## 2 Relevant Principles
### Spectroscopic Schemes  
High-resolution and high-SNR spectroscopic schemes are very helpful to achieve a higher accuracy of the hyperfine splitting measurement, so as to deduce accurate hyperfine interaction constants. (here, the magnetic dipole hyperfine interaction constant \(A\) and the electrical quadrupole hyperfine interaction constant \(B\) for the Cs \(5p^{6}7d\)\({}^{2}D_{5/2}\) excited state). A DROP spectroscopic scheme using the ladder-type three-level system (\(5p^{6}6s\)\({}^{2}S_{1/2}\)-\(5p^{6}6p\)\({}^{2}P_{3/2}\)-\(5p^{6}7d\)\({}^{2}D_{5/2}\)) was employed to measure the hyperfine splitting in the Cs \(5p^{6}7d\)\({}^{2}D_{5/2}\) excited state. An 852 nm laser was used as the probe laser to detect the population change of the ground state, reflecting the hyperfine spectrum between excited states. At the same time, we also studied the optical-optical double-resonance (OODR) absorption spectroscopic scheme and used a 698 nm probe laser to detect the population change of intermediate state to reflect the hyperfine spectrum between excited states. The OODR spectroscopic signal was related to the population between \(5p^{6}6p\)\({}^{2}P_{3/2}\) (\(F^{\prime}=5\)) and \(5p^{6}7d\)\({}^{2}D_{5/2}\) (\(F^{\prime}=4\), 5, 6), while \(5p^{6}6p\)\({}^{2}P_{3/2}\) (\(F^{\prime}=5\)) had a large spontaneous decay rate (\(\Gamma=2\pi\times 5.22\) MHz); thus, atoms were not easy to inhabit in the \(5p^{6}6p\)\({}^{2}P_{3/2}\) (\(F^{\prime}=5\)) state. On the contrary, this accelerated the DROP process, which rapidly reduced the population of atoms in the ground state (\(F=4\)), thereby leading to a relatively higher SNR, which was helpful to more accurately measure the hyperfine splitting interval. In the three-level system shown in Figure 1, when the coupling laser and the probe laser were in resonance, under the weak laser approximation, the polarizability of the medium could be expressed as follows [32]:  
\[\chi(v)dv=\frac{4i\hslash g_{21}^{2}/\varepsilon_{0}}{\gamma_{21}-i\Delta_{p} -i\frac{\omega_{p}}{c}v+\frac{\Omega_{c}^{2}/4}{\gamma_{31}-i\left(\Delta_{p} +\Delta_{c}\right)-i\left(\omega_{p}\mp\omega_{c}\right)\triangleright/c}}N(v)dv, \tag{2}\]  
where \(\omega_{p}\) is the frequency of the probe laser, \(\omega_{c}\) is the frequency of the coupling laser, \(\Delta_{p}\) is the detuning of the probe laser corresponding to the frequency of the \(5p^{6}6s\)\({}^{2}S_{1/2}\) (\(F=4\))-\(5p^{6}6p\)\({}^{2}P_{3/2}\) (\(F^{\prime}=5\)) transition, \(\Delta_{c}\) is the detuning of the coupling laser corresponding to the frequency of the \(5p^{6}6p\)\({}^{2}P_{3/2}\) (\(F^{\prime}=5\))-\(5p^{6}7d\)\({}^{2}D_{5/2}\) (\(F^{\prime\prime}=6\)) transition, the decay rate is \(\gamma_{ij}=\left(G_{i}+G_{j}\right)/2\), \(g_{21}\) is the dipole moment matrix element for the \(5p^{6}6s\)\({}^{2}S_{1/2}\) (\(F=4\))-\(5p^{6}6p\)\({}^{2}P_{3/2}\) (\(F^{\prime}=5\)) transition, \(\Omega_{c}=2\gamma_{32}E_{c}\) is the Rabi frequency of the coupling laser, \(\Gamma_{i}\) is the natural linewidth of level \(i\), \(N\) is the density of Cs atoms, \(v\) is the velocity of Cs atoms, \(E_{c}\) represents the amplitude of the coupling laser field, and \(c\) is the speed of the laser [32]. For the counter-propagating (CTP) configuration, the difference of narrow linewidth is attributed to the atomic coherence between the ground and upper states driven by the two coherent laser fields via a common intermediate state, resulting in an EIT effect. In Equation (2), the term \(-i\Big{(}\omega_{p}-\omega_{c}\Big{)}v/c\) corresponds to CTP configuration, making it easy to achieve EIT signals in the two-photon Doppler-free configuration, as the linewidth of the signal is narrow due to the atomic coherence effect. On the contrary, for the co-propagating (CP) configuration, the term \(-i\Big{(}\omega_{p}+\omega_{c}\Big{)}v/c\) cannot be ignored. The Doppler background basically overwhelms the EIT signals [33].