3.4 Matrix of Discriminants
3.4.1 Optimized Broadband
A set of optics for the optimized broadband interferometer was derived in Section 2.3.2 from the bench. m program. Best overall sensitivity to neutron-neutron binary coa- lescence was found with an interferometer whose ITM transmittances are 0.5% and a signal mirror transmittance of 5%. The power mirror is chosen to couple 99% of the input carrier power into the interferometer, erring on the over-coupled side. This sets the power mirror transmittance at roughly 8.5%. This number can vary somewhat depending on the actual losses in the interferometer. The detuning phase to optimize the frequency response is 0.09 rad.
RF frequencies f
=
27 MHz and 3f=
81 MHz are used (this matches with the tabletop experiment in Chapter 4) to satisfy theh =
3!1 signal extraction scheme, as well as to keep the RF frequencies below 100 MHz. A free spectral range of 18 MHz will resonate the PM sidebands and the single RF sideband, as well as setting the power cavity on anti-resonance for the carrier. The length corresponding to this is 8.328 m. The signal cavity, at broadband, is chosen to be consistent with a free spectral range of 81/5=
16.2 MHz, corresponding to 9.253 m. With a detuning of 0.09 radians, the length shift necessary is 2.65 em, setting the signal cavity length to 9.279 m.The asymmetry which optimally couples the 81 MHz sideband is, from Eq. (3.34), 1.83 em. Given this, the matrix of discriminants is calculated and presented in Ta- ble 3.1. First, it can be noted that the comparison between the analytical formulas and the results of Twiddle is quite good. Disparities can be attributed to assumptions about the sensitivity of the RF sidebands to the arm cavities, as in the <I>_ signal in the reflected and pickoff 81 MHz ports, as well as the <I>+ signal in the 54 MHz ports.
The discrepancy of the <Ps signals in the reflected and pickoff 81 ports can be at- tributed to the assumption of the lack of sensitivity of the non-resonant RF sideband in the <Ps cavity.
The signals for the arm cavity degrees of freedom are without competition. The presence of other signals will contaminate the performance of these degrees of freedom,
II
<~>+ <I>_Refl 81 2400 0 -1.7 -0.51 0.37 2400 -0.0006 -1.7 -0.51 0.34
Pick 81 96000 0 54 -71 51
96000 -0.09 54 -71 47
Dark 81 0 -77 0 -0.097 0
0 -77 0 -0.097 0
Refl 54 0 0 -0.25 -0.014 -0.35 0 0 -0.25 -0.015 -0.35
Pick 54 0 0 -6.3 -0.057 9.6
0 0 -6.4 -0.055 9.7
Table 3.1: Matrix of discriminants for an optimized broadband interferometer. Num- bers in bold are predictions from Twiddle, while normal text numbers are generated from the analytical formulas of this Chapter.
but it's noted that the coupling is typically small; less than part per thousand.
The
<P-
signal is not the dominant signal in any port. However, in the pickoff81 port, only the <I>+ matrix element is larger. Analysis of an ill-conditioned 2 x 2 submatrix similar to the one defined by the <I>+/¢- matrix elements in the reflected and pickoff 81 signal ports was a large part of the thesis of Regehr. [32] He showed that such a plant can be stable, and how to set specifications for loop gains such that system performance degradation due to the cross coupling is minimized. Some of his results are repeated in Appendix C, and will be returned to later.
The SSB output signals, which are used for ¢+ and ¢5 , are also somewhat ill- conditioned. It might be expected that this would occur due to the fact that, analyzed independently, the signal cavity has a higher finesse than the power cavity. This implies that the phase gain is larger for ¢5 , and so it would dominate the ¢+ signal.
This matrix has been derived assuming an asymmetry which optimally couples the resonant PM RF sideband to the dark port. It was noted earlier that this probably isn't the best choice, due to the sensitive dependence of the state of that RF sideband with respect to variation in losses. In response, the asymmetry should be changed to reduce the transmittance to some non-maximum, but still acceptable level. The dependence of the matrix terms on the asymmetry will be examined.
The signal cavity is fairly non-resonant for the upper PM RF sideband and the SSB. Variation of the asymmetry, which changes the reflectivity of the front mirror of this cavity, doesn't affect these sidebands much. Increasing the asymmetry does decrease the transmittance of the coupled cavity for the lower PM RF sideband and lowers the effective finesse of both the power and signal cavities. This also increases the amount of reflected light, while decreasing the light at the pickoff. The internal phase gains for the lower PM RF sideband with respect to the power and signal cav- ities then stay relatively the same, because both the derivative and the transmission function decrease. The external phase gains, however, decrease because the deriva- tives decrease, while the transmission function increases. The asymmetry can be in- creased enough such that, in reflection, N'_,</J. decreases below N'_,<P+ - N~,<P+ ~ N~,<P+
(see Eq. (3.80) and Eq. (3.72)). Thus, the ¢+ signal can be made dominant in the reflected SSB output port, diagonalizing the control of¢+ and ¢8• Table 3.2 shows this trend.
- I I -I s ' ' '
1.8 640
+
96 1000 24- 48 41 98%3.0 20
+
95 77 15- 48 58 82%6.0 1
+
96 16 5- 48 88 33%Table 3.2: Variation of phase gains with asymmetry. Overall signs haven't been preserved. The final column is the transmittance of the lower PM RF sideband to the dark port.
Increasing the asymmetry increases the diagonalization of the plant for ¢+ and
¢s, but it also decreases the RF sideband power at the dark port. The question arises: how much power is needed at the dark port? For the proposed schemes, the power recycling factor is roughly 15. It's anticipated that advanced LIGO II will have about 120 Watts of carrier power incident on the power mirror, which implies (conservatively) about 2000 Watts incident on the beamsplitter. Experiments in Garching have shown that contrast defects on the order of 10-4 can be achieved. [30]
This estimates that about 0.2 Watts of "junk" light will be incident on the photodiode.
It would be desired that the power due to the RF sidebands be at least a factor of
10 more than this. With a transmittance of 100%, this implies a modulation depth of at least 0.27. At the 30% transmittance with a 6 em asymmetry, the necessary modulation depth increases to about 0.5. The decrease in carrier amplitude, and hence, gravitational wave sensitivity, is about 4%, with a corresponding decrease in the probed volume of space of about 11%. This is just on the edge of being acceptable.
There are other factors which come into play, however. One is the fact that LIGO II proposes to use an output mode cleaner, which can improve the contrast by a factor of approximately 100. [52] If this is the case, then the 6 em asymmetry with a smaller modulation depth will certainly be acceptable.
The matrix with the 6 em asymmetry is shown in Table 3.3. The optical and
II
<P+Ref!. 81 170 0 0.0030 -0.26 0.017
cp_
170 -0.0003 0.0033 -0.26 0.017Pick 81 37000 0 -16 -37 2.4
<P+ 37000 -0.05 -16 -37 2.4
Dark 81 0 71 0 0.089 0
<P_ 0 71 0 0.089 0
Ref!. 54 0 0 0.83 0.017 0.11
¢+ 0 0 0.84 0.015 0.12
Pick 54 0 0 -1.7 -0.040 -3.1
¢s 0 0 -1.7 -0.034 3.2
Table 3.3: Matrix of discriminants for an optimized broadband interferometer, with the optimized asymmetry of 6 em.
physical parameters used to generate the matrix of Table 3.3 are given in Table 3.4.
The specifications for the acceptable RMS of residual fluctuations in ¢+ and <P + are given by the deviation which degrades the carrier power stored by 1%. The re- quirements on ¢s aren't as clear. Two things are affected by a fluctuation in ¢s : a decrease in RF sideband power at the dark port, and a change in the transfer func- tion. Fluctuations in RF sideband power don't affect the sensitivity very much, since both the shot noise and signal scale as the RF sideband amplitude. A requirement of
< 1% is probably too strict. The changes in the transfer function will place a stricter
requirement on the control of ¢5 • It's taken as a specification that the fluctuation of
Power recycling mirror Tprc
=
8.3% Arm cavity power 780kW Signal mirror Tsem=
5.0% PM RF sideband frequency 81 MHzITM ~tm
=
0.5% PM modulation depth 0.2 radAverage coating loss 37.5 ppm SSB sideband frequency 27MHz ITM substrate loss 480 ppm SSB input power 1.25
w
Power recycling gain 17 Power recycling length 8.328 m Detuning phase 0.09 rad Signal cavity length 9.279 m
Input power 125
w
Asymmetry 6.0 emTable 3.4: Optical and physical parameters for optimized broadband interferometer.
Average coating loss includes ETM transmittance.
the transfer function at any frequency point be less than 1% in the bandwidth of in- terest. For the optimized broadband interferometer, the common mode requirements are given in the following table.
I
Degree of freedomII
Residual length Residual phaseq,+ 5£+ ~ 8 X 10 ·H m &<P+ ~ 5 x 10 -orad
¢+ &l+ ~ 1.3 X 10-!J m &¢+ ~ 0.008 rad
c/Js &ls ~ 8.5 X 10 ·l l m &¢s ~ 0.0005 rad
Table 3.5: Common mode residual requirements for the optimized broadband inter- ferometer.
First, it's noted that the ¢+/¢s submatrix has been adequately diagonalized. The non-diagonal elements of the matrix do couple noise from one control loop into the other. This is examined in Appendix C. The degradation of the performance in each loop is roughly the scaling factor between the two elements in a row - roughly a factor of 1/7 worse in the¢+ loop and a factor 1/2 worse in the c/Js loop. The required performance of these two loops, however, is not terribly strict. In fact, the strictest requirement, for c/Js, is only a little smaller than the¢+ requirement for LIGO !.[53]
Not only has the ¢+/ c/Js submatrix been fairly well diagonalized, but the state of the
c/J-
degree of freedom has improved as well, although it is still ill-conditioned.The performance of such a coupled system is not significantly degraded with the establishment of a "gain hierarchy," discussed in [32] and Appendix C, in which the gain of the <P+ loop is larger than the
q; _
loop by the coupling factor. The couplingfactor to <P+, which was r v 5000 in the reflected 81 port, is now down to about 600.
The loop gain in <P + is typically very large, and it's anticipated that a factor of 1000 between these two loop gains is acceptable, since a similar ratio is achieved in ill-conditioned <P +I¢+ sub matrix in the the LIGO I design. [54]