Observation of trapped light within the radiation continuum

Journal name:
Nature
Volume:
499,
Pages:
188–191
Date published:
DOI:
doi:10.1038/nature12289
Received
Accepted
Published online

The ability to confine light is important both scientifically and technologically. Many light confinement methods exist, but they all achieve confinement with materials or systems that forbid outgoing waves. These systems can be implemented by metallic mirrors, by photonic band-gap materials1, by highly disordered media (Anderson localization2) and, for a subset of outgoing waves, by translational symmetry (total internal reflection1) or by rotational or reflection symmetry3, 4. Exceptions to these examples exist only in theoretical proposals5, 6, 7, 8. Here we predict and show experimentally that light can be perfectly confined in a patterned dielectric slab, even though outgoing waves are allowed in the surrounding medium. Technically, this is an observation of an ‘embedded eigenvalue’9—namely, a bound state in a continuum of radiation modes—that is not due to symmetry incompatibility5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16. Such a bound state can exist stably in a general class of geometries in which all of its radiation amplitudes vanish simultaneously as a result of destructive interference. This method to trap electromagnetic waves is also applicable to electronic12 and mechanical waves14, 15.

At a glance

Figures

  1. Predictions of the theory.
    Figure 1: Predictions of the theory.

    a, Diagram of the photonic crystal (PhC) slab. b, Calculated band structure. The yellow shaded area indicates the light cone of the surrounding medium, in which there is a continuum of radiation modes in free space. The trapped state is marked with a red circle, and the TM1 band is marked with a green line. Inset: the first Brillouin zone. c, d, Normalized radiative lifetime Qr of the TM1 band calculated from FDTD (c); values along the Γ–X direction are shown in d. Below the light cone there is no radiation mode to couple to (that is, total internal reflection), so Qr is infinite. However, at discrete points inside the light cone, Qr also goes to infinity. e, Electric-field profile Ez of the trapped state, plotted on the y = 0 slice. f, g, Amplitudes of the s- and p-polarized outgoing plane waves for the TM1 band (f); cp along the Γ–X direction is shown in g. Black circles in f indicate k points at which both cs and cp are zero.

  2. Fabricated PhC slab and the measurement setup.
    Figure 2: Fabricated PhC slab and the measurement setup.

    a, Schematic layout of the fabricated structure. The device is immersed in a liquid, index-matched to silica at 740nm. b, c, SEM images of the structure in top view (b) and side view (c). The inset to b shows an image of the whole PhC. d, Diagram of the setup for reflectivity measurements. BS, beamsplitter; SP, spectrometer.

  3. Detection of resonances from reflectivity data.
    Figure 3: Detection of resonances from reflectivity data.

    a, Top: experimentally measured specular reflectivity for p-polarized light along Γ–X. The crucial feature of interest is the resonance, which shows up as a thin faint line (emphasized by white arrows) extending from the top left corner to the bottom right corner. Disappearance of the resonance feature near 35° indicates a trapped state with no leakage. Bottom: slices at three representative angles, with close-ups near the resonance features. b, Calculated p-polarized specular reflectivity using the rigorous coupled-wave analysis (RCWA) method20 with known refractive indices and measured layer thickness. c, Top: diagram for the scattering process in temporal CMT, which treats the resonance A and the incoming and outgoing plane waves sm± as separate entities weakly coupled to each other. Bottom: reflectivity given by the analytical CMT expression; the resonance frequency and lifetimes, which are the only unknowns in the CMT expression, are fitted from the experimental data in a.

  4. Quantitative evidence on the disappearance of leakage.
    Figure 4: Quantitative evidence on the disappearance of leakage.

    a, b, Normalized radiative lifetime Qr extracted from the experimentally measured reflectivity spectrum (a) and the RCWA-calculated reflectivity spectrum (b). The black solid line shows the prediction from FDTD.

References

  1. Joannopoulos, J. D., Johnson, S. G., Winn, J. N. & Meade, R. D. Photonic Crystals: Molding the Flow of Light 2ndedn (Princeton Univ. Press, 2008)
  2. Lagendijk, A., van Tiggelen, B. & Wiersma, D. S. Fifty years of Anderson localization. Phys. Today 62, 2429 (2009)
  3. Plotnik, Y. et al. Experimental observation of optical bound states in the continuum. Phys. Rev. Lett. 107, 183901 (2011)
  4. Lee, J. et al. Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs. Phys. Rev. Lett. 109, 067401 (2012)
  5. Watts, M. R., Johnson, S. G., Haus, H. A. & Joannopoulos, J. D. Electromagnetic cavity with arbitrary Q and small modal volume without a complete photonic bandgap. Opt. Lett. 27, 17851787 (2002)
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  8. Hsu, C. W. et al. Bloch surface eigenstates within the radiation continuum. Light Sci. Applics 2, e84 http://dx.doi.org/10.1038/lsa.2013.40 (in the press)
  9. Hislop, P. D. & Sigal, I. M. Introduction to Spectral Theory: with Applications to Schrödinger Operators (Springer, 1996)
  10. von Neumann, J. & Wigner, E. Über merkwürdige diskrete Eigenwerte. Phys. Z. 30, 465467 (1929)
  11. Stillinger, F. H. & Herrick, D. R. Bound states in the continuum. Phys. Rev. A 11, 446454 (1975)
  12. Friedrich, H. & Wintgen, D. Interfering resonances and bound states in the continuum. Phys. Rev. A 32, 32313242 (1985)
  13. Zhang, J. M., Braak, D. & Kollar, M. Bound states in the continuum realized in the one-dimensional two-particle hubbard model with an impurity. Phys. Rev. Lett. 109, 116405 (2012)
  14. Porter, R. & Evans, D. Embedded Rayleigh–Bloch surface waves along periodic rectangular arrays. Wave Motion 43, 2950 (2005)
  15. Linton, C. M. & McIver, P. Embedded trapped modes in water waves and acoustics. Wave Motion 45, 1629 (2007)
  16. Krüger, H. On the existence of embedded eigenvalues. J. Math. Anal. Appl. 395, 776787 (2012)
  17. Taflove, A. & Hagness, S. C. Computational Electrodynamics: the Finite-difference Time-domain Method 3rd edn (Artech House, 2005)
  18. Johnson, S. G. & Joannopoulos, J. D. Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis. Opt. Express 8, 173190 (2001)
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  20. Liu, V. & Fan, S. S4: a free electromagnetic solver for layered periodic structures. Comput. Phys. Commun. 183, 22332244 (2012)

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Author information

  1. These authors contributed equally to this work.

    • Chia Wei Hsu &
    • Bo Zhen

Affiliations

  1. Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

    • Chia Wei Hsu,
    • Bo Zhen,
    • Jeongwon Lee,
    • Song-Liang Chua,
    • Steven G. Johnson,
    • John D. Joannopoulos &
    • Marin Soljačić
  2. Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

    • Chia Wei Hsu
  3. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

    • Steven G. Johnson

Contributions

C.W.H., B.Z., S.-L.C., J.D.J. and M.S. conceived the idea of this study. C.W.H. performed numerical simulations. C.W.H. and B.Z. conducted the measurement and analysis. J.L. fabricated the sample. S.G.J. proposed the Fourier-coefficient explanation. M.S. and J.D.J. supervised the project. C.W.H. wrote the paper with input from all authors.

Competing financial interests

The authors declare no competing financial interests.

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Supplementary information

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  1. Supplementary Information (172 KB)

    This file contains Supplementary Equations, Supplementary Discussion, additional references and Supplementary Figures 1 and 2.

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