Dispersion Engineering for Vertical Microcavities Using Subwavelength Gratings

Zhaorong Wang, Bo Zhang, and Hui Deng

Phys. Rev. Lett. 114, 073601 – Published 17 February 2015

Abstract

We show that the energy-momentum dispersion of a vertical semiconductor microcavity can be modified by design using a high-index-contrast subwavelength grating (SWG) as a cavity mirror. We analyze the angular dependence of the reflection phase of the SWG to illustrate the principles of dispersion engineering. We show examples of engineered dispersions such as ones with much reduced or increased energy density of states and one with a double-well-shaped dispersion. This method of dispersion engineering is compatible with maintaining a high cavity quality factor and incorporating fully protected active media inside the cavity, thus enabling the creation of new types of cavity quantum electrodynamics systems.

Received 23 September 2014

DOI:https://doi.org/10.1103/PhysRevLett.114.073601

Authors & Affiliations

Zhaorong Wang, Bo Zhang, and Hui Deng^*

Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA

^*Corresponding author. dengh@umich.edu

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Vol. 114, Iss. 7 — 20 February 2015

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Figure 1
(a) Schematic of an SWG-DBR hybrid vertical cavity. The SWG followed by an air gap and one high-index DBR layer comprise the top mirror. We use ${Al}_{0.15} {Ga}_{0.85} As$ (refractive index $n_{r} = 3.58$ ) for the grating bars and high-index DBR layers, and AlAs ( $n_{r} = 3.02$ ) for the low-index DBR and cavity layers. (b) Cross section of an SWG and the wave vectors inside and outside the SWG. The SWG is treated as a WGA between input plane $z = 0$ and output plane $z = t_{g}$ . The light outside the WGA is the superposition of diffraction modes, with only the zero-order mode propagating for an SWG and the higher-order ones evanescent. $ϕ_{0}$ is the reflection phase of the zero-order wave. (c) The $β - ω$ dispersions of the transverse-magnetic (TM) WGA modes in an SWG with a duty cycle $η = 65 %$ , for incidence angles of 0° (blue line), 15° (pink), and 30° (cyan). Dash-dotted lines mark modes that cannot be excited. The zeroth WGA modes at different angles almost overlap with the ${TM}_{0}$ mode. The higher modes shift with the incidence angle, leading to large changes in the reflection phase. The gray shade marks the dual-mode regime at normal incidence. The black dashed lines are the dispersions of light in homogeneous air and grating-bar dielectric medium.
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Figure 2
$t_{g} - ω$ maps of the reflectance [(a) and (c)] and reflection phase [(b) and (d)] of an SWG with $η = 65 %$ for the TM polarization, under normal incidence [(a) and (b)] and $θ_{0} = 30 °$ oblique incidence [(b) and (d)]. The black dash-dotted lines in (a) and (b) show the dual-mode regime defined by $ω_{c 2}$ and $ω_{c 4}$ obtained in Fig. 1. The dispersions of the dual WGA modes are plotted as the two sets of white dashed and dash-dotted lines in all four figures, using the approximated Fabry-Perot resonance condition of $β t_{g} = π$ . These lines overlap well with the zero-reflectance (blue) stripes in (a) and (c). Broadband high-reflectance regions (red) can be found between those lines. Each point on the figure corresponds to one SWG design. An example is marked by the $white + symbol$ , which has a phase shift of $\sim 0.4 π$ over 30° while maintaining high reflectance ( $> 0.995$ ). The large phase shift is caused by the large WGA-mode shift, as seen by comparing the dash-dotted white lines in (b) and (d).
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Figure 3
(a) Comparison of the angular dependence of the reflection phase of two SWGs with a DBR. $ϕ_{0}^{'}$ is the shifted reflection phase that starts with zero at normal incidence. (b) Energy dispersions of cavities with the SWGs and DBR as in (a) as the top mirror and a bottom DBR with 30 $λ / 2$ pairs. The linewidths of the cavity resonances $δ (ℏ ω)$ are shown as the shades, to indicate the quality factors of the cavities $Q = ω / δ ω$ . The linewidth corresponding to $Q = 10^{3}$ is marked. The curvature of dispersion is proportional to the effective mass defined as $m^{*} \equiv ℏ (d^{2} ω / d k^{2})^{- 1}$ . We obtain at $k_{∥} \sim 0$ an effective mass $m^{*} \approx 3 \times 10^{- 5} m_{e}$ for the DBR-DBR cavity, where $m_{e}$ is the mass of an electron. In comparison, $m^{*} \approx 1 \times 10^{- 5} m_{e}$ for SWG1-DBR, $m^{*} \approx - 20 \times 10^{- 5} m_{e}$ for SWG2-DBR. (c) Energy dispersions of SWG1-SWG1 and SWG2-SWG2 cavities compared to the DBR-DBR cavity, showing more substantial tuning of the dispersion than in SWG-DBR cavities. At $k_{∥} \sim 0$ , we obtain $m^{*} \approx 0.3 \times 10^{- 5} m_{e}$ for SWG1-SWG1, and $m^{*} \approx - 0.6 \times 10^{- 5} m_{e}$ for the SWG2-SWG2 cavity. (d) A double-well-shaped dispersion for TM-polarized light in the SWG3-SWG3 cavity. The materials used in the cavities are given in Sec. 2. All dimensions are scaled to give a resonance of 1.55 eV at normal incidence. The structural parameters are as follows: SWG1: $Λ = 539 nm$ , $t_{g} = 350 nm$ , $η = 0.31$ , transverse electric polarization; SWG2: $Λ = 328 nm$ , $t_{g} = 557 nm$ , $η = 0.65$ , TM polarization; SWG3: $Λ = 300 nm$ , $t_{g} = 584 nm$ , $η = 0.62$ , TM polarization.
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