Theory and experiments of disorder-induced resonance shifts and mode-edge broadening in deliberately disordered photonic crystal waveguides

Nishan Mann, Alisa Javadi, P. D. García, Peter Lodahl, and Stephen Hughes

Phys. Rev. A 92, 023849 – Published 26 August 2015

Abstract

We study both theoretically and experimentally the effects of introducing deliberate disorder in a slow-light photonic crystal waveguide on the photon density of states. We introduce a theoretical model that includes both deliberate disorder through statistically moving the hole centers in the photonic crystal lattice and intrinsic disorder caused by fabrication imperfections. We demonstrate a disorder-induced mean blueshift and an overall broadening of the photonic density of states for deliberate disorder values ranging 0–12 nm. By comparing with measurements obtained from a GaAs photonic crystal waveguide, we find very good agreement between theory and experiment. These results highlight the importance of carefully including local field effects for modeling high-index contrast perturbations and demonstrate the efficiency of our perturbative approach for modeling disorder-induced changes in the density of states. Our work also demonstrates the importance of using asymmetric dielectric polarizabilities for positive and negative dielectric perturbations when modeling a perturbed dielectric interface in photonic crystal platforms. Finally, we also show examples of disorder-induced resonances that can appear for various instances of disorder.

Received 13 May 2015

DOI:https://doi.org/10.1103/PhysRevA.92.023849

Authors & Affiliations

Nishan Mann^1,*, Alisa Javadi², P. D. García², Peter Lodahl², and Stephen Hughes¹

¹Department of Physics, Queen's University, Kingston, Ontario, Canada, K7L 3N6
²Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark

^*nmann@physics.queensu.ca

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Vol. 92, Iss. 2 — August 2015

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Images

Figure 1
(a) Schematic of a disordered $W 1$ with ideal air holes solid dark (black) circles], disordered holes [light (red) circles], and the background slab (gray). (b) Schematic of a hole center shift with the centers marked by solid light (orange) circles and the direction of perturbation given by the solid-black arrow. The magnitude of the perturbation is denoted by $Δ r$ and the azimuthal direction or angle of the perturbation is given by $ϕ$ . Dashed arrows indicate the shifts of the air-slab interface with dashed-light (red) and dashed-dark (black) arrows representing positive and negative shifts, respectively.
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Figure 2
(a) Photonic band structure (in units of wave vector vs vacuum wavelength) of the ideal $W 1$ showing the fundamental (solid) and higher order (dashed) guided modes. The light line is shown in black. (b) Mean frequency shift $E [Δ ω]$ for six disordered samples for the fundamental guided mode. (c) RMS frequency shift $\sqrt{E [Δ ω^{2}]}$ representing the standard deviation if and only if the mean frequency shift is zero. (d) Standard deviation of $Δ ω$ given a nonzero mean frequency shift (b). In all three graphs, the intrinsic disorder is kept fixed at $0.005 a (1.2 nm)$ while the external disorder is varied as follows: $0 a$ [solid light gray (cyan)], $0.01 a (2.4 nm)$ [dashed light gray (green)], $0.02 a (4.8 nm)$ [solid medium gray (red)], $0.03 a (7.2 nm)$ [dashed medium gray (magenta)], $0.04 a (9.6 nm)$ [solid dark gray (blue)], $0.05 a (12 nm)$ (dashed black).
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Figure 3
(a) and (b) Experimental normalized intensity spectra obtained by scanning along the waveguide position for two different amounts of extrinsic disorder as indicated in the figure. (c) Experimental spatially integrated intensity for varying degrees of extrinsic disorder as labeled in the figure. (d) Calculated ensemble averaged DOS (normalized) for the fundamental waveguide mode for the six disordered samples in (c). The red-dashed line in the lowest disorder case is given by $A \frac{λ_{\min}^{2}}{λ^{2}}$ with $A = 0.2$ and represents the qualitative contribution of radiation modes to the DOS. The amount of intrinsic disorder in all samples is $σ_{i} = 0.005 a (1.2 nm)$ .
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Figure 4
Top: Projected LDOS values of a $y$ -oriented dipole centered at the antinode of $E_{y}$ , computed for ten instances of disorder [dark solid (blue)] with internal and external disorder values of $σ_{i} = 0.005 a (1.2 nm), σ_{e} = 0.02 a (4.8 nm)$ respectively. For reference, the LDOS with no extrinsic disorder [light solid (red)] and ideal mode-edge [dashed light (red) vertical line] are also shown. All LDOS instances are normalized to their own LDOS peak. The length of the waveguides was kept fixed at $7.2 μ m$ (30 unit cells). Bottom: As highlighted by the black markers $(+)$ on the Projected LDOS instance, starting from above (right) and going below (left) the ideal mode edge, disorder-induced localized mode intensity $| E_{y} |^{2}$ is shown.
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