Time Domain Analysis on a Driven, Damped Simple Pendulum has produced a method for implementing gravity.

COMSOL made this simple task a cryptic one: the following methods had previously failed:

Previous Frequency Domain testing lead to unwanted oscillations of all loads.

Prescribed accelerations at first seemed to create a constant gravity, but instead incorrectly constrained net acceleration to the inputted amount

Methodology:

1) An (approximately) impulse displacement was applied in the horizontal direction. The pendulum bob's displacement was observed for varying pendulum lengths.

2) The drive and response displacements vs. time were FFT'd to produce transfer functions.

3) The fundamental frequencies were inverted, squared, and plotted against frequency.

4) Since the graph is linear with an R^2 of over 0.99, it is reasonable to assume that gravity is properly acting as a restoration force.

For the sake of future users, I have decided to periodically add tips and tricks in using COMSOL that I have figured out, most probably after hours of circuitous efforts. They will always be listed under the new COMSOL Tips category.

Today's topic: Intrusions

COMSOL has a very user-friendly interface for taking objects from 2D to 3D using the "extrusion" feature. But suppose one wants to design an object which contains screw holes or some other indentation. I've found that creating "punctures" in COMSOL is either impossible or very complicated.

Instead, COMSOL encourages users to always "add" to the object. In other words, one must form the lowest level first, then build layers sequentially on top using new work plane and boolean difference operators. This will probably be a bit clearer with an example:

1) First, create the planar projection in a work plane:

2) Extrude the first layer only in the regular fashion:

3) Add a new work plane which is offset in the z-direction to the deepest point of the intrusion.

4) Now, create the shape of the intrusion in this new work plane.

5) Use the Boolean "Difference" to let COMSOL know that, on this plane, the object has a hole.

6) Extrude once more from the second work plane to complete the intrusion.

Wed. 7/7: COMSOL Busbar tutorials; began stack design; began base; Viton rubber research

Thurs. 7/8: Completed Viton rubber research; updated materials; finished designing the base layer

Fri. 7/9: Research model coupling papers; extensive eLog entry about base design and troubleshooting

Sun. 7/11: Played around with Busbar to find first eigenfrequency; continued crashing COMSOL

Mon. 7/12: Intrusions in COMSOL eLog tutorial entry; research eigenfrequency analysis; successfully got first eigenmode of rectangular bar

Tues. 7/13: Updated Poisson ratio of Viton and subsequently succeeded in running eigenfrequency tests on base stack layer. Systematic Perturbation Tests were documented in the most recent elog entry. Discussed results with Rana and decided this didn't make sense. Analytical study required.

Wed. 7/14: Went over to machine shop to experimentally extrapolate spring constant of Viton. Calculations to be done in the afternoon.

In this experiment, we used a feedback control to create a stable trap for a NdFeB permanent magnet. The block diagram is the following:

The displacement of the magnet is sensed by the Hall-effect sensor, whose output voltage is proportional to the magnetic flux produced

by the permanent magnet. It has a flat response within the frequencies we are interested in. It is driven by a 5 V power supplier and its

output has a DC voltagle of 2.5 V. We subtracted the DC voltage and used the resulting signal as the error signal. This was

simply achieved by using two channels "A" and "B". The output is "A-B" with a gain equal to one. We then put the error

signal into another SR560 as a low-pass filter with a gain of 100 above 30 Hz. We used the "DC" coupling modes in both

preamplifers. The output is then used to drive a coil. The coil has a dimension of 1.5 inch in diameter and 2 inch in length.

The inductance of the coil is around 0.5 H and the resistance is 4.7 Om. Therefore, it has a corner frequency aournd 10/2pi Hz.

The coil has a iron core inside to enhance the DC force to the permanent magnet. The low-frequency 1/f response of the magnet is produced

by the eddy current damping of the aluminum plane that is below the magnet. This 1/f response is essential for a stable configuration. In the

next stage, we will remove the aluminum plane, and instead we will use a filter to create similar transfer function. At high-frequencies, it behaves as

a free-mass and has a 1/f^2 response. Finally, the magnet is stably levitated.

I am using SR785 Spectrum Analyzer now and also tomorrow.
I will put it back on Sunday. If anyone needs it during the weekend,
please just take it and I can reset it by myself later. Thanks.

I measured the open-loop transfer function of the magnetic levitation system.

The schematic block diagram for this measurement is the following:

I injected a signal at a level of 20mV between two preamplifiers, and the corresponding open-loop

transfer function is given by B/A. I took a picture of the resulting measurement, because

I encountered some difficulties to save the data to the computer via the wireless network.

The bode plots for the transfer function shown on the screen is the following:

I am puzzled with the zero near 10 Hz. I think it should come from the mechanical response function, because there is no zero in the transfer functions

of the preamplifer and the coil itself. I am not sure at the moment.

The corresponding configuration of the levitated magnet is

1. Why do all of the BNCs have no GND connection? Each should have the individual cables to the ground. Each signal line and the corresponding ground line should be twisted together.

2. This looks the (usual) oscillation of the V-I conversion stage but I can't tell anything as I don't have the circuit diagram of the whole circuit.

3. In a certain case, putting some capacitance at the feedback may help. Read P.11 of the data sheet of LT1125. Try to put some capacitors from 20pF to some larger one whether it changes the situation or not. I suppose the bandwidth of your sensor can be ~1kHz. So putting a capacitance less than 10nF still has no effect to the servo.

1. They are all connected to the box which has a single connection to the board ground. If I connect each of them to the ground, there would be many small loops

of ground. Of course, I should have replaced all the connectors such that the they are disconnected to the box as point out by Robert.

2. The oscillation disappears after I add 5 nF capacitor in parallel to the existing resistor. Thank you very much for pointing this out.

We did some quick DC balancing of the MC2 coil drivers to reduce the l2a coupling. We updated the gains in the C1:SUS-MC2_UL/UR/LR/LLCOIL to be 1, -0.99, 0.937,-0.933, respectively. The previous values were 1, -1, 1, -1.

The procedures are the following:

Lock IMC.

Drive UL+LR and change the gain of LR to zero pitch.

Drive UR+LL and change the gain of LL to zero pitch.

Lastly, drive all 4 coils and change UR & LR together to zero yaw.

We used C1:SUS-MC2_LOCKIN1_OSC to create the excitations at 33 Hz w/ 30,000 cts. The angular error signals were derived from IMC WFSs.

While this time we did things by hand, in the future it should be automated as the procedure is sufficiently straightforward.

My old scheme was flawed as I used pitch as the readback. The pitch signal could not distinguish the cross-coupling due to coil imbalance and that due to the natural suspension L2P. A new scheme based on yaw alone has been developed and will be integrated into ifo_test. For now we revert the C1:SUS-MC2_UL/UR/LR/LLCOIL gains back to 1, -1, 1, -1.

Quote:

[Yehonathan, Hang]

We did some quick DC balancing of the MC2 coil drivers to reduce the l2a coupling. We updated the gains in the C1:SUS-MC2_UL/UR/LR/LLCOIL to be 1, -0.99, 0.937,-0.933, respectively. The previous values were 1, -1, 1, -1.

The procedures are the following:

Lock IMC.

Drive UL+LR and change the gain of LR to zero pitch.

Drive UR+LL and change the gain of LL to zero pitch.

Lastly, drive all 4 coils and change UR & LR together to zero yaw.

We used C1:SUS-MC2_LOCKIN1_OSC to create the excitations at 33 Hz w/ 30,000 cts. The angular error signals were derived from IMC WFSs.

While this time we did things by hand, in the future it should be automated as the procedure is sufficiently straightforward.

After updating the 40 m finesse file to incorporate the new SRC length (and the removal of SR2), we find that the current SRM radius curvature is fine. Thus a replacement of SRM is NOT required.

Basically, the new one-way SRC gouy phase is 11.1 deg according to Finesse, which is very close to the previous value of 10.8 deg. Thus the transmode spacing should be essentially the same.

In the first attached plot is the mode content calculated with Finesse. Here we have first offset DARM by 1m deg and misaligned the SRM by 10 urad. From the top to bottom we show the amplitude of the carrier fields, f1, and f2 sidebands, respectively. The red vertical line is the nominal operating point (thanks Koji for pointing out that we do signal recycling instead of extraction now). No direct co-resonance for the low-order TEM modes. (Note that the HOMs appeared to also have peaks at \phi_srm = 0. This is just because the 00 mode is resonant and thus the seed for the HOMs is greater. )

We can also consider a clean case without mode interactions in the second plot. Indeed we don't see co-resonances of high order modes.

We proposed a few BHD mode-matching telescope designs and then preformed a few monte-carlo experiments to see how the imperfections would change the story. We assumed a 2 mm (1-sigma) error on the location of the components and 1% (1-sigma) fractional error on the RoC of the curved mirrors. The angle of incidence has not yet been taken into account (no astigmatism at the moment but will be included in the follow-up study.)

For the LO path things are mostly fine. We can use LO1 and LO2 as the actuators (Sec. 2.2 of the note), and when errors are taken into account more than 90% of times we can still achieve 98% mode matching. The gouy phase separation between LO1 and LO2 > 34 deg for 90% of the time, which corresponds to a condition number of the sensing matrix of ~ 3.

The situation is more tricky for the AS path. While the telescopes are usually robust against 2 or 3 mm of positional error, the 1% RoC does affect the performance quite significantly. In the note we choose two best-performing ones but still only 50% of the time they can maintain a power-overlap of > 99%. In fact, the 1% RoC error assumed should be quite optimistic... Not sure if we could achieve this in reality.

One potential way out is to ignore the MM for the first round of BHD. Here anyway we only need to test the ISC schemes. Then in the second round when we have the whole BHD board suspended, we can then use AS1 and the BHD board as the actuators. This might be able to make things more forgiving if we don't need to shrink the AS beam very fast so that it could be separated from AS4 in gouy phase.

As Rana suggested, we present the scattering plot of the AS path mode matching for various variables. The plot is for the AS path, Plan 2 (whose params we summarize at the end of this entry).

In the corner plot, we color-coded each realization according to the mode matching. We use (purple, olive, grey) for (MM>0.99, 0.98<MM<=0.99, MM<=0.98), respectively. From the plot, we can see that it is most sensitive to the RoC of AS1. The plot also shows that we can compensate for some of the MM errors if we adjust the distance between AS1-AS3 (note that AS2 is a flat mirror). The telescope is quite robust to other errors.

The compensation requirement is further shown in the second plot. To correct for the 1% RoC error of AS1, we typically need to adjust AS1-AS3 distance by ~ 1 cm (if we want to go back to MM=1; the window for >0.99 MM spans also about 1 cm). This should be doable because the nominal distance between AS1-AS3 is 115 cm.

The story for plan1 is similar and thus not shown here.

I think the conclusion is that if the AS1 RoC error is not significantly more than 1%, then with some adjustment of the AS1-AS3 distance (~ 1 cm), we could find a solution that simultaneously makes the AS path mode-matching better than 99% for the t- and s-planes.

The requirement of the LO path is less strict and the current plan using LO1-LO2 actuation should work.

We computed the required actuation range for the telescope design in elog:15357. The result is summarized in the table below. Here we assume we misalign an IFO mirror by 1 urad, and then compute how many urad do we need to move the (AS1, AS4) or (LO1, LO2) mirrors to simultaneously correct for the two gouy phases.

Actuation requirement in urad per urad misalignment

[urad/urad]

ITMX

ITMY

ETMX

ETMY

BS

PRM

PR2

PR3

SR3

SRM

AS1

1.9

2.1

-5.0

-5.5

0.5

0.5

-0.3

0.2

0.1

0.6

AS4

2.9

2.0

-8.8

-5.5

-5.9

-0.7

1.3

-0.7

-0.5

0.7

LO1

-4.0

-3.9

11.0

10.4

1.9

-0.4

-0.2

0.1

0.0

-1.1

LO2

-5.0

-3.7

15.1

10.4

8.7

0.8

1.9

1.1

0.7

-1.3

The most demanding ifo mirrors are the ETMs and the BS, for every 1 urad misalignment the telescope needs to move 10-15 urad to correct for that. However, it is unlikely for those mirrors to move more 100 nrad for a locked ifo with ASC engaged. Thus a few urad actuation should be sufficient. For the recycling mirrors, every 1 urad misalignment also requires ~ 1 urad actuation.

As a result, if we could afford 10 urad actuation range for each telescope suspension, then the gouy phase separations we have should be fine.

We looked at the oplev spectra from gps 1274418500 for 512 sec. This should be a period when the ifo was locked in the PRFPMI state according to elog:15348. We just focused on the yaw data for now. Please see the attached plots. The solid traces are for the ASD, and the dotted ones are the cumulative rms. The total rms for each mirror is also shown in the legend.

I am now confused... The ITMs looked somewhat reasonable in that at least the < 1 Hz motion was suppressed. The total rms is ~ 0.1 urad, which was what I would expect naively (~ x100 times worse than aLIGO).

There seems to be no low-freq suppression on the ETMs though... Is there no arm ASC at the moment???

We consider the astigmatism effects of the stock options. The conclusions are:

1. For the AS path, the stock should work fine for the phase-one of BHD, if we could tolerate a few percent MM loss. The window for length adjustment to achieve >99% MM for both s and t is only 1 mm for 1% RoC error (compared to ~ 1 cm in the customized case).

2. The LO path seemed tricky. As LO3 & LO4 are both significantly curved (RoC<=0.5 m), the non-zero angle of incidence makes the astigmatism quite sever. For the t-plane the nominal MM can be 0.98, yet for the s-plane, the nominal MM is only 0.72. We could move things around to achieve a MM ~ 0.85, which is probably fine for the phase-one implementation but not long term.

Details:

Attachments 1-3 are for the AS path; 4-6 are for the LO path.

1 & 4. Marginalized MM distribution for the AS/LO paths. Here we assumed 5 mm positional error and 1% fractional RoC error. Due to the astigmatism, the nominal s-plane MM is only 0.72 for the LO path.

2 & 5. Scattering plots for the AS/LO paths. We color coded the points as the following: pink: MM>0.99; olive: 0.98<MM<=0.99; grey: MM<=0.98. For the AS path, MM is mostly sensitive to the AS1 RoC and can be adjusted by changing AS1-AS3 distance. For the LO path, the LO3 RoC and LO3-LO4 distance are most critical for the MM.

3 & 6. Assuming +- 1% AS1 (LO3) fractional RoC error, how much can we compensate for it using AS1-AS3 (LO3-LO4) distance. For the AS path, there exists a ~ 1 mm window where the MM for s and t can simultaneously > 99%. For the LO path, the best we can do is to make s and t both ~ 85%.

Quote:

Summary

For the initial phase of BHD testing, we recently discussed whether the mode-matching telescopes could be built with 100% stock optics. This would allow the optical system to be assembled more quickly and cheaply at a stage when having ultra-low loss and scattering is less important. I've looked into this possibility and conclude that, yes, we do have a good stock optics option. It in fact achieves comprable performance to our optimized custom-curvature design [ELOG 15357]. I think it is certainly sufficient for the initial phase of BHD testing.

Vendor

It turns out our usual suppliers (e.g., CVI, Edmunds) do not have enough stock options to meet our requirements. This is for two reasons:

For sufficient LO1-LO2 (AS1-AS4) Gouy phase separation, we require a very particular ROC range for LO1 (AS1) of 5-6 m (2-3 m).

We also require a 2" diameter for the suspended optics, which is a larger size than most vendors stock for curved reflectors (for example, CVI has no stock 2" options).

However I found that Lambda Research Optics carries 1" and 2" super-polished mirror blanks in an impressive variety of stock curvatures. Even more, they're polished to comprable tolerances as I had specificied for the custom low-scatter optics [DCC E2000296]: irregularity < λ/10 PV, 10-5 scratch-dig, ROC tolerance ±0.5%. They can be coated in-house for 1064 nm to our specifications.

From modeling Lambda's stock curvature options, I find it still possible to achieve mode-matching of 99.9% for the AS beam and 98.6% for the LO beam, if the optics are allowed to move ±1" from their current positions. The sensitivity to the optic positions is slightly increased compared to the custom-curvature design (but by < 1.5x). I have not run the stock designs through Hang's full MC corner-plot analysis which also perturbs the ROCs [ELOG 15339]. However for the early BHD testing, the sensitivity is secondary to the goal of having a quick, cheap implementation.

Stock-Part Telescope Designs

The following tables show the best telescope designs using stock curvature options. It assumes the optics are free to move ±1" from their current positions. For comparison, the values from the custom-curvature design are also given in parentheses.

AS Path

The AS relay path is shown in Attachment 1:

AS1-AS4 Gouy phase separation: 71°

Mode-matching to OMC: 99.9%

Optic

ROC (m)

Distance from SRM AR (m)

AS1

2.00 (2.80)

0.727 (0.719)

AS2

Flat (Flat)

1.260 (1.260)

AS3

0.20 (-2.00)

1.864 (1.866)

AS4

0.75 (0.60)

2.578 (2.582)

LO Path

The LO relay path is shown in Attachment 2:

LO1-LO2 Gouy phase separation: 67°

Mode-matching to OMC: 98.6%

Optic

ROC (m)

Distance from PR2 AR (m)

LO1

5.00 (6.00)

0.423 (0.403)

LO2

1000 (1000)

2.984 (2.984)

LO3

0.50 (0.75)

4.546 (4.596)

LO4

0.15 (-0.45)

4.912 (4.888)

Ordering Information

I've created a new tab in the BHD procurement spreadsheet ("Stock MM Optics Option") listing the part numbers for the above telescope designs, as well as their fabrication tolerances. The total cost is $2.8k + the cost of the coatings (I'm awaiting a quote from Lambda for the coatings). The good news is that all the curved substrates will receive the same HR/AR coatings, so I believe they can all be done in a single coating run.

We explore bilinear SRCL to DARM noise coupling mechanisms, and show two cases that by doing BHD readout the noise performance can be improved. In the first case, the bilinear piece is due to residual DHARD motion (see also LHO:45823), and it matters mostly for the low-frequency (<100 Hz) part, and in the second piece the bilinear piece is due to residual SRCL fluctuation and it matters mostly for the a few x 100 Hz part. Details are below:

=================================================

General Model:

We can write the SRCL to DARM transfer function as (Evan Hall's thesis, eq. 2.29)

The first term in (2) is due to residual DARM motion dx_D. This term does not depends on the DC value of DARM offset <x_D> and thus does not depend on doing BHD or DC readout. On the other hand, the typical residual DARM motion is 1 fm << 1 pm of DARM offset. Since the current feedforward reduction factor is about 10 (see both Den Martynov's thesis and Evan Hall's thesis), clearly we are not limited by the residual DARM motion.

The second term is due to the change in the arm finesse, which can be affected by, e.g., the alignment fluctuation (both increasing the loss due to scattering into 01/10 modes and affecting the spot positon and hence changing the losses), and is likely to be the reason why we see the effect being modulated by DHARD.

The last term in (2) is due to the residual SRCL fluctuation and is important for the ~ a few x 100 Hz band.

=================================================

DHARD effects.

As argued above, the DHARD affects the SRCL -> DARM coupling as it changes the finesse in the arm cavity (through scattering into 01/10 modes; in finesse we cannot directly simulate the effects due to spot hitting a rougher location).

Since in the second term of eq. (2) the LF part depends on the DARM DC offset <x_D>, this effect can be improved by going from DC readout to BHD.

To simulate it in finesse, at a fixed DARM DC offset, we compute the SRCL->DARM transfer functions at different DHARD offsets, and then numerically compute the derivative \partial Z_s2d / \partial \theta_{DH}. Then multiplying this derivative with the rms value of DHARD fluctuation \theta_{DH} we then know the expected bilinear coupling piece.

The result is shown in the first attached plot. Here we have assumed a flat SRCL noise of 5e-16 m/rtHz for simplicity (see PRD 93, 112004, 2016). We do not account for the loop effects which further reduces the high frequency components for now. The residual DHARD RMS is assumed to be 1 nrad.

In the first plot, from top to bottom we show the SRCL noise projection at different DARM DC offsets of (0.1, 1, 10) pm. Since the DHARD alignment only affects the arm finesse starting at quadratic order, it thus matters what DC offset in DHARD we assume. In each pannel, the blue trace is for no DC offset in DHARD and the orange one for a 5 nrad DC offset. As a reference, the A+ sensitivity is shown in grey trace in each plot as a reference.

We can see if there is a large DC offset in DHARD (a few nrad) and we still do DC readout with a few pm of DARM offset, then the bilinear piece of SRCL can still contaminate the sensitivity in the 10-100 Hz band (bottom panel; orange trace). On the other hand, if we do BHD, then the SRCL noise should be down by ~ x100 even compared to with the top panel.

(A 5 nrad of DC offset in DHARD coupled with 1 nrad RMS would cause about 0.5% RIN in the arms. This is somewhat greater than the typically measured RIN which is more like <~ 0.2%. See the second plot).

=================================================

SRCL effect.

Similarly we can consider the SRCL->DARM coupling due to residual SRCL rms. The approach is very similar to what we did above for DHARD. I.e., we compute Z_s2d at fixed DARM offset and for different SRCL offsets, then we numerically evaluate \partial Z_s2d / \partial dphi_S. A residual SRCL rms of 0.1 nm is then used to generate the projection shown in the third figure.

Unlike the DHARD effect, the bilinear SRCL piece does not depend on the DC SRCL detuning (for the 50-500 Hz part). It does still depends on the DARM DC offset and therefore could be improved by BHD.

Since we do not include the LP of the SRCL loop in this plot, the HF noise at 1 kHz is artifical as it can be easily filtered out. However, the LP will not be very strong around 100-300 Hz for a SRCL UGF ~ 30 Hz, and thus doing BHD could still have some small improvements for this effect.

What: Anchal and I measured the XARM OLTF last Thursday.

Goal: 1. measure the 2 zeros and 2 poles in the analog whitening filter, and potentially constrain the cavity pole and an overall gain.

2. Compare the parameter distribution obtained from measurements and that estimated analytically from the Fisher matrix calculation.

3. Obtain the optimized excitation spectrum for future measurements.

How: we inject at C1:SUS-ETMX_LSC_EXC so that each digital count should be directly proportional to the force applied to the suspension. We read out the signal at C1:SUS-ETMX_LSC_OUT_DQ. We use an approximately white excitation in the 50-300 Hz band, and intentionally choose the coherence to be only slightly above 0.9 so that we can get some statistical error to be compared with the Fisher matrix's prediction. For each measurement, we use a bandwidth of 0.25 Hz and 10 averages (no overlapping between adjacent segments).

The 2 zeros and 2 poles in the analog whitening filter and an overall gain are treated as free parameters to be fitted, while the rest are taken from the model by Anchal and Paco (elog:16363). The optical response of the arm cavity seems missing in that model, and thus we additionally include a real pole (for the cavity pole) in the model we fit. Thus in total, our model has 6 free parameters, 2 zeros, 3 poles, and 1 overall gain.

The analysis codes are pushed to the 40m/sysID repo.

Fig. 1 shows one measurement. The gray trace is the data and the olive one is the maximum likelihood estimation. The uncertainty for each frequency bin is shown in the shaded region. Note that the SNR is related to the coherence as

SNR^2 = [coherence / (1-coherence)] * (# of average),

and for a complex TF written as G = A * exp[1j*Phi], one can show the uncertainty is given by

\Delta A / A = 1/SNR, \Delta \Phi = 1/SNR [rad].

Fig. 2. The gray contours show the 1- and 2-sigma levels of the model parameters using the Fisher matrix calculation. We repeated the measurement shown in Fig. 1 three times, and the best-fit parameters for each measurement are indicated in the red-crosses. Although we only did a small number of experiments, the amount of scattering is consistent with the Fisher matrix's prediction, giving us some confidence in our analytical calculation.

One thing to note though is that in order to fit the measured data, we would need an additional pole at around 1,500 Hz. This seems a bit low for the cavity pole frequency. For aLIGO w/ 4km arms, the single-arm pole is about 40-50 Hz. The arm is 100 times shorter here and I would naively expect the cavity pole to be at 3k-4k Hz if the test masses are similar.

Fig. 3. We then follow the algorithm outlined in Pintelon & Schoukens, sec. 5.4.2.2, to calculate how we should change the excitation spectrum. Note that here we are fixing the rms of the force applied to the suspension constant.

Fig. 4 then shows how the expected error changes as we optimize the excitation. It seems in this case a white-ish excitation is already decent (as the TF itself is quite flat in the range of interest), and we only get some mild improvement as we iterate the excitation spectra (note we use the color gray, olive, and purple for the results after the 0th, 1st, and 2nd iteration; same color-coding as in Fig. 3).

Yesterday afternoon Paco and I measured the PRM L2P transfer function. We drove C1:SUS-PRM_LSC_EXC with a white noise in the 0-10 Hz band (effectively a white, longitudinal force applied to the suspension) and read out the pitch response in C1:SUS-PRM_OL_PIT_OUT. The local damping was left on during the measurement. Each FFT segment in our measurement is 32 sec and we used 8 non-overlapping segments for each measurement. The empirically determined results are also compared with the Fisher matrix estimation (similar to elog:16373).

Results:

Fig. 1 shows one example of the measured L2P transfer function. The gray traces are measurement data and shaded region the corresponding uncertainty. The olive trace is the best fit model.

Note that for a single-stage suspension, the ideal L2P TF should have two zeros at DC and two pairs of complex poles for the length and pitch resonances, respectively. We found the two resonances at around 1 Hz from the fitting as expected. However, the zeros were not at DC as the ideal, theoretical model suggested. Instead, we found a pair of right-half plane zeros in order to explain the measurement results. If we cast such a pair of right-half plane zeros into (f, Q) pair, it means a negative value of Q. This means the system does not have the minimum phase delay and suggests some dirty cross-coupling exists, which might not be surprising.

Fig. 2 compares the distribution of the fitting results for 4 different measurements (4 red crosses) and the analytical error estimation obtained using the Fisher matrix (the gray contours; the inner one is the 1-sigma region and the outer one the 3-sigma region). The Fisher matrix appears to underestimate the scattering from this experiment, yet it does capture the correlation between different parameters (the frequencies and quality factors of the two resonances).

One caveat though is that the fitting routine is not especially robust. We used the vectfit routine w/ human intervening to get some initial guesses of the model. We then used a standard scipy least-sq routine to find the maximal likelihood estimator of the restricted model (with fixed number of zeros and poles; here 2 complex zeros and 4 complex poles). The initial guess for the scipy routine was obtained from the vectfit model.

Fig. 3 shows how we may shape our excitation PSD to maximize the Fisher information while keeping the RMS force applied to the PRM suspension fixed. In this case the result is very intuitive. We simply concentrate our drive around the resonance at ~ 1 Hz, focusing on locations where we initially have good SNR. So at least code is not suggesting something crazy...

Fig. 4 then shows how the new uncertainty (3-sigma contours) should change as we optimize our excitation. Basically one iteration (from gray to olive) is sufficient here.

We will find a time very recently to repeat the measurement with the optimized injection spectrum.

We tried to compare the parameter estimation uncertainties of white vs. optimal excitation. We drove C1:SUS-PRM_LSC_EXC with "Normal" excitation and digital gain of 700.

For the white noise exciation, we simply put a butter("LowPass",4,10) filter to select out the <10 Hz band.

For the optimal exciation, we use butter("BandPass",6,0.3,1.6) gain(3) notch(1,20,8) to approximate the spectral shape reported in elog:16384. We tried to use awg.ArbitraryLoop yet this function seems to have some bugs and didn't run correctly; an issue has been submitted to the gitlab repo with more details. We also noticed that in elog:16384, the pitch motion should be read out from C1:SUS-PRM_OL_PIT_IN1 instead of the OUT channel, as there are some extra filters between IN1 and OUT. Consequently, the exact optimal exciation should be revisited, yet we think the main result should not be altered significantly.

While a more detail analysis will be done later offline, we post in the attached plot a comparison between the white (blue) vs optimal (red) excitation. Note in this case, we kept the total force applied to the PRM the same (as the RMS level matches).

Under this simple case, the optimal excitation appears reasonable in two folds.

First, the optimization tries to concentrate the power around the resonance. We would naturally expect that near the resonance, we would get more Fisher information, as the phase changes the fastest there (i.e., large derivatives in the TF).

Second, while we move the power in the >2 Hz band to the 0.3-2 Hz band, from the coherence plot we see that we don't lose any information in the > 2 Hz region. Indeed, even with the original white excitation, the coherence is low and the > 2 Hz region would not be informative. Therefore, it seems reasonable to give up this band so that we can gain more information from locations where we have meaningful coherence.

We report here the analysis results for the measurements done in elog:16388.

Figs. 1 & 2 are respectively measurements of the white noise excitation and the optimized excitation. The shaded region corresponds to the 1-sigma uncertainty at each frequency bin. By eyes, one can already see that the constraints on the phase in the 0.6-1 Hz band are much tighter in the optimized case than in the white noise case.

We found the transfer function was best described by two real poles + one pair of complex poles (i.e., resonance) + one pair of complex zeros in the right-half plane (non-minimum phase delay). The measurement in fact suggested a right-hand pole somewhere between 0.05-0.1 Hz which cannot be right. For now, I just manually flipped the sign of this lowest frequency pole to the left-hand side. However, this introduced some systematic deviation in the phase in the 0.3-0.5 Hz band where our coherence was still good. Therefore, a caveat is that our model with 7 free parameters (4 poles + 2 zeros + 1 gain as one would expect for an ideal signal-stage L2P TF) might not sufficiently capture the entire physics.

In Fig. 3 we showed the comparison of the two sets of measurements together with the predictions based on the Fisher matrix. Here the color gray is for the white-noise excitation and olive is for the optimized excitation. The solid and dotted contours are respectively the 1-sigma and 3-sigma regions from the Fisher calculation, and crosses are maximum likelihood estimations of each measurement (though the scipy optimizer might not find the true maximum).

Note that the mean values don't match in the two sets of measurements, suggesting potential bias or other systematics exists in the current measurement. Moreover, there could be multiple local maxima in the likelihood in this high-D parameter space (not surprising). For example, one could reduce the resonant Q but enhance the overall gain to keep the shoulder of a resonance having the same amplitude. However, this correlation is not explicit in the Fisher matrix (first-order derivatives of the TF, i.e., local gradients) as it does not show up in the error ellipse.

In Fig. 4 we show the further optimized excitation for the next round of measurements. Here the cyan and olive traces are obtained assuming different values of the "true" physical parameter, yet the overall shapes of the two are quite similar, and are close to the optimized excitation spectrum we already used in elog:16388.

We did a few quick XARM oltf measurements. We excited C1:LSC-ETMX_EXC with a broadband white noise upto 4 kHz. The timestamps for the measurements are: 1318199043 (start) - 1318199427 (end).

We will process the measurement to compute the cavity pole and analog filter poles & zeros later.

One goal of our sysID study is to improve the aLIGO L2A feedforward. Our algorithm currently improves only the statistical uncertainty and assumes the systematic errors are negligible. However, I am currently baffled by how to fit a (nearly) realistic suspension model...

My test study uses the damped aLIGO QUAD suspension model. From the Matlab model I extract the L2 drive in [N] to L3 pitch in [rad] transfer function (given by a SS model with the A matrix having a shape of 103x103). I then tried to use VectFIT to fit the noiseless TF. After removing nearby z-p pairs (defined by less than 0.2 times the lowest pole frequency) and high-frequency zeros, I got a model with 6 complex pole pairs and 4 complex zero pairs (21 free parameters in total). I also tried to fit the TF (again, noiseless) with an MCMC algorithm assuming the underlying model has the same number of parameters as the VectFIT results.

Please see the first attached plots for a comparison between the fitted models and the true one. In the second plot, we show the fractional residual

| TF_true - TF_fit | / | TF_true |,

and the inverse of this number gives the saturating SNR at each frequency. I.e., when the statistical SNR is more than the saturating value, we are then limited by systematic errors in the fitting. And so far, disappointingly I can only get an SNR of 10ish for the main resonances...

I wonder if people know better ways to reduce this fitting systematic... Help is greatly appreciated!

We have been discussing how does the parameter estimation depends on the length per FFT segment. In other words, after we collected a series of data, would it be better for us to divide it into many segments so that we have many averages, or should we use long FFT segments so that we have more frequency bins?

My conclusions are that:

1). We need to make sure that the segment length is long enough with T_seg > min[ Q_i / f_i ], where f_i is the resonant frequency of the i'th resonant peak and the Q_i its quality factor.

2). Once 1) is satisfied, the result depends weakly on the FFT length. There might be a weak hint preferring a longer segment length (i.e., want more freq bins than more averages) though.

To reach the conclusion, I performed the following numerical experiment.

I considered a simple pendulum with resonant frequency f_1 = 0.993 Hz and Q_1 = 6.23. The value of f_1 is chosen such that it is not too special to fall into a single freq bin. Additionally, I set an overall gain of k=20. I generated T_tot = 512 s of data in the time domain and then did the standard frequency domain TF estimation. I.e., I computed the CSD between excitation and response (with noise) over the PSD of the excitation. The spectra of excitation and noise in the readout channel are shown in the first plot.

In the second plot, I showed the 1-sigma errors from the Fisher matrix calculation of the three parameters in this problem, as well as the determinant of the error matrix \Sigma = inv(Fisher matrix). All quantities are plotted as functions of the duration per FFT segment T_seg. The red dotted line is [Q_1/f_1], i.e., the time required to resolve the resonant peak. As one would expect, if T_seg <~ (Q_1/f_1), we cannot resolve the dynamics of the system and therefore we get nonsense PE results. However, once T_seg > (Q_1/f_1), the PE results seem to be just fluctuating (as f_1 does not fall exactly into a single bin). Maybe there is a small hint that longer T_seg is better. Potentially, this might be due to that we lose less information due to windowing? To be investigated further...

I also showed the Fisher estimation vs. MCMC results in the last two plots. Here each dot is an MCMC posterior. The red crosses are the true values, and the purple contours are the results of the Fisher calculations (3-sigma contours). The MCMC results showed similar trends as the Fisher predictions and the results for T_seg = (32, 64, 128) s all have similar amounts of scattering << the scattering of the T_seg=8 s results. Though somehow it showed a biased result. In the third plot, I manually corrected the mean so that we could just compare the scattering. The fourth plot showed the original posterior distribution.

1. In the error propogation equation, it should be \Delta \Theta = -H^{-1} M \Delta \Lambda, instead of the fractional error.

2. For the astro parameters, in general you would need t_c for the time of coalescence and \phi_c for the phase. See, e.g., https://ui.adsabs.harvard.edu/abs/1994PhRvD..49.2658C/abstract.

3. Fig. 1 looks very nice to me, yet I don't understand Fig. 3... Why would phase or amplitude uncertainties at 30 Hz affect the tidal deformability? The tide should be visible only > 500 Hz.

4. For BBH, we don't measure individual spin well but only their mass-weighted sum, \chi_eff = (m_1*a_1 + m_2*a_2)/(m_1 + m_2). If you treat S1z and S2z as free parameters, your matrix is likely degenerate. Might want to double-check. Also, for a BBH, you don't need to extend the signal much higher than \omega ~ 0.4/M_tot ~ 10^4 Hz * (Ms/M_tot). So if the total mass is ~ 100 Ms, then the highest frequency should be ~ 100 Hz. Above this number there is no signal.

I modified astroFisherLib.py to include these parameters. Please note that their meaning is that we don't know when the signal happens and at which phase it merges.

It does not mean the time & phase from a reference frequency to the merger. This part is not free to vary because it is fixed by the intrinsic parameters.

It might be good to have a quick scan through the Cutler & Flanagan 94 paper to better understand their physical meanings.

To use a razorblade to measure beam waist at multiple points along the optical axis, so as to later extrapolate the modal profile of the entire beam. This information will then be used to effectively couple AUX laser light to fibers for use in the frequency offset locking apparatus.

Data Acquisition

1) Step the micrometer-controlled razorblade across the beam at a given value of Z, along optical axis, in the plane orthogonal to it (arbitrarily called X).

2) At each value of X, record the corresponding output of a photodiode, (Thorlabs PD A55) here given in mV.

3) Repeat process at multiple points along Z

Analysis

Data from each iteration in the X were fitted to the error function shown below.

V(x) = A*(erf((x-m)/s)+c)

In the Y, they were fitted to:

V(x) = -A*(erf((x-m)/s)+c)

'A' corresponds to an amplitude, 'm' to a mean, 's' to a σ, and 'c' to an offset.

(Only because in Y measurements, the blade progressed toward eclipsing the beam, as opposed to in the X where it progressively revealed the beam.

These fits can be solved for x = (erf^{-1}((V/A)-c)*s)+m1 which can be calculated at the points (V_{max}/e^{2}) and (V_{max}*(1-1/e^{2})). The difference between these points will yield beam waist, w(z).

To use a razorblade to measure beam waist at four points along the optical axis, so as to later extrapolate the waist. This information will then be used to effectively couple AUX laser light to fibers for use in the frequency offset locking apparatus.

Data Acquisition

1) Step the micrometer-controlled razorblade across the beam at a given value of Z, along optical axis, in the plane orthogonal to it (arbitrarily called X).

2) At each value of X, record the corresponding output of a photodiode, (Thorlabs PD A55) here given in mV.

3) Repeat in Y plane at the same value of Z

4) Repeat process at multiple points along Z

Analysis

Data from each iteration were fitted to the error function shown below.

y(x) = (.5*P)*(1-erf((sqrt(2)*(x-x0))/wz))

'P' corresponds to peak power, 'x0' to the corresponding value of x (or y, as the case may be), and 'wz' to the spot size at the Z value in question.

The spot sizes from the four Z values were then fit to:

y(x) = w0*sqrt(1+((x*x)/(zr*zr)))

Where 'w0' corresponds to beam waist, and 'zr' to Rayleigh Range.

Conclusion

This yielded a Y-Waist of 783.5 um, and an X-Waist of 915.2 um.

The respective Rayleigh ranges were 2.965e+05 um (Y) and 3.145e+05 um (X).

Next

I will do the same analysis with light from the optical cables, which information I will then use to design a telescope to effectively couple the beams.

I was finally able to get a reasonable measurement for the beam waist(s) of the spare NPRO.

Methods

I used a razorblade setup, pictured below, to characterize the beam waist of the spare 1064nm NPRO after a lens (PLCX-25.4-38.6-UV-1064) in order to subsequently calculate the overall waist of the beam. The setup is pictured below:

After many failed attempts, this was the apparatus we (Manasa, Eric Q, Koji, and I) arrived with. The first lens after the laser was installed to focus the laser, because it's true waist was at an inaccessible location. Using the lens as the origin for the Z axis, I was able to determine the waist of the beam after the lens, and then calculate the beam waist of the laser itself using the equation wf = (lambda*f)/(pi*wo) where wf is the waist after the lens, lambda the wavelength of the laser, f the focal legth of the lens (75.0 mm in this case) and wo the waist before the lens.

We put the razorblade, second lens (to focus the beam onto the photodiode (Thorlabs PDA255)), and the PD with two attenuating filters with optical density of 1.0 and 3.0, all on a stage, so that they could be moved as a unit, in order to avoid errors caused by fringing effects caused by the razorblade.

I took measurements at six different locations along the optical axis, in orthogonal cross sections (referred to as X and Y) in case the beam turned to be elliptical, instead of perfectly circular in cross section. These measurements were carried out in 1" increments, starting at 2" from the lens, as measured by the holes in the optical table.

Analysis

Once I had the data, each cross section was fit to V(x) = (.5*Vmax)*(1-erf((sqrt(2)*(x-x0))/wz))+c, which corresponds to the voltage supplied to the PD at a particular location in x (or y, as the case may be). Vmax is the maximum voltage supplied, x0 is an offset in x from zero, wz is the spot size at that location in z, and c is a DC offset (ie the voltage on the PD when the laser is fully eclipsed.) These fits may all be viewed in the attached .zip file.

The spot sizes, extracted as parameters of the previous fits, were then fit to the equation which describes the propagation of the spot radius, w(z) = wo*sqrt(1+((z-b)/zr)^2)+c, w(z) = w0*sqrt(1+((((z-b)*.000001064)^2)/((pi*w0^2)^2))) where wo corresponds to beam waist, b is an offset in the z. Examples of these fits can be viewed in the attached .zip file.

Finally, since the waists given by the fits were the waists after a lens, I used the equation wf = (lambda*f)/(pi*wo), described above, to determine the waist of the beam before the lens.

Plots

note: I was not able to open the first measurement in the X plane (Z = 2in). The rest of the plots have been included in the body of the elog, as per Manasa's request.

Conclusion

The X Waist after the lens (originally yielded from fit parameters) was 90.8 27.99 ± .14 um. The corresponding Y Waist was 106.2 30.22 ± .11 um.

After adjustment for the lens, the X Waist was 279.7 907.5 ± 4.5 um and the Y Waist was 239.2 840.5 ± 3.0 um.

edit: After making changes suggested by koji, these were the new results of the fits.

Attachments

Attached you should be able to find the razor blade schematic, all of the fits, along with code used to generate them, plus the matlab workspace containing all the necessary variables.

NOTE: Rana brought to my attention that my error bars need to be adjusted, which I will do as soon as possible.

-I continued to struggle with the razorblade beam analysis, though after a sixth round of measurements, and a lot of fiddling around with fit parameters in matlab, there seems to be a light at the end of the tunnel.

Next Week:

-I plan to check my work with the beamscan tomorrow (wednesday) morning

-Further characterize the light from the fibers, and set up the collimator

-Design and hopefully construct the telescope that will focus the beam into the collimator

Materials:

- Razorblade setup or beamscan (preferably beamscan)

Today, I borrowed the beam profiler from Brian (another SURF) in order to double check my razor blade measurement figures, using the below setup:

Measurements are included in the a la mode code that is attached entitled beamfit.m. The beam fitting application yielded me waists (after the lens) of 35.44 um in the x plane, and 33.26 um in the y plane. These are both within 3 um of the measurements I found using the razor blade method. (I moved and resized the labels for the waists in the figure below for readability purposes.)

I then plugged these waists back into ALM, in addition to the lens specifications, to determine waist size and location of the NPRO, which turned out to be 543 um in the x located at Z = 1.160m, and 536 um in the y, located at 1.268m. These measurements are based upon zero at the waist after the lens, and the positive direction being back toward the NPRO.

The only systemic difference between these measurement and my original razor blade measurements was that I had taken the focal length of the lens as 75mm, which is advertised on the manufacturer's site. However, the more detailed specs revealed that the focal length was 85.8mm at 1064nm, which made a difference of about 400 um for the final waist determination.

I designed this telescope to couple the 1064 NPRO into the PM980 fiber, using lenses from the Thorlabs LSB04-C kit.

The collimator is a CFC-2X-C, which has a variable focus length (2.0, 4.6, 7.5, and 11.0 mm) which gives corresponding angles of divergence of 0.298, 0.130, 0.79, and 0.054 degrees by the formula theta = (180*MFD) / (pi*f).

Then, using these values I calculated the spot size of a beam collimated by the CFC-2X-C, using f = w / tan(theta) where w is the spot size. This gave a value of 10.4 um.

I used this value (10.4 um) as a target waist for the telescope system, with the NPRO waist as a seed, at the origin.

It consists of two lenses, one located at Z = 77cm f = 50cm, and the second located at Z = 85.88 cm f = 2.54cm, which yields a waist of 13um at Z = 88.32cm, (which is where the collimator would go) for an overlap of .974.

Note that the telescope is so far "downrange" from the NPRO waist because it's a virtual waist, and the NPRO itself is located at about Z = 73cm.

I used a la mode to make a design for the coupling telescope with a 3.3um target waist, that included the collimator in the overall design. The plot is below, and code is attached.

The components are as follows:

label z (m) type parameters

----- ----- ---- ----------

lens1 0.7681 lens focalLength: 0.5000

lens2 0.8588 lens focalLength: 0.0350

collimator 0.8832 lens focalLength: 0.0020

The z coordinates are as measured from the beam waist of the NPRO (the figure on the far left of the plot).

Moving forward, this setup will be used to couple the NPRO (more specifically, the AUX lasers) light into the SM 980 fibers, as well as to help characterize the fibers themselves.

Ultimately, this will be a key component in the Frequency Offset Locking project that Akhil and I are working on, as it will transport the AUX light to the PSL, where the two beams will be beaten with each other to generate the input signal to the PID control loop, which will actuate the temperature servos of the AUX lasers.

To design an optical setup (telescope / lens) to couple 1064nm NPRO light into PANDA PM980 fibers in order to characterize the fibers for further use in the frequency offset locking setup.

Design

Calculations

The beam waist of the NPRO was determined as 233um 6cm in front of the NPRO. This was used as the seed waist in ALM.

The numerical aperture of the fiber was given as 0.12, which allowed me to calculate the maximum angle of light it would accept, with respect to the optical axis, as NA = sin(theta) where theta is that angle.

Given that the coupler has a focal length of 2mm, I used the formula r = f * tan(theta), to yield a "target waist" for efficient coupling into the fiber. This ended up being 241.7um.

Since there was not a huge difference between the natural beam width of the NPRO and our target waist, I had no need for multiple lenses.

I used 230um as a target waist for a la mode, to leave myself some room for error while coupling. This process gave me a beam profile with a lens (f=0.25m), and a target waist of 231um, located 38.60cm from the coupling lens

I have attached ALM code, as well as the beam profile image. Note that the profile takes zero to be the location of the NPRO waist.

Next Steps

After this setup is assembled, and light is coupled into the fibers, we will use it to run various tests to the fiber, for further use in FOL. First of all, we wish to measure the coupling efficiency, which is the purpose of the powermeter in the above schematic. We will measure optical power before and after the fibers, hoping for at least ~%60 coupling. Next is the polarization extinction ratio measurement, for which we will control the input polarization to the fibers, and then measure what proportion of that polarization remains at the output of the fiber.

To couple the spare NPRO into our Panda PM980 fibers, in order to carry out tests to characterize the fibers, in order to use them in FOL.

Design

Manasa and I spent this morning building the setup to couple NPRO light into the fibers. We used two steering mirrors to precisely guide the beam into the coupler (collimator).

We also attached the lens to a moveable stage (in the z axis), so the setup could be fine tuned to put the beam waist precisely at the photodiode.

The fiber was attached to a fiber-coupled powermeter, so I would be able to tell the coupling efficiency.

Methods

During alignment, the NPRO was operating at 1.0 amps, roughly half of nominal current (2.1A).

I first placed the coupler at the distance that I believed the target waist of 231um to be.

Using the steering mirrors and the stage that holds the couple, I aligned the axes of the coupler and the beam.

Finally, I used the variable stage that the lens is attached to to fine tune the location of the target waist.

Results

Once I was getting readings on the powermeter (~0.5nW), the laser was turned up to nominal current of 2.1A.

At this point, I and getting 120nW through the fiber.

While far from "good" coupling, it is enough to start measuring some fiber characteristics.

Moving Forward

Tomorrow, I hope to borrow the beam profiler once again so as to measure the fiber mode.

Beyond this, I will be taking further measurements of the Polarization Extinction Ratio, the Frequency Noise within the fiber, and the effects of a temperature gradient upon the fiber.

Once these measurements are completed, the fiber will have been characterized, and will be ready for implementation in FOL.

Steve and I moved some things around in the 1X2 rack in order to make room (roughly 6") for the electronics box that will contain rf frequency counters, ADC, and Raspberry Pi's for use in the Frequency Offset Locking apparatus

Picture

Occurrences

First, we killed power by removing the fuse that the boxes we were moving were running through.

Then, we moved the boxes. I dropped//lost a washer. It didn't seem to cause any problems, so no further attempts to locate it were made.

The fuse was reinstalled, and everything was reconnected.

Moving Forward

We are now working on putting together the electronics box, which will contain ADC, and raspberry pi's. The frequency counters will be mounted on the front of the box.

Once complete, it will be installed for use in FOL.

We wanted to measure the mode coming out of the fibers, so we can later couple it to experimental setups for measuring different noise sources within the fiber. i.e. Polarization Extinction Ratio, Frequency Noise, Temperature Effects.

Methods

I used the beamscan mounted on a micrometer stage in order to measure the spot sizes of the fiber coupled light at different points along the optical axis, in much the same way as in the razorblade setup I used earlier in the summer.

Analysis

I entered my data (z coordinates, spot size in x, spot size in y) into a la mode to obtain the beam profile (waist size, location)

Code is attached in .zip file.

Moving Forward

After I took these measurements, Manasa pointed out that I need points over a longer distance. (These were taken over the range of the micrometer stage, which is 0.5 inches.)

I will be coming in to the 40m early on Monday to make these measurements, since precious beamscan time is so elusive.

Eventually, we will use this measurement to design optical setups to characterize Polarization Extinction Ratio, Frequency Noise, and temperature effects of the fibers, for further use in FOL.

The idea was to measure the profile of the light coming out of the fiber, so we could have knowledge of it for further design of measurement apparatuses, for characterization of the fibers' properties.

Methods

The method was the same as the last time I tried to measure the fiber mode.

This time I moved the beam profiler in a wider range along the z-axis.

Additionally, I adjusted the coupling until it gave ~1mW through the fiber, so the signal was high enough to be reliably detectable.

Measurements were taken in both X and Y transections of the beam.

The range of movement was limited by the aperture of the beam profiler, which cuts off at 9mm. My measurements stop at 8.3mm, as the next possible measurement was beyond the beam profiler's range.

Analysis

I entered my data into A La Mode, which gave me a waist of 5um, at a location of z = -0.0071 m, that is to say, 7.1mm inside the fiber.

Note that in the plot, data points and fits overlap, and so are sometimes hard to distinguish from each other.

Code is attached.

Moving Forward

Using this data, I will begin designing setups to measure fiber characteristics, the first of which being Polarization Extinction Ratio.

Eventually, the data collected from these measurements will be put to use in the frequency offset locking setup.

The idea was to measure the profile of the light coming out of the fiber, so we could have knowledge of it for further design of measurement apparatuses, for characterization of the fibers' properties.

Methods

The method was the same as the last time I tried to measure the fiber mode.

This time I moved the beam profiler in a wider range along the z-axis.

Additionally, I adjusted the coupling until it gave ~1mW through the fiber, so the signal was high enough to be reliably detectable.

Measurements were taken in both X and Y transections of the beam.

The range of movement was limited by the aperture of the beam profiler, which cuts off at 9mm. My measurements stop at 8.3mm, as the next possible measurement was beyond the beam profiler's range.

Analysis

I entered my data into A La Mode, which gave me a waist of 5um, at a location of z = -0.0071 m, that is to say, 7.1mm inside the fiber.

Note that in the plot, data points and fits overlap, and so are sometimes hard to distinguish from each other.

Code is attached.

Moving Forward

Using this data, I will begin designing setups to measure fiber characteristics, the first of which being Polarization Extinction Ratio.

Eventually, the data collected from these measurements will be put to use in the frequency offset locking setup.

Edit

The previous data were flawed, in that they were taken in groups of three, as I had to move the micrometer stage which held the beamscan between holes in the optical table.

In order to correct for this, I clamped a straightedge (ruler) to the table, so I could more consistently align the profiler with the beam axis.

These data gave a waist w_o = 4um, located 6mm inside the fiber. While these figures are very close to what I would expect (3.3um at the end of the fiber) the fitting still isn't as good as I would like.

The fit given by ALM is below.

Moving Forward

I would like to get a stage//rail so I can align the axes of the beam and profiler more consistently.

I would also like to use an aperture the more precisely align the profiler aperture with the beam axis.

Once these measurements have been made, I can begin assembling the setup to measure the Polarization Extinction Ratio of the fiber.

I repeated this process once more, this time using the computer controlled stage that the beam profiler is designed to be mounted to.

These data//fitting appears to be within error bars. The range of my measurements was limited when the beam width was near the effective aperture of the profiler.

This latest trial yielded a waist of 4um, located 2.9 mm inside the fiber for the X profile, and 3.0mm inside the fiber for the Y profile.

Code is attached in fiberModeMeasurement4.zip. Note that the z=0 point is defined as the end of the fiber.