Amorphous Photonic Lattices: Band Gaps, Effective Mass, and Suppressed Transport

Mikael Rechtsman, Alexander Szameit, Felix Dreisow, Matthias Heinrich, Robert Keil, Stefan Nolte, and Mordechai Segev

Phys. Rev. Lett. 106, 193904 – Published 13 May 2011

Abstract

We study, experimentally and numerically, amorphous photonic lattices and the existence of band gaps therein. Our amorphous system comprises 2D waveguides distributed randomly according to a liquidlike model responsible for the absence of Bragg peaks, as opposed to ordered lattices with disorder which always exhibit Bragg peaks. In amorphous lattices the bands comprise localized states, but we find that defect states residing in the gap are more localized than the localization length of states within the band. Finally, we show how the concept of effective mass carries over to amorphous photonic lattices.

Received 12 October 2010

DOI:https://doi.org/10.1103/PhysRevLett.106.193904

Authors & Affiliations

Mikael Rechtsman¹, Alexander Szameit¹, Felix Dreisow², Matthias Heinrich², Robert Keil², Stefan Nolte², and Mordechai Segev¹

¹Physics Department and Solid State Institute, Technion, 32000 Haifa, Israel
²Institute of Applied Physics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany

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Issue

Vol. 106, Iss. 19 — 13 May 2011

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Images

Figure 1
Theoretical results. (a) Refractive index profile, $n (x, y)$ , of an amorphous waveguide structure. The index varies from 1.45 to $1.45 + 9 \times 10^{- 4}$ . (b) Modulus of the Fourier transform of this structure, which shows no Bragg peaks. (c) $n (x, y)$ for a square lattice with superimposed uncorrelated disorder. (d) Modulus of the Fourier transform of (c), displaying clear Bragg peaks. Note that (b) and (d) were averaged over many different realizations to highlight the Bragg peaks.Reuse & Permissions
Figure 2
Amorphous photonic lattices: experiments. (a) Microscope image of the input facet of the waveguide lattice. (b) Modulus of the Fourier transform of (a), showing no Bragg peaks. (c) Eigenvalue spectrum of the amorphous lattice at $λ = 633 nm$ , showing a band gap. Inset: size of the band gap as a function of $λ$ . (d) Variation of band gap size as a function of standard deviation of the spacing between waveguides.Reuse & Permissions
Figure 3
(a),(b) Eigenvalue spectrum of the amorphous lattice with a defect waveguide at $λ = 633 nm$ and $λ = 875 nm$ : (a) contains a defect mode in its gap (indicated by arrow), while (b) has no gap and therefore no defect mode. (c),(d) Experimental images of the light intensity at output facet, showing a highly localized defect mode in the gap in (c), but not in (d).Reuse & Permissions
Figure 4
(a) Experimental ensemble-averaged output beam for light launched into the defect waveguide. Ensemble averaging does not affect the shape of the defect mode. (b) Experimental ensemble-average output beam for 633 nm light input on 30 different nondefect waveguides in different local environments of the disordered pattern. The ensemble-average wave packet exhibits Anderson localization. (c) Semilog light intensity cross sections for the experiments of (a) and (b). (d) Numerical results for curves corresponding to those in (c) derived from beam-propagation simulations. The defect state is always narrower than the Anderson localization length.Reuse & Permissions
Figure 5
Quantifying the effective mass through deflection. Top, middle, bottom rows: deflection of a wave packet taken from the top edge of the first band, bottom edge of first band, and top edge of the second band, respectively. Left, middle, and right column: deflection vs propagation distance, input light distribution, and output light distribution, respectively. The deflections in (c), (f), (k), indicate positive, negative, and positive effective mass, respectively. All results are ensemble averaged over 100 realizations of the amorphous pattern.Reuse & Permissions

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