6.3 Sensing
6.3.5 Discriminants
The derivative of a signal with respect to optic motion is called a discriminant. By writing down the derivatives of all the signal outputs with respect to motion of all the optics, one can evaluate the matrix of discriminants, also called a sensing matrix (see [83, 43, 84] for detailed discussions and
derivations). Each element of the sensing matrix is the optical gain of a given signalSi in units of watts/meter with respect to the corresponding motion of a degree of freedomLj. The matrix thus connects optic motion to signal output,
S~ =M~L. (6.1)
In general, despite the best efforts of interferometer designers, the matrix can be well populated by off-diagonal terms. Moreover, all the elements are non-linear functions of the microscopic positions of every optic in the interferometer (i.e., the location in configuration space), and they also will vary with audio frequency ωa. Writing down a useful expression for a sensing matrix is thus a difficult task; so, to actually calculate a sensing matrix, we turn to the interferometer simulation tool Optickle (cf. appendixF). An example sensing matrix,Mex, is shown in table 6.3; this matrix gives the DC signal gains, with the interferometer at its operating point for DC readout (there is a 50 pm DARM offset). This particular matrix is calculated in the canonical basis (i.e., MICH, PRCL, etc.), but it could also of course be written in the optic basis (BS, PRM, etc.); these bases are related in a straightforward manner by the output matrix (more in section6.4.1). We can see by inspecting table6.3that there are significant off-diagonal terms, and this is after the sensing scheme described here was designed with an effort to make this matrix as diagonal as possible.
6.3.5.1 Frequency dependence
The sensing matrix can vary with audio frequency ωa. To illustrate the variation with frequency, figure6.4 shows the first row of Mex (magnitude only) as a function of frequency. We choose this row because, as the signal which is chosen to sense DARM (and hence GW) motion, it is a priori interesting—we need to know how motion of all the degrees of freedom might show up in the signal we are going to use as the GW channel. Inspection reveals that the curves in figure 6.4 are not trending at low frequency towards the values shown in the DC matrix (table6.3). This is the result of radiation pressure modifying the interferometer dynamics, an effect included in figure 6.4 but not table6.3. This is indicated in they-axis units of figure6.4, which are watts/meter*, where the asterisk is used to indicate such modified dynamics (which is by radiation pressure in this case, but could also be by the control system). We choose this curious unit because it allows us to focus on what is interesting to us, which is ultimately the response of the interferometer signal outputs to the various disturbances which displace the optics. Thus, the unit meter* does not indicate actual optic motion—it indicates what the optic motion would have been in the absence of modification. This allows us to simply input disturbances, and see the actual resulting signal, naturally including the effects of radiation pressure (or the control system, or both).
PRCL SRCL MICH CARM DARM OMC DC -6.5e+03 -3.7e+03 3.9e+04 -2.4e+06 3.1e+07 AS Q2 5.5e+04 -1.2e+05 -1.8e+05 7.8e+07 -1.4e+08 REFL I1 -2.6e+07 -6.9e+05 5e+05 -2.9e+09 1.5e+08 REFL Q1 3.1e+06 1.5e+05 -2.2e+06 2.1e+08 -1.2e+07 REFL I2 3.3e+06 -5.8e+06 -2.7e+05 3.7e+09 -1.2e+08 REFL Q2 -3.9e+05 3.6e+04 -3.4e+05 2.7e+08 -4.2e+07 REFL IM 1e+07 2e+06 -8.4e+05 8.8e+05 -1e+05 REFL QM 2.1e+06 2.2e+06 4.1e+03 -8.3e+04 -3e+04 REFL IP -9.6e+06 -1.8e+06 -3.6e+05 -1.5e+04 2.7e+03 REFL QP -2.5e+06 -2.3e+06 1.1e+04 -1.3e+04 3.7e+03 AS IM 2.9e+05 -5.3e+05 5.5e+04 -5.1e+04 7.5e+04 AS QM -4.1e+05 -1.2e+05 4.8e+04 1.4e+05 2.7e+04 AS IP 8.6e+04 -5.5e+05 -6.4e+04 -9.8e+02 6.9e+02 AS QP -5.6e+05 2.5e+05 -4.8e+03 -2.2e+02 -5e+02 POB IM 6.9e+03 -2.2e+04 2.9e+03 8e+03 2.1e+02 POB QM -1.9e+04 -1.8e+03 -1.6e+03 3.1e+03 -1.3e+02 POB IP -1.3e+04 2.6e+04 2.6e+03 43 -39
POB QP 2e+04 1.2e+03 -9.9e+02 -14 -2
TRX DC -1.6e+05 3.7e+04 -6.6e+04 -6.3e+07 -5.3e+07 TRY DC -1.7e+05 -2.8e+04 3.1e+04 -6.7e+07 2.5e+07
Table 6.3: Example DC matrix of discriminants, computed for the 40 m. The signals are indicated by port, demodulation phase, and demodulation frequency. The P and M indicate f2+f1 and f2-f1, respectively.
The units are Watts/meter. The largest magnitude element in each row is bolded.
101 102 103 104
101 102 103 104 105 106 107 108 109
Frequency [Hz]
Discriminant [Watts/meter*]
Discriminants for OMC Transmission
101 102 103 104
101 102 103 104 105 106 107 108 109
Frequency [Hz]
Discriminant [Watts/meter*]
Discriminants for OMC Transmission
DARM CARM SRCL MICH PRCL
Figure 6.4: Frequency dependent magnitude of the OMC DC row of the matrix of discriminants. The units are Watts/meter*, where the asterisk indicates the optic motion has been modified by radiation pressure.
This plot shows the open-loop response (no control system).
6.3.5.2 Position dependence
The sensing matrix (including, of course, the frequency response) can also vary with microscopic optic position. In general, each element of the sensing matrix is a non-linear function of the position of all the interferometer optics. This is illustrated in figure 6.5, which plots the DC value of the signals inMex as the signal recycling mirror is swept through 0.2 microns, centered on the operating point. The corresponding table column SRCL is the derivative of these signals at SRCL = 0. Similar plots can be made for all the columns ofMex.
−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1
−80
−60
−40
−20 0 20 40 60 80
SRCL [λ]
Signal [mW]
−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1
−80
−60
−40
−20 0 20 40 60 80
SRCL [λ]
Signal [mW]
OMC DC AS Q2 REFL I1 REFL Q1 REFL I2 REFL Q2 REFL IM REFL QM REFL IP REFL QP AS IM AS QM AS IP AS QP POB IM POB QM POB IP POB QP TRX DC TRY DC
Figure 6.5: Output of signals as SRCL is varied.
6.3.5.3 Example matrix at operating point
The signals used at the operating point are a subset of the full sensing matrix; these are listed in table 6.4. This table shows the philosophy of the signal extraction scheme: the arm cavity signals (CARM and DARM) are sensed using single-demodulation signals (or the OMC transmitted light for DARM, when in DC readout), and the short degrees of freedom (MICH, PRCL, SRCL) are sensed using double demodulation signals. For each of the short degrees of freedom, all of the available DDM signals are used. This is done by measuring and inverting a subset of the sensing matrix, a process described in section8.5.
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.