Abstract

Nonreciprocal light propagation is important in many applications, ranging from optical telecommunications to integrated photonics. A simple way to achieve optical nonreciprocity is to use the nonlinear interaction between counterpropagating light in a Kerr medium. Within a ring resonator, this leads to spontaneous symmetry breaking, resulting in light of a given frequency circulating in one direction, but not in both directions simultaneously. In this work, we demonstrate that this effect can be used to realize optical isolators and circulators based on a single ultra-high-Q microresonator. We obtain isolation of >24dB and develop a theoretical model for the power scaling of the attainable nonreciprocity.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Nonreciprocal light propagation [1] is expected to play an important role in future photonic networks and optical data processing. A challenge for integrated photonic circuits is the realization of nonreciprocal elements like isolators and circulators [2]. Most optical isolators and circulators are based on the magneto-optic Faraday effect, which requires the integration of (electro)magnets [37]. Additional and compelling ways to achieve optical nonreciprocity include specially designed waveguides [8,9], optomechanical systems [1014], parity-time symmetry breaking in resonator systems [15,16], indirect interband photonic transitions [17,18], and Brillouin scattering [19,20]. Other systems include spatially asymmetric “optical diodes” based on microresonator systems [21]. This device works based on asymmetric thermal resonance frequency shifts, with the disadvantage that it would suffer from dynamically reciprocal backward transmission in the spectral vicinity of the forward-transmitted light [22].

In this work we demonstrate a novel and simple isolator and circulator based on the intrinsic nonreciprocity of the Kerr nonlinearity in a single whispering gallery microresonator. The underlying Kerr-nonreciprocity corresponds to the physical effect that counterpropagating light induces twice the Kerr shift compared to the same amount of co-propagating light [23,24]. In a passive microresonator this leads to spontaneous symmetry-breaking between clockwise (cw) and counterclockwise (ccw) modes, which has been recently demonstrated [25,26]. Here, we make use of this Kerr-nonlinearity-induced nonreciprocity in a microresonator in an add–drop configuration to realize a novel type of optical isolator and circulator [27]. In contrast to the case of spontaneous symmetry breaking, we intentionally set a microresonator into a state where it can only support the unidirectional propagation of light at a given frequency. Our measurements of the power scaling of the isolation are in excellent agreement with theoretical calculations based on the interplay of the Kerr nonlinearity with the modes of an ultra-high-Q microresonator.

Sending equal amounts of light in counterpropagating directions into a passive microresonator with χ(3) nonlinearity leads to spontaneous symmetry breaking of the cw and ccw modes [25,26]. In addition, this effect induces optical nonreciprocity such that the cw and ccw modes of a microresonator acquire different resonance frequencies, as shown in Fig. 1(a).

 

Fig. 1. Kerr nonreciprocity in a microresonator and the experimental setup for an isolator. (a) Schematic of the nonreciprocity. Light is coupled in the ccw direction through an ultra-high-Q microresonator in an add–drop configuration. The presence of the ccw light induces a Kerr shift that is twice as strong for cw circulating light, which leads to a resonance frequency splitting (shown in the lower panel). Consequently, backward-propagating light (cw) cannot pass through the resonator. (b) The experimental setup for the characterization of an isolator based on the Kerr nonreciprocity. Laser light is coupled through an ultra-high-Q silica microrod resonator using two tapered optical fibers. A fiber mirror simulates an optical circuit that reflects 100% of the incident light, and the polarization is adjusted to maximize the power coupled into the resonator in the backward direction. Photodiodes monitor the input, transmission, and return powers of the laser, while the laser power is adjusted with a variable attenuator. Circ = fiber-optical circulator, PD = photodiode.

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In the biased state (with the input laser on) the coupled resonator breaks Lorentz reciprocity, and consequently light at the input frequency can only circulate in one direction. Only backward-propagating light (or noise) that falls within the spectrally well-separated cw resonance could get back to the input port (obeying dynamic reciprocity [22]). If desired, this narrow backward-transmitting window can be closed by using a second isolator with a slightly different mode splitting in series. Once the resonator is in a symmetry-broken state with unidirectional light propagation, the power sent in the backward direction toward the resonator can even exceed the forward-propagating power without changing the state of the resonator [26].

2. Results

In our experiments, we use a fused silica microrod resonator (quality factor Q=1.5×108, diameter=1mm) coupled to two tapered optical fibers in an add–drop configuration [see setup in Fig. 1(b)]. Light from a diode laser at 1550 nm is sent into the microresonator in the ccw direction. The microresonator mode passively locks itself to the laser frequency, making the system insensitive to laser frequency drifts [28]. The light is then coupled out of the resonator through the second tapered fiber and directed to an optical circuit of interest. Any reflection from this circuit cannot couple back into the resonator, because it is not resonant in the cw direction, and is thus transmitted through the tapered fiber to an absorber. A common application for optical isolators is to prevent backreflections in optical circuits from disturbing the operation of a laser. One main advantage of optically induced isolation via the Kerr effect is the increase in isolation with optical power (as higher power leads to a larger frequency splitting between the cw and ccw resonator modes). We analyze the power scaling of the isolation and simulate a worst-case scenario where all of the transmitted power is reflected back to the resonator by using a fiber mirror. The behavior of the setup is characterized with photodiodes that monitor the input, transmitted, and reflected optical power levels [see Fig. 1(b)]. Figure 2 shows the experimental results for insertion loss and isolation versus the launched optical power. The power Pccwin corresponds to the power entering the input tapered fiber, Pout is the power exiting the second tapered fiber, and Pback is the power that passes through the resonator in the backward direction. The data in Fig. 2(a) shows an initial increase in backward-propagating light with input power. At the point where the Kerr nonreciprocity starts to split the cw and ccw resonance frequencies, the isolation kicks in and reduces the backward-propagating power, which can be seen at around 10 mW of input power in Fig. 2(a). At higher powers, the optical isolation reaches 24 dB [Fig. 2(b)]. The insertion loss of the setup (transmission in the forward direction) is around 5 dB. Note that the threshold power is comparably high as a result of the large diameter and large mode volume of the resonators in these proof-of-principle experiments. The threshold powers could be reduced significantly and the attainable isolation improved by using smaller resonators. This is illustrated in Fig. 2(c), which shows the calculated isolation in resonators made from different materials.

 

Fig. 2. Kerr-nonlinearity-induced isolation at 1550 nm, theory and experiment. (a) The measurement of the return power Pback versus the input power Pccwin with a theoretical fit. For low input powers, the return power is proportional to the input power, as expected for a linear system. For high powers, the return power decreases due to the nonreciprocal Kerr effect. (b) The insertion loss Pout/Pccwin and isolation Pback/Pccwin versus input power. We measure an insertion loss of around 7 dB and a maximum isolation in excess of 24 dB. (c) The predicted isolation for waveguide resonators of various materials, assuming an effective mode cross-sectional area of 4μm2 and a resonator diameter of 100 μm. Our measurement (SiO2 rod resonator, diameter 1 mm) is included for comparison. The respective Q-factors for the calculations are 109 for magnesium fluoride and calcium fluoride [29,30], 5×107 for silicon nitride [31], and 5×108 for SiO2. Our calculations show isolation at sub-milliwatt power levels and even down to tens of microwatts in these currently available microresonator systems.

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The theoretical curves in Fig. 2 are calculated using a model that takes into account the interaction between counterpropagating light via the Kerr nonlinearity. The optical powers Pcw and Pccw coupled into the resonator in the cw and ccw directions are given by [26]

Pccw=ηinPccwin1+(δγ+1P0(Pccw+2Pcw))2,
Pcw=ηinPcwin1+(δγ+1P0(Pcw+2Pccw))2.

Here, Pccwin and Pcwin are the powers launched in counterpropagating directions into the coupling waveguides, ηin is the incoupling efficiency from the waveguides to the resonator, and δ is the detuning of the laser frequency from the cold cavity resonance with a half-linewidth γ. We use a symmetrically coupled resonator, which means that ηin=4k(γk)/γ2 is the same for both waveguides, with k being the coupling coefficient between the resonator and each waveguide. The terms (Pccw+2Pcw)/P0 and (Pcw+2Pccw)/P0 correspond to the normalized nonreciprocal Kerr resonance frequency shifts. The factor of 2 in the dependence on the respective counterpropagating power is important for the isolator/circulator to work, which enables the isolation via interaction of counterpropagating light in a single resonator (compared to previous work on coupled microresonator systems [21]). P0 is a resonator-specific constant given by

P0=π2n02dAeff(γk)QL2n2λγ,
for refractive index n0 at wavelength λ, nonlinear refractive index n2, resonator diameter d, effective mode area Aeff, and loaded quality factor QL. The output and return powers are given by Pout=ηoutPccw and Pback=ηoutPcw, with the outcoupling coefficient ηout=k/(γk). Assuming that the laser is on resonance with the ccw mode, the insertion loss Pout/Pccwin will be equal to the through-coupling efficiency ηthru=ηinηout=4k2/γ2. Assuming that the rest of the optical circuit has a reflectivity R such that Pcwin=RPout, we thus find that the isolation I=Pback/Pcwin is given by
I=ηthru1+(PccwinηinP0(1IR))2.

Note that the insertion loss of the microresonator-isolator system can be decreased by increasing the coupling to the tapered fibers. However, this would lead to a decreased Q-factor and a trade-off between the attainable isolation and insertion loss.

The resonance frequency difference in our system is on the order of 10 linewidths, corresponding to 50 MHz, and could be increased to GHz ranges by using microresonators with higher optical nonlinearity or by operating higher above the threshold power. The resonance frequency difference in terms of cavity linewidths is comparable to existing electromagnetic microresonator systems with larger absolute resonance frequency differences [32]. In terms of applications, our system could be used to isolate chip-integrated continuous-wave laser diodes (with subsequent modulation if used in telecom systems).

In addition to operating as an isolator, a microresonator in an add-drop configuration can be used to realize a three-port circulator [27], as shown in Fig. 3(a). Light sent into port 1 is transmitted via the resonator to port 2. The light circulating counterclockwise around the resonator causes the cw resonance frequency to be shifted away. This means that light of the same frequency entering port 2 cannot couple into the resonator and is instead transmitted to port 3. This was tested by splitting light from a laser between port 1 (404 mW) and port 2 (87 mW), and measuring the output powers with fast photodiodes. The transmission coefficients, i.e., the output powers normalized by the relevant input power, are plotted in Fig. 3(b) while sweeping the laser frequency across the resonance.

 

Fig. 3. Microresonator-based circulator. (a) A schematic of the setup. Light is split between ports 1 and 2, with slightly more light going into port 1, so that the resonator is set into the ccw state. Photodiodes measure the optical output power from each port. (b) The transmission Tij (from port i to port j) as the laser frequency is swept through the resonance.

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In our microresonator-based circulator we observe 47% transmission from port 1 to port 2 (corresponding to 3.3 dB insertion loss), and a directivity T23/T21 of >19dB while the microresonator is thermally locked to the laser. The oscillations in T23 result from interference with light from port 1 that is backscattered within the resonator. From the amplitude of these oscillations, we obtain a value of 33dB for the (unwanted) transmission coefficient T13. This is comparable with values for commercial magneto-optic circulators, and may be further improved by reducing the backscattering through surface defects of the resonator, which will also increase the Q-factor and thereby the isolation. In order to achieve stable circulation for larger laser frequency changes than those supported by the thermal self-locking of the microresonator, the resonator could be temperature controlled and/or the laser could be injection-locked to the resonance.

3. Conclusion

In conclusion, we have demonstrated an optical isolator and circulator based on the intrinsic nonreciprocity of the Kerr effect in an optical microresonator. The biased microresonator breaks Lorentz reciprocity and our results show increasing isolation with optical power, which is supported by the theoretical model based on the nonreciprocal χ(3)-nonlinearity. In our experiments, we achieve more than 20 dB isolation. The threshold power in our millimeter-sized rod resonators is around 10 mW, which could be significantly reduced to tens of microwatts by using smaller resonators with higher finesse and higher nonlinearity. In chip-based optical devices, such isolators would be ideal for reducing backreflections into integrated laser diodes, especially given the increasing isolation at higher optical powers. Aside from the fundamental physics of the intrinsic Kerr nonreciprocity in ring resonators, our work provides a new and simple route for realizing optical isolators and circulators for applications in integrated photonic circuits.

Funding

H2020 Marie Skłodowska-Curie Actions (MSCA) (748519, CoLiDR); National Physical Laboratory Strategic Research; National Natural Science Foundation of China (NSFC) (61435002, 61675015); H2020 European Research Council (ERC) (756966); Engineering and Physical Sciences Research Council (EPSRC).

Acknowledgment

X. Z. acknowledges support from the Chinese Scholarship Council and Natural Science Foundation of China. J. M. S. acknowledges funding via a Royal Society of Engineering fellowship. L. D. B. and M. T. M. W. acknowledge funding from EPSRC through the Centre for Doctoral Training in Applied Photonics.

 

See Supplement 1 for supporting content.

REFERENCES

1. R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004). [CrossRef]  

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]  

3. 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, 758–762 (2011). [CrossRef]  

4. H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B 22, 240–253 (2005). [CrossRef]  

5. N. Kono, K. Kakihara, K. Saitoh, and M. Koshiba, “Nonreciprocal microresonators for the miniaturization of optical waveguide isolators,” Opt. Express 15, 7737–7751 (2007). [CrossRef]  

6. Y. Shoji, T. Mizumoto, H. Yokoi, I.-W. Hsieh, and R. M. Osgood Jr., “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92, 071117 (2008). [CrossRef]  

7. 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, 1429–1437 (2017). [CrossRef]  

8. L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011). [CrossRef]  

9. K. Gallo, G. Assanto, K. Parameswaran, and M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001). [CrossRef]  

10. K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13, 465–471 (2017). [CrossRef]  

11. S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102, 213903 (2009). [CrossRef]  

12. F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016). [CrossRef]  

13. Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016). [CrossRef]  

14. Z. Wang, L. Shi, Y. Liu, X. Xu, and X. Zhang, “Optical nonreciprocity in asymmetric optomechanical couplers,” Sci. Rep. 5, 8657 (2015). [CrossRef]  

15. L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014). [CrossRef]  

16. B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014). [CrossRef]  

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

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

19. X. Huang and S. Fan, “Complete all-optical silica fiber isolator via stimulated Brillouin scattering,” J. Lightwave Technol. 29, 2267–2275 (2011). [CrossRef]  

20. M. Kang, A. Butsch, and P. St. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5, 549–553 (2011). [CrossRef]  

21. 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]  

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

23. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2006).

24. A. E. Kaplan and P. Meystre, “Directionally asymmetrical bistability in a symmetrically pumped nonlinear ring interferometer,” Opt. Commun. 40, 229–232 (1982). [CrossRef]  

25. Q. T. Cao, H. M. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. H. Gong, and Y. F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033901 (2017). [CrossRef]  

26. L. Del Bino, J. M. Silver, S. L. Stebbings, and P. Del’Haye, “Symmetry breaking of counter-propagating light in a nonlinear resonator,” Sci. Rep. 7, 43142 (2017). [CrossRef]  

27. P. Pintus, F. D. Pasquale, and J. E. Bowers, “Integrated TE and TM optical circulators on ultra-low-loss silicon nitride platform,” Opt. Express 21, 5041–5052 (2013). [CrossRef]  

28. T. Carmon, L. Yang, and K. J. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express 12, 4742–4750 (2004). [CrossRef]  

29. I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102, 043902 (2009). [CrossRef]  

30. A. A. Savchenkov, V. S. Ilchenko, F. Di Teodoro, P. M. Belden, W. T. Lotshaw, A. B. Matsko, and L. Maleki, “Generation of Kerr combs centered at 4.5 um in crystalline microresonators pumped with quantum-cascade lasers,” Opt. Lett. 40, 3468–3471 (2015). [CrossRef]  

31. Y. Xuan, Y. Liu, L. Varghese, A. J. Metcalf, X. Xue, P.-H. Wang, K. Han, J. A. Jaramillo-Villegas, S. Kim, F. Li, J. Wang, B. Niu, M. Teng, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultra-high-Q silicon nitride micro-resonators for low-power frequency comb initiation,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2016), paper JW2A.75.

32. D. Huang, P. Pintus, C. Zhang, Y. Shoji, T. Mizumoto, and J. E. Bowers, “Electrically driven and thermally tunable integrated optical isolators for silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 22, 271–278 (2016). [CrossRef]  

References

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  1. R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004).
    [Crossref]
  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]
  3. 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, 758–762 (2011).
    [Crossref]
  4. H. Dotsch, N. Bahlmann, O. Zhuromskyy, M. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B 22, 240–253 (2005).
    [Crossref]
  5. N. Kono, K. Kakihara, K. Saitoh, and M. Koshiba, “Nonreciprocal microresonators for the miniaturization of optical waveguide isolators,” Opt. Express 15, 7737–7751 (2007).
    [Crossref]
  6. Y. Shoji, T. Mizumoto, H. Yokoi, I.-W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92, 071117 (2008).
    [Crossref]
  7. 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, 1429–1437 (2017).
    [Crossref]
  8. L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
    [Crossref]
  9. K. Gallo, G. Assanto, K. Parameswaran, and M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
    [Crossref]
  10. K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13, 465–471 (2017).
    [Crossref]
  11. S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102, 213903 (2009).
    [Crossref]
  12. F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016).
    [Crossref]
  13. Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
    [Crossref]
  14. Z. Wang, L. Shi, Y. Liu, X. Xu, and X. Zhang, “Optical nonreciprocity in asymmetric optomechanical couplers,” Sci. Rep. 5, 8657 (2015).
    [Crossref]
  15. L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
    [Crossref]
  16. B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
    [Crossref]
  17. H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 033901 (2012).
    [Crossref]
  18. Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
    [Crossref]
  19. X. Huang and S. Fan, “Complete all-optical silica fiber isolator via stimulated Brillouin scattering,” J. Lightwave Technol. 29, 2267–2275 (2011).
    [Crossref]
  20. M. Kang, A. Butsch, and P. St. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5, 549–553 (2011).
    [Crossref]
  21. 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]
  22. Y. Shi, Z. Yu, and S. Fan, “Limitations of nonlinear optical isolators due to dynamic reciprocity,” Nat. Photonics 9, 388–392 (2015).
    [Crossref]
  23. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2006).
  24. A. E. Kaplan and P. Meystre, “Directionally asymmetrical bistability in a symmetrically pumped nonlinear ring interferometer,” Opt. Commun. 40, 229–232 (1982).
    [Crossref]
  25. Q. T. Cao, H. M. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. H. Gong, and Y. F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033901 (2017).
    [Crossref]
  26. L. Del Bino, J. M. Silver, S. L. Stebbings, and P. Del’Haye, “Symmetry breaking of counter-propagating light in a nonlinear resonator,” Sci. Rep. 7, 43142 (2017).
    [Crossref]
  27. P. Pintus, F. D. Pasquale, and J. E. Bowers, “Integrated TE and TM optical circulators on ultra-low-loss silicon nitride platform,” Opt. Express 21, 5041–5052 (2013).
    [Crossref]
  28. T. Carmon, L. Yang, and K. J. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express 12, 4742–4750 (2004).
    [Crossref]
  29. I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102, 043902 (2009).
    [Crossref]
  30. A. A. Savchenkov, V. S. Ilchenko, F. Di Teodoro, P. M. Belden, W. T. Lotshaw, A. B. Matsko, and L. Maleki, “Generation of Kerr combs centered at 4.5 um in crystalline microresonators pumped with quantum-cascade lasers,” Opt. Lett. 40, 3468–3471 (2015).
    [Crossref]
  31. Y. Xuan, Y. Liu, L. Varghese, A. J. Metcalf, X. Xue, P.-H. Wang, K. Han, J. A. Jaramillo-Villegas, S. Kim, F. Li, J. Wang, B. Niu, M. Teng, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultra-high-Q silicon nitride micro-resonators for low-power frequency comb initiation,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2016), paper JW2A.75.
  32. D. Huang, P. Pintus, C. Zhang, Y. Shoji, T. Mizumoto, and J. E. Bowers, “Electrically driven and thermally tunable integrated optical isolators for silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 22, 271–278 (2016).
    [Crossref]

2017 (4)

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, 1429–1437 (2017).
[Crossref]

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13, 465–471 (2017).
[Crossref]

Q. T. Cao, H. M. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. H. Gong, and Y. F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033901 (2017).
[Crossref]

L. Del Bino, J. M. Silver, S. L. Stebbings, and P. Del’Haye, “Symmetry breaking of counter-propagating light in a nonlinear resonator,” Sci. Rep. 7, 43142 (2017).
[Crossref]

2016 (3)

D. Huang, P. Pintus, C. Zhang, Y. Shoji, T. Mizumoto, and J. E. Bowers, “Electrically driven and thermally tunable integrated optical isolators for silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 22, 271–278 (2016).
[Crossref]

F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

2015 (3)

Z. Wang, L. Shi, Y. Liu, X. Xu, and X. Zhang, “Optical nonreciprocity in asymmetric optomechanical couplers,” Sci. Rep. 5, 8657 (2015).
[Crossref]

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

A. A. Savchenkov, V. S. Ilchenko, F. Di Teodoro, P. M. Belden, W. T. Lotshaw, A. B. Matsko, and L. Maleki, “Generation of Kerr combs centered at 4.5 um in crystalline microresonators pumped with quantum-cascade lasers,” Opt. Lett. 40, 3468–3471 (2015).
[Crossref]

2014 (2)

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[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]

P. Pintus, F. D. Pasquale, and J. E. Bowers, “Integrated TE and TM optical circulators on ultra-low-loss silicon nitride platform,” Opt. Express 21, 5041–5052 (2013).
[Crossref]

2012 (2)

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

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 (4)

X. Huang and S. Fan, “Complete all-optical silica fiber isolator via stimulated Brillouin scattering,” J. Lightwave Technol. 29, 2267–2275 (2011).
[Crossref]

M. Kang, A. Butsch, and P. St. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5, 549–553 (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, 758–762 (2011).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

2009 (3)

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102, 213903 (2009).
[Crossref]

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

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102, 043902 (2009).
[Crossref]

2008 (1)

Y. Shoji, T. Mizumoto, H. Yokoi, I.-W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92, 071117 (2008).
[Crossref]

2007 (1)

2005 (1)

2004 (2)

2001 (1)

K. Gallo, G. Assanto, K. Parameswaran, and M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
[Crossref]

1982 (1)

A. E. Kaplan and P. Meystre, “Directionally asymmetrical bistability in a symmetrically pumped nonlinear ring interferometer,” Opt. Commun. 40, 229–232 (1982).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2006).

Alù, A.

F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016).
[Crossref]

Assanto, G.

K. Gallo, G. Assanto, K. Parameswaran, and M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
[Crossref]

Ayache, M.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[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]

Bahlmann, N.

Belden, P. M.

Bender, C. M.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

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, 758–762 (2011).
[Crossref]

Bowers, J. E.

Butsch, A.

M. Kang, A. Butsch, and P. St. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5, 549–553 (2011).
[Crossref]

Cao, Q. T.

Q. T. Cao, H. M. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. H. Gong, and Y. F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033901 (2017).
[Crossref]

Carmon, T.

Chang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Chen, X.

Q. T. Cao, H. M. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. H. Gong, and Y. F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033901 (2017).
[Crossref]

Chen, Y.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

Chen, Y.-F.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Clerk, A. A.

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13, 465–471 (2017).
[Crossref]

Del Bino, L.

L. Del Bino, J. M. Silver, S. L. Stebbings, and P. Del’Haye, “Symmetry breaking of counter-propagating light in a nonlinear resonator,” Sci. Rep. 7, 43142 (2017).
[Crossref]

Del’Haye, P.

L. Del Bino, J. M. Silver, S. L. Stebbings, and P. Del’Haye, “Symmetry breaking of counter-propagating light in a nonlinear resonator,” Sci. Rep. 7, 43142 (2017).
[Crossref]

Di Teodoro, F.

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, 758–762 (2011).
[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]

Dong, C. H.

Q. T. Cao, H. M. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. H. Gong, and Y. F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033901 (2017).
[Crossref]

Dong, C.-H.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

Dotsch, H.

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]

Fainman, Y.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[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]

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

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]

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

X. Huang and S. Fan, “Complete all-optical silica fiber isolator via stimulated Brillouin scattering,” J. Lightwave Technol. 29, 2267–2275 (2011).
[Crossref]

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

Fang, K.

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13, 465–471 (2017).
[Crossref]

Fejer, M.

K. Gallo, G. Assanto, K. Parameswaran, and M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
[Crossref]

Feng, L.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[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]

Gallo, K.

K. Gallo, G. Assanto, K. Parameswaran, and M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
[Crossref]

Ge, L.

Q. T. Cao, H. M. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. H. Gong, and Y. F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033901 (2017).
[Crossref]

Gerhardt, R.

Gianfreda, M.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Gong, Q. H.

Q. T. Cao, H. M. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. H. Gong, and Y. F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033901 (2017).
[Crossref]

Grudinin, I. S.

I. S. Grudinin, A. B. Matsko, and L. Maleki, “Brillouin lasing with a CaF2 whispering gallery mode resonator,” Phys. Rev. Lett. 102, 043902 (2009).
[Crossref]

Guo, G.-C.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

Hammer, M.

Han, K.

Y. Xuan, Y. Liu, L. Varghese, A. J. Metcalf, X. Xue, P.-H. Wang, K. Han, J. A. Jaramillo-Villegas, S. Kim, F. Li, J. Wang, B. Niu, M. Teng, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultra-high-Q silicon nitride micro-resonators for low-power frequency comb initiation,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2016), paper JW2A.75.

Hertel, P.

Hsieh, I.-W.

Y. Shoji, T. Mizumoto, H. Yokoi, I.-W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92, 071117 (2008).
[Crossref]

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, 758–762 (2011).
[Crossref]

Hua, S.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Huang, D.

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, 1429–1437 (2017).
[Crossref]

D. Huang, P. Pintus, C. Zhang, Y. Shoji, T. Mizumoto, and J. E. Bowers, “Electrically driven and thermally tunable integrated optical isolators for silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 22, 271–278 (2016).
[Crossref]

Huang, J.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref]

Huang, X.

Ilchenko, V. S.

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]

Jaramillo-Villegas, J. A.

Y. Xuan, Y. Liu, L. Varghese, A. J. Metcalf, X. Xue, P.-H. Wang, K. Han, J. A. Jaramillo-Villegas, S. Kim, F. Li, J. Wang, B. Niu, M. Teng, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultra-high-Q silicon nitride micro-resonators for low-power frequency comb initiation,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2016), paper JW2A.75.

Jiang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

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, 758–762 (2011).
[Crossref]

Jiang, X.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Jing, H.

Q. T. Cao, H. M. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. H. Gong, and Y. F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033901 (2017).
[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]

Kakihara, K.

Kang, M.

M. Kang, A. Butsch, and P. St. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5, 549–553 (2011).
[Crossref]

Kaplan, A. E.

A. E. Kaplan and P. Meystre, “Directionally asymmetrical bistability in a symmetrically pumped nonlinear ring interferometer,” Opt. Commun. 40, 229–232 (1982).
[Crossref]

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, 758–762 (2011).
[Crossref]

Kim, S.

Y. Xuan, Y. Liu, L. Varghese, A. J. Metcalf, X. Xue, P.-H. Wang, K. Han, J. A. Jaramillo-Villegas, S. Kim, F. Li, J. Wang, B. Niu, M. Teng, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultra-high-Q silicon nitride micro-resonators for low-power frequency comb initiation,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2016), paper JW2A.75.

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, 758–762 (2011).
[Crossref]

Kono, N.

Koshiba, M.

Leaird, D. E.

Y. Xuan, Y. Liu, L. Varghese, A. J. Metcalf, X. Xue, P.-H. Wang, K. Han, J. A. Jaramillo-Villegas, S. Kim, F. Li, J. Wang, B. Niu, M. Teng, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultra-high-Q silicon nitride micro-resonators for low-power frequency comb initiation,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2016), paper JW2A.75.

Lei, F.

B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Li, F.

Y. Xuan, Y. Liu, L. Varghese, A. J. Metcalf, X. Xue, P.-H. Wang, K. Han, J. A. Jaramillo-Villegas, S. Kim, F. Li, J. Wang, B. Niu, M. Teng, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultra-high-Q silicon nitride micro-resonators for low-power frequency comb initiation,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2016), paper JW2A.75.

Li, G.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

Lipson, M.

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

S. Manipatruni, J. T. Robinson, and M. Lipson, “Optical nonreciprocity in optomechanical structures,” Phys. Rev. Lett. 102, 213903 (2009).
[Crossref]

Lira, H.

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

Liu, R. S.

Q. T. Cao, H. M. Wang, C. H. Dong, H. Jing, R. S. Liu, X. Chen, L. Ge, Q. H. Gong, and Y. F. Xiao, “Experimental demonstration of spontaneous chirality in a nonlinear microresonator,” Phys. Rev. Lett. 118, 033901 (2017).
[Crossref]

Liu, Y.

Z. Wang, L. Shi, Y. Liu, X. Xu, and X. Zhang, “Optical nonreciprocity in asymmetric optomechanical couplers,” Sci. Rep. 5, 8657 (2015).
[Crossref]

Y. Xuan, Y. Liu, L. Varghese, A. J. Metcalf, X. Xue, P.-H. Wang, K. Han, J. A. Jaramillo-Villegas, S. Kim, F. Li, J. Wang, B. Niu, M. Teng, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultra-high-Q silicon nitride micro-resonators for low-power frequency comb initiation,” in Conference on Lasers and Electro-Optics (Optical Society of America, 2016), paper JW2A.75.

Long, G. L.

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Vanwolleghem, M.

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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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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, 1429–1437 (2017).
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Z. Wang, L. Shi, Y. Liu, X. Xu, and X. Zhang, “Optical nonreciprocity in asymmetric optomechanical couplers,” Sci. Rep. 5, 8657 (2015).
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Zhang, Y.-L.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
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Zhuromskyy, O.

Zou, C.-L.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
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Zou, X.-B.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
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Appl. Phys. Lett. (2)

Y. Shoji, T. Mizumoto, H. Yokoi, I.-W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92, 071117 (2008).
[Crossref]

K. Gallo, G. Assanto, K. Parameswaran, and M. Fejer, “All-optical diode in a periodically poled lithium niobate waveguide,” Appl. Phys. Lett. 79, 314–316 (2001).
[Crossref]

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

D. Huang, P. Pintus, C. Zhang, Y. Shoji, T. Mizumoto, and J. E. Bowers, “Electrically driven and thermally tunable integrated optical isolators for silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 22, 271–278 (2016).
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J. Lightwave Technol. (2)

J. Opt. Soc. Am. B (1)

Nat. Commun. (1)

F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016).
[Crossref]

Nat. Photonics (7)

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photonics 8, 524–529 (2014).
[Crossref]

M. Kang, A. Butsch, and P. St. J. Russell, “Reconfigurable light-driven opto-acoustic isolators in photonic crystal fibre,” Nat. Photonics 5, 549–553 (2011).
[Crossref]

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Supplementary Material (1)

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Figures (3)

Fig. 1.
Fig. 1. Kerr nonreciprocity in a microresonator and the experimental setup for an isolator. (a) Schematic of the nonreciprocity. Light is coupled in the ccw direction through an ultra-high-Q microresonator in an add–drop configuration. The presence of the ccw light induces a Kerr shift that is twice as strong for cw circulating light, which leads to a resonance frequency splitting (shown in the lower panel). Consequently, backward-propagating light (cw) cannot pass through the resonator. (b) The experimental setup for the characterization of an isolator based on the Kerr nonreciprocity. Laser light is coupled through an ultra-high-Q silica microrod resonator using two tapered optical fibers. A fiber mirror simulates an optical circuit that reflects 100% of the incident light, and the polarization is adjusted to maximize the power coupled into the resonator in the backward direction. Photodiodes monitor the input, transmission, and return powers of the laser, while the laser power is adjusted with a variable attenuator. Circ = fiber-optical circulator, PD = photodiode.
Fig. 2.
Fig. 2. Kerr-nonlinearity-induced isolation at 1550 nm, theory and experiment. (a) The measurement of the return power Pback versus the input power Pccwin with a theoretical fit. For low input powers, the return power is proportional to the input power, as expected for a linear system. For high powers, the return power decreases due to the nonreciprocal Kerr effect. (b) The insertion loss Pout/Pccwin and isolation Pback/Pccwin versus input power. We measure an insertion loss of around 7 dB and a maximum isolation in excess of 24 dB. (c) The predicted isolation for waveguide resonators of various materials, assuming an effective mode cross-sectional area of 4μm2 and a resonator diameter of 100 μm. Our measurement (SiO2 rod resonator, diameter 1 mm) is included for comparison. The respective Q-factors for the calculations are 109 for magnesium fluoride and calcium fluoride [29,30], 5×107 for silicon nitride [31], and 5×108 for SiO2. Our calculations show isolation at sub-milliwatt power levels and even down to tens of microwatts in these currently available microresonator systems.
Fig. 3.
Fig. 3. Microresonator-based circulator. (a) A schematic of the setup. Light is split between ports 1 and 2, with slightly more light going into port 1, so that the resonator is set into the ccw state. Photodiodes measure the optical output power from each port. (b) The transmission Tij (from port i to port j) as the laser frequency is swept through the resonance.

Equations (4)

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Pccw=ηinPccwin1+(δγ+1P0(Pccw+2Pcw))2,
Pcw=ηinPcwin1+(δγ+1P0(Pcw+2Pccw))2.
P0=π2n02dAeff(γk)QL2n2λγ,
I=ηthru1+(PccwinηinP0(1IR))2.

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