Abstract
Periodic structures have a large influence on propagating waves. This holds for various types of waves over a large range of length scales: from electrons in atomic crystals1 and light in photonic crystals2,3,4 to acoustic waves in sonic crystals5. The eigenstates of these waves are best described with a band structure, which represents the relation between the energy and the wavevector (k). This relation is usually not straightforward: owing to the imposed periodicity, bands are folded into every Brillouin zone, inducing splitting of bands and the appearance of bandgaps. As a result, exciting phenomena such as negative refraction6,7, auto-collimation of waves8,9 and low group velocities10,11,12 arise. k-space investigations of electronic eigenstates have already yielded new insights into the behaviour of electrons at surfaces and in novel materials13,14,15,16. However, for a complete characterization of a structure, an understanding of the mutual coupling of eigenstates is also essential. Here, we investigate the propagation of light pulses through a photonic crystal structure using a near-field microscope17,18. Tracking the evolution of the photonic eigenstates in both k-space and time allows us to identify individual eigenstates and to uncover their dynamics and coupling to other eigenstates on femtosecond timescales even when co-localized in real space and time.
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References
Bloch, F. Über die quantenmechanik der elektronen in kristallgittern. Z. Physik 52, 555–600 (1928).
Soukoulis, C. M. (ed.) Photonic Crystals and Light Localization in the 21st Century (NATO Science Series, Kluwer Academic, Dordrecht, 2001).
Fujita, M., Takahashi, S., Tanaka, Y., Asano, T. & Noda, S. Simultaneous inhibition and redistribution of spontaneous light emission in photonic crystals. Science 308, 1296–1298 (2005).
Koenderink, A. F., Kafesaki, M., Soukoulis, C. M. & Sandoghdar, V. Spontaneous emission in the near field of two-dimensional photonic crystals. Opt. Lett. 30, 3210–3212 (2005).
Cervera, F. et al. Refractive acoustic devices for airborne sound. Phys. Rev. Lett. 88, 023902 (2002).
Hu, X. H. et al. Superlensing effect in liquid surface waves. Phys. Rev. E 69, 030201 (2004).
Feng, L. et al. Acoustic backward-wave negative refractions in the second band of a sonic crystal. Phys. Rev. Lett. 96, 014301 (2006).
Mahieu, G. et al. Direct evidence for shallow acceptor states with nonspherical symmetry in GaAs. Phys. Rev. Lett. 94, 026407 (2005).
Lu, Z. L. et al. Experimental demonstration of self-collimation inside a three-dimensional photonic crystal. Phys. Rev. Lett. 96, 173902 (2006).
Feynman, R. P. et al. Mobility of slow electrons in a polar crystal. Phys. Rev. 127, 1004–1017 (1962).
Gersen, H. et al. Real-space observation of ultraslow light in photonic crystal waveguides. Phys. Rev. Lett. 94, 073903 (2005).
Vlasov, Y. A., O’Boyle, M., Hamann, H. F. & McNab, S. J. Active control of slow light on a chip with photonic crystal waveguides. Nature 438, 65–69 (2005).
Crommie, M. F., Lutz, C. P. & Eigler, D. M. Imaging standing waves in a 2-dimensional electron gas. Nature 363, 524–527 (1993).
Shen, Z. X., Spicer, W. E., King, D. M., Dessau, D. S. & Wells, B. O. Photoemission-studies of high-Tc superconductors—the superconducting gap. Science 267, 343–350 (1995).
Lemay, S. G. et al. Two-dimensional imaging of electronic wavefunctions in carbon nanotubes. Nature 412, 617–620 (2001).
McElroy, K. et al. Relating atomic-scale electronic phenomena to wave-like quasiparticle states in superconducting Bi2Sr2CaCu2O8+δ . Nature 422, 592–596 (2003).
Betzig, E. & Trautman, J. K. Near-field optics—microscopy, spectroscopy, and surface modification beyond the diffraction limit. Science 257, 189–195 (1992).
Bozhevolnyi, S. I., Volkov, V. S., Devaux, E., Laluet, J. Y. & Ebbesen, T. W. Channel plasmon subwavelength waveguide components including interferometers and ring resonators. Nature 440, 508–511 (2006).
Balistreri, M. L. M., Gersen, H., Korterik, J. P., Kuipers, L. & van Hulst, N. F. Tracking femtosecond laser pulses in space and time. Science 294, 1080–1082 (2001).
Profunser, D. M., Wright, O. B. & Matsuda, O. Imaging ripples on phononic crystals reveals acoustic band structure and Bloch harmonics. Phys. Rev. Lett. 97, 055502 (2006).
Russell, P. St. J. Optics of Floquet–Bloch waves in dielectric gratings. Appl. Phys. B 39, 231–246 (1986).
Gersen, H. et al. Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides. Phys. Rev. Lett. 94, 123901 (2005).
Engelen, R. J. P. et al. Local probing of Bloch mode dispersion in a photonic crystal waveguide. Opt. Express 13, 4457–4464 (2005).
Dolling, G., Enkrich, C., Wegener, M., Soukoulis, C. M. & Linden, S. Simultaneous negative phase and group velocity of light in a metamaterial. Science 312, 892–894 (2006).
Rousse, A. et al. Non-thermal melting in semiconductors measured at femtosecond resolution. Nature 410, 65–68 (2001).
Greiner, M., Mandel, O., Esslinger, T., Hansch, T. W. & Bloch, I. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature 415, 39–44 (2002).
Siwick, B. J., Dwyer, J. R., Jordan, R. E. & Miller, R. J. D. An atomic-level view of melting using femtosecond electron diffraction. Science 302, 1382–1385 (2003).
Shigekawa, H., Takeuchi, O. & Aoyama, M. Development of femtosecond time-resolved scanning tunneling microscopy for nanoscale science and technology. Sci. Technol. Adv. Mater. 6, 582–588 (2005).
Damascelli, A. et al. Fermi surface, surface states, and surface reconstruction in Sr2RuO4 . Phys. Rev. Lett. 85, 5194–5197 (2000).
Szymanski, P., Garrett-Roe, S. & Harris, C. B. Time- and angle-resolved two-photon photoemission studies of electron localization and solvation at interfaces. Prog. Surf. Sci. 78, 1–39 (2005).
Acknowledgements
We thank M. Bonn for helpful discussions.
This work is part of the research program of the ‘Stichting voor Fundamenteel Onderzoek der Materie (FOM)’, which is financially supported by the ‘Nederlandse organisatie voor Wetenschappelijk Onderzoek (NWO)’.
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Y.S., N.I. and K.A. provided the sample. R.J.P.E. carried out the near-field measurements. R.J.P.E., H.G. and L.K. carried our the k-space analysis. R.J.P.E., Y.S., K.A. and L.K. planned the project. All authors contributed in discussions.
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Engelen, R., Sugimoto, Y., Gersen, H. et al. Ultrafast evolution of photonic eigenstates in k-space. Nature Phys 3, 401–405 (2007). https://doi.org/10.1038/nphys576
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DOI: https://doi.org/10.1038/nphys576
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