Abstract

We study light transmission and reflection from an integrated microresonator device, formed by a circular microresonator coupled to a bus waveguide, with an embedded S-shaped additional crossover waveguide element that selectively couples counter-propagating modes in a propagation-direction-dependent way. The overall shape of the device resembles a “taiji” symbol, hence its name. While Lorentz reciprocity is preserved in transmission, the peculiar geometry allows us to exploit the non-Hermitian nature of the system to obtain high-contrast unidirectional reflection with negligible reflection for light incident in one direction and a significant reflection in the opposite direction.

© 2020 Chinese Laser Press

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  1. 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, 579–582 (2013).
    [Crossref]
  2. M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
    [Crossref]
  3. Y. H. Ja, “A spectacles-shaped optical fibre ring resonator with two couplers,” Opt. Quantum Electron. 23, 379–389 (1991).
    [Crossref]
  4. S. Kharitonov and C. Brés, “Dual-emission band all-fiber laser based on theta cavity with thulium- and holmium-doped fibers,” in Optical Fiber Communications Conference and Exhibition (OFC) (2017), pp. 1–3.
  5. S. Kharitonov and C. S. Brés, “Isolator-free unidirectional thulium-doped fiber laser,” Light Sci. Appl. 4, e340 (2015).
    [Crossref]
  6. J. P. Hohimer, G. A. Vawter, and D. C. Craft, “Unidirectional operation in a semiconductor ring diode laser,” Appl. Phys. Lett. 62, 1185–1187 (1993).
    [Crossref]
  7. L. Zhou, T. Ye, and J. Chen, “Coherent interference induced transparency in self-coupled optical waveguide-based resonators,” Opt. Lett. 36, 13–15 (2011).
    [Crossref]
  8. Z. Xu, Y. Luo, Q. Sun, C. Mou, Y. Li, P. P. Shum, and D. Liu, “Light velocity control in monolithic microfiber bridged ring resonator,” Optica 4, 945–950 (2017).
    [Crossref]
  9. M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
    [Crossref]
  10. S. Biasi, F. Ramiro Manzano, F. Turri, P. E. Larre, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Hermitian and non-Hermitian mode coupling in a micro-disk resonator due to stochastic surface roughness scattering,” IEEE Photon. J. 11, 6101114 (2018).
    [Crossref]
  11. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed., Wiley Series in Pure and Applied Optics (Wiley Interscience, 2007).
  12. A. Trenti, M. Borghi, S. Biasi, M. Ghulinyan, F. Ramiro-Manzano, G. Pucker, and L. Pavesi, “Thermo-optic coefficient and nonlinear refractive index of silicon oxynitride waveguides,” AIP Adv. 8, 025311 (2018).
    [Crossref]
  13. S. Chen, Q. Yan, Q. Xu, Z. Fan, and J. Liu, “Optical waveguide propagation loss measurement using multiple reflections method,” Opt. Commun. 256, 68–72 (2005).
    [Crossref]
  14. C. Castellan, S. Tondini, M. Mancinelli, C. Kopp, and L. Pavesi, “Reflectance reduction in a whiskered SOI star coupler,” IEEE Photon. Technol. Lett. 28, 1870–1873 (2016).
    [Crossref]
  15. G. Ren, S. Chen, Y. Cheng, and Y. Zhai, “Study on inverse taper-based mode transformer for low loss coupling between silicon wire waveguide and lensed fiber,” Opt. Commun. 284, 4782–4788 (2011).
    [Crossref]
  16. M. Ghulinyan, F. Ramiro-Manzano, N. Prtljaga, R. Guider, I. Carusotto, A. Pitanti, G. Pucker, and L. Pavesi, “Oscillatory vertical coupling between a whispering-gallery resonator and a bus waveguide,” Phys. Rev. Lett. 110, 163901 (2013).
    [Crossref]

2019 (1)

M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref]

2018 (3)

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

S. Biasi, F. Ramiro Manzano, F. Turri, P. E. Larre, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Hermitian and non-Hermitian mode coupling in a micro-disk resonator due to stochastic surface roughness scattering,” IEEE Photon. J. 11, 6101114 (2018).
[Crossref]

A. Trenti, M. Borghi, S. Biasi, M. Ghulinyan, F. Ramiro-Manzano, G. Pucker, and L. Pavesi, “Thermo-optic coefficient and nonlinear refractive index of silicon oxynitride waveguides,” AIP Adv. 8, 025311 (2018).
[Crossref]

2017 (1)

2016 (1)

C. Castellan, S. Tondini, M. Mancinelli, C. Kopp, and L. Pavesi, “Reflectance reduction in a whiskered SOI star coupler,” IEEE Photon. Technol. Lett. 28, 1870–1873 (2016).
[Crossref]

2015 (1)

S. Kharitonov and C. S. Brés, “Isolator-free unidirectional thulium-doped fiber laser,” Light Sci. Appl. 4, e340 (2015).
[Crossref]

2013 (2)

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, 579–582 (2013).
[Crossref]

M. Ghulinyan, F. Ramiro-Manzano, N. Prtljaga, R. Guider, I. Carusotto, A. Pitanti, G. Pucker, and L. Pavesi, “Oscillatory vertical coupling between a whispering-gallery resonator and a bus waveguide,” Phys. Rev. Lett. 110, 163901 (2013).
[Crossref]

2011 (2)

L. Zhou, T. Ye, and J. Chen, “Coherent interference induced transparency in self-coupled optical waveguide-based resonators,” Opt. Lett. 36, 13–15 (2011).
[Crossref]

G. Ren, S. Chen, Y. Cheng, and Y. Zhai, “Study on inverse taper-based mode transformer for low loss coupling between silicon wire waveguide and lensed fiber,” Opt. Commun. 284, 4782–4788 (2011).
[Crossref]

2005 (1)

S. Chen, Q. Yan, Q. Xu, Z. Fan, and J. Liu, “Optical waveguide propagation loss measurement using multiple reflections method,” Opt. Commun. 256, 68–72 (2005).
[Crossref]

1993 (1)

J. P. Hohimer, G. A. Vawter, and D. C. Craft, “Unidirectional operation in a semiconductor ring diode laser,” Appl. Phys. Lett. 62, 1185–1187 (1993).
[Crossref]

1991 (1)

Y. H. Ja, “A spectacles-shaped optical fibre ring resonator with two couplers,” Opt. Quantum Electron. 23, 379–389 (1991).
[Crossref]

Alù, A.

M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref]

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, 579–582 (2013).
[Crossref]

Bandres, M. A.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

Biasi, S.

S. Biasi, F. Ramiro Manzano, F. Turri, P. E. Larre, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Hermitian and non-Hermitian mode coupling in a micro-disk resonator due to stochastic surface roughness scattering,” IEEE Photon. J. 11, 6101114 (2018).
[Crossref]

A. Trenti, M. Borghi, S. Biasi, M. Ghulinyan, F. Ramiro-Manzano, G. Pucker, and L. Pavesi, “Thermo-optic coefficient and nonlinear refractive index of silicon oxynitride waveguides,” AIP Adv. 8, 025311 (2018).
[Crossref]

Borghi, M.

A. Trenti, M. Borghi, S. Biasi, M. Ghulinyan, F. Ramiro-Manzano, G. Pucker, and L. Pavesi, “Thermo-optic coefficient and nonlinear refractive index of silicon oxynitride waveguides,” AIP Adv. 8, 025311 (2018).
[Crossref]

Brés, C.

S. Kharitonov and C. Brés, “Dual-emission band all-fiber laser based on theta cavity with thulium- and holmium-doped fibers,” in Optical Fiber Communications Conference and Exhibition (OFC) (2017), pp. 1–3.

Brés, C. S.

S. Kharitonov and C. S. Brés, “Isolator-free unidirectional thulium-doped fiber laser,” Light Sci. Appl. 4, e340 (2015).
[Crossref]

Carusotto, I.

S. Biasi, F. Ramiro Manzano, F. Turri, P. E. Larre, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Hermitian and non-Hermitian mode coupling in a micro-disk resonator due to stochastic surface roughness scattering,” IEEE Photon. J. 11, 6101114 (2018).
[Crossref]

M. Ghulinyan, F. Ramiro-Manzano, N. Prtljaga, R. Guider, I. Carusotto, A. Pitanti, G. Pucker, and L. Pavesi, “Oscillatory vertical coupling between a whispering-gallery resonator and a bus waveguide,” Phys. Rev. Lett. 110, 163901 (2013).
[Crossref]

Castellan, C.

C. Castellan, S. Tondini, M. Mancinelli, C. Kopp, and L. Pavesi, “Reflectance reduction in a whiskered SOI star coupler,” IEEE Photon. Technol. Lett. 28, 1870–1873 (2016).
[Crossref]

Chen, J.

Chen, S.

G. Ren, S. Chen, Y. Cheng, and Y. Zhai, “Study on inverse taper-based mode transformer for low loss coupling between silicon wire waveguide and lensed fiber,” Opt. Commun. 284, 4782–4788 (2011).
[Crossref]

S. Chen, Q. Yan, Q. Xu, Z. Fan, and J. Liu, “Optical waveguide propagation loss measurement using multiple reflections method,” Opt. Commun. 256, 68–72 (2005).
[Crossref]

Cheng, Y.

G. Ren, S. Chen, Y. Cheng, and Y. Zhai, “Study on inverse taper-based mode transformer for low loss coupling between silicon wire waveguide and lensed fiber,” Opt. Commun. 284, 4782–4788 (2011).
[Crossref]

Christodoulides, D. N.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

Craft, D. C.

J. P. Hohimer, G. A. Vawter, and D. C. Craft, “Unidirectional operation in a semiconductor ring diode laser,” Appl. Phys. Lett. 62, 1185–1187 (1993).
[Crossref]

Doerr, C. 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, 579–582 (2013).
[Crossref]

Eich, M.

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, 579–582 (2013).
[Crossref]

Fan, S.

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, 579–582 (2013).
[Crossref]

Fan, Z.

S. Chen, Q. Yan, Q. Xu, Z. Fan, and J. Liu, “Optical waveguide propagation loss measurement using multiple reflections method,” Opt. Commun. 256, 68–72 (2005).
[Crossref]

Freude, W.

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, 579–582 (2013).
[Crossref]

Ghulinyan, M.

S. Biasi, F. Ramiro Manzano, F. Turri, P. E. Larre, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Hermitian and non-Hermitian mode coupling in a micro-disk resonator due to stochastic surface roughness scattering,” IEEE Photon. J. 11, 6101114 (2018).
[Crossref]

A. Trenti, M. Borghi, S. Biasi, M. Ghulinyan, F. Ramiro-Manzano, G. Pucker, and L. Pavesi, “Thermo-optic coefficient and nonlinear refractive index of silicon oxynitride waveguides,” AIP Adv. 8, 025311 (2018).
[Crossref]

M. Ghulinyan, F. Ramiro-Manzano, N. Prtljaga, R. Guider, I. Carusotto, A. Pitanti, G. Pucker, and L. Pavesi, “Oscillatory vertical coupling between a whispering-gallery resonator and a bus waveguide,” Phys. Rev. Lett. 110, 163901 (2013).
[Crossref]

Guider, R.

M. Ghulinyan, F. Ramiro-Manzano, N. Prtljaga, R. Guider, I. Carusotto, A. Pitanti, G. Pucker, and L. Pavesi, “Oscillatory vertical coupling between a whispering-gallery resonator and a bus waveguide,” Phys. Rev. Lett. 110, 163901 (2013).
[Crossref]

Harari, G.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

Hohimer, J. P.

J. P. Hohimer, G. A. Vawter, and D. C. Craft, “Unidirectional operation in a semiconductor ring diode laser,” Appl. Phys. Lett. 62, 1185–1187 (1993).
[Crossref]

Ja, Y. H.

Y. H. Ja, “A spectacles-shaped optical fibre ring resonator with two couplers,” Opt. Quantum Electron. 23, 379–389 (1991).
[Crossref]

Jalas, D.

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, 579–582 (2013).
[Crossref]

Joannopoulos, J. D.

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, 579–582 (2013).
[Crossref]

Khajavikhan, M.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

Kharitonov, S.

S. Kharitonov and C. S. Brés, “Isolator-free unidirectional thulium-doped fiber laser,” Light Sci. Appl. 4, e340 (2015).
[Crossref]

S. Kharitonov and C. Brés, “Dual-emission band all-fiber laser based on theta cavity with thulium- and holmium-doped fibers,” in Optical Fiber Communications Conference and Exhibition (OFC) (2017), pp. 1–3.

Kopp, C.

C. Castellan, S. Tondini, M. Mancinelli, C. Kopp, and L. Pavesi, “Reflectance reduction in a whiskered SOI star coupler,” IEEE Photon. Technol. Lett. 28, 1870–1873 (2016).
[Crossref]

Larre, P. E.

S. Biasi, F. Ramiro Manzano, F. Turri, P. E. Larre, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Hermitian and non-Hermitian mode coupling in a micro-disk resonator due to stochastic surface roughness scattering,” IEEE Photon. J. 11, 6101114 (2018).
[Crossref]

Li, Y.

Liu, D.

Liu, J.

S. Chen, Q. Yan, Q. Xu, Z. Fan, and J. Liu, “Optical waveguide propagation loss measurement using multiple reflections method,” Opt. Commun. 256, 68–72 (2005).
[Crossref]

Luo, Y.

Mancinelli, M.

C. Castellan, S. Tondini, M. Mancinelli, C. Kopp, and L. Pavesi, “Reflectance reduction in a whiskered SOI star coupler,” IEEE Photon. Technol. Lett. 28, 1870–1873 (2016).
[Crossref]

Melloni, A.

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, 579–582 (2013).
[Crossref]

Miri, M.-A.

M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363, eaar7709 (2019).
[Crossref]

Mou, C.

Parto, M.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

Pavesi, L.

A. Trenti, M. Borghi, S. Biasi, M. Ghulinyan, F. Ramiro-Manzano, G. Pucker, and L. Pavesi, “Thermo-optic coefficient and nonlinear refractive index of silicon oxynitride waveguides,” AIP Adv. 8, 025311 (2018).
[Crossref]

S. Biasi, F. Ramiro Manzano, F. Turri, P. E. Larre, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Hermitian and non-Hermitian mode coupling in a micro-disk resonator due to stochastic surface roughness scattering,” IEEE Photon. J. 11, 6101114 (2018).
[Crossref]

C. Castellan, S. Tondini, M. Mancinelli, C. Kopp, and L. Pavesi, “Reflectance reduction in a whiskered SOI star coupler,” IEEE Photon. Technol. Lett. 28, 1870–1873 (2016).
[Crossref]

M. Ghulinyan, F. Ramiro-Manzano, N. Prtljaga, R. Guider, I. Carusotto, A. Pitanti, G. Pucker, and L. Pavesi, “Oscillatory vertical coupling between a whispering-gallery resonator and a bus waveguide,” Phys. Rev. Lett. 110, 163901 (2013).
[Crossref]

Petrov, A.

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, 579–582 (2013).
[Crossref]

Pitanti, A.

M. Ghulinyan, F. Ramiro-Manzano, N. Prtljaga, R. Guider, I. Carusotto, A. Pitanti, G. Pucker, and L. Pavesi, “Oscillatory vertical coupling between a whispering-gallery resonator and a bus waveguide,” Phys. Rev. Lett. 110, 163901 (2013).
[Crossref]

Popovic, M.

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, 579–582 (2013).
[Crossref]

Prtljaga, N.

M. Ghulinyan, F. Ramiro-Manzano, N. Prtljaga, R. Guider, I. Carusotto, A. Pitanti, G. Pucker, and L. Pavesi, “Oscillatory vertical coupling between a whispering-gallery resonator and a bus waveguide,” Phys. Rev. Lett. 110, 163901 (2013).
[Crossref]

Pucker, G.

A. Trenti, M. Borghi, S. Biasi, M. Ghulinyan, F. Ramiro-Manzano, G. Pucker, and L. Pavesi, “Thermo-optic coefficient and nonlinear refractive index of silicon oxynitride waveguides,” AIP Adv. 8, 025311 (2018).
[Crossref]

M. Ghulinyan, F. Ramiro-Manzano, N. Prtljaga, R. Guider, I. Carusotto, A. Pitanti, G. Pucker, and L. Pavesi, “Oscillatory vertical coupling between a whispering-gallery resonator and a bus waveguide,” Phys. Rev. Lett. 110, 163901 (2013).
[Crossref]

Ramiro Manzano, F.

S. Biasi, F. Ramiro Manzano, F. Turri, P. E. Larre, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Hermitian and non-Hermitian mode coupling in a micro-disk resonator due to stochastic surface roughness scattering,” IEEE Photon. J. 11, 6101114 (2018).
[Crossref]

Ramiro-Manzano, F.

A. Trenti, M. Borghi, S. Biasi, M. Ghulinyan, F. Ramiro-Manzano, G. Pucker, and L. Pavesi, “Thermo-optic coefficient and nonlinear refractive index of silicon oxynitride waveguides,” AIP Adv. 8, 025311 (2018).
[Crossref]

M. Ghulinyan, F. Ramiro-Manzano, N. Prtljaga, R. Guider, I. Carusotto, A. Pitanti, G. Pucker, and L. Pavesi, “Oscillatory vertical coupling between a whispering-gallery resonator and a bus waveguide,” Phys. Rev. Lett. 110, 163901 (2013).
[Crossref]

Ren, G.

G. Ren, S. Chen, Y. Cheng, and Y. Zhai, “Study on inverse taper-based mode transformer for low loss coupling between silicon wire waveguide and lensed fiber,” Opt. Commun. 284, 4782–4788 (2011).
[Crossref]

Ren, J.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

Renner, H.

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, 579–582 (2013).
[Crossref]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed., Wiley Series in Pure and Applied Optics (Wiley Interscience, 2007).

Segev, M.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

Shum, P. P.

Sun, Q.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed., Wiley Series in Pure and Applied Optics (Wiley Interscience, 2007).

Tondini, S.

C. Castellan, S. Tondini, M. Mancinelli, C. Kopp, and L. Pavesi, “Reflectance reduction in a whiskered SOI star coupler,” IEEE Photon. Technol. Lett. 28, 1870–1873 (2016).
[Crossref]

Trenti, A.

A. Trenti, M. Borghi, S. Biasi, M. Ghulinyan, F. Ramiro-Manzano, G. Pucker, and L. Pavesi, “Thermo-optic coefficient and nonlinear refractive index of silicon oxynitride waveguides,” AIP Adv. 8, 025311 (2018).
[Crossref]

Turri, F.

S. Biasi, F. Ramiro Manzano, F. Turri, P. E. Larre, M. Ghulinyan, I. Carusotto, and L. Pavesi, “Hermitian and non-Hermitian mode coupling in a micro-disk resonator due to stochastic surface roughness scattering,” IEEE Photon. J. 11, 6101114 (2018).
[Crossref]

Vanwolleghem, M.

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, 579–582 (2013).
[Crossref]

Vawter, G. A.

J. P. Hohimer, G. A. Vawter, and D. C. Craft, “Unidirectional operation in a semiconductor ring diode laser,” Appl. Phys. Lett. 62, 1185–1187 (1993).
[Crossref]

Wittek, S.

M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren, M. Segev, D. N. Christodoulides, and M. Khajavikhan, “Topological insulator laser: experiments,” Science 359, eaar4005 (2018).
[Crossref]

Xu, Q.

S. Chen, Q. Yan, Q. Xu, Z. Fan, and J. Liu, “Optical waveguide propagation loss measurement using multiple reflections method,” Opt. Commun. 256, 68–72 (2005).
[Crossref]

Xu, Z.

Yan, Q.

S. Chen, Q. Yan, Q. Xu, Z. Fan, and J. Liu, “Optical waveguide propagation loss measurement using multiple reflections method,” Opt. Commun. 256, 68–72 (2005).
[Crossref]

Ye, T.

Yu, Z.

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Figures (9)

Fig. 1.
Fig. 1. Sketch of the taiji microresonator: EinL and EinR (EoutL and EoutR) are input (output) field amplitudes at the left and right facet, respectively, while Ee1 and Ee2 are the amplitudes of fields emitted as radiative dissipation; κi and ti, with i=1,2,3, are the coupling and transmission amplitudes at the different beamsplitting regions indicated by the gray squares. The gray dashed lines define the spatial size of the different segments.
Fig. 2.
Fig. 2. Panels (a) and (b): numerical results for the field intensity in the taiji microresonator with light incident from the left and right, respectively. The geometrical dimensions are in μm. The frequency is resonant with the ring and the bus waveguide is critically coupled. The color plot shows the electric field amplitude in V/m. It is noteworthy that only light incident from the right excites the S waveguide. This highlights the non-symmetrical behavior of light reflection. Panels (c) and (d): transmitted (blue dots) and reflected intensity as a function of the incident wavelength for light incident from the left (black dots) and from the right (green dots). The red lines display the fitting results employing the analytical model.
Fig. 3.
Fig. 3. Panels (a) and (b) show the optical micrograph and the SEM image of the top and the cross-section view of a taiji microresonator, respectively. Panel (c): sketch of the experimental setup.
Fig. 4.
Fig. 4. Experimental spectra of the (a) transmitted and (b), (c) reflected intensities as a function of the incident wavelength. The blue lines show the experimental measurements while the red lines display the fitting results employing the analytical model. The bottom panels show the zoom of the transmitted (Zoom 1) and reflected (Zoom 2, 3) intensities for the resonance highlighted by the vertical dashed lines.
Fig. 5.
Fig. 5. Intensity as a function of the wavelength computed with the Eqs. (7) and (8) using the parameters of Table 1 (Appendix B) at the resonant wavelengths (λi). Precisely, the red squares are the transmitted intensity, the upward light blue triangles are the reflected intensity for light incident from right and the downward blue triangles are the reflected intensity for light incident from left. The light blue and blue dashed lines denote the average of the resonant values for the |rR(λi)|2 and the |rL(λi)|2, respectively.
Fig. 6.
Fig. 6. Map of the fields within the device, used to calculate the scattering matrix elements when light enters from the left. Labels Em, with m=1,,6, represent complex amplitudes of the guided fields propagating in the device. Labels Ee1 and Ee2 indicate the modes that are radiated into the cladding. ti and κi, where i=1,2,3, are the transmission and coupling amplitudes at the different beamsplitting regions.
Fig. 7.
Fig. 7. Map of the fields within the device, used to calculate the scattering matrix elements when light enters from the right. Again Em, with m integers, are complex amplitudes of guided-mode fields, while Ee1 and Ee2 indicate the amplitudes of the modes that are radiated into the cladding. ti and κi, where i=1,2,3, are the transmission and coupling amplitudes at the different beamsplitting regions.
Fig. 8.
Fig. 8. Results of the simulation of the ring-bus waveguide coupling region of the taiji. Plotted curves represent the power transmission to either the bus waveguide or the ring, as a function of their mutual separation. The inset shows the distribution of electric field amplitude in the system in V/m, for a chosen distance of 335 nm. Geometrical dimensions are in μm.
Fig. 9.
Fig. 9. Results of the simulation of the ring-S-shaped waveguide coupling region of the taiji. Plotted curves represent the power transmission to either the ring or the S-shaped branch, as a function of their mutual separation. The inset shows the distribution of electric field amplitude in the system in V/m, for a chosen distance of 289 nm. Geometrical dimensions are in μm.

Tables (1)

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Table 1. Parameters Used in the Fitting Process of the Experimental Data Shown in Fig. 4

Equations (22)

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(EinLEoutL)=M(EoutREinR),
MFR=1τR(1ρRρR1)MFL=1τL(1ρLρL1),
MPj=(eiθj00eiθj),
Mtaiji=(A1A2A3A4)=(1t1t2t3eiγpt1t2t3eiγp2κ12κ2κ3t3eiγ(z2+2z3+z4)(1t1t2t3eiγp)(t1t2t3eiγp)0t1t2t3eiγp1t1t2t3eiγp),
(EoutLEoutR)=S(EinLEinR).
Staiji=(rtaijiLttaijiRttaijiLrtaijiR)=(0t1t2t3eiγp1t1t2t3eiγpt1t2t3eiγp1t1t2t3eiγp2κ12κ2κ3t3eiγ(z2+2z3+z4)(1t1t2t3eiγp)2),
t=τRτLA1eiΘ+ρRρLA4eiΘ++ρRA2eiΘρLA3eiΘ,
rL=ρRA4eiΘ+ρLA1eiΘ+ρRρLA2eiΘ+A3eiΘA1eiΘ+ρRρLA4eiΘ++ρRA2eiΘρLA3eiΘ,
rR=ρRA1eiΘ+ρLA4eiΘ++A2eiΘρLρRA3eiΘA1eiΘ+ρRρLA4eiΘ++ρRA2eiΘρLA3eiΘ.
EoutR=t1EinL+iκ1E6,Ee1=iκ2E2,Ee2=iκ3E4,E1=iκ1EinL+t1E6,E2=eiγz1E1,E3=t2E2,E4=eiγz2E3,                      E5=t3E4,          E6=eiγz3E5.
EoutREinL=ttaijiL=t1κ12t2t3eiγp1t1t2t3eiγp=t1t2t3eiγp1t1t2t3eiγp.
E1EinL=iκ11t1t2t3eiγp,
ttaijiL=t1+iκ1t2t3eiγpE1EinL.
Ee1LEin=κ1κ2eiγz11t1t2t3eiγp=iκ2eiγz1E1Ein,
Ee2LEin=κ1t2κ3eiγ(z1+z2)1t1t2t3eiγp=iκ3t2eiγ(z1+z2)E1Ein.
EoutL=t1EinR+iκ1E15,EoutR=iκ1E2,Ee1=t2E14+iκ2E12,Ee2=t3E8+iκ3E6,E1=iκ1EinR+t1E15,E2=eiγz3E4,E3=eiγz3E1,E4=iκ3E8+t3E6,E5=t3E3,E6=eiγz2E10,E7=iκ3E3,E8=eiγz4E13,E9=eiγz2E5,E10=iκ2E14+t2E12,E11=t2E9,E12=eiγz1E16,E13=iκ2E9,E14=eiγz4E7,E15=eiγz1E11,E16=t1E2.
EoutLEinR=ttaijiR=t1κ12t2t3eiγp1t1t2t3eiγp=t1t2t3eiγp1t1t2t3eiγp,EoutREinR=rtaijiR=2κ12κ2κ3t3eiγ(z2+2z3+z4)(1t1t2t3eiγp)2.
E2EinR=2iκ1κ2κ3t3eiγ(z2+2z3+z4)(1t1t2t3eiγp)2,
EoutREinR=iκ1E2EinR.
Ee1REin=κ1κ3t2eiγ(z3+z4)1t1t2t3eiγp+2κ1κ22κ3t1t3eiγ(z1+z2+2z3+z4)(1t1t2t3eiγp)2=iκ3t2eiγ(z3+z4)E1Ein+iκ2t1eiγz3E2Ein,
Ee2REin=κ1κ2(t32κ32)eiγ(z2+z3+z4)1t1t2t3eiγp2iκ1κ2κ32t1t2t3eiγ(z1+2z2+2z3+z4)(1t1t2t3eiγp)2=iκ2(t32κ32)eiγ(z2+z3+z4)E1Ein+κ3t1t2eiγ(z1+z2)E2Ein,
Mtaiji=1ttaijiL(1rtaijiRrtaijiLdet[S]).