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  • Letter
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Enhanced sensitivity at higher-order exceptional points

Nature volume 548, pages 187–191 (2017)Cite this article

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An Erratum to this article was published on 30 November 2017

This article has been updated

Abstract

Non-Hermitian degeneracies, also known as exceptional points, have recently emerged as a new way to engineer the response of open physical systems, that is, those that interact with the environment. They correspond to points in parameter space at which the eigenvalues of the underlying system and the corresponding eigenvectors simultaneously coalesce1,2,3. In optics, the abrupt nature of the phase transitions that are encountered around exceptional points has been shown to lead to many intriguing phenomena, such as loss-induced transparency4, unidirectional invisibility5,6, band merging7,8, topological chirality9,10 and laser mode selectivity11,12. Recently, it has been shown that the bifurcation properties of second-order non-Hermitian degeneracies can provide a means of enhancing the sensitivity (frequency shifts) of resonant optical structures to external perturbations13. Of particular interest is the use of even higher-order exceptional points (greater than second order), which in principle could further amplify the effect of perturbations, leading to even greater sensitivity. Although a growing number of theoretical studies have been devoted to such higher-order degeneracies14,15,16, their experimental demonstration in the optical domain has so far remained elusive. Here we report the observation of higher-order exceptional points in a coupled cavity arrangement—specifically, a ternary, parity–time-symmetric photonic laser molecule—with a carefully tailored gain–loss distribution. We study the system in the spectral domain and find that the frequency response associated with this system follows a cube-root dependence on induced perturbations in the refractive index. Our work paves the way for utilizing non-Hermitian degeneracies in fields including photonics, optomechanics10, microwaves9 and atomic physics17,18.

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Figure 1: Parity–time-symmetric coupled cavity systems that support exceptional points.
Figure 2: Bifurcations of complex eigenfrequencies around a third-order exceptional point.
Figure 3: Binary parity–time-symmetric system operating around a second-order exceptional point.
Figure 4: Response of a ternary parity–time-symmetric system biased at a third-order exceptional point.

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Change history

  • 08 November 2017

    Please see accompanying Erratum (http://doi.org/10.1038/nature24024). In this Letter, the y-axis labels of the inset to Fig. 3a should have been 102 rather than 101 (top label) and 101 rather than 100 (bottom label). This error has been corrected online.

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Acknowledgements

We acknowledge financial support from the Office of Naval Research (ONR; N00014-16-1-2640), the National Science Foundation (NSF; ECCS-1454531, DMR-1420620), the Air force Office of Scientific Research (AFOSR; FA9550-14-1-0037) and the Army Research Office (ARO; W911NF-16-1-0013). This work was also partially funded by the Qatar National Research Fund (NPRP 9-020-1-006). R.E.-G. acknowledges support from the Henes Center for Quantum Phenomena at Michigan Tech University. H.H. thanks J. Boroumand from UCF for helping with the SEM images.

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Authors and Affiliations

  1. CREOL, The College of Optics and Photonics, University of Central Florida, Orlando, 32816, Florida, USA

    Hossein Hodaei, Absar U. Hassan, Steffen Wittek, Hipolito Garcia-Gracia, Demetrios N. Christodoulides & Mercedeh Khajavikhan

  2. Department of Physics and Henes Center for Quantum Phenomena, Michigan Technological University, Houghton, 49931, Michigan, USA

    Ramy El-Ganainy

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Contributions

H.H., M.K. and D.N.C. conceived and designed the experiments, H.H. and H.G.-G. fabricated the samples, H.H. and S.W. performed the experiments, H.H., A.U.H., M.K., R.E.-G. and D.N.C. analysed the data, and H.H., M.K., D.N.C., R.E.-G. and A.U.H. wrote the paper.

Corresponding author

Correspondence to Mercedeh Khajavikhan.

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Competing interests

The authors declare no competing financial interests.

Additional information

Reviewer Information Nature thanks K. Bliokh, M. Rechtsman, Y.-F. Xiao and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Figure 1 Fabrication steps in realizing parity–time-symmetric photonic molecules.

a, Schematic of the fabrication process. b, A microscope image of the fabricated metallic micro-heaters. c, The heaters are electrically connected to the pins of the header via wire bonding. d, The photonic molecule systems are accessible for measurement from the back side through a hole in the header.

Extended Data Figure 2 Measurement station for characterizing sensitivity at higher-order exceptional points.

A schematic of the micro-photoluminescence experimental set-up. BS, beam splitter; NIR, near-infrared; NA, numerical aperture.

Extended Data Figure 3 Pump distribution.

The intensity profile of the pump beam at the sample without (a) and with (b) the image of the knife edge. By adjusting the intensity of the pump beam and the position of the knife edge, the desired gain/loss distribution is realized.

Extended Data Figure 4 Sample imaging.

a, The intensity profile of three coupled micro-ring resonators when they all pumped equally. b, The associated heaters imaged on the measurement station using a broadband near-infrared source.

Extended Data Figure 5 Characterizing the heat induced detuning.

a, b, The change in resonance frequency (Δω) of the intentionally decoupled cavities as a function of I2 (heater power) in the binary structure (a) and the ternary structure (b). The applied perturbation varies linearly with the power of the heater, and is imposed differentially on the micro-rings with respect to their distance from the active heater. The colours of the rings are used for distinction purposes and not as a representation of gain or loss. In all cases, solid circles indicate measured data and solid lines denote linear fits to the frequency response.

Extended Data Figure 6 Effect of coupling on enhancing sensitivity.

a, b, The wavelength splitting as a function of the differential perturbation applied to the gain cavity, for parity–time systems with different coupling coefficients (solid curves depict square-root simulations) and for a single micro-ring (straight line), on linear (a) and logarithmic (b) scales. In all six cases, filled circles indicate measured values and the error bars represent the uncertainty in the measurement because of the resolution limit of the spectrometer.

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Hodaei, H., Hassan, A., Wittek, S. et al. Enhanced sensitivity at higher-order exceptional points. Nature 548, 187–191 (2017). https://doi.org/10.1038/nature23280

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