DOF Port Demod
SRCL PO, REFL, AS DD
PRCL PO, REFL, AS DD
MICH PO, REFL, AS DD
CARM REFL f2 I
DARM AS f2Q
OMC DC
Table 6.4: Signal port and frequency selection for the 40 m length sensing and control scheme. The DD indicates a double-demodulation (DDM) signal; the PRCL, MICH, SRCL degrees of freedom are sensed through a combination of all the available DDM signals.
6.3.5.4 SPOB
There is one additional signal which has not been discussed, but is extremely useful—the sideband product in the power recycling cavity. This signal is extracted from the PO port, and is the result of a demodulation at 2×f1. It is directly proportional to the RF sideband power, and so provides a convenient measure of the RF sideband power buildup. It is used in lock acquisition (more in chapter8).
6.3.5.5 Non-resonant sideband
One significant drawback of the 40 m scheme in general is the lack of a non-resonant sideband. Such a sideband, which would be nearly totally reflected from the power recycling mirror, would provide a stable phase reference for sensing common mode signals with no sideband imbalance. It can also provide a better shot noise limited SNR for common mode signals for interferometers with high PRM reflectivity, since no sideband light is ‘lost’ by being coupled into the interferometer. It could also be used in a double-demodulation scheme for a detuned interferometer, which would benefit from a stable phase reference. In short, this is a good idea that simplifies many aspects of length sensing and control; the only drawbacks are the (slight) reduction in carrier laser power and the additional complication of getting it through the input mode cleaner. All interferometer LSC designs should include a non-resonant sideband.
input multiple output (MIMO) optical plant before applying servo filters. In the diagram, the optical plant is represented by a picture of an interferometer, with 8 inputs (the core optics length actuation signals, plus the a mode cleaner length actuator signal which represents control of the laser frequency—see section5.7 for details about the common mode servo) and 31 outputs (the analog output signals which are either direct photodiode powers or outputs from I&Q demodulators). These signals are then digitally acquired and fed to the LSC computer. In the LSC computer, they undergo a basis transformation in the input matrix (ideally, this is the inverse of the sensing matrixM; more in the next section), are digitally filtered (the feedback filter in the feedback control loops) in the canonical degrees of freedom, and the resulting control signals are transformed to the optic basis before being sent to the digital suspension controllers, where they are converted to analog signals and fed to the optic coil drivers.
6.4.1 Matrices and Bases
Figure 6.6 shows 8 degrees of freedom (DARM, MICH, PRCL, CARM, SRCL, XARM, YARM, MC) available as choices for feedback filtering; this is to allow flexibility during lock acquisition (which is covered in detail in chapter 8). The interferometer, of course only has 5 length degrees of freedom—in normal operation we do not use XARM, YARM, or CARM (MC is used as a laser frequency feedback, cf. section5.7). The physical meaning of these degrees of freedom is determined by the output matrix, which converts the control basis (MICH, PRCL, etc.) to the optic basis (BS, PRM, ETMX, etc.). We have purposely named the control basis to follow the canonical basis (cf.
figure3.10), because ideally they would be the same. In practice, they are not the same, for a few reasons. This can lead to confusion; in this thesis, references to the 40 m will always refer to the actual control basis unless otherwise specified. The one typically used is in table 6.5, where the actual values used which differ from the canonical basis are in parentheses. The only meaningful choice is how to control the MICH. We could actuate on the arm cavity mirrors (both ITMs and both ETMs), but we choose instead to actuate on the beamsplitter, and also feed the signal to the recycling mirrors to keep the recycling cavity lengths constant.
ETMX ETMY BS PRM SRM MC2
SRCL 1
PRCL 1
MICH 1 √12 (.588) √12 (0.18)
DARM 1 -1
CARM 1 1
MC 1
Table 6.5: The typical 40 m LSC output matrix. The actual values used in the control system are noted in parentheses when they differ significantly from the canonical basis.
laser
Optical Plant
Coil Drivers Input
Matrix 22 x 8 + additional calculated
signals
DOF Filtering
DARM MICH PRCL CARM SRCL XARM YARM MC
Output Matrix 8 x 8
Digital Suspension
Control Computers
I & Q Demodulators
DC photodiode
signals RF photodiode
signals
signal conditioning (AI/dewhite) Digital to
Analog Convertors Analog to
Digital Convertors
signal conditioning
(AA/white)
Length Sensing and Control Computer
x8 signals x7 signals
x24 signals
x9
Figure 6.6: A simplified block diagram of the length sensing and control system. The optical plant is depicted here as an 8-input, 31-output analog system; this plant is a highly cross-coupled MIMO (multiple input multiple output) system. The length sensing and control subsystem block diagram shows the basic strategy, which is to combine the acquired signals in an input matrix, apply filtering in a certain basis, and then use the output matrix to convert the output/control signals to the actuator basis.
The first cause of the difference between the control and canonical bases is essentially a boot- strapping problem. Ensuring that the control basis is exactly the canonical basis requires precise calibration of the mirror actuators; this is not difficult for the ETMs (it is part of the calibration procedure in chapter7). It is problematic, however, for the PRM and SRM. These optics can only be interferometrically sensed as part of a coupled system (i.e., they cannot be a part of a single DOF subset of the interferometer). In order to get a precise calibration, then, that coupled system would need a perfectly diagonal sensing matrix. Unfortunately, the sensing matrix at the 40 m is not really diagonal. This problem could probably be circumvented with additional hardware to determine accurate calibrations, but this was not deemed to be worth the effort. A reasonable technique is just to assume the BS, PRM, and SRM have the same calibration, and use simple geometry (the
knowledge that the BS is at a 45◦ angle to the PRM and SRM) to determine the output matrix.
This then only requires the determination of the relative signs of the mirror actuators (which can be different depending on how the magnets were attached).
The second reason the control basis is different from the canonical basis has to do with lock acquisition (more in chapter 8). The particular lock acquisition protocol developed and regularly used at the 40 m worked better with the control basis in table6.5, which is quite different from the canonical basis. This could be due to a radically different actuator strength in the SRM; it is not currently known if that is the case. In any case, attempts to write the canonical values into the output matrix proved unsuccessful: lock acquisition was slower, and the locks achieved were not stable.
6.4.1.1 Input Matrix
The input matrix connects the photodiode signals to the feedback filters for the degrees of freedom.
In an ideal system, the input matrix would be the inverse of the sensing matrix M; then the error signals for the control loops would only be sensitive to motion of their own degree of freedom. In principle, we should exactly measure the full sensing matrix M at every frequency, invert it, and use that as the input matrix. In practice, we do not do this for a few reasons. First, it is difficult;
at best, we generally measure and invert it at one frequency. Even then, we do not measure and invert the whole thing. At the 40 m, this measurement and inversion is only done for the MICH, PRCL, and SRCL degrees of freedom. The measurement and inversion of the input matrix (the part which connects the DDM signals to these DOFs) is done during the lock acquisition bootstrapping, described in section8.5. The actual reason for doing this is purely to reduce cross-talk between these control loops; this complicates the analysis of control loop gain and stability, because the other loops become part of the plant for each loop (see section E.4.1.1). The effect of imperfect input matrix diagonalization is residual cross talk between the control loops.
The second reason we do not invert the complete matrix is sensing noise. It is simpler to use the signals with the most gain, and simply exclude signals with much lower SNR rather than add them to the mix.
6.4.2 Feedforward Corrections
In addition to the servo loops which control the canonical degrees of freedom, it is also sometimes necessary to have correction paths that feed control signals from the auxiliary degrees of freedom directly to DARM. This can be seen by inspecting figure6.4, where we can see that all the canonical degrees of freedom of the interferometer have some optical coupling to the output mode cleaner transmission, which is the DARM sensing signal. A similar statement can be made for RF readout.
This optical coupling represents a problem, since noise in these auxiliary degrees of freedom can pollute the DARM signal. The control system, of course is supposed to suppressdisplacement noise for these auxiliary DOFs, but it will not suppress sensing noise: it will instead impose the sensing noise on the auxiliary degree of freedom, and this will pollute the DARM signal via the optical coupling (M).
The solution, when the auxiliary degrees of freedom are sensing noise limited (usually the most important sensing noise in this case is shot noise, but we will see in chapter 9 that light source noise can also be a problem), is to simultaneously send the feedback signal to DARM, appropriately filtered and scaled. This is called feedforward loop correction, and is done in LIGO for MICH and PRCL. It has also been done at the 40 m for MICH in a power-recycled FPMI state, where it is especially simple, as the coupling factor is a constantπ/2F.
Figure6.4shows that in a detuned interferometer, the feedforward paths will need some filtering in order to be effective, especially for the SRCL and PRCL loops. These were never implemented because these length couplings never limited the noise of the interferometer, as other noise sources were dominant (specifically, digital to analog conversion noise for the test mass actuators).
Although sophisticated frequency-dependent feedforward loop corrections have not been tested for a detuned interferometer at the 40 m, they will prove crucial in Advanced LIGO, not just to reduce the effects shot of noise in the auxiliary loops, but as the measurements in chapter 9 will show, also for laser and oscillator noise.
The 40 m digital control system is currently being significantly upgraded (cf. section 5.10); the implementation of feedforward techniques should be a high priority. One technique that will be tested is the use of Wiener analysis (see [85] for a description) to determine the optimal filters for these correction paths.