## Abstract

The brightness theorem—brightness is nonincreasing in passive systems—is a foundational conservation law, with applications ranging from photovoltaics to displays, yet it is restricted to the field of ray optics. For general linear wave scattering, we show that power per scattering channel generalizes brightness, and we derive power-concentration bounds for systems of arbitrary coherence. The bounds motivate a concept of “wave étendue” as a measure of incoherence among the scattering-channel amplitudes and which is given by the rank of an appropriate density matrix. The bounds apply to nonreciprocal systems that are of increasing interest, and we demonstrate their applicability to maximal control in nanophotonics, for metasurfaces and waveguide junctions. Through inverse design, we discover metasurface elements operating near the theoretical limits.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

The “brightness theorem” states that optical radiance cannot increase in passive ray-optical systems [1]. It is a consequence of a phase-space conservation law for optical étendue, which is a measure of the spatial and angular spread of a bundle of rays and has had a wide-ranging impact: it dictates the upper bounds to solar-energy concentration [2,3] and fluorescent-photovoltaic efficiency [3], it is a critical design criterion for projectors and displays [4], and it undergirds the theory of nonimaging optics [5]. Yet a generalization to electromagnetic radiance is not possible, as coherent wave interference can yield dramatic radiance enhancements. A natural question is whether Maxwell’s equations, and more general wave-scattering physics, exhibit related conservation laws.

In this paper, we develop analogous conservation laws for power flow through the scattering channels that comprise the bases of linear scattering matrices. By a density-matrix framework more familiar to quantum settings, we derive bounds on power concentration in scattering channels, determined by the coherence of the incident field. The ranks of the density matrices for the incoming and outgoing fields play the role of étendue, and maximal eigenvalues dictate maximum possible power concentration. For the specific case of a purely incoherent excitation of $N$ incoming channels, power cannot be concentrated onto fewer than $N$ outgoing channels, which in the ray-optical limit simplifies to the classical brightness theorem. In resonant systems described by temporal coupled-mode theory, the number of coupled resonant modes additionally restricts the flow of wave étendue through the system. The bounds require only passivity and apply to nonreciprocal systems. We discuss their ramifications in nanophotonics—for the design of metasurfaces, waveguide multiplexers, random-media transmission, and more––while noting that the bounds apply more generally to scattering in acoustic, quantum, and other wave systems.

*Background*: Optical rays exist in a four-dimensional phase space determined by their position and momentum values in a plane transverse to their propagation direction. Optical étendue [5] denotes the phase-space volume occupied by a ray bundle. In ideal optical systems, phase-space evolution is governed by Liouville’s theorem, and thus radiance and étendue are invariants of the propagation. A differential ray bundle propagating through area $\mathrm{d}A$ and solid angle $\mathrm{d}\mathrm{\Omega}$, in a medium of refractive index $n$ and tilted at an angle $\theta $, has an étendue of ${n}^{2}\mathrm{cos}\theta \mathrm{d}A\mathrm{d}\mathrm{\Omega}$. Figure 1(a) depicts étendue conservation in ray-optical systems and the consequent trade-off between spatial ($\mathrm{d}A$) and angular ($\mathrm{d}\mathrm{\Omega}$) concentration. Electromagnetic radiance is intensity per unit area per unit solid angle, which in ray optics is proportional to the flux per unit étendue. By étendue invariance, in tandem with energy conservation, ray-optical brightness cannot increase. In nonideal systems, étendue can decrease when rays are reflected or absorbed, but any such reduction is accompanied by power loss, and the theorem still applies.

Extending radiometric concepts such as radiance into wave systems with coherence, beyond ray optics, has been the subject of considerable study [6–15]. Wigner functions can represent generalized phase-space distributions in such settings, and they are particularly useful for “first-order optics,” i.e., paraxial approximations, spherical waves, etc. Yet the Wigner function and similar approaches cannot simultaneously satisfy all necessary properties of a generalized radiance [8,12,14]. This does not preclude the possibility for a Wigner-function-based brightness theorem—indeed, this represents an interesting open question [16]—but we circumvent the associated challenges by recognizing power transported on scattering channels as the “brightness” constraints in general wave-scattering systems.

*Concentration bounds*: Consider generic linear wave scattering in which some set of input waves ${\psi}_{\mathrm{in}}$ are coupled to a set of output waves ${\psi}_{\mathrm{out}}$ (in domains that may be overlapping or disjoint) through a scattering operator $\mathcal{S}$ as follows: ${\psi}_{\mathrm{out}}=\mathcal{S}{\psi}_{\mathrm{in}}$. We assume that the scattering process is not amplifying but does not have to be reciprocal or unitary. To describe the scattering process in a finite-dimensional basis, we adapt the formalism developed in Refs. [17–21]. As is well established in classical and quantum scattering theory [22–25], the operator $\mathcal{S}$ comprises two contributions: a “direct” (background) contribution from waves that travel from input to output without the scatterer present, which we denote with the operator $\mathcal{D}$, and a “scattered-field” contribution from waves that are scattered from input to output only in the presence of the scatterer, which we denote with the operator $\mathcal{T}$ (as in “T matrix” approaches [26–29]). A key insight of Refs. [17–21] is that the $\mathcal{T}$ operator is compact (in fact, it is a Hilbert–Schmidt operator, by the integrability of the squared Frobenius norm of its kernel), which means that one can accurately represent it by a *finite*-dimensional singular-value decomposition

*not*necessarily compact—for example, $\mathcal{D}$ for scattering within a spherical domain is the identity operator [23,24,26]—and thus does not have the same natural decomposition. Nevertheless, we can still project the input and output states onto $\mathcal{V}$ and $\mathcal{U}$, respectively. Such a representation will necessarily miss an infinite number of input states with a nontrivial direct-process contribution, but by definition those states will have no interaction with the scatterer, and thus they have no consequence on power-concentration bounds or on the definition of a wave étendue. We include the direct process at all in order to naturally incorporate interference effects between the direct and scattering processes. Thus, for any scattering problem, the columns of $\mathcal{V}$ and $\mathcal{U}$ define our scattering channels, within which our input and output waves can be decomposed, as follows: where ${\mathbf{c}}_{\mathrm{in}}$ and ${\mathbf{c}}_{\mathrm{out}}$ are the vector coefficients of the excitations on these channels as shown in Fig. 1(b). The scattering

*matrix*connects ${\mathbf{c}}_{\mathrm{in}}$ to ${\mathbf{c}}_{\mathrm{out}}$ and can be found by starting with the definition of the $\mathcal{S}$ operator, ${\psi}_{\mathrm{out}}=\mathcal{U}{\mathbf{c}}_{\mathrm{out}}=\mathcal{S}{\psi}_{\mathrm{in}}=\mathcal{SV}{\mathbf{c}}_{\mathrm{in}}$, and then taking the inner product with $\mathcal{U}$ to find

Perfectly coherent excitations allow for arbitrarily large modal concentration (e.g., through phase-conjugate optics [30,31]), but the introduction of incoherence incurs restrictions. To describe the coherence of incoming waves, we use a density matrix ${\mathit{\rho}}_{\mathrm{in}}$ [32] that is the ensemble average (hereafter denoted by $\u27e8\xb7\u27e9$, over the source of incoherence) of the outer product of the incoming wave amplitudes, written as

For inputs defined by some ${\mathit{\rho}}_{\mathrm{in}}$, how much power can flow into a single output channel, or more generally into a linear combination given by a unit vector $\widehat{\mathbf{u}}$? If we denote ${\widehat{\mathbf{u}}}^{\u2020}{\mathbf{c}}_{\mathrm{out}}$ as ${\mathbf{c}}_{\mathrm{out},\widehat{\mathbf{u}}}$, then the power through $\widehat{\mathbf{u}}$ is $\u27e8{|{\mathbf{c}}_{\mathrm{out},\widehat{\mathbf{u}}}|}^{2}\u27e9={\widehat{\mathbf{u}}}^{\u2020}{\mathit{\rho}}_{\mathrm{out}}\widehat{\mathbf{u}}={\widehat{\mathbf{u}}}^{\u2020}\mathbf{S}{\mathit{\rho}}_{\mathrm{in}}{\mathbf{S}}^{\u2020}\widehat{\mathbf{u}}$. The quantity $\u27e8{|{\mathbf{c}}_{\mathrm{out},\widehat{\mathbf{u}}}|}^{2}\u27e9$ is a quadratic form in ${\mathit{\rho}}_{\mathrm{in}}$, such that its maximum value is dictated by its largest eigenvalue [33], ${\lambda}_{\mathrm{max}}$, leading to the inequality

*coherent*scattering for a new input: ${\mathbf{c}}_{\mathrm{in}}={\mathbf{S}}^{\u2020}\widehat{\mathbf{u}}$. For this input field, the incoming power is ${\widehat{\mathbf{u}}}^{\u2020}{\mathbf{SS}}^{\u2020}\widehat{\mathbf{u}}$, while the outgoing power in the unit vector $\widehat{\mathbf{u}}$ is ${|{\widehat{\mathbf{u}}}^{\u2020}{\mathbf{c}}_{\mathrm{out}}|}^{2}={({\widehat{\mathbf{u}}}^{\u2020}{\mathbf{SS}}^{\u2020}\widehat{\mathbf{u}})}^{2}$. Enforcing the inequality that the outgoing power in $\widehat{\mathbf{u}}$ must be no larger than the (coherent) total incoming power, we immediately have the identity ${\widehat{\mathbf{u}}}^{\u2020}{\mathbf{SS}}^{\u2020}\widehat{\mathbf{u}}\le 1$. (We provide an alternative proof in Supplement 1.) Inserting into Eq. (6), we arrive at the bound

Just like the ray-optical brightness theorem [1,35], our scattering-channel bounds can alternatively be understood as a consequence of the second law of thermodynamics. If it were possible to concentrate incoherent excitations of multiple channels, then one could filter out all other channels and create a scenario with a cold body on net sending energy to a warm body [20]. The partially coherent case is not as physically intuitive, but the application of such thermodynamic reasoning could be applied to the modes that diagonalize ${\mathit{\rho}}_{\mathrm{in}}$, and then a basis transformation would yield Eq. (7).

*Wave étendue*: Equations (7) and (8) imply that the incoherent excitation of $N$ inputs cannot be fully concentrated to fewer than $N$ outputs. This motivates the identification of “wave étendue” as the number of incoherent excitations on any subset of channels (incoming, outgoing, etc.). For a density matrix $\mathit{\rho}$, one can count independence by the matrix *rank* and define étendue = $\mathrm{rank}(\mathit{\rho})$.

To understand the evolution of wave étendue through the scattering process, we reconsider the singular-value decomposition (SVD) of Eq. (1). The matrix $\mathrm{\Sigma}$ is a square matrix with dimensions $N\times N$, where $N$ is the number of well-coupled pairs of input and output scattering channels. Since the singular values are nonzero, we know that $\mathrm{\Sigma}$ is full rank. The scattering matrix $\mathbf{S}$ is the sum of $\mathrm{\Sigma}$ and the direct-process matrix, and its rank will be $N$ minus the number of coherent perfect absorber (CPA) states, $N-{N}_{\mathrm{CPA}}$, where the CPA states arise if the direct process exactly cancels a scattered wave, yielding perfect absorption [36,37]. (Technically, these may be partial-CPA states, exhibiting perfect cancellation of the direct fields only on the range of the $\mathcal{T}$ operator.) The density matrix ${\rho}_{\mathrm{in}}$ is a representation of the incoming excitations on the basis $\mathcal{V}$ of Eq. (1), and thus it cannot have rank greater than $N$ itself. By the relation ${\mathit{\rho}}_{\mathrm{out}}=\mathbf{S}{\mathit{\rho}}_{\mathrm{in}}{\mathbf{S}}^{\u2020}$ and the matrix-product inequality $\mathrm{rank}(AB)\le \mathrm{min}(\mathrm{rank}(A),\mathrm{rank}(B))$ (Ref. [33]), the rank of ${\mathit{\rho}}_{\mathrm{out}}$ must lie within bounds given by the rank of ${\mathit{\rho}}_{\mathrm{in}}$ minus the number of CPA states and the rank of ${\mathit{\rho}}_{\mathrm{in}}$ itself as follows:

*Metasurface design*: To probe the channel-concentration bounds, we consider control of diffraction orders through complex metasurfaces for potential applications such as augmented-reality optics [38,39] and photovoltaic concentrators [40–42]. Figure 2(a) depicts a designable gradient refractive-index profile with a period of $2\lambda $ and a thickness of $0.5\lambda $. (Such an element could be one unit cell within a larger, non-periodic metasurface [43–46].) For incoherent excitation of $N$ diffraction orders, Eq. (8) dictates that the maximum average efficiency of concentrating light into a single output order ($+1$) cannot be greater than $1/N$ [dashed lines in Fig. 2(c)]. For $s$-polarized light incoherently incident from orders 0 (red); $-1$, 0 (green); $-1$, 0, $+1$ (blue); and $-2$, $-1$, 0, $+1$ (purple) (20 deg angle of incidence for the zeroth order), we use adjoint-based “inverse design” [47–54] (Supplement 1) to discover optimal refractive-index profiles of the four metasurfaces shown in Fig. 2(b). (Broader angular control and binary refractive-index profiles could be generated through standard optimization augmentations [50,51], but here we emphasize the brightness-theorem consequences.) The transmission spectrum was computed by the Fourier modal method [55] with a freely available software package [56]. In Fig. 2(b), as the number of incoherent channels excited increases from 1 to 4, the average efficiency of the optimal structures decreases from 95.5% to 24.9%. (In Supplement 1 we show that optimal blazed gratings fall far short of the bounds.) We also probe the effects of partial coherence by varying the coherence between two input orders, per the density matrix in Fig. 2(a). By Eq. (7), maximum concentration is determined by the largest eigenvalue of ${\mathit{\rho}}_{\mathrm{in}}$, which is $1-c/2$, where $c$ is the coherence parameter. Figure 2(c) shows inverse-designed structures for $c=0.2,0.4,0.6,0.8,1$, with unique structures optimizing the response depending on the coherence of the excitation. All of the structures maximize efficiency in the incoherent $c=0$ case because the eigenvalues of the density matrix are degenerate, and thus transmission of any state is optimal.

*Étendue transmission*: An important related scenario to consider is one in which the direct (background) process is ignored, with the focus solely on interactions with scatterers. Instead of the input and output fields considered above, the relevant decomposition is instead into *incident* and *scattered* fields. Using the same terminology as in the input-output scattering operator, we can connect the incident and scattered fields by a $\mathcal{T}$ operator [26–29] ${\psi}_{\mathrm{scat}}=\mathcal{T}{\psi}_{\mathrm{inc}}$. Again, as shown in Refs. [17–21], the $\mathcal{T}$ operator is compact and defines incoming- and scattered-wave bases by its SVD, Eq. (1). Furthermore, to align with various applications described below, we will specify a set of ${N}_{\mathrm{trans}}\le \mathrm{rank}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathcal{U}$ desirable “transmission” channels that are a subset of the scattered-field channels defined by $\mathcal{U}$. To understand transmission flow into these channels, we will define our finite-dimensional $\mathbf{T}$ matrix as the restriction of $\mathcal{T}$ onto this subset of scattered-field channels. The matrix $\mathbf{T}$ connects the incoming-field channels to the transmission channels. The “transmission” terminology, partially meant to avoid further overload of the word “scattering,” is intended simply to represent the flow of energy through a system, enabled by interactions with a scatterer. For a planar or periodic scatterer, both reflected and transmitted waves would be part of this generalized “transmission” process, as long as they differ from the direct free-space process.

We define “étendue transmission” as the number of incoherent excitations that can successfully be transmitted through scatterer interactions onto the transmission channels. Equation (8) dictates that at least $N$ output channels are excited for $N$ orthogonal inputs, and indeed this result is proven in the incoherent case in Ref. [20] through an SVD of the $\mathbf{T}$ matrix (denoted therein by “$\mathbf{S}$”). If the number of transmission channels, ${N}_{\mathrm{trans}}$, is less than $\mathrm{rank}({\mathit{\rho}}_{\mathrm{inc}})$, where ${\mathit{\rho}}_{\mathrm{inc}}$ is the incident-wave density matrix, then the incoherent excitations cannot all be concentrated onto the transmission channels, and some power must necessarily be scattered into other scattering channels.

Resonance-assisted transmission, in which resonances couple the incident and transmission channels, introduces an additional constraint: the number of resonant modes (resonances) $M$ coupled to the relevant channels. Resonant modes are not scattering channels; instead, they are the quasi-normal modes (QNMs) of the scatterer, subject to outgoing boundary conditions. (Quasi-normal modes have been extensively studied and applied to various scattering systems for the last decade [57–59], and in the limit of closed systems and self-adjoint Maxwell operators they reduce to conventional guided and standing-wave modes [60].) We consider systems where the interaction with resonant modes can be described by temporal coupled mode theory (TCMT) [61–63], wherein the scattering process is encoded in an $M\times M$ matrix $\mathrm{\Omega}$, comprising the real and imaginary parts of the resonant-mode resonant frequencies, and a matrix $\mathbf{K}$, denoting channel–mode coupling. In TCMT, the $T$ matrix for the resonance-assisted transmission component is (Supplement 1) $\mathbf{T}=-i{\mathbf{K}}_{\mathrm{trans}}{(\mathrm{\Omega}-\omega )}^{-1}{\mathbf{K}}_{\mathrm{inc}}^{T}$, where ${\mathbf{K}}_{\mathrm{trans}}$ and ${\mathbf{K}}_{\mathrm{inc}}$ are the ${N}_{\mathrm{trans}}\times M$ and ${N}_{\mathrm{inc}}\times M$ submatrices of $\mathbf{K}$ denoting modal couplings to the transmission and incident channels, respectively.

The maximum (average) power flow into a single transmission output channel is subject to the bounds of Eqs. (7) and (8), now in terms of the density matrix ${\mathit{\rho}}_{\mathrm{inc}}$. The matrix ${\mathit{\rho}}_{\mathrm{trans}}$ equals $\mathbf{T}{\mathit{\rho}}_{\mathrm{inc}}{\mathbf{T}}^{\u2020}$. By recursive application of the matrix-rank inequality used above, we can see that

*minimum*of the numbers of incident inputs, resonant modes, and transmission channels. As depicted in Fig. 3, transmission channels and resonant modes act like apertures [35] in restricting the flow of étendue through a system.

We may also consider total transmission onto all ${N}_{\mathrm{trans}}$ transmission channels, i.e., $\sum _{i}\u27e8{|{\mathbf{c}}_{\mathrm{trans},i}|}^{2}\u27e9$. Since the transmission onto a single output is bounded above by ${\lambda}_{\mathrm{max}}({\mathit{\rho}}_{\mathrm{inc}})$, the total power is bounded above by the sum of the first $\mathrm{rank}({\mathit{\rho}}_{\mathrm{trans}})$ eigenvalues (Supplement 1) as follows:

We apply the resonance-assisted-transmission bounds to TCMT models of waveguide multiplexers as depicted in Fig. 4. There has been significant interest [52,64–68] in the design of compact junctions for routing light. In Figs. 4(a)–4(c), we consider “input” waveguides and “output” waveguides coupled by a resonant scattering system. For two input waveguides, we consider three scenarios: (a) one output and two resonances, (b) one resonance and two outputs, and (c) two resonances and two outputs. In each case, a highly controlled coherent excitation can, through appropriate design of the resonator, yield perfect transmission on resonance at the output port. But a *robust* design, impervious to noise or other incoherence, may be required, and such noise would introduce incoherence that is subject to the bound of Eq. (11). In each case, we optimize TCMT model parameters (cf. Supplement 1) to maximize transmission for all phase differences between two inputs. Device (c) maintains perfect transmission, whereas devices (a) and (b) are highly sensitive to noise as predicted by the restrictions to étendue flow.

The channel-concentration bounds and wave-étendue concept generalize classical ray-optical ideas to general wave scattering. In addition to the nanophotonic design problems considered here, there are numerous potential applications. First, they resolve how to incorporate polarization into ray-optical étendue, showing unequivocally that polarizing unpolarized light requires doubling classical étendue, an uncertain conjecture in display design [69,70]. Moreover, the natural incorporation of nonreciprocity into the bounds is of particular relevance given the emerging interest in nonreciprocal photonics [71–73] and acoustics [74], and it places constraints on many of these systems (extensions to time-modulated TCMT systems should be possible). Another additional application space is in random-scattering theory [25,75–77]. For opaque optical media comprising random scatterers, there is significant interest both in controlling the scattering channels (e.g., wavefront shaping) as well as studying the effects of partial coherence on scattering and absorption [78]. Our work shows that fully coherent excitations are optimal for maximal concentration on the fewest possible states. A related question that remains open is whether partial coherence might be optimal for total transmission. The framework developed herein may lead to fundamental limits to control in such systems.

## Funding

Air Force Office of Scientific Research (FA9550-17-1-0093).

## Acknowledgment

We thank Erik Shipton and Scott McEldowney for helpful discussions regarding étendue in display systems. H. Z. and O. D. M. were supported by the Air Force Office of Scientific Research.

See Supplement 1 for supporting content.

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