# Ripple Suppression in Broadband Microwave Photonic Phase Shifter Frequency Response
## 2 Phase Shifter Amplitude Variation and Phase Deviation
\begin{array}{c}-k_{1}J_{0}(a_{1}) \sin(\omega_{c}t)-k_{1}J_{1}(a_{1})\sin[(\omega_{c}+\omega_{RF})t]+k_{1}J_{1}(a _{1})\sin[(\omega_{c}-\omega_{RF})t]\\ +k_{2}J_{0}(a_{1})\sin(\omega_{c}t)-k_{2}J_{1}(a_{1})\sin[(\omega_{c}+\omega_{ RF})t]+k_{2}J_{1}(a_{1})\sin[(\omega_{c}-\omega_{RF})t]\\ -k_{3}J_{0}(a_{2})\cos(\omega_{c}t)+k_{3}J_{1}(a_{2})\sin[(\omega_{c}+\omega_{ RF})t+\theta]+k_{3}J_{1}(a_{2})\sin[(\omega_{c}-\omega_{RF})t-\theta]\\ +k_{4}J_{0}(a_{2})\cos(\omega_{c}t)+k_{4}J_{1}(a_{2})\sin[(\omega_{c}+\omega_{ RF})t+\theta]+k_{4}J_{1}(a_{2})\sin[(\omega_{c}-\omega_{RF})t-\theta]\\ \end{array} \right] \tag{3}\]  
The electric field amplitude of the wanted sideband \(A_{us}\), i.e., the frequency shifted light, at an angular frequency \(\omega_{c}-\omega_{RF}\), the unwanted sideband \(A_{us}\) at \(\omega_{c}+\omega_{RF}\), and the residual carrier \(A_{rc}\) at \(\omega_{c}\) at the DPMZM-based OFS output can be obtained from Equation (3) and are given by:  
\[A_{us}=\frac{E_{in}\sqrt{L_{OFS}}}{2\sqrt{2}}\sqrt{ \begin{array}{c}\left[k_ {1}J_{1}(a_{1})+k_{2}J_{1}(a_{1})+\left(k_{3}J_{1}(a_{2})+k_{4}J_{1}(a_{2}) \right)\cos\theta\right]^{2}\\ +\left[-(k_{3}J_{1}(a_{2})+k_{4}J_{1}(a_{2}))\sin\theta\right]^{2}\end{array} } \tag{4}\]  
\[A_{uns}=\frac{E_{in}\sqrt{L_{OFS}}}{2\sqrt{2}}\sqrt{ \begin{array}{c}\left[- k_{1}J_{1}(a_{1})-k_{2}J_{1}(a_{1})+\left(k_{3}J_{1}(a_{2})+k_{4}J_{1}(a_{2}) \right)\cos\theta\right]^{2}\\ +\left[\left(k_{3}J_{1}(a_{2})+k_{4}J_{1}(a_{2})\right)\sin\theta\right]^{2} \end{array} } \tag{5}\]  
\[A_{rc}=\frac{E_{in}\sqrt{L_{OFS}}}{2\sqrt{2}}\sqrt{\left(-k_{1}J_{0}(a_{1})+k _{2}J_{0}(a_{1})\right)^{2}+\left(-k_{3}J_{0}(a_{2})+k_{4}J_{0}(a_{2})\right)^ {2}} \tag{6}\]  
where \(L_{OFS}\) is the OFS insertion loss. In the ideal case, \(k_{1}=k_{2}=k_{3}=k_{4}=1\) and the \(90^{\circ}\) hybrid coupler has no amplitude and phase imbalance, i.e., \(a_{1}=a_{2}\) and \(\theta=0^{\circ}\), thus the unwanted sideband and the residual carrier are cancelled at the OFS output. Figure 2: (**a**) Amplitude and (**b**) phase responses of three Marki Microwave 4 to 40 GHz bandwidth \(90^{\circ}\) hybrid couplers measured at the coupler \(90^{\circ}\) output port with reference to the coupler \(0^{\circ}\) output port. The power of the residual carrier and the unwanted sideband relative to the frequency shifted light power can be obtained from Equations (4)-(6). It can be seen from Figure 3 that the ratio of the frequency shifted light power to the unwanted sideband power is 15.2 dB and the ratio of the frequency shifted light power to the residual carrier power is 17 dB when the two RF signals into the DPMZM have a \(\pm 10^{\circ}\) phase difference from 90\({}^{\circ}\) and a 2.5 dB power difference. Note that the MPPS in Reference [5] has an OPS in parallel with the OFS. The OPS only reduces the carrier by around 2 dB due to its insertion loss. Therefore, the residual carrier at the OFS output is much smaller than the carrier at the OPS output. Hence, it has a negligible effect on the MPPS performance. On the other hand, the unwanted sideband cannot be neglected. The OFS- and OPS-based MPPS output electric field with the inclusion of the unwanted sideband at \(\omega_{c}+\omega_{RF}\) is given by:  
\[E_{o}(t)=E_{in}\left[ \begin{array}{c}\sqrt{\frac{L_{OPS}}{2}}\cos(\omega_{ c}t+\gamma)\\ +\sqrt{\frac{L_{OPS}}{2}}\frac{A_{max}}{2}\cos((\omega_{c}+\omega_{RF})t+\varphi _{uns})\\ +\sqrt{\frac{L_{OPS}}{2}}\frac{A_{max}}{2}\cos((\omega_{c}-\omega_{RF})t+ \varphi_{us})\end{array} \right] \tag{7}\]  
where \(L_{OPS}\) is the OPS insertion loss, \(\gamma=\pi V_{DC}/V_{\pi}\) is the optical phase shift introduced by the OPS, \(V_{DC}\) is the DC voltage into the OPS, \(V_{\pi}\) is the OPS switching voltage, and \(\phi_{uns}\) and \(\phi_{uns}\) are the unwanted and wanted sideband optical phases, respectively. The photocurrent at the RF signal angular frequency \(\omega_{RF}\) can be obtained from Equation (7) and is written as:  
\[