Abstract

Faraday rotation is arguably the most widely studied nonreciprocal phenomenon, observable in the form of polarization rotation as waves travel in magneto-optical materials. It is at the basis of the realization of various forms of isolators and circulators. However, magneto-optical materials are bulky and difficult to integrate, and inherently associated with losses. Here we leverage nonlinear phenomena in conventional optical crystals to mimic Faraday rotation in a resonant cavity excited with two pumps with opposite circular polarizations and slightly detuned frequencies. The effect can be observed in basic resonant structures with no need for material resonances or tailored dispersion, arbitrary frequency of the pumps, and low loss, yielding an interesting path towards the realization of compact magnet-free nonreciprocal optical elements.

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

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References

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  1. D. Huang, P. Pintus, C. Zhang, P. Morton, Y. Shoji, T. Mizumoto, and J. E. Bowers, “Dynamically reconfigurable integrated optical circulators,” Optica 4, 23–30 (2017).
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    [Crossref]
  3. N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
    [Crossref]
  4. T. Dinc, M. Tymchenko, A. Nagulu, D. Sounas, A. Alu, and H. Krishnaswamy, “Synchronized conductivity modulation to realize broadband lossless magnetic-free non-reciprocity,” Nat. Commun. 8, 795 (2017).
    [Crossref]
  5. A. Kord, D. L. Sounas, and A. Alu, “Magnet-less circulators based on spatiotemporal modulation of bandstop filters in a delta topology,” IEEE Trans. Microw. Theory Tech. 66, 911–926 (2018).
    [Crossref]
  6. D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11, 774–783 (2017).
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    [Crossref]
  9. D. B. Sohn, S. Kim, and G. Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photonics 12, 91–97 (2018).
    [Crossref]
  10. F. Ruesink, J. P. Mathew, M.-A. Miri, A. Alù, and E. Verhagen, “Optical circulation in a multimode optomechanical resonator,” Nat. Commun. 9, 1798 (2018).
    [Crossref]
  11. S. Lepri and G. Casati, “Asymmetric wave propagation in nonlinear systems,” Phys. Rev. Lett. 106, 164101 (2011).
    [Crossref]
  12. L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
    [Crossref]
  13. I. V. Shadrivov, V. A. Fedotov, D. A. Powell, Y. S. Kivshar, and N. I. Zheludev, “Electromagnetic wave analogue of an electronic diode,” New J. Phys. 13, 33025 (2011).
    [Crossref]
  14. H. Zhou, K.-F. Zhou, W. Hu, Q. Guo, S. Lan, X.-S. Lin, and A. V. Gopal, “All-optical diodes based on photonic crystal molecules consisting of nonlinear defect pairs,” J. Appl. Phys. 99, 123111 (2006).
    [Crossref]
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    [Crossref]
  16. D. L. Sounas and A. Alù, “Fundamental bounds on the operation of Fano nonlinear isolators,” Phys. Rev. B 97, 115431 (2018).
    [Crossref]
  17. Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9, 388–392 (2015).
    [Crossref]
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    [Crossref]
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    [Crossref]
  20. Y. P. Svirko and N. I. Zheludev, “Coherent and incoherent pump-probe specular inverse Faraday effect in media with instantaneous nonlinearity,” J. Opt. Soc. Am. B 11, 1388–1393 (1994).
    [Crossref]
  21. R. W. Boyd, Nonlinear Optics (Academic, 2008).
  22. S. V. Popov, Y. P. Svirko, and N. I. Zheludev, “Coherent and incoherent specular inverse Faraday effect: χ(3) measurements in opaque materials,” Opt. Lett. 19, 13–15 (1994).
    [Crossref]
  23. C. L. Tang and H. Rabin, “Selection rules for circularly polarized waves in nonlinear optics,” Phys. Rev. B 3, 4025–4034 (1971).
    [Crossref]
  24. R. Duggan, J. del Pino, E. Verhagen, and A. Alù, “Optomechanically induced birefringence and optomechanically induced Faraday effect,” Phys. Rev. Lett. 123, 023602 (2019).
    [Crossref]
  25. M. Lawrence and J. A. Dionne, “Nanoscale nonreciprocity via photon-spin-polarized stimulated Raman scattering,” Nat. Commun. 10, 3297 (2019).
    [Crossref]
  26. M. S. Wismer, M. I. Stockman, and V. S. Yakovlev, “Ultrafast optical Faraday effect in transparent solids,” Phys. Rev. B 96, 224301 (2017).
    [Crossref]
  27. 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]
  28. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
    [Crossref]
  29. E. Collett, Field Guide to Polarization (SPIE, 2005).
  30. A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

2019 (2)

R. Duggan, J. del Pino, E. Verhagen, and A. Alù, “Optomechanically induced birefringence and optomechanically induced Faraday effect,” Phys. Rev. Lett. 123, 023602 (2019).
[Crossref]

M. Lawrence and J. A. Dionne, “Nanoscale nonreciprocity via photon-spin-polarized stimulated Raman scattering,” Nat. Commun. 10, 3297 (2019).
[Crossref]

2018 (7)

L. Del Bino, J. M. Silver, M. T. M. Woodley, S. L. Stebbings, X. Zhao, and P. Del’Haye, “Microresonator isolators and circulators based on the intrinsic nonreciprocity of the Kerr effect,” Optica 5, 279–282 (2018).
[Crossref]

D. B. Sohn, S. Kim, and G. Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photonics 12, 91–97 (2018).
[Crossref]

F. Ruesink, J. P. Mathew, M.-A. Miri, A. Alù, and E. Verhagen, “Optical circulation in a multimode optomechanical resonator,” Nat. Commun. 9, 1798 (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, “Fundamental bounds on the operation of Fano nonlinear isolators,” Phys. Rev. B 97, 115431 (2018).
[Crossref]

T. Mizumoto, R. Baets, and J. E. Bowers, “Optical nonreciprocal devices for silicon photonics using wafer-bonded magneto-optical garnet materials,” MRS Bull. 43(6), 419–424 (2018).
[Crossref]

A. Kord, D. L. Sounas, and A. Alu, “Magnet-less circulators based on spatiotemporal modulation of bandstop filters in a delta topology,” IEEE Trans. Microw. Theory Tech. 66, 911–926 (2018).
[Crossref]

2017 (4)

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

T. Dinc, M. Tymchenko, A. Nagulu, D. Sounas, A. Alu, and H. Krishnaswamy, “Synchronized conductivity modulation to realize broadband lossless magnetic-free non-reciprocity,” Nat. Commun. 8, 795 (2017).
[Crossref]

D. Huang, P. Pintus, C. Zhang, P. Morton, Y. Shoji, T. Mizumoto, and J. E. Bowers, “Dynamically reconfigurable integrated optical circulators,” Optica 4, 23–30 (2017).
[Crossref]

M. S. Wismer, M. I. Stockman, and V. S. Yakovlev, “Ultrafast optical Faraday effect in transparent solids,” Phys. Rev. B 96, 224301 (2017).
[Crossref]

2015 (1)

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

2014 (1)

N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
[Crossref]

2013 (1)

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]

2012 (1)

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

2011 (2)

I. V. Shadrivov, V. A. Fedotov, D. A. Powell, Y. S. Kivshar, and N. I. Zheludev, “Electromagnetic wave analogue of an electronic diode,” New J. Phys. 13, 33025 (2011).
[Crossref]

S. Lepri and G. Casati, “Asymmetric wave propagation in nonlinear systems,” Phys. Rev. Lett. 106, 164101 (2011).
[Crossref]

2009 (1)

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

2006 (1)

H. Zhou, K.-F. Zhou, W. Hu, Q. Guo, S. Lan, X.-S. Lin, and A. V. Gopal, “All-optical diodes based on photonic crystal molecules consisting of nonlinear defect pairs,” J. Appl. Phys. 99, 123111 (2006).
[Crossref]

2003 (1)

1994 (2)

1971 (1)

C. L. Tang and H. Rabin, “Selection rules for circularly polarized waves in nonlinear optics,” Phys. Rev. B 3, 4025–4034 (1971).
[Crossref]

1966 (1)

P. S. Pershan, J. P. van der Ziel, and L. D. Malmstrom, “Theoretical discussion of the inverse Faraday effect, Raman scattering, and related phenomena,” Phys. Rev. 143, 574–583 (1966).
[Crossref]

1965 (1)

J. P. van der Ziel, P. S. Pershan, and L. D. Malmstrom, “Optically-induced magnetization resulting from the inverse Faraday effect,” Phys. Rev. Lett. 15, 190–193 (1965).
[Crossref]

Alu, A.

A. Kord, D. L. Sounas, and A. Alu, “Magnet-less circulators based on spatiotemporal modulation of bandstop filters in a delta topology,” IEEE Trans. Microw. Theory Tech. 66, 911–926 (2018).
[Crossref]

T. Dinc, M. Tymchenko, A. Nagulu, D. Sounas, A. Alu, and H. Krishnaswamy, “Synchronized conductivity modulation to realize broadband lossless magnetic-free non-reciprocity,” Nat. Commun. 8, 795 (2017).
[Crossref]

Alù, A.

R. Duggan, J. del Pino, E. Verhagen, and A. Alù, “Optomechanically induced birefringence and optomechanically induced Faraday effect,” Phys. Rev. Lett. 123, 023602 (2019).
[Crossref]

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

F. Ruesink, J. P. Mathew, M.-A. Miri, A. Alù, and E. Verhagen, “Optical circulation in a multimode optomechanical resonator,” Nat. Commun. 9, 1798 (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]

N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
[Crossref]

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]

Baets, R.

T. Mizumoto, R. Baets, and J. E. Bowers, “Optical nonreciprocal devices for silicon photonics using wafer-bonded magneto-optical garnet materials,” MRS Bull. 43(6), 419–424 (2018).
[Crossref]

Bahl, G.

D. B. Sohn, S. Kim, and G. Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photonics 12, 91–97 (2018).
[Crossref]

Bowers, J. E.

T. Mizumoto, R. Baets, and J. E. Bowers, “Optical nonreciprocal devices for silicon photonics using wafer-bonded magneto-optical garnet materials,” MRS Bull. 43(6), 419–424 (2018).
[Crossref]

D. Huang, P. Pintus, C. Zhang, P. Morton, Y. Shoji, T. Mizumoto, and J. E. Bowers, “Dynamically reconfigurable integrated optical circulators,” Optica 4, 23–30 (2017).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2008).

Caloz, C.

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]

Casati, G.

S. Lepri and G. Casati, “Asymmetric wave propagation in nonlinear systems,” Phys. Rev. Lett. 106, 164101 (2011).
[Crossref]

Collett, E.

E. Collett, Field Guide to Polarization (SPIE, 2005).

Del Bino, L.

del Pino, J.

R. Duggan, J. del Pino, E. Verhagen, and A. Alù, “Optomechanically induced birefringence and optomechanically induced Faraday effect,” Phys. Rev. Lett. 123, 023602 (2019).
[Crossref]

Del’Haye, P.

Dinc, T.

T. Dinc, M. Tymchenko, A. Nagulu, D. Sounas, A. Alu, and H. Krishnaswamy, “Synchronized conductivity modulation to realize broadband lossless magnetic-free non-reciprocity,” Nat. Commun. 8, 795 (2017).
[Crossref]

Dionne, J. A.

M. Lawrence and J. A. Dionne, “Nanoscale nonreciprocity via photon-spin-polarized stimulated Raman scattering,” Nat. Commun. 10, 3297 (2019).
[Crossref]

Duggan, R.

R. Duggan, J. del Pino, E. Verhagen, and A. Alù, “Optomechanically induced birefringence and optomechanically induced Faraday effect,” Phys. Rev. Lett. 123, 023602 (2019).
[Crossref]

Estep, N. A.

N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
[Crossref]

Fan, L.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Fan, S.

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

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

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2003).
[Crossref]

Fedotov, V. A.

I. V. Shadrivov, V. A. Fedotov, D. A. Powell, Y. S. Kivshar, and N. I. Zheludev, “Electromagnetic wave analogue of an electronic diode,” New J. Phys. 13, 33025 (2011).
[Crossref]

Gopal, A. V.

H. Zhou, K.-F. Zhou, W. Hu, Q. Guo, S. Lan, X.-S. Lin, and A. V. Gopal, “All-optical diodes based on photonic crystal molecules consisting of nonlinear defect pairs,” J. Appl. Phys. 99, 123111 (2006).
[Crossref]

Guo, Q.

H. Zhou, K.-F. Zhou, W. Hu, Q. Guo, S. Lan, X.-S. Lin, and A. V. Gopal, “All-optical diodes based on photonic crystal molecules consisting of nonlinear defect pairs,” J. Appl. Phys. 99, 123111 (2006).
[Crossref]

Hagness, S.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Hu, W.

H. Zhou, K.-F. Zhou, W. Hu, Q. Guo, S. Lan, X.-S. Lin, and A. V. Gopal, “All-optical diodes based on photonic crystal molecules consisting of nonlinear defect pairs,” J. Appl. Phys. 99, 123111 (2006).
[Crossref]

Huang, D.

Joannopoulos, J. D.

Kim, S.

D. B. Sohn, S. Kim, and G. Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photonics 12, 91–97 (2018).
[Crossref]

Kivshar, Y. S.

I. V. Shadrivov, V. A. Fedotov, D. A. Powell, Y. S. Kivshar, and N. I. Zheludev, “Electromagnetic wave analogue of an electronic diode,” New J. Phys. 13, 33025 (2011).
[Crossref]

Kord, A.

A. Kord, D. L. Sounas, and A. Alu, “Magnet-less circulators based on spatiotemporal modulation of bandstop filters in a delta topology,” IEEE Trans. Microw. Theory Tech. 66, 911–926 (2018).
[Crossref]

Krishnaswamy, H.

T. Dinc, M. Tymchenko, A. Nagulu, D. Sounas, A. Alu, and H. Krishnaswamy, “Synchronized conductivity modulation to realize broadband lossless magnetic-free non-reciprocity,” Nat. Commun. 8, 795 (2017).
[Crossref]

Lan, S.

H. Zhou, K.-F. Zhou, W. Hu, Q. Guo, S. Lan, X.-S. Lin, and A. V. Gopal, “All-optical diodes based on photonic crystal molecules consisting of nonlinear defect pairs,” J. Appl. Phys. 99, 123111 (2006).
[Crossref]

Lawrence, M.

M. Lawrence and J. A. Dionne, “Nanoscale nonreciprocity via photon-spin-polarized stimulated Raman scattering,” Nat. Commun. 10, 3297 (2019).
[Crossref]

Lepri, S.

S. Lepri and G. Casati, “Asymmetric wave propagation in nonlinear systems,” Phys. Rev. Lett. 106, 164101 (2011).
[Crossref]

Lin, X.-S.

H. Zhou, K.-F. Zhou, W. Hu, Q. Guo, S. Lan, X.-S. Lin, and A. V. Gopal, “All-optical diodes based on photonic crystal molecules consisting of nonlinear defect pairs,” J. Appl. Phys. 99, 123111 (2006).
[Crossref]

Malmstrom, L. D.

P. S. Pershan, J. P. van der Ziel, and L. D. Malmstrom, “Theoretical discussion of the inverse Faraday effect, Raman scattering, and related phenomena,” Phys. Rev. 143, 574–583 (1966).
[Crossref]

J. P. van der Ziel, P. S. Pershan, and L. D. Malmstrom, “Optically-induced magnetization resulting from the inverse Faraday effect,” Phys. Rev. Lett. 15, 190–193 (1965).
[Crossref]

Mathew, J. P.

F. Ruesink, J. P. Mathew, M.-A. Miri, A. Alù, and E. Verhagen, “Optical circulation in a multimode optomechanical resonator,” Nat. Commun. 9, 1798 (2018).
[Crossref]

Miri, M.-A.

F. Ruesink, J. P. Mathew, M.-A. Miri, A. Alù, and E. Verhagen, “Optical circulation in a multimode optomechanical resonator,” Nat. Commun. 9, 1798 (2018).
[Crossref]

Mizumoto, T.

T. Mizumoto, R. Baets, and J. E. Bowers, “Optical nonreciprocal devices for silicon photonics using wafer-bonded magneto-optical garnet materials,” MRS Bull. 43(6), 419–424 (2018).
[Crossref]

D. Huang, P. Pintus, C. Zhang, P. Morton, Y. Shoji, T. Mizumoto, and J. E. Bowers, “Dynamically reconfigurable integrated optical circulators,” Optica 4, 23–30 (2017).
[Crossref]

Morton, P.

Nagulu, A.

T. Dinc, M. Tymchenko, A. Nagulu, D. Sounas, A. Alu, and H. Krishnaswamy, “Synchronized conductivity modulation to realize broadband lossless magnetic-free non-reciprocity,” Nat. Commun. 8, 795 (2017).
[Crossref]

Niu, B.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Pershan, P. S.

P. S. Pershan, J. P. van der Ziel, and L. D. Malmstrom, “Theoretical discussion of the inverse Faraday effect, Raman scattering, and related phenomena,” Phys. Rev. 143, 574–583 (1966).
[Crossref]

J. P. van der Ziel, P. S. Pershan, and L. D. Malmstrom, “Optically-induced magnetization resulting from the inverse Faraday effect,” Phys. Rev. Lett. 15, 190–193 (1965).
[Crossref]

Pintus, P.

Popov, S. V.

Powell, D. A.

I. V. Shadrivov, V. A. Fedotov, D. A. Powell, Y. S. Kivshar, and N. I. Zheludev, “Electromagnetic wave analogue of an electronic diode,” New J. Phys. 13, 33025 (2011).
[Crossref]

Qi, M.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Rabin, H.

C. L. Tang and H. Rabin, “Selection rules for circularly polarized waves in nonlinear optics,” Phys. Rev. B 3, 4025–4034 (1971).
[Crossref]

Ruesink, F.

F. Ruesink, J. P. Mathew, M.-A. Miri, A. Alù, and E. Verhagen, “Optical circulation in a multimode optomechanical resonator,” Nat. Commun. 9, 1798 (2018).
[Crossref]

Shadrivov, I. V.

I. V. Shadrivov, V. A. Fedotov, D. A. Powell, Y. S. Kivshar, and N. I. Zheludev, “Electromagnetic wave analogue of an electronic diode,” New J. Phys. 13, 33025 (2011).
[Crossref]

Shen, H.

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

Shi, Y.

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

Shoji, Y.

Silver, J. M.

Sohn, D. B.

D. B. Sohn, S. Kim, and G. Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photonics 12, 91–97 (2018).
[Crossref]

Soric, J.

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

N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
[Crossref]

Sounas, D.

T. Dinc, M. Tymchenko, A. Nagulu, D. Sounas, A. Alu, and H. Krishnaswamy, “Synchronized conductivity modulation to realize broadband lossless magnetic-free non-reciprocity,” Nat. Commun. 8, 795 (2017).
[Crossref]

Sounas, D. L.

A. Kord, D. L. Sounas, and A. Alu, “Magnet-less circulators based on spatiotemporal modulation of bandstop filters in a delta topology,” IEEE Trans. Microw. Theory Tech. 66, 911–926 (2018).
[Crossref]

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]

N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
[Crossref]

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Stebbings, S. L.

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M. S. Wismer, M. I. Stockman, and V. S. Yakovlev, “Ultrafast optical Faraday effect in transparent solids,” Phys. Rev. B 96, 224301 (2017).
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Tang, C. L.

C. L. Tang and H. Rabin, “Selection rules for circularly polarized waves in nonlinear optics,” Phys. Rev. B 3, 4025–4034 (1971).
[Crossref]

Tymchenko, M.

T. Dinc, M. Tymchenko, A. Nagulu, D. Sounas, A. Alu, and H. Krishnaswamy, “Synchronized conductivity modulation to realize broadband lossless magnetic-free non-reciprocity,” Nat. Commun. 8, 795 (2017).
[Crossref]

van der Ziel, J. P.

P. S. Pershan, J. P. van der Ziel, and L. D. Malmstrom, “Theoretical discussion of the inverse Faraday effect, Raman scattering, and related phenomena,” Phys. Rev. 143, 574–583 (1966).
[Crossref]

J. P. van der Ziel, P. S. Pershan, and L. D. Malmstrom, “Optically-induced magnetization resulting from the inverse Faraday effect,” Phys. Rev. Lett. 15, 190–193 (1965).
[Crossref]

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L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

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R. Duggan, J. del Pino, E. Verhagen, and A. Alù, “Optomechanically induced birefringence and optomechanically induced Faraday effect,” Phys. Rev. Lett. 123, 023602 (2019).
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L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
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L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

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M. S. Wismer, M. I. Stockman, and V. S. Yakovlev, “Ultrafast optical Faraday effect in transparent solids,” Phys. Rev. B 96, 224301 (2017).
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H. Zhou, K.-F. Zhou, W. Hu, Q. Guo, S. Lan, X.-S. Lin, and A. V. Gopal, “All-optical diodes based on photonic crystal molecules consisting of nonlinear defect pairs,” J. Appl. Phys. 99, 123111 (2006).
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T. Dinc, M. Tymchenko, A. Nagulu, D. Sounas, A. Alu, and H. Krishnaswamy, “Synchronized conductivity modulation to realize broadband lossless magnetic-free non-reciprocity,” Nat. Commun. 8, 795 (2017).
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F. Ruesink, J. P. Mathew, M.-A. Miri, A. Alù, and E. Verhagen, “Optical circulation in a multimode optomechanical resonator,” Nat. Commun. 9, 1798 (2018).
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M. Lawrence and J. A. Dionne, “Nanoscale nonreciprocity via photon-spin-polarized stimulated Raman scattering,” Nat. Commun. 10, 3297 (2019).
[Crossref]

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).
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Nat. Electron. (1)

D. L. Sounas, J. Soric, and A. Alù, “Broadband passive isolators based on coupled nonlinear resonances,” Nat. Electron. 1, 113–119 (2018).
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Nat. Photonics (4)

Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9, 388–392 (2015).
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D. B. Sohn, S. Kim, and G. Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photonics 12, 91–97 (2018).
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D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11, 774–783 (2017).
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Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[Crossref]

Nat. Phys. (1)

N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
[Crossref]

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I. V. Shadrivov, V. A. Fedotov, D. A. Powell, Y. S. Kivshar, and N. I. Zheludev, “Electromagnetic wave analogue of an electronic diode,” New J. Phys. 13, 33025 (2011).
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Opt. Lett. (1)

Optica (2)

Phys. Rev. (1)

P. S. Pershan, J. P. van der Ziel, and L. D. Malmstrom, “Theoretical discussion of the inverse Faraday effect, Raman scattering, and related phenomena,” Phys. Rev. 143, 574–583 (1966).
[Crossref]

Phys. Rev. B (3)

C. L. Tang and H. Rabin, “Selection rules for circularly polarized waves in nonlinear optics,” Phys. Rev. B 3, 4025–4034 (1971).
[Crossref]

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

M. S. Wismer, M. I. Stockman, and V. S. Yakovlev, “Ultrafast optical Faraday effect in transparent solids,” Phys. Rev. B 96, 224301 (2017).
[Crossref]

Phys. Rev. Lett. (3)

J. P. van der Ziel, P. S. Pershan, and L. D. Malmstrom, “Optically-induced magnetization resulting from the inverse Faraday effect,” Phys. Rev. Lett. 15, 190–193 (1965).
[Crossref]

S. Lepri and G. Casati, “Asymmetric wave propagation in nonlinear systems,” Phys. Rev. Lett. 106, 164101 (2011).
[Crossref]

R. Duggan, J. del Pino, E. Verhagen, and A. Alù, “Optomechanically induced birefringence and optomechanically induced Faraday effect,” Phys. Rev. Lett. 123, 023602 (2019).
[Crossref]

Science (1)

L. Fan, J. Wang, L. T. Varghese, H. Shen, B. Niu, Y. Xuan, A. M. Weiner, and M. Qi, “An all-silicon passive optical diode,” Science 335, 447–450 (2012).
[Crossref]

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R. W. Boyd, Nonlinear Optics (Academic, 2008).

E. Collett, Field Guide to Polarization (SPIE, 2005).

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

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Figures (6)

Fig. 1.
Fig. 1. (a), (b) Allowed four-wave interactions for a linear signal (ωs) with a right-handed circularly polarized (↻) pump (ωp) in an isotropic medium. (c), (d) Allowed interactions with oppositely polarized pumps with frequency spacing ωp1ωp2=Δ. (e), (f) Model of the resonator model in (e) transmission mode and (f) reflection mode.
Fig. 2.
Fig. 2. (a), (b) Time-domain envelope amplitudes for a+,0(t), a,s(t); (c), (d) time-domain phases for the fundamentals a+,0(t), for cavities with ω0=104, |D|=107, and (a), (c) Δ=2 or (b), (d) Δ=20. The time is normalized to the low-detuning envelope oscillation period T0=2π|D|ω02.
Fig. 3.
Fig. 3. Transmission magnitudes for a symmetric cavity with Q=5×104 and silicon-like nonlinearity (see main text for details). Iin is the power in one of the (equally strong) two-pump beams, in W/cm2. This definition is used in all subsequent plots. The pump beams are detuned as Δ=8γ. The pumps break the degeneracy between LH and RH resonances, which can create polarization rotation (Tyx0).
Fig. 4.
Fig. 4. (a), (d) Rotation angle, (b), (e) ellipticity, and (c), (f) fraction of power reflected at the input frequency for (a)–(c) Δ=γ and (d)–(f) Δ=20γ with Q=5×104. The black lines indicate the contours for θ=(±40°,±50°), around the typically interesting value of 45°.
Fig. 5.
Fig. 5. (a) Polarization rotation angle and (b) reflection at the fundamental harmonic for probes at the resonance center as a function of pump detuning and power, with Q=5×104 and silicon-like nonlinearity. Non-unitary reflection coefficients are due to conversion to frequency sidebands.
Fig. 6.
Fig. 6. FDTD results for (a) polarization and ellipticity angle and (b) fundamental amplitude as a function of pump power. The cavity is filled with a silicon-like nonlinear material (n=3.4, χ(3)=3×1018m2/V2), considering a detuning Δ8γ, and a coupling constant D2χ(3)I1I2cn3ε0ω0, due to the reflection of the pump beams. Ipump is in W/cm2.

Equations (12)

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χLRLL(ωs;ωp,ωp,ωs)=χLLRL(ωs;ωp,ωp,ωs),
χLLRL(ωs;ωp,ωp,ωs)=χLLRL(ωs;ωp,ωp,ωs).
E1=B1(z)e^+exp[i(ωp+Δ/2)t],E2=B2(z)e^exp[i(ωpΔ/2)t],
E±=a±(t)Am(z)e^±exp(iωmt),
t(a+(t)e^++a(t)e^)=exp(iωmt)Am*(z)12iω0ε2PNLt2dz.
PNL=ε0χ(3)(E·E)E,
PNL=4ε0χ(3)Am(z)exp(iωmt)[I(z)(a+(t)e^++a(t)e^)+e^+K(z)a(t)exp(iΔt)+e^K*(z)a+(t)exp(iΔt)],
a+(t)t=iDexp(iΔt)[2i(ω0+Δ)a(t)t(ω0+Δ)2a(t)],a(t)t=iD*exp(iΔt)[2i(ω0Δ)a+(t)t(ω0Δ)2a+(t)],
a±,0(t)=e±iΔt2[cos(12xt)iΔxsin(12xt)],a±,s(t)=e±iΔt2[2D(ω0±Δ)2xsin(12xt)],
a+(t)t=iDeiΔt[2i(ω0+Δ)a(t)t(ω0+Δ)2a(t)]±γa+(t)+γSin,+a(t)t=iD*eiΔt[2i(ω0Δ)a+(t)t(ω0Δ)2a+(t)]±γa(t)+γSin,.
S21,±(Ω)=γ(γ+i(ΩΔ))(γ+i(ΩΔ))(γ+iΩ)+|D|2(ω02Δ2)(ω0+2ΩΔ)2,
S11,±(Ω)=1+2γ(γ+i(ΩΔ))(γ+i(ΩΔ))(γ+iΩ)+|D|2(ω02Δ2)(ω0+2ΩΔ)2.