Controllable optical Bloch oscillation in planar graded optical waveguide arrays

M. J. Zheng, J. J. Xiao, and K. W. Yu

Phys. Rev. A 81, 033829 – Published 17 March 2010

Abstract

Optical Bloch oscillation is studied theoretically in planar graded optical waveguide arrays with nearest-neighbor couplings. The gradient in the propagation constants can be achieved with the eletro-optical effect. We identify a variety of normal modes (called gradons) in the waveguide arrays with the aid of a phase diagram. Moreover, the localization properties of the normal modes are characterized and the transitions among these modes are obtained from a picture of overlapping bands. The existence of Bloch oscillation and other oscillations are confirmed by using the field-evolution analysis with various input Gaussian beams. From the results, we obtain a correspondence between gradon localization and Bloch oscillation. This study can be extended to more general waveguide arrays in higher dimensions and with further neighbor couplings. The results offer great potential applications in controlling wave propagation by means of graded materials and graded systems, which can be used to explore the tunability of light manipulation and applied to design suitable optical devices.

Received 4 December 2009

DOI:https://doi.org/10.1103/PhysRevA.81.033829

Authors & Affiliations

M. J. Zheng¹, J. J. Xiao², and K. W. Yu^1,3,*

¹Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, People’s Republic of China
²Department of Electronic and Information Engineering and Key Laboratory of Network Oriented Intelligent Computation, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, People’s Republic of China
³Institute of Theoretical Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, People’s Republic of China

^*kwyu@phy.cuhk.edu.hk

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Vol. 81, Iss. 3 — March 2010

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Images

Figure 1
Schematic diagrams for the planar graded optical waveguide arrays and the input Gaussian beam described by Eq. (7), with the input transverse wave numbers (a) $k_{0} = 0$ and (b) $k_{0} \neq 0$ . The light propagates along the axis of the waveguide, i.e., the $z$ direction. The waveguide is labeled as $n$ ( $n = 1, 2, \dots, N$ ). The graded propagation constant $α n$ varies linearly with $n$ . The boundary conditions of the two cases are as follows: (a) $k_{0} = 0$ . A plane wave inputs at the $z = 0$ boundary. (b) $k_{0} \neq 0$ . The additional optical path differences due to the dielectric blocks placed in front of the input channels lead to the required phase differences $\exp [- {ik}_{0} (n - n_{0})]$ between the $n$ th and $n_{0}$ th waveguide at the $z = 0$ boundary.Reuse & Permissions
Figure 2
Pseudodispersion relation between $β$ and the supermode index number $m$ for (a) $α = 0.5$ when $β (N, π) > β (1, 0)$ and (c) $α = 0.05$ when $β (N, π) < β (1, 0)$ . Panels (b) and (d) are the corresponding density of states (DOS) for (a) and (c), respectively.Reuse & Permissions
Figure 3
Phase diagram for the planar graded optical waveguide arrays with $N = 50$ waveguides. There are four regions of different gradon modes in the phase diagram as follows: soft gradon (S), hard gradon (H), soft-hard gradon (SH), and unbounded modes (U). The corresponding oscillations for input beams consist of different gradon modes are as follows: LR, left-end reflection; RR, right-end reflection; BW, breathing-wave-like oscillations; BO, Bloch oscillation. The left-and-right-end reflection (LRR) in the U region is not shown here. The distribution of a component for each kind of oscillation is marked on the corresponding positions on the phase diagram.Reuse & Permissions
Figure 4
Typical mode patterns of (a) and (b) soft-hard gradon (SH), (c) soft gradon (S), and (d) hard gradon (H), corresponding to the dominant modes of beams, which undergo Bloch oscillation, breathing-wave-like oscillation, left-end reflection, and right-end reflection, respectively. The unit of the model amplitude is arbitrary, while the abscissa denotes the waveguide index.Reuse & Permissions
Figure 5
Evolution of various excitation beams shown by the contour plots of $| ψ (z) |^{2}$ on the propagation distance-waveguide domain. The rescaled propagation distance is dimensionless, while the abscissa denotes the waveguide index. (a) Bloch oscillation ( $α = 0.5$ , $σ = 3$ , $k = 0.0$ ), (b) breathing-wave-like oscillation ( $α = 0.3$ , $σ = 0.02$ , $k = 0.8 π$ ), (c) reflection from the left end ( $α = 0.1$ , $σ = 3$ , $k = 0.9 π$ ), and (d) reflection from the right end ( $α = 0.05$ , $σ = 3$ , $k = 0.0$ ).Reuse & Permissions
Figure 6
Evolution of various excitation beams in $k$ -space shown by the contour plots of $| ϕ (k) |^{2}$ in the reciprocal position-propagation distance (i.e., $k$ - $z$ ) domain. The rescaled propagation distance is dimensionless. (a) Bloch oscillation ( $α = 0.5$ , $σ = 3$ , $k = 0.0$ ), (b) breathing-wave-like oscillation ( $α = 0.3$ , $σ = 0.02$ , $k = 0.8 π$ ), (c) reflection from the left end ( $α = 0.1$ , $σ = 3$ , $k = 0.9 π$ ), and (d) reflection from the right end ( $α = 0.05$ , $σ = 3$ , $k = 0.0$ ).Reuse & Permissions

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