 \sigma_x  \hat{\bf c}_{0\omega} \text{cosh}(r_{\omega})\text{sinh}(r_{\omega})(0-\text{cos}\,\theta_{\omega}) + \hat{\bf d}_{0\omega}[\text{cosh}^0(r_{\omega})-\text{sinh}^0(r_{\omega})\text{cos}\,\theta_{\omega}] 