TAMA Modecleaner alignment error signals
Version 1.3
G. Heinzel, K. Arai, NAO Mitaka
January 11, 2001

1

Introduction

The effects of mirror misorientations1 in interferometers with three or more mirrors
cannot easily be determined analytically in the general case. Therefore a Mathematica program for numerical 3-D ray tracing was written. It uses geometrical optics tofind the axis of the eigenmode for a given combination of reflecting and/or refracting
plane and spherical surfaces. A similar program is described and printed in Appendix
E3 of MPQ243. The term ‘beam’ in this text refers to the geometrical axis of a beam,
without taking into account the transverse shape or optical phase.

2

The modecleaner cavities

Figure 1 shows a schematic view of the TAMA modecleaner cavity seen from above,
together with the coordinate system adopted in this section.
The cavity consists of two flat mirrors (Ma and Mb ) that are separated by a relatively
short distance (20 cm), and a curved mirror Mc with radius of curvature R = 15 m,
which is at a distance L = 9.738 m from the (center between the) flat mirrors. The
beam enters through Ma and travels clockwise to Mb , Mc , Ma , etc.
Me is the location of the photodetector behind the end mirror (in the following
called ‘end detector’ for short), which is at a distance of 2.08 m behind mirror Mc .
There are four beams of interest leaving the cavity. The main output beam is ‘Outb ’which goes to the interferometer and the stabilization of which is the main purpose of
the modecleaner. There are two beams coming from Ma : the directly reflected input
beam and the beam ‘Outa ’ which is a fraction of the cavity eigenmode. The cavity iswell aligned to the incoming beam (which we consider fixed), if these two beams are
perfectly superimposed. By taking two quadrant diodes with two different lens systems
and appropriately demodulating their outputs, we can obtain four independent error
signals, similar to the case of a simple two-mirror Fabry-Perot cavity. The longitudinallocking signal is also obtained from these two interfering beams (using the PoundDrever-Hall scheme).
1
To avoid confusion, we call a mirror or other component misoriented in this section, if its angularorientation differs from its reference orientation. The resulting movement of beams (e.g. cavity eigenmodes) will be called misalignment. An interferometer is called well-aligned if there are neither
misorientations nor misalignments, i.e. components as well as beams are in their reference positions.

1

In
Ma

Mc

Me

Outa

Outc
x

Mb
Outb
z

y
Figure 1: Schematic diagram of the TAMA modecleaner cavity seen from above.

In the three-mirror cavity, there are two additional degrees of freedom. They arelinear combinations of movements of all three mirrors, as computed below. We willcall them “neutral modes” because they have no influence on either the interference of
the incoming beam with the cavity eigenmode nor on the outgoing beam. They can beused to control the spot position on the far mirror Mc . This is described in more detailbelow.
In the ray-tracing program, we first compute the well-aligned case (i.e., all mirrors
are hit in their center) as reference. We call Pa , Pb and Pc the points where the axis of
the eigenmode intersects the mirrors Ma , Mb and Mc , respectively. After misorienting
one particular mirror by the small angle ε, we recompute the eigenmode axis, compare
it with the well-aligned case and divide the difference by ε. The main results are:
• The shifts of the spots Pa , Pb and Pc .
• The angles γa , γb and γc between the beams ‘Outa ’, ‘Outb ’, ‘Outc ’ and theirrespective references. For the vertical misalignments which are considered separately, we call these angles δa , δb and δc , respectively. We also compute the angle
γd (δd ) between the directly reflected incoming beam and its reference for thecase that Ma is misoriented.
• For ‘Outa ’ in the case of misorienting Ma , we also compute the angle γa0 = γa −γd ,which is the angle between the beam leaving the cavity and the directly reflected
beam, because this is the angle between the interfering wavefronts that is detectedby the quadrant diode.
• As described above we finally compute the angle θ which describes the ‘character’of the misalignment at the waist. It is given by θw = arctan(γa0 zR /∆zwaist ).
The waist of the cavity eigenmode is located halfway between the mirrors Ma and2

Mb . Its Rayleigh range is given by
s


LRT
LRT
zR =
R−
= 7.126 m,
2
2

(1)

where LRT /2 is one half of the round-trip distance:p
LRT /2 = L2 + d2 /4 + d/2 = 9.8385 m.

3(2)

Horizontal misalignments:

By ‘horizontal’ misalignments we mean that a mirror is rotated around the vertical axis,
i.e. beam spots move horizontally (in the plane of the modecleaner cavity). Angles arecounted as positive if a mirror is rotated clockwise, if seen from above (as in Figure 1).
We introduce the common- and differential mode motion of mirrors Ma and Mb
by defining angles α+ and α− . Furthermore we introduce the “neutral” mode αn as
follows:Ma
αa = α+αa = α−αa = αn

α+
α−
αn

Mb
αb = α+αb = −α−αb = αnMc
αc = 0αc = 0αc = 0.701αn(3)αa + αb,2
αa − αb
α− =
,
2

α+ =

(4)
(5)

The results of the raytracing program are given in Table 1.
Cause
αa
αb
αc
α+
α−
αn
Cause
αa
αb
αc
α+
α−
αn

Pa
∆x
−9.740 m · αa
9.538 m · αb0.288 m · αc−0.202 m · α+−19.278 m · α−0.000 m · αnPb∆x
9.538 m · αa−9.740 m · αb0.288 m · αc−0.202 m · α+19.278 m · α−0.000 m · αnPa∆z9.840 m · αa−9.636 m · αb−0.291 m · αc
0.204 m · α+
19.477 m · α−0.000 m · αnPb∆z
9.637 m · αa−9.841 m · αb
0.291 m · αc−0.204 m · α+
19.477 m · α−0.000 m · αn

waist
∆z
9.739 m · αa−9.739 m · αb0.000 m · αc
0.000 m · α+
19.477 m · α−0.000 m · αnOutbγb1.019 αa1.019 αb−2.906 αc2.039 α+0.000 α−0.000 αnOuta
γa0.981 αa−1.019 αb2.906 αc−0.039 α+
2.000 α−2.000 αnPc∆x−0.291 m · αa−0.291 m · αb28.596 m · αc−0.581 m · α+0.000 m · α−
19.478 m · αn

Out0a
0
γa
= γa − γd−1.019 αa−1.019 αb2.906 αc−2.039 α+
0.000 α−0.000 αnOutc
γc−1.019 αa0.981 αb2.906 αc−0.039 α+−2.000 α−
2.000 αnd. refl.
γd2 · αa0 · αb0 · αc2 · α+
2 · α−
2 · αnθw−36.72◦36.72◦−90.00◦90.00◦
0.00◦
—
End
∆x
−2.411 m · αa1.749 m · αb34.642 m · αc−0.662 m · α+−4.160 m · α−23.639 m · αnTable 1: Results of the ray-tracing program for horizontal misalignments of the TAMA modecleaner.

3

4

Vertical misalignments:

In the modecleaners, the horizontal and vertical axes are not similar. The results of theraytracing program for vertical misalignments are given in Table 2. Note, for example,
that a small vertical tilt βa of the input mirror Ma (which is hit under approximately 45◦
from the incoming beam) causes a deflection of the reflected beam by only δd = 1.43βa
as compared to γd = 2αa in the horizontal case. Another example is the tilt of Mc
which, if horizontal, causes a pure angular misalignment at the waist. A vertical tiltof Mc , on the other hand, shifts the cavity eigenmode downwards parallely, withoutchanging any angles. Angles are now counted as positive when the mirror normal
moves downward from the reference direction.
As before we introduce the common- and differential mode motion of mirrors Ma
and Mb by defining angles β+ and β− . Furthermore we introduce the “neutral” mode
βn as follows:

β+
β−
βn

Ma
βa = β+βa = β−βa = βnMb
βb = β+βb = −β−βb = βnMc
βc = 0βc = 0βc = −0.4986βn

βa + βb,2
βa − βb
β− =
,
2
β+ =

(6)

(7)
(8)

The results of the raytracing program are given in Table 2.

5

Degrees of freedom

The most important alignment task is to superimpose the axis of the cavity eigenmodewith the axis of the incoming beam. This requires the control of four degrees of freedom.
For this purpose, in the differential wavefront sensing method, we place two quadrant
detectors with different lens systems in the beam reflected from Ma . The interference
between the directly reflected incoming beam, which is phase modulated at an RF
frequency, and the beam ‘Outa ’ leaking out of the cavity contains enough information
to lock the cavity longitudinally and to obtain alignment error signals for those four
degrees of freedom that determine the superposition of the incoming beam and thecavity eigenmode.
In particular, we now assume all mirrors to be slightly misoriented and computethe combined signals which are obtained by demodulating the outputs of two quadrant
detectors, one (called XI ) with Φ = 0◦ and the other one (called XQ ) with Φ = 90◦ ofextra phase shift. We scale parallel shifts ∆y or ∆z with the appropriate factor zR andobtain for horizontal misalignments:
XI = −1.019 αa − 1.019 αb + 2.906 αc = −1.019 α+ + 2.906 αc ,(9)

XQ = 1.366 αa − 1.366 αb = 1.366 α− ,

(10)

4

Cause
βa
βb
βc
β+
β−
βn
Cause
βa
βb
βc
β+
β−
βn

Pa
∆y
−3.670 m · βa−3.810 m · βb−15.000 m · βc−7.480 m · β+0.141 m · β−0.000 m · βnPb∆y−3.810 m · βa−3.670 m · βb−15.000 m · βc−7.480 m · β+−0.141 m · β−0.000 m · βn

waist
∆y
−3.740 m · βa−3.740 m · βb−15.000 m · βc−7.480 m · β+0.000 m · β−0.000 m · βnOutbδb
0.704 βa−0.704 βb0.000 βc
0.000 β+1.407 β−0.000 βnOuta
δa−0.718 βa−0.704 βb0.000 βc−1.421 β+−0.014 β−−1.421 βnPc∆y−10.661 m · βa−10.661 m · βb−15.000 m · βc−21.323 m · β+0.000 m · β−−13.843 m · βnOut0a0δa = δa − δd0.704 βa−0.704 βb0.000 βc
0.000 β+1.407 β−0.000 βnOutc
δc0.704 βa0.718 βb0.000 βc1.421 β+−0.014 β−1.421 βnθw
−53.27◦53.27◦
0.0◦
0.0◦
90.0◦
—

d. refl.
δd1.42 · βa
0.00 · βb0 · βc−1.421 · β+−1.421 · β−−1.420 · βnEnd∆y−12.125 m · βa−12.155 m · βb−15.000 m · βc−24.280 m · β+0.030 m · β−−16.800 m · βnTable 2: Results of the ray-tracing program for vertical misalignments of the TAMA modecleaner.

and for vertical misalignments:
YI = 0.7035 βa − 0.7035 βb = 0.7035 β− ,(11)

YQ = −0.524 βa − 0.524 βb − 2.105 βc = −0.524 β+ − 2.105 βc .

(12)Using simple linear algebra we also find the “neutral” modes from these results.
They are given in tables (3) and (6) above.
By inspecting tables 1 and 2 one finds that the main effect of these “neutral modes”
is to shift the spot position on the far end mirror Mc . Hence their control is not
absolutely essential for the basic function of the modecleaner. In the present situation,
there is neither feedback to Mc nor is the spot position on Mc monitored. One problem
with this approach might be that according to equations (9) to (12), a small motion ofMc translates into error signals at the reflected light port which is 3. . . 4 times larger
than the error signal for a motion of the front mirrors of comparable amplitued. Thishuge error signal is then fed back the the front mirrors which are forced to compensate
Mc ’s misorientation although they are not very efficient actuators for that purpose.
Furthermore, such feedback shifts the spot position on Mc which due to the curved
mirror surface again causes the whole circle to start. It is possible that some of thepresent dynamic range problems may be relieved by sensing and feeding back all degrees
of freedom.
Because the main effect of the “neutral modes” is to shift the spot position on thefar end mirror Mc , the most straightforward approach is to sense this spot position(with a DC quadrant detector looking at the transmitted light). In tables 1 and 2
above, this shift is computed as ∆x or ∆y of point Pc, respectively.
However, the detector is located at a distance e = 2.08 m behind Mc , and the spot5

position detected is hence a linear combination2 of ∆x or ∆y of point Pc, respectively
with the angle δc .
This shift of the beam spot on the photodetector is also calculated by the programand printed as “End ∆y” in Tables 1 and 2.
If we also scale it by the factor zR for consistency (although the way of detection isdifferent and hence an arbitrary scale factor appears anyway), and call the error signals
thus obtained XE and YE , we get
XE = −0.3383 αa + 0.2455 αb + 4.8613 αc= −0.0464 α+ − 0.2919 α− + 4.8613 αc ,YE = −1.701 βa − 1.705 βb − 2.105 βc= −1.703 β+ + 0.002 β− − 2.105 βc .

(13)
(14)
(15)
(16)

We see that for the horizontal direction, the additional error signal is dominatedby the misorientation of Mc and could directly be fed back to that mirror, while forthe vertical direction the signals are more mixed. However, they are still sufficiently
linearly independent such that by inversion of the well-conditioned matrix error signals
for all three mirrors can be found. This matrix inversion yields
αa =XE −1.6726 XI
αb =XE −1.6726 XI
αc = 0.7015 XE −0.0319 XI+1.4273 XQ−XQ+0.1498 XQ

(17)

βa =
−YE +1.6785 YI
βb =−YE −1.6785 YI
βc = 0.4987 YE −0.0015 YI+YQ+YQ−1.6186 YQ(18)

where scaling factors of 1/0.301462 and 1/0.42418, respectively, have been applied
to make the most common coefficients unity. Note that the present alignment system
uses only a 2 × 2 matrix without XE , YE , αc and βc .

6Actuator transfer functions

The transfer functions from all actuators to all sensors have been measured by K. Arai.
The results were fitted with LISO and are shown on the following pages.
2

Many thanks to K. Arai for noting this point.

6

6.1

Input Mirror Yaw coil (“m1xcoil”)

20

0

10

-45

0

-135
dB

-20
-180
-30
-225
-40

Phase [Degree]

-90

-10

-270

-50

-315

-60
-70
0.1

-360
1

10
Frequency [Hz]aamc/T1220_1.BOD dB
aamc/T1220_1.BOD Phase

100

TF dB
TF Phase

20

Pi Yaw error

T1220_1.BOD

pole0:f = 1.6798911 +- 75.2m (4.48%)
pole0:q = 636.47668m +- 21.6m (3.39%)
zero0:f = 3.1323094 +- 124.6m (3.98%)
zero0:q = 969.43027m +- 64m (6.6%)
pole1:f = 3.7858238 +- 25.9m (0.684%)
pole1:q = 3.3333594 +- 177.3m (5.32%)
factor = 5.0043609 +- 40.31m (0.805%)

0

10

-45

0
-90
-135

dB

-20
-30

-180

-40

-225

Phase [Degree]

-10

-50
-270
-60
-315

-70
-80

-360
1

10
Frequency [Hz]aamc/T1220_2.BOD dB
aamc/T1220_2.BOD Phase

100

TF dB
TF Phase

20

Pi/2 Yaw error

T1220_2.BOD

pole0:f = 1.3598055 +- 43.77m (3.22%)
pole0:q = 830.24256m +- 19.97m (2.41%)
zero0:f = 2.4762582 +- 72.09m (2.91%)
zero0:q = 842.25894m +- 47.99m (5.7%)
pole1:f = 3.7674492 +- 19.53m (0.518%)
pole1:q = 3.711112 +- 128m (3.45%)
factor = 3.2644629 +- 29.84m (0.914%)

0

10

-45

0
-90
-135

dB

-20
-30

-180

-40

-225

-50
-270
-60
-315

-70
-80

-360
1

10

100

Frequency [Hz]aamc/T1220_3.BOD dB
aamc/T1220_3.BOD Phase

TF dB
TF Phase

Weighted Averages:
pole0:f = 1.372388
pole0:q = 0.755353
pole1:f = 3.777486
pole1:q = 3.455169
zero0:f = 2.508392
zero0:q = 0.840509

7

Phase [Degree]-10

End QPD DC Yaw

T1220_3.BOD

pole0:f = 1.1584155 +- 66.98m (5.78%)
pole0:q = 948.86053m +- 53.28m (5.61%)
zero0:f = 2.0163492 +- 120.4m (5.97%)
zero0:q = 642.40736m +- 78.67m (12.2%)
pole1:f = 3.8080309 +- 46.85m (1.23%)
pole1:q = 3.0342956 +- 189.4m (6.24%)
factor = 2.7204019 +- 63.32m (2.33%)

Output Mirror Yaw coil (“m2xcoil”)

20

0

10

-45

0

-90

-10
-135
dB

-20
-180
-30
-225
-40

Phase [Degree]

6.2

-270

-50

-315

-60
-70

-360
1

10
Frequency [Hz]aamc/T1220_4.BOD dB
aamc/T1220_4.BOD Phase

TF dB
TF Phase

20

Pi Yaw error

T1220_4.BOD

pole0:f = 864.55282m +- 31.63m (3.66%)
pole0:q = 801.30084m +- 22.58m (2.82%)
zero0:f = 1.7498418 +- 67.69m (3.87%)
zero0:q = 436.75937m +- 27.63m (6.33%)
pole1:f = 3.7556255 +- 19.05m (0.507%)
pole1:q = 3.6996359 +- 118m (3.19%)
factor = 2.4554303 +- 33.68m (1.37%)

180

10
135

90

dB

-10
-20

45

-30
0

Phase [Degree]

0

-40
-45
-50
-60

-90
1

10
Frequency [Hz]aamc/T1220_5.BOD dB
aamc/T1220_5.BOD Phase

TF dB
TF Phase

20

180

10

135

0

Pi/2 Yaw error

T1220_5.BOD

pole0:f = 1.3974404 +- 26.7m (1.91%)
pole0:q = 792.95797m +- 11.9m (1.5%)
zero0:f = 2.5625299 +- 44.11m (1.72%)
zero0:q = 877.60669m +- 27.81m (3.17%)
pole1:f = 3.630812 +- 9.79m (0.27%)
pole1:q = 3.9492515 +- 79.85m (2.02%)
factor = -3.5256953 +- -21.65m (0.614%)

45
dB

-20
0
-30
-45
-40
-90

-50

-135

-60
-70

-180
1

10
Frequency [Hz]aamc/T1220_6.BOD dB
aamc/T1220_6.BOD Phase

TF dB
TF Phase

Weighted Averages:
pole0:f = 1.204886
pole0:q = 0.804331
pole1:f = 3.661688
pole1:q = 3.842627
zero0:f = 2.316205
zero0:q = 0.669030

8Phase [Degree]

90

-10

End QPD DC Yaw

T1220_6.BOD

pole0:f = 1.5309455 +- 68.42m (4.47%)
pole0:q = 1.0838276 +- 56.89m (5.25%)
zero0:f = 2.2872224 +- 93.83m (4.1%)
zero0:q = 1.0726809 +- 108.7m (10.1%)
pole1:f = 3.7250662 +- 31.92m (0.857%)
pole1:q = 3.6033149 +- 192.1m (5.33%)
factor = -2.1236194 +- -32.95m (1.55%)

6.3

Input Mirror Pitch PZT (“m1ypzt”)
0

180

-10

135

-20
90
45

dB

-40
-50

0

-60

-45

Phase [Degree]

-30

-70

Pi Pitch error

T1220_7.BOD

-90
-80

pole0:f = 8.721352 +- 31.18m (0.357%)
pole0:q = 6.3320232 +- 330.2m (5.22%)
pole1:f = 20.249388 +- 100.9m (0.498%)
pole1:q = 11.065166 +- 1.296 (11.7%)
factor = 62.862325m +- 1.561m (2.48%)

-135

-90
-100

-180
1

10
Frequency [Hz]aamc/T1220_7.BOD dB
aamc/T1220_7.BOD Phase

TF dB
TF Phase

10

180

0

135

-10

-30

45

dB

-40
0
-50
-45

-60
-70

Phase [Degree]

90

-20

Pi/2 Pitch error

T1220_8.BOD

-90

-80

pole0:f = 8.6836151 +- 12.55m (0.145%)
pole0:q = 11.122052 +- 405.8m (3.65%)
pole1:f = 20.153226 +- 42.54m (0.211%)
pole1:q = 20.638458 +- 1.887 (9.15%)
factor = 84.273794m +- 1.359m (1.61%)

-135

-90
-100
10
Frequency [Hz]aamc/T1220_8.BOD dB
aamc/T1220_8.BOD Phase

-180
100

TF dB
TF Phase

20

180

10

135

0
90
45

dB

-20
-30

0

-40

-45

-50

Phase [Degree]-10

End QPD DC Pitch

T1220_9.BOD

-90
-60
-135

-70
-80
1

10

-180
100

Frequency [Hz]aamc/T1220_9.BOD dB
aamc/T1220_9.BOD Phase

TF dB
TF Phase

Weighted Averages:
pole0:f = 8.688076
pole0:q = 9.340454
pole1:f = 20.08673
pole1:q = 16.34634

9

pole0:f = 8.6874272 +- 10.29m (0.118%)
pole0:q = 11.243448 +- 337.4m (3%)
pole1:f = 20.031176 +- 32.46m (0.162%)
pole1:q = 19.667483 +- 1.306 (6.64%)
factor = 590.1168m +- 7.742m (1.31%)

6.4

Output Mirror Pitch PZT (“m2ypzt”)

10

0

0

-45

-10
-90
-135

dB

-30
-40

-180

-50

-225

Phase [Degree]

-20

-60

Pi Pitch error

T1220_10.BOD

-270
-70

pole0:f = 8.9312815 +- 8.936m (0.1%)
pole0:q = 11.488715 +- 297.2m (2.59%)
pole1:f = 20.561491 +- 19.32m (0.094%)
pole1:q = 29.210305 +- 1.681 (5.76%)
factor = -110.87858m +- -1.238m (1.12%)

-315

-80
-90
1

10
Frequency [Hz]aamc/T1220_10.BOD dB
aamc/T1220_10.BOD Phase

-360
100

TF dB
TF Phase

0

180

-10

135

-20
90
45

dB

-40
-50

0

-60

-45

Phase [Degree]

-30

-70

Pi/2 Pitch error

T1220_11.BOD

-90
-80

pole0:f = 8.947484 +- 15.86m (0.177%)
pole0:q = 9.3417483 +- 354.6m (3.8%)
pole1:f = 20.507774 +- 30.04m (0.146%)
pole1:q = 27.878288 +- 2.424 (8.69%)
factor = 37.419176m +- 645.2u (1.72%)

-135

-90
-100
1

10
Frequency [Hz]aamc/T1220_11.BOD dB
aamc/T1220_11.BOD Phase

-180
100

TF dB
TF Phase

20

180

10

135

0
90
45

dB

-20
-30

0

-40

-45

-50

Phase [Degree]-10

End QPD DC Pitch

T1220_12.BOD

-90
-60
-135

-70
-80
1

10

-180
100

Frequency [Hz]aamc/T1220_12.BOD dB
aamc/T1220_12.BOD Phase

TF dB
TF Phase

Weighted Averages:
pole0:f = 8.934858
pole0:q = 11.15256
pole1:f = 20.48184
pole1:q = 28.21300

10

pole0:f = 8.9347062 +- 5.266m (0.0589%)
pole0:q = 11.469713 +- 173.4m (1.51%)
pole1:f = 20.448379 +- 11.75m (0.0575%)
pole1:q = 27.958598 +- 926.9m (3.32%)
factor = 688.79131m +- 4.501m (0.653%)

6.5

Input Mirror Pitch coil (“m1ycoil”)
0

0

-10

-45

-20

-40

-135

dB

-50
-180
-60
-225

-70
-80

Phase [Degree]

-90

-30

-270

-90
-315

-100
-110
10

-360
1000

100
Frequency [Hz]aamc/T1220_13.BOD dB
aamc/T1220_13.BOD Phase

TF dB
TF Phase

0

Pi Pitch error

T1220_13.BOD

pole0:f = 8.7707005 +- 19.38m (0.221%)
pole0:q = 10.585199 +- 541.9m (5.12%)
zero0:f = 14.532192 +- 125.6m (0.864%)
zero0:q = 15.263638 +- 4.742 (31.1%)
pole1:f = 20.291927 +- 33.79m (0.167%)
pole1:q = 27.515739 +- 2.803 (10.2%)
factor = 43.351314m +- 975.7u (2.25%)

0

-10

-45

-20
-90
-135

dB

-40
-50

-180

-60

-225

Phase [Degree]

-30

-70
-270
-80
-315

-90
-100
10

100

-360
1000

Frequency [Hz]aamc/T1220_14.BOD dB
aamc/T1220_14.BOD Phase

TF dB
TF Phase

20

Pi/2 Pitch error

T1220_14.BOD

pole0:f = 8.7581187 +- 23.51m (0.268%)
pole0:q = 10.768086 +- 656.8m (6.1%)
zero0:f = 14.220014 +- 291m (2.05%)
zero0:q = 16.341334 */ 1.856
pole1:f = 20.299641 +- 47.29m (0.233%)
pole1:q = 23.289386 +- 3.274 (14.1%)
factor = 86.457836m +- 3.091m (3.58%)

0

10

-45

0

-20

-135

dB

-30
-180
-40
-225

-50
-60

-270

-70
-315

-80
-90
10

100

-360
1000

Frequency [Hz]aamc/T1220_15.BOD dB
aamc/T1220_15.BOD Phase

TF dB
TF Phase

Weighted Averages:
pole0:f = 8.757994
pole0:q = 11.90231
pole1:f = 20.31437
pole1:q = 27.83178
zero0:f = 14.49367
zero0:q = 12.04289

11

Phase [Degree]

-90

-10

End QPD DC Pitch

T1220_15.BOD

pole0:f = 8.7555137 +- 8.536m (0.0975%)
pole0:q = 12.695277 +- 333.3m (2.63%)
zero0:f = 14.497233 +- 67.74m (0.467%)
zero0:q = 11.732884 +- 1.393 (11.9%)
pole1:f = 20.321918 +- 16.99m (0.0836%)
pole1:q = 28.889025 +- 1.515 (5.24%)
factor = 478.86074m +- 5.221m (1.09%)

6.6

Output Mirror Pitch coil (“m2ycoil”)
10

180

0

135

-10

-30

45

dB

-40
0
-50
-45

-60
-70

Phase [Degree]

90

-20

-90

-80
-135

-90
-100
10

100
Frequency [Hz]aamc/T1220_16.BOD dB
aamc/T1220_16.BOD Phase

-180
1000

TF dB
TF Phase

0

Pi Pitch error

T1220_16.BOD

pole0:f = 8.9343651 +- 27.53m (0.308%)
pole0:q = 14.281851 +- 1.225 (8.58%)
zero0:f = 14.399736 +- 235m (1.63%)
zero0:q = 10.829207 +- 272.2m (2.51%) > MAX
pole1:f = 20.60976 +- 44.38m (0.215%)
pole1:q = 33.383364 +- 5.186 (15.5%)
factor = -94.29963m +- -3.853m (4.09%)

180

-10

135

-20

-40

45

dB

-50
0
-60
-45

-70
-80

Phase [Degree]

90

-30

-90

-90
-135

-100
-110
10

100
Frequency [Hz]aamc/T1220_17.BOD dB
aamc/T1220_17.BOD Phase

-180
1000

TF dB
TF Phase

20

Pi/2 Pitch error

T1220_17.BOD

pole0:f = 8.9970043 +- 32.66m (0.363%)
pole0:q = 15.935303 +- 1.807 (11.3%)
zero0:f = 14.374722 +- 317.3m (2.21%)
zero0:q = 10.838184 +- 241.2m (2.23%) > MAX
pole1:f = 20.577195 +- 61.19m (0.297%)
pole1:q = 33.687723 +- 6.336 (18.8%)
factor = 49.476234m +- 2.697m (5.45%)

0

10

-45

0

-20

-135

dB

-30
-180
-40
-225

-50
-60

-270

-70
-315

-80
-90
10

100

-360
1000

Frequency [Hz]aamc/T1220_18.BOD dB
aamc/T1220_18.BOD Phase

TF dB
TF Phase

Weighted Averages:
pole0:f = 8.949924
pole0:q = 11.66273
pole1:f = 20.58340
pole1:q = 26.94568
zero0:f = 14.29287
zero0:q = 10.79691

12

Phase [Degree]

-90

-10

End QPD DC Pitch

T1220_18.BOD

pole0:f = 8.9471133 +- 10.88m (0.122%)
pole0:q = 11.299087 +- 344.5m (3.05%)
zero0:f = 14.280006 +- 68.37m (0.479%)
zero0:q = 10.79691 +- 1.226 (11.4%)
pole1:f = 20.579641 +- 17.91m (0.087%)
pole1:q = 26.245019 +- 1.311 (4.99%)
factor = 761.14975m +- 10.98m (1.44%)

6.7

End Mirror Yaw Coil (“m3xcoil”)

30

0

20
-45

0

-90

dB

-10
-135
-20
-30

-180

Phase [Degree]

10

-40
-225

Pi Yaw error

-50
-60

-270
1

10
Frequency [Hz]TF dB
TF Phase

20

180

0

135

45
dB

-40
0
-60
-45
-80

Phase [Degree]

90

-20

-90

-100

Pi/2 Coil Yaw error

-135

-120
10
Frequency [Hz]100aamc/T1220_20.BOD dB
aamc/T1220_20.BOD Phase

-180
1000

0

40

-45

20

-90

0

-135

-20

-180

-40

-225

-60

-270

-80

-315

-100
1

pole0:f = 2.3769774 +- 87.34m (3.67%)
pole0:q = 3.1497205 +- 1.04 (33%)
factor = -270.8383m +- -49.8m (18.4%)

TF dB
TF Phase

60

10

100

-360
1000

Frequency [Hz]aamc/T1220_21.BOD dB
aamc/T1220_21.BOD Phase

TF dB
TF Phase

Weighted Averages:
pole0:f = 1.8826161
pole0:q = 1.3989960

13

T1220_20.BOD

Phase [Degree]1dB

pole0:f = 1.8713374 +- 11.13m (0.595%)
pole0:q = 1.4166323 +- 29.77m (2.1%)
factor = 16.61447 +- 181.6m (1.09%)100

aamc/T1220_19.BOD dB
aamc/T1220_19.BOD Phase

T1220_19.BOD

End QPD DC Yaw

T1220_21.BOD

pole0:f = 1.8896638 +- 8.812m (0.466%)
pole0:q = 1.3891876 +- 22.04m (1.59%)
factor = 82.249734 +- 641.4m (0.78%)

End Mirror Pitch PZT (“m3ypzt”)
0

180

-10

135

-20

90

-30

45

-40

0

-50

-45

-60

-90

-70

-135

-80

Phase [Degree]

dB

6.8

10
Frequency [Hz]aamc/T1220_22.BOD dB
aamc/T1220_22.BOD Phase

TF dB
TF Phase

10

T1220_22.BOD

pole0:f = 6.4108286 +- 18.1m (0.282%)
pole0:q = 8.4989131 +- 459.5m (5.41%)
pole1:f = 15.438905 +- 94.27m (0.611%)
pole1:q = 10.134435 +- 1.324 (13.1%)
factor = -85.75237m +- -2.02m (2.36%)-180
1

Pi Pitch error

180

0

135

-10

-30

45

dB

-40
0
-50
-45

-60
-70

Phase [Degree]

90

-20

-80

T1220_23.BOD

pole0:f = 6.4501997 +- 17.03m (0.264%)
pole0:q = 7.2412406 +- 310.6m (4.29%)
pole1:f = 15.548908 +- 63.98m (0.411%)
pole1:q = 12.259952 +- 1.299 (10.6%)
factor = 216.30985m +- 4.018m (1.86%)

-135

-90
-100

-180
1

10
Frequency [Hz]TF dB
TF Phase

40

180

20

135

0

90

-20

45

-40

0

-60

-45

-80

-90

-100

-135

-120

-180
1

10

100

Frequency [Hz]aamc/T1220_24.BOD dB
aamc/T1220_24.BOD Phase

TF dB
TF Phase

Weighted Averages:
pole0:f = 6.436784
pole0:q = 8.052757
pole1:f = 15.49933
pole1:q = 10.58777

14

Phase [Degree]

aamc/T1220_23.BOD dB
aamc/T1220_23.BOD PhasedB

Pi/2 Pitch error-90

End QPD DC Pitch

T1220_24.BOD

pole0:f = 6.441543 +- 11.96m (0.186%)
pole0:q = 8.6539705 +- 308.9m (3.57%)
pole1:f = 15.47822 +- 62.81m (0.406%)
pole1:q = 10.059454 +- 849.4m (8.44%)
factor = 904.141m +- 14m (1.55%)

End Mirror Pitch coil (“m3ycoil”)
0

0

-10

-45
-90

-20

-135
dB

-30
-180
-40
-225
-50

Phase [Degree]

6.9

-270

-60

-315

-70

-360
10
Frequency [Hz]aamc/T1220_25.BOD dB
aamc/T1220_25.BOD Phase

TF dB
TF Phase

20

Pi Pitch error

T1220_25.BOD

pole0:f = 6.4562745 +- 12.24m (0.19%)
pole0:q = 10.880457 +- 519.8m (4.78%)
zero0:f = 13.118087 +- 141.4m (1.08%)
zero0:q = 10.177679 +- 2.379 (23.4%)
pole1:f = 15.621772 +- 66.15m (0.423%)
pole1:q = 16.829252 +- 2.755 (16.4%)
factor = 73.69469m +- 1.956m (2.65%)

180

10

135

0
90
45

dB

-20
-30

0

-40

-45

Phase [Degree]

-10

-50
-90
-60
-135

-70
-80

-180
10

100
Frequency [Hz]aamc/T1220_26.BOD dB
aamc/T1220_26.BOD Phase

TF dB
TF Phase

30

Pi/2 Pitch error

T1220_26.BOD

pole0:f = 6.4394746 +- 4.254m (0.0661%)
pole0:q = 11.47595 +- 191.8m (1.67%)
zero0:f = 13.214751 +- 53.88m (0.408%)
zero0:q = 8.8553113 +- 623m (7.04%)
pole1:f = 15.652186 +- 26.16m (0.167%)
pole1:q = 15.990668 +- 937.6m (5.86%)
factor = -433.8639m +- -3.66m (0.843%)

180

20

135

10
90
45

dB

-10
-20

0

-30

-45

-40
-90
-50
-135

-60
-70

-180
10

100
Frequency [Hz]aamc/T1220_27.BOD dB
aamc/T1220_27.BOD Phase

TF dB
TF Phase

Weighted Averages:
pole0:f = 6.439799
pole0:q = 11.48701
pole1:f = 15.65282
pole1:q = 16.45207
zero0:f = 13.19353
zero0:q = 9.075651

15

Phase [Degree]0

End QPD DC Pitch

T1220_27.BOD

pole0:f = 6.4356973 +- 6.681m (0.104%)
pole0:q = 11.732925 +- 311.2m (2.65%)
zero0:f = 13.167754 +- 85.29m (0.648%)
zero0:q = 9.613809 +- 1.201 (12.5%)
pole1:f = 15.663831 +- 37.03m (0.236%)
pole1:q = 17.752179 +- 1.652 (9.31%)
factor = -1.1676377 +- -14.64m (1.25%)

7

Calibration Factors

Both actuators and sensors have calibration factors which are not yet known. In order
to find them (and test the above theroretical model), the LISO-fitted factors from thetransfer functions were fitted to simple model

factor = Aoptical × Gactuator × Gsensor .

(19)

where all measurements were fitted simultaneously to one set of calibration constants Gi .
For that purpose a simple C program (“prod.c”) was written which uses the NelderMead Simplex algorithm to minimize2

χ (G1 , . . . , Gm ) =

nX
data

wi fi − f˜i (G1 , . . . , Gm )

2

(20)

i=1

where fi are the measured factors, f˜i their approximations from Equation 19, wi =1/σi2 are the weights of each data point derived from their standard deviation (takenfrom the LISO fit), and Gi the unknown calibration constants.

16

As input the following data were used:fmeas.
5.0043609
3.2644629
2.7204019
2.4554303−3.5256953−2.1236194
62.862325 × 10−3
84.273794 × 10−3
590.1168 × 10−3−110.87858 × 10−3
37.419176 × 10−3
688.79131 × 10−3
43.351314 × 10−3
86.457836 × 10−3
478.86074 × 10−3
−94.29963 × 10−3
49.476234 × 10−3
761.14975 × 10−3
16.61447−270.8383 × 10−3
82.249734−85.75237 × 10−3
216.30985 × 10−3
904.141 × 10−3
73.69469 × 10−3−433.8639 × 10−3
−1.1676377

σi Aoptical Gactuator
40.31 × 10−3 −1.019 m1xcoil
29.84 × 10−3
1.366 m1xcoil
63.32 × 10−3 −.3383 m1xcoil
33.68 × 10−3 −1.019 m2xcoil
21.65 × 10−3 −1.366 m2xcoil
32.95 × 10−3.2454 m2xcoil−3
1.561 × 10
0.7035 m1ypzt
1.359 × 10−3 −0.524 m1ypzt
7.742 × 10−3 −1.701 m1ypzt
1.238 × 10−3 −0.7035 m2ypzt
645.2 × 10−6 −0.524 m2ypzt
4.501 × 10−3 −1.705 m2ypzt
975.7 × 10−6
0.7035 m1ycoil−33.091 × 10
−0.524 m1ycoil
5.221 × 10−3 −1.701 m1ycoil
3.853 × 10−3 −0.7035 m2ycoil
2.697 × 10−3 −0.524 m2ycoil
10.98 × 10−3 −1.705 m2ycoil
181.6 × 10−3
2.906 m3xcoil
49.8 × 10−30 m3xcoil−3
641.4 × 10
4.8613 m3xcoil
2.02 × 10−30 m3ypzt
4.018 × 10−3 −2.105 m3ypzt
14 × 10−3 −2.105 m3ypzt
1.956 × 10−3
0. m3ycoil−33.66 × 10−2.105 m3ycoil
14.64 × 10−3 −2.105 m3ycoil

Gsensor
xi
xq
xe
xi
xq
xe
yi
yq
ye
yi
yq
ye
yi
yq
ye
yi
yq
ye
xi
xq ∗ ∗∗xeyi ∗ ∗∗yqyeyi ∗ ∗∗
yq
ye

(21)

The lines marked *** at the end were excluded from the fit, because the theoretically
expected factor was zero (and the measured one was indeed small).

17

The resulting calibration constants are:xi
= 0.339034
xq=−0.256563
yi=−0.361489
yq= 0.326227
m1xcoil= −10.3098
m2xcoil= −8.37122
m1ycoil= −0.2823
m2ycoil= −0.44523
m1ypzt =−0.346921
m2ypzt = −0.40358
m3xcoil= 16.9186
m3ypzt =−0.425773
m3ycoil= 0.556748(22)

The residuals were:
σi
f˜
ratio
ratio
fmeas.
5.004360.040313.5618 1.40501
2.95 dB
3.26446
0.02984
3.61324 0.903472 −0.88 dB
2.72040.06332
3.48782 0.779972 −2.16 dB
2.455430.03368
2.89205 0.849027 −1.42 dB−3.5257
0.02165 −2.93382 1.20174
1.60 dB−2.12362
0.03295−2.0543 1.03374
0.29 dB
0.0628623 0.001561 0.0882244 0.712527 −2.94 dB
0.0842738 0.001359 0.0593036 1.42106
3.05 dB
0.590112 1.00001
0.00 dB
0.590117 0.007742−0.110879 0.001238 −0.102633 1.08034
0.67 dB
0.0374192 0.0006452 0.0689892 0.542392 −5.31 dB
0.688791 0.004501
0.688104
1.001
0.01 dB
0.0433513 0.0009757 0.0717908 0.603856 −4.38 dB
0.0864578 0.003091 0.0482571 1.79161
5.06 dB
0.478861 0.005221
0.480192 0.997228 −0.02 dB−0.0942996 0.003853 −0.113225 0.83285 −1.59 dB
0.0494762 0.002697
0.076109 0.650071 −3.74 dB
0.761150.01098
0.759117 1.00268
0.02 dB
16.61450.1816
16.6688 0.99674 −0.03 dB
82.2497
0.641482.2466 1.00004
0.00 dB
0.21631 0.004018
0.292382 0.73982 −2.62 dB
0.904141
0.014
0.896252
1.00880.08 dB−0.433864
0.00366 −0.382323 1.13481
1.10 dB−1.16764
0.01464 −1.17195 0.996316 −0.03 dB

18(23)