evan.hall@LIGO.ORG - posted 20:45, Sunday 09 March 2014 - last comment - 12:10, Wednesday 12 March 2014(10642)
PRC Length Measurement
(Evan H, Ed D, Stefan B and Dave O)(Evan H, Ed D, Stefan B and Dave O)
(Evan H, Ed D, Stefan B and Dave O)
We measured the length of the PRC by injecting the light from a 250 mW auxiliary NPRO (Lightwave) through the back of IM4 towards the PRM. The NPRO was phased locked to the main PSL carrier by observing the beat between these two lasers from the path IO_Forward using a New Focus 1811.
Phase Locking
The error signal was obtained by feeding the signal from the 1811 into the RF port of a double-balanced mixer. The LO of the mixer was driven with the source port of a network analyzer (HP4395A). The reulting IF was low-passed at 1.9 MHz and then fed into the input of a Newport LB1005 servo box. The settings on the box were 3kHz P-I corner, 50 dB low-frequency gain limit, and 4-0 on the gain knob. The output of this box was sent to the fast PZT input of the NPRO. The output was also attenuated by 4x10^-4, summed in with a DC trimpot voltage, and then and sent to the NPRO's slow temperature input. Once fast lock was acquired, the gain was increased until a small oscillation was observed, and then the gain was backed off. Then the LB1005 was switched from "LFGL" to "lock on", and the slow loop was switched on.
FSR Measurements
We measured the transfer function which takes the network analyzer's LO drive to REFLAIR_B_RF. The magnitude and phase of the this TF gives the complex reflectance function of the PRC. We measured over a series bands including 32.4 – 32.6 MHz, 68.8 – 69 MHz and 102.6 – 102.8 MHz. These bands were analysed when the auxiliary laser was locked above and below the PSL carrier.
The settings on the network analyzer were: sweep time of 500 s, 801 samples, IF bandwidth of 1kHz, and a span of 200 kHz.
Because of the sensitivity of the PLL servo, for each measurement we started the sweep on the network analyzer and then brought the auxiliary/PSL beat note into lock. Throughout these measurements, PRMI was sideband locked.
Results
We obtained the frequency response plots in the following order +102.7 MHz, +32.5 MHz, -32.5 MHz, -102.7 MHz (twice), -68.9 MHz, and +68.8 MHz.
Plots of the frequency response are attached. These magnitude data were fitted to a Lorentzian to determine the exact frequency of the resonances. The results were as follows:
FSR number Resonance frequency HWHM frequency
-39.5 102.701020 MHz +/- 180 Hz 26.7 kHz +/- 400 Hz
-39.5 102.700690 MHz +/- 190 Hz 27.3 kHz +/- 500 Hz
-26.5 68.900360 MHz +/- 150 Hz 24.0 kHz +/- 400 Hz
-12.5 32.499990 MHz +/- 90 Hz 20.56 kHz +/- 180 Hz
12.5 32.501860 MHz +/- 90 Hz 20.5 kHz +/- 200 Hz
26.5 68.904100 MHz +/- 170 Hz 25.3 kHz +/- 400 Hz
39.5 102.705400 MHz +/- 190 Hz 29.2 kHz +/- 500 Hz
Already from these numbers we can see that the PSL carrier does not appear to be perfectly antiresonant; there is an offset of about 1400 Hz. Note also there is a systematic disagreement in the numbers for the HWHM frequency. Taking a nominal value of 25 kHz for the HWHM and 2.6 MHz for the FSR gives a finesse of 50.
We then plotted these with FSR number on the horizontal axis and resonance frequency on the vertical axis. The residuals do not show a random behaviour; there appears to be additional structure not captured in this model. The slope of the line gives the FSR of the PRC, and the offset is 1400 Hz +/- 300 Hz, indicating the offset from antiresonance.
At the present time we can say that the FSR of the PRC is 2.600075 MHz +/- 26 Hz. This corresponds to a PRC length of 57.651 m +/- 1 mm. This measurement is limited by our residuals, and we are currently investigating this.
In estimating the uncertainty in the FSR, we note that the largest residual is for the 102.7 MHz measurement, and is equal to about 1 kHz. This fractional uncertainty is 1x10^-5; using this as the fractional uncertainty on the FSR gives 26 Hz. This dominates over the purely statistical error given by the fitting algorithm (11 Hz). Propagating this 26 Hz uncertainty forward to the PRC length gives 57.6507 m +/- 0.6 mm. To be conservative, we quote the uncertainty to the nearest 1 mm.
We measured the length of the PRC by injecting the light from a 250 mW auxiliary NPRO (Lightwave) through the back of IM4 towards the PRM. The NPRO was phased locked to the main PSL carrier by observing the beat between these two lasers from the path IO_Forward using a New Focus 1811.
Phase Locking
The error signal was obtained by feeding the signal from the 1811 into the RF port of a double-balanced mixer. The LO of the mixer was driven with the source port of a network analyzer (HP4395A). The reulting IF was low-passed at 1.9 MHz and then fed into the input of a Newport LB1005 servo box. The settings on the box were 3kHz P-I corner, 50 dB low-frequency gain limit, and 4-0 on the gain knob. The output of this box was sent to the fast PZT input of the NPRO. The output was also attenuated by 4x10^-4, summed in with a DC trimpot voltage, and then and sent to the NPRO's slow temperature input. Once fast lock was acquired, the gain was increased until a small oscillation was observed, and then the gain was backed off. Then the LB1005 was switched from "LFGL" to "lock on", and the slow loop was switched on.
FSR Measurements
We measured the transfer function which takes the network analyzer's LO drive to REFLAIR_B_RF. The magnitude and phase of the this TF gives the complex reflectance function of the PRC. We measured over a series bands including 32.4 – 32.6 MHz, 68.8 – 69 MHz and 102.6 – 102.8 MHz. These bands were analysed when the auxiliary laser was locked above and below the PSL carrier.
The settings on the network analyzer were: sweep time of 500 s, 801 samples, IF bandwidth of 1kHz, and a span of 200 kHz.
Because of the sensitivity of the PLL servo, for each measurement we started the sweep on the network analyzer and then brought the auxiliary/PSL beat note into lock. Throughout these measurements, PRMI was sideband locked.
Results
We obtained the frequency response plots in the following order +102.7 MHz, +32.5 MHz, -32.5 MHz, -102.7 MHz (twice), -68.9 MHz, and +68.8 MHz.
Plots of the frequency response are attached. These functions were fitted to a simple cavity model to determine the exact frequency of the resonances. The results were as follows:
FSR number Resonance frequency
-39.5 102.701020 MHz +/- 180 Hz
-39.5 102.700690 MHz +/- 190 Hz
-26.5 68.900360 MHz +/- 150 Hz
-12.5 32.499990 MHz +/- 90 Hz
12.5 32.501860 MHz +/- 90 Hz
26.5 68.904100 MHz +/- 170 Hz
39.5 102.705400 MHz +/- 190 Hz
Already from these numbers we can see that the PSL carrier does not appear to be perfectly antiresonant; there is an offset of about 1400 Hz.
We then plotted these with FSR number on the horizontal axis and resonance frequency on the vertical axis. The residuals do not show a random behaviour; there appears to be additional structure not captured in this model.
At the present time we can say that the FSR of the PRC is 2.600075 MHz +/- 26 Hz. This corresponds to a PRC length of 57.6507 m +/- 0.6 mm. This measurement is limited by our residuals, and we are currently investigating this.
Non-image files attached to this report
Comments related to this report
Assuming the frequency calibration of the network analyzer is accurate, we can compare the measured PRC length with the measured mode cleaner length. This was measured in alog 9679.
| Parameter | Value | Unit |
|---|---|---|
| FSRPRC | 2.600075 | MHz |
| LPRC | 57.6508 | m |
| FSRMC | 9.099173 | MHz |
| LMC | 16.473612 | m |
| FSRMC / 3.5 - FSRPRC | -306 | Hz |
| (1 - FSRMC / 3.5 FSRPRC) LPRC | 6.8 | mm |
Compared with the modeclaner, the power recycling cavity is about 7 mm too short. The other way around, the modecleaner is about 2 mm too long.
We hooked up the network analyzer to the timing comparator/frequency counter and set it to 40 MHz sharp at 0 dBm. The readback value was dead on, occasionally we would read 1 Hz higher. Conclusion: the frequency of the sweep is no more than 1 Hz off, even at 100 MHz.
I've redone the fits using both the magnitude and the phase. The fitting function is now the usual Fabry–Pérot reflectance function, with a complex magnitude to allow for global amplitude rescaling and global phase offset.
| Nominal FSR | Frequency (Hz) |
| −39.5 | −102 701 040 ± 200 |
| −39.5 | −102 700 900 ± 220 |
| −26.5 | −68 900 600 ± 180 |
| −12.5 | −32 500 080 ± 100 |
| 12.5 | 32 501 720 ± 110 |
| 26.5 | 68 903 940 ± 200 |
| 39.5 | 102 704 700 ± 200 |
The linear fit now gives an FSR of (2 600 073 ± 9) Hz. This is consistent with the previous fit, and anyway the total error is still dominated by some systematic, as seen by the fact that the residuals are excessively large.
Taking a systematic 400 Hz uncertainty on the residual for the 12.5 FSR measurement gives a systematic uncertainty of 32 Hz on the PRC FSR. Propagating foward gives (57.6508 ± 0.0007) m.
Non-image files attached to this comment