Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Exceptional points in lossy media lead to deep polynomial wave penetration with spatially uniform power loss

Nature Nanotechnology volume 17, pages 583–589 (2022)Cite this article

Subjects

Abstract

Waves entering a spatially uniform lossy medium typically undergo exponential intensity decay, arising from either the energy loss of the Beer–Lambert–Bouguer transmission law or the evanescent penetration during reflection. Recently, exceptional point singularities in non-Hermitian systems have been linked to unconventional wave propagation. Here, we theoretically propose and experimentally demonstrate exponential decay free wave propagation in a purely lossy medium. We observe up to 400-wave deep polynomial wave propagation accompanied by a uniformly distributed energy loss across a nanostructured photonic slab waveguide with exceptional points. We use coupled-mode theory and fully vectorial electromagnetic simulations to predict deep wave penetration manifesting spatially constant radiation losses through the entire structured waveguide region regardless of its length. The uncovered exponential decay free wave phenomenon is universal and holds true across all domains supporting physical waves, finding immediate applications for generating large, uniform and surface-normal free-space plane waves directly from dispersion-engineered photonic chip surfaces.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Linear wave penetration and uniform energy losses across a periodically structured medium with EP-containing band diagrams.
Fig. 2: CMT and numerical analysis of the light radiated by a periodically structured SiNx slab waveguide.
Fig. 3: Experimental observation of the deep linear wave penetration with decay-free radiation profiles.
Fig. 4: Radiation loss profiles in non-Hermitian periodically structured media.

Similar content being viewed by others

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

References

  1. Miri, M.-A. & Alù, A. Exceptional points in optics and photonics. Science 363, eaar7709 (2019).

    Article  CAS  Google Scholar 

  2. Özdemir, Ş., Rotter, S., Nori, F. & Yang, L. Parity–time symmetry and exceptional points in photonics. Nat. Mater. 18, 783–798 (2019).

    Article  Google Scholar 

  3. Peng, B. et al. Loss-induced suppression and revival of lasing. Science 346, 328–332 (2014).

    Article  CAS  Google Scholar 

  4. Peng, B. et al. Chiral modes and directional lasing at exceptional points. Proc. Natl Acad. Sci. USA 113, 6845–6850 (2016).

    Article  CAS  Google Scholar 

  5. Brandstetter, M. et al. Reversing the pump dependence of a laser at an exceptional point. Nat. Commun. 5, 4034 (2014).

    Article  CAS  Google Scholar 

  6. Hodaei, H., Miri, M.-A., Heinrich, M., Christodoulides, D. N. & Khajavikhan, M. Parity-time–symmetric microring lasers. Science 346, 975–978 (2014).

    Article  CAS  Google Scholar 

  7. Feng, L., Wong, Z. J., Ma, R.-M., Wang, Y. & Zhang, X. Single-mode laser by parity-time symmetry breaking. Science 346, 972–975 (2014).

    Article  CAS  Google Scholar 

  8. Chen, W., Özdemir, Ş. K., Zhao, G., Wiersig, J. & Yang, L. Exceptional points enhance sensing in an optical microcavity. Nature 548, 192–196 (2017).

    Article  CAS  Google Scholar 

  9. Hodaei, H. et al. Enhanced sensitivity at higher-order exceptional points. Nature 548, 187–191 (2017).

    Article  CAS  Google Scholar 

  10. Guo, A. et al. Observation of P T-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009).

    Article  CAS  Google Scholar 

  11. Rüter, C. E. et al. Observation of parity–time symmetry in optics. Nat. Phys. 6, 192–195 (2010).

    Article  Google Scholar 

  12. Peng, B. et al. Parity–time-symmetric whispering-gallery microcavities. Nat. Phys. 10, 394–398 (2014).

    Article  CAS  Google Scholar 

  13. Feng, L. et al. Demonstration of a large-scale optical exceptional point structure. Opt. Express 22, 1760–1767 (2014).

    Article  CAS  Google Scholar 

  14. Regensburger, A. et al. Parity–time synthetic photonic lattices. Nature 488, 167–171 (2012).

    Article  CAS  Google Scholar 

  15. Doppler, J. et al. Dynamically encircling an exceptional point for asymmetric mode switching. Nature 537, 76–79 (2016).

    Article  CAS  Google Scholar 

  16. Poli, C., Bellec, M., Kuhl, U., Mortessagne, F. & Schomerus, H. Selective enhancement of topologically induced interface states in a dielectric resonator chain. Nat. Commun. 6, 1–5 (2015).

    Article  Google Scholar 

  17. Weimann, S. et al. Topologically protected bound states in photonic parity–time-symmetric crystals. Nat. Mater. 16, 433–438 (2017).

    Article  CAS  Google Scholar 

  18. Shen, H., Zhen, B. & Fu, L. Topological band theory for non-Hermitian Hamiltonians. Phys. Rev. Lett. 120, 146402 (2018).

    Article  CAS  Google Scholar 

  19. Yoon, J. W. et al. Time-asymmetric loop around an exceptional point over the full optical communications band. Nature 562, 86–90 (2018).

    Article  CAS  Google Scholar 

  20. Lin, Z. et al. Unidirectional invisibility induced by PT-symmetric periodic structures. Phys. Rev. Lett. 106, 213901 (2011).

    Article  Google Scholar 

  21. Feng, L. et al. Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical frequencies. Nat. Mater. 12, 108–113 (2013).

    Article  CAS  Google Scholar 

  22. Zhen, B. et al. Spawning rings of exceptional points out of Dirac cones. Nature 525, 354–358 (2015).

    Article  CAS  Google Scholar 

  23. Lee, S.-G. & Magnusson, R. Band flips and bound-state transitions in leaky-mode photonic lattices. Phys. Rev. B 99, 045304 (2019).

    Article  CAS  Google Scholar 

  24. Daxhelet, X. & Kulishov, M. Theory and practice of long-period gratings: when a loss becomes a gain. Opt. Lett. 28, 686–688 (2003).

    Article  Google Scholar 

  25. Kulishov, M., Jones, H. & Kress, B. Analysis of PT-symmetric volume gratings beyond the paraxial approximation. Opt. Express 23, 9347–9362 (2015).

    Article  CAS  Google Scholar 

  26. Kulishov, M., Kress, B. & Slavík, R. Resonant cavities based on parity-time-symmetric diffractive gratings. Opt. Express 21, 9473–9483 (2013).

    Article  Google Scholar 

  27. Makris, K. G., Musslimani, Z. H., Christodoulides, D. N. & Rotter, S. Constant-intensity waves and their modulation instability in non-Hermitian potentials. Nat. Commun. 6, 1–7 (2015).

    Article  Google Scholar 

  28. Makris, K., Krešić, I., Brandstötter, A. & Rotter, S. Scattering-free channels of invisibility across non-Hermitian media. Optica 7, 619–623 (2020).

    Article  Google Scholar 

  29. Makris, K. G., Ge, L. & Türeci, H. Anomalous transient amplification of waves in non-normal photonic media. Phys. Rev. X 4, 041044 (2014).

    Google Scholar 

  30. Rivet, E. et al. Constant-pressure sound waves in non-Hermitian disordered media. Nat. Phys. 14, 942–947 (2018).

    Article  CAS  Google Scholar 

  31. Hardy, A., Welch, D. F. & Streifer, W. Analysis of second-order gratings. IEEE J. Quantum Electron. 25, 2096–2105 (1989).

    Article  CAS  Google Scholar 

  32. Kazarinov, R. & Henry, C. Second-order distributed feedback lasers with mode selection provided by first-order radiation losses. IEEE J. Quantum Electron. 21, 144–150 (1985).

    Article  Google Scholar 

  33. Hsu, C. W., Zhen, B., Stone, A. D., Joannopoulos, J. D. & Soljačić, M. Bound states in the continuum. Nat. Rev. Mater. 1, 16048 (2016).

    Article  CAS  Google Scholar 

  34. Koshelev, K., Lepeshov, S., Liu, M., Bogdanov, A. & Kivshar, Y. Asymmetric metasurfaces with high-Q resonances governed by bound states in the continuum. Phys. Rev. Lett. 121, 193903 (2018).

    Article  Google Scholar 

  35. Yesilkoy, F. et al. Ultrasensitive hyperspectral imaging and biodetection enabled by dielectric metasurfaces. Nat. Photonics 13, 390–396 (2019).

    Article  CAS  Google Scholar 

  36. Kim, S. et al. Photonic waveguide to free-space Gaussian beam extreme mode converter. Light Sci. Appl. 7, 72 (2018).

    Article  Google Scholar 

  37. Yulaev, A. et al. Metasurface-integrated photonic platform for versatile free-space beam projection with polarization control. ACS Photonics 6, 2902–2909 (2019).

    Article  CAS  Google Scholar 

  38. Hummon, M. T. et al. Photonic chip for laser stabilization to an atomic vapor with 10−11 instability. Optica 5, 443–449 (2018).

    Article  CAS  Google Scholar 

  39. Berman, P. R. Atom Interferometry (Academic Press, 1997).

Download references

Acknowledgements

A.Y. acknowledges support under the Professional Research Experience Program (PREP), funded by the National Institute of Standards and Technology and administered through the Department of Chemistry and Biochemistry, University of Maryland. We also thank T. LeBrun, H. Lezec, J. Liu, J. A. Liddle and E. Secula for reading the manuscript and making insightful comments. The research was performed in part at the NIST Center for Nanoscale Science and Technology. This work is partially supported by the NIST-on-a-chip programme.

Author information

Author notes

  1. These authors contributed equally: Alexander Yulaev, Sangsik Kim.

Authors and Affiliations

  1. Physical Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, MD, USA

    Alexander Yulaev, Sangsik Kim, Qing Li, Daron A. Westly, Brian J. Roxworthy, Kartik Srinivasan & Vladimir A. Aksyuk

  2. Department of Chemistry and Biochemistry, University of Maryland, College Park, MD, USA

    Alexander Yulaev

  3. Department of Electrical and Computer Engineering, Texas Tech University, Lubbock, TX, USA

    Sangsik Kim

  4. Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA, USA

    Qing Li

  5. Aeva, Inc., Mountain View, CA, USA

    Brian J. Roxworthy

Authors
  1. Alexander Yulaev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sangsik Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qing Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Daron A. Westly
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Brian J. Roxworthy
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Kartik Srinivasan
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Vladimir A. Aksyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

A.Y. and S.K. contributed to the project equally. S.K. conceived the idea. A.Y. and S.K. performed FEM and FDTD simulations. D.A.W. and A.Y. fabricated samples. Q.L., B.J.R. and K.S. contributed to the experimental device design and fabrication process development. A.Y. and S.K. characterized samples. A.Y. and V.A.A. developed theoretical analysis. A.Y., S.K. and V.A.A. contributed to the data interpretation. V.A.A. supervised the project. The manuscript was written through the extensive contributions of all authors. All authors have approved the final version of the manuscript.

Corresponding authors

Correspondence to Alexander Yulaev, Sangsik Kim or Vladimir A. Aksyuk.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Nanotechnology thanks Stefan Rotter and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

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

Extended data

Extended Data Fig. 1 Free-space beam angle and intensity dependences on the optical wavelength.

(a) The orange curve shows the radiation angle as a function of the wavelength for DC = 0.527. The black dashed line indicates 0°. (b) The corresponding band diagram (BD) for infinite grating. The dashed blue arrows depict the wavelength tuning. Once the wavelength crosses the EPs in the BD (panel b), the outcoupling angle experiences high variation about the chip’s normal, that is noticeable as a double kink in the wavelength dependence of the outcoupled angle (panel a).

Extended Data Fig. 2 Spatially uniform and non-uniform energy losses in non-Hermitian periodically structured media excited from one side.

FEM simulated time-averaged power flow along the uniform periodically structured waveguide in the positive z-direction (a) and averaged |E| over one grating period (low-pass spatial filtering) (b) vs. the coordinate z for the optical frequency tuned to the EP wavelength (red curve) and detuned (black curve) from the EP wavelength. The waveguide is excited by a wave incident from the left side at z = 10 µm.

Extended Data Fig. 3 Linear radiative energy losses experimentally observed in 45 µm, 100 µm, 180 µm, and 250 µm long gratings with ≈ 50 % DC.

Insets depict top-view optical images of the projected free-space beams. The saturated ≈ 20 µm wide spike at the grating input is due to the fabrication imperfections related to the electron lithography proximity effect (Supplementary Fig. 10).

Extended Data Fig. 4 Sample fabrication flowchart.

The partially etched gratings and supporting photonic structures are fabricated by sequential electron beam lithography patterning and etching of a silicon nitride layer, followed by silicon dioxide cladding deposition, as described in the Sample Fabrication section of the Methods.

Extended Data Fig. 5 Experimental apparatus.

Light from a tunable laser was fiber coupled through an attenuator (A) and polarization controller (PC) to the inverted taper waveguide couplers on the grating chip (device under test, DUT). Grating emission was imaged by a microscope camera. Free space linear polarizer P was used for polarization selective imaging to tune the input fiber polarization to transverse electric. Images were captured at the grating plane with additional images captured above the grating plane to assess emission angle.

Supplementary information

Supplementary Information

Supplementary discussion and Figs. 1–13.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yulaev, A., Kim, S., Li, Q. et al. Exceptional points in lossy media lead to deep polynomial wave penetration with spatially uniform power loss. Nat. Nanotechnol. 17, 583–589 (2022). https://doi.org/10.1038/s41565-022-01114-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41565-022-01114-3

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing