Photonic realization of topologically protected bound states in domain-wall waveguide arrays

J. P. Lee-Thorp, I. Vukićević, X. Xu, J. Yang, C. L. Fefferman, C. W. Wong, and M. I. Weinstein

Phys. Rev. A 93, 033822 – Published 10 March 2016

Abstract

We present an analytical theory of topologically protected photonic states for the two-dimensional Maxwell equations for a class of continuous periodic dielectric structures, modulated by a domain wall. We further numerically confirm the applicability of this theory for three-dimensional structures.

Received 2 January 2015

DOI:https://doi.org/10.1103/PhysRevA.93.033822

Physics Subject Headings (PhySH)

Photonics

Waveguides

Topology

Atomic, Molecular & Optical

Authors & Affiliations

J. P. Lee-Thorp^1,*, I. Vukićević^1,†, X. Xu^2,‡, J. Yang^3,§, C. L. Fefferman^4,∥, C. W. Wong^3,¶, and M. I. Weinstein^5,#

¹Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York, USA
²Department of Computer Science, Columbia University, New York, New York, USA
³Mesoscopic Optics and Quantum Electronics Laboratory, University of California, Los Angeles, California and Department of Mechanical Engineering, Columbia University, New York, New York, USA
⁴Department of Mathematics, Princeton University, Princeton, New Jersey, USA
⁵Department of Applied Physics and Applied Mathematics and Department of Mathematics, Columbia University, New York, New York, USA

^*jpl2154@columbia.edu
^†iv2143@columbia.edu
^‡xx2153@columbia.edu
^§yangjh@seas.ucla.edu
^∥cf@math.princeton.edu
^¶cheewei.wong@ucla.edu
^#miw2103@columbia.edu

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 93, Iss. 3 — March 2016

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Spectra of $H^{δ} = - \partial_{x}^{2} + V_{e} (x) + δ κ (δ x) W_{o} (x)$ . (a) $δ = 0$ . First four dispersion curves $k_{x} \mapsto E_{b} (k_{x})$ (gray) and continuous spectrum (black) of $H^{0} = - \partial_{x}^{2} + V_{e} (x)$ . Dirac points at $k_{x, ☆} = π, E_{☆} \approx π^{2}, {(3 π)}^{2}$ . (b) Spectra of $H^{δ}$ with $κ (X) = tanh (X)$ . Defect mode energy $E^{δ}$ (red dot) lies in the spectral gap about $E_{☆} = π^{2}$ . (c) For a range of $δ > 0$ , continuous spectrum (black) of $H^{δ}$ with domain-wall function $κ (X) = 5 tanh (X) + 10 e^{- X^{2}}$ . Red curves are topologically protected bifurcations, seeded by effective Dirac equations. Blue curves are (nonprotected) band-edge bifurcations, seeded by effective Schrödinger equations. (d) Spectrum of $h^{δ} = H^{δ} + δ^{2} G (δ x)$ , with $G (X) = 3 e^{- X^{2}}$ and $κ (X)$ as in (c). Bifurcations from Dirac points (red) persist while those from band edges (blue) may be destroyed when subjected to localized perturbations.
Reuse & Permissions
Figure 2
Greater robustness of the operating frequency of protected modes when compared with conventional nonprotected modes, illustrated by the spectra of $h_{γ}^{δ} = - \partial_{x}^{2} + V_{e} (x) + δ κ (γ δ x) W_{o} (x) + δ^{2} G (γ δ x)$ near the first spectral gap. (a) Protected mode; $G \equiv 0$ and $κ (X) = tanh (X)$ . (b) Nonprotected mode; $κ \equiv 1$ and $G (X) = 3 e^{- X^{2}}$ .
Reuse & Permissions
Figure 3
2D Maxwell problem. (a) Periodic $(δ = 0)$ 2D silicon-air waveguide array with lattice constant (period) $a$ . (b) DWM $(δ = 1)$ 2D silicon-air array. Left and right asymptotic $2 a -periodic$ structures are phase shifted relative to one another. Modulated structures consisting of five waveguide channels (three shown here) on either side of the defect region were used for numerical computations. (c) TM dispersion curves $ω (k_{x})$ (gray) and frequency spectra (black) of the first few TM bands of the periodic 2D waveguide array $(δ = 0)$ , with $k_{z} = 2 π / 10$ fixed. Linear band crossings (Dirac points) occur at $k_{x} = π / a$ . (d) For the 2D DWM waveguide array $(δ = 1)$ , bound state energies (red dot) bifurcate into spectral gaps about Dirac points of the periodic (unmodulated) structure. (e) Dispersion curves $ω (k_{z})$ of the first few TM bands for the 2D DWM waveguide. Eigenvalue curves (red), corresponding to guided modes, lie in between bands of continuous spectrum (black).
Reuse & Permissions
Figure 4
On-axis $z$ propagation of electric field $E (x, z)$ through (a) periodic and (b) DWM 2D waveguide arrays.
Reuse & Permissions
Figure 5
3D Maxwell problem. (a) Periodic $(δ = 0)$ 3D silicon waveguide with lattice constant ( $x -direction$ period) $a$ and height ( $y$ direction) $0.25 a$ , and silicon dioxide cladding (not shown). Height exaggerated for clarity. (b) DWM $(δ = 1)$ 3D silicon waveguide with height $0.25 a$ , and silicon dioxide cladding. Computations were carried out on DWM structures with five waveguide channels on either side of the defect region. (c) Electric field ${|E|}^{2}$ of the localized mode with $ω = 4.0 c / a^{}$ and $k_{z} = 8.3 / a^{}$ [blue square in (e)], over a fixed $z$ slice of the DWM waveguide. (d) Mean relative deviation of GVD vs wavelength $λ$ , of the DWM (solid curves) and regular channel (dotted curves) waveguides when subjected to thermal (purple) and geometric (cyan) perturbations. (e) Dispersion curves $ω (k_{z})$ of the first few TM bands of the (unperturbed) 3D DWM waveguide. Inset shows the associated GVD of lower frequency protected guided modes.
Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information

Photonic realization of topologically protected bound states in domain-wall waveguide arrays

J. P. Lee-Thorp, I. Vukićević, X. Xu, J. Yang, C. L. Fefferman, C. W. Wong, and M. I. Weinstein

Phys. Rev. A 93, 033822 – Published 10 March 2016

Abstract

Physics Subject Headings (PhySH)

Authors & Affiliations

Article Text (Subscription Required)

References (Subscription Required)

Issue

Access Options

Physical Review A

covering atomic, molecular, and optical physics and quantum information

Photonic realization of topologically protected bound states in domain-wall waveguide arrays

J. P. Lee-Thorp, I. Vukićević, X. Xu, J. Yang, C. L. Fefferman, C. W. Wong, and M. I. Weinstein

Phys. Rev. A 93, 033822 – Published 10 March 2016

Abstract

Physics Subject Headings (PhySH)

Authors & Affiliations

Article Text (Subscription Required)

References (Subscription Required)

Issue

Access Options

Authorization Required

Other Options

Download & Share

Images

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Log In

Search

Article Lookup

Paste a citation or DOI

Enter a citation