We investigate the modal properties of a beam carrying orbital angular momentum (OAM) generated by a circular array (ring) of multiple micro-ring emitters (rings) analytically and via simulation. In such a “ring-of-rings” structure, $N$ emitters generate $N$ optical vortex beams with the same OAM-order ${{l}_0}$ at the same wavelength. The output beam is a coherent combination of the $N$ vortex beams located at different azimuthal positions, having the same radial displacement. We derive an analytical expression for the output optical field and calculate the OAM-order power spectrum of the generated beam. The analytical expression and simulation results show that (1) the OAM spectrum of the output beam composes equidistant OAM spectral components, symmetrically surrounding ${{l}_0}$ with a spacing equal to $N$; (2) the envelope of the OAM spectrum broadens with an increased radius of the circular array or the value of ${{l}_0}$; and (3) the OAM components of the generated beam could be tuned either by changing the value of ${{l}_0},$ corresponding to different spectrum envelopes, or by adding different linear phase delays to the micro-ring emitters, which does not affect the envelope of the OAM spectrum.

中文翻译：

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由多个环形谐振器发射器的圆形阵列产生的携带 OAM 的光束的模态特性

我们通过分析和模拟研究了由多个微环发射器（环）的圆形阵列（环）产生的携带轨道角动量（OAM）的光束的模态特性。在这样的“环中”结构中，$N$发射器在相同波长下产生具有相同 OAM 阶${{l}_0}$ 的$N$光学涡旋光束。输出光束是$N$的相干组合位于不同方位角位置的涡旋光束具有相同的径向位移。我们推导出输出光场的解析表达式并计算生成光束的 OAM 阶功率谱。解析表达式和仿真结果表明：（1）输出光束的OAM光谱由等距的OAM光谱分量组成，对称围绕${{l}_0}$，间距等于$N$；(2) OAM频谱的包络随着圆形阵列半径的增加或${{l}_0}$的值而变宽；(3) 生成光束的 OAM 分量可以通过改变${{l}_0},$ 对应不同的频谱包络，或者通过给微环发射器增加不同的线性相位延迟，不影响OAM频谱的包络。