SCIENCE CHINA Information Sciences, Volume 63 , Issue 6 : 160409(2020) https://doi.org/10.1007/s11432-020-2888-2

A design method for high fabrication tolerance integrated optical mode multiplexer

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  • ReceivedFeb 1, 2020
  • AcceptedApr 23, 2020
  • PublishedMay 11, 2020



This work was supported by National Natural Science Foundation of China (Grant Nos. 61635001, 61535002), Major International Cooperation and Exchange Program of the National Natural Science Foundation of China (Grant No. 61120106012), and Beijing Municipal Science $\&$ Technology Commission (Grant No. Z19110004819006).



The derivation of the adjusted coupling equation

According to CMT, the field of the whole tapered structure could be written as the sum of egien-modes in different waveguides: \begin{equation}\left\{\begin{array}{l} \tilde{E}=A(z) \tilde{E}_{1}+B(z) \tilde{E}_{2}, \\ \tilde{H}=A(z) \tilde{H}_{1}+B(z) H_{2}. \end{array}\right. \tag{11}\end{equation}

Combining (A1) with Maxwell's equations, we can get \begin{equation}\begin{aligned} \frac{{\rm d} A}{{\rm d} z} &+\frac{{\rm d} B}{{\rm d} z} \frac{\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} u_{z} \cdot\left(\tilde{E}_{1}^{*} \times \widetilde{H}_{2}+\tilde{E}_{2} \times \tilde{H}_{1}^{*}\right) {\rm d} x {\rm d} y}{\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} u_{z} \cdot\left(\tilde{E}_{1}^{*} \times \tilde{H}_{1}+\tilde{E}_{1} \times \tilde{H}_{1}^{*}\right) {\rm d} x {\rm d} y} \\ &+{\rm j} A \frac{\omega \varepsilon_{0} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty}\left(N^{2}-N_{1}^{2}\right) \tilde{E}_{1}^{*} \cdot \tilde{E}_{1} {\rm d} z {\rm d} y}{\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} u_{z} \cdot\left(\tilde{E}_{1}^{*} \times \widetilde{H}_{1}+\tilde{E}_{1} \times \tilde{H}_{1}^{*}\right) {\rm d} x {\rm d} y} \\ &+{\rm j} A \frac{\omega \varepsilon_{0} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty}\left(N^{2}-N_{2}^{2}\right) \tilde{E}_{1}^{*} \cdot \tilde{E}_{2} {\rm d} x {\rm d} y}{\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} u_{z} \cdot\left(\tilde{E}_{1}^{*} \times \widetilde{H}_{1}+\tilde{E}_{1} \times \widetilde{H}_{1}^{*}\right) {\rm d} x {\rm d} y}=0, \end{aligned} \tag{12}\end{equation}

\begin{equation}\begin{aligned} \frac{{\rm d} B}{{\rm d} z} &+\frac{{\rm d} A}{{\rm d} z} \frac{\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} u_{z} \cdot\left(\tilde{E}_{2}^{*} \times \tilde{H}_{2}+\tilde{E}_{2} \times \tilde{H}_{1}^{*}\right) {\rm d} x {\rm d} y}{\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} u_{z} \cdot\left(\tilde{E}_{1}^{*} \times \tilde{H}_{1}+\tilde{E}_{1} \times \tilde{H}_{1}^{*}\right) {\rm d} x {\rm d} y} \\ &+{\rm j} A \frac{\omega \varepsilon_{0} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty}\left(N^{2}-N_{1}^{2}\right) \tilde{E}_{1}^{*} \cdot \tilde{E}_{1} {\rm d} x {\rm d} y}{\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} u_{z} \cdot\left(\tilde{E}_{1}^{*} \times \widetilde{H}_{1}+\tilde{E}_{1} \times \widetilde{H}_{1}^{*}\right) {\rm d} x {\rm d} y} \\ &+{\rm j} A \frac{\omega \varepsilon_{0} \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty}\left(N^{2}-N_{2}^{2}\right) E_{1}^{*} \cdot \tilde{E}_{2} {\rm d} x {\rm d} y}{\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} u_{z} \cdot\left(\tilde{E}_{1}^{*} \times \widetilde{H}_{1}+\tilde{E}_{1} \times \widetilde{H}_{1}^{*}\right) {\rm d} x {\rm d} y}=0. \end{aligned} \tag{13}\end{equation}

For a tapered ADC structure, the eigen modes in the two waveguide are given by \begin{equation}\left\{\begin{array}{l} \displaystyle\tilde{E}_{p}=E_{p}(z) \exp \left[-{\rm j} \int \beta_{p}({z}) \mathrm{d} {z}\right], \\ \displaystyle\widetilde{H}_{p}=H_{p}(z) \exp \left[-{\rm j} \int \beta_{p}({z}) \mathrm{d} {z}\right]. \end{array}\right. \tag{14}\end{equation}

Substituting (A4) into (A2) and (A3), the coupling equations can be expressed as \begin{equation}\begin{aligned} \frac{{\rm d} A}{{\rm d} z}&+ c_{12}(z) \frac{{\rm d} B}{{\rm d} z} \exp \left[-{\rm j} \int\left(\beta_{2}(z)-\beta_{1}(z)\right) {\rm d} z\right]+{\rm j} \chi_{1}(z) A \\ &+{\rm j} \kappa_{12}(z) B \exp \left[-{\rm j} \int\left(\beta_{2}(z)-\beta_{1}(z)\right) {\rm d} z\right]=0, \end{aligned} \tag{15}\end{equation} \begin{equation}\begin{aligned} \frac{{\rm d} B}{{\rm d} z}&+ c_{21}(z) \frac{{\rm d} A}{{\rm d} z} \exp \left[+{\rm j} \int\left(\beta_{2}(z)-\beta_{1}(z)\right) {\rm d} z\right]+{\rm j} \chi_{2}(z) B \\ &+{\rm j} \kappa_{21}(z) A \exp \left[+{\rm j} \int\left(\beta_{2}(z)-\beta_{1}(z)\right) {\rm d} z\right]=0. \end{aligned} \tag{16}\end{equation}

Ignoring $c_{p~q}$ and $\chi_{p}$, we can get the simplified coupling equations: \begin{equation}\begin{array}{l} \displaystyle\frac{{\rm d} A}{{\rm d} z}=-{\rm j} \kappa_{12}(z) B\operatorname{exp}\left[-{\rm j} \int\left(\beta_{2}(z)-\beta_{1}(z)\right) {\rm d} z\right], \\ \displaystyle\frac{{\rm d} B}{{\rm d} z}=-{\rm j} \kappa_{21}(z) A\operatorname{exp}\left[+{\rm j} \int\left(\beta_{2}(z)-\beta_{1}(z)\right) {\rm d} z\right]. \end{array} \tag{17}\end{equation}


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  • Figure 1

    (Color online) (a) Schematic of the ADC based mode multiplexer; (b) detailed geometry parameters of the coupling region in a non-tapered ADC; (c) detailed geometry parameters of the coupling region in a tapered ADC.

  • Table 1  

    Table 1The geometric parameters of designed devices

    $W_{bs}$ (nm) $W_{be}$ (nm) $L_{c}$ ($\mu$m)
    $\mathrm{{\rm~TE}_{0}\&{\rm~TE}_{1}}$ 761 879 54.08
    $\mathrm{{\rm~TE}_{0}\&{\rm~TE}_{2}}$ 1222 1318 51.54
    $\mathrm{{\rm~TE}_{0}\&{\rm~TE}_{5}}$ 2208 2428 58.23