Abstract

In this paper we achieve non-reciprocity in a silicon optical ring resonator, by introducing two small time-modulated perturbations into the ring. Isolators are designed using this time-perturbed ring, side-coupled to waveguides. The underlying operation of the time-modulated ring and isolator is analyzed using Temporal Coupled Mode Theory (TCMT). The TCMT is used to find the angular distance, phase difference and thickness of the two time-modulated points on the ring resonator and also to find and justify the optimum values for the modulation frequency and amplitude, which yields maximum isolation in the isolator arrangements. Insight into the major players that determine isolation are also presented, with the aid of TCMT. Our proposed structure is much simpler to implement compared to other ring-based optical isolators, as it does not require spatio-temporal modulation, or large regions with modulation, but only two point perturbations on the ring. All results are obtained using realistic values of modulation and validated using an in-house full-wave solver. We achieve 21 dB isolation and −0.25 dB insertion loss at the telecommunication wavelengths.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  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]
  2. D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11(12), 774–783 (2017).
    [Crossref]
  3. D. Bharadia, E. McMilin, and S. Katti, “Full duplex radios,” in ACM SIGCOMM computer communication review, vol. 43 (ACM, 2013), pp. 375–386.
  4. H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall, 1984).
  5. B. J. Stadler and T. Mizumoto, “Integrated magneto-optical materials and isolators: a review,” IEEE Photonics J. 6(1), 1–15 (2014).
    [Crossref]
  6. E. Ishida, K. Miura, Y. Shoji, H. Yokoi, T. Mizumoto, N. Nishiyama, and S. Arai, “Amorphous-si waveguide on a garnet magneto-optical isolator with a te mode nonreciprocal phase shift,” Opt. Express 25(1), 452–462 (2017).
    [Crossref]
  7. D. Jalas, A. Petrov, M. Krause, J. Hampe, and M. Eich, “Resonance splitting in gyrotropic ring resonators,” Opt. Lett. 35(20), 3438–3440 (2010).
    [Crossref]
  8. N. Kono, K. Kakihara, K. Saitoh, and M. Koshiba, “Nonreciprocal microresonators for the miniaturization of optical waveguide isolators,” Opt. Express 15(12), 7737–7751 (2007).
    [Crossref]
  9. M.-C. Tien, T. Mizumoto, P. Pintus, H. Kromer, and J. E. Bowers, “Silicon ring isolators with bonded nonreciprocal magneto-optic garnets,” Opt. Express 19(12), 11740–11745 (2011).
    [Crossref]
  10. P. Pintus, D. Huang, C. Zhang, Y. Shoji, T. Mizumoto, and J. E. Bowers, “Microring-based optical isolator and circulator with integrated electromagnet for silicon photonics,” J. Lightwave Technol. 35(8), 1429–1437 (2017).
    [Crossref]
  11. L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
    [Crossref]
  12. 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(2), 023902 (2008).
    [Crossref]
  13. M. Soljačić, C. Luo, J. D. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. 28(8), 637–639 (2003).
    [Crossref]
  14. L. Fan, L. T. Varghese, J. Wang, Y. Xuan, A. M. Weiner, and M. Qi, “Silicon optical diode with 40 db nonreciprocal transmission,” Opt. Lett. 38(8), 1259–1261 (2013).
    [Crossref]
  15. Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3(2), 91–94 (2009).
    [Crossref]
  16. K. Fang, Z. Yu, and S. Fan, “Photonic aharonov-bohm effect based on dynamic modulation,” Phys. Rev. Lett. 108(15), 153901 (2012).
    [Crossref]
  17. D. L. Sounas, C. Caloz, and A. Alu, “Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials,” Nat. Commun. 4(1), 2407 (2013).
    [Crossref]
  18. D. L. Sounas and A. Alu, “Angular-momentum-biased nanorings to realize magnetic-free integrated optical isolation,” ACS Photonics 1(3), 198–204 (2014).
    [Crossref]
  19. D. Correas-Serrano, A. Alu, and J. Gomez-Diaz, “Magnetic-free nonreciprocal photonic platform based on time-modulated graphene capacitors,” Phys. Rev. B 98(16), 165428 (2018).
    [Crossref]
  20. Y. Shi, Q. Lin, M. Minkov, and S. Fan, “Nonreciprocal optical dissipation based on direction-dependent rabi splitting,” IEEE J. Sel. Top. Quantum Electron. 24(6), 1–7 (2018).
    [Crossref]
  21. K. Tsakmakidis, L. Shen, S. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. Boyd, “Breaking lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356(6344), 1260–1264 (2017).
    [Crossref]
  22. I. Cardea, D. Grassani, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstrating a non-reciprocal optical resonator for unlimited time-bandwidth performance,” arXiv preprint arXiv:1903.08386 (2019).
  23. D. Grassani, I. Cardea, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstration of ultra-high time-bandwidth product in a non-reciprocal fiber-optic system,” in Frontiers in Optics / Laser Science, (Optical Society of America, 2018), p. JTu3A.32.
  24. A. Mock, D. Sounas, and A. Alù, “Magnet-free circulator based on spatiotemporal modulation of photonic crystal defect cavities,” ACS Photonics 6(8), 2056–2066 (2019).
    [Crossref]
  25. A. Zarif., M. Memarian., and K. Mehrany., “Rigorous derivation of temporal coupled mode theory expressions for travelling and standing wave resonators coupled to optical waveguides,” in Proceedings of the 7th International Conference on Photonics, Optics and Laser Technology - Volume 1: PHOTOPTICS, INSTICC (SciTePress, 2019), pp. 201–208.
  26. R. Fleury, D. Sounas, and A. Alù, “Non-reciprocal optical mirrors based on spatio-temporal acousto-optic modulation,” J. Opt. 20(3), 034007 (2018).
    [Crossref]

2019 (1)

A. Mock, D. Sounas, and A. Alù, “Magnet-free circulator based on spatiotemporal modulation of photonic crystal defect cavities,” ACS Photonics 6(8), 2056–2066 (2019).
[Crossref]

2018 (3)

R. Fleury, D. Sounas, and A. Alù, “Non-reciprocal optical mirrors based on spatio-temporal acousto-optic modulation,” J. Opt. 20(3), 034007 (2018).
[Crossref]

D. Correas-Serrano, A. Alu, and J. Gomez-Diaz, “Magnetic-free nonreciprocal photonic platform based on time-modulated graphene capacitors,” Phys. Rev. B 98(16), 165428 (2018).
[Crossref]

Y. Shi, Q. Lin, M. Minkov, and S. Fan, “Nonreciprocal optical dissipation based on direction-dependent rabi splitting,” IEEE J. Sel. Top. Quantum Electron. 24(6), 1–7 (2018).
[Crossref]

2017 (4)

K. Tsakmakidis, L. Shen, S. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. Boyd, “Breaking lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356(6344), 1260–1264 (2017).
[Crossref]

D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11(12), 774–783 (2017).
[Crossref]

E. Ishida, K. Miura, Y. Shoji, H. Yokoi, T. Mizumoto, N. Nishiyama, and S. Arai, “Amorphous-si waveguide on a garnet magneto-optical isolator with a te mode nonreciprocal phase shift,” Opt. Express 25(1), 452–462 (2017).
[Crossref]

P. Pintus, D. Huang, C. Zhang, Y. Shoji, T. Mizumoto, and J. E. Bowers, “Microring-based optical isolator and circulator with integrated electromagnet for silicon photonics,” J. Lightwave Technol. 35(8), 1429–1437 (2017).
[Crossref]

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 (2)

B. J. Stadler and T. Mizumoto, “Integrated magneto-optical materials and isolators: a review,” IEEE Photonics J. 6(1), 1–15 (2014).
[Crossref]

D. L. Sounas and A. Alu, “Angular-momentum-biased nanorings to realize magnetic-free integrated optical isolation,” ACS Photonics 1(3), 198–204 (2014).
[Crossref]

2013 (2)

L. Fan, L. T. Varghese, J. Wang, Y. Xuan, A. M. Weiner, and M. Qi, “Silicon optical diode with 40 db nonreciprocal transmission,” Opt. Lett. 38(8), 1259–1261 (2013).
[Crossref]

D. L. Sounas, C. Caloz, and A. Alu, “Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials,” Nat. Commun. 4(1), 2407 (2013).
[Crossref]

2012 (1)

K. Fang, Z. Yu, and S. Fan, “Photonic aharonov-bohm effect based on dynamic modulation,” Phys. Rev. Lett. 108(15), 153901 (2012).
[Crossref]

2011 (2)

M.-C. Tien, T. Mizumoto, P. Pintus, H. Kromer, and J. E. Bowers, “Silicon ring isolators with bonded nonreciprocal magneto-optic garnets,” Opt. Express 19(12), 11740–11745 (2011).
[Crossref]

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

2010 (1)

2009 (1)

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

2008 (1)

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(2), 023902 (2008).
[Crossref]

2007 (1)

2003 (1)

Altug, H.

K. Tsakmakidis, L. Shen, S. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. Boyd, “Breaking lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356(6344), 1260–1264 (2017).
[Crossref]

I. Cardea, D. Grassani, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstrating a non-reciprocal optical resonator for unlimited time-bandwidth performance,” arXiv preprint arXiv:1903.08386 (2019).

D. Grassani, I. Cardea, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstration of ultra-high time-bandwidth product in a non-reciprocal fiber-optic system,” in Frontiers in Optics / Laser Science, (Optical Society of America, 2018), p. JTu3A.32.

Alu, A.

D. Correas-Serrano, A. Alu, and J. Gomez-Diaz, “Magnetic-free nonreciprocal photonic platform based on time-modulated graphene capacitors,” Phys. Rev. B 98(16), 165428 (2018).
[Crossref]

D. L. Sounas and A. Alu, “Angular-momentum-biased nanorings to realize magnetic-free integrated optical isolation,” ACS Photonics 1(3), 198–204 (2014).
[Crossref]

D. L. Sounas, C. Caloz, and A. Alu, “Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials,” Nat. Commun. 4(1), 2407 (2013).
[Crossref]

Alù, A.

A. Mock, D. Sounas, and A. Alù, “Magnet-free circulator based on spatiotemporal modulation of photonic crystal defect cavities,” ACS Photonics 6(8), 2056–2066 (2019).
[Crossref]

R. Fleury, D. Sounas, and A. Alù, “Non-reciprocal optical mirrors based on spatio-temporal acousto-optic modulation,” J. Opt. 20(3), 034007 (2018).
[Crossref]

D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11(12), 774–783 (2017).
[Crossref]

Arai, S.

Bharadia, D.

D. Bharadia, E. McMilin, and S. Katti, “Full duplex radios,” in ACM SIGCOMM computer communication review, vol. 43 (ACM, 2013), pp. 375–386.

Bi, L.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

Bowers, J. E.

Boyd, R.

K. Tsakmakidis, L. Shen, S. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. Boyd, “Breaking lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356(6344), 1260–1264 (2017).
[Crossref]

Boyd, R. W.

I. Cardea, D. Grassani, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstrating a non-reciprocal optical resonator for unlimited time-bandwidth performance,” arXiv preprint arXiv:1903.08386 (2019).

D. Grassani, I. Cardea, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstration of ultra-high time-bandwidth product in a non-reciprocal fiber-optic system,” in Frontiers in Optics / Laser Science, (Optical Society of America, 2018), p. JTu3A.32.

Brès, C.-S.

D. Grassani, I. Cardea, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstration of ultra-high time-bandwidth product in a non-reciprocal fiber-optic system,” in Frontiers in Optics / Laser Science, (Optical Society of America, 2018), p. JTu3A.32.

I. Cardea, D. Grassani, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstrating a non-reciprocal optical resonator for unlimited time-bandwidth performance,” arXiv preprint arXiv:1903.08386 (2019).

Caloz, C.

D. L. Sounas, C. Caloz, and A. Alu, “Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials,” Nat. Commun. 4(1), 2407 (2013).
[Crossref]

Cardea, I.

I. Cardea, D. Grassani, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstrating a non-reciprocal optical resonator for unlimited time-bandwidth performance,” arXiv preprint arXiv:1903.08386 (2019).

D. Grassani, I. Cardea, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstration of ultra-high time-bandwidth product in a non-reciprocal fiber-optic system,” in Frontiers in Optics / Laser Science, (Optical Society of America, 2018), p. JTu3A.32.

Correas-Serrano, D.

D. Correas-Serrano, A. Alu, and J. Gomez-Diaz, “Magnetic-free nonreciprocal photonic platform based on time-modulated graphene capacitors,” Phys. Rev. B 98(16), 165428 (2018).
[Crossref]

Deng, X.

K. Tsakmakidis, L. Shen, S. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. Boyd, “Breaking lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356(6344), 1260–1264 (2017).
[Crossref]

Dionne, G. F.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

Eich, M.

Fabbri, S. J.

D. Grassani, I. Cardea, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstration of ultra-high time-bandwidth product in a non-reciprocal fiber-optic system,” in Frontiers in Optics / Laser Science, (Optical Society of America, 2018), p. JTu3A.32.

I. Cardea, D. Grassani, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstrating a non-reciprocal optical resonator for unlimited time-bandwidth performance,” arXiv preprint arXiv:1903.08386 (2019).

Fan, L.

Fan, S.

Y. Shi, Q. Lin, M. Minkov, and S. Fan, “Nonreciprocal optical dissipation based on direction-dependent rabi splitting,” IEEE J. Sel. Top. Quantum Electron. 24(6), 1–7 (2018).
[Crossref]

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

K. Fang, Z. Yu, and S. Fan, “Photonic aharonov-bohm effect based on dynamic modulation,” Phys. Rev. Lett. 108(15), 153901 (2012).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3(2), 91–94 (2009).
[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(2), 023902 (2008).
[Crossref]

M. Soljačić, C. Luo, J. D. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. 28(8), 637–639 (2003).
[Crossref]

Fang, K.

K. Fang, Z. Yu, and S. Fan, “Photonic aharonov-bohm effect based on dynamic modulation,” Phys. Rev. Lett. 108(15), 153901 (2012).
[Crossref]

Fleury, R.

R. Fleury, D. Sounas, and A. Alù, “Non-reciprocal optical mirrors based on spatio-temporal acousto-optic modulation,” J. Opt. 20(3), 034007 (2018).
[Crossref]

Gomez-Diaz, J.

D. Correas-Serrano, A. Alu, and J. Gomez-Diaz, “Magnetic-free nonreciprocal photonic platform based on time-modulated graphene capacitors,” Phys. Rev. B 98(16), 165428 (2018).
[Crossref]

Grassani, D.

D. Grassani, I. Cardea, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstration of ultra-high time-bandwidth product in a non-reciprocal fiber-optic system,” in Frontiers in Optics / Laser Science, (Optical Society of America, 2018), p. JTu3A.32.

I. Cardea, D. Grassani, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstrating a non-reciprocal optical resonator for unlimited time-bandwidth performance,” arXiv preprint arXiv:1903.08386 (2019).

Hampe, J.

Haus, H. A.

H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall, 1984).

Hu, J.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

Huang, D.

Ishida, E.

Jalas, D.

Jiang, P.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

Joannopoulos, J. D.

Kakihara, K.

Katti, S.

D. Bharadia, E. McMilin, and S. Katti, “Full duplex radios,” in ACM SIGCOMM computer communication review, vol. 43 (ACM, 2013), pp. 375–386.

Kim, D. H.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

Kimerling, L. C.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

Kono, N.

Koshiba, M.

Krause, M.

Kromer, H.

Lin, Q.

Y. Shi, Q. Lin, M. Minkov, and S. Fan, “Nonreciprocal optical dissipation based on direction-dependent rabi splitting,” IEEE J. Sel. Top. Quantum Electron. 24(6), 1–7 (2018).
[Crossref]

Luo, C.

McMilin, E.

D. Bharadia, E. McMilin, and S. Katti, “Full duplex radios,” in ACM SIGCOMM computer communication review, vol. 43 (ACM, 2013), pp. 375–386.

Mehrany., K.

A. Zarif., M. Memarian., and K. Mehrany., “Rigorous derivation of temporal coupled mode theory expressions for travelling and standing wave resonators coupled to optical waveguides,” in Proceedings of the 7th International Conference on Photonics, Optics and Laser Technology - Volume 1: PHOTOPTICS, INSTICC (SciTePress, 2019), pp. 201–208.

Memarian., M.

A. Zarif., M. Memarian., and K. Mehrany., “Rigorous derivation of temporal coupled mode theory expressions for travelling and standing wave resonators coupled to optical waveguides,” in Proceedings of the 7th International Conference on Photonics, Optics and Laser Technology - Volume 1: PHOTOPTICS, INSTICC (SciTePress, 2019), pp. 201–208.

Minkov, M.

Y. Shi, Q. Lin, M. Minkov, and S. Fan, “Nonreciprocal optical dissipation based on direction-dependent rabi splitting,” IEEE J. Sel. Top. Quantum Electron. 24(6), 1–7 (2018).
[Crossref]

Miura, K.

Mizumoto, T.

Mock, A.

A. Mock, D. Sounas, and A. Alù, “Magnet-free circulator based on spatiotemporal modulation of photonic crystal defect cavities,” ACS Photonics 6(8), 2056–2066 (2019).
[Crossref]

Nishiyama, N.

Petrov, A.

Pintus, P.

Qi, M.

Ross, C.

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

Saitoh, K.

Schulz, S.

K. Tsakmakidis, L. Shen, S. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. Boyd, “Breaking lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356(6344), 1260–1264 (2017).
[Crossref]

Schulz, S. A.

I. Cardea, D. Grassani, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstrating a non-reciprocal optical resonator for unlimited time-bandwidth performance,” arXiv preprint arXiv:1903.08386 (2019).

D. Grassani, I. Cardea, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstration of ultra-high time-bandwidth product in a non-reciprocal fiber-optic system,” in Frontiers in Optics / Laser Science, (Optical Society of America, 2018), p. JTu3A.32.

Shen, L.

K. Tsakmakidis, L. Shen, S. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. Boyd, “Breaking lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356(6344), 1260–1264 (2017).
[Crossref]

Shi, Y.

Y. Shi, Q. Lin, M. Minkov, and S. Fan, “Nonreciprocal optical dissipation based on direction-dependent rabi splitting,” IEEE J. Sel. Top. Quantum Electron. 24(6), 1–7 (2018).
[Crossref]

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

Shoji, Y.

Soljacic, M.

Sounas, D.

A. Mock, D. Sounas, and A. Alù, “Magnet-free circulator based on spatiotemporal modulation of photonic crystal defect cavities,” ACS Photonics 6(8), 2056–2066 (2019).
[Crossref]

R. Fleury, D. Sounas, and A. Alù, “Non-reciprocal optical mirrors based on spatio-temporal acousto-optic modulation,” J. Opt. 20(3), 034007 (2018).
[Crossref]

Sounas, D. L.

D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11(12), 774–783 (2017).
[Crossref]

D. L. Sounas and A. Alu, “Angular-momentum-biased nanorings to realize magnetic-free integrated optical isolation,” ACS Photonics 1(3), 198–204 (2014).
[Crossref]

D. L. Sounas, C. Caloz, and A. Alu, “Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials,” Nat. Commun. 4(1), 2407 (2013).
[Crossref]

Stadler, B. J.

B. J. Stadler and T. Mizumoto, “Integrated magneto-optical materials and isolators: a review,” IEEE Photonics J. 6(1), 1–15 (2014).
[Crossref]

Tien, M.-C.

Tsakmakidis, K.

K. Tsakmakidis, L. Shen, S. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. Boyd, “Breaking lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356(6344), 1260–1264 (2017).
[Crossref]

Tsakmakidis, K. L.

I. Cardea, D. Grassani, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstrating a non-reciprocal optical resonator for unlimited time-bandwidth performance,” arXiv preprint arXiv:1903.08386 (2019).

D. Grassani, I. Cardea, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstration of ultra-high time-bandwidth product in a non-reciprocal fiber-optic system,” in Frontiers in Optics / Laser Science, (Optical Society of America, 2018), p. JTu3A.32.

Upham, J.

K. Tsakmakidis, L. Shen, S. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. Boyd, “Breaking lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356(6344), 1260–1264 (2017).
[Crossref]

I. Cardea, D. Grassani, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstrating a non-reciprocal optical resonator for unlimited time-bandwidth performance,” arXiv preprint arXiv:1903.08386 (2019).

D. Grassani, I. Cardea, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstration of ultra-high time-bandwidth product in a non-reciprocal fiber-optic system,” in Frontiers in Optics / Laser Science, (Optical Society of America, 2018), p. JTu3A.32.

Vakakis, A. F.

K. Tsakmakidis, L. Shen, S. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. Boyd, “Breaking lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356(6344), 1260–1264 (2017).
[Crossref]

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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(2), 023902 (2008).
[Crossref]

Wang, J.

Wang, Z.

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(2), 023902 (2008).
[Crossref]

Weiner, A. M.

Xuan, Y.

Yokoi, H.

Yu, Z.

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

K. Fang, Z. Yu, and S. Fan, “Photonic aharonov-bohm effect based on dynamic modulation,” Phys. Rev. Lett. 108(15), 153901 (2012).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3(2), 91–94 (2009).
[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(2), 023902 (2008).
[Crossref]

Zarif., A.

A. Zarif., M. Memarian., and K. Mehrany., “Rigorous derivation of temporal coupled mode theory expressions for travelling and standing wave resonators coupled to optical waveguides,” in Proceedings of the 7th International Conference on Photonics, Optics and Laser Technology - Volume 1: PHOTOPTICS, INSTICC (SciTePress, 2019), pp. 201–208.

Zhang, C.

Zheng, X.

K. Tsakmakidis, L. Shen, S. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. Boyd, “Breaking lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356(6344), 1260–1264 (2017).
[Crossref]

ACS Photonics (2)

D. L. Sounas and A. Alu, “Angular-momentum-biased nanorings to realize magnetic-free integrated optical isolation,” ACS Photonics 1(3), 198–204 (2014).
[Crossref]

A. Mock, D. Sounas, and A. Alù, “Magnet-free circulator based on spatiotemporal modulation of photonic crystal defect cavities,” ACS Photonics 6(8), 2056–2066 (2019).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

Y. Shi, Q. Lin, M. Minkov, and S. Fan, “Nonreciprocal optical dissipation based on direction-dependent rabi splitting,” IEEE J. Sel. Top. Quantum Electron. 24(6), 1–7 (2018).
[Crossref]

IEEE Photonics J. (1)

B. J. Stadler and T. Mizumoto, “Integrated magneto-optical materials and isolators: a review,” IEEE Photonics J. 6(1), 1–15 (2014).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. (1)

R. Fleury, D. Sounas, and A. Alù, “Non-reciprocal optical mirrors based on spatio-temporal acousto-optic modulation,” J. Opt. 20(3), 034007 (2018).
[Crossref]

Nat. Commun. (1)

D. L. Sounas, C. Caloz, and A. Alu, “Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials,” Nat. Commun. 4(1), 2407 (2013).
[Crossref]

Nat. Photonics (4)

L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photonics 5(12), 758–762 (2011).
[Crossref]

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

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

D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11(12), 774–783 (2017).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. B (1)

D. Correas-Serrano, A. Alu, and J. Gomez-Diaz, “Magnetic-free nonreciprocal photonic platform based on time-modulated graphene capacitors,” Phys. Rev. B 98(16), 165428 (2018).
[Crossref]

Phys. Rev. Lett. (2)

K. Fang, Z. Yu, and S. Fan, “Photonic aharonov-bohm effect based on dynamic modulation,” Phys. Rev. Lett. 108(15), 153901 (2012).
[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(2), 023902 (2008).
[Crossref]

Science (1)

K. Tsakmakidis, L. Shen, S. Schulz, X. Zheng, J. Upham, X. Deng, H. Altug, A. F. Vakakis, and R. Boyd, “Breaking lorentz reciprocity to overcome the time-bandwidth limit in physics and engineering,” Science 356(6344), 1260–1264 (2017).
[Crossref]

Other (5)

I. Cardea, D. Grassani, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstrating a non-reciprocal optical resonator for unlimited time-bandwidth performance,” arXiv preprint arXiv:1903.08386 (2019).

D. Grassani, I. Cardea, S. J. Fabbri, J. Upham, R. W. Boyd, H. Altug, S. A. Schulz, K. L. Tsakmakidis, and C.-S. Brès, “Demonstration of ultra-high time-bandwidth product in a non-reciprocal fiber-optic system,” in Frontiers in Optics / Laser Science, (Optical Society of America, 2018), p. JTu3A.32.

A. Zarif., M. Memarian., and K. Mehrany., “Rigorous derivation of temporal coupled mode theory expressions for travelling and standing wave resonators coupled to optical waveguides,” in Proceedings of the 7th International Conference on Photonics, Optics and Laser Technology - Volume 1: PHOTOPTICS, INSTICC (SciTePress, 2019), pp. 201–208.

D. Bharadia, E. McMilin, and S. Katti, “Full duplex radios,” in ACM SIGCOMM computer communication review, vol. 43 (ACM, 2013), pp. 375–386.

H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall, 1984).

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Figures (9)

Fig. 1.
Fig. 1. (a) Ring resonator with two discrete time-modulated perturbations, (b) time-perturbed ring resonator side coupled to waveguides. Angular position and thickness of the modulated perturbations are $\phi _1,\phi _2$ and $\delta$ , respectively.
Fig. 2.
Fig. 2. Zeroth order harmonics of two supermodes versus normalized modulation amplitude for $f_{m1}$ (dotted), $f_{m2}$ (solid) and $f_{m3}$ (dash-dotted) where $f_{m1}$ > $f_{m2}$ > $f_{m3}$ . Blue and red represent the c.c.w. and c.w. harmonics, respectively. $f_0$ is the resonance frequency of the static ring resonator.
Fig. 3.
Fig. 3. Spectrum of the modulated ring supermodes for (a) the first supermode and (b) the second supermode. Both supermodes consist of c.c.w. (blue) and c.w. (red) mode amplitudes. It is evident that the first supermode (a) has a dominant c.c.w. spectral content, while (b) has a dominant c.w. spectral content. $f_0$ is the resonance frequency of the static ring resonator.
Fig. 4.
Fig. 4. (a) Frequency-splitting between two supermodes and, (b) ratio of c.c.w. to the c.w. mode at the central frequency of supermode spectrum, versus modulation frequency and amplitude. Black and cyan curves represent the splitting of $BW$ and $f_m-BW$ , respectively and dashed yellow line indicates modulation amplitude of $\Delta \varepsilon _m=5\times 10^{-3}\varepsilon _s$ .
Fig. 5.
Fig. 5. (a) Zeroth order harmonic of power transmission of different ports for excitation from port 1 and, (b) from port 2. Solid and dash-dotted lines represent multi-frequency FDFD and TCMT results, respectively. (c) Electric field profile for excitation at frequency $f_0+0.25f_m$ from port 2 and, (d) at $f_0+0.25f_m$ from port 1.
Fig. 6.
Fig. 6. (a) Spectral content of transmission to different ports for excitation at the central frequency of $f_{1}=f_0-0.25f_m$ from port 1 and, (b) for excitation at the central frequency of $f_{2}=f_0+0.25f_m$ from port 2. Dotted and solid lines represent TCMT and multi-frequency FDFD results, respectively.
Fig. 7.
Fig. 7. (a) $|S_{21}|^2$ (blue) and $|S_{12}|^2$ (red) and, (b) Isolation versus modulation frequency in the structure with two waveguides for $\Delta \varepsilon _m=5\times 10^{-3}\varepsilon _s$ . Maximum isolation occurs at $f_m=74.2$ GHz.
Fig. 8.
Fig. 8. (a) $|S_{21}|^2$ (blue) and $|S_{12}|^2$ (red) and, (c) isolation versus modulation frequency in one-waveguide structure for $\Delta \varepsilon _m=5\times 10^{-3}\varepsilon _s$ . Maximum isolation occurs at $f_m=19.5$ GHz. (b) and (d) the same parameters for resonator with higher intrinsic loss, with $\varepsilon _s = 12-2.052\times 10^{-5}j$ . Maximum isolation occurs at $f_m=76$ GHz.
Fig. 9.
Fig. 9. Two-port transmission response of the isolator, $|S_{21}|^2$ (blue) and $|S_{12}|^2$ (red) are shown for (a) the optimum point of Fig. 8(c) ( $f_m=19.5$ GHz and $\Delta \varepsilon _m=5\times 10^{-3}\varepsilon _s$ ), and (b) the optimum point of Fig. 8(d) ( $f_m=76$ GHz and $\Delta \varepsilon _m=5\times 10^{-3}\varepsilon _s$ ).

Equations (35)

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Δ ε ( t ) = { Δ ε m 1 cos ( ω m t + Θ 1 ) ϕ 1 δ / 2 ϕ ϕ 1 + δ / 2 Δ ε m 2 cos ( ω m t + Θ 2 ) ϕ 2 δ / 2 ϕ ϕ 2 + δ / 2
E = a c ( t ) e c e j ω 0 t + a c c ( t ) e c c e j ω 0 t + c . c . H = a c ( t ) h c e j ω 0 t + a c c ( t ) h c c e j ω 0 t + c . c .
d a ( t ) dt = j M ( t ) a ( t )
a ( t ) = [ a c ( t ) a c c ( t ) ] T
M ( t ) = [ ω 0 2 κ s + e j ω m t ω 0 2 κ s + e j ω m t ω 0 2 κ m + e j ω m t ω 0 2 κ m e j ω m t ω 0 2 κ m e j ω m t ω 0 2 κ m + e j ω m t ω 0 2 κ s + e j ω m t ω 0 2 κ s + e j ω m t ]
κ s + = 1 4 π ε s ϕ 1 δ / 2 ϕ 1 + δ / 2 Δ ε m 1 | e c | 2 d ϕ e j Θ 1 + 1 4 π ε s ϕ 2 δ / 2 ϕ 2 + δ / 2 Δ ε m 2 | e c | 2 d ϕ e j Θ 2
κ m + = 1 4 π ε s ϕ 1 δ / 2 ϕ 1 + δ / 2 Δ ε m 1 e c c . e c d ϕ e j Θ 1 + 1 4 π ε s ϕ 2 δ / 2 ϕ 2 + δ / 2 Δ ε m 2 e c c . e c d ϕ e j Θ 2
κ m = 1 4 π ε s ϕ 1 δ / 2 ϕ 1 + δ / 2 Δ ε m 1 e c c . e c d ϕ e j Θ 1 + 1 4 π ε s ϕ 2 δ / 2 ϕ 2 + δ / 2 Δ ε m 2 e c c . e c d ϕ e j Θ 2
a c ( t ) = n = a n e j ( ω + n ω m ) t
a c c ( t ) = n = b n e j ( ω + n ω m ) t
j ω [   a n b n ] = j [   A B B A ] [   a n b n ]
A = [ N ω m ω 0 2 κ s + 0 ω 0 2 κ s + ω 0 2 κ s + 0 ω 0 2 κ s + N ω m ]
B = [ 0 ω 0 2 κ m 0 ω 0 2 κ m + ω 0 2 κ m 0 ω 0 2 κ m + 0 ]
B = [ 0 ω 0 2 κ m + 0 ω 0 2 κ m ω 0 2 κ m + 0 ω 0 2 κ m 0 ]
κ m + = α e j Θ 1 + β e j Θ 2
κ m = α e j Θ 1 + β e j Θ 2
α = 1 4 π ε s ϕ 1 δ / 2 ϕ 1 + δ / 2 Δ ε m 1 e c c . e c d ϕ
β = 1 4 π ε s ϕ 2 δ / 2 ϕ 2 + δ / 2 Δ ε m 2 e c c . e c d ϕ
α = 1 4 π ε s ϕ 1 δ / 2 ϕ 1 + δ / 2 Δ ε m exp ( 2 j l ϕ ) d ϕ = Δ ε m 4 l π ε s ( exp ( 2 j l ϕ 1 ) sin ( l δ ) )
β = 1 4 π ε s ϕ 2 δ / 2 ϕ 2 + δ / 2 Δ ε m exp ( 2 j l ϕ ) d ϕ = Δ ε m 4 l π ε s ( exp ( 2 j l ϕ 2 ) sin ( l δ ) )
| κ m | >> | κ m + |
| κ m + | >> | κ m |
( Θ 2 Θ 1 ) + ( ϕ β ϕ α ) = 2 n π
( Θ 2 Θ 1 ) + ( ϕ β ϕ α ) = ( 2 n + 1 ) π
Θ 2 Θ 1 = π 2 2 l ϕ 2 = 2 l ϕ 1 + 2 n π + π 2
Θ 2 Θ 1 = π 2 2 l ϕ 2 = 2 l ϕ 1 + 2 n π + π 2
δ = π 2 l
d a ( t ) d t = ( j M ( t ) Γ ) a ( t ) + K T s +
K T = [ 0 κ 2 0 κ 4 κ 1 0 κ 3 0 ]
Γ = [ Γ 11 0 0 Γ 22 ]
s = C s + + D a
C = [ 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 ]
D T = [ d 1 0 d 3 0 0 d 2 0 d 4 ]
j ω [   a n b n ] = ( j [   A B B A ] + [   Γ n 0 0 Γ n ] ) [   a n b n ] + [   0 K 2 0 K 4 K 1 0 K 3 0 ] [   s + 1 s + 2 s + 3 s + 4 ]
[   s 1 s 2 s 3 s 4 ] = [   0 I 0 0 I 0 0 0 0 0 0 I 0 0 I 0 ] [   s + 1 s + 2 s + 3 s + 4 ] + [   D 1 0 0 D 2 D 3 0 0 D 4 ] [   a n b n ]

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