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from Gerginov et al. [16, 25]. The two-photon tran- sition center-of-gravity frequencies ν$_{6S}$$_{1}$$_{/}$$_{2}$$_{:}$$_{n}$ ′′ L ′′ J ′′ and the hyperfine coupling constants A$_{n}$ ′′ L ′′ J ′′ and B$_{n}$ ′′ L ′′ J ′′ were extracted from the experimental data by least-squares comparison of the experimental resonant repetition-rate frequencies with the resonant repetition-rate frequencies determined from the calculated spectra. 

The resonant repetition rate for each of the peaks in both the experimental and calculated spectra were deter- mined from a least-squares fit. Fitting the spectra was necessary in order to account for the overlap of the dif- ferent peaks. The calculated spectra were assumed to have a Lorentzian line shape. However, the experimental spectra were found to be well described by a line shape of a Gaussian coupled to a Lorentzian, 

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where C G i and C L i are the Gaussian and Lorentzian amplitudes, respectively, γ$_{G}$ and γ$_{L}$ are the Gaussian and Lorentzian widths, f$_{i}$ is the line center of the i th peak, and C B is a constant base-line offset. We at- tribute the deviation of the experimental line shape from the expected Lorentzian line shape to residual misalign- ment of the two beams, which we discuss further be- low. We designate the resonant repetition-rate frequency for the i th peak determined by fitting the calculated spectra for a given ν$_{6S}$$_{1}$$_{/}$$_{2}$$_{:}$$_{n}$ ′′ L ′′ J $_{′′}$, A$_{n}$ ′′ L ′′ J $_{′′}$, and B$_{n}$ ′′ L ′′ J ′′ by f calc i ( ν$_{6S}$$_{1}$$_{/}$$_{2}$$_{:}$$_{n}$ ′′ L ′′ J $_{′′}$, A$_{n}$ ′′ L ′′ J $_{′′}$, B$_{n}$ ′′ L ′′ J ′′ ) . The resonant repetition-rate frequencies determined from the experi- mental spectra we designate as f expt i . The best fit val- ues for two-photon frequency and the hyperfine constants were determined by minimizing the weighted χ 2 function 

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with respect to ν$_{6S}$$_{1}$$_{/}$$_{2}$$_{:}$$_{n}$ ′′ L ′′ J $_{′′}$, A$_{n}$ ′′ L ′′ J $_{′′}$, and B$_{n}$ ′′ L ′′ J $_{′′}$. Here σ expt i is the standard uncertainty in f expt i determined by the fit to the experimental data. For N peaks and all three parameters ( ν$_{6S}$$_{1}$$_{/}$$_{2}$$_{:}$$_{n}$ ′′ L ′′ J $_{′′}$, A$_{n}$ ′′ L ′′ J $_{′′}$, and B$_{n}$ ′′ L ′′ J $_{′′}$) one would expect the minimum value of the χ 2 to be N − 3. We consistently observed larger values of the χ $^{2}$. We attribute this uncertainty to residual misalign- ment, which will be discussed in more detail below. In order to account for this larger χ 2 value, we normalize the χ 2 function so that its minimum corresponds to the ex- pected value. We then determine the uncertainty in the two-photon optical frequency and the hyperfine coupling constants by finding the values of ν$_{6S}$$_{1}$$_{/}$$_{2}$$_{:}$$_{n}$ ′′ L ′′ J $_{′′}$, A$_{n}$ ′′ L ′′ J $_{′′}$, and B$_{n}$ ′′ L ′′ J ′′ where the normalized χ 2 function increases by one from the point at which it is a minimum. This amounts to increasing the uncertainty to make the fre- quencies extracted from the different peaks self consis- tent. 

The spectral range of the comb allowed the 8S$_{1}$$_{/}$$_{2}$ state to be studied by use of transitions through either the 6P$_{1}$$_{/}$$_{2}$ (895 nm and 761 nm) or 6P$_{3}$$_{/}$$_{2}$ (852 nm and 795 nm) states. This redundancy provided a check for con- sistency of the measurement and the parameter extrac- tion method. A complete study of the systematics was 

## B. Results for the 8S 1 / 2 state 

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performed only for the transitions that occurred through the 6P$_{3}$$_{/}$$_{2}$ states. 

Figure 6 shows the spectrum for the 6S$_{1}$$_{/}$$_{2}$ → 6P$_{1}$$_{/}$$_{2}$ → 8S$_{1}$$_{/}$$_{2}$ transition along with the cal- culated spectrum. The spectrum was collected by use of a 890 nm filter with a 10 nm passband for F1 and 

