Bifurcation phenomena and control analysis in class-B laser system with delayed feedback

Abstract

In this paper, we study dynamics in delayed class-B laser system, with particular attention focused on Hopf and double Hopf bifurcations. Firstly, we identify the critical values for stability switches, Hopf and double Hopf bifurcations and derive the normal forms near the Hopf and double Hopf bifurcations critical points. By analyzing local dynamics near bifurcation critical points, we show how the delayed feedback control parameters effect the dynamical behaviors of the system. Furthermore, detailed numerical analysis using MATLAB extends the local bifurcation analysis to a global picture, and stable windows are observed as we change control parameter. Namely, even for parameter values not chosen in the neighborhood of the Hopf bifurcation critical points, two families of stable periodic solutions, which are resulted from Hopf bifurcation, exist in a large region of delay, and they merge into a family of stable and globally existed periodic solutions. Finally, by choosing proper control parameters, numerical simulations, including stable equilibrium, stable periodic solutions and stable quasiperiodic solutions are presented to demonstrate the theoretical results. Therefore, in accordance with above theoretical analysis, reasonable lasers with proper control parameters can be designed in order to achieve various applications.

Appendices

Appendix 1

$$\begin{aligned} q_1&= d_0\left[ \hbox {i}\beta _0+\frac{\hbox {i}k(\omega -1)}{(1+\gamma )\beta _0} +\frac{\hbox {i}k(\omega -1)\gamma \hbox {e}^{-\hbox {i}\beta _0\tau _0}}{(1+\gamma )\beta _0}\right] ,\\ q_2&= -\frac{\gamma \tau _0d_0^2\hbox {i}k^2(\omega -1)}{(1+\gamma )\beta _0^3} \left( \hbox {e}^{-{\mathrm{i}}\beta _0\tau _0}-\hbox {e}^{{\mathrm{i}}\beta _0\tau _0}\right) \\&\quad \left[ \frac{\omega -1}{1+\gamma } +\frac{\gamma (\omega -1)\hbox {e}^{-{\mathrm{i}}\beta _0\tau _0}}{1+\gamma }-\hbox {i}\beta _0-\omega \right] \\&\quad -\,\gamma \tau _0d_0h_{11}^{(2)}-\gamma \tau _0d_0h_{20}^{(2)}\\&\quad +\,\frac{2\tau _0d_0\bar{d}_0\hbox {i}k^2}{3\beta _0^3} \left[ \hbox {i}\beta _0+\omega -\frac{\omega -1}{1+\gamma } -\frac{\gamma (\omega -1)\hbox {e}^{{\mathrm{i}}\beta _0\tau _0}}{1+\gamma }\right] \\&\quad \left[ \hbox {i}\beta _0-\omega +\frac{\omega -1}{1+\gamma } +\frac{\gamma (\omega -1)\hbox {e}^{-{\mathrm{i}}\beta _0\tau _0}}{1+\gamma }\right] \\&\quad +\,\frac{\gamma ^2\tau _0d_0\bar{d}_0\hbox {i}k^2(\omega -1)^2}{(1+\gamma )^2\beta _0^3} \left( \hbox {e}^{-{\mathrm{i}}\beta _0\tau _0}-\hbox {e}^{{\mathrm{i}}\beta _0\tau _0}\right) ^2\\&\quad +\,\frac{\tau _0d_0\hbox {i}k}{\beta _0}\left[ \frac{\omega -1}{1+\gamma } +\frac{\gamma (\omega -1)\hbox {e}^{-{\mathrm{i}}\beta _0\tau _0}}{1+\gamma }-\hbox {i}\beta _0-\omega \right] C_{11}^{(1)}\\&\quad -\,\frac{\tau _0d_0\hbox {i}k}{\beta _0}\left[ \frac{\omega -1}{1+\gamma } +\frac{\gamma (\omega -1)\hbox {e}^{{\mathrm{i}}\beta _0\tau _0}}{1+\gamma }-\hbox {i}\beta _0-\omega \right] C_{20}^{(1)}\\&\quad +\,\tau _0d_0\left[ \frac{(\hbox {i}\beta _0+\omega )(1+\gamma )}{\omega -1}-1\right] \left( C_{11}^{(2)}+C_{20}^{(2)}\right) , \end{aligned}$$

$$\begin{aligned} C_{11}^{(1)}&= \frac{p_{12}f_{11}^{(1)}(1+\gamma )}{k\tau _0(\omega -1)},\\ C_{11}^{(2)}&= \frac{1}{\tau _0(1+\gamma )}\left( f_{11}^{(1)}-p_{11}f_{11}^{(1)}-\omega \tau _0C_{11}^{(1)}\right) \\&\quad + \frac{\gamma f_{11}^{(1)}k(\omega -1)}{(1+\gamma )^2\beta _0^2\tau _0}\left[ d_0\left( \hbox {e}^{-{\mathrm{i}}\beta _0\tau _0}-1\right) + \bar{d}_0\left( \hbox {e}^{{\mathrm{i}}\beta _0\tau _0}-1\right) \right] ,\\ \end{aligned}$$

$$\begin{aligned} C_{20}^{(1)}&= \frac{2\hbox {i}\beta _0 (f_{20}^{(1)}-p_{11}f_{20}^{(1)}- p_{12}f_{20}^{(2)})-f_{20}^{(2)} -\gamma \hbox {e}^{-2{{\mathrm{i}}}\beta _0\tau _0}(f_{20}^{(2)}-p_{21}f_{20}^{(1)}-p_{22}f_{20}^{(2)})+p_{21}f_{20}^{(1)}+p_{22}f_{20}^{(2)}}{2\hbox {i}\beta _0\tau _0\left[ 2\hbox {i}\beta _0+\omega +k(\omega -1)(1+\gamma )\right] }\\&\quad -\frac{\gamma \hbox {e}^{-2{\mathrm{i}}\beta _0\tau _0}f_{20}^{(1)}\left[ 3k(\omega -1)d_0\left( \hbox {e}^{{\mathrm{i}}\beta _0\tau _0}-1\right) - k(\omega -1)\bar{d}_0\left( \hbox {e}^{3{\mathrm{i}}\beta _0\tau _0}-1\right) \right] }{3\beta _0^2\tau _0(1+\gamma )\left[ 2\hbox {i}\beta _0+\omega +k(\omega -1)(1+\gamma )\right] }\\&\quad -\frac{\gamma \hbox {e}^{-2{\mathrm{i}}\beta _0\tau _0}f_{20}^{(2)}\left[ 3(\omega +\hbox {i}\beta _0)d_0\left( \hbox {e}^{{\mathrm{i}}\beta _0\tau _0}-1\right) - (\omega -\hbox {i}\beta _0)\bar{d}_0\left( \hbox {e}^{3{\mathrm{i}}\beta _0\tau _0}-1\right) \right] }{3\beta _0^2\tau _0\left[ 2\hbox {i}\beta _0+\omega +k(\omega -1)(1+\gamma )\right] },\\ C_{20}^{(2)}&= \frac{(1+\gamma )f_{20}^{(2)}+k\tau _0(\omega -1)C_{20}^{1}-(1+\gamma )p_{21}f_{20}^{(1)}-(1+\gamma )p_{22}f_{20}^{(2)}}{2(1+\gamma )\hbox {i}\beta _0\tau _0}, \end{aligned}$$

$$\begin{aligned} h_{11}^{(2)}&= C_{11}^{(2)}-\frac{f_{11}^{(1)}k(\omega -1)}{(1+\gamma )\beta _0^2\tau _0} \left[ d_0\left( \hbox {e}^{-{\mathrm{i}}\beta _0\tau _0}-1\right) \right. \\&\quad \left. +\, \bar{d}_0\left( \hbox {e}^{{\mathrm{i}}\beta _0\tau _0}-1\right) \right] ,\\ h_{20}^{(2)}&= \hbox {e}^{-2{\mathrm{i}}\beta _0\tau _0}C_{20}^{(2)}+ \hbox {e}^{-2{\mathrm{i}}\beta _0\tau _0}f_{20}^{(1)}k(\omega -1)\\&\quad \left[ \frac{d_0\left( \hbox {e}^{{\mathrm{i}}\beta _0\tau _0}-1\right) }{(1+\gamma )\beta _0^2\tau _0}- \frac{\bar{d}_0\left( \hbox {e}^{3{\mathrm{i}}\beta _0\tau _0}-1\right) }{3(1+\gamma )\beta _0^2\tau _0}\right] \\&\quad +\,\hbox {e}^{-2{\mathrm{i}}\beta _0\tau _0}f_{20}^{(2)} \left[ \frac{(\omega +\hbox {i}\beta _0)d_0\left( \hbox {e}^{{\mathrm{i}}\beta _0\tau _0}-1\right) }{\beta _0^2\tau _0}\right. \\&\left. - \frac{(\omega -\hbox {i}\beta _0)\bar{d}_0\left( \hbox {e}^{3{\mathrm{i}}\beta _0\tau _0}-1\right) }{3\beta _0^2\tau _0}\right] , \end{aligned}$$

with

$$\begin{aligned} d_0&= \left( 2-\frac{\hbox {i}\omega }{\beta _0}+\frac{\hbox {i}k\tau _0(\omega -1)\gamma }{(1+\gamma )\beta _0} \hbox {e}^{-{\mathrm{i}}\beta _0\tau _0}\right) ^{-1},\,p_{11}=d_0+\bar{d}_0,\\&\quad p_{12}=\frac{d_0(\hbox {i}\beta _0+\omega )(1+\gamma )}{k(\omega -1)}+\frac{\bar{d}_0(\omega -\hbox {i}\beta _0)(1+\gamma )}{k(\omega -1)},\\ p_{21}&= \frac{\hbox {i}k(\omega -1)\bar{d}_0}{(1+\gamma )\beta _0}-\frac{\hbox {i}k(\omega -1)d_0}{(1+\gamma )\beta _0},\\&\quad p_{22}=\frac{\hbox {i}\bar{d}_0(\omega -\hbox {i}\beta _0)}{\beta _0}-\frac{\hbox {i}d_0(\omega +\hbox {i}\beta _0)}{\beta _0},\\ f_{20}^{(1)}&= \frac{\tau _0\hbox {i}k(\omega -1)}{\beta _0(1+\gamma )}\left( 1+\gamma \hbox {e}^{-\hbox {i}\beta _0\tau _0}\right) ,\\ f_{20}^{(2)}&= -\frac{k^2\tau _0\hbox {i}(\omega -1)}{\beta _0(1+\gamma )},\\ f_{11}^{(1)}&= \frac{\gamma \tau _0\hbox {i}k(\omega -1)}{\beta _0(1+\gamma )} \left( \hbox {e}^{-{\mathrm{i}}\beta _0\tau _0}-\hbox {e}^{{\mathrm{i}}\beta _0\tau _0}\right) . \end{aligned}$$

Appendix 2

For simplicity, we define function $f_1(x)\!=\!\frac{\omega -1}{1+\gamma _c}+\hbox {i}x-\omega$ and $f_2(x)=\frac{\gamma _c\hbox {e}^{{\mathrm{i}}x\tau _c}(\omega -1)}{1+\gamma _c}$, then,

$$\begin{aligned} P_{11}&= \frac{k^2 d_1 d_2\tau _c}{\beta _1 (2\beta _1-\beta _2)}(f_1(-\beta _2) +f_2(-\beta _1))\\&\quad \left[ f_1(-\beta _1)\left( \frac{1}{\beta _2}-\frac{1}{\beta _1}\right) +\frac{f_2(-\beta _2)}{\beta _2}-\frac{f_2(\beta _1)}{\beta _1}\right] \\&\quad -\gamma _cd_1\tau _c\left( h_{1100}^{(2)}+h_{2000}^{(2)}\right) \\&\quad -\frac{k^2 d_1\bar{d}_2\tau _c}{\beta _1 (2\beta _1+\beta _2)}(f_1(\beta _2)+f_1(-\beta _1))\\&\quad \left[ f_1(-\beta _1)\left( \frac{1}{\beta _2}+\frac{1}{\beta _1}\right) +\frac{f_2(\beta _1)}{\beta _1}+\frac{f_2(\beta _2)}{\beta _2}\right] \\&\quad -\frac{2k^2 d_1\bar{d}_1\tau _c}{3\beta _1^3}\left( f_1(-\beta _1)\right. \\&\quad \left. +\,f_2(\beta _1)) (f_1(\beta _1)+f_2(-\beta _1)\right) \\&\quad +\frac{d_1\tau _c\hbox {i}k}{\beta _1}(f_1(-\beta _1)+f_2(-\beta _1))C_{1100}^{(1)}\\&\quad -\frac{d_1\tau _c\hbox {i}k}{\beta _1} (f_1(-\beta _1)+f_2(\beta _1))C_{2000}^{(1)}\\&\quad +\,d_1\tau _c\left[ \frac{(\hbox {i}\beta _1+\omega )(1+\gamma _c)}{\omega -1}\!-\!1\right] \left( C_{1100}^{(2)}+C_{2000}^{(2)}\right) , \end{aligned}$$

$$\begin{aligned} P_{12}&= \frac{d_1^2k^2\tau _c}{\beta _2}\left[ \left( \frac{1}{\beta _1}+\frac{1}{\beta _2}\right) f_1(-\beta _1) +\frac{f_2(-\beta _1)}{\beta _1}+\frac{f_2(-\beta _2)}{\beta _2}\right] \\&\quad \left[ f_1(-\beta _1)\left( \frac{1}{\beta _1}-\frac{1}{\beta _2}\right) +\frac{f_2(-\beta _1)}{\beta _1}-\frac{f_1(\beta _2)}{\beta _2}\right] \\&\quad -\frac{d_1^2k^2\tau _c}{\beta _2}\left[ \left( \frac{1}{\beta _1}-\frac{1}{\beta _2}\right) f_1(-\beta _1)\right. \\&\quad \left. +\,\frac{f_2(-\beta _1)}{\beta _1}-\frac{f_2(\beta _2)}{\beta _2}\right] \left[ f_1(-\beta _1)\left( \frac{1}{\beta _1}\right. \right. \\&\quad \left. \left. +\,\frac{1}{\beta _2}\right) +\frac{f_2(-\beta _1)}{\beta _1}+\frac{f_2(-\beta _2)}{\beta _2}\right] \\&\quad +\frac{d_1\bar{d}_1k^2\tau _c}{2\beta _1-\beta _2}\left[ \left( \frac{1}{\beta _1}-\frac{1}{\beta _2}\right) f_1(\beta _1)\right. \\&\quad \left. +\,\frac{f_2(-\beta _1)}{\beta _1}-\frac{f_2(\beta _2)}{\beta _2}\right] \left[ f_1(-\beta _1)\left( \frac{1}{\beta _2}-\frac{1}{\beta _1}\right) \right. \\&\quad \left. +\,\frac{f_2(-\beta _2)}{\beta _2}-\frac{f_2(\beta _1)}{\beta _1}\right] \\&\quad -\frac{d_1\bar{d}_1k^2\tau _c}{2\beta _1+\beta _2}\left[ \left( \frac{1}{\beta _1}+\frac{1}{\beta _2}\right) f_1(\beta _1) +\frac{f_2(-\beta _1)}{\beta _1}+\frac{f_2(-\beta _2)}{\beta _2}\right] \\&\quad \left[ f_1(-\beta _1)\left( \frac{1}{\beta _2}+\frac{1}{\beta _1}\right) +\frac{f_2(\beta _1)}{\beta _1}+\frac{f_2(\beta _2)}{\beta _2}\right] \\ \end{aligned}$$

$$\begin{aligned}&\qquad \quad +\,\frac{2 d_1 d_2k^2\tau _c}{\beta _1-2\beta _2}\left[ \left( \frac{1}{\beta _1}-\frac{1}{\beta _2}\right) f_1(-\beta _2) +\frac{f_2(-\beta _1)}{\beta _1}\right. \\&\qquad \quad \left. -\frac{f_2(\beta _2)}{\beta _2}\right] \left( \frac{f_1(-\beta _1)}{\beta _2}+\frac{f_2(-\beta _2)}{\beta _2}\right) \\&\qquad \quad -\frac{2 d_1\bar{d}_2k^2\tau _c}{\beta _1+2\beta _2}\left( \frac{f_1(-\beta _1)}{\beta _2}+\frac{f_2(\beta _2)}{\beta _2}\right) \\&\qquad \quad \left[ f_1(\beta _2)\left( \frac{1}{\beta _1}+\frac{1}{\beta _2}\right) +\frac{f_2(-\beta _1)}{\beta _1}+\frac{f_2(-\beta _2)}{\beta _2}\right] \\&\qquad \quad +\,\frac{d_1\tau _c\hbox {i}k}{\beta _1}(f_1(-\beta _1)+f_2(-\beta _1))C_{0011}^{(1)}\\&\qquad \quad +\,\frac{d_1\tau _c\hbox {i}k}{\beta _2} (f_1(-\beta _1)+f_2(-\beta _2))C_{1001}^{(1)}\\&\qquad \quad -\,\,\frac{d_1\tau _c\hbox {i}k}{\beta _2} (f_1(-\beta _1)+f_2(\beta _2))C_{1010}^{(1)}\\&\qquad \quad +\,\,d_1\tau _c\left[ \frac{(\hbox {i}\beta _1+\omega )(1+\gamma _c)}{\omega -1}-1\right] \left( C_{0011}^{(2)}+C_{1001}^{(2)}+C_{1010}^{(2)}\right) \\&\qquad \quad -\,\,\gamma _cd_1\tau _c\left( h_{0011}^{(2)}+h_{1001}^{(2)}+h_{1010}^{(2)}\right) , \end{aligned}$$

$$\begin{aligned} P_{21}&= \frac{2d_1^2k^2\tau _c}{\beta _1(\beta _2-2\beta _1)}\left[ \left( \frac{1}{\beta _2}-\frac{1}{\beta _1}\right) f_1(-\beta _1)\right. \\&\quad \left. +\,\frac{f_2(-\beta _2)}{\beta _2}-\frac{f_2(\beta _1)}{\beta _1}\right] (f_1(-\beta _2)+f_2(-\beta _1))\\&\quad -\,\frac{2\bar{d}_1 d_2k^2\tau _c}{\beta _1(\beta _2+2\beta _1)}\left[ \left( \frac{1}{\beta _2}+\frac{1}{\beta _1}\right) f_1(\beta _1)\right. \\&\left. \left. +\frac{f_2(-\beta _1)}{\beta _1}+\frac{f_2(-\beta _2)}{\beta _2}\right) \right] (f_1(-\beta _2)+f_2(\beta _1))\\&\quad -\,\frac{d_2^2k^2\tau _c}{\beta _1}\left[ \left( \frac{1}{\beta _2} -\frac{1}{\beta _1}\right) f_1(-\beta _2)+\frac{f_2(-\beta _2)}{\beta _2}\right. \\&\quad \left. \left. -\,\frac{f_2(\beta _1)}{\beta _1}\right) \right] \left[ f_1(-\beta _2)\left( \frac{1}{\beta _2}+\frac{1}{\beta _1}\right) +\frac{f_2(-\beta _1)}{\beta _1}\right. \\&\left. +\,\frac{f_2(-\beta _2)}{\beta _2}\right] +\frac{d_2^2k^2\tau _c}{\beta _1}\left[ \left( \frac{1}{\beta _1}+\frac{1}{\beta _2}\right) f_1(-\beta _2)\right. \\&\quad \left. +\,\frac{f_2(-\beta _1)}{\beta _1}+\frac{f_2(-\beta _2)}{\beta _2}\right] \left[ f_1(-\beta _2)\left( \frac{1}{\beta _2}-\frac{1}{\beta _1}\right) \right. \\&\quad \left. +\,\frac{f_2(-\beta _2)}{\beta _2}-\frac{f_2(\beta _1)}{\beta _1}\right] \\ \end{aligned}$$

$$\begin{aligned}&\qquad \quad +\,\frac{d_2\bar{d}_2k\tau _c}{2\beta _2-\beta _1}\left[ \left( \frac{1}{\beta _2}-\frac{1}{\beta _1}\right) f_1(\beta _2) +\frac{f_2(-\beta _2)}{\beta _2}\right. \\&\qquad \quad \left. \left. -\,\frac{f_2(\beta _1)}{\beta _1}\right) \right] \left[ \left( \frac{1}{\beta _1}-\frac{1}{\beta _2}\right) f_1(-\beta _2)\right. \\&\qquad \quad \left. +\,\frac{f_2(-\beta _1)}{\beta _1}-\frac{f_2(\beta _2)}{\beta _2}\right] \\&\qquad \quad -\,\frac{d_2\bar{d}_2k^2\tau _c}{\beta _1+2\beta _2}\left[ \left( \frac{1}{\beta _1}+\frac{1}{\beta _2}\right) f_1(\beta _2) +\frac{f_2(-\beta _1)}{\beta _1}\right. \\&\qquad \quad \left. +\,\frac{f_2(-\beta _2)}{\beta _2}\right] \left[ f_1(-\beta _2)\left( \frac{1}{\beta _1}+\frac{1}{\beta _2}\right) \right. \\&\qquad \quad \left. +\,\frac{f_2(\beta _1)}{\beta _1} +\frac{f_2(\beta _2)}{\beta _2}\right] \\&\qquad \quad +\,\frac{d_2\tau _c\hbox {i}k}{\beta _1}(f_1(-\beta _2)+f_2(-\beta _1))C_{0110}^{(1)}\\&\qquad \quad -\,\frac{d_2\tau _c\hbox {i}k}{\beta _1} (f_1(-\beta _2)+f_2(\beta _2))C_{1010}^{(1)}\\&\qquad \quad +\,\frac{d_2\tau _c\hbox {i}k}{\beta _2} (f_1(-\beta _2)+f_2(-\beta _2))C_{1100}^{(1)}\\&\qquad \quad +\,d_2\tau _c\left[ \frac{(\hbox {i}\beta _2+\omega )(1+\gamma _c)}{\omega -1}-1\right] \left( C_{0110}^{(2)}+C_{1010}^{(2)}\right. \\&\qquad \quad \left. +\,C_{1100}^{(2)}\right) -\gamma _cd_2\tau _c\left( h_{0110}^{(2)}+h_{1010}^{(2)}+h_{1100}^{(2)}\right) , \end{aligned}$$

$$\begin{aligned} P_{22}&= \frac{d_1 d_2k^2\tau _c}{\beta _2 (2\beta _2-\beta _1)}(f_1(-\beta _1)+f_2(-\beta _2))\\&\quad \left( f_1(-\beta _2)\left( \frac{1}{\beta _1}-\frac{1}{\beta _2}\right) +\frac{f_2(-\beta _1)}{\beta _1}-\frac{f_2(\beta _2)}{\beta _2}\right) \\&\quad -\,\frac{\bar{d}_1 d_2k^2\tau _c}{\beta _2(\beta _1+2\beta _2)}(f_1(\beta _1)+f_2(-\beta _2))\\&\quad \left[ \left( \frac{1}{\beta _1}+\frac{1}{\beta _2}\right) f_1(-\beta _2) +\frac{f_2(\beta _1)}{\beta _1}+\frac{f_2(\beta _2)}{\beta _2}\right] \\&\quad -\,\frac{2 d_2\bar{d}_2k^2\tau _c}{3\beta _2^3}(f_1(-\beta _2)+f_2(\beta _2))(f_1(\beta _2)\\&\quad +\,f_2(-\beta _2)) +\frac{d_2\tau _c\hbox {i}k}{\beta _2}(f_1(-\beta _2)+f_2(-\beta _2))C_{0011}^{(1)}\\&\quad -\,\frac{d_2\tau _c\hbox {i}k}{\beta _2} (f_1(-\beta _2)+f_2(\beta _2))C_{0020}^{(1)}\\&\quad +\,d_2\tau _c\left[ \frac{(\hbox {i}\beta _2+\omega )(1+\gamma _c)}{\omega -1}-1\right] \\&\quad \left( C_{0011}^{(2)}+C_{0020}^{(2)}\right) -\gamma _cd_2\tau _c\left( h_{0011}^{(2)}+h_{0020}^{(2)}\right) , \end{aligned}$$

with

$$\begin{aligned} a_{11}&= d_1+\bar{d}_1+d_2+\bar{d}_2,\\ a_{12}&= \frac{d_1(\omega +\hbox {i}\beta _1)(1+\gamma _c)}{k(\omega -1)}+\frac{\bar{d}_1(\omega -\hbox {i}\beta _1)(1+\gamma _c)}{k(\omega -1)}\\&\quad +\,\frac{d_2(\omega +\hbox {i}\beta _2)(1+\gamma _c)}{k(\omega -1)}+\frac{\bar{d}_2(\omega -\hbox {i}\beta _2)(1+\gamma _c)}{k(\omega -1)},\\ a_{21}&= -\frac{d_1\hbox {i}k(\omega -1)}{\beta _1(1+\gamma _c)}+\frac{\bar{d}_1\hbox {i}k(\omega -1)}{\beta _1(1+\gamma _c)}\\&\quad -\,\frac{d_2\hbox {i}k(\omega -1)}{\beta _2(1+\gamma _c)}+\frac{\bar{d}_2\hbox {i}k(\omega -1)}{\beta _2(1+\gamma _c)},\\ a_{22}&= -\frac{d_1\hbox {i}(\omega +\hbox {i}\beta _1)}{\beta _1}+\frac{\bar{d}_1\hbox {i}(\omega -\hbox {i}\beta _1)}{\beta _1}\\&\quad -\,\frac{d_2\hbox {i}(\omega +\hbox {i}\beta _2)}{\beta _2}+\frac{\bar{d}_2\hbox {i}(\omega -\hbox {i}\beta _2)}{\beta _2},\\ b_{2000}^{(1)}&= \frac{\tau _c\hbox {i}k\left( 1+\gamma _c\hbox {e}^{-{\mathrm{i}}\beta _1\tau _c}\right) (\omega -1)}{\beta _1(1+\gamma _c)},\\ b_{2000}^{(2)}&= -\frac{\tau _c\hbox {i}k^2(\omega -1)}{\beta _1(1+\gamma _c)},\\ b_{1100}^{(1)}&= \frac{\tau _c\hbox {i}k\gamma _c(\omega -1) \left( \hbox {e}^{-{\mathrm{i}}\beta _1\tau _c}-\hbox {e}^{{\mathrm{i}}\beta _1\tau _c}\right) }{\beta _1(1+\gamma _c)}, b_{1100}^{(2)}=0,\\ b_{1010}^{(1)}&= \frac{\tau _c\hbox {i}k(\omega -1)}{1+\gamma _c}\left[ \frac{1}{\beta _1} +\frac{1}{\beta _2}+\gamma _c\left( \frac{\hbox {e}^{-{\mathrm{i}}\beta _1\tau _c}}{\beta _1}+\frac{\hbox {e}^{-{\mathrm{i}}\beta _2\tau _c}}{\beta _2}\right) \right] ,\\&\quad b_{1010}^{(2)}=-\frac{\tau _c\hbox {i}k^2(\omega -1)}{1+\gamma _c}\left( \frac{1}{\beta _1}+\frac{1}{\beta _2}\right) ,\\ \end{aligned}$$

$$\begin{aligned} b_{1001}^{(1)}&= \frac{\tau _c\hbox {i}k(\omega -1)}{1+\gamma _c}\left[ \frac{1}{\beta _1} -\frac{1}{\beta _2}+\gamma _c\left( \frac{\hbox {e}^{-{\mathrm{i}}\beta _1\tau _c}}{\beta _1}-\frac{\hbox {e}^{{\mathrm{i}}\beta _2\tau _c}}{\beta _2}\right) \right] ,\\&\quad b_{1001}^{(2)}=-\frac{\tau _c\hbox {i}k^2(\omega -1)}{1+\gamma _c}\left( \frac{1}{\beta _1}-\frac{1}{\beta _2}\right) ,\\ b_{0200}^{(1)}&= -\frac{\tau _c\hbox {i}k(\omega -1)}{\beta _1(1+\gamma _c)}\left( 1+\gamma _c\hbox {e}^{{\mathrm{i}}\beta _1\tau _c}\right) ,\\&\quad b_{0200}^{(2)}=\frac{\tau _c\hbox {i}k^2(\omega -1)}{\beta _1(1+\gamma _c)},\\ b_{0110}^{(1)}&= \frac{\tau _c\hbox {i}k(\omega -1)}{1+\gamma _c}\left[ \frac{1}{\beta _2} -\frac{1}{\beta _1}+\gamma _c\left( \frac{\hbox {e}^{-{\mathrm{i}}\beta _2\tau _c}}{\beta _2}-\frac{\hbox {e}^{{\mathrm{i}}\beta _1\tau _c}}{\beta _1}\right) \right] ,\\&\quad b_{0110}^{(2)}=-\frac{\tau _c\hbox {i}k^2(\omega -1)}{1+\gamma _c}\left( \frac{1}{\beta _2}-\frac{1}{\beta _1}\right) ,\\ b_{0101}^{(1)}&= -\frac{\tau _c\hbox {i}k(\omega -1)}{1+\gamma _c}\left[ \frac{1}{\beta _1} +\frac{1}{\beta _2}+\gamma _c\left( \frac{\hbox {e}^{{\mathrm{i}}\beta _1\tau _c}}{\beta _1}+\frac{\hbox {e}^{{\mathrm{i}}\beta _2\tau _c}}{\beta _2}\right) \right] ,\\&\quad b_{0101}^{(2)}=\frac{\tau _c\hbox {i}k^2(\omega -1)}{1+\gamma _c}\left( \frac{1}{\beta _1}+\frac{1}{\beta _2}\right) ,\\ b_{0020}^{(1)}&= \frac{\tau _c\hbox {i}k(\omega -1)}{\beta _2(1+\gamma _c)}\left( 1+\gamma _c\hbox {e}^{-{\mathrm{i}}\beta _2\tau _c}\right) ,\\&\quad b_{0020}^{(2)}=-\frac{\tau _c\hbox {i}k^2(\omega -1)}{\beta _2(1+\gamma _c)},\\ b_{0002}^{(1)}&= -\frac{\tau _c\hbox {i}k(\omega -1)}{\beta _2(1+\gamma _c)}\left( 1+\gamma _c\hbox {e}^{{\mathrm{i}}\beta _2\tau _c}\right) ,\\&\quad b_{0002}^{(2)}=\frac{\tau _c\hbox {i}k^2(\omega -1)}{\beta _2(1+\gamma _c)},\\ b_{0011}^{(1)}&= \frac{\tau _c\hbox {i}k\gamma _c(\omega -1) \left( \hbox {e}^{-{\mathrm{i}}\beta _2\tau _c}-\hbox {e}^{{\mathrm{i}}\beta _2\tau _c}\right) }{\beta _2(1+\gamma _c)}, b_{0011}^{(2)}=0,\\ \end{aligned}$$

$$\begin{aligned} C_{1100}^{(1)}&= -\frac{a_{21}(1+\gamma _c)b_{1100}^{(1)}}{k(\omega -1)},\\ C_{1100}^{(2)}&= \frac{b_{1100}^{(1)}-a_{11}b_{1100}^{(1)}-\omega C_{1100}^{(1)}}{1+\gamma _c}+\frac{\gamma _ck(\omega -1)b_{1100}^{(1)}}{\tau _c(1+\gamma _c)^2}\\&\quad \left[ \frac{Re(d_1\hbox {e}^{-{\mathrm{i}}\beta _1\tau _c}-d_1)}{\beta _1^2}+ \frac{Re(d_2\hbox {e}^{-{\mathrm{i}}\beta _2\tau _c}-d_2)}{\beta _2^2}\right] ,\\ C_{2000}^{(1)}&= \frac{(1+\gamma _c)\left( b_{2000}^{(2)}-a_{21}b_{2000}^{(1)}-a_{22}b_{2000}^{(2)}-2\hbox {i}\beta _1\tau _cC_{2000}^{(2)}\right) }{k(\omega -1)}, \end{aligned}$$

$$\begin{aligned} C_{2000}^{(2)}&= \frac{k(\omega -1)(b_{2000}^{(1)}-a_{11}b_{2000}^{(1)}-a_{12}b_{2000}^{(2)}) -(\omega +2\hbox {i}\beta _1\tau _c)(1+\gamma _c)\left( b_{2000}^{(2)}-a_{21}b_{2000}^{(1)}-a_{22}b_{2000}^{(2)}\right) }{k(\omega -1)\left( 1+\gamma _c\hbox {e}^{-2{\mathrm{i}}\beta _1\tau _c}\right) -2\hbox {i}\beta _1\tau _c(\omega +2\hbox {i}\beta _1\tau _c)(1+\gamma _c)}\\&\quad -\,\frac{k^2\gamma _c\hbox {e}^{-2{\mathrm{i}}\beta _1\tau _c}(\omega -1)b_{2000}^{(1)}}{k(\omega -1)\left( 1+\gamma _c\hbox {e}^{-2{\mathrm{i}}\beta _1\tau _c}\right) -2\hbox {i}\beta _1\tau _c(\omega +2\hbox {i}\beta _1\tau _c)(1+\gamma _c)}\\&\quad \left[ \frac{d_1(\omega -1)\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _1^2\tau _c} - \frac{\bar{d}_1(\omega -1)\left( \hbox {e}^{3{\mathrm{i}}\beta _1\tau _c}-1\right) }{3(1+\gamma _c)\beta _1^2\tau _c}-\frac{d_2(\omega -1)\left( \hbox {e}^{2{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _2\tau _c(\beta _2-2\beta _1)} -\frac{\bar{d}_2(\omega -1)\left( \hbox {e}^{2{\mathrm{i}}\beta _1\tau _c+\hbox {i}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _2\tau _c(\beta _2+2\beta _1)}\right] \\&\quad -\,\frac{k\gamma _c\hbox {e}^{-2{\mathrm{i}}\beta _1\tau _c}(\omega -1)b_{2000}^{(2)}}{k(\omega -1)\left( 1+\gamma _c\hbox {e}^{-2{\mathrm{i}}\beta _1\tau _c}\right) -2\hbox {i}\beta _1\tau _c(\omega +2\hbox {i}\beta _1\tau _c)(1+\gamma _c)}\\&\quad \left[ \frac{d_1(\omega +\hbox {i}\beta _1)\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c}-1\right) }{\beta _1^2\tau _c}- \frac{\bar{d}_1(\omega -\hbox {i}\beta _1)\left( \hbox {e}^{3{\mathrm{i}}\beta _1\tau _c}-1\right) }{3\beta _1^2\tau _c}-\frac{d_2(\omega +\hbox {i}\beta _2)\left( \hbox {e}^{2{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}-1\right) }{\beta _2\tau _c(\beta _2-2\beta _1)} -\frac{\bar{d}_2(\omega -\hbox {i}\beta _2)\left( \hbox {e}^{2{\mathrm{i}}\beta _1\tau _c+\hbox {i}\beta _2\tau _c}-1\right) }{\beta _2\tau _c(\beta _2+2\beta _1)}\right] , \end{aligned}$$

$$\begin{aligned} C_{1010}^{(1)}&= \frac{(1+\gamma _c)\left[ b_{1010}^{(2)}-a_{21}b_{1010}^{(1)}-a_{22}b_{1010}^{(2)} -(\hbox {i}\beta _1\tau _c+\hbox {i}\beta _2\tau _c)C_{1010}^{(2)}\right] }{k(\omega -1)},\\ C_{1010}^{(2)}&= \frac{k(\omega -1)(b_{1010}^{(1)}-a_{11}b_{1010}^{(1)}-a_{12}b_{1010}^{(2)}) -(\omega +\hbox {i}\beta _1\tau _c+\hbox {i}\beta _2\tau _c)(1+\gamma _c)(b_{1010}^{(2)}-a_{22}b_{1010}^{(2)}-a_{21}b_{1010}^{(1)})}{k(\omega -1)\left( 1+\gamma _c\hbox {e}^{-{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}\right) -(\hbox {i}\beta _1\tau _c+\hbox {i}\beta _2\tau _c+\omega )(\hbox {i}\beta _1\tau _c+\hbox {i}\beta _2\tau _c)(1+\gamma _c)}\\&\quad -\,\frac{k\gamma _c\hbox {e}^{-{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}(\omega -1)b_{1010}^{(1)}}{k^2(\omega -1)^2\left( 1+\gamma _c\hbox {e}^{-{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}\right) -(\hbox {i}\beta _1\tau _c+\hbox {i}\beta _2\tau _c+\omega )(\hbox {i}\beta _1+\hbox {i}\beta _2)\tau _c(1+\gamma _c)}\\&\quad \left[ \frac{d_1\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _1\beta _2\tau _c}- \frac{\bar{d}_1\left( \hbox {e}^{2{\mathrm{i}}\beta _1\tau _c+\hbox {i}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _1\tau _c(2\beta _1+\beta _2)} +\frac{d_2\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _1\beta _2\tau _c}- \frac{\bar{d}_2\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c+2\hbox {i}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _2\tau _c(\beta _1+2\beta _2)}\right] \\&\quad -\,\frac{k\gamma _c\hbox {e}^{-{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}(\omega -1)b_{1010}^{(2)}}{k(\omega -1)\left( 1+\gamma _c\hbox {e}^{-{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}\right) -(\hbox {i}\beta _1\tau _c+\hbox {i}\beta _2\tau _c+\omega )(\hbox {i}\beta _1+\hbox {i}\beta _2)\tau _c(1+\gamma _c)}\\&\quad \left[ \frac{d_1(\omega +\hbox {i}\beta _1)\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c}-1\right) }{\beta _1\beta _2\tau _c} - \frac{\bar{d}_1(\omega -\hbox {i}\beta _1)\left( \hbox {e}^{2{\mathrm{i}}\beta _1\tau _c+\hbox {i}\beta _2\tau _c}-1\right) }{\beta _1\tau _c(2\beta _1+\beta _2)}+\frac{d_2(\omega +\hbox {i}\beta _2)\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c}-1\right) }{\beta _1\beta _2\tau _c} -\frac{\bar{d}_2(\omega -\hbox {i}\beta _2)\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c+2\hbox {i}\beta _2\tau _c}-1\right) }{\beta _2\tau _c(\beta _1+2\beta _2)}\right] , \end{aligned}$$

$$\begin{aligned} C_{1001}^{(1)}&= \frac{(1+\gamma _c)\left[ b_{1001}^{(2)}-a_{21}b_{1001}^{(1)}-a_{22}b_{1001}^{(2)} -(\hbox {i}\beta _1\tau _c-\hbox {i}\beta _2\tau _c)C_{1001}^{(2)}\right] }{k(\omega -1)},\\ C_{1001}^{(2)}&= \frac{k(\omega -1)(b_{1001}^{(1)}-a_{11}b_{1001}^{(1)}-a_{12}b_{1001}^{(2)}) -(\omega +\hbox {i}\beta _1\tau _c-\hbox {i}\beta _2\tau _c)(1+\gamma _c)\left( b_{1001}^{(2)}-a_{21}b_{1001}^{(1)}-a_{22}b_{1001}^{(2)}\right) }{k(\omega -1)\left( 1+\gamma _c\hbox {e}^{{\mathrm{i}}\beta _2\tau _c-\hbox {i}\beta _1\tau _c}\right) -(\hbox {i}\beta _1\tau _c-\hbox {i}\beta _2\tau _c+\omega )(\hbox {i}\beta _1\tau _c-\hbox {i}\beta _2\tau _c)(1+\gamma _c)}\\&\quad +\frac{k^2(\omega -1)^2\gamma _c\hbox {e}^{{\mathrm{i}}\beta _2\tau _c-\hbox {i}\beta _1\tau _c}b_{1001}^{(1)}}{k(\omega -1)\left( 1+\gamma _c\hbox {e}^{{\mathrm{i}}\beta _2\tau _c-\hbox {i}\beta _1\tau _c}\right) -(\hbox {i}\beta _1\tau _c-\hbox {i}\beta _2\tau _c+\omega )(\hbox {i}\beta _1-\hbox {i}\beta _2)\tau _c(1+\gamma _c)}\\&\quad \left[ \frac{d_1\left( \hbox {e}^{-{\mathrm{i}}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _1\beta _2\tau _c} - \frac{\bar{d}_1\left( \hbox {e}^{2{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _1\tau _c(\beta _2-2\beta _1)}+\frac{d_2\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c-2\hbox {i}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _2\tau _c(2\beta _2-\beta _1)}+ \frac{\bar{d}_2\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _1\beta _2\tau _c}\right] \\&\quad +\frac{k\gamma _c\hbox {e}^{{\mathrm{i}}\beta _2\tau _c-\hbox {i}\beta _1\tau _c}(\omega -1)b_{1001}^{(2)}}{k(\omega -1)\left( 1+\gamma _c\hbox {e}^{{\mathrm{i}}\beta _2\tau _c-\hbox {i}\beta _1\tau _c}\right) -(\hbox {i}\beta _1\tau _c-\hbox {i}\beta _2\tau _c+\omega )(\hbox {i}\beta _1-\hbox {i}\beta _2)\tau _c(1+\gamma _c)}\\&\quad \left[ \frac{d_1(\omega +\hbox {i}\beta _1)\left( \hbox {e}^{-{\mathrm{i}}\beta _2\tau _c}-1\right) }{\beta _1\beta _2\tau _c}- \frac{\bar{d}_1(\omega -\hbox {i}\beta _1)\left( \hbox {e}^{2{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}-1\right) }{\beta _1\tau _c(\beta _2-2\beta _1)}+\frac{d_2(\omega +\hbox {i}\beta _2)\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c-2\hbox {i}\beta _2\tau _c}-1\right) }{\beta _2\tau _c(2\beta _2-\beta _1)}+\frac{\bar{d}_2(\omega -\hbox {i}\beta _2)\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c}-1\right) }{\beta _1\beta _2\tau _c}\right] , \end{aligned}$$

$$\begin{aligned} C_{0110}^{(1)}&= \frac{(1+\gamma _c)\left[ b_{0110}^{(2)}-a_{21}b_{0110}^{(1)}-a_{22}b_{0110}^{(2)} +(\hbox {i}\beta _1\tau _c-\hbox {i}\beta _2\tau _c)C_{0110}^{(2)}\right] }{k(\omega -1)}\\ C_{0110}^{(2)}&= \frac{k(\omega -1)(b_{0110}^{(1)}-a_{11}b_{0110}^{(1)}-a_{12}b_{0110}^{(2)}) -(\omega -\hbox {i}\beta _1\tau _c+\hbox {i}\beta _2\tau _c)(1+\gamma _c)\left( b_{0110}^{(2)}-a_{21}b_{0110}^{(1)}-a_{22}b_{0110}^{(2)}\right) }{k(\omega -1)\left( 1+\gamma _c\hbox {e}^{{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}\right) -(\hbox {i}\beta _1\tau _c-\hbox {i}\beta _2\tau _c+\omega )(\hbox {i}\beta _2\tau _c-\hbox {i}\beta _1\tau _c)(1+\gamma _c)}\\&\quad +\frac{\gamma _ck^2\hbox {e}^{{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}(\omega -1)^2b_{0110}^{(1)}}{k(\omega -1)\left( 1+\gamma _c\hbox {e}^{{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}\right) -(\hbox {i}\beta _1\tau _c-\hbox {i}\beta _2\tau _c+\omega )(\hbox {i}\beta _2\tau _c-\hbox {i}\beta _1\tau _c)(1+\gamma _c)}\\&\quad \left[ \frac{d_1\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c-2\hbox {i}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _1\tau _c(2\beta _1-\beta _2)} +\frac{\bar{d}_1\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _1\beta _2\tau _c} +\frac{d_2\left( \hbox {e}^{-{\mathrm{i}}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _1\beta _2\tau _c}- \frac{\bar{d}_2\left( \hbox {e}^{2{\mathrm{i}}\beta _2\tau _c-\hbox {i}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _2\tau _c(\beta _1-2\beta _2)}\right] \\&\quad +\frac{k\gamma _c\hbox {e}^{{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}(\omega -1)b_{0110}^{(2)}}{k(\omega -1)\left( 1+\gamma _c\hbox {e}^{{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}\right) -(\hbox {i}\beta _1\tau _c-\hbox {i}\beta _2\tau _c+\omega )(\hbox {i}\beta _2\tau _c-\hbox {i}\beta _1\tau _c)(1+\gamma _c)}\\&\quad \left[ \frac{d_1(\omega +\hbox {i}\beta _1)\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c-2\hbox {i}\beta _1\tau _c}-1\right) }{\beta _1\tau _c(2\beta _1-\beta _2)}+ \frac{\bar{d}_1(\omega -\hbox {i}\beta _1)\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c}-1\right) }{\beta _1\beta _2\tau _c}+\frac{d_2(\omega +\hbox {i}\beta _2)\left( \hbox {e}^{-{\mathrm{i}}\beta _1\tau _c}-1\right) }{\beta _1\beta _2\tau _c}-\frac{\bar{d}_2(\omega -\hbox {i}\beta _2)\left( \hbox {e}^{2{\mathrm{i}}\beta _2\tau _c-\hbox {i}\beta _1\tau _c}-1\right) }{\beta _2\tau _c(\beta _1-2\beta _2)}\right] , \end{aligned}$$

$$\begin{aligned} C_{0011}^{(1)}&= -\frac{a_{21}(1+\gamma _c)b_{0011}^{(1)}}{k(\omega -1)},\\ C_{0011}^{(2)}&= \frac{b_{0011}^{(1)}-a_{11}b_{0011}^{(1)}-\omega C_{0011}^{(1)}}{1+\gamma _c} +\frac{\gamma _ckb_{0011}^{(1)}(\omega -1)}{\tau _c(1+\gamma _c)^2}\\&\quad \left[ \frac{Re(d_1\hbox {e}^{-{\mathrm{i}}\beta _1\tau _c}-d_1)}{\beta _1^2} +\frac{Re(d_2\hbox {e}^{-{\mathrm{i}}\beta _2\tau _c}-d_2)}{\beta _2^2}\right] ,\\ C_{0020}^{(1)}&= \frac{(1+\gamma _c)\left[ b_{0020}^{(2)}-a_{21}b_{0020}^{(1)}-a_{22}b_{0020}^{(2)} -2\hbox {i}\beta _2\tau _cC_{0020}^{(2)}\right] }{k(\omega -1)}, \end{aligned}$$

$$\begin{aligned} C_{0020}^{(2)}&= \frac{k(\omega -1)(b_{0020}^{(1)}-a_{11}b_{0020}^{(1)}-a_{12}b_{0020}^{(2)}) -(\omega +2\hbox {i}\beta _2\tau _c)(1+\gamma _c)\left( b_{0020}^{(2)}-a_{21}b_{0020}^{(1)}-a_{22}b_{0020}^{(2)}\right) }{k(\omega -1)\left( 1+\gamma _c\hbox {e}^{-2{\mathrm{i}}\beta _2\tau _c}\right) -2\hbox {i}\beta _2\tau _c(2\hbox {i}\beta _2\tau _c+\omega )(1+\gamma _c)}\\&\quad +\frac{\gamma _ck^2\hbox {e}^{-2{\mathrm{i}}\beta _2\tau _c}(\omega -1)^2b_{0020}^{(1)}}{k(\omega -1)\left( 1+\gamma _c\hbox {e}^{-2{\mathrm{i}}\beta _2\tau _c}\right) -2\hbox {i}\beta _2\tau _c(2\hbox {i}\beta _2\tau _c+\omega )(1+\gamma _c)}\\&\quad \left[ \frac{d_1\left( \hbox {e}^{2{\mathrm{i}}\beta _2\tau _c-\hbox {i}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _1\tau _c(\beta _1-2\beta _2)}+ \frac{\bar{d}_1\left( \hbox {e}^{2{\mathrm{i}}\beta _2\tau _c+\hbox {i}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _1\tau _c(\beta _1+2\beta _2)}-\frac{d_2\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _2^2\tau _c}+ \frac{\bar{d}_2\left( \hbox {e}^{3{\mathrm{i}}\beta _2\tau _c}-1\right) }{(1+\gamma _c)3\beta _2^2\tau _c}\right] \\&\quad +\frac{k\gamma _c\hbox {e}^{-2{\mathrm{i}}\beta _2\tau _c}(\omega -1)b_{0020}^{(2)}}{k(\omega -1)\left( 1+\gamma _c\hbox {e}^{-2{\mathrm{i}}\beta _2\tau _c}\right) -2\hbox {i}\beta _2\tau _c(2\hbox {i}\beta _2\tau _c+\omega )(1+\gamma _c)}\\&\quad \left[ \frac{d_1(\omega +\hbox {i}\beta _1)\left( \hbox {e}^{2{\mathrm{i}}\beta _2\tau _c-\hbox {i}\beta _1\tau _c}-1\right) }{\beta _1\tau _c(\beta _1-2\beta _2)}+ \frac{\bar{d}_1(\omega -\hbox {i}\beta _1)\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c+2\hbox {i}\beta _2\tau _c}-1\right) }{\beta _1\tau _c(\beta _1+2\beta _2)}-\frac{d_2(\omega +\hbox {i}\beta _2)\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c}-1\right) }{\beta _2^2\tau _c}+ \frac{\bar{d}_2(\omega -\hbox {i}\beta _2)\left( \hbox {e}^{3{\mathrm{i}}\beta _2\tau _c}-1\right) }{3\beta _2^2\tau _c}\right] , \end{aligned}$$

$$\begin{aligned} h_{2000}^{(2)}&= \hbox {e}^{-2{\mathrm{i}}\beta _1\tau _c}C_{2000}^{(2)}+k(\omega -1)\hbox {e}^{-2{\mathrm{i}}\beta _1\tau _c}b_{2000}^{(1)}\\&\quad \left[ \frac{d_1\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _1^2\tau _c} -\frac{\bar{d}_1\left( \hbox {e}^{3{\mathrm{i}}\beta _1\tau _c}-1\right) }{3(1+\gamma _c)\beta _1^2\tau _c} - \frac{d_2\left( \hbox {e}^{2{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _2\tau _c(\beta _2-2\beta _1)}- \frac{\bar{d}_2\left( \hbox {e}^{2{\mathrm{i}}\beta _1\tau _c+\hbox {i}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _2\tau _c(2\beta _1+\beta _2)}\right] \\&\quad +\,\hbox {e}^{-2{\mathrm{i}}\beta _1\tau _c}b_{2000}^{(2)} \left[ \frac{d_1(\omega +\hbox {i}\beta _1)\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c}-1\right) }{\beta _1^2\tau _c} - \frac{\bar{d}_1(\omega -\hbox {i}\beta _1)\left( \hbox {e}^{3{\mathrm{i}}\beta _1\tau _c}-1\right) }{3\beta _1^2\tau _c}\right. \\&\quad \left. -\, \frac{d_2(\omega +\hbox {i}\beta _2)\left( \hbox {e}^{2{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}-1\right) }{\beta _2\tau _c(\beta _2-2\beta _1)} - \frac{\bar{d}_2(\omega -\hbox {i}\beta _2)\left( \hbox {e}^{2{\mathrm{i}}\beta _1\tau _c+\hbox {i}\beta _2\tau _c}-1\right) }{\beta _2\tau _c(2\beta _1+\beta _2)}\right] , \end{aligned}$$

$$\begin{aligned} h_{1100}^{(2)}&= C_{1100}^{(2)}-k(\omega -1)b_{1100}^{(1)} \left[ \frac{d_1\left( \hbox {e}^{-{\mathrm{i}}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _1^2\tau _c}\right. \\&\quad +\, \frac{\bar{d}_1\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _1^2\tau _c}+ \frac{d_2\left( \hbox {e}^{-{\mathrm{i}}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _2^2\tau _c}\\&\quad \left. +\,\frac{\bar{d}_2\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _2^2\tau _c}\right] ,\\ h_{1010}^{(2)}&= \hbox {e}^{-{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}C_{1010}^{(2)} +k(\omega -1)\hbox {e}^{-{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}b_{1010}^{(1)}\\&\quad \left[ \frac{d_1\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _1\beta _2\tau _c}- \frac{\bar{d}_1\left( \hbox {e}^{2{\mathrm{i}}\beta _1\tau _c+\hbox {i}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _1\tau _c(2\beta _1+\beta _2)}\right. \\&\quad \left. +\, \frac{d_2\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _1\beta _2\tau _c}- \frac{\bar{d}_2\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c+2\hbox {i}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _2\tau _c(\beta _1+2\beta _2)}\right] \\&\quad +\,\hbox {e}^{-{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}b_{1010}^{(2)} \left[ \frac{d_1(\omega +\hbox {i}\beta _1)\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c}-1\right) }{\beta _1\beta _2\tau _c}\right. \\&\quad -\, \frac{\bar{d}_1(\omega -\hbox {i}\beta _1)\left( \hbox {e}^{2{\mathrm{i}}\beta _1\tau _c+\hbox {i}\beta _2\tau _c}-1\right) }{\beta _1\tau _c(2\beta _1+\beta _2)}\\&\quad +\, \frac{d_2(\omega +\hbox {i}\beta _2)\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c}-1\right) }{\beta _1\beta _2\tau _c}\\&\quad \left. -\, \frac{\bar{d}_2(\omega -\hbox {i}\beta _2)\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c+2\hbox {i}\beta _2\tau _c}-1\right) }{\beta _2\tau _c(\beta _1+2\beta _2)}\right] , \end{aligned}$$

$$\begin{aligned} h_{1001}^{(2)}&= \hbox {e}^{{\mathrm{i}}\beta _2\tau _c-\hbox {i}\beta _1\tau _c}C_{1001}^{(2)} -k(\omega -1)\hbox {e}^{{\mathrm{i}}\beta _2\tau _c-\hbox {i}\beta _1\tau _c}b_{1001}^{(1)}\\&\quad \left[ \frac{d_1\left( \hbox {e}^{-{\mathrm{i}}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _1\beta _2\tau _c}- \frac{\bar{d}_1\left( \hbox {e}^{2{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _1\tau _c(\beta _2-2\beta _1)}\right. \\&\quad \left. +\, \frac{d_2\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c-2\hbox {i}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _2\tau _c(2\beta _2-\beta _1)}+ \frac{\bar{d}_2\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _1\beta _2\tau _c}\right] \\&\quad -\,\hbox {e}^{{\mathrm{i}}\beta _2\tau _c-\hbox {i}\beta _1\tau _c}b_{1001}^{(2)} \left[ \frac{d_1(\omega +\hbox {i}\beta _1)\left( \hbox {e}^{-{\mathrm{i}}\beta _2\tau _c}-1\right) }{\beta _1\beta _2\tau _c}\right. \\&\quad -\, \frac{\bar{d}_1(\omega -\hbox {i}\beta _1)\left( \hbox {e}^{2{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}-1\right) }{\beta _1\tau _c(\beta _2-2\beta _1)}\\&\quad +\,\frac{d_2(\omega +\hbox {i}\beta _2)\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c-2\hbox {i}\beta _2\tau _c}-1\right) }{\beta _2\tau _c(2\beta _2-\beta _1)}\\&\quad \left. +\, \frac{\bar{d}_2(\omega -\hbox {i}\beta _2)\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c}-1\right) }{\beta _1\beta _2\tau _c}\right] , \end{aligned}$$

$$\begin{aligned} h_{0110}^{(2)}&= \hbox {e}^{{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}C_{0110}^{(2)} -k(\omega -1)\hbox {e}^{{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}b_{0110}^{(1)}\\&\quad \left[ \frac{d_1\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c-2\hbox {i}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _1\tau _c(2\beta _1-\beta _2)}+ \frac{\bar{d}_1\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _1\beta _2\tau _c}\right. \\&\quad \left. +\, \frac{d_2\left( \hbox {e}^{-{\mathrm{i}}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _1\beta _2\tau _c}- \frac{\bar{d}_2\left( \hbox {e}^{2{\mathrm{i}}\beta _2\tau _c-\hbox {i}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _2\tau _c(\beta _1-2\beta _2)}\right] \\&\quad -\,\hbox {e}^{{\mathrm{i}}\beta _1\tau _c-\hbox {i}\beta _2\tau _c}b_{0110}^{(2)} \left[ \frac{d_1(\omega +\hbox {i}\beta _1)\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c-2\hbox {i}\beta _1\tau _c}-1\right) }{\beta _1\tau _c(2\beta _1-\beta _2)}\right. \\&\quad +\, \frac{\bar{d}_1(\omega -\hbox {i}\beta _1)\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c}-1\right) }{\beta _1\beta _2\tau _c}\\&\quad +\, \frac{d_2(\omega +\hbox {i}\beta _2)\left( \hbox {e}^{-{\mathrm{i}}\beta _1\tau _c}-1\right) }{\beta _1\beta _2\tau _c}\\&\quad \left. -\, \frac{\bar{d}_2(\omega -\hbox {i}\beta _2)\left( \hbox {e}^{2{\mathrm{i}}\beta _2\tau _c-\hbox {i}\beta _1\tau _c}-1\right) }{\beta _2\tau _c(\beta _1-2\beta _2)}\right] , \end{aligned}$$

$$\begin{aligned} h_{0020}^{(2)}&= \hbox {e}^{-2{\mathrm{i}}\beta _2\tau _c}C_{0020}^{(2)} -k(\omega -1)\hbox {e}^{-2{\mathrm{i}}\beta _2\tau _c}b_{0020}^{(1)}\\&\quad \left[ \frac{d_1\left( \hbox {e}^{2{\mathrm{i}}\beta _2\tau _c-\hbox {i}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _1\tau _c(\beta _1-2\beta _2)}\right. \\&\quad +\, \frac{\bar{d}_1\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c+2\hbox {i}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _1\tau _c(\beta _1+2\beta _2)}\\&\quad \left. -\, \frac{d_2\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _2^2\tau _c}+ \frac{\bar{d}_2\left( \hbox {e}^{3{\mathrm{i}}\beta _2\tau _c}-1\right) }{3(1+\gamma _c)\beta _2^2\tau _c}\right] \\&\quad -\,\hbox {e}^{-2{\mathrm{i}}\beta _2\tau _c}b_{0020}^{(2)} \left[ \frac{d_1(\omega +\hbox {i}\beta _1)\left( \hbox {e}^{2{\mathrm{i}}\beta _2\tau _c-\hbox {i}\beta _1\tau _c}-1\right) }{\beta _1\tau _c(\beta _1-2\beta _2)}\right. \\&\quad +\, \frac{\bar{d}_1(\omega -\hbox {i}\beta _1)\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c+2\hbox {i}\beta _2\tau _c}-1\right) }{\beta _1\tau _c(\beta _1+2\beta _2)}\\&\quad -\, \frac{d_2(\omega +\hbox {i}\beta _2)\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c}-1\right) }{\beta _2^2\tau _c}\\&\quad \left. +\, \frac{\bar{d}_2(\omega -\hbox {i}\beta _2)\left( \hbox {e}^{3{\mathrm{i}}\beta _2\tau _c}-1\right) }{3\beta _2^2\tau _c}\right] , \end{aligned}$$

$$\begin{aligned} h_{0011}^{(2)}&= C_{0011}^{(2)} -k(\omega -1)b_{0011}^{(1)}\\&\quad \left[ \frac{d_1\left( \hbox {e}^{-{\mathrm{i}}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _1^2\tau _c}+ \frac{\bar{d}_1\left( \hbox {e}^{{\mathrm{i}}\beta _1\tau _c}-1\right) }{(1+\gamma _c)\beta _1^2\tau _c}\right. \\&\quad \left. +\, \frac{d_2\left( \hbox {e}^{-{\mathrm{i}}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _2^2\tau _c}+ \frac{\bar{d}_2\left( \hbox {e}^{{\mathrm{i}}\beta _2\tau _c}-1\right) }{(1+\gamma _c)\beta _2^2\tau _c}\right] . \end{aligned}$$