This post reports on the results from SRCL dither measurement I ran in January, briefly reported in alog 82248. It's taken me a long time to write up this report because I spent significant time processing the results in different ways to try to account for some of the possible pitfalls of this measurement. The overall problem is that, in O4, this measurement has traditionally reported very low arm power compared to our other methods of estimating arm power (for example, see Craig's work in 66860). I have been doing an exhaustive study to understand why that might be.
A full derivation of this measurement can be found in Craig's dissertation Section 3.2.2, which also includes references to work by Daniel and Kiwamu that orignally inspired this method. To summarize, the idea of the measurement is that dithering the SRM creates differential amplitude sidebands due to the radiation pressure coupling in the SRC. This response can be readout as a transfer function from the differential arm length to the relative intensity noise on the arm transmission QPDs. The resulting transfer function takes a simple form of DARM/RIN = alpha/f^2. The arm power is calculated via P_arm = 1/2 * alpha * pi^2 * M * c (M = test mass mass, c = speed of light)
Here are some possible issues with the measurement:
- Achieving good SNR is challenging due to the noise in the TMS QPDs. When driving a SRCL line that is ~10^3-4 above the DARM noise in amplitude, the line height in the TMS QPDs is only about ~10 above the noise in amplitude. Line height uncertainty and bias scale as 1/SNR (in PSD). While this is still a small effect on the order of a few percent for the TMS QPDs, I have other reasons to believe that it is significant enough to make a difference in this measurement.
- A good accounting of the uncertainty and bias is a necessary aspect of this measurement, especially considering that the fit error in Craig's results has always been very small. For example, in 66860, his results include an uncertainty that is less than a percent. Craig's SRCL dither analysis code, found here, uses scipy curvefit and does not include measurement uncertainty in the fit. The reported error comes from the fit error only.
- Dan Brown ran a SRCL dither measurement model in finesse, found here, and he shows that applying a differential test mass curvature change (something that might occur due to to thermal effects), could deviate the DARM/TMS RIN transfer function from 1/f^2 at low frequency. This appears to be a very subtle effect, but it could cause a misestimation of the arm power.
In order to fully study this properly, I ran a bayesian inference on my measurement data with both the amplitude and power law as free parameters. I wanted to confirm that the 1/f^2 trend agrees with the data, and if it agrees, get the arm power estimate (note: if the power law is different from -2, the overall amplitude of the fit cannot be used to measure the arm power). In the process of setting up the inference, I set about making sure that the uncertainty on the line height in both DARM and the TMS QPDs was being correctly calculated.
Uncertainty and Bias:
Appendix E of Craig's thesis is a nice reference for line uncertainty.
- The line height bias is measured as 1/SNR, following E.4 and E.5. "So even with an SNR of 100, you will expect a 1% bias in the calibration line." Line height bias in the TMS QPDs from these measurements ranges from less than a percent to 7%, depending on how noisy the QPD is.
- The line height uncertainty is shown in E.25, 1/(2*N_avg * SNR) *(1 + 1/SNR)
- I pulled the DARM calibration uncertainty from the time nearest to my measurement, which provides an uncertainty and bias in the DARM estimate (a measurement taken at GPS time 1420853061)
- Overall, the uncertainty is still very small, but accounting for the bias in both the DARM and TMS QPD line height estimates increases the estimate of the transfer function by a few percent
Testing frequency dependence:
I set up a bayesian inference using a model that fits the frequency dependence and "power", assuming a gaussian distribution of my uncertainty. Following Craig's discussion in his thesis appendix E, with SNR > 5 the ASD distribution of a line can be well-approxmiated by a gaussian distribution. I assume a flat prior, with possible powers ranging from 0-1000 kW, and possible power laws from -3 to -1. I fix the inference to assume the same arm power in each arm, so it uses both A and B QPDs in each arm to fit one X arm power and one Y arm power. I then fix the frequency dependence to be the same for both arms and both QPDs in each arm.
The results from this inference do not favor a slope of -2, with a fit that gives m=-2.016 +-0.014 (95% CI).
However, fixing the slope to -2, following the model, gives the following power results (95% CI):
X arm power = 319.6 +- 2.8 kW
Y arm power = 303.5 +- 2.4 kW
These are the highest power results achieved with this method during O4. However, they are still very low compared to what we know about the interferometer, namely that our PRG at full power is about 50, and we have about 56-57 W of power on the PRM, which predicts about 360 kW of arm power, assuming an arm gain of 260. Craig, Sheila, and I have all done work to verify these numbers before and during O4. This result also indicates a significant mismatch in the arm powers; a surprising result since our pre-O4 test mass replacement of ITMY should have made the arms more well-matched to each other than in O3.
Some commentary:
One aspect of this measurement that is very constraining on the slope of the transfer function is the overall uncertainty on each transfer function measurement point, which is very narrow. This makes sense, since we are achieving fairly good SNR overall. However, while processing the data I did notice that there is modulation of the DARM/RIN transfer function, that is on the order of a Hz to a few Hz. My guess is that this is coming from the ASC modulating DARM, SRCL, or both. I'm not sure overall what effect this could have on the estimation of the transfer function or the uncertainty. Returning to the results from Dan's finesse model, the sparseness of points in this measurement also makes it harder to determine if the slope has diverged from -2 at lower frequency due to a different effect, such as some differential mismatch in the test mass radius of curvatures.
If we choose to use these measurement results, they would certainly place a lower bound on our possible arm power, which is compatible with some of Sheila's quantum noise modeling work (see 82097). However, those models require minimal or no readout losses to achieve such a low arm power, which is incompatible with some of our other results measuring readout loss, such as the work Jennie Wright has been doing.
Back to the measurement itself, we could try to improve the result slightly by integrating longer at each point, and measuring more points to get a better idea of the slope. Instead of running Craig's swept sine measurement, I injected several lines by hand because I found it easier to verify that we were achieving good SNR this way. I'm not sure if there is a way to overcome the modulation of the injection.
