I determined the OLG TF of the whole PMC loop, and TFs from servo paths and optical path.
We want to modify the PMC servo to optimize the PMC loop, so we have to know what are the TFs from part where we can modify,
and where we can't (optical path).
The whole TF is measured before, but I remeasured again just to make sure that there won't be any problem from the laser.
How I measure the whole TF is [here].
I measured the OLG TF from the PMC servo.
The results agree well with the LISO model, see fig 1.
The pole (in LISO model)around 100kHz comes from non ideal behavior of PA85.
When I switch to ideal opamp model, the response is flat.
Optical TF = Whole TF - Servo TF.
The Optical TF won't be modified. It will be used to compute the whole TF after the PMC servo modification.
The measurement at low frequency does not look nice because the signal was suppressed by the gain.
But the TF around UGF still looks fine to work with.
Why is the optical TF not (kinda) flat?
Why does the PZT actuator completely ignored?
You need to talk to me tomorrow afternoon when I am in ATF.
I measured the OLG TF from the PMC servo. The results agree well with the LISO model, see fig 1.
Sorry for the confusion, PZT actuator is included in the optical TF.
The plot on fig2 below shows the TF of PZT part, offset by 1 dB to match the misnomer optical path TF.
Thus, the real optical TF is rather flat with magnitude~ 1 dB, the phase shift is 180 degree,
and the modifiable TF (LISO model) is plot on fig1. This plot has not taken the gain from the slider into account yet.
What are the units of the vert axes?
Separate the open loop gain into three part:
- Optical Gain, Unit [V/m] or [V/Hz], usually flat or simple low path shape
- Servo Filter Gain, Unit [V/V], various shape
- Actuator Gain, Unit [m/V] or [Hz/V], flat or low path filter like up to kHz~100kHz (depending on the time constant of the RC filter),
mechanical resonances above that freq region, which usually determin the highest UGF.
You can change the servo gain by modifying the circuit.
You can change the optical gain by changing the amount of the light in the cavity / on the PD as well as changing the cavity finesse etc.
You can change the actuator gain by replacing the actuator.
I got the calibration from [here]
1) DC ext channel on PMC servo: 32.82 MHz/ V
The DC gain between DC ext channel and the voltage at PZT is 27.65 dB (x24.13),
so the Actuator gain will be 32.82/24.13 = 1.36 MHz/ V;
The plot on fig1 is the Transfer function of the PZT actuator in MHz/ Volt.
The liso plot, [fig1] offset by 30.5 dB, match the result from the measurement.
This means that the gain from AD602 is 30.5 dB, even though the gain slider says 30dB.
Assuming that from DC to 100kHz, the TF from optic is flat.
The OLG TF measurement must equal The TF from servo(From LISO) + gain slider(30.5 dB) + PZT(LISO) + optics(flat offset)
The offset in the plot is 25.5 dB. With the 30.5 dB from gain slider, TF from optics is -5dB flat, with 180 degree phase shift see fig2. [add calibration from Hz -> V] [plot2]
The result from previous entry which gives the optic's TF to be flat at 1 dB is wrong because I did not use the whole TF from the servo
when I compare the model and the measurement, so I missed -6 dB from AD797.
- Actuator Gain, Unit [m/V] or [Hz/V], flat or low path filter like up to kHz~100kHz (depending on the time constant of the RC filter),
mechanical resonances above that freq region, which usually determin the highest UGF.
I measured the slope of the error signal vs RF voltage, the result is plotted below
To increase the overall gain of the PMC loop, one thing we can do is changing the slope of the error signal.
This will increase the gain on the optical path of the loop.
So I measured the slope of the error signal, this information will allow me to know how much
gain I would get from each RF setting. The slope increases as the RF voltage increases, until V_RF ~ 8 V.
The error signal does not change at all when V_RF on the slider is between 8 to 10 [Max] V, and
there is no saturation in the signal.
Note: I use the oscilloscope to measure the slope around the center of the error signal, by
measureing dt and dV to get dV/dt around the center, (this can be converted to dV/dHz by the sideband)
but the result has large deviation, so I measure the pk-pk in stead, and divide that by the
cavity FWHM = 3.8 MHz which corresponds to the peak-peak of the signal. to get the average slope.
It will be lower than the actual value
but I'll keep it for now.
I plot the TF from each stage in the PMC loop and plot below.
1) Servo (+ gain slider)[V/V], From the mixer output to the output of PA85. The amplitude can be added upto 30.5 dB by the gain slider setup
2) PZT [V/V]. From PA85 V output to V at PZT. This includes the last R in the servo (R44 = 64.3k Ohm) and C_pzt (0.23 uF).
3) Opt [V/V]. This includes the PMC and the frequency discriminator part up to the signal to the mixer.
The PMC converts V -> Hz [1.36 MHz/V]. The PMC pole is 1.9 MHz, so I
assume that it is flat at the region of interest (1-100kHz). The frequency discriminator convert Hz->V, and assuming flat response for now.
Thus the total unit of this part is [V/V] too. I'll separate this part into PMC and+ RFPD later.
I measure the OLG TF of the PMC with 3 different RF and gain slider settings. I plot the OLG TF of each setup and identify their UGF.
I increase the RF as a first step to optimize PMC loop w/o modifying the circuit. This will increase the TF of the optical path.
The setting are
First I adjust the RF power to reach where I can adjust the stability by changing the gain slider.
RF V above (6 or 7) the gain is too large, even with smallest gain slider, the signal is not stable and the PMC_RCTRANSPD drops from maximum.
So RF V ends up around 5.5 V. this makes the gain slider sit around 10 -15 where I obtain maximum stability.
The gain slider should not to be set too low because of stability problem. The voltage supply for the opamp should be > +10V.
The gain slider setups are chosen to obtain the maximum stability and maximum power out put (PMC_RCTRANSPD.)
c and d have the same RF V, I change the gain to see if there would be any significant change in the performance, and
the data will be used for RF V calibration (how much gain we got from RF V adj).
Now we have some room to increase the gain once we lower the power, but
I have to understand why increasing the gain slider makes signal unstable.
The phase margin seems to be ok. It might be the slope of the TF at UGF that causes instability.
I checked that the optical gain in PMC loop increases as the power in the sideband increases. The result is 10.7 dB/V.
This measurement is for checking how much gain (in optical path) will we get from changing power in the side bands.
The excitation is sent to EXT DC channel on PMC. Reference signal is at HV mon, response is picked up at Mix mon.
This TF includes PZT and OPT paths, PZT TF should remain the same independent from the side band power.
I vary the RF voltage, and adjust the gain slider for maximum stability. The gain setup should not matter
in the TF part we are measuring as long as the loop is stable.
I measured the gain at 3 different frequencies, 290.8 Hz, 1.035 kHz, 5.09 kHz where the TF look reasonable and smooth.
(The loop UGF is ~ 500-900 Hz, Thus the data at 1k and 5 kHz are nicer than that of 290 Hz)
the slopes from each fit are
The results are fairly linear in our region (RF between 4.8 to 5.9 V). The gain slider for this voltage range is between 13 - 20 dB.
At higher RF voltage, PMC_RCTRANSPD starts to drop significantly.
At lower RF voltage, the gain is too low.
This means we can increase the gain in OPT TF up to 10 dB by adjusting RF voltage (increase side band power)
The current plot for PMC's OLG TF is plotted below. The RF V is 6V, Gain slider is 14 dB.
The UGF is 820 Hz with phase margin (PM) = 180 - 53 = 127 degree.
At higher gain slider setup, the system starts to oscillate. One possible cause is the peak near 10^4 Hz which
might be the PZT's resonance frequency.
Without the notch the total gain we can increase will be limited by the peak.
I'll make a notch to damp it down.
The current spec will be ~20dB notch at 12.5 kHz, FWHM ~1kHz
From current setup, the optical TF should be + 16.5 dB flat, and the gain added by the gain slider is +14 dB.
previous setup we have opt TF = -5 dB and +30 dB from gain slider.
So we have improved the overall gain by ~5 dB and to UGF increased from 530 to 830 Hz.
The RefCav pole is 37 kHz, not 37 MHz.
To minimize the RFAM, you just look at the PMC REFL PD with the PMC unlocked and adjust the waveplate to minimize the peak. Before doing this, make sure that there is no signal on the PD with the light blocked.
Ah right, that's embarassing. I'll try that.
corrected the PMC transPD calibration. Old values for EGUF/EGUL fields in the record were 74.1/-74.1.
The corrected values are 133/-133. Value now reflects what i measure with the Thorlabs power meter.
Changed both entries in the database file.
Just a thought....
If you're going to post LISO models (the .fil files), it might be handy to also include a sketch of the circuit, or a link to the circuit.
Already done. Sorry about that.
For EOM HV:
You need a single zero (or double zero) in order to make the bend of the magnitude curve at high freq.
However, this adds the phase advance at around the cut off freq although you actually have the phase delay.
This means that you need additional double pole just above the measurement freq. This pole may have relatively
high Q so that it can cancel the phase advance by the zero.
For EOM circuit:
A zero is missing at around 10^5 Hz. Also a pole or several poles are necessary to realize the magnitude curve and the phase delay at the high freq edge.
Limit the freq range from 10^5 instead of rediculous 10^-9.
Replace one or two single poles by a double pole.
started realigning everything from scratch and calibrate all channels right. PMC optical power channels need re-calibration.
input beam to PMC: 28.9mW
PMC transmitted beam: 23.4mw
transmission trough curved mirror: 1.63mW
--> only 81% visibility (87% including back mirror leakage), but should be enough for what we want to do. Don't know the loss mechanism (didn't investigate)
A reminder for Frank about the setup,
thanks, for the info.
I only did the PMC so far as i has to fix the daq and add the new channels. What's the problem with beam height of the isolator? Is the beam too low or the mount too high? Do you know?
The V-block's height is a bit too high. The beam height is very close to 3".
thanks for the info.
I only did the PMC so far as i had to fix the daq and add the new channels. What's the problem with beam height of the isolator? Is the beam too low or the mount too high? Do you know?
As mention in the previous corresponding entry, the height of the base for the faraday isolator is not correct. I removed the thing from the table and measure its height again in order to have it fixed. It has to be cut by 3/128 inch from the base (see figure below). The groove for the isolator is well leveled. I got the same height for both ends. I'll bring it back to machine shop tomorrow.
There are other mechanical parts I need to fix:
I'll try to have these parts before we open the chamber to change the seismic stage.
Today when Sarah and I tried to align the beam to PMC, we lost PMC lock and could not bring it back. So we investigated it. The cause of the lost lock is not yet exactly concluded, but we can lock pmc back at ~ 40mW with usual stability.
What we did before we lost lock:
What we did to check what was wrong with the PMC servo:
I got PMC drawing from Dmass, this will be similar to gyro's steel PMC. I'll submit the work to machine shop soon.
The drawing is on svn full PMC assemble can be found at ATF:1543.There are spare mirrors in PSL that can be used. I still have to look for a PZT.
The round trip length is 0.33 cm. this corresponds to FSR = 454.45 MHz. If I want to be able to scan through 2 FSR, the displacement range of the PZT will be dL = 2*FSR * L / f, where L = 0.33m, f = c/lambda. dL ~ 1um.
[with Zach and Dmass] We discussed about the stainless steel pmc design and here are the list of what should be modified.
The drawing can be found, on svn.
PZT can be ordered from www.pi.ws.
The requirements for PZT from (LIGO-xx), are (A) pzt range = 2.7 FSR, for 0 - 375 V, (B) resonant frequency at 10kHz or above.
Zach is using model P-016.10H. The displacement is 15 um (with 1000V), OD = 16mm, ID = 8mm, L = 15mm, resonant frequency = 67kHz. Assuming the pzt is linear, the displacement will be 5.6 um for 375V, this corresponds to 11 FSR for our cavity (FSR = 454 MHz). I don't know if this will cause some locking problem or not, or it might just give us an extra gain in the pmc loop.
If I follow the requirement, the displacment of 5um @ 1000V will be enough for us (model P-016.00H), but the length of the PZT will be 7 mm, and I think we have to fix the drawing accordingly.
Above, an excerpt from the pzt catalog, the full one can be found HERE.
Kriten sent me solidwork part files for the steel pmc. I'm checking all the parts and will decide what material we want to use.
She reported that a ss pmc will have the first body mode at 780 Hz, while an aluminum one will have the first body mode at 1kHz. But we have to take thermal expansion, stiffness into account. here are some material properties
I think the thermal expansion will be a problem, but their thermal expansion coefficients are not that different. I don't know about stiffness of the body. I'll ask someone about this. Otherwise Al might be a better material if we look for higher resonant frequency.
The PMC round trip is ~ 0.32m. The end mirror has ROC = 1.0m. The spotsize is 384 micron. The end mirror has radius ~2.5 mm. The clipping loss will be ~ 1*10^-43 on the curve mirror, and much smaller at the flat mirrors. The number seems very small but I think it is correct.
This is just a simple integration for power of the beam P(a) where a = radius of the mirror (2.5mm). The total loss on all three mirrors per one trip is definitely way below 1ppm.
I made a simulation for PMC body mode, and found out that for Al PMC, the first body mode is 1kHz. And 780 Hz for stainless steel pmc.
November 27, 2012
It is desirable for the first body mode of the PMC to be at or above 1000 Hz in order to provide consistent length for the cavity.
Above you can see the first mode shape of the PMC. The colors represent the displacement - deep blue indicates no motion, while red indicates the greatest amount of motion. The animation of this mode shape shows the PMC spacer rocking transversely on the PMC base. The PMC base does not move at all.
One question that came up is whether ANSYS is importing the geometry file at the correct size. According to the scale on the screen, it is the right size. However, when the material is changed to resemble fused silica, the lowest body mode is 998 Hz, which is about an order of magnitude lower than expected. This indicates some other error, possibly in importing the structure into ANSYS.
/more to come
Above you can see the first mode shape of the PMC. The colors represent the displacement - deep blue indicates no motion, while red indicates the gr
This does not look like a longitudinal mode. Do you have the frequency for the first longitudinal mode(along cavity length)? the first longitudinal mode should look like this ( this model has no fixed boundary condition, just a block in space).
Kristen and Norna came to ATF for impact-hammering of the metal PMC in the gyro setup
The PMC was tested & lowest resonant frequency was 330 Hz; FEA model was adjusted to new frequency of 441 Hz
Results from December 18, 2012
The PMC in 058B W. Bridge was secured with several dog clamps to the laser table. This table is not as stiff as the table in the Modal Lab in Downs, but was thought to be sufficient for this test. Testing was done with the B&K system, using a laser vibrometer for the accelerometer and the small 8206 B&K hammer for excitation. Below is a representation of the axis for this test, to understand where the PMC was excited and measured.
Measurements and excitations were approximately at the center of the corresponding face, as indicated in the image.
Below is a graph of the results of these measurements. You can see that the lowest resonant frequency is at 330 Hz.
Next I will update the ANSYS model to be more accurate, hopefully showing about 330 Hz as the lowest mode.
FEA Model in ANSYS:
Briefly: the previous model reported in the elog was changed by refining the mesh on the slots/cones and the bearings. This would allow for that portion to behave more accurately. The contacts were left as Program-controlled (any other control seemed to overestimate the contact, raising the predicted resonant frequencies).
Below is an image of the lowest mode, at 441 Hz. The arrow indicates the motion - the mode is roughly a transverse flagpole mode.
Now that the model has been made more accurate, steps can be taken to raise the resonant frequency. While the initial goal that was mentioned to me of 1kHz is improbable, there are certainly ways to raise the frequency and damp the modes that are problematic.
We are interested in the longitudinal mode along the Y direction. That is the only one which is problematic for the servo. Please remeasure so that you excite Y and measure Y and then model the first longitudinal mode.
The other modes are interesting, but they're not the main thing we care about.
Longitudinal Mode frequency @ 420 Hz
Monday, January 28, 2013
Tests of resonant frequency specifically for longitudinal mode done on PMC because the piezo will only affect this length. Originally the PMC spacer was constrained in motion by clamps on the PMC base, as shown below.
Data was taken with an oscilloscope and laser vibrometer - the B&K system was not functioning correctly - and the PMC spacer was excited by hitting it with a marker. The PMC was hit near the endcap, along the long axis of the PMC, as shown in the below image.
The results in the time domain are shown below and indicate a resonant frequency of 450 Hz. A next step is to fit this data to a decaying sine wave.
When the above clamps were loosened to allow for more motion, the frequency dropped to closer to 350 Hz.
I asked Comsol for the eigenfrequencies of a simplified PMC body. The outer dimensions are as in the design document (6.89″ × 2.375″ × 2.00″), and the borehole has a uniform diameter of 1.188″ (instead of stepping down to a smaller diameter part-way through the body).
Comsol says the lowest mode for silica is at 8.3 kHz, and for stainless steel the lowest mode is at 7.2 kHz. For this simulation the body is assumed to be completely free; I didn't add 3-point contacts or anything like that.
The lowest longitudinal mode for silica is 16.4 kHz, and for steel is 14.2 kHz.
I use COMSOL to find the first longitudinal mode of a stainless steel PMC, it is about 16 kHz. I'll find an analytical solution and compare them to make sure that the FEA result gives us a reasonable answer or not.
The FEA result in psl:1088 does not show the right body mode of the PMC. The frequency of 440 Hz is from some weird mode as seen from the figure in the entry. Evan checked the body mode of a simplified steel PMC, and I also check independently. Our results agree quite well that the first longitudinal mode is at ~16kHz.
However, this does not answer what we measured in PSL:1097, where the longitudinal motion is around 300Hz. I checked the body frequency of the base blocked and it is even higher than the PMC body modes' frequencies (this should be expected since the base is even bulkier).
Note: I just learned from Zach that the PMC in GYRO setup does not have 3-point support. It just sits on the base block. But this has not given me any clues about the possible modes yet.
I'm writing some background and requirement for the PMC[coming soon]
I compared results between COMSOL and analytical solution. The first longitudinal mode from both results are comparable.
Peter sent me a note from Dennis about PMC longitudinal mode calculation. Dennis mentioned about a book by Young&Roark (here), so I looked it up and see how to estimate body mode frequencies of a simple block/beam. I tried a simple geometry, a 0.1x0.1x0.175 (m) block. According to the book, cf situation 7b, table16.1 page 771, the first longitudinal mode is
f1 = (1.57/2*pi) * sqrt ( E/ rho*L^2), ), rho is the mass density of the material (2202 kg/m^3, for SiO2), E is the Young's modulus (72 GPa), L is the length of the block ( I use L = 0.175/2 because 7b situation is a uniform bar vibrates along its longitudinal axis, with upper end fixed, lower end free. This is similar to a whole beam resonate freely on both end because its center will be fix. Thus, to use the formula for our case, we have to use half length of the beam).
The analytical solution and COMSOL give f1 ~ 16 kHz.
It is very strange that, according to COMSOL simulation, when the cross sectional area of the block is changed to 0.01x0.01 m^2 instead of 0.1x0.1 m^2, the frequency of the longitudinal mode does not change that much (still close to 16kHz. However, from the analytical solution, the frequency should drop by a factor of 10 ( around 165 Hz).
I'm going to think about this a bit more, but at this point, I think my COMSOL model is not correct. Might be some kind of bdy conditions that I'm missing.
I think the analytical formula in terms of rho is going to be (1.57/2*pi) * sqrt(E / rho * L^2), since the Roark formula is (1.57/2*pi) * sqrt(A* E * g / w * L^2) and the weight per unit length is w = m * g / L = rho * A * g. With your values for L, A, E, and rho, this gives f1 = 16 kHz. Since A does not appear in the analytical formula, this also explains why changing the area in the Comsol model doesn't change the frequency.
I compared results between COMSOL and analytical solution. The first longitudinal mode from both results differ by an order of magnitude!!
f1 = (1.57/2*pi) * sqrt ( AE/ rho*L^2), where A is the cross section area (0.1x0.1), rho is the mass density of the material (2202 kg/m^3, for SiO2), E is the Young's modulus (72 GPa), L is the length of the block ( I use L = 0.175/2 because 7b situation is a uniform bar vibrates along its longitudinal axis, with upper end fixed, lower end free. This is similar to a whole beam resonate freely on both end because its center will be fix. Thus, to use the formula for our case, we have to use half length of the beam).
The analytical solution gives f1 = 1.6 kHz ,while COMSOL result is ~ 16 kHz.
good catch! Thanks. Then both analytical and FEA results are the same. So our COMSOL results for PMC should be valid, the first body for a stainless steel PMC, see psl:1131,at 16 kHz is reasonable.
I calculated some requirement for the beam jitter at the output of the PMC. A rough estimate shows that we need the angular stability at the PMC about half nano radian so that the frequency noise of the beam locked to the refcav is less than 10-2 Hz/rtHz.
PMC also reduces beam jitters from the laser, so that the beam alignment to the cavity is kept centered. Since the laser is locked to the reference cavity, any misalignment of the input beam will cause the beam to sense the change of the cavity length.
So vibration that shakes the PMC will change the alignment of the output beam. With stiff material, the seismic induced deformation of the PMC will be reduced.
Eavn is working on COMSOL to find out the angular tilt of the output beam due to PMC sagging. Optimum support points will be determined to minimize beam jitter due to seismic.
I ran another Comsol simulation with a simplified version of the PMC spacer. This time I put fixed constraints on two circular regions on the sides of the PMC near where it was clamped for the ringdown measurement. Comsol says the spacer has a mode where it twists about these clamp points, and the frequency of the mode is 270 Hz.
These are plots of the sagging of the front and back mirrors as a function of the longitudinal positions of the mounting holes (these positions are measured from the back of the PMC). The first plot is a coarse search, and the second is more targeted toward a region of lower sagging.
I generated these plots by taking Tara's Comsol model of the PMC body, assigning fixed displacement to the three mounting holes, and assigning a body load to the PMC body equal to the weight of the steel. Then, I extracted the displacements of four points on the front edge and four points on the back edge of the PMC borehole (these edges are where the faces of the mirrors will make contact with the body). I then took some cross-products with these points in order to get the unit normals that would result when the mirrors are placed against the deformed body. I then compute the angle between the deformed unit normals and the undeformed unit normals to get the sag of the mirrors in radians.
I'm a bit uneasy about how precision is handled in the Comsol/Matlab combination used to generate these plots. The Comsol GUI has no problem reporting displacements all the way down to 10^-24 meters, but anything smaller than 10^-15 meters or so gets truncated to exactly 0 when the results are reported in Matlab. When propagated through to the sagging computation, this means any sagging smaller than 10^-8 radians or so also gets rounded to exactly 0. You can see in the second set of plots that there are large swaths of exactly the same light blue and periwinkle, which seems to indicate a low level of precision in the computation. There's probably some obvious Comsol/Matlab setting that I'm missing, but I haven't been able to find it so far.
Regardless, it appears there is an optimum range of hole placements for the PMC body: 10 cm for the front holes and 3 cm for the back holes, give or take a centimeter or so.
[Rana, Tara, Evan, Eric, Nic]
We are designing a PMC, to do that we should be able to answer some fundamental questions about a PMC.
Why do we want a PMC?
What should we consider in the design of the cavity?
What should we consider in the design of the spacer?
Considerations for PMC design:
RXA: In general, all of these considerations need some sort of quantitative detail. Make a DeBra Matrix so that we can evaluate.
I added an endcap to Tara's steel PMC Comsol model and looked at the eigenmoedes for a 3-point contact. The lowest mode is a rolling mode at 2.0 kHz, followed by other modes at 3.0 kHz and 3.5 kHz. The first longitudinal stretching mode is at 16 kHz. The rectangular part of the spacer for this steel PMC has dimensions 5" x 2.6 " x 2" and a cavity length of 32 cm (first picture).
I also looked at a beefed up version of the spacer, with dimensions 10" x 4.6" x 2" and a cavity length of 78 cm (second picture). The lowest mode is again a rolling mode at 1.4 kHz, followed by other modes at 2.0 kHz, 2.2 kHz, and so on. The first longitudinal stretching mode is at 9.0 kHz. So it looks like if we want a longer cavity, we can almost double two of the spacer dimensions without shifting the resonances down significantly.
If we use a 10" x 4.6" x 2" spacer but go with a 4-mirror bow-tie design (third picture), we can get something closer to a 1.1 m cavity length. Comsol gives a lowest mode at 1.4 kHz, followed by modes at 2.4 kHz, 2.6 kHz, etc.
Some requirements for the PMC:
For intensity filtering. The modulation frequencies for the refcavs is ~ 15-25 MHz, we want the intensity fluctuation at this frequency to be shot noise limited. We have to determine what should be the frequency pole. Intensity noise around 1MHz - 30MHz will be ~ 1/f^2, see the paper by Harb etal, eq1 and fig9, get the paper from psl:1156. Under the assumption that RIN remains constant, at 20MHz the laser will already by shot noise limited (@ 1mW input). laser intensity noise / shot noise ~ 0.16. (laser intensity noise here means intensity noise from spontaneous emission/ pump-source intensity noise/ dipole fluctuation noise/ noise from intra cavity losses, any thing except shot noise)
Thie pole can change with the cavity length and Finesse, [ Finesse = FSR/(2*cavity Pole)] , so our choices for mirror reflectivity, cavity length will affect this number as well. So for a fixed set of mirrors (fixed finesse), longer perimeter means lower cavity pole, but the cavity will be more susceptible to acoustic coupling.
==First longitudinal body mode==
It should be at high frequency ( for high UGF servo). The shorter the length, the higher the frequency. See PSL:1134.
For a stable cavity, g factor has to be between 0 and 1. Another reason: We should choose g-factor such that HOMs do not coincide with other cavity axial modes (FSR apart). For a ring cavity with 2 curve mirror R1,and R2, g = (1- p/R1) x (1 - p/R2) where p is the round trip length. (For 3-mirror cavity, g = (1 - p/(R))^2 . See HOM calculation.
we want a solid, bulk shape PMC, not thin long one. This will make the PMC less susceptible to acoustic noise.
==Higher order mode suppression==
Other transverse modes will be suppressed by a factor of (1-r)^2 / (1 +r^2 -2rcos(2*pi* dfmn/ FSR) where dfmn is the gouy phase shift of m+n mode, r =r1*r2*r3.. (reflectivity of each mirror in the cavity) see evan's note. Transverse modes of the output of the NPRO can be found by scanning the PMC and measure the transmitted beam. Other modes beside TEM00, will be reflected back from the refcav and incident on the RFPD. This will cause the mode mismatch and increase shot noise level. Usually, higher r (higher Finesse), will suppress more HOMs.
==Build up power==:
= Pin x Finesse/ pi. CVI mirrorsfor high damage threshold power have maximum power for cw around 10MW/cm2. So I use this number as an upper limit for the power threshold. Assuming the power input is ~ 30 mW, average spotsize is 350 um. This gives ~ 8W/cm2. So Finesse can be up to ~ 3e6. (10 MW/cm2 > (Finesse/pi) x 8 W/cm2) .
Let's see some of the designs that are available. Then we can decide which one we should modify to suit our requirement.
For a 3-mirror cavity with a single curved mirror, the g-factor is (1-p/R)^2; there is no factor of 2 in the denominator because for a ring cavity the overall cavity length is equal to the round-trip length.
Also, I think we should shoot for a transmission of at least 90%. If this is going to be for general lab use, then there will probably be situations where people want a good power throughput. The input power might be as high as 2 W if used, e.g., at the 40m with one of those Innolight Mephistos.
I computed the occurrence of higher-order modes up to order m + n = 20 as a function of g factor for a ring cavity.
In the first set of plots of plots, I've fixed the cavity half-length L and chosen several values of modulation frequency fPDH. In the second set of plots, I've fixed fPDH and chosen several values of L. Green is the carrier, red is the lower sideband, and blue is the upper sideband. The takeaway messages from these plots are that
So I think we should go for as low a crystal frequency as possible that is consistent with having shot-noise limited intensity and a high loop speed. I know the number 20 MHz has been thrown around as the lowest reasonable PDH frequency, but I don't understand quantitatively why this is.
This and some other PMC design issues are now in the SVN trunk under docs/modecleaner_design/
Considerations for PMC design is corrected and updated
I can't take it anymore: what the $#@& is a Debra matrix??