Abstract

A cornerstone equation of optics – Snell's law – relates the angles of incidence and refraction for light passing through an interface between two media. It is built on two fundamental constraints: the conservation of tangential momentum and the conservation of energy. By relaxing the classical Snell’s law photon momentum conservation constraint when using space-gradient phase discontinuity, optical metasurfaces enabled an entirely new class of ultrathin optical devices. Here, we show that by eradicating the photon energy conservation constraint when introducing time-gradient phase discontinuity, we can further empower the area of flat photonics and obtain a new genus of optical devices. With this approach, classical Snell’s relations are developed into a more universal form not limited by Lorentz reciprocity, hence, meeting all the requirements for building magnetic-free optical isolators. Furthermore, photons experience inelastic interaction with time-gradient metasurfaces, which modifies photonic energy eigenstates and results in a Doppler-like wavelength shift. Consequently, metasurfaces with both space- and time-gradients can have a strong impact on a plethora of photonic applications and provide versatile control over the physical properties of light.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
  42. G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315(5815), 1120–1122 (2007).
    [Crossref] [PubMed]

2015 (1)

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9(6), 388–392 (2015).
[Crossref]

2014 (6)

A. Shaltout, J. Liu, V. M. Shalaev, and A. V. Kildishev, “Optically Active Metasurface with Non-Chiral Plasmonic Nanoantennas,” Nano Lett. 14(8), 4426–4431 (2014).
[Crossref] [PubMed]

Y. Yang, W. Wang, P. Moitra, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “Dielectric meta-reflectarray for broadband linear polarization conversion and optical vortex generation,” Nano Lett. 14(3), 1394–1399 (2014).
[Crossref] [PubMed]

C. Pfeiffer and A. Grbic, “Bianisotropic Metasurfaces for Optimal Polarization Control: Analysis and Synthesis,” Phys. Rev. Appl. 2(4), 044011 (2014).
[Crossref]

W. T. Chen, K.-Y. Yang, C.-M. Wang, Y.-W. Huang, G. Sun, I.-D. Chiang, C. Y. Liao, W.-L. Hsu, H. T. Lin, S. Sun, L. Zhou, A. Q. Liu, and D. P. Tsai, “High-efficiency broadband meta-hologram with polarization-controlled dual images,” Nano Lett. 14(1), 225–230 (2014).
[Crossref] [PubMed]

Y. Montelongo, J. O. Tenorio-Pearl, C. Williams, S. Zhang, W. I. Milne, and T. D. Wilkinson, “Plasmonic nanoparticle scattering for color holograms,” Proc. Natl. Acad. Sci. U.S.A. 111(35), 12679–12683 (2014).
[Crossref] [PubMed]

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(9), 701–705 (2014).
[Crossref]

2013 (11)

D. L. Sounas, C. Caloz, and A. Alù, “Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials,” Nat. Commun. 4, 2407 (2013).
[Crossref] [PubMed]

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4, 2807 (2013).
[Crossref]

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13(9), 4269–4274 (2013).
[Crossref] [PubMed]

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref] [PubMed]

A. Pors, M. G. Nielsen, R. L. Eriksen, and S. I. Bozhevolnyi, “Broadband Focusing Flat Mirrors Based on Plasmonic Gradient Metasurfaces,” Nano Lett. 13(2), 829–834 (2013).
[Crossref] [PubMed]

X. Ni, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin, planar, Babinet-inverted plasmonic metalenses,” Light Sci. Appl. 2(4), e72 (2013).
[Crossref]

A. Shaltout, V. Shalaev, and A. Kildishev, “Homogenization of bi-anisotropic metasurfaces,” Opt. Express 21(19), 21941–21950 (2013).
[Crossref] [PubMed]

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339(6126), 1405–1407 (2013).
[Crossref] [PubMed]

N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-optical metamaterial route to spin-controlled photonics,” Science 340(6133), 724–726 (2013).
[Crossref] [PubMed]

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, 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(8), 579–582 (2013).
[Crossref]

2012 (6)

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12(9), 4932–4936 (2012).
[Crossref] [PubMed]

X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C.-W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun. 3, 1198 (2012).
[Crossref] [PubMed]

S. Larouche, Y.-J. Tsai, T. Tyler, N. M. Jokerst, and D. R. Smith, “Infrared metamaterial phase holograms,” Nat. Mater. 11(5), 450–454 (2012).
[Crossref] [PubMed]

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12(12), 6328–6333 (2012).
[Crossref] [PubMed]

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109(3), 033901 (2012).
[Crossref] [PubMed]

2011 (4)

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

A. Pors, M. G. Nielsen, G. Della Valle, M. Willatzen, O. Albrektsen, and S. I. Bozhevolnyi, “Plasmonic metamaterial wave retarders in reflection by orthogonally oriented detuned electrical dipoles,” Opt. Lett. 36(9), 1626–1628 (2011).
[Crossref] [PubMed]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

N. Shitrit, I. Bretner, Y. Gorodetski, V. Kleiner, and E. Hasman, “Optical spin Hall effects in plasmonic chains,” Nano Lett. 11(5), 2038–2042 (2011).
[Crossref] [PubMed]

2010 (1)

A. R. Katko, S. Gu, J. P. Barrett, B.-I. Popa, G. Shvets, and S. A. Cummer, “Phase conjugation and negative refraction using nonlinear active metamaterials,” Phys. Rev. Lett. 105(12), 123905 (2010).
[Crossref] [PubMed]

2009 (4)

E. Plum, X.-X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
[Crossref] [PubMed]

E. Plum, J. Zhou, J. Dong, V. Fedotov, T. Koschny, C. Soukoulis, and N. Zheludev, “Metamaterial with negative index due to chirality,” Phys. Rev. B 79(3), 035407 (2009).
[Crossref]

S. L. Prosvirnin and N. I. Zheludev, “Analysis of polarization transformations by a planar chiral array of complex-shaped particles,” J. Opt. A, Pure Appl. Opt. 11(7), 074002 (2009).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3(2), 91–94 (2009).
[Crossref]

2008 (2)

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321(5893), 1172–1176 (2008).
[Crossref] [PubMed]

J. B. Pendry, “Time reversal and negative refraction,” Science 322(5898), 71–73 (2008).
[Crossref] [PubMed]

2007 (1)

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-field time reversal,” Science 315(5815), 1120–1122 (2007).
[Crossref] [PubMed]

2005 (1)

S. L. Prosvirnin and N. I. Zheludev, “Polarization effects in the diffraction of light by a planar chiral structure,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(33 Pt 2B), 037603 (2005).
[Crossref] [PubMed]

2004 (1)

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004).
[Crossref] [PubMed]

2003 (1)

E. Hasman, V. Kleiner, G. Biener, and A. Niv, “Polarization dependent focusing lens by use of quantized Pancharatnam–Berry phase diffractive optics,” Appl. Phys. Lett. 82(3), 328–330 (2003).
[Crossref]

2002 (1)

1988 (1)

V. Fedoseyev, “Conservation laws and transverse motion of energy on reflection and transmission of electromagnetic waves,” J. Phys. Math. Gen. 21(9), 2045–2059 (1988).
[Crossref]

Aieta, F.

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12(9), 4932–4936 (2012).
[Crossref] [PubMed]

N. Yu, F. Aieta, P. Genevet, M. A. Kats, Z. Gaburro, and F. Capasso, “A broadband, background-free quarter-wave plate based on plasmonic metasurfaces,” Nano Lett. 12(12), 6328–6333 (2012).
[Crossref] [PubMed]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Albrektsen, O.

Alegre, T. P.

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

Alù, A.

D. L. Sounas, C. Caloz, and A. Alù, “Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials,” Nat. Commun. 4, 2407 (2013).
[Crossref] [PubMed]

Antoniou, N.

J. Lin, P. Genevet, M. A. Kats, N. Antoniou, and F. Capasso, “Nanostructured holograms for broadband manipulation of vector beams,” Nano Lett. 13(9), 4269–4274 (2013).
[Crossref] [PubMed]

Aspelmeyer, M.

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

Baets, R.

D. Jalas, A. Petrov, M. Eich, W. Freude, S. Fan, Z. Yu, R. Baets, M. Popovic, 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(8), 579–582 (2013).
[Crossref]

S. Fan, R. Baets, A. Petrov, Z. Yu, J. D. Joannopoulos, W. Freude, A. Melloni, M. Popović, M. Vanwolleghem, D. Jalas, M. Eich, M. Krause, H. Renner, E. Brinkmeyer, and C. R. Doerr, “Comment on “Nonreciprocal light propagation in a silicon photonic circuit”,” Science 335(6064), 38 (2012).
[Crossref] [PubMed]

Bai, B.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

X. Chen, L. Huang, H. Mühlenbernd, G. Li, B. Bai, Q. Tan, G. Jin, C.-W. Qiu, S. Zhang, and T. Zentgraf, “Dual-polarity plasmonic metalens for visible light,” Nat. Commun. 3, 1198 (2012).
[Crossref] [PubMed]

Barrett, J. P.

A. R. Katko, S. Gu, J. P. Barrett, B.-I. Popa, G. Shvets, and S. A. Cummer, “Phase conjugation and negative refraction using nonlinear active metamaterials,” Phys. Rev. Lett. 105(12), 123905 (2010).
[Crossref] [PubMed]

Biener, G.

E. Hasman, V. Kleiner, G. Biener, and A. Niv, “Polarization dependent focusing lens by use of quantized Pancharatnam–Berry phase diffractive optics,” Appl. Phys. Lett. 82(3), 328–330 (2003).
[Crossref]

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H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109(3), 033901 (2012).
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X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4, 2807 (2013).
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Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9(6), 388–392 (2015).
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E. Plum, X.-X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
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H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109(3), 033901 (2012).
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Science (8)

X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339(6126), 1405–1407 (2013).
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N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-optical metamaterial route to spin-controlled photonics,” Science 340(6133), 724–726 (2013).
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N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
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Figures (4)

Fig. 1
Fig. 1 Schematic of a light beam incident on a space-time gradient metasurface with angle of incidence θ i , reflected beam with angle of reflection θ r , and transmitted (refracted) beam with angle of transmission (refraction) θ t .
Fig.2
Fig.2 (a) Light reflected from a space-gradient metasurface inducing tangential momentum discontinuity. The dashed red line represents the reflected beam without the metasurface effect. (b) Light reflected from a time-gradient metasurface inducing energy discontinuity and changing the isofrequency curve. The dashed red line represents the reflected beam without the metasurface effect. (c) Time-reversal test of a space-gradient metasurface. Red dashed line denotes reciprocal propagation of light. (d) Performance of a time-gradient metasurface in time-reversal with the dashed green and blue lines denoting the nonreciprocal traces of the incident and reflected beams respectively.
Fig. 3
Fig. 3 (a) Schematics of reflection angle from a time-gradient metasurface. (b) reflection angle in time-reversal.
Fig. 4
Fig. 4 (a,b) Schematics of an optical isolator with uni-directional light flow using a time-varying metasurface and two high quality resonators. (c,d) An optical isolator with same input/output frequency using two time-varying metasurfaces and a high quality resonator.

Equations (12)

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

E=a e iψ
ω= ψ t
k=ψ
ψ s = ψ i + ψ ms,s ,s={ r,t }.
ω s = ω i ψ ms,s / t ,s={ r,t };
k s,x = k i,x + ψ ms,s / x ,s={ r,t },
k r sin θ r = k i sin θ i + ψ ms,r x
k t sin θ t = k i sin θ i + ψ ms,t x
k s = n s ω s c = n s c ( ω i ψ ms,s t ),s={ r,t },
ksin θ 1 =( k+Δk )sin θ 2
( k+Δk )sin θ 2 =( k+2Δk )sin θ 3
sin θ 3 = sin θ 1 1+ 2Δk k = sin θ 1 1+ 2Δω ω

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