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Photon transport and localization in optical superlattices

P. Hsieh1*, C.Chung2, J.F. McMillan1, M. Lu3, N. C. Panoiu4, and C. W. Wong1* 1Optical Nanostructures Laboratory, Columbia University, New York, NY 10027, USA

2Center for Micor/Nano Science and Technology, National Cheng Kung University, Tainan, Taiwan 3Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY 11973, U.S.A

4Photonics Group, Department of Electronic and Electrical Engineering, University College London, Torrington Place, London WC1E 7JE, UK Author email-address: ph2285@columbia.edu; cww2104@columbia.edu

Abstract: Here we examine the photon transport and collimation in optical superlattices, involving transverse guided resonances and disorder-induced localization. With increasing structural disorder, we observe the crossover from cascaded guided resonances into transverse localization modes. © 2011 Optical Society of America OCIS codes: (230.5298) Photonic crystals; (160.3918) Metamaterials

1. 3D FDTD simulations for disordered superlattices: Beam diffraction can be suppressed by structural disorder, intrinsic to nanofabrication processes, despite the fact that geometric randomness generally degrades optical functionalities. This phenomenon of arrested diffraction [1-4] and wave transport in disordered systems, also called Anderson localization [5-10]. Figure 1 shows the nanofabricated photonic crystal (PhC) superlattices with PhC layered sections of thickness d1, consisting of two-dimensional hexagonal lattice of holes of different shapes, separated by homogeneous sections of thickness d2 for a superperiod lattice Λ = d1 + d2. In the case of the circular-hole superlattices (CHS), the hole-to-lattice ratio (r/a) is 0.29 with a lattice constant a of 500 nm. To introduce structural disorder, three other structures were also nanofabricated with heptagonal-hole superlattices (HHS; ~ 2% disorder), square-hole superlattices (SHS; ~ 6% disorder), and triangular-hole superlattices (THS; ~ 13% disorder). All devices are fabricated in silicon-on-insulator with 320 nm device thickness t, with 20 superperiods, and the incoming transverse-magnetic-like (TM-like) polarization light is coupled into the superlattice via a single-mode waveguide with width w of 450 nm. A one-to-four beam splitter couples the single input waveguide to each of the four superlattices simultaneously to ensure equal coupling. The thickness of the homogeneous section is chosen to satisfy d2/d1=0.18. Significantly, from the superlattice band structure, our d2/d1=0.18 design allows flat bands (highlighted in red) to appear near the normalized frequencies of ~ 0.314 and 0.327 (in non-dimensional units of a/2c; centered around 0.322) at the kx ~ 0 high symmetry point. These flat bands represent leaky guided resonances (located outside the light cone) [3, 6] which propagate transversely in the 1D homogeneous dielectric region that forms a 1D PhC waveguide (in dashed white lines of Figure 1b to 1d) separating the PhC lattice with the designed |E|2-field profile shown in Figure 1f insets. The evanescent Bloch modes of the PhC lattice couple to these guided resonances and are resonantly amplified when tunneling from one PhC section to the adjacent one. The mutual coupling of the two leaky guided resonances (one at the input interface and one at the output interface) in the transverse waveguides gives rise to the fine mode splitting seen in the highlighted-red bands of Figure 1f.

2. Collimating guiding resonance: We employed in these calculations fully 3D finite-difference time-domain (FDTD) simulations, performed across the 1500 nm to 1600 nm spectral domain with 5 nm spectral separation. The blue regions in Figure 2 illustrate the spatial and spectral regions of tightest collimation; for our designed CHS, these are within the 1545 nm to 1555 nm band. Surprisingly, the HHS, SHS, and THS structures show larger bandwidths for collimation versus the CHS. This is attributed to the interplay of inhomogeneous band broadening of the guided resonances induced by disorder, and localization through time-independent disorder in quasi-two-dimensional lattice. Encouraged by the theoretical predictions, we examine the far-field infrared scattering for 900 wavelengths (1530 nm to 1620 nm at 100 pm spectral resolution), for each of the superlattices. For the circular-hole superlattice, the most effective collimation (ec) is observed around 1546 nm and 1554 nm. Figure 3a next illustrates the electromagnetic propagation for the HHS and SHS disordered superlattices at the ec wavelengths, selected from the 2,700 scattering images. Collimation is observed even in these disordered superlattices. The most effective ec wavelengths are determined to be ~ 1550 nm, 1555 nm and 1580 nm respectively. At other wavelengths, the beam diverges significantly from the input excitation in the disordered superlattices. Concurrently the large disorder superlattices such as triangular and square realizations show shorter transmission lengths due to increased disorder scattering losses.

Figure 1 | Periodic and disordered photonic crystal superlattices. a, Example of nanofabricated silicon photonic crystal superlattice with 20 superperiods and single-mode input waveguide. b, Periodic superlattice with circular holes. c, Structural disorder is introduced by replacing the circular holes with heptagonal ones (~ 2% structural disorder), and rotating them with an angle prescribed by a uniform random distribution. d, Square holes (~ 6% structural disorder). e, Triangular holes (~ 13% structural disorder). f, Band structure of the circular-hole superlattice,

Figure 2 | Dispersive-propagation numerical maps of the ordered and disordered superlattices. The plotted effective beam width (blue with the tightest spatial extent; red the widest) is determined from the near-field spatial distribution of the field intensity, computed from 3D FDTD numerical simulations. For the circular-hole superlattice, collimation is observed to be centered at 1550 nm (top). The heptagonal-hole (middle), square-hole (bottom), superlattices show larger collimation bandwidths compared to the circular-hole superlattices.

FM3C.3.pdf CLEO:2014 © 2014 OSA

3. Experimental observations of transverse Anderson localization: Figures 3c-e show examples of the superlattice wave transport at different wavelengths. The physical nature of the electromagnetic propagation is revealed by the slope of the function when represented on a log-log scale. As illustrated in the log-log plots of Figures 3f to 4i, the asymptotic dependence of the experimental is of the form ∝ , where the slope is a power exponent determined from linear fitting. For the CHS in Figure 3f, we observe values up to 0.24 at the longer wavelengths, but with a near-zero slope of ~ 0.05 between 1546 nm to 1554 nm (ec – 4 nm to ec + 4 nm, with ec taken at 1550 nm), corresponding to an ~ 8 nm collimation bandwidth and arising from guided resonances without disorder. In the presence of ~ 2% and ~ 6% structural disorder (HHS and SHS, respectively), however, the measured log-log plots of show significantly different spectral dependence. The slope decreases significantly in the HHS between 1546 nm to 1567 nm (ec – 4 nm to ec + 17 nm) and in the SHS between 1551 nm to 1572 nm (ec – 4 nm to ec + 17 nm). In both superlattices the near-zero values of ~ 0.05 are thus achieved within a 21 nm collimation bandwidth, sizably larger than the CHS and suggestive of a disorder-induced mechanism to enhance the collimation bandwidth. In the presence of ~ 13% structural disorder (THS), the analyzed collimation bandwidth is even larger, namely ~ 31 nm (1576 nm to 1604 nm; ec – 7 nm to ec + 24 nm). The observed near-zero bandwidths are also larger than (and outside) the bandwidths of computed regular CHS without disorder. We also note that, to characterize the effects of disorder, an ensemble average is needed over different realizations of disorder; in our superlattices the ensemble average is self-consistently performed as the beam propagates over 20 sections of the superlattices – each of the superperiods having the same level of disorder but with different random distribution of the holes. From the experimental results, an ~ 3.9 increase in collimation bandwidth is achieved in the presence of disorder as compared to the periodic CHS. This phenomenon of disorder-induced enhanced beam collimation is reminiscent of transverse localization. In this regime for all the disordered superlattices, the beam is localized and its divergence is halted by the ensemble disorder in the superlattice, subjected only to statistical fluctuations in the scattering sites. We also note that the observed near-zero propagation regime in the disordered superlattices cannot solely arise from a “smeared-out” of a flat equifrequency surface contour as that will only result in increased beam divergence.

4. Conclusion: In summary we examined the transport of photons in optical superlattices with controllable disordered states. For the circular superlattices without appreciable disorder, a new type of collimation based on cascaded guided resonances at the superlattice interface is observed with highly dispersive features and supported by numerical modeling. With increasing disorder, collimation in random heptagonal, square and triangular superlattices are observed for the first time, even with an increased bandwidth and tighter collimation than regular superlattices, and aided by transverse localization. Transport in the disordered superlattices achieves arrested spatial divergence, departing from diffusive and ballistic transport regimes, and are verified by the power-law scaling observations, exponential and asymmetric intensity profiles in the localization regimes. This study supports the understanding of photon localization in chip-scale quasi-two-dimensional lattices and the control of photons at the mesoscopic scale.

5. References [1] S. Kocaman et. al, Nature Photonics 5, 499 (2011). [2] P. T. Rakish et. at, Nature Mat 5, 93 (2006). [3] V. Mocella, et al. Phys. Rev. Lett. 102, 133902 (2009). [4] J. Arlandis, et al. Phys. Rev. Lett. 108, 037401 (2012). [5] P. Anderson. Phy. Rev. 109, 1492 (1958). [6] H. De Raedt, et al. Phy. Rev. Let. 62, 47 (1989). [7] Y. Lahini, et al. Phy. Rev. Let. 100, 013906 (2008). [8] T. Schwartz, et al. Nat. 446, 52 (2007). [9] M. Segev, et al. Nat. Photon. 7, 197 (2013). [10] V. Savona, et al. Phys. Rev. B 83, 085301 (2011).

Figure 3 | Disorder-induced enhanced beam collimation at the onset of transverse localization. a, CHS, HHS, and SHS at ec (top to bottom). b, Spectra of superlattice collimation spot size of circular-hole superlattice at z = 49 m. Filled dots are experimental measurements (FWHM); open circles are from numerical simulations. c-e, Far-field infrared scattering image for superlattices with (c) circular, (c) random heptagonal, and (e) random square scatters, each at ec + 8 nm (left), ec + 14 nm (middle), and ec + 17 nm (right), respectively. The beam diverges in the circular-hole superlattice but shows collimation in the heptagonal- and square-hole superlattices. f-i, Log-log plots of the experimentally derived effective beam width versus propagation distance for (f) circular-hole, (g), heptagonal-hole, (h) square-hole, and (i) triangular-hole superlattices, determined from the full-width half-maximum of the far-field infrared scattering, within the spectral region ec - 4 to ec + 24 nm. The red solid lines represent the approximate wavelength for most effective collimation ec Inset: distributions of slope.