# Ripple Suppression in Broadband Microwave Photonic Phase Shifter Frequency Response
## 2 Phase Shifter Amplitude Variation and Phase Deviation  
Figure 1 shows the structure of a conventional MPPS formed by an optical frequency shifter (OFS) and an optical phase shifter (OPS) connected in parallel. The OPS shifts the carrier phase. The OFS shifts the carrier frequency by using the single sideband suppressed carrier (SSB-SC) modulation technique to generate only one RF signal modulation sideband with the same frequency as the modulator input RF signal frequency and to eliminate the carrier. The carrier from the OPS beats with the RF signal modulation sideband from the OFS at the photodetector. This generated an RF signal with the same phase shift as the optical phase shift introduced by the OPS. The OFS can be implemented by driving a dual-parallel Mach Zehnder modulator (DPMZM) by a pair of RF signals with 90\({}^{\circ}\) phase difference from a 90\({}^{\circ}\) hybrid coupler [8]. In order for the MPPS to have a wide bandwidth, the 90\({}^{\circ}\) hybrid coupler needs to be able to operate over a wide frequency range. Broadband 90\({}^{\circ}\) hybrid couplers are commercially available. The frequency responses of three commercial broadband 90\({}^{\circ}\) hybrid couplers were measured as shown in Figure 2. The three couplers were manufactured by Marki Microwave and have the same model number of QH-0440. The operating frequency of the couplers specified in the datasheet is 4-40 GHz. Figure 2 shows that the maximum amplitude and phase imbalance of the three 90\({}^{\circ}\) hybrid couplers are 2 dB and 8.5\({}^{\circ}\), respectively. It can be seen from the figure that large amplitudes and phase imbalances appear at frequencies above 30 GHz. This results in an unwanted RF modulation sideband and a residual carrier at the OFS output, which can be seen from the following analysis. Referring to Figure 1, when an RF signal with an angular frequency \(\omega_{RF}\) is applied to the OFS, the electric field at the output of MZM\({}_{1}\) and MZM\({}_{2}\) are given by:  
\[E_{MZM_{1}}=\frac{E_{in}\sqrt{t_{ff}}}{4}[k_{1}\cos(\omega_{c}t+a_{1}\sin( \omega_{RF}t+\beta_{b1}))+k_{2}\cos(\omega_{c}t-a_{1}\sin(\omega_{RF}t-\beta_{ b1}))] \tag{1}\]  
\[E_{MZM_{2}}=\frac{E_{in}\sqrt{t_{ff}}}{4}\Bigg{[} \begin{array}{c}k_{3}\cos (\omega_{c}t+a_{2}\sin((\omega_{RF}t+90^{\circ}+\theta)+\beta_{b2}))\\ +k_{4}\cos(\omega_{c}t-a_{2}\sin((\omega_{RF}t+90^{\circ}+\theta)-\beta_{b2}) )\end{array} \Bigg{]} \tag{2}\]  
where \(E_{in}\) is the electric field amplitude at the MPPS input, \(t_{ff}\) is the insertion loss of MZM\({}_{1}\) and MZM\({}_{2}\), \(k_{i}\) is the optical transmittance for light passing through the \(i\)th arm of the DPMZM, \(a_{1}\) and \(a_{2}\) are the modulation index of the top and bottom Mach Zehnder modulator (MZM), respectively, \(\beta_{bi}\) is the optical phase shift induced by the bias voltage \(V_{i}\) into MZM\({}_{i}\), and \(\theta\) is the phase deviation from 90\({}^{\circ}\) of the two RF signals into the DPMZM. MZM\({}_{1}\) and MZM\({}_{2}\) are biased at the null point, i.e., \(\beta_{b1}=\beta_{b2}\)  
Figure 1: Schematic of a conventional microwave photonic phase shifter (MPPS) that involves a 90\({}^{\circ}\) hybrid coupler [5]. \(\omega_{c}\) is the angular carrier frequency and \(\omega_{RF}\) is the angular RF signal frequency. \(=\pi/2\), and the main MZM of the DPMZM is biased at the quadrature to realize SSB-SC modulation. Neglecting the second and higher order sidebands, the DPMZM output electric field can be written as:  
\[E_{DPMZM}=\frac{E_{in}}{4\sqrt{2}}\left[