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 dA and solid angle dΩ, in a medium of refractive index n and tilted at an angle θ, has an étendue of n2cosθdAdΩ. Figure 1(a) depicts étendue conservation in ray-optical systems and the consequent trade-off between spatial (dA) and angular (dΩ) 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.

 

Fig. 1. (a) In ray optics, there is a trade-off in spatial and angular concentration of rays, by virtue of étendue conservation and the brightness theorem. (b) For general wave scattering, the scattering channels comprise the phase space. In ideal systems, the phase-space volumes are conserved: Aout=Ain in (a), and Nout=Nin in (b), where N denotes the number of excited channels (filled circles) or, more generally, the rank of the respective density matrix ρ.

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Extending radiometric concepts such as radiance into wave systems with coherence, beyond ray optics, has been the subject of considerable study [615]. 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 ψin are coupled to a set of output waves ψout (in domains that may be overlapping or disjoint) through a scattering operator S as follows: ψout=Sψin. 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. [1721]. As is well established in classical and quantum scattering theory [2225], the operator 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 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 T (as in “T matrix” approaches [2629]). A key insight of Refs. [1721] is that the 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

T=UΣV,
where U and V define orthonormal bases under an appropriate inner product ,, and the restriction to finite dimensions is possible by retaining only those singular vectors corresponding to nonzero singular values, i.e., “well-coupled channels” [21]. The direct-process operator D is not necessarily compact—for example, 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 V and 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 V and U define our scattering channels, within which our input and output waves can be decomposed, as follows:
ψin=Vcin,
ψout=Ucout,
where cin and cout are the vector coefficients of the excitations on these channels as shown in Fig. 1(b). The scattering matrix connects cin to cout and can be found by starting with the definition of the S operator, ψout=Ucout=Sψin=SVcin, and then taking the inner product with U to find
cout=U,SVScin=Scin.
We take our inner product to be a power normalization, such that cincin and coutcout represent the total incoming and outgoing powers, respectively.

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 ρin [32] that is the ensemble average (hereafter denoted by ·, over the source of incoherence) of the outer product of the incoming wave amplitudes, written as

ρin=cincin.
The incoherence of the outgoing channels is represented in the corresponding outgoing-wave density matrix
ρout=coutcout=SρinS.
The matrices ρin and ρout represent density operators projected onto the U and V bases. Both matrices are Hermitian and positive semidefinite.

For inputs defined by some ρin, how much power can flow into a single output channel, or more generally into a linear combination given by a unit vector u^? If we denote u^cout as cout,u^, then the power through u^ is |cout,u^|2=u^ρoutu^=u^SρinSu^. The quantity |cout,u^|2 is a quadratic form in ρin, such that its maximum value is dictated by its largest eigenvalue [33], λmax, leading to the inequality

|cout,u^|2λmax(ρin)(u^SSu^).
To bound the term in parentheses, u^SSu^, we consider coherent scattering for a new input: cin=Su^. For this input field, the incoming power is u^SSu^, while the outgoing power in the unit vector u^ is |u^cout|2=(u^SSu^)2. Enforcing the inequality that the outgoing power in u^ must be no larger than the (coherent) total incoming power, we immediately have the identity u^SSu^1. (We provide an alternative proof in Supplement 1.) Inserting into Eq. (6), we arrive at the bound
|cout,u^|2λmax(ρin).
Equation (7) is a key theoretical result of this paper. It states that for a system whose incoming power flow and coherence are described by a density matrix ρin, the maximum concentration of power is the largest eigenvalue of that density matrix. For a coherent input (akin to quantum-mechanical “pure states” [34]), there is a single nonzero eigenvalue, equal to 1, such that all of the power can be concentrated into a single channel. For equal incoherent excitation of N independent incoming states, the density matrix is diagonal with all nonzero eigenvalues equal to 1/N, in which case
|cout,u^|21N.
Equation (8) is less general than Eq. (7) but provides intuition and is a closer generalization of the ray-optical brightness theorem. Since the average output power per independent state must be less than or equal to 1/N, at least N independent outgoing states must be excited, or a commensurate amount of power must be lost to dissipation. In reciprocal systems, this bound follows from reversibility. In Supplement 1, we prove that Eq. (8) simplifies to the ray-optical brightness theorem for continuous plane-wave channels in homogeneous media.

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 ρin, 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 ρ, one can count independence by the matrix rank and define étendue = rank(ρ).

To understand the evolution of wave étendue through the scattering process, we reconsider the singular-value decomposition (SVD) of Eq. (1). The matrix Σ is a square matrix with dimensions N×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 Σ is full rank. The scattering matrix S is the sum of Σ and the direct-process matrix, and its rank will be N minus the number of coherent perfect absorber (CPA) states, NNCPA, 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 T operator.) The density matrix ρin is a representation of the incoming excitations on the basis V of Eq. (1), and thus it cannot have rank greater than N itself. By the relation ρout=SρinS and the matrix-product inequality rank(AB)min(rank(A),rank(B)) (Ref. [33]), the rank of ρout must lie within bounds given by the rank of ρin minus the number of CPA states and the rank of ρin itself as follows:

rank(ρin)NCPArank(ρout)rank(ρin).
Equation (9) defines the maximum diversity possible in the evolution of wave étendue in linear scattering systems. For lossless systems—or more generally any system without CPA states—we must have NCPA=0, in which case Eq. (9) is a conservation law stating that the density-matrix rank is always conserved. (In Supplement 1 we show that this simplifies to the classical wave-étendue conservation law in the ray-optics limit.) Figure 1(b) depicts this rank-defined (channel-counting) definition of wave étendue. In wave-scattering systems, phase space is defined by distinct scattering channels, without recourse to the position and momentum unique to free-space states.

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 [4042]. Figure 2(a) depicts a designable gradient refractive-index profile with a period of 2λ and a thickness of 0.5λ. (Such an element could be one unit cell within a larger, non-periodic metasurface [4346].) 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” [4754] (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 ρin, which is 1c/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.

 

Fig. 2. (a) Periodic metasurface element to be designed for maximal power in the +1 transmission diffraction order (yellow). We consider incoherent excitations among the four incident orders, with a diagonal density matrix, as well as partially coherent excitations between the 0 and 1 order, represented by an off-diagonal term with coherence parameter c. Inverse-designed metasurfaces closely approaching the coherence- and channel-dependent bounds are shown in (b) for incoherent excitations among up to four channels, and in (c) for partially coherent excitations between two channels. [Designs in (c) are all optimal for the fully incoherent case because ρin is a constant multiple of the identity matrix. This should not be considered a generic phenomenon when excitation powers are unevenly distributed.]

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Fig. 3. Étendue, defined as the rank of wave-scattering density matrices, is restricted in resonance-assisted transmission processes by the number of transmission channels and channel-coupled resonances in the process.

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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 T operator [2629] ψscat=Tψinc. Again, as shown in Refs. [1721], the 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 NtransrankU desirable “transmission” channels that are a subset of the scattered-field channels defined by U. To understand transmission flow into these channels, we will define our finite-dimensional T matrix as the restriction of T onto this subset of scattered-field channels. The matrix 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 T matrix (denoted therein by “S”). If the number of transmission channels, Ntrans, is less than rank(ρinc), where ρinc 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 [5759], 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) [6163], wherein the scattering process is encoded in an M×M matrix Ω, comprising the real and imaginary parts of the resonant-mode resonant frequencies, and a matrix K, denoting channel–mode coupling. In TCMT, the T matrix for the resonance-assisted transmission component is (Supplement 1) T=iKtrans(Ωω)1KincT, where Ktrans and Kinc are the Ntrans×M and Ninc×M submatrices of 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 ρinc. The matrix ρtrans equals TρincT. By recursive application of the matrix-rank inequality used above, we can see that

rank(ρtrans)min(rank(ρinc),M,Ntrans).
The number of orthogonal outputs is less than or equal to the 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 Ntrans transmission channels, i.e., i|ctrans,i|2. Since the transmission onto a single output is bounded above by λmax(ρinc), the total power is bounded above by the sum of the first rank(ρtrans) eigenvalues (Supplement 1) as follows:

i|ctrans,i|2i=1min(rank(ρinc),M,Ntrans)λi,
where the eigenvalues are indexed in descending order. For incoherent excitation of the Ninc channels, λi(ρinc)=1/Ninc for all i, and the term on the right of Eq. (11) simplifies to min(Ninc,M,Ntrans)/Ninc. In resonance-assisted transmission scenarios, Eq. (11) represents an additional constraint on power flow: in addition to the number of output channels, total power flow is further constrained by the number of distinct resonant modes that interact with them. Étendue restrictions anywhere in the transmission process necessarily generate scattering to channels other than the desired transmission channels.

We apply the resonance-assisted-transmission bounds to TCMT models of waveguide multiplexers as depicted in Fig. 4. There has been significant interest [52,6468] 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.

 

Fig. 4. Robustness of waveguide junctions is susceptible to étendue restrictions. For two input channels, we consider (a) one output, (b) one mode, and (c) no restrictions. (e)–(g) Transmission for (a)–(c) with input phase angles in θ=[0,π/2]. (d) Transmission as a function of phase, on resonance. Case (c) is designed to be almost perfectly insensitive to phase; such designs are impossible in cases (a) and (b).

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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 [7173] 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 [25,7577]. 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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54. N. Aage, E. Andreassen, B. S. Lazarov, and O. Sigmund, “Giga-voxel computational morphogenesis for structural design,” Nature 550, 84–86 (2017). [CrossRef]  

55. M. Moharam and T. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981). [CrossRef]  

56. V. Liu and S. Fan, “S4: a free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183, 2233–2244 (2012). [CrossRef]  

57. C. Sauvan, J.-P. Hugonin, I. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013). [CrossRef]  

58. P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018). [CrossRef]  

59. P. Lalanne, W. Yan, A. Gras, C. Sauvan, J.-P. Hugonin, M. Besbes, G. Demésy, M. Truong, B. Gralak, F. Zolla, and A. Nicolet, “Quasinormal mode solvers for resonators with dispersive materials,” J. Opt. Soc. Am. A 36, 686–704 (2019). [CrossRef]  

60. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

61. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003). [CrossRef]  

62. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

63. W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004). [CrossRef]  

64. Y. Ding, J. Xu, F. Da Ros, B. Huang, H. Ou, and C. Peucheret, “On-chip two-mode division multiplexing using tapered directional coupler-based mode multiplexer and demultiplexer,” Opt. Express 21, 10376–10382 (2013). [CrossRef]  

65. R. Ji, L. Yang, L. Zhang, Y. Tian, J. Ding, H. Chen, Y. Lu, P. Zhou, and W. Zhu, “Five-port optical router for photonic networks-on-chip,” Opt. Express 19, 20258–20268 (2011). [CrossRef]  

66. A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vucković, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9, 374–377 (2015). [CrossRef]  

67. B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4  μm2 footprint,” Nat. Photonics 9, 378–382 (2015). [CrossRef]  

68. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998). [CrossRef]  

69. S. A. Lerner and B. Dahlgrenn, “Etendue and optical system design,” Proc. SPIE 6338, 633801 (2006). [CrossRef]  

70. F. Fournier and J. Rolland, “Design methodology for high brightness projectors,” J. Display Technol. 4, 86–91 (2008). [CrossRef]  

71. K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics 6, 782–787 (2012). [CrossRef]  

72. L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8, 701–705 (2014). [CrossRef]  

73. D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11, 774–783 (2017). [CrossRef]  

74. S. A. Cummer, J. Christensen, and A. Alù, “Controlling sound with acoustic metamaterials,” Nat. Rev. Mater. 1, 16001 (2016). [CrossRef]  

75. A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012). [CrossRef]  

76. C. W. Hsu, S. F. Liew, A. Goetschy, H. Cao, and A. D. Stone, “Correlation-enhanced control of wave focusing in disordered media,” Nat. Phys. 13, 497–502 (2017). [CrossRef]  

77. C. W. Hsu, A. Goetschy, Y. Bromberg, A. D. Stone, and H. Cao, “Broadband coherent enhancement of transmission and absorption in disordered media,” Phys. Rev. Lett. 115, 223901 (2015). [CrossRef]  

78. F. Bigourdan, R. Pierrat, and R. Carminati, “Enhanced absorption of waves in stealth hyperuniform disordered media,” Opt. Express 27, 8666–8682 (2019). [CrossRef]  

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    [Crossref]
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    [Crossref]
  77. C. W. Hsu, A. Goetschy, Y. Bromberg, A. D. Stone, and H. Cao, “Broadband coherent enhancement of transmission and absorption in disordered media,” Phys. Rev. Lett. 115, 223901 (2015).
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    [Crossref]

2019 (2)

2018 (3)

Z. Lin, B. Groever, F. Capasso, A. W. Rodriguez, and M. Lončar, “Topology-optimized multilayered metaoptics,” Phys. Rev. Appl. 9, 044030 (2018).
[Crossref]

M. A. Shameli and L. Yousefi, “Absorption enhancement in thin-film solar cells using an integrated metasurface lens,” J. Opt. Soc. Am. B 35, 223–230 (2018).
[Crossref]

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, and J.-P. Hugonin, “Light interaction with photonic and plasmonic resonances,” Laser Photon. Rev. 12, 1700113 (2018).
[Crossref]

2017 (7)

N. Aage, E. Andreassen, B. S. Lazarov, and O. Sigmund, “Giga-voxel computational morphogenesis for structural design,” Nature 550, 84–86 (2017).
[Crossref]

C. W. Hsu, S. F. Liew, A. Goetschy, H. Cao, and A. D. Stone, “Correlation-enhanced control of wave focusing in disordered media,” Nat. Phys. 13, 497–502 (2017).
[Crossref]

S. Rotter and S. Gigan, “Light fields in complex media: mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

D. A. Miller, L. Zhu, and S. Fan, “Universal modal radiation laws for all thermal emitters,” Proc. Natl. Acad. Sci. USA 114, 4336–4341 (2017).
[Crossref]

D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11, 774–783 (2017).
[Crossref]

M. Khorasaninejad and F. Capasso, “Metalenses: versatile multifunctional photonic components,” Science 358, eaam8100 (2017).
[Crossref]

D. G. Baranov, A. Krasnok, T. Shegai, A. Alù, and Y. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2, 17064 (2017).
[Crossref]

2016 (1)

S. A. Cummer, J. Christensen, and A. Alù, “Controlling sound with acoustic metamaterials,” Nat. Rev. Mater. 1, 16001 (2016).
[Crossref]

2015 (5)

C. W. Hsu, A. Goetschy, Y. Bromberg, A. D. Stone, and H. Cao, “Broadband coherent enhancement of transmission and absorption in disordered media,” Phys. Rev. Lett. 115, 223901 (2015).
[Crossref]

A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10, 937–943 (2015).
[Crossref]

B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4  μm2 footprint,” Nat. Photonics 9, 378–382 (2015).
[Crossref]

A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vucković, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9, 374–377 (2015).
[Crossref]

J. S. Price, X. Sheng, B. M. Meulblok, J. A. Rogers, and N. C. Giebink, “Wide-angle planar microtracking for quasi-static microcell concentrating photovoltaics,” Nat. Commun. 6, 6223 (2015).
[Crossref]

2014 (4)

L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8, 701–705 (2014).
[Crossref]

D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345, 298–302 (2014).
[Crossref]

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13, 139–150 (2014).
[Crossref]

V. Ganapati, O. D. Miller, and E. Yablonovitch, “Light trapping textures designed by electromagnetic optimization for subwavelength thick solar cells,” IEEE J. Photovoltaics 4, 175–182 (2014).
[Crossref]

2013 (5)

Y. Ding, J. Xu, F. Da Ros, B. Huang, H. Ou, and C. Peucheret, “On-chip two-mode division multiplexing using tapered directional coupler-based mode multiplexer and demultiplexer,” Opt. Express 21, 10376–10382 (2013).
[Crossref]

C. Sauvan, J.-P. Hugonin, I. Maksymov, and P. Lalanne, “Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators,” Phys. Rev. Lett. 110, 237401 (2013).
[Crossref]

C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Opt. Express 21, 21693–21701 (2013).
[Crossref]

M. Mishchenko and P. Martin, “Peter waterman and t-matrix methods,” J. Quantum Spectrosc. Radiat. Transfer 123, 2–7 (2013).
[Crossref]

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popović, A. Melloni, J. D. Joannopoulos, M. Vanwolleghem, C. R. Doerr, and H. Renner, “What is—and what is not—an optical isolator,” Nat. Photonics 7, 579–582 (2013).
[Crossref]

2012 (5)

V. Liu and S. Fan, “S4: a free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183, 2233–2244 (2012).
[Crossref]

D. A. Miller, “All linear optical devices are mode converters,” Opt. Express 20, 23985–23993 (2012).
[Crossref]

K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics 6, 782–787 (2012).
[Crossref]

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012).
[Crossref]

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics 6, 474–479 (2012).
[Crossref]

2011 (4)

2010 (1)

Y. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
[Crossref]

2008 (1)

2007 (1)

2006 (3)

O. Cakmakci and J. Rolland, “Head-worn displays: a review,” J. Disp. Technol. 2, 199–216 (2006).
[Crossref]

S. A. Lerner and B. Dahlgrenn, “Etendue and optical system design,” Proc. SPIE 6338, 633801 (2006).
[Crossref]

T. Levola, “Diffractive optics for virtual reality displays,” J. Soc. Inf. Disp. 14, 467–475 (2006).
[Crossref]

2004 (1)

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[Crossref]

2003 (2)

O. Sigmund and J. Søndergaard Jensen, “Systematic design of phononic band-gap materials and structures by topology optimization,” Philos. Trans. R. Soc. London, Ser. A 361, 1001–1019 (2003).
[Crossref]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[Crossref]

2001 (2)

2000 (1)

1998 (3)

A. Jameson, L. Martinelli, and N. A. Pierce, “Optimum aerodynamic design using the Navier-Stokes equations,” Theor. Comput. Fluid Dyn. 10, 213–237 (1998).
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A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vucković, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9, 374–377 (2015).
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J. S. Price, X. Sheng, B. M. Meulblok, J. A. Rogers, and N. C. Giebink, “Wide-angle planar microtracking for quasi-static microcell concentrating photovoltaics,” Nat. Commun. 6, 6223 (2015).
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Shegai, T.

D. G. Baranov, A. Krasnok, T. Shegai, A. Alù, and Y. Chong, “Coherent perfect absorbers: linear control of light with light,” Nat. Rev. Mater. 2, 17064 (2017).
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N. Aage, E. Andreassen, B. S. Lazarov, and O. Sigmund, “Giga-voxel computational morphogenesis for structural design,” Nature 550, 84–86 (2017).
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J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).
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S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. Haus, “Channel drop tunneling through localized states,” Phys. Rev. Lett. 80, 960–963 (1998).
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Y. Chong, L. Ge, H. Cao, and A. D. Stone, “Coherent perfect absorbers: time-reversed lasers,” Phys. Rev. Lett. 105, 053901 (2010).
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R. Winston, J. C. Miñano, and P. G. Benitez, Nonimaging Optics (Elsevier, 2005).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. (a) In ray optics, there is a trade-off in spatial and angular concentration of rays, by virtue of étendue conservation and the brightness theorem. (b) For general wave scattering, the scattering channels comprise the phase space. In ideal systems, the phase-space volumes are conserved: Aout=Ain in (a), and Nout=Nin in (b), where N denotes the number of excited channels (filled circles) or, more generally, the rank of the respective density matrix ρ.
Fig. 2.
Fig. 2. (a) Periodic metasurface element to be designed for maximal power in the +1 transmission diffraction order (yellow). We consider incoherent excitations among the four incident orders, with a diagonal density matrix, as well as partially coherent excitations between the 0 and 1 order, represented by an off-diagonal term with coherence parameter c. Inverse-designed metasurfaces closely approaching the coherence- and channel-dependent bounds are shown in (b) for incoherent excitations among up to four channels, and in (c) for partially coherent excitations between two channels. [Designs in (c) are all optimal for the fully incoherent case because ρin is a constant multiple of the identity matrix. This should not be considered a generic phenomenon when excitation powers are unevenly distributed.]
Fig. 3.
Fig. 3. Étendue, defined as the rank of wave-scattering density matrices, is restricted in resonance-assisted transmission processes by the number of transmission channels and channel-coupled resonances in the process.
Fig. 4.
Fig. 4. Robustness of waveguide junctions is susceptible to étendue restrictions. For two input channels, we consider (a) one output, (b) one mode, and (c) no restrictions. (e)–(g) Transmission for (a)–(c) with input phase angles in θ=[0,π/2]. (d) Transmission as a function of phase, on resonance. Case (c) is designed to be almost perfectly insensitive to phase; such designs are impossible in cases (a) and (b).

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

T=UΣV,
ψin=Vcin,
ψout=Ucout,
cout=U,SVScin=Scin.
ρin=cincin.
ρout=coutcout=SρinS.
|cout,u^|2λmax(ρin)(u^SSu^).
|cout,u^|2λmax(ρin).
|cout,u^|21N.
rank(ρin)NCPArank(ρout)rank(ρin).
rank(ρtrans)min(rank(ρinc),M,Ntrans).
i|ctrans,i|2i=1min(rank(ρinc),M,Ntrans)λi,

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