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

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References

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2019 (2)

2018 (3)

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]

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

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]

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]

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

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

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

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]

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]

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. 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]

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]

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]

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]

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

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

2013 (5)

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]

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]

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. 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]

2012 (5)

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

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

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)

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

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

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]

A. Jameson, L. Martinelli, and N. A. Pierce, “Optimum aerodynamic design using the Navier-Stokes equations,” Theor. Comput. Fluid Dyn. 10, 213–237 (1998).
[Crossref]

S. Withington and J. A. Murphy, “Modal analysis of partially coherent submillimeter-wave quasi-optical systems,” IEEE Trans. Anntenas Propag. 46, 1651–1659 (1998).
[Crossref]

1995 (1)

1993 (1)

1990 (1)

G. Smestad, H. Ries, R. Winston, and E. Yablonovitch, “The thermodynamic limits of light concentrators,” Sol. Energy Mater. 21, 99–111 (1990).
[Crossref]

1982 (1)

1981 (1)

1979 (1)

1978 (1)

A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. 14, 650–660 (1978).
[Crossref]

1968 (1)

1965 (1)

P. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[Crossref]

Aage, N.

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

Alonso, M. A.

Alù, A.

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

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

Andreassen, E.

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

Apresyan, L.

L. Apresyan and Y. A. Kravtsov, Radiation Transfer (Routledge, 2019).

Arbabi, A.

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]

Babinec, T. M.

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]

Baets, R.

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]

Bagheri, M.

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]

Baranov, D. G.

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]

Benitez, P. G.

R. Winston, J. C. Miñano, and P. G. Benitez, Nonimaging Optics (Elsevier, 2005).

Besbes, M.

Bhargava, S.

Bigourdan, F.

Boyd, R. W.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).

Boyd, S.

Brennesholtz, M. S.

M. S. Brennesholtz and E. H. Stupp, Projection Displays, 2nd ed. (Wiley, 2008).

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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,