# Ripple Suppression in Broadband Microwave Photonic Phase Shifter Frequency Response
## 2 Phase Shifter Amplitude Variation and Phase Deviation
\begin{array}{ll}I=&\Re P_{in}\frac{\sqrt{L_{OPS}}}{4}\sqrt{\left(A_{uns} \sin(\gamma-\varphi_{uns})+A_{us}\sin(\gamma-\varphi_{us})\right)^{2}+\left(A _{uns}\cos(\gamma-\varphi_{uns})+A_{us}\cos(\gamma-\varphi_{uns})\right)^{2} }.\\ &\sin(\omega_{RF}t+\varphi)\end{array}  \tag{8}\]  
where \(\Re\) is the photodiode responsivity, \(P_{in}\) is the optical power into the MPPS, and \(\varphi\) is the output RF signal phase, which is given by:  
\[\varphi=\tan^{-1}\frac{A_{uns}\cos(\gamma-\varphi_{uns})+A_{us}\cos(\gamma- \varphi_{us})}{A_{uns}\sin(\gamma-\varphi_{uns})+A_{us}\sin(\gamma-\varphi_{ us})} \tag{9}\]  
It can be seen from Equations (8) and (9) that the MPPS output RF signal amplitude and phase are dependent on \(A_{us}\), \(A_{uns}\), \(\phi_{ws}\), and \(\phi_{uns}\). The presence of the unwanted sideband with amplitude \(A_{uns}\) and phase \(\phi_{uns}\) is due to the amplitude and phase imbalance of the 90\({}^{\circ}\) hybrid coupler used in the MPPS. This shows that the 90\({}^{\circ}\) hybrid coupler amplitude and phase imbalance affects the MPPS output RF signal. Figure 4 shows the deviation from the desired phase shift and the amplitude variation of the MPPS as a function of the wanted to unwanted sideband power ratio. The figure shows that the amount of MPPS amplitude variation and phase deviation are different for different phase shifts. The \(\pm 90^{\circ}\) phase shifts have the largest phase deviation, and the \(\pm 45^{\circ}\) and \(\pm 135^{\circ}\) phase shifts have the largest amplitude variation. The MPPS has a maximum amplitude variation of 3.8 dB and a maximum phase deviation of \(\pm 16^{\circ}\) when the unwanted sideband is 15 dB below the wanted sideband. A 90\({}^{\circ}\) hybrid coupler with low amplitude and phase imbalance is required to minimize the unwanted sideband amplitude in order to reduce the MPPS amplitude variation and phase deviation. However, such couplers cannot operate in a wide frequency range, which limits the MPPS bandwidth. Suppressing the unwanted sideband can reduce the MPPS amplitude variation and phase deviation, which can be achieved by inserting an optical filter at the optical modulator output. Since the unwanted sideband has a high amplitude only at the frequencies far away from the optical carrier due to the 90\({}^{\circ}\) hybrid coupler has large amplitudes and phase imbalances at high frequencies as shown in Figure 2, an expensive sharp edge roll-off optical filter is not required to suppress this unwanted sideband. The optical filter is designed to pass the wanted sideband from the OFS and the carrier from the OPS but suppress the unwanted sideband at high frequencies, e.g., >30 GHz. Such filters can be implemented by using autoregressive ring lattice architecture in planar waveguides [9] or by designing the reflection spectra of a fiber Bragg grating [10]. Figure 4: **(a)** Maximum phase deviation and (**b**) maximum amplitude variation of the phase shifter output RF signal versus the wanted to unwanted sideband power ratio for different phase shifts. Figure 3: Ratio of the frequency shifted light power to the residual carrier power (**dotted lines**) and the ratio of the frequency shifted light power to the unwanted sideband power (**solid lines**) versus the two RF signal phase difference from 90\({}^{\circ}\). The two RF signals into the dual-parallel Mach Zehnder modulator (DPMZM) have 1.5 dB (**red**), 2.5 dB (**blue**), and 3.5 dB (**black**) power differences. The modulation index is 0.2. The loss of the OFS and OPS-based MPPS, which is defined as the ratio of the photodetector output RF signal power to the RF signal power into the MPPS, was simulated using VPItransmissionMaker photonic simulation software. The simulation results show that the MPPS has an 8.3 dB extra loss compared with a conventional fiber optic link, which is formed by a laser source, a quadrature-biased MZM, and a photodetector, when the two systems have the same 7 dBm average optical power into the photodetector.