## 2 Frequency Factors and Constraints
### Readout Displacement Noise and Sideband Frequency Constraints  
The Taiji program requires that the total readout displacement noise be as small as possible, which includes carrier and first-order sideband readout displacement noise. Moreover, based on the experience of the LISA mission [13; 14; 18], the first-order sideband readout displacement noise cannot be higher than \(1/10\) of the total readout displacement noise. The power ratio between the first-order sidebands and carrier could be expressed by the ratio of the squares of first- and zero-orders of the Bessel functions of the first kind:  
\[\frac{P_{\mathrm{sideband}}}{P_{\mathrm{carrier}}}=\frac{J_{1}(m)^{2}}{J_{0}(m)^{ 2}}, \tag{1}\]  
where \(J_{0}(m)\) and \(J_{1}(m)\) denote the zero- and first-orders of the Bessel functions of the first kind, respectively, and \(m\) is the modulation depth. Research on the LISA mission [14; 18] indicates that this power ratio should be in the range of 5% to 10%, and that the corresponding modulation depth \(m\) is in the range of [0.44, 0.61], as shown in Figure 1.  
Therefore, the optimization objective and constraints are  
\[ \begin{split}& Min(\delta x^{\mathrm{total}})\\ &\left\{\begin{array}{l}\delta x^{\mathrm{sideband}}<\frac{1}{ 10}\delta x^{\mathrm{total}}\\ 0.44<m<0.61\end{array}\right.,\\ \end{split}  \tag{2}\]  
Figure 1: Carrier, first-order sideband (normalized power over the modulation depth \(m\)), and the ratio between the first-order sideband and carrier. where \(\delta x^{\rm sideband}\) and \(\delta x^{\rm total}\) indicate the sideband and total readout displacement noise, respectively. The expressions of the readout displacement noise of the carrier and first-order sideband and the total readout displacement noise are [13]  
\[\left\{ \begin{array}{l}\delta x^{\rm carrier}=\frac{\lambda}{2\pi}\frac{1}{l_ {0}(m)^{2}}\delta\phi_{\rm total}\\ \\ \delta x^{\rm sideband}=\frac{1}{\sqrt{2}}\frac{\lambda}{2\pi}\frac{f_{\rm fluct }}{f_{\rm USO}}\frac{1}{l_{1}(m)^{2}}\delta\phi_{\rm total}\\ \\ \delta x^{\rm total}=\delta x^{\rm sideband}+\delta x^{\rm carrier}\end{array} \right., \tag{3}\]  
respectively, where \(\lambda\) is the laser wavelength, which is 1064 nm in the Taiji program; \(f_{\rm het}\) is the beat-note frequency; and \(f_{\rm USO}\) is the USO frequency. Given that two identical sidebands are generated on the left- and right-hand sides of the carrier after modulation by EOM, a factor of \(1/\sqrt{2}\) must be added to the calculation of the sideband frequency noise. \(\delta\phi_{\rm total}\) is the total readout phase noise and three components, which is shown as follows [13; 19]:  
\[\left\{ \begin{array}{l}\delta\phi_{\rm total}=\sqrt{\delta\phi_{\rm SN}^{2} +\delta\phi_{\rm RIN}^{2}+\delta\phi_{\rm EN}^{2}}\\ \\ \delta\phi_{\rm SN}=\sqrt{\frac{2\epsilon(P_{\rm back}+P_{\rm receive})}{R_{\rm pd }\rm het}P_{\rm local}P_{\rm receive}}\\ \\ \delta\phi_{\rm RIN}=\rm{RIN}\sqrt{\frac{P_{\rm Best}^{2}+P_{\rm receive}^{2}}{2 \epsilon_{\rm het}P_{\rm local}P_{\rm receive}}}\\ \\ \delta\phi_{\rm EN}=\frac{\sqrt{2N_{\rm pd}}}{R_{\rm pd}}\sqrt{\frac{\delta P_{ \rm pd}^{2}+2\pi C_{\rm pd}/\rm{upper}\delta U_{\rm pd}}{\epsilon_{\rm het}P_{ \rm local}P_{\rm receive}}}\end{array} \right., \tag{4}\]  
where \(\delta\phi_{\rm SN}\), \(\delta\phi_{\rm RIN}\), and \(\delta\phi_{\rm EN}\) are the shot, RIN, and electronic noise, respectively; and \(P_{\rm local}\) and \(P_{\rm receive}\) represent the local laser power and laser power received by the telescope, respectively. \(P_{\rm receive}\)[12] is calculated as  
\[P_{\rm receive}=0.4073\times\frac{\pi^{2}D^{4}P_{\rm tel}\epsilon_{\rm opt}}{8 L^{2}\lambda^{2}}. \tag{5}\]  
For the Taiji program, the explanation and values of other parameters in Equations (4) and (5) are listed in Table 1.  
In Equation (3), \(\delta x^{\rm sideband}\) is positively correlated with \(f_{\rm het}\) and negatively correlated with \(f_{\rm USO}\).