Abstract

The time–bandwidth limit inherently relates the lifetime of a resonance and its spectral bandwidth, with direct implications for the maximum storage time of a pulse versus its frequency content. Recently, it has been argued that nonreciprocal cavities may overcome this constraint by breaking the strict equality of their incoupling and outcoupling coefficients. Here, we study the implications of nonreciprocity on resonant linear, time-invariant cavities and derive general relations, stemming from microscopic reversibility, that govern their dynamics. We show that nonreciprocal cavities do not provide specific advantages in terms of the time–bandwidth limit, but enable other attractive properties for nanophotonic systems.

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

Full Article  |  PDF Article
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References

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  3. M. Ayata, Y. Fedoryshyn, W. Heni, B. Baueuerle, A. Josten, M. Zahner, U. Koch, Y. Salamin, C. Hoessbacher, C. Haffner, D. L. Elder, L. R. Dalton, and J. Leuthold, “High-speed plasmonic modulator in a single metal layer,” Science 358, 630–632 (2017).
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  48. A. D. Bresler, “On the TEn0 modes of a ferrite slab loaded rectangular waveguide and the associated thermodynamic paradox,” IEEE Trans. Microw. Theory Tech. 8, 81–95 (1960).
    [Crossref]
  49. H. Seidel, “Ferrite slabs in transverse electric mode wave guide,” J. Appl. Phys. 28, 218–226 (1957).
    [Crossref]
  50. M. Kales, “Topics in guided-wave propagation in magnetized ferrites,” Proc. IRE 44, 1403–1409 (1956).
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    [Crossref]
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    [Crossref]

2018 (6)

M. Minkov and S. Fan, “Localization and time-reversal of light through dynamic modulation,” Phys. Rev. B 97, 60301 (2018).
[Crossref]

D. L. Sounas, J. Soric, and A. Alù, “Broadband passive isolators based on coupled nonlinear resonances,” Nat. Electron. 1, 113–119 (2018).
[Crossref]

M. Lawrence, D. R. Barton, and J. A. Dionne, “Nonreciprocal flat optics with silicon metasurfaces,” Nano Lett. 18, 1104–1109 (2018).
[Crossref]

D. L. Sounas and A. Alù, “Fundamental bounds on the operation of Fano nonlinear isolators,” Phys. Rev. B 97, 115431 (2018).
[Crossref]

L. Zhu, Y. Guo, and S. Fan, “Theory of many-body radiative heat transfer without the constraint of reciprocity,” Phys. Rev. B 97, 094302 (2018).
[Crossref]

M. Tsang, “Quantum limits on the time–bandwidth product of an optical resonator,” Opt. Lett. 43, 150–153 (2018).
[Crossref]

2017 (4)

K. L. Tsakmakidis, L. Shen, S. A. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. W. Boyd, “Breaking Lorentz reciprocity to overcome the time–bandwidth limit in physics and engineering,” Science 356, 1260–1264 (2017).
[Crossref]

M. Marvasti and B. Rejaei, “Formation of hotspots in partially filled ferrite-loaded rectangular waveguides,” J. Appl. Phys. 122, 233901 (2017).
[Crossref]

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

M. Ayata, Y. Fedoryshyn, W. Heni, B. Baueuerle, A. Josten, M. Zahner, U. Koch, Y. Salamin, C. Hoessbacher, C. Haffner, D. L. Elder, L. R. Dalton, and J. Leuthold, “High-speed plasmonic modulator in a single metal layer,” Science 358, 630–632 (2017).
[Crossref]

2016 (2)

K. Liu, A. Torki, and S. He, “One-way surface magnetoplasmon cavity and its application for nonreciprocal devices,” Opt. Lett. 41, 800–803 (2016).
[Crossref]

L. Zhu and S. Fan, “Persistent directional current at equilibrium in nonreciprocal many-body near field electromagnetic heat transfer,” Phys. Rev. Lett. 117, 134303 (2016).
[Crossref]

2015 (5)

L. Shen, Y. You, Z. Wang, and X. Deng, “Backscattering-immune one-way surface magnetoplasmons at terahertz frequencies,” Opt. Express 23, 950–962 (2015).
[Crossref]

L. Shen, X. Zheng, and X. Deng, “Stopping terahertz radiation without backscattering over a broad band,” Opt. Express 23, 11790–11798 (2015).
[Crossref]

C. T. Phare, Y.-H. Daniel Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30  GHz bandwidth,” Nat. Photonics 9, 511–514 (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]

S. Lannebère and M. G. Silveirinha, “Optical meta-atom for localization of light with quantized energy,” Nat. Commun. 6, 8766 (2015).
[Crossref]

2014 (2)

D. L. Sounas and A. Alù, “Angular-momentum-biased nanorings to realize magnetic-free integrated optical isolation,” ACS Photon. 1, 198–204 (2014).
[Crossref]

U. K. Chettiar, A. R. Davoyan, and N. Engheta, “Hotspots from nonreciprocal surface waves,” Opt. Lett. 39, 1760–1763 (2014).
[Crossref]

2013 (4)

Y. Hadad and B. Z. Steinberg, “One-way optical waveguides for matched non-reciprocal nanoantennas with dynamic beam scanning functionality,” Opt. Express 21, A77–A83 (2013).
[Crossref]

R. R. Grote, J. B. Driscoll, and R. M. Osgood, “Integrated optical modulators and switches using coherent perfect loss,” Opt. Lett. 38, 3001–3004 (2013).
[Crossref]

A. R. Davoyan and N. Engheta, “Theory of wave propagation in magnetized near-zero-epsilon metamaterials: evidence for one-way photonic states and magnetically switched transparency and opacity,” Phys. Rev. Lett. 111, 257401 (2013).
[Crossref]

N. M. Estakhri and A. Alù, “Physics of unbounded, broadband absorption/gain efficiency in plasmonic nanoparticles,” Phys. Rev. B 87, 205418 (2013).
[Crossref]

2012 (3)

O. Luukkonen, U. K. Chettiar, and N. Engheta, “One-way waveguides connected to one-way loads,” IEEE Antennas Wireless Propag. Lett. 11, 1398–1401 (2012).
[Crossref]

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. W. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photon. Rev. 6, 145–177 (2012).
[Crossref]

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, 033901 (2012).
[Crossref]

2011 (1)

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5, 758–762 (2011).
[Crossref]

2010 (1)

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4, 518–526 (2010).
[Crossref]

2008 (3)

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[Crossref]

E. Verhagen, A. Polman, and L. Kuipers, “Nanofocusing in laterally tapered plasmonic waveguides,” Opt. Express 16, 45–57 (2008).
[Crossref]

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[Crossref]

2007 (4)

M. G. Silveirinha and N. Engheta, “Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using near-zero metamaterials,” Phys. Rev. B 76, 245109 (2007).
[Crossref]

W. M. Green, M. J. Rooks, L. Sekaric, and Y. A. Vlasov, “Ultra-compact, low RF power, 10  Gb/s silicon Mach–Zehnder modulator,” Opt. Express 15, 17106 (2007).
[Crossref]

D. A. B. Miller, “Fundamental limit to linear one-dimensional slow light structures,” Phys. Rev. Lett. 99, 203903 (2007).
[Crossref]

Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nat. Phys. 3, 406–410 (2007).
[Crossref]

2006 (1)

D. F. Pile and D. K. Gramotnev, “Adiabatic and nonadiabatic nanofocusing of plasmons by tapered gap plasmon waveguides,” Appl. Phys. Lett. 89, 2004–2007 (2006).
[Crossref]

2005 (2)

M. Gerken and D. A. B. Miller, “Limits to the performance of dispersive thin-film stacks,” Appl. Opt. 44, 3349–3357 (2005).
[Crossref]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[Crossref]

2004 (3)

M. F. Yanik and S. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004).
[Crossref]

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]

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004).
[Crossref]

1999 (2)

C. Manatolou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add–drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[Crossref]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
[Crossref]

1991 (1)

P. R. McIsaac, “Mode orthogonality in reciprocal and nonreciprocal waveguides,” IEEE Trans. Microw. Theory Tech. 39, 1808–1816 (1991).
[Crossref]

1987 (1)

R. E. Camley, “Nonreciprocal surface waves,” Surf. Sci. Rep. 7, 103–187 (1987).
[Crossref]

1972 (1)

J. J. Brion, R. F. Wallis, A. Hartstein, and E. Burstein, “Theory of surface magnetoplasmons in semiconductors,” Phys. Rev. Lett. 28, 1455–1458 (1972).
[Crossref]

1960 (1)

A. D. Bresler, “On the TEn0 modes of a ferrite slab loaded rectangular waveguide and the associated thermodynamic paradox,” IEEE Trans. Microw. Theory Tech. 8, 81–95 (1960).
[Crossref]

1959 (1)

H. Gamo, “On passive one-way systems,” IRE Trans. Circuit Theory 6, 283–298 (1959).
[Crossref]

1957 (1)

H. Seidel, “Ferrite slabs in transverse electric mode wave guide,” J. Appl. Phys. 28, 218–226 (1957).
[Crossref]

1956 (1)

M. Kales, “Topics in guided-wave propagation in magnetized ferrites,” Proc. IRE 44, 1403–1409 (1956).
[Crossref]

1955 (2)

B. Lax and K. J. Button, “New ferrite mode configurations and their applications,” J. Appl. Phys. 26, 1186–1187 (1955).
[Crossref]

H. Carlin, “On the physical realizability of linear non-reciprocal networks,” Proc. IRE 43, 608–616 (1955).
[Crossref]

1945 (1)

H. B. G. Casimir, “On Onsager’s principle of microscopic reversibility,” Rev. Mod. Phys. 17, 343–350 (1945).
[Crossref]

Altug, H.

K. L. Tsakmakidis, L. Shen, S. A. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. W. Boyd, “Breaking Lorentz reciprocity to overcome the time–bandwidth limit in physics and engineering,” Science 356, 1260–1264 (2017).
[Crossref]

Alù, A.

D. L. Sounas and A. Alù, “Fundamental bounds on the operation of Fano nonlinear isolators,” Phys. Rev. B 97, 115431 (2018).
[Crossref]

D. L. Sounas, J. Soric, and A. Alù, “Broadband passive isolators based on coupled nonlinear resonances,” Nat. Electron. 1, 113–119 (2018).
[Crossref]

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

D. L. Sounas and A. Alù, “Angular-momentum-biased nanorings to realize magnetic-free integrated optical isolation,” ACS Photon. 1, 198–204 (2014).
[Crossref]

N. M. Estakhri and A. Alù, “Physics of unbounded, broadband absorption/gain efficiency in plasmonic nanoparticles,” Phys. Rev. B 87, 205418 (2013).
[Crossref]

Ayata, M.

M. Ayata, Y. Fedoryshyn, W. Heni, B. Baueuerle, A. Josten, M. Zahner, U. Koch, Y. Salamin, C. Hoessbacher, C. Haffner, D. L. Elder, L. R. Dalton, and J. Leuthold, “High-speed plasmonic modulator in a single metal layer,” Science 358, 630–632 (2017).
[Crossref]

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2, 465–473 (2008).
[Crossref]

Barton, D. R.

M. Lawrence, D. R. Barton, and J. A. Dionne, “Nonreciprocal flat optics with silicon metasurfaces,” Nano Lett. 18, 1104–1109 (2018).
[Crossref]

Baueuerle, B.

M. Ayata, Y. Fedoryshyn, W. Heni, B. Baueuerle, A. Josten, M. Zahner, U. Koch, Y. Salamin, C. Hoessbacher, C. Haffner, D. L. Elder, L. R. Dalton, and J. Leuthold, “High-speed plasmonic modulator in a single metal layer,” Science 358, 630–632 (2017).
[Crossref]

Bi, L.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5, 758–762 (2011).
[Crossref]

Boyd, R. W.

K. L. Tsakmakidis, L. Shen, S. A. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. W. Boyd, “Breaking Lorentz reciprocity to overcome the time–bandwidth limit in physics and engineering,” Science 356, 1260–1264 (2017).
[Crossref]

Bresler, A. D.

A. D. Bresler, “On the TEn0 modes of a ferrite slab loaded rectangular waveguide and the associated thermodynamic paradox,” IEEE Trans. Microw. Theory Tech. 8, 81–95 (1960).
[Crossref]

Brion, J. J.

J. J. Brion, R. F. Wallis, A. Hartstein, and E. Burstein, “Theory of surface magnetoplasmons in semiconductors,” Phys. Rev. Lett. 28, 1455–1458 (1972).
[Crossref]

Burstein, E.

J. J. Brion, R. F. Wallis, A. Hartstein, and E. Burstein, “Theory of surface magnetoplasmons in semiconductors,” Phys. Rev. Lett. 28, 1455–1458 (1972).
[Crossref]

Button, K. J.

B. Lax and K. J. Button, “New ferrite mode configurations and their applications,” J. Appl. Phys. 26, 1186–1187 (1955).
[Crossref]

Cai, H.

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It is important to note a conceptual difference in coupled-mode theory when dealing with nonreciprocal systems: if ki corresponds to the coupling coefficient of a given forward mode to the resonance, then usually di is the coupling coefficient to the backward version of the same mode. However, in nonreciprocal systems, a subtler definition of these coefficients is required, as the waveguide might be unidirectional or have largely different propagation properties in the two directions. Here, we assume (without loss of generality) that the port supports at least one forward or backward mode, and at most both. If there is no forward or backward mode, then we set ki=0 or di=0, respectively.

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Supplementary Material (1)

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» Supplement 1       Supplementary derivations

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Figures (4)

Fig. 1.
Fig. 1. (a) In reciprocal systems, the rates at which energy is transferred into and out of a cavity (shown as ρ in and ρ out , respectively) are equal. Reference [22] suggested that this may not be necessarily the case in nonreciprocal systems, enabling nonreciprocal cavities to beat the time–bandwidth limit by incoupling light over a large bandwidth (proposed to be proportional to ρ in ), while decaying slowly (proportional to ρ out ). (b) A visual legend for the coupled-mode theory coefficients and amplitudes that appear throughout the equations in this work. In this schematic, we consider a cavity coupled to at least two ports, i and j , with a single incident wave from port i .
Fig. 2.
Fig. 2. (a) Schematic of the waveguide/cavity geometry under consideration (see Supplement 1 for additional details). (b) Fourier transform of the electric field inside the cavity at the center and outside of the cavity at the termination [red and blue crosses in Fig. 1(a), respectively], with the unidirectional bandwidth of the waveguide shown in gray. Inset: ring down of the electric field component E x inside the cavity, in perfect agreement with the line width obtained from the spectrum. (c) Electric field intensity in and near the cavity at 1.52 THz when excited from inside with a magnetic line source I m , highlighting the dissipative mechanism at the Si/InSb/PEC corner. (d, e) Electric field intensity in and near the cavity at (d) 1.52 THz and (e) 1.4 THz when excited from the port. In these cases, the hotspot is also clearly visible.
Fig. 3.
Fig. 3. (a) Bounds on | k i | as a function of | C i i | for one of the cavities discussed in the main text, which has | d i | 2 / d d = 0.53 . (b) Bounds in the cases that | d i | 2 / d d = 0.1 and | d i | 2 / d d = 0.9 .
Fig. 4.
Fig. 4. (a) For the proper reflection amplitude, there are positions in a nonreciprocal waveguide with perfect destructive interference in the magnetic field, as visible in the magnetic field intensity shown here (black is low and white is high field intensity). (b) The required direct reflection coefficient C = 0.4 is determined by the field ratio of the forward and backward power-normalized mode profiles. (c) A color plot of Re ( H z ) , demonstrating that a cavity with its opening aperture (indicated with a circle) at the position of destructive interference cannot be excited. (d) However, due to the nonreciprocal nature of the system, if the resonance is excited from inside the cavity, it does decay into the port. (e, f) By reversing the direction of the biasing magnetic field, the cavity can be excited from the port but not decay into it. The color scale is clipped at 1/25th of the scale in (c) and (d) to enhance the visibility of the fields in the waveguide.

Equations (11)

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d d t a = ( i ω 0 γ r γ i ) a + k T s + ,
s = Cs + + d a .
2 γ r = d d .
a ( ω ) = k T s + i ( ω 0 ω ) γ r γ i .
C ˜ = C T ,
d ˜ = k ,
k ˜ = d ,
γ ˜ r = γ r ,
C T d * = k ,
d d = k k .
| | C i i d i | 1 | C i i | 2 | d j | | | k i | | C i i d i | + 1 | C i i | 2 | d j | ,

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