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- Photonics Research
- Vol. 6, Issue 5, A38 (2018)

Abstract

1. INTRODUCTION

Photonic ring and disk resonators have received significant attention because they are widely used for both fundamental research as well as practical applications in optical communications and signal processing ^{[1]}. A method to realize high optical quality factors is by confining light to a “defect” region defined by distributed Bragg reflectors. Such resonators are adopted for semiconductor lasers and have been studied with various techniques such as conformal mapping, coupled-wave approach, and transfer-matrix method ^{[2–10]}. Previously, much research attention was given to the radial modes, and the effort was focused on designing the structures for larger modal discrimination between modes of different radial orders. In order to achieve single-mode operation, it is equally important to achieve large discrimination between different azimuthal modes. The existing approaches to overcome the challenge include adoption of radially chirped Bragg gratings ^{[7]} and azimuthal modulation ^{[8]}.

“Parity–time (PT) symmetry” was first introduced by Bender ^{[11]}. Due to the similarity between the Schrödinger equation and the Helmholtz equation, the concept of PT symmetry has been adopted in optics ^{[12–14]}. The PT-symmetric and PT-broken phases are separated by the exceptional point, where the system’s eigenvalues coalesce. Interesting phenomena by mode management around the exceptional points include single-mode lasers ^{[15–20]}, reversing pump dependence ^{[21–23]}, orbital angular momentum lasers ^{[24]}, and unidirectional reflection ^{[25,26]}.

In 2016, we proposed PT-symmetric circular Bragg lasers with radially periodic gratings. We employed a transfer-matrix method to calculate their reflection and transmission coefficients and analyzed their modal properties ^{[16]}. We found enhanced modal discrimination among the radial modes. However, the eigenwaves in circular Bragg lasers take the form of Hankel functions, whose zeros are not equidistantly distributed. In order to achieve perfect phase matching in the radial direction, the phase of the circular gratings of the laser structure has to follow that of the Hankel functions, rendering nonperiodicity (or chirping) in the radial direction. Here, we develop a coupled-mode approach for designing a PT-symmetric circular Bragg laser with the radially chirped gratings. We also investigate the effect of an additional azimuthal modulation of the refractive index and analyze the threshold conditions for different azimuthal modes. The rest of this paper is structured as follows. In Section

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2. PT-SYMMETRIC CIRCULAR BRAGG LASERS WITH RADIALLY CHIRPED GRATINGS

We previously proposed a laser structure with its complex refractive index in the radial direction distributed periodically. We used a transfer-matrix method to analyze its threshold gain ^{[16]} and found that the threshold gain for different radial modes could have large difference. However, because circular Bragg lasers with radially periodic gratings cannot obtain large modal discrimination between the azimuthal modes ^{[2]}, here we propose to apply the concept of PT symmetry to radially chirped gratings, as shown in Fig.

Figure 1.(a) Schematic of the PT-symmetric circular Bragg laser. (b) Schematic of the azimuthally modulated circular Bragg laser with radial PT symmetry. (c) Radial profile of the refractive index of the two types of lasers, showing the real (

Let us consider a specific azimuthal mode with perturbation in the radial direction. Erdogan ^{[2]}. In contrast with Ref. ^{[2]}, here we focus our analysis on the TM modes whose major field component is ^{[8]}: where

The above equations are typical coupled equations for chirped circular Bragg gratings. With ^{[8]}.

Let us suppose the refractive index is

Next, by applying the boundary conditions that the

Now that we have designed radially chirped PT-symmetric circular Bragg lasers, we are ready to analyze the properties of their azimuthal modes. Due to different phase shifts in the approximation of even- and odd-order Hankel functions, the odd azimuthal modes are unlikely to be excited in lasers designed for even azimuthal modes ^{[2]}. By using the transfer-matrix method, which was verified by numerical solutions of the coupled equations, we calculated the normalized threshold gain (

Figure 2.(a) Normalized threshold gain (

As shown in Fig.

With the superior properties, the PT-symmetric circular Bragg lasers can be fabricated on a III–V wafer. Selective etching and metal deposition can modulate, respectively, the real (^{[15,24]}. The effect of a small deviation away from the exceptional point was also studied, where the imaginary part (

3. CIRCULAR BRAGG LASERS WITH RADIAL PT SYMMETRY AND AZIMUTHAL MODULATION

In the previous section, a coupled-mode approach has been employed to analyze and design laser structures with modulation only in the radial direction. Because azimuthal modulation is usually employed to control azimuthal modes, in this section we propose to impose additional modulation in the azimuthal direction, as illustrated in Fig. ^{[8]}, under the weak perturbation condition, the responsible terms in the Fourier expansion of ^{[2]}, we can analyze the structure with coupled-mode theory. For the PT-symmetric structure with refractive index defined in Eq. (

We compared the reflectivity of the same azimuthal mode for two laser structures (without and with azimuthal modulation) by solving the coupled-mode equations numerically. The two structures share the same ^{[27]}.

We also simulated the two laser structures in Lumerical by an FDTD method ^{[28]}. Figure

Figure 3.(a) Output spectra for two circular Bragg laser structures with radial PT symmetry. The upper corresponds to the one without azimuthal modulation, while the lower corresponds to the one with azimuthal modulation. (b)

4. CONCLUSION

We have adopted a coupled-mode approach to design and analyze circular Bragg lasers with radial PT symmetry. We analyzed and compared the threshold gain for different azimuthal modes of circular Bragg lasers with radially chirped gratings for both conventional and PT-symmetric structures. We have shown that the PT-symmetric radially chirped circular Bragg lasers have higher modal discrimination. Further suppression to unwanted modes is obtained by introducing an additional azimuthal modulation to the circular Bragg lasers with radial PT symmetry. With an intrinsic broadside circular aperture, such lasers will lend themselves to a variety of applications in integrated photonic and optoelectronic circuitry as well as fiber-optic communication.

Ziyao Feng, Jingwen Ma, Zejie Yu, Xiankai Sun. Circular Bragg lasers with radial PT symmetry: Design and analysis with a coupled-mode approach[J]. Photonics Research, 2018, 6(5): A38

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