## 2 Experiment  
In this study, an external-cavity diode laser at 1542 nm (RIO PLANEX) was used for spectroscopy and laser frequency stabilization. The output of the laser light was injected into an Er-doped fiber amplifier for power amplification. The amplified laser output was sent to a periodically poled lithium niobate (PPLN) waveguide [7] for the third-harmonic generation (THG) at 514 nm. The generated THG light was used for Doppler-free spectroscopy of molecular iodine based on modulation transfer spectroscopy [12; 13]. The iodine spectrometer contained a 30 cm-long iodine cell. In the modulation transfer spectroscopy, a pump beam with phase modulation was used to saturate the iodine absorption. The phase modulation was set to 220 kHz using an acousto-optic modulator. A probe laser beam was used to detect the saturation dip through modulation transfer [12; 13]. The modulation on  
Figure 1: Frequency atlas of the \({}^{127}\)I\({}_{2}\) absorption lines (blue and green lines) near the tripled frequency of acetylene lines (red lines) at 514 nm. The blue dashed and dot-dashed lines are the iodine lines investigated in Refs. [8; 10], respectively. The green lines are the iodine lines investigated in the present study. The relative intensity of iodine lines was taken from Ref. [11]. The intensity of the acetylene lines is not on the same scale as the iodine lines. the pump beam was transferred to the probe beam only when saturation occurred. The probe beam was sent to a photodetector to generate first-derivative-like modulation transfer signals through demodulation. Laser frequency stabilization was achieved by using the observed modulation transfer signal as an error signal. Possible error signal offset from feedback electronics was minimized by adjusting the lock point to the averaged level of the off-resonance spectral baseline. Servo control was achieved by the feedback control of the laser current and temperature. For the laser frequency measurement, we conducted a beat frequency measurement between the frequency-locked laser and Er-doped fiber comb [15]. An iodine-stabilized Nd:YAG laser [16] was used as an optical frequency reference for the Er-doped fiber comb. Further details of the experimental setup and method can be found elsewhere [8; 10]. Figure 2 shows the observed hyperfine structures of the R(106)50-0, R(100)49-0, R(84)47-0, R(59)45-0, P(82)47-0, and P(71)46-0 lines of \({}^{127}\)I\({}_{2}\). The laser frequency was swept to observe hyperfine structures by gradually changing the laser temperature. The cold-finger temperature of the iodine cell was maintained at \(-\)10 \({}^{\circ}\)C, which corresponds to an iodine pressure of approximately 1.4 Pa. The optical powers of the pump and probe beams were 1.8 and 0.28 mW, respectively, except for the P(71)46-0 line, where the probe beam power was 0.15 mW. The diameters of both the beams were approximately 1.7 mm. The hyperfine transitions of the R(84)47-0 and R(59)45-0 lines are shown as the \"a\" and \"b\" series, respectively, in the same graph [Figure 2c]. For the R(106)50-0, R(100)49-0, P(82)47-0 and R(84)47-0 lines, whose ground state \(J^{\prime\prime}\) is even (106, 100, 82, and 84, respectively), 15 hyperfine transitions were observed [Figure 2a,b,d, and \"a\" series of Figure 2c, respectively]. On the other hand, 21 hyperfine transitions were observed for the P(71)46-0 and R(59)45-0 lines, whose ground state \(J^{\prime\prime}\) is odd (71 and 59, respectively) [Figure 2e and \"b\" series of Figure 2c]. The weak components in the spectra of the R(106)50-0, R(100)49-0, and P(82)47-0 lines belonged to nearby rovibrational lines. Several close hyperfine transitions that were not resolved in Figure 2 were resolved on an enlarged scale and could be locked separately: (i) the a\({}_{2}\) transition of the R(100)49-0 line and the nearby transition of another line, (ii) the b\({}_{7}\) and b\({}_{8}\) transitions of the R(59)45-0 line, and (iii) the a\({}_{6}\) and a\({}_{7}\) transitions of the P(82)47-0 line.