Takano
Abstract
I established a scheme to estimate the nonlinear coupling noise from OMC length fluctuation to OMC transmission noise.
Detail
I tried to estimate the nonlinear noise from OMC length fluctuation to OMC trans by considering the property of the cavity. The procedure is:
- Estimation of HWHM from the peak value of the spectra during injection tests
- Calibration of the error signal and estimaiton of the free running spectrum
- Estimation of the sensor noise level of length error signal coming from the breathing noise floor above a few hundred Hz
- Estimation of the residual length fluctuation of OMC with the estimated sensor noise level and the open loop transfer function of OMC length control loop
- Calculation of the nonlinear noise by convolving the power spectrum of the residual length fluctuation
To confirm that this scheme works well, I used measurement data with changing the dither amplitude and tried to explain these results.
Estimation of HWHM
First, I estimated the HWHM of OMC by comparing the peak height of data when shaker injection tests were performed on July 19th. I took data of the injection data at {80,, 81, 82, 83, 91, 92, 93} Hz to compare the peak value of trans signal at the 2nd harmonics of the injected frequency with the peak height of error signal at the injection frequency.
Using the optical gain (5.1e9 [cnt/m]) and DC power (28.9 [mW]), the amplitude level of error and trans signal is calibrated as shown in Figure1. I fitted the tilt of the curve and estimated the HWHM of OMC with the theoretical relationship between them. The result is 4.85(2)e-10 m.
Calibration of the free running spectrum and estimation of the sensor noise level
Next, I calibrated the spectrum of the error signal and estimated the free running spectrum of the length fluctuation. For the estimation, I used data about the open loop transfer function measurement and modeled it as plotted in Figure2. The calibration results of the free run spectra are shown in Figure3. Apparently the sensing noise level depends on the dither amplitude.
Using these results, I estimated the sensing noise level for each dither amplide case. I averaged the spectrum within the frequeny band where the spectrum is limited by some white noise and adopted the average of the sensor noise level.
Estimation of the residual length fluctuation
Next, I went back to the spectrum signal and tried to estimate the residual motion. Since the sensor noise is suppressed by ~ G above UGF, the spectrum of the actual residual fluctuation is better than that of the error signal. To estimate the actual residual fluctuation, I divided the frequency region roughly into two where the sensor noise is dominant and not. For the freuqency band where the sensor noise is not dominant, I made a model sprctrum by adding a sensor noise model (white noise times the average I estimated above) to the error signal spectrum. For the freuqency band where the sensor noise is dominant, I adopted the original error signal. For both cases, the sensor noise is multiplied by G/(1+G) to reflect the suppression of the sensor noise for the actual residual fluctuation. The estimated spectra of the residual motion with the different dither amplitudes are shown in Figure4.
Calculation of the nonlinear noise by convolution
Finally, I calculated the nonlinear noise from OMC residual length fluctuation to OMC transmission power. The quadratic nonlinear noise can be estimated by a convolution of the linear spectra. With thr estimated HWHM if OMC, the nonlinear contribution of OMC length is calculated as shown in Figure5. Here, the different color means the different dither amplitude condition. The calculated nonlinear noises are plotted as dashed lines, and the nonlinear noises are pretty matched to the excess noise of OMC trans for small dither amplitude cases.
Therefore, I concluded that this scheme works well.
Next
Assuming some vibration coupling to OMC length, I will estimate the nonlinear noise contributed from OMC length fluctuation which is so far buried with the sensing noise. If possible I want to put some requirement for OMC vibration by this analysis.
Notes
The estimated HWHM of OMC (4.85(2)e-10 m) is larger by ~ 1.5 times than the previous estimation. Some assumptions (Finesse of OMC, optical gain, ...) could be wrong.
