” physical constraints and thereby make analysis
feasible: working in the geometric optics regime; maximizing modal
contributions without regards for geometric constraints; maximizing material
response by application of optical theorems based on passivity. Over the past
few years, a collection of work has formalized the notion of physical
relaxations through the mathematical language of optimization theory, and it
is this perspective that will dominate the subsequent text.

Before moving to this topic proper, it is important to recognize that there
are other closely related lines of investigation. With regards to antenna
design, a great deal of progress has been made deriving flexible limits to
various aspects of antenna performance by formulating the limits as the
solutions of convex optimizations over possible current distributions of
particular antenna geometries [103, 104, 105, 106]. For problems where an
ideal target field distribution $\hat{\mathbf{E}}(\mathbf{x})$ is known, the
design may be specified as minimizing the norm squared deviation
$\lVert\mathbf{E}-\hat{\mathbf{E}}\rVert^{2}$ subject to the constraint of
Maxwell’s equations $\mathbb{M}\mathbf{E}=i\omega\mu_{0}\mathbf{J}$, with the
Maxwell operator $\mathbb{M}=\curl\curl\,-k_{0}^{2}$ [10] and $\mathbf{J}$
being fixed sources of the problem. Both the field distribution
$\mathbf{E}(\mathbf{x})$ and material distribution $\epsilon(\mathbf{x})$ are
then treated as optimization degrees of freedom, resulting in a non-convex
optimization problem where finding the minimum possible deviation is
computationally difficult [44]. Nevertheless, much along the lines of what
will be done below, the minimum deviation can be bounded by the global optimum
of the convex Lagrangian dual relaxation [107], giving a limit on how closely
$\hat{\mathbf{E}}$ can be realized in practice; an analogous procedure was
used to obtain bounds on minimum achievable mode volumes of dielectric
resonators, given constraints on device size and material [108]. More broadly,
the method can be extended to any separable functions of the field at
different positions
$f(\mathbf{E})=\sum_{\mathbf{x}}f_{\mathbf{x}}(\mathbf{E}(\mathbf{x}))$, which
are of great relevance to any design problem concerning the actualizing of
some specific field transformations. For other types of objectives, especially
if there is no rigorous way to assert that some particular field solution is
optimal, it may be difficult to actually evaluate the form of the dual
function and thereby obtain limits [44].

## Technical description

Scattering preliminaries—As noted above, the core idea motivating the study of
physical limits is to extract attributes that apply to many realizable
instances (devices with different material parameters, structural parameters,
etc.) by means of some relaxation: when the space of possible designs (or
solutions) is too complex to be characterized directly, as is almost always
the case, the only tractable approach to understanding the degree to which
universal properties may be controlled is to equate large groups of designs by
purposefully filtering out certain details. To this end, the perspective
offered by scattering theory is quite helpful. First, by working from the
basic definitions of scattering theory, the relationship between the structure
of the potential and the polarization density generated by a particular
excitation becomes more apparent. Second, as scattering descriptions innately
lead to integral formulations, the constraints of any scattering theory are
naturally organized into a spatial hierarchy that meshes well with both
physical intuition and optimization. These two aspects, taken together,
establish the crucial link between the features that may be imparted to waves
through material structuring, and the standard forms and techniques of
optimization theory. Throughout the following text, the constitutive relations
$\textbf{D}\left(\textbf{x},\omega\right)=\epsilon_{0}\mathbb{V}\left(\textbf{x},\omega\right)\cdot\textbf{E}\left(\textbf{x},\omega\right)$
and
$\textbf{B}\left(\textbf{x},\omega\right)=\mu_{0}\textbf{H}\left(\textbf{x},\omega\right)$
will be assumed for simplicity. However, much of the subsequent development
can be carried out in greater generality, e.g. magnetic media, non-reciprocal
media, etc., c.f. Ref. [42].

Figure 3: Schematic of scattering theory. The basis of scattering theory
broadly rests on the existence, for each particular object (scattering
potential), of an exact relation (the $\mathbb{T}$-operator) between incident
(initial or free) electromagnetic fields and total (generated or net)
polarization currents. Once this relation is known, the total electromagnetic
field (the solution of Maxwell’s equations) is given by applying the Green’s
function of the free (empty) domain $\mathbb{G}_{0}$ to the total current
associated with the incident field.

At the finest level of detail that the design of a photonic device may be
described (“true physics”), Maxwell’s wave equation associates each unique
inhomogeneous dielectric profile $\mathbb{V}\left(\textbf{x},\omega\right)$
within a given domain $\Omega$ (assumed to enforce outgoing scattered fields
[109, 110]), with a unique differential equation,

$\nabla\times\nabla\times\textbf{E}\left(\textbf{x},\omega\right)-k_{0}^{2}~{}\mathbb{V}\left(\textbf{x},\omega\right)\cdot\textbf{E}\left(\textbf{x},\omega\right)=i\omega\mu_{0}~{}\textbf{J}^{f}\left(\textbf{x},\omega\right),$
(1)

where $k_{0}=\omega/c=2\pi/\lambda$ and $\lambda$ is the free space
wavelength. Unless $\mathbb{V}\left(\textbf{x},\omega\right)$ exhibits special
symmetries, equations like (1) do not typically have complete closed form
solutions, meaning that it is usually not possible to fully analyze how
changes to the potential $\mathbb{V}$ alter the total field E beyond local
expansions (derivatives) [111]. Nevertheless, this does not mean that the
relation between $\mathbb{V}$ and E is completely opaque. In particular,
suppose that $\textbf{J}^{f}\left(\textbf{x},\omega\right)=\textbf{0}$, and
that the boundary conditions on $\Omega$ are set to describe an incident
(incoming) electromagnetic field with electric component
$\textbf{E}^{i}\left(\textbf{x},\omega\right)$. Introducing
$\mathbb{I}_{{}_{\Omega}}\left(\textbf{x},\omega\right)$ as the identity
operator, $\mathbb{I}_{{}_{\Omega}}\cdot\textbf{A}=\textbf{A}$, Eq. (1) may be
rewritten as

$\displaystyle\nabla\times\nabla\times\textbf{E}\left(\textbf{x},\omega\right)-k_{0}^{2}~{}\mathbb{I}_{{}_{\Omega}}\left(\textbf{x},\omega\right)\cdot\textbf{E}\left(\textbf{x},\omega\right)=$
$\displaystyle
k_{0}^{2}~{}\left(\mathbb{V}\left(\textbf{x},\omega\right)-\mathbb{I}_{{}_{\Omega}}\left(\textbf{x},\omega\right)\right)\cdot\textbf{E}\left(\textbf{x},\omega\right).$
(2)

The scheme of Eq. (2) is no different than that of Eq. (1) as applied to
vacuum and the polarization “source” $\left(-ik_{0}/Z\right)$
$\left(\mathbb{V}\left(\textbf{x},\omega\right)-\mathbb{I}\left(\textbf{x},\omega\right)\right)\cdot\textbf{E}\left(\textbf{x},\omega\right)$,
and as such, Eq. (2) implies an implicit integral relation on E (a volume
integral formulation [112, 113]) that, while offering a distinct conceptual
perspective [114] and possible computational advantages [115, 116, 117], is
functionally equivalent to Maxwell’s equations. Setting
$\mathbb{R}\left(\textbf{x},\omega\right)=\mathbb{V}\left(\textbf{x},\omega\right)-\mathbb{I}\left(\textbf{x},\omega\right)$
and decomposing $\textbf{E}\left(\textbf{x},\omega\right)$ as
$\textbf{E}\left(\textbf{x},\omega\right)=\textbf{E}^{i}\left(\textbf{x},\omega\right)+\textbf{E}^{s}\left(\textbf{x},\omega\right)$
(with the $i$ superscript standing for “incident” and the $s$ superscript for
“scattered”), regardless of what the shape specified by
$\mathbb{R}\left(\textbf{x},\omega\right)$ actually is,
$\textbf{E}\left(\textbf{x},\omega\right)$ must obey the self-referential
(Lippmann-Schwinger [118] or Liouville-Neumann series relation [119, 120])
222It is equally possible to derive results completely analogous to Eqs.
(5)–(8) using scattered electromagnetic fields [44], as opposed to the
polarization current density perspective used here.

$\displaystyle\textbf{E}^{s}\left(\textbf{x},\omega\right)$
$\displaystyle=\int\limits_{{}_{\Omega}}\\!\\!d\textbf{x}^{\prime}~{}\mathbb{G}_{0}\left(\textbf{x},\textbf{x}^{\prime},\omega\right)\cdot\mathbb{R}\left(\textbf{x}^{\prime},\omega\right)\cdot\textbf{E}\left(\textbf{x}^{\prime},\omega\right)$
(3)
$\displaystyle=\int\limits_{{}_{\Omega}}\\!\\!\mathbb{G}_{0}\cdot\mathbb{R}\cdot\textbf{E}^{i}+\iint\limits_{{}_{\Omega}}\\!\\!\mathbb{G}_{0}\cdot\mathbb{R}\cdot\mathbb{G}_{0}\cdot\mathbb{R}\cdot\textbf{E}^{i}+\dots,$

where, taking r to be $k_{0}\left(\textbf{x}-\textbf{x}^{\prime}\right)$,
$r=\lVert\textbf{r}\rVert$, $\hat{\textbf{r}}\otimes\hat{\textbf{r}}$ to be
the vector outer product of
$\hat{\textbf{r}}=\textbf{r}/\lVert\textbf{r}\rVert$ with itself, and
$\overline{\textbf{id}}$ the $3\times 3$ vector identity matrix,

$\displaystyle\mathbb{G}_{0}\left(\textbf{x}-\textbf{x}^{\prime},\omega\right)=$
$\displaystyle\frac{k_{0}^{3}}{4\pi}\frac{e^{ir}}{r}\Bigg{[}\left(1+\frac{ir-1}{r^{2}}\right)\overline{\textbf{id}}~{}-\left(1+3\frac{ir-1}{r^{2}}\right)\hat{\textbf{r}}\otimes\hat{\textbf{r}}\Bigg{]},$
(4)

is the vacuum Green’s function for the left hand side of Eq. (2) 333The
Green’s function (fundamental solution) of a linear differential equation is
the solution of the differential equations for the Dirac distribution, i.e.
$\left(1/k_{0}^{2}\right)\nabla\times\nabla\times\mathbb{G}_{0}\left(\textbf{x},\textbf{x}^{\prime},\omega\right)-\mathbb{G}_{0}\left(\textbf{x},\textbf{x}^{\prime},\omega\right)=\bm{\delta}\left(\textbf{x}-\textbf{x}^{\prime}\right)$
in our notation. The solution of the differential equation for any
inhomogeneous source term is given by convolution with the Green’s function
[123].. (Note that an the additional factor of $k_{0}^{2}$ is included in Eq.
(4) compared to its usual definition [123]. This is done so that every length
that appears in the use of the Green’s function inside volume integrals is
defined relative to the wavelength.) The mathematical form of Eq. (3)
abstractly shows that
$\textbf{J}\left(\textbf{x},\omega\right)=-\frac{ik_{0}}{Z}\mathbb{R}\left(\textbf{x},\omega\right)\cdot\textbf{E}\left(\textbf{x},\omega\right)$
must be regarded as the _total_ polarization current density generated in
response to $\textbf{E}^{i}\left(\textbf{x},\omega\right)$, and that Eq. (2)
may be equivalently stated (in Fredholm integral form [113, 124]) as

$\displaystyle\textbf{J}\left(\textbf{x},\omega\right)-\int\limits_{{}_{\Omega}}\\!\\!d\textbf{x}^{\prime}~{}\mathbb{R}\left(\textbf{x},\omega\right)\cdot\mathbb{G}_{0}\left(\textbf{x},\textbf{x}^{\prime},\omega\right)\cdot\textbf{J}\left(\textbf{x}^{\prime},\omega\right)=$
$\displaystyle-\frac{ik_{0}}{Z}~{}\mathbb{R}\left(\textbf{x},\omega\right)\cdot\textbf{E}^{i}\left(\textbf{x}^{\prime},\omega\right)=\textbf{J}^{i}\left(\textbf{x},\omega\right),$
(5)

where $\bm{\mathrm{J}}^{i}$ is the “initial” polarization current density
setup in response to the initial electric field, see Fig. 3.

Eqs. (3) and (5) rest at the foundation of scattering theory and definition of
the $\mathbb{T}$-operator [125, 120] ($\mathbb{S}$-matrix [126, 127, 128]).
Directly, via Eq. (3), every incident electric field is related to a specific
polarization current density (the polarization that it generates) by

$\displaystyle\mathbb{I}_{{}_{\mathsf{R}}}\textbf{E}^{i}\left(\textbf{x},\omega\right)=$
(6)
$\displaystyle\\!\int\limits_{\mathsf{R}}\\!d\textbf{x}^{\prime}~{}\left(\bm{\delta}\left(\textbf{x}-\textbf{x}^{\prime}\right)-\mathbb{G}_{0}\left(\textbf{x},\textbf{x}^{\prime},\omega\right)\cdot\mathbb{R}\left(\textbf{x}^{\prime},\omega\right)\right)\cdot\textbf{E}\left(\textbf{x}^{\prime},\omega\right)=$
$\displaystyle\\!\frac{iZ}{k_{0}}\int\limits_{\mathsf{R}}\\!d\textbf{x}^{\prime}~{}\left(\mathbb{R}^{-1}\left(\textbf{x}^{\prime},\omega\right)-\mathbb{G}_{0}\left(\textbf{x},\textbf{x}^{\prime},\omega\right)\right)\cdot\textbf{J}\left(\textbf{x}^{\prime},\omega\right),$

where $\mathsf{R}$ is the subdomain of $\Omega$ where
$\mathbb{R}\left(\textbf{x},\omega\right)\neq\textbf{0}$, the “material”
extent of the scattering potential, and
$\mathbb{R}^{-1}\left(\textbf{x},\omega\right)$ is the pseudo inverse of
$\mathbb{R}$, the inverse over the subdomain where $\mathbb{R}$ is nonzero.
Because $\mathbb{R}$ and $\mathbb{G}$ describe causal, passive, linear system
responses, the total linear operator relating J to $\textbf{E}^{i}$ within
$\mathsf{R}$ must be “invertible” [129, 130]. Accordingly, Eq. (3) delineates
the existence of the inverse relation

$\displaystyle\textbf{J}\left(\textbf{x},\omega\right)=-\frac{ik_{0}}{Z}\int_{{}_{\mathsf{R}}}d\textbf{x}^{\prime}~{}\mathbb{T}\left(\textbf{x},\textbf{x}^{\prime},\omega\right)\cdot\textbf{E}^{i}\left(\textbf{x}^{\prime},\omega\right)\Rightarrow$
(7)
$\displaystyle\mathbb{I}_{{}_{\mathsf{R}}}=\int\limits_{\mathsf{R}}d\textbf{x}^{\prime}~{}\left(\mathbb{R}^{-1}\left(\textbf{x}^{\prime},\omega\right)-\mathbb{G}_{0}\left(\textbf{x},\textbf{x}^{\prime},\omega\right)\right)\cdot\mathbb{T}\left(\textbf{x}^{\prime},\textbf{x}^{\prime\prime},\omega\right),$

defining for each unique scattering potential $\mathbb{V}$ a unique linear
response function $\mathbb{T}$ that relates any incident field
$\textbf{E}^{i}$ to the polarization current density J that self-consistently
solves Maxwell’s equations through Eq. (3). The operator relation governing
the $\mathbb{T}$-operator, the Green’s function, and $\mathbb{V}$ given in Eq.
(7), like Eqs. (3) and (6), is fully equivalent to Maxwell’s equations and
serve as an advantageous starting point for deriving conserved quantities.

From Eq. (6), relatively little must be done to reframe the determination of
an optimal scattering object $\mathbb{V}$ in terms of the polarization current
density. First, by integrating against the characteristic function of any
other _known_ subdomain $\mathsf{P}$ of $\Omega$, together with a local
“polarization” projection matrix $\mathbb{P}\left(\textbf{x},\omega\right)$—a
linear response that does not mix distinct spatial points—the integration
domains appearing in Eq. (5) are shifted from $\mathsf{P}$ to
$\mathsf{R}\cap\mathsf{P}$, standing for the common spatial volume of
$\mathsf{R}$ and $\mathsf{P}$. Next, by applying this result against the
conjugate of $\textbf{J}\left(\textbf{x},\omega\right)$ itself, for _any_
$\mathbb{P}\left(\textbf{x},\omega\right)$ Eq. (6) is transformed into

$\int\limits_{{}_{\mathsf{P}}}\\!\\!d\textbf{x}~{}~{}\textbf{J}^{*}\left(\textbf{x},\omega\right)\cdot\mathbb{P}\left(\textbf{x},\omega\right)\cdot\\\
\left(\underline{\mathbb{R}}^{-1}\left(\textbf{x},\omega\right)\cdot\textbf{J}\left(\textbf{x},\omega\right)-\int\limits_{{}_{\Omega}}\mathbb{G}_{0}\left(\textbf{x}-\textbf{x}^{\prime},\omega\right)\cdot\textbf{J}\left(\textbf{x}^{\prime},\omega\right)\right)=\\\
-\frac{ik_{0}}{Z}\int\limits_{{}_{\mathsf{P}}}\\!\\!d\textbf{x}~{}\textbf{J}^{*}\left(\textbf{x},\omega\right)\cdot\mathbb{P}\left(\textbf{x},\omega\right)\cdot\textbf{E}^{i}\left(\textbf{x}\right);$
(8)

where, crucially, the dependence of the domains of integration on the spatial
structure of $\mathbb{R}\left(\textbf{x},\omega\right)$ has been removed.
Based on the fact that $\mathbb{R}\left(\textbf{x},\omega\right)=\textbf{0}$
implies that $\textbf{J}\left(\textbf{x},\omega\right)=\textbf{0}$, the
content of Eq. (8) is unchanged for _any_ choice of response function
$\underline{\mathbb{R}}^{-1}$ on $\Omega$ so long as
$\underline{\mathbb{R}}^{-1}\left(\textbf{x},\omega\right)=\mathbb{R}^{-1}\left(\textbf{x},\omega\right)$
when $\textbf{x}\in\mathsf{R}$. If $\mathbb{V}\left(\textbf{x},\omega\right)$
may only take on a single $3\times 3$ matrix form distinct from 0, as is
usually true in photonics when designing a device composed of a single
material 444The same conclusion, with minor modifications, also holds if the
design domain $\Omega$ is split into a collection of subdomains, and in each
subdomain any possible structuring must be carried out in a single known
material., then Eq. (8), with $\bm{\chi}\left(\omega\right)$ denoting the
electric susceptibility matrix of the material, may be further simplified by
setting
$\underline{\mathbb{R}}^{-1}\left(\textbf{x},\omega\right)=\bm{\chi}^{-1}\left(\textbf{x},\omega\right)$,
see Eq. (10) below. The only quantity in Eq. (8) that is not typically known
from the outset of a design problem is
$\textbf{J}\left(\textbf{x},\omega\right)$. At the same time, because the
physics of Maxwell’s equation is incorporated through $\mathbb{G}_{0}$, the
true behaviour of the dielectric scattering potential is incorporated through
$\bm{\chi}^{-1}$, and the self-consistency of viewing
$\textbf{J}\left(\textbf{x},\omega\right)$ as the electric polarization
current density resulting from the electromagnetic field given by
$\textbf{E}^{i}\left(\textbf{x},\omega\right)$ is incorporated by association
with Eq. (3), _any_ vector field that respects Eq. (8) for all possible
choices of $\mathbb{P}\left(\textbf{x},\omega\right)$ actually defines an
effective medium scattering structure [43] (a mix between the material
properties asserted by $\mathbb{V}\left(\textbf{x},\omega\right)$ and the
background).

Optimization bounds—The observation that every constraint of the form given by
Eq. (8) applies to _any_ structure of a given material within $\Omega$ implies
that a great number of common photonic objectives can be stated as
quadratically constrained quadratic programs (QCQPs) [44], and, in turn,
bounded using standard relaxation techniques from optimization theory [107].
The connection between Eq. (8) and QCQPs is most easily seen by switching to a
more compact notation. Making use of the fact that integration may be viewed
as an inner product for fields or functions [132] 555Technically, the integral
is an inner product for almost everywhere equal equivalences classes of
functions and fields. However, because we are primarily concerned with finite
numerical representations, here the distinction is not important., with
$\left<\textbf{A}\big{|}\textbf{B}\right>=\int
d\bm{\mathrm{x}}~{}\textbf{A}^{*}(\bm{\mathrm{x}})\cdot\textbf{B}(\bm{\mathrm{x}})$
denoting the standard complex-conjugate inner product, Eq. (8) may be written
in bra-ket form as

$\left<\textbf{T}\right|\mathbb{P}\left(\bm{\chi}^{-1}-\mathbb{G}_{0}\right)\left|\textbf{T}\right>=\left<\textbf{T}\right|\mathbb{P}\left|\textbf{E}^{i}\right>,$
(9)

with
$\left|\textbf{T}\right>=\frac{iZ}{k_{0}}\left|\textbf{J}\right>=\mathbb{T}\left|\textbf{E}^{i}\right>$,
leading to the following equivalent adjoint quadratic constraint equations

$\displaystyle\imaginary\left(\left<\textbf{E}^{i}\right|\mathbb{P}\left|\textbf{T}\right>\right)-\left<\textbf{T}\right|\left[\left(\bm{\chi}^{-1\dagger}-\mathbb{G}_{0}^{\dagger}\right)\mathbb{P}\right]^{\mathsf{A}}\left|\textbf{T}\right>=0,$
$\displaystyle\real\left(\left<\textbf{E}^{i}\right|\mathbb{P}\left|\textbf{T}\right>\right)-\left<\textbf{T}\right|\left[\left(\bm{\chi}^{-1\dagger}-\mathbb{G}_{0}^{\dagger}\right)\mathbb{P}\right]^{\mathsf{S}}\left|\textbf{T}\right>=0,$
(10)

In these expressions, and the proceeding text, $\mathsf{S}$ superscripts will
be used to mark the Hermitian (symmetric) part of the contained linear
response function
$\textbf{M}^{\mathsf{S}}=\left(\textbf{M}+\textbf{M}^{\dagger}\right)/2$, and
$\mathsf{A}$ superscripts will be used to denote the anti-symmetric part,
$\textbf{M}^{\mathsf{A}}=\left(\textbf{M}-\textbf{M}^{\dagger}\right)/2i$, so
that, like a complex number,
$\textbf{M}=\textbf{M}^{\mathsf{S}}+i~{}\textbf{M}^{\mathsf{A}}$ 666Given our
freedom in choosing $\mathbb{P},$ there is no difference between $\mathbb{P}$
and $\mathbb{P}^{\dagger}$..

When $\mathbb{P}$ is set to the domain identity $\mathbb{I}_{{}_{\Omega}}$,
the first relation of Eq. (10) is a statement of the conservation of _real_
power within the domain [40]: the power drawn by the polarization current from
the field, the inner product
$\imaginary\left(\left<\textbf{E}^{i}\big{|}\textbf{T}\right>\right)$, must
equal the sum of the power lost by the polarization current to material
extinction [135],

$\left<\textbf{T}\right|\left(\bm{\chi}^{-1\dagger}\right)^{\mathsf{A}}\left|\textbf{T}\right>=\\!\int\limits_{\Omega}\\!\\!d\textbf{x}~{}\textbf{T}^{*}\left(\textbf{x}\right)\cdot\frac{\imaginary\left[\mathbb{V}\left(\textbf{x},\omega\right)\right]~{}}{\lVert\mathbb{V}\left(\textbf{x},\omega\right)\rVert^{2}}\cdot\textbf{T}\left(\textbf{x}\right)$
(11)

and to outgoing radiation [136]

$\left<\textbf{T}\right|\mathbb{G}_{0}^{\mathsf{A}}\left|\textbf{T}\right>=\\!\iint\limits_{\Omega}\\!\\!d\textbf{x}^{\prime}d\textbf{x}~{}\textbf{T}^{*}\left(\textbf{x}^{\prime}\right)\cdot\imaginary\left[\mathbb{G}_{0}\left(\textbf{x}^{\prime},\textbf{x},\omega\right)\right]\cdot\textbf{T}\left(\textbf{x}\right).$
(12)

The second relation contained in Eq. (10), as illustrated further below, makes
the related statement that _reactive_ power must also be conserved, see Refs.
[137, 39, 111]. Together, these equalities impose minimum requirements on the
characteristics of $\mathbb{V}$ and spatial extent of the domain
$\Omega$—through $\mathbb{G}_{0}$ which is limited to $\Omega$—needed to
achieve resonant response. That is, depending on the material the device will
be made of and the spatial volume that it may possibly occupy, there may be
strong limits on the degree to which the amplitude and phase of J relative to
$\textbf{E}^{i}$ may be tuned.

Recalling from Eq. (3) that the field and induced polarization currents are
linearly related by
$\left|\textbf{E}\right>=\left|\textbf{E}^{i}\right>+\frac{iZ}{k_{0}}\mathbb{G}_{0}\left|\textbf{J}\right>$,
it follows that any physical process described by quadratic field
terms—including the fundamental time-averaged power-transfer quantities of
absorption
$k_{0}\left<\textbf{E}\right|\bm{\chi}^{\mathsf{A}}\left|\textbf{E}\right>/2Z$,
extraction $\real\left(\left<\textbf{E}^{i}\big{|}\textbf{J}\right>\right)/2$
and scattering
$Z\left<\textbf{J}\right|\mathbb{G}_{0}^{\mathsf{A}}\left|\textbf{J}\right>/2k_{0}$,
which rest as the basic figures of merit for the design of antennas [138, 139,
106], light trapping devices [63, 140, 141, 142, 143, 144], and optoelectronic
coupling [145, 146, 147, 9]—can be considered as a quadratic objective. In
this language, taking $\mathtt{f}_{0}\left(\left|\textbf{T}\right>\right)$ to
denote some quadratic function of the polarization, and K a complete set of
constraints 777Practically, the size of K in numeric simulation is set by the
size of the fields. the goal of maximizing (resp. minimizing) any such
objective through material structuring may be formulated as

$\displaystyle\max_{\left|\textbf{T}\right>}~{}\left(\text{resp.}\min_{\left|\textbf{T}\right>}\right)~{}\mathtt{f}_{0}\left(\left|\textbf{T}\right>\right)$
$\displaystyle\text{such that}~{}\forall k\in\text{K}$
$\displaystyle\real\left(\left<\textbf{E}\right|\mathbb{P}_{k}\left|\textbf{T}\right>\right)-\left<\textbf{T}\right|\left[\mathbb{P}_{k}\left(\bm{\chi}^{-1\dagger}-\mathbb{G}_{0}^{\dagger}\right)\right]^{\mathsf{S}}\left|\textbf{T}\right>=0,$
$\displaystyle\imaginary\left(\left<\textbf{E}\right|\mathbb{P}_{k}\left|\textbf{T}\right>\right)-\left<\textbf{T}\right|\left[\mathbb{P}_{k}\left(\bm{\chi}^{-1\dagger}-\mathbb{G}_{0}^{\dagger}\right)\right]^{\mathsf{A}}\left|\textbf{T}\right>=0,$
(13)

which is the form of a quadratically constrained quadratic program (QCQP).
Because the enforcement of fewer constraints always leads to maxima of greater
or equal value (resp. minima of equal or smaller value) in any optimization,
the imposition of any collection of constraints that can be formed from K may
be used to construct a relaxed QCQP that contains the feasible set of Eq.
(13)—an optimization with maxima (resp. minima) at least as large (resp.
small) as Eq. (13) 888A feasible field is a field that respects every imposed
constraint. The feasible set of an optimization statement is the collection of
all feasible fields.. Any bound on such a relaxed program is necessarily a
bound on Eq. (13), and so, by applying any additional relaxation such as
Lagrange duality or semi-definite programming [44], it is possible to obtain
limits on the physically realizable values of
$\mathtt{f}_{0}\left(\left|\textbf{T}\right>\right)$ that universally apply to
any possible material structure within $\Omega$, c.f. Refs. [39, 111, 40, 41,
42, 150, 151, 43, 44]. As highlighted by the expositive examples below, the
extent to which these limits incorporate various physical phenomena may be
tuned by selecting, either by intuition or algorithm [40], the collection of
constraints ($\mathbb{P}_{k}$ projections) that are concurrently imposed, and,
in contrast to many traditional approaches to limits, where individual
components of an expression are bounded and then subsequently summed or
composed to form a global bound, the optimization framework of Eq. (13)
properly describes interactions between constraints.

Although no further refinements of Eq. (13) will be examined hereafter, it
should be noted that this basic optimization bounds approach can, and in
certain cases should, be extended in at least two meaningful ways. First, by
moving to complex frequencies as described in Ref. [18] and Ref. [152], it is
possible to adapt Eq. (13) to treat both broadband and temporal problems.
Detailed accounts of these modifications can be found in Refs. [42, 153].
Second, in situations involving multiple incident fields or scattering
objectives, including applications to multi-functional devices—design
objectives like optical multiplexing [154, 155, 156], meta-optic imaging
components [157, 158, 37, 36] and optical computing [159, 160, 161]—it is
necessary to broaden the scope of the quadratic equalities included in Eq.
(13) to properly account for the additional challenges presented by the need
to engineer multiple field transformations within a common structure. A full
account of these alterations is given in Ref. [43].

## Representative scattering limits

In order to build intuition and provide context, the ensuing section reviews
three increasingly complex tutorial applications of Eq. (13) to set
fundamental upper bounds on optical response. Beginning with the conservation
of _real power_ , focusing on the equivalent problems of maximizing thermal
emission or net absorption, the modal characteristics of
$\mathbb{G}_{0}^{\mathsf{A}}$ in relation to Eq. (10) are shown to reproduce
familiar asymptotic results from quasi-statics and ray optics. However,
because optimization limits do not rely on the validity of such approximate
forms, calculated bounds are also seen to be meaningfully applicable to
intermediate and hitherto inaccessible wavelength-scale regimes. Next, by
further imposing that _reactive power_ be conserved, simultaneously enforcing
the two constraints in Eq. (10) through $\mathbb{P}=\mathbb{I}_{{}_{\Omega}}$,
limits on achievable scattering cross sections are found to anticipate
conditions on the size of the design domain under which resonant response is
possible for a given material choice. Finally—as exemplified through
calculations of bounds on scattering cross sections, radiative emission from a
dipolar source in the near field of body, and power splitting—the set of
integral relations contained in the relaxations of Eq. (13) has the effect of
defining the degree to which the physics of scattering theory is enforced, and
correspondingly, the number and types of integral constraints imposed in
calculating optimization bounds function as complements to the different
number and types of optimization degrees of freedom that may be used in
structural optimization. For proper comparison against realizable devices,
constraints must be selected in a manner that “resolves” the wave physics of
the problem, e.g., accurate limits on phenomena dominated by rapidly decaying
(evanescent) fields require a greater number of local constraints. In almost
all of these representative applications, objective values obtained through
structural “topology” (or “density”) optimization are found to come within an
order of magnitude of their determined limit values [19].

Real power conservation—As a first application of Eq. (13), we review how the
conservation of real power sets an upper bound on thermal radiation and, by
reciprocity, angle-integrated absorption. At a microscopic level, thermal
emission results from stochastically fluctuating electrical currents in matter
[162, 163], with the precise relationship between temperature, energy
dissipation, and field fluctuations in an object in equilibrium determined by
the fluctuation-dissipation theorem [164]. The basis for such a relation may
be intuitively understood from Brownian motion [165]. A particle traveling in
a fluid experiences a dissipation of its net velocity due to collisions with
the constituent particles of the surrounding fluid. Complementarily, these
collisions impart momentum, causing fluctuations in the position of the
particle about its average location, $\langle x^{2}\rangle=2Dt$, where $t$ is
the elapsed time and $D$ is the diffusion coefficient, which by the Stokes-
Einstein relation $D=k_{B}T/\gamma$ is inversely proportional to the the drag
(dissipation) coefficient $\gamma$ [166]. An analogous relation is seen in the
Nyquist formula for Johnson noise, $\langle V^{2}\rangle=4Rk_{B}Td\nu$, where
$V$ is the voltage between the terminals of an open circuit (e.g. a conductive
wire), $R$ the electrical resistance, and $d\nu$ a frequency interval [167].
The fluctuation-dissipation theorem generalizes and formalizes these
observations. For the electromagnetic settings considered here [123],

$\displaystyle\underbrace{\left\\{J_{i}(\mathbf{x},\omega)J_{j}^{*}(\mathbf{x}^{\prime},\omega^{\prime})\right\\}_{T}}_{\text{fluctuation}}=$
(14)
$\displaystyle\frac{\omega\epsilon_{0}}{2\pi}\coth\left(\frac{\hslash\omega}{2k_{B}T}\right)\underbrace{\bm{\chi}_{ij}^{\mathsf{A}}(\mathbf{x},\omega)}_{\text{dissipation}}~{}\delta_{ij}~{}\delta(\mathbf{x}-\mathbf{x}^{\prime})~{}\delta(\omega-\omega^{\prime}),$

where $\left\\{\dots\right\\}$ denotes a thermal ensemble average:
Fluctuations in the current density are point correlated and proportional to
the dissipative part of the electric susceptibility (the optical
conductivity).

Figure 4: Bounds on angle-integrated absorption based on the conservation of
real power. Panel a illustrates the results of Eq. (19) as a function of the
size (radius $R/\lambda$) of a spherically bounded design volume for several
values of the material factor
$\zeta=\lVert\chi\rVert^{2}/\imaginary\left(\chi\right)$. Notably, in
transitioning between very small design volumes, $R\ll\lambda$, and very large
design volumes, $R\gg\lambda$, Eq. (19) is seen to smoothly blend the familiar
$\propto V$ and $\propto A$ limits of quasi-static and ray-optic (blackbody)
approximations. Above panel a, a collection of representative use cases of
radiative absorption are included to give a sense of the range of length
scales covered below. Working from left to right, these images are taken from
Ref. [168], Ref. [135], Ref. [147], Ref. [142], and Ref. [169]. Panel b
depicts a comparison of $\Phi_{opt}$ against performance values obtained by
computational design methods for two set domain sizes, $R/\lambda=1/20$ for
metals ($\real\left(\chi\right)<0$) and $R/\lambda=1/2$ for dielectrics
($\real\left(\chi\right)>0$). Unless additional constraints are included,
findings for dielectric materials limited to small domains, and findings for
metals in large domains, exhibit substantially larger disagreement.

Exploiting this relation and the incoherent nature of the fluctuations, the
net emitted power may be expressed as a sum over independent radiative
channels. Generally, the instantaneous power emitted by a current source is

$P_{rad}=-\int_{\mathsf{R}}\mathbf{J}(\mathbf{x},t)\cdot\mathbf{E}(\mathbf{x},t)$
(15)

where the minus sign results from the convention of emitted power. Switching
to the spectral domain 999The convention used here is
$\mathbf{J}(\mathbf{x},t)=\int_{-\infty}^{\infty}d\omega~{}e^{-i\omega
t}\mathbf{J}(\mathbf{x},\omega)$., and assuming that the collection of
fluctuating dipolar sources distributed throughout the body satisfy Eq. (14),
the thermal power radiated by a body held at a constant temperature $T$ (c.f.
[171, 125, 136]) is given by

$\displaystyle\\{P_{rad}\\}_{T}$
$\displaystyle=\int\limits_{0}^{\infty}d\omega~{}\Pi(\omega,T)~{}\underbrace{\frac{2}{\pi}\text{Tr}\,[\mathbb{G}_{0}^{\mathsf{A}}(\mathbb{T}^{\mathsf{A}}-\mathbb{T}^{\dagger}\mathbb{G}_{0}^{\mathsf{A}}\mathbb{T})]}_{\Phi(\omega)}$
(16)

where
$\Pi(\omega,T)=\frac{\hslash\omega}{\exp(\hslash\omega/\left(k_{B}T\right))-1}$
is the Planck thermal occupation function, and $\Phi(\omega)$ the
corresponding angle-integrated spectral transfer function (absorption or
emission) of the body; the Tr symbol denotes a trace over both the position
and polarization indices of the dipole sources, i.e. the complete set of
indices of the enclosed operators.

In the breakup of Eq. (16), the
$\mathbb{T}^{\mathsf{A}}-\mathbb{T}^{\dagger}\mathbb{G}_{0}^{\mathsf{A}}\mathbb{T}$
term contained in $\Phi$ constitutes an algebraic description of absorption,
Eq. (11), expressed as the subtraction of radiated power
$\mathbb{T}^{\dagger}\mathbb{G}_{0}^{\mathsf{A}}\mathbb{T}$, Eq. (12), from
the total extracted (extinction) power $\mathbb{T}^{\mathsf{A}}$. This
association is no accident: as a consequence of reciprocity, evaluating the
trace over a (delocalized) basis of waves incident on the body changes the
interpretation of $\Phi$ from the net emitted power due to dipolar sources
within the object (thermal emission) to the net power absorbed in the body due
to incident plane waves (angle-integrated absorption), but the algebraic form
of $\Phi$ remains unaltered. Because $\mathbb{G}_{0}^{\mathsf{A}}$ describes
how outgoing radiation carries power away from an object into the surrounding
environment [129, 136], a natural basis in which to evaluate the trace is the
eigenmode expansion

$\displaystyle\mathbb{G}_{0}^{\mathsf{A}}=\sum_{n}\rho_{n}\left|\mathbf{Q}_{n}\right>\left<\mathbf{Q}_{n}\right|,$
(17)

with each of the radiative coefficients $\rho_{n}$ nonnegative by passivity.
Setting $\left|\mathbf{T}_{n}\right>=\mathbb{T}\left|\mathbf{Q}_{n}\right>$
101010Up to a constant, $\left|\mathbf{T}_{n}\right>$ is the polarization
current resulting from the $n$-th radiative mode., $\Phi$ becomes

$\Phi=\frac{2}{\pi}\sum_{n}\rho_{n}\left(\imaginary\left[\left<\mathbf{Q}_{n}|\mathbf{T}_{n}\right>\right]-\left<\mathbf{T}_{n}\right|\mathbb{G}_{0}^{\mathsf{A}}\left|\mathbf{T}_{n}\right>\right).$
(18)

Even without the imposition of a single constraint, the form of Eq. (18)
places fairly strong restrictions on the extent to which the net absorption
(resp. emission) cross section of an object can be enhanced compared to its
geometric cross section [136]. In order to optimize absorption it is clear
from Eq. (18) that each radiative mode must generate a strong polarization:
$\imaginary\left[\left<\mathbf{Q}_{n}|\mathbf{T}_{n}\right>\right]$ is the
extracted power. However, the generation of these currents necessarily leads
to radiative losses,
$\left<\mathbf{T}_{n}\right|\mathbb{G}_{0}^{\mathsf{A}}\left|\mathbf{T}_{n}\right>$,
which grow relatively in strength as the size of the domain increases through
the growth of the $\rho_{n}$ radiative coupling coefficients [136, 97].

As a first example of optimization bounds, we analyze the maximization of
$\Phi$ subject to the constraint that real power is conserved:

$\displaystyle\max_{\\{\left|\mathbf{T}_{n}\right>\in\Omega\\}}~{}\frac{2}{\pi}\sum_{n}\rho_{n}\left(\imaginary\left[\left<\mathbf{Q}_{n}|\mathbf{T}_{n}\right>\right]-\left<\mathbf{T}_{n}\right|\mathbb{G}_{0}^{\mathsf{A}}\left|\mathbf{T}_{n}\right>\right)$
$\displaystyle\text{such that}~{}\forall n$
$\displaystyle\imaginary\left[\left<\mathbf{Q}_{n}|\mathbf{T}_{n}\right>\right]-\left<\mathbf{T}_{n}\right|\left(\bm{\chi}^{-1\dagger}-\mathbb{G}_{0}^{\dagger}\right)^{\mathsf{A}}\left|\mathbf{T}_{n}\right>=0,$
(19)

effectively the simplest version of Eq. (13). Due to the form of $\Phi$, the
only difference between the objective and constraint in Eq. (19) is the
material loss term $\lVert\mathbf{T}_{n}\rVert^{2}/\zeta$, with the factor
$\zeta\equiv\lVert\chi\rVert^{2}/\imaginary\left[\chi\right]$ quantifying the
maximum magnitude that the polarization current density can achieve relative
to the incident electric field [173]. Simply, to maintain equilibrium, the net
(integrated) power extracted by the object at each frequency must be perfectly
balanced by the sum of two possible loss mechanisms: the absorption of power
into material degrees of freedom ($\bm{\chi}^{-1\dagger\mathsf{A}}$), here
considered to be an infinitely large thermal bath [174], and power re-radiated
(scattered or reflected) back into the ambient environment
($\mathbb{G}_{0}^{\mathsf{A}}$).

Applying the relaxation of Lagrange duality (c.f. Refs. [107, 175, 67, 111,
44]), the optimal objective value of Eq. (19) can be expressed as

$\displaystyle\Phi_{opt}=\frac{1}{2\pi}\sum_{n}\begin{cases}~{}1&\zeta\geq\frac{1}{\rho_{n}}\\\
\frac{4\zeta\rho_{n}}{(1+\zeta\rho_{n})^{2}}&\zeta<\frac{1}{\rho_{n}}\\\
\end{cases}\leftrightarrow\begin{cases}~{}1&\tau_{m,n}\geq\tau_{r,n}\\\
\frac{4\tau_{m,n}\tau_{r,n}}{(\tau_{r,n}+\tau_{m,n})^{2}}&\tau_{m,n}<\tau_{r,n}\\\
\end{cases},$ (20)

with $\leftrightarrow$ marking associations, $\tau_{r,n}\leftrightarrow
1/\rho_{n}$ and $\tau_{m,n}\leftrightarrow\zeta$, with a coupled-mode
analysis, Box. 1. The surprising simplicity of $\Phi_{opt}$ as arising from a
sum over independent channel contributions follows from the fact that, absent
other scattering constraints (beside real power conservation), the optimal
bound polarization currents end up becoming diagonal in the basis of radiation
states (the eigenbasis of $\mathbb{G}_{0}^{\mathsf{A}}$),
$\ket{\bm{\mathrm{T}}_{n}}\approx c_{n}\left|\mathbf{Q}_{n}\right>$
(neglecting non-radiative terms that yield minor modifications), with the
maximum bound polarization response $\lVert
c_{n}\rVert\leq\min\left\\{\frac{1}{2\rho_{n}},\zeta\right\\}$ in each
channel.

A trio of plots of $\Phi_{opt}$, bounding angle-integrated absorption
(equivalently emission) for bodies enclosed in a spherical ball of radius $R$,
are shown in Fig. 4. Beyond the exact quantitative predictions appearing in a,
and the excellent agreement with computationally designed structures appearing
in b (see Refs. [116, 136] for details), it is seen that the mere conservation
of net real power is sufficient for Eq. (19) to inherently reproduce
fundamental quasi-static and blackbody behavior. In the limit of a small
design volume, $\zeta\rho_{n}\ll 1$ (resp. $\tau_{m,n}\ll\tau_{r,n}$) for all
$n$, $\Phi_{opt}$ is seen to exhibit a volumetric scaling consistent with the
assumption that the magnitude of all generated polarization currents can grow
as large as material loss allows: as the volume grows, so does the available
power in each channel, and hence so should the polarization response. However,
due to the necessary coupling of these currents with radiative states,
volumetric growth cannot persist indefinitely. Eventually, in each index of
Eq. (18), growth in $\rho_{n}$ (resp. decay in $\tau_{r,n}$) causes radiative
losses to discordantly overwhelm net extracted power if the magnitude of
$\left|\mathbf{T}_{n}\right>$ (resp. the material lifetime $\tau_{m,n}$)
becomes too large, leading the associated channel (index) to enter the
saturation condition of Eq. (20), visible in Fig. 4 as the onset of steps. As
an increasing number of channels saturate, volumetric scaling begins to
asymptotically transition to area scaling, regardless of the supposed value of
$\zeta$. Directly, the black-body limit equating absorption and geometric
cross sections appears out of Eq. (19), irrespective of assumed material
properties.

The behaviors of the $\rho_{n}$ radiative expansion coefficients also have
interesting implications for antenna design [105, 176]. For any excitation
contained within $\Omega$, the largest radiative coefficient $\rho_{n}$ in Eq.
(17) sets a lower bound on the radiative lifetime, $\tau_{r}$, and
consequently, a lower bound on the radiative quality factor
$Q_{r}=\omega\tau_{r}/2$. In the limit of a deeply subwavelength design
volume, the largest $\rho_{n}$ (the dipole coefficient) scales
$\propto\left(k_{0}R\right)^{3}$, setting a lower bound on the radiative
quality factor $\propto\left(k_{0}R\right)^{-3}$ [62, 177, 178]. This lower
bound on $Q_{r}$, known as the Chu limit
($Q_{\mathrm{rad}}\geq\frac{1}{k_{0}R}+\frac{1}{(k_{0}R)^{3}}$ [178]),
correspondingly sets an upper limit on antenna bandwidth that restricts
processing speed [179].

Finally, as suggested above, the sum expression for $\Phi_{opt}$ given by Eq.
(20) is completely analogous to a modal description of absorption by non-
interacting excitations and, as one might expect, under this analogy the
channel saturation condition of $\zeta\rho_{n}=1$ is exactly the rate-matching
condition $\tau_{r,n}=\tau_{m,n}$ between radiative and material lifetimes.
Consequently, it is accurate to interpret Eq. (20) as a model wherein an
idealized object independently extracts power from each of the radiative
(multipole) states in a physically optimal way, with $\rho_{n}$ and $\zeta$
setting limits on the associated coupled-mode radiative and dissipation rates
for every channel. As discussed in greater detail below, this perspective
implicitly assumes that resonant response is achievable in each individual
channel, and that attaining resonant response in one channel has no
implications for any other. Neither assumption is typically true in practice.

Figure 5: Bounds on scattering cross sections based on the conservation of
total power. Panels a and b, taken from Ref. [111], depict the results of Eq.
(22) as a function of the size (radius $R/\lambda$) of a spherically bounded
design volume for two representative values of
$\real\left[\chi\left(\omega\right)\right]$ near optical frequencies (a near
plasmonic metal a and a strong dielectric b.) Dashed lines result from only
imposing the conservation of real power; solid lines result from additionally
imposing the conservation of reactive power. The dots appearing in both panels
mark scattering cross sections achieved in actual geometries discovered by
numeric (inverse) design, for $\chi=-10+i10^{-1}$ and $\chi=16+i10^{-6}$
respectively. For the metal structures in a, aligned cross-hatched dots result
from binarizing the discovered permittivity profiles, which are otherwise
allowed to take on “gray-scale” values [19]. Two sample structures are shown
as insets, with the planewave incident from the more solid side of both
designs, from the left in a, from the right in b, and aligned along the left-
right symmetry axis. Again, in the limit that $R/\lambda\gg 1$ or
$R/\lambda\ll 1$ the calculated bounds approach the scaling predictions of ray
optics (geometric cross sections) and quasi-static scattering ($\propto
V^{2}$), irrespective of the electric susceptibility. The inclusion of
reactive power conservation for strong metals
($\real\left[\chi\left(\omega\right)\right]\ll-3$) confined to small design
volumes ($R/\lambda\ll 1$) shows that the structuring needed to achieve
resonant scattering away from the plasmon condition of
$\real\left[\chi\left(\omega\right)\right]=-3$ reduces achievable material
scaling characteristics—the appearance of “dilution factors”. The of effect
total power conservation on obtained limits for dielectric materials confined
to subwavelength domains is more radical—causing the appearance of “resonance
gaps”—leading to the general conclusion that scattering cross section
enhancements surpassing $\approx 200$ should not be expected for near optical
frequencies.

Total power conservation—Given the remarkable agreement observed in Fig. 4b
between the absorption characteristics of structures obtained by computational
techniques and the associated bounds, it is natural to wonder whether related
conclusions can be drawn for all basic scattering quantities; namely, to what
extent are the results of current inverse methods explained by the basic
necessity of conserving power?

Recalling that the power scattered from an initial electric field
$|\mathbf{E}^{i}\rangle$ at a single frequency $\omega$ is [120, 111]

$\displaystyle P_{sct}$
$\displaystyle=\frac{k_{0}}{2Z}\left<\mathbf{E}^{i}\right|\left[\mathbb{T}^{\mathsf{A}}-\mathbb{T}^{\dagger}\left(\mathbb{V}^{-1\dagger}\right)^{\mathsf{A}}\mathbb{T}\right]\left|\mathbf{E}^{i}\right>,$
(21)

the problem of maximizing the scattering cross section of an object contained
within a design volume $\Omega$ becomes the optimization statement

$\displaystyle\max_{\left|\mathbf{T}\right>\in\Omega}~{}\frac{k_{0}}{2Z}\left[\imaginary\left(\left<\mathbf{E}^{i}|\mathbf{T}\right>\right)-\left<\mathbf{T}\right|\bm{\chi}^{-1\dagger\mathsf{A}}\left|\mathbf{T}\right>\right]$
such that
$\displaystyle\imaginary\left(\left<\textbf{E}^{i}|\textbf{T}\right>\right)-\left<\textbf{T}\right|\left(\bm{\chi}^{-1\dagger}-\mathbb{G}_{0}^{\dagger}\right)^{\mathsf{A}}\left|\textbf{T}\right>=0,$
$\displaystyle\real\left(\left<\textbf{E}^{i}|\textbf{T}\right>\right)-\left<\textbf{T}\right|\left(\bm{\chi}^{-1\dagger}-\mathbb{G}_{0}^{\dagger}\right)^{\mathsf{S}}\left|\textbf{T}\right>=0,$
(22)

where $|\mathbf{E}^{i}\rangle$ is the electric field of an incident plane wave
and, as before,
$\left|\mathbf{T}\right>=\mathbb{T}\left|\mathbf{E}^{i}\right>$ so that
$\left(-ik_{0}/Z\right)\left|\mathbf{T}\right>=\left|\mathbf{J}\right>$ is the
resulting electric polarization current density in the object. If only the
conservation of real power is imposed, the solution of Eq. (22) closely
mirrors the coupled-mode expression for integrated emission given by Eq. (20).
However, as is clear from a comparison of the solid (enforcing total power
conservation) and dashed (enforcing only real power conservation) lines of
Fig. 5 (particularly b), the inclusion of the global reactive power
constraint, the final line of Eq. (22), leads to substantially tighter limits
111111Similar limit tightening also occurs when the conservation of reactive
power is enforced for thermal radiation as applied to dielectric materials and
subwavelength design domains., and under the imposition of this additional
constraint, barring the single channel asymptotic examined in Box. 1, the
solution of Eq. (22) does not have a simple semi-analytic form. The physical
mechanism underlying these differences is the appearance of phase information.
Paralleling the well-known response characteristics of a simple harmonic
oscillator, when only the conservation of real power is enforced there is a
bound on the relative _magnitude_ of $\left|\textbf{T}\right>$ set by material
absorption and radiative emission. All information about the relative phase
offset of the response, partially set by
$\real\left[\chi\left(\omega\right)\right]$, is ignored. Once the need to
conserve reactive power is taken into account, physical restrictions are
placed on both the relative _magnitude_ and _phase_ of the response [137, 39,
111]. These restrictions can either limit, or even exclude, resonant response
121212Here, by resonant response, we mean that, under sufficiently small
changes to the material properties of a design, there is a field observable
that scales roughly as
$\propto\zeta=\lVert\chi\left(\omega\right)\rVert^{2}/\imaginary\left[\chi\left(\omega\right)\right]$.
in certain situations.

Box 1: Single-channel asymptotics (antenna cross sections)

Figure 6: Schematic of coupled-mode theory. The figure, adapted from Ref.
[182], illustrates the basic coupled-mode analysis of resonant scattering
discussed within. To each mode (channel) $n$ of some scattering object,
represented by the amplitude $a_{n}$, there are decay channels associated with
radiative emission, the radiative lifetime $\tau_{r,n}$, and material
absorption, the material lifetime $\tau_{m,n}$. Whenever an incident field
overlaps with the mode indexed by $n$, power is transferred from the incident
field, $s_{+n}$, into the mode $a_{n}$, defining “ports” into and out of the
channel. Supposing that each mode $n$ is orthogonal to every other (e.g., if
the modes are the radiative solutions of a small domain) the scattering cross
section of the object can then be defined by summing over $n$.

As a means of gaining further insight, it is useful to examine the solutions
of Eq. (22) in the single-channel asymptotic (quasistatic) regime of small
domain sizes $R\ll\lambda$ through the lens of modal decompositions and
coupled-mode theory—a type of time-dependent perturbation theory (also known
as Breit–Wigner scattering theory [183]) born out of the assumption that the
optical response of an object may be described as a sum of weakly interacting
resonant modes coupled to one another and/or incident fields via radiative
channels [184]. Following the work of Hamam et al. [182] along the lines
described below Fig. 6, through a combination of real power conservation and
time-reversal symmetry, it can be shown that the amplitude $a_{n}$ of each
such mode obeys the equation
$da_{n}/dt=\left[-i\omega_{n}-\left(\frac{1}{\tau_{r,n}}+\frac{1}{\tau_{m,n}}\right)\right]~{}a_{n}+\sqrt{\frac{2}{\tau_{r,n}}}~{}s_{+n},$
where $\omega_{n}$ denotes the frequency of oscillation of mode $n$. Solving
for $a_{n}\left(t\right)$, using the scattering cross section formula
$P_{sct}/I_{0}=2\lVert a_{n}\rVert^{2}/\left(\tau_{m,n}\lVert
a_{n}\rVert^{2}\right)$, as applied to a $\left(2n+1\right)$-fold degenerate
state (e.g. the total angular momentum number $n$ of radiative solutions in
spherical coordinates), one finds

$\frac{\sigma_{sct,n}}{\sigma_{geo}}=2\left(2n+1\right)\frac{\left(1/\tau_{r,n}\right)^{2}}{\left(\omega-\omega_{n}\right)^{2}+\left(\frac{1}{\tau_{m,n}}+\frac{1}{\tau_{r,n}}\right)^{2}}\left(\frac{\lambda}{2\pi
R}\right)^{2},$ (23)

with $\sigma_{geo}=\pi R^{2}$ denoting the geometric cross section of a sphere
of radius $R$. Intuitively, an expression like Eq. (23) can also be considered
as a bound, set by the conservation of real power, on maximum scattering:
assuming that each channel can be designed to have the same resonance
frequency, the parametric relations imply a set of rate matching conditions
(e.g., $\tau_{r,n}=\tau_{m,n}$ in Eq. (23)) for maximizing each channel’s
contribution to scattering at some wavelength $\lambda$. While providing a
predictive and conceptual model of scattering, with meaningful insights into a
wide range of applications c.f. Refs. [185, 186, 187, 188, 189, 190, 191,
192], such modal descriptions face significant challenges in setting
quantitative limits. Explicitly, it is seldom clear (1) how many channels
should be considered, i.e. where the sum should be cut-off to avoid divergence
without being overly restrictive [193, 194]; (2) what range of parameter
values are possible for each channel; and (3) to what level the parameters are
connected; i.e. does material structuring allow for independent parameter
tuning.

When Eq. (22) is considered on a sufficiently small ball, the decomposition of
an incident planewave source into radiative multipoles approximately
terminates at the ($n=1$) dipole fields. Under this quasistatic condition, Eq.
(22) can be solved semi-analytically [111], leading to the single-channel
$R\ll\lambda$ asymptotics

$\frac{\sigma_{sct}^{opt}}{\sigma_{geo}}\leq\frac{3}{2}\begin{cases}\frac{\rho_{1}^{2}}{\left(\frac{1}{3}+\frac{\real\left(\chi\right)}{~{}\lVert\chi\rVert^{2}}\right)^{2}+\left(\rho_{1}+\frac{\imaginary\left(\chi\right)}{~{}\lVert\chi\rVert^{2}}\right)^{2}}&\real\left(\chi\right)>-3\\\
\frac{\left[\rho_{1}\lVert\chi\rVert/\real\left(\chi\right)\right]^{2}}{\left[\rho_{1}+\frac{\imaginary\left(\chi\right)}{3\lVert\real\left(\chi\right)\rVert}\right]^{2}}&\real\left(\chi\right)\leq-3\end{cases}$
(24)

where $\rho_{1}=\frac{2}{9}\left(2\pi R/\lambda\right)^{3}$ is the radiative
coefficient of the three-fold degenerate dipole channel, Eq. (17). Up to a
factor of $4$, which as originally explained in Ref. [75] occurs because
optimal scattering simultaneously implies optimal absorption, there is again a
clear association between Eq. (24) and the coupled-mode theory expression
given by Eq. (23). So long as $\imaginary\left[\chi\left(\omega\right)\right]$
is small compared to the absolute value of
$\real\left[\chi\left(\omega\right)\right]$, Eq. (24) is consistently
interpreted in terms of coupled mode parameters as

$\displaystyle\omega-\omega_{1}$ $\displaystyle\leftrightarrow
1/\real\left(\chi\right)-\left(-1/3\right)~{}~{}~{}~{}\tau_{r,1}\leftrightarrow
1/\rho_{1}$ $\displaystyle\tau_{m,1}$
$\displaystyle=\begin{cases}\lVert\chi\rVert^{2}/\imaginary\left(\chi\right),&\real\left(\chi\right)>-3\\\
3\,\lVert\real\left(\chi\right)\rVert/\imaginary\left(\chi\right),&\real\left(\chi\right)\leq-3\end{cases}$
(25)

With regards to optimal response, Eq. (24) shows that “truly” resonant
response ($\imaginary\left(\chi\right)/\lVert\chi\rVert^{2}$ scaling) is only
possible when $\real\left[\chi\left(\omega\right)\right]=-3$, the localized
plasmon resonance of a spherical nanoparticle [123]. For all other material
choices, Eq. (24) implies that, when a resonance is possible, some amount of
material structuring is needed to achieve a maximized scattering cross
section, and, in undertaking this structuring, the potential for field
enhancement is reduced. In the case of dielectrics and weak metals
($\real\left(\chi\right)>-3$) “resonant” scattering is simply not possible. In
the case of plasmonic metals ($\real\left(\chi\right)<-3$), resonant response
is only possible through a “dilution” of the effective electric susceptibility
to the plasmon condition, Fig. 5a; see Ref. [111] for additional details.
Most notably, as confirmed by Fig. 5b, when confined to a spherical
subwavelength domain, the largest possible scattering cross section that can
be achieved by structuring a dielectric material is _exactly_ the $\propto
V^{2}$ Rayleigh scattering of a ball [195] (see Box. 1 and Ref. [111] for
further details).

The inclusion of reactive power also implies drastic alterations to the
mathematical model and interpretation of Eq. (22) as compared to Eq. (19).
Precisely, the sum form of $\Phi_{opt}$ given by Eq. (20) arises because both
the objective and constraint of Eq. (19) are diagonalized in an eigenbasis of
$\mathbb{G}_{0}^{\mathsf{A}}$ 131313It should be kept in mind that this
description often accurately anticipates the performance of structures
discovered by computational methods.. When further physical constraints are
imposed (e.g., global reactive power or the local constraints introduced in
the next set of examples) simultaneous diagonalization is rarely possible. In
part indicating the richness of the physics being described, the symmetric and
anti-symmetric components of the Green’s function,
$\mathbb{G}_{0}^{\mathsf{S}}$ and $\mathbb{G}_{0}^{\mathsf{A}}$, do not share
a common eigenbasis [197, 41]. Hence, whenever response characteristics are
not dominated by the conservation of real power, it is generally not possible
to describe scattering phenomena in terms of an orthogonal basis of weakly
interacting modes. Once both aspects of the Green’s function are included,
radiative channels are mixed both among themselves and with non-radiative
states. Relatedly, although the content of Box. 1 stands an exception, there
is typically no simple way to use optimization bounds as means of doing
parameter extraction for coupled mode theory.

Local constraints—Extrapolating from the last two examples, enforcing that
total power must be conserved is found to capture almost all relevant physical
effects that limit achievable scattering characteristics for propagating
waves—far-field applications like maximizing planewave absorption, thermal
radiation [136], and scattering cross sections [111, 39, 40, 198]. However,
when applied to objectives governed by rapidly varying (e.g., evanescent)
fields or multiple length scales, these coarse characterizations of the
integral-wave relations may miss important aspects of the problem, leading to
bounds with little connection to reality [41, 42]. To remedy this issue,
additional physics in the form of localized constraints incorporating higher
spatial resolution must be included. Specifically, contrasting Eq. (22) with
the system of equations that must be solved in order to approximate Maxwell’s
equations numerically, it is clear that imposing global power conservation (a
total of two constraints) cannot possibly capture all relevant wave features:
the use of $\mathbb{P}=\mathbb{I}_{{}_{\Omega}}$ in Eq. (10) only guarantees
that scattering theory is true _on average_ over the volume of the domain. At
any spatial point, the solution resulting from an optimization problem
statement like Eq. (22) may violate Eq. (6), so long as these violations
cancel each other when integrated over the entire scattering domain.

Figure 7: Impact of localized constraints. The figure shows upper bounds under
the enforcement of local constraints on a the minimum radius $R$ of a
cylindrical design domain necessary to encode a “power-splitting” $\mathbb{S}$
matrix, evenly distributing power from a single plane-wave input into $2n+1$
outgoing radiative channels (Ref. [42]); b plane-wave scattering as described
in the previous tutorial (Ref. [41]); and c radiative Purcell enhancement for
an electric dipole source in the near field of an object (Ref. [41]). Lighter-
colored lines in b and c result from imposing the global conservation of real
and reactive power. Like-colored darker lines are obtained by enforcing local
conservation constraints over radial shell subdomains (varying distribution
with application). As in previous plots, dots in b and c mark actual
enhancement values achieved by structures discovered by computational methods.
Views of one such design are given below c. The logarithmic color maps
included below b indicate the local violation of the conservation of reactive
power for $1,2$, and $4$ evenly spaced shell subregion constraints. The
schematic shown below a conceptualizes the engineering of a device to realize
a particular scattering matrix. In all three panels, additional constraints
lead to increasingly realistic field features and limit values.

Owing to the nature of the relaxation techniques commonly used to solve
optimization bounds (c.f. Refs. [41, 44]), the spatial oscillation of such
local violations of “true” physics tend to track the spatial oscillations of
the incident field, as seen in Fig. 7b, and accordingly, by imposing localized
projections ($\mathbb{P}=\mathbb{I}_{\mathsf{P}}$ where
$\mathsf{P}\subset\Omega$) in Eq. (10) targeting a specific region of
violation, it is usually possible to guide optimization limits towards
increasingly physical characteristics: the vacuum Green’s function
$\mathbb{G}_{0}$ does not propagate all information equally, but rather blurs
rapid field fluctuations as the point of observation moves away from the
source. As such, for each design problem, there are characteristic lengths
below which differences between field solutions have no pragmatic relevance.
If independent constraints enforce “averaged” physics on a grid finer than the
smallest of these length scales, then the solution of Eq. (13) should differ
little from what is realizable in practice [199]. Directly, the number and
distribution of local constraints can be viewed as “tuning knobs” enforcing
physics at the expense of computational complexity.

An exemplification of these ideas is illustrated by the minimum-radius limits
on a two-dimensional power splitter, distributing power from an incident wave
equally between $2n+1$ cylindrical wave channels, depicted in Fig. 7a (Ref.
[42]). In asserting only the conservation of global power, the minimum
diameter asymptotes to $\sim\lambda/3$, a size at which it begins to become
possible for a non-physical response oscillating near
$\lambda/\sqrt{\real(\chi)+1}$ to satisfy global power conservation while
maintaining large local violations. Opposingly, when local constraints are
added, this asymptote disappears and the required radius (suggesting
increasing device complexity) begins to grow steadily with the number of power
divisions desired. In Fig 7b, the strict use of global constraints similarly
suggests that shortly after attaining a resonance criteria of
$R\gtrsim\lambda/5$, by structuring a material with $\real(\chi)=16$ within a
spherical ball of radius $R$, it is possible to enhance the scattering cross
section inversely proportional to material loss ($\imaginary(\chi)$). No such
scaling is found in computationally synthesized structures, marked in the
figure by dots, and under the imposition of $8$ (evenly spaced) radial shell
constraints this nonphysical feature all but vanishes.

Stemming from the need to properly describe sub-wavelength field
characteristics, local constraints are also generally required to formulate
relevant limits on near field phenomena. Following Fig. 7c—bounds on the
maximization of radiative Purcell enhancement for a dipolar current source
separated from an arbitrary device by a distance $d$, again contained within a
spherical ball of radius $R$—when only global constraints are considered
enhancement is seen to scale $\propto 1/\imaginary\chi$ when $d\ll\lambda$
(light lines), as is characteristic of material loss limited resonant response
[75]. Upon inclusion of localized constraints defined over concentric
spherical shells (dark lines), the maximal radiation is observed to drop by
several orders of magnitude and no longer grows with decreasing
$\imaginary\chi$, confirming the challenge of achieving optimal polarization
fields capable of resonantly scattering evanescent fields into propagating
waves. Conversely, even with many more localizing shells (see Ref. [41] for
details), limits on larger domains and stronger dielectrics show no such
saturation and exhibit $\propto d^{-3}$ diverging growth—implying that
arbitrarily high angular momentum states can be out-coupled with nearly fixed
efficiency. Because of the resolution arguments stated above, if additional
non-symmetric localized constraints were to be imposed, particularly in the
region of the design volume nearest the dipole, tighter limits may be
anticipated.

Figure 8: From performance bounds to structural optimization. Through the
common denominator of spatial divisions, a complementary relationship exists
between inverse design and performance bounds. In inverse design, the non-
convex optimization problem of structural optimization is heuristically
approximated by particular designs representing local optima, with resulting
device performance generally increasing with additional structural degrees of
freedom. In optimization bounds, a coarse grained version of the same
optimization problem is bounded by exploiting some convex relaxation in order
to determine a value that must be respected by all allowed designs, with
resulting limits generally becoming tighter under the use of additional
constraints. Consequently, as the number and resolution of the local
constraints going into a bounds QCQP is increased, its solution fields
typically become increasingly better approximations of the polarization field
of a globally optimal structure. This observation suggests a potential,
presently unexplored, workflow for the computational design of photonic
devices: evaluate the limits of an application to a tightness wherein the
approximate field distribution can be extracted and used as a starting point
for the discovery of near optimal structures.

## Outlook

As exemplified by the preceding discussion and representative examples, the
nascent development of a methodology combining physical constraints with
optimization theory has already proven to be a remarkably versatile and
fruitful framework for understanding and computing electromagnetic limits.
Still, a panorama of challenges and opportunities remain. First, the core
ideas behind the method are by no means restricted to phenomena governed by
Maxwell’s equations, and have direct analogues in other domains of wave
physics, such as acoustics and quantum mechanics. And even more broadly, the
general concept of obtaining performance limits by rigorously bounding a
simpler problem through relaxed physics is applicable to essentially any field
of engineering. (See Ref. [153] for recent applications to quantum control.)
Second, as it currently stands, the framework requires physical objectives to
be formulated as quadratic functions of the polarization fields, which
excludes various photonic problems, including nonlinear processes (such as the
Kerr effect) or objectives which may be more naturally expressed as
eigenproblems (such as the maximization of bandgaps [200] or the engineering
of Dirac points [201]. Yet another avenue for further research, particularly
relevant to device applications, is the incorporation of fabrication
constraints. Presently, the limit framework considers any possible structure
that may fit within the design region without regard for existing
nanofabrication constraints such as minimum feature sizes and material
connectivity.

Moreover, it is not yet understood whether, or under what conditions, the
convex relaxation techniques that are used to obtain optimization limits can
be guaranteed to solve their associated QCQPs, i.e., whether the limit is
equal to the true QCQP solution or just some larger (resp. smaller) value.
Non-affine equality constraints such as the conservation of real and reactive
power are non-convex, and QCQPs with non-convex constraints are not generally
thought to be solvable by any convex relaxation [202, 44]. So far, however, a
vast majority of investigations have found that calculated limit fields are in
fact optimal solutions of the initial optimization problem statement [39, 38,
41], with inclusion of large numbers of local constraints only resulting in
numerical ill-conditioning. Such guarantees are not merely a theoretical
exercise. So long as the underlying QCQP is actually solved, through the
addition of increasingly finer localized constraints progressively better
approximations to an optimal realizable polarization field emerge from bounds
computations, and this information could be leveraged to great effect in
inverse design. For instance, it could be used as a starting point for adjoint
optimization to recover a near optimal structure, see Fig. 8, or as a guide
for the design parameters that should be considered to possibly realize
improved performance characteristics. Furthermore, it is likely not necessary
to enforce local constraints down to the computational pixel level to make use
of these potentially powerful connections; a coarse distribution may be enough
to extract an approximate optimal structure. Indeed, the onset of ill-
conditioning with finer local constraints suggests that, depending on the
design problem, there is a characteristic length scale beyond which more
detailed structuring becomes unnecessary. Relatedly, while the bounds
computation is convex, it is not necessarily easy to solve numerically.
Besides ill-conditioning resulting from the imposition of large numbers of
local constraints, numerical instabilities also occur in systems with low loss
(e.g., semi-transparent media). To address this, it may be possible to
formulate alternate constraints better suited to handle low-loss/lossless
design problems; one example is given by Ref. [198], though the proposed
constraints appear to provide non-trivial limits only for small design regions
and low dielectric contrasts.

The large scale of most practical photonics problems also poses a challenge:
current demonstrations of the framework are restricted to 2D [198, 42, 44, 43]
or highly symmetric domains in 3D, exploiting efficient spectral basis
representations [39, 38, 68] to limit the size of the associated system
matrices. Beyond these early proof-of-concept explorations, there is much room
for development of standardized packages for evaluating limits (possibly in
conjunction with inverse methods) on 3D photonic devices by exploiting
general-purpose techniques—numerical Maxwell solvers employing localized
bases—and open-source optimization methods [203].

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