## 5 Experimental Results and Discussion
two-photon natural linewidth, laser linewidth, and power and collision broadening effects. The mean frequency splitting intervals between the Cs 5\(p^{67}\)_d_\({}^{2}D_{5/2}\) (_F\({}^{\prime\prime}\)_ = 6, 5, 4) hyperfine levels were 10.60(17) MHz (5\({}^{\prime\prime}\)-6\({}^{\prime\prime}\)) and 8.54(15) MHz (4\({}^{\prime\prime}\)-5\({}^{\prime\prime}\)), respectively. Table 1 and Figure 9 show the mean values and errors of the energy-level intervals obtained through multiple fitting. The magnetic dipole hyperfine interaction constant \(A\) and the electrical quadrupole hyperfine interaction constant \(B\) for the Cs 5\(p^{67}\)_d_\({}^{2}D_{5/2}\) excited state were determined as a function of the splitting interval of the spectra, expressed as follows:  
\[\text{HFS}_{6^{\prime\prime}-5^{\prime\prime}}=6A+\frac{18}{35}B=-10.60(17)\ \text{MHz}, \tag{3}\]  
\[\text{HFS}_{5^{\prime\prime}-4^{\prime\prime}}=5A+\frac{3}{70}B=-8.54(15)\ \text{MHz}. \tag{4}\]  
According to Equations (3) and (4), the magnetic dipole hyperfine interaction constant \(A\) was calculated to be \(-1.70(03)\) MHz and the electrical quadrupole hyperfine interaction constant \(B\) was calculated to be \(-0.77(58)\) MHz for the Cs 5\(p^{67}\)_d_\({}^{2}D_{5/2}\) excited state. These results are in good agreement with the previous reported results [27; 28; 29; 34], as shown in Table 2. The \(A\) coupling constant agrees well with previous measurements (e.g., reference [34], which demonstrated an experiment using direct frequency-comb spectroscopy). In reference [27], a positive value was also obtained for the constant \(B\) for the Cs 5\(p^{67}\)_d_\({}^{2}D_{5/2}\) state. This discrepancy could have arisen from the overlap of multiple peaks in that study. The hyperfine splitting intervals of the Cs 5\(p^{67}\)_d_\({}^{2}D_{5/2}\) state were too small and could only \begin{table}
\begin{tabular}{c c} \hline \hline
**Hyperfine Splitting** & **Measured Splitting Values (MHz)** \\ \hline
5\(p^{67}\)_d_\({}^{2}D_{5/2}\) (_F\({}^{\prime\prime}\)_ = 6)–(_F\({}^{\prime\prime}\)_ = 5): \(\text{HFS}_{6^{\prime\prime}-5^{\prime\prime}}\) & \(-10.60(17)\) \\
5\(p^{67}\)_d_\({}^{2}D_{5/2}\) (_F\({}^{\prime\prime}\)_ = 5)–(_F\({}^{\prime\prime}\)_ = 4): \(\text{HFS}_{5^{\prime\prime}-4^{\prime\prime}}\) & \(-8.54(15)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Measured values of hyperfine splitting of the Cs 5\(p^{67}\)_d_\({}^{2}D_{5/2}\) state. Figure 9: The measured values of the hyperfine splitting intervals for the Cs 5\(p^{67}\)_d_\({}^{2}D_{5/2}\) excited state. The horizontal red lines represent the mean values of the hyperfine splitting. The ranges between the two horizontal blue dashed lines donate the statistical errors. (**a**) Hyperfine splitting between (_F\({}^{\prime\prime}\)_ = 6) and (_F\({}^{\prime\prime}\)_ = 5) levels of the Cs 5\(p^{67}\)_d_\({}^{2}D_{5/2}\) state; (**b**) Hyperfine splitting between (_F\({}^{\prime\prime}\)_ = 5) and (_F\({}^{\prime\prime}\)_ = 4) levels of the Cs 5\(p^{67}\)_d_\({}^{2}D_{5/2}\) state. The histograms of the hyperfine splitting intervals are shown in (**c**,**d**). be barely resolved. In reference [29], a study of the Cs 5\(p^{6}\)6s \({}^{2}S_{1/2}\)-5\(p^{6}\)7\(d\)\({}^{2}D_{3/2,\,5/2}\) electric quadrupole transition in a thermal Cs vapor was reported using a single laser with a wavelength of 767 nm, also giving a positive value for the constant \(\mathbf{B}\). The exact determination of the constant \(\mathbf{B}\) using laser absorption spectroscopy is very difficult and subject to high uncertainty. The experimental error mainly resulted from the spectral resolution limitation caused by the wide linewidth of the obtained spectra and the accuracy of the function generator driving the EOPM. Considering the error of the system, the frequency locking accuracy of the 852 nm laser needs to be improved. The nonlinear frequency scanning of the 698 nm coupling laser also affected the experimental results. A larger scanning range led to an obvious nonlinear effect.