[Tanaka, Fujimoto, Saito]

To observe the beat signal with the RFPD installed at OMC REFL, the vertical axis of the spectrum analyzer was set to a linear scale, and the number of frame averages was increased to make the peak height and frequency easier to identify. During the observation, the beat frequency occasionally shifted toward lower frequencies, sometimes as often as once every few seconds. The No. 3 sub-laser used in this experiment (as identified in the JGW DOC documentation) is known to exhibit frequency-noise events that increase its RMS frequency noise approximately once every 5 s to 2 min, and the occurrence rate closely matched that of the observed beat-frequency shifts. Therefore, these frequency shifts are considered to originate from the frequency noise of the sub-laser. The beat-signal amplitude also fluctuated, which is believed to be caused by fluctuations of PRX. Accordingly, when the beat signal was stable, the LO frequency was varied, and the minimum and maximum frequencies at which the beat-signal amplitude reached its maximum were measured around 160 MHz, 140 MHz, −160 MHz, and −140 MHz. Fitting these measurements yielded a PRX length of 68.27 ± 0.01 m, compared with the design value of 68.2563 m.
 

• The sub-laser was injected into PRX, the PLL was engaged, and the beat signal was observed with the RFPD installed at OMC REFL. The objective was to determine the minimum and maximum frequencies at which the beat-signal amplitude reached its maximum. For this purpose, the vertical axis of the spectrum analyzer was set to a linear scale, and the number of frame averages was increased to improve the visibility of the peak height and frequency. By varying the PLL LO frequency, both the resonance point (Fig. 1) and the anti-resonance point (Fig. 2) were observed. At the anti-resonance point, however, the beat-signal peak appeared to split into two peaks.
   
• During the measurements, the beat frequency occasionally shifted toward lower frequencies, sometimes as often as once every few seconds. The beat signal in the PLL path exhibited the same frequency shift. Initially, fluctuations of the Moku:Lab LO signal were suspected, so the LO source was switched from the Moku:Lab to the function generator that had originally been used. However, no change was observed. Fluctuations of the main laser were also considered, and an additional control loop was applied to suppress them, but this likewise produced no improvement. On the other hand, the No. 3 sub-laser used in this experiment (as described in the JGW DOC documentation) is known to exhibit frequency-noise events that increase its RMS frequency noise approximately once every 5 s to 2 min, and the occurrence rate closely matched that of the observed beat-frequency shifts. Therefore, the observed frequency shifts are considered to originate from the frequency noise of the sub-laser. The beat-signal amplitude also fluctuated, which is believed to be caused by fluctuations of PRX.
   

• Therefore, when the beat signal was stable, the LO frequency was varied, and the minimum and maximum frequencies at which the beat-signal amplitude reached its maximum were measured around 160 MHz, 140 MHz, −160 MHz, and −140 MHz. Negative frequencies correspond to the case where the sub-laser frequency is lower than the main-laser frequency. The measured values are listed below.

  Minimum    Maximum
  162.363 MHz    162.442 MHz
  140.366 MHz    140.494 MHz
  −140.683 MHz    −140.555 MHz (assuming ±0.064 MHz around −140.619 MHz)
  −160.435 MHz    −160.307 MHz (assuming ±0.064 MHz around −160.371 MHz)

  The midpoint between the minimum and maximum frequencies was then calculated for each measurement. Each midpoint was divided by the FSR calculated from the PRX design length of 68.2563 m. The resulting values were rounded to the nearest integers, and the measured frequencies were fitted with the linear function AN+B, where A and B are fitting parameters and N is the corresponding integer. The fitting results are shown in Fig. 3 and are summarized below:

  A = 2.1957 ± 0.0004 MHz
  B = −0.08 ± 0.03 MHz

  Since A corresponds to the FSR, the PRX length calculated from the fitted FSR is

  Fitted PRX length: 68.27 ± 0.01 m
  Design value: 68.2563 m

[Tanaka, Hirose, Fujimoto, Saito]

Following the same procedure as in the previous measurement (klog:37185), the lengths of PRY, SRY, and SRX were measured. The differences between the measured and design values were 0.6 ± 1.2 cm for PRY, 1.5 ± 1.3 cm for SRY, and 2.8 ± 1.6 cm for SRX. Therefore, the PRY measurement is consistent with the design value within the measurement uncertainty of 1.2 cm, whereas the differences for SRY and SRX exceed their respective uncertainties, suggesting that their actual lengths may differ from the design values.
 

• As in the previous measurement (klog:37185), the sub-laser was injected into PRY, SRY, and SRX, the PLL was engaged, and the beat signal was observed with the RFPD installed at OMC REFL. The minimum and maximum frequencies at which the beat-signal amplitude reached its maximum were measured. The results are summarized below.

  PRY
  Minimum    Maximum
  161.501 MHz    161.583 MHz
  138.351 MHz    138.485 MHz (assuming ±67 kHz around 138.418 MHz)
  −140.999 MHz    −140.865 MHz (assuming ±67 kHz around −140.932 MHz)
  −161.783 MHz    −161.649 MHz (assuming ±67 kHz around −161.716 MHz)
  SRY
  Minimum    Maximum
  161.371 MHz    161.492 MHz
  140.6095 MHz    140.7305 MHz (assuming ±60.5 kHz around 140.67 MHz)
  −141.0055 MHz    −140.8845 MHz (assuming ±60.5 kHz around −140.945 MHz)
  −161.7595 MHz    −161.6385 MHz (assuming ±60.5 kHz around −161.699 MHz)
  SRX
  Minimum    Maximum
  −160.384 MHz    −160.244 MHz (assuming ±70 kHz around −160.314 MHz)
  −140.65 MHz    −140.51 MHz (assuming ±70 kHz around −140.58 MHz)
  162.316 MHz    162.456 MHz (assuming ±70 kHz around 162.386 MHz)
  140.33 MHz    140.47 MHz (assuming ±70 kHz around 140.400 MHz)

  During the measurements, the sub-laser temperature was changed significantly when switching the beat frequency from +160 MHz to −160 MHz. Under these conditions, the beat frequency observed at OMC REFL fluctuated much more frequently, suggesting that the fluctuations become significant until the sub-laser temperature stabilizes. In addition, after switching from +140 MHz to −140 MHz during the SRY measurement, the frequency fluctuations did not subside. However, when the MCE feedback was enabled during the subsequent SRX measurement, the fluctuations were noticeably reduced. This suggests that fluctuations of the main laser also contribute to the beat-frequency instability.
   

• For each cavity, the midpoint between the measured minimum and maximum frequencies was calculated and divided by the corresponding FSR calculated from the design lengths of 64.9265 m (PRY), 64.9264 m (SRY), and 68.2562 m (SRX). The resulting values were rounded to the nearest integers, and the measured frequencies were fitted with the linear function AN+B, where A and B are fitting parameters and N is the corresponding integer. The fitting results are as follows.

  PRY (Fig. 1)
  A = 2.30892 ± 0.00044 MHz
  B = −0.091 ± 0.030 MHz
  SRY (Fig. 2)
  A = 2.30818 ± 0.00046 MHz
  B = −0.136 ± 0.030 MHz
  SRX (Fig. 3)
  A = 2.19520 ± 0.00051 MHz
  B = −0.076 ± 0.035 MHz

  Since A corresponds to the FSR, the cavity lengths obtained from the fitted FSR values are

  PRY
  Fitted length: 64.921 ± 0.012 m
  Design length: 64.9265 m
  Difference (Fitted − Design): −0.6 ± 1.2 cm
  SRY
  Fitted length: 64.941 ± 0.013 m
  Design length: 64.9264 m
  Difference (Fitted − Design): 1.5 ± 1.3 cm
  SRX
  Fitted length: 68.284 ± 0.016 m
  Design length: 68.2562 m
  Difference (Fitted − Design): 2.8 ± 1.6 cm

  Therefore, the measured PRY length is consistent with the design value within the measurement uncertainty of 1.2 cm. In contrast, the differences between the measured and design values for SRY and SRX exceed their respective uncertainties, suggesting that their actual cavity lengths may differ from the design values.