Attached is a notebook to calculate the thermal lensing in a michelson interferometer.

Update to 11481. 

Improved how the cavity bandwidth is calculated and cleaned everything up. Attached is the python notebook used to calculate this. Data can be found in \Nextcloud\GQuEST\B102\Output Filter Cavity\Cavity Piezo Calculations\cav_piezo_data\ . The associated spectra is also in this directory. These plots now show the piezo TF for the PA44M3KW glued to a mirror and in daniels compressed design, PA44LEW (small thorlabs piezo), and cylindrical Noliac piezo.

cavity_piezo_transfer_function_setup.py and lock_and_TF.py will create the required moku configurations in 11481 and 11449 respectively. Note that until this is resolved, you must put output3 in the correct spot manually, as well as configure output3 settings manually.

Found a glitch in the moku I think people should be aware of.

I don't know what causes it but there have been multiple instances where the moku thinks its outputing the correct voltage but there is no signal coming out of the output port. Ex: While measuring several transfer functions today, I measured a TF with and without the high voltage amplifier (same set up as 11481). I took a TF using the high voltage amplifier, unpluged both BNCs from the HVA, connected them in order to bypass it, and went back to take the transfer function. Functionally nothing should change when doing this but when I attempted it, the transfer function was garbage. I eventually realized that even though it said there was an excitation signal on, nothing was actually coming out of the output (confirmed with a volt meter and oscilliscope). You even see the proper voltage at the closest test point to the output, but still no signal actually coming out of the output. Closing the moku app and reopening it did not fix it. Exiting out of the multi-instrument mode, opening up any random function in single instrument mode (I opened the oscilliscope), and opening MIM again fixed the issue.

Just putting this here in case people encounter this bug. Or maybe if anyone has an explanation for this please feel free to comment below. This may be worth a moku instruments forum post as well.

[Torrey, Ian, Lee, Daniel]

We cleaned the end cubes (which are in B102) with a clean room wipe and isopropanol since I there was some residual grime from my cleaning in B150 (or it was introduced after I cleaned, maybe during the move). We then took the top of the end cubes off and looked at the inside to look for the internal hole pattern on the bottom flange. In the SolidWorks model, they had 4 #10-32 tapped holes. Unfortunately, one cube has no holes and the other cubes has just one hole (see attached photos). One of the bend cubes (in B150) has a flange taken off and covered in foil, so I went to check the inside of that cube. I couldn't see the bottom since a mirror base was still in the cube.

In order to hold the mounts for the Laser Filter Cavity mirrors, we have 3 options. The first is to buy a new bottom flange with tapped holes. I couldn't find this product, but I did ask ~6 companies if it exists.

The second option is to machine holes ourselves. I think I could do this myself or we could pay someone to do it.

The third option is a more creative solution from Lee to hold a 6" diameter circular breadboard with friction. I don't love this idea because it's inevitably going to come lose.

I have revived my old finesse model of the four filter cavities to try to model the transfer function of power through the cavities.

cavity_modes.pdf (and the .png version) shows a single cavity with a laser going into it at 1550 nm while being phase modulated at 15.6 MHz. the cavity length is offset by 15.6 MHz. each peak is calculated individually and then combined where each peak represents an n,m mode. Since the Gouy phase of the cavity is about 120 degrees, only the third modes should get through. The heights of the modes have been given a per mode attenuation of  (1 / (m + n + 1)**2) and 0.001 times that for every non third mode. These values are arbitrary and will be updated in the future. This single cavity goes up to 5 TEM modes.

Work on this modeling is ongoing.

I attempted to improve the quality of lock on the output filter cavity using the piezo by simultaneously agressively notching 100dB at 3.5kHz - 5kHz using the digital filter box and also locking the cavity while bypassing the HV amplifier to the piezo. This can be done by scanning on the laser frequency but still engaging the slow lock. No visible improvements were made. The transfer functions showed a much more linear regime compared to those posted in 11449, but the UGF was still quite low (<1kHz). I can post the TFs if anyones curious but don't have them as of writing this.

In order to see whether some of the resonances in the output filter cavities are due to a mechanical self resonance in the long axis, here is a simple calculation to see whether it's possible:

\[\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{EA/L}{\rho AL}} = \frac{1}{L} \sqrt{\frac{E}{\rho}} \]

\[f = \frac{\omega}{2\pi} = \frac{1}{2\pi L} \sqrt{\frac{E}{\rho}} = \frac{1}{2\pi \ 0.5 m} \sqrt{\frac{69 GPa}{2700 kg/m^3}} = 1600 Hz\]

Here, \omega is the angular frequency, k is the spring constant, m is the mass, E is Young's modulus, A is the cross sectional area, L is the length, \rho is the density, and f is the frequency.

It appears that the worst resonances in the cavity spectra are closer to 4 kHz, so either this is too simple a model or that resonance is due to something else.

[Ian, Torrey, Daniel]

We moved the 2 end cubes (not bend cubes) from B150 to B102. Since the end cubes bases have screws sticking out, they are resting on my custom base plates flipped upsidedown. The next steps are to take the bottoms off, put on my custom base, clean the insides, and move them to their final location. We will then start assembling the rest of the vacuum equipment for the LFC.

[Lee, Daniel]

Lee has updated the model of bulk acoustic wave noise in GWINC (I believe to account more accurately for shear modes), but the results seem to indicate more bulk acoustic wave noise than the Holometer measured or I have modeled with COMSOL. Attached are 4 figures. The first is the measured Holometer data (fig 12.) The second is a plot from the GWINC model of the Holometer. The differences between the GQuEST model and Holometer model are arm length, laser wavelength, arm power, mirror size, mirror material, and mirror spot size. The Holometer end mirrors were modeled as 0.5" thick, 1" radius and the beamsplitter as 0.5" thick, 1.5" radius. Both were made from fused silica at 294 K. The beam size on the end mirrors is 5 mm. The calculated noise in GWINC is a maybe bit higher in this model, but it is fairly close.

I also did 2 COMSOL simulations with a 2 mm thick, 12 mm side length mirror with a beam waist of 2 mm with 1e6 Q Silicon. Both simulations had 14 mesh layers in the beam axis. For the first simulation, there were 42 mesh layers on each transverse side. For the second simulation, there were 84 mesh layers on each transverse side (making the mesh elements cubes). The minimum value of the graphs is nearly identical. Comparing the peaks is less meaningful because of aliasing due to the course frequency sampling of 100 kHz. The fact that the minima are so close implies that previous COMSOL simulations, which showed minimal noise from shear modes outside of certain peaks, were not limited by the transverse mesh density. Lee points out that the Krylov-space inversion solver might drop modes since it is a reduced-order solver. I am therefore running an eigenmode solver right now.

[Torrey, Ian, Daniel, Sander, Lee]

Introduction

We have been struggling for a while to figure out a configuration for a stable cavity lock using only the piezo mirrors. In an effort to troubleshoot, we want to take a transfer function of just the piezo (h_p). We can't do this directly while the cavity is aligned and locked so we have to be a little clever about it. Below is a how we go about doing this.

Set up and methods

In the moku multi-instrument mode we can set up a combination of transfer functions to achieve the piezo transfer function. As seen in this, we lock the cavity with the laser. The digital filter box is used as a summer. Output 1 is used to control the DC modulation port of the ULN15TK laser and Output 2 is used for the cavity piezo. Output 3 is the 50 MHz signal used for demod. Input 1 is the newport 1811 high bandwidth PD in reflection of the cavity. Input 2 is the GE lower bandwidth PD in transmission of the cavity. Two transfer functions are set up in this configuration:

1. Fast controller (laser frequency controller) divided by excitation to laser frequency.
2. Fast controller (laser frequency controller) divided by excitation to the laser piezo.

We can reduce this diagram to a clearer picture of the control systems in play using a Signal Flow Graph (alternate picture to the more common block diagrams). This reduction can be seen as signal_flow.pdf. From this diagram we can see the open loop gain G can be written as \[G = H_1*F*H_2*\alpha.\] We can subsequently reduce our individual transfer functions into smaller diagrams, seen in reduction_A.pdf and reduction_B.pdf. From these it is a little easier to write down the equations for our transfer functions in terms of G and individual components. From reduction_A.pdf we see that,

\[T_p = V_{LF} \] and \[V_{VL} =  A + G V_{LF}.\] This means \[\frac{T_p}{A} = \frac{1}{1-G}.\] For now we will call this measurement one, or \[M_1 = \frac{T_p}{A} = \frac{1}{1-G}.\] Similarly from reduction_B.pdf,

\[T_p = \frac{H_p}{H_2} \frac{G}{1-G} B.\] Again lets call this measurement two, so \[M_2 = \frac{T_p}{B} = \frac{H_p}{H_2} \frac{G}{1-G}.\]

Take the ratio of M2 and M1:

\[\frac{M_2}{M_1} = G \frac{H_P}{H_2}\]. Substitute the above expression for G and solve for H₂,

\[H_p = \frac{M_2}{M_1} * \frac{1}{H_1 F \alpha}\].

This is a nice form as every variable can be obtained experimentally or is known already.

1. M1 and M2 are our measured transfered functions.
2. F is the filter used in the digital filter box which at the time of measurement 0 db for all frequencies so trivially F=1.
3. H_1 is the shape of the fast controller of the laser lock box. This is slightly annoying to recreate as there is no way to download the shape of the loop directly from the fast controller of the laser lock box. I get around this by recreating it in python using scipy. Most scipy functions seem to want the order of the filter and cutoff frequency; cutoff frequency being a problem as the controller shape is defined by the UGF and not cutoff frequency (ex: fast_controller_example.png). A conversion can be made but slightly annoying. Jeff is currently very close to being done with a script that will take these 6 inputs and recreate the shape of the controller in python.
4. Alpha is the conversion from Hz into Volts from the laser frequency to the PDH lock. This is found by the ratio of the amplitude of the error signal in the PDH lock and the cavity bandwidth. The error_signal amplitude can be found trivially by putting one of the moku test points at the error signal. The cavity bandwidth can be derived or measured in a few different ways. The method we went with was to scan the laser frequency to see 0,0 peaks in the TRANS PD. This peak will have some width, from which the FWHM in time (delta t). In order to convert this to a frequency we do the following conversion:

\[\mathrm{BW(Hz)}  = \Delta t * \frac{2 A}{T} * \frac{2 mA}{V} \frac{5 pm}{20 mA} \frac{c}{\lambda^2}\]

• 2A/T converts the width of the peak from time to volts.
• 2 mA}/V is the conversion of the DC modulation quoted on the thorlabs website from input voltage to pump current modulation. 
• 5 pm/20 mA is how much the laser wavelength changes given some amount of change in pump current. Note that this is only true in the linear regime and is not true everywhere. For small changes, as is the case here, it is true.
• c/\lambda^2is the derivative df/d lambda of f=c/lambda to convert from meters to frequency.

From the data collected this yields approximately 307 kHz bandwidth for the cavity with the low reflectivity mirrors (R ~~ 99%).

Results

The final result of \[H_p = \frac{M_2}{M_1} * \frac{1}{H_1 F \alpha}\] yields result.png. I think there is a scaling factor off somewhere but the shape makes sense. Also something to look into is the low quality of the data at low frequencies, of which Lee has given me ideas on how to correct this. We cannot simply drive these things harder.

1) The LaserLockBox instrument is currently not fully configurable in Multi-Instument mode, the API does not allow you to configure the modulation output.  LaserLockBox in MultiInstrument

2) If you set filters with the API and then connect to the Moku via iPad, it resets the filters to default. Switching from API to GUI

I tried to remove the residue on the end cubes that was left over from the tape. I used isopropanol last week and acetone this week, but neither worked. I therefore covered the residue in Kapton so that the residue does not contaminate the lab space. See attached photo.

I removed the 2" long, 5/16-24 set screws from the 10" to 8" Flange Size Zero Length Reducer. The set screws do not have a hex drive (perhaps they are more accurately called studs), and ~16 of the 20 required pliers to remove. This caused a bit of silver plating to come off, but it was easily removed with an air duster. The ends of the screws furthest away from the flange were quite dirty and/or lost their silver coating. See attached photo.

I then used isopropanol and Kim wipes to clean the non vacuum part of the reducer. Some gunk made it into the part where the copper gasket lies outside the knife edge. I cleaned this to the best of my abilities. See attached photo.

Torrey and Daniel October 3, 2023

Daniel 3D printed a cover for the ULN15TK Seeder Laser that makes it very difficult to accidentally turn on or off the seeder. Torrey printed and attached warning labels conveying that one should not turn the seeder off so that the amplifier isn't damaged. If we are really paranoid, Daniel could 3D print a cover with a tighter fit.