text:
40m QIL Cryo_Lab CTN SUS_Lab TCS_Lab OMC_Lab CRIME_Lab FEA ENG_Labs OptContFac Mariner WBEEShop
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ID Date Author Type Category Subject
2740   Mon Mar 28 18:00:43 2022 KojiSafetyCleanlinessLab flooding

2741   Tue Mar 29 09:13:20 2022 AidanSafetyCleanlinessLab flooding

Facilities placed a blower and dehumidifier in B265B. I checked the airflow and the air around the tables is comparitively still. The North table is covered and the South table is over pressurized by HEPA filters, so there should be little risk of dust being stirred up.

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2742   Wed Mar 30 10:46:49 2022 RadhikaSafetyCleanlinessLab flooding

This morning, facilities removed all the porous ceiling panels that had been soaked/damaged by water (in B265B: above WS1 and WS2, see Attachments 1+2; In B265A, see Attachment 3). Specifically in B265A, an enclosure was created (Attachment 4) and a dehumidifier was placed inside. All monitors/equipment underneath the panels were thoroughly covered, and the floors were swept up afterward.

No work was done above the North table in the QIL. I asked about it and facilities said they would look into it, but it wasn't on the schedule for today. A member of facilities also pointed out that the sink in the QIL was running black liquid (Attachment 5). It looks like soil/dirt entered the water pipes? This seemed to also be outside of their scope for today.

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2743   Wed Mar 30 16:25:06 2022 KojiSafetyCleanlinessLab flooding

Muddy Waters is not new, but if the facility can fix it we'd take it.

2744   Fri Apr 1 13:53:02 2022 RadhikaSafetyCleanlinessLab flooding

Facilities will be returning on Monday 4/4 between 8-9 AM to remove all ceiling panels above the workstations in B265B (QIL). Replacement of the panels is not yet scheduled, but in the meantime the open ceiling will be covered and the workstations will still be accessible.

2745   Mon Apr 4 17:05:20 2022 RadhikaSafetyCleanlinessLab flooding

Pictures attached. WS1 and WS2 have been turned back on, since the replacement for the ceiling panels will not arrive for another few weeks according to Facilities.

 Quote: Facilities will be returning on Monday 4/4 between 8-9 AM to remove all ceiling panels above the workstations in B265B (QIL). Replacement of the panels is not yet scheduled, but in the meantime the open ceiling will be covered and the workstations will still be accessible.

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2011   Tue Jan 5 14:48:08 2016 ranaSafetyLab InfrastructureHVACheating and cooling

As noticed by Kate a few times last year, the north side of the lab has hot air comiing out of the HEPA vents and the south side has cold air. This seems to be a problem with the setpoints for the sensors or the hot water actuators.

Let's remember to call physical plant after the current roof leaking situation settles down.

2140   Mon Jul 17 22:19:36 2017 DhruvaSummaryWOPOLast Week in WOPO

1. I have achieved 50% fibre coupling efficiency for the 532nm beam. I have attached the simulation of the estimated solution in ABCD and the profiles after the first two lenses.

Solution - 50cm lens at 73.4cm, 35cm lens at 105.4cm 17.5cm at 150.2 cm.

The beam profiles approximately agree wth the given solution.

2. I measured the attenuation of 1064nm light in the 532 nm single mode patch cable. These are the reaadings I took of input 1064nm power vs output power.

5mW   -0.07uW

10mW - 0.10uW

20mW- 0.12uW

30mW - 0.14uW

40mW - 0.15uW

50mW - 0.26uW

I will later calculate how this would lead to a loss in measured squeezing.

3. While trying to characterise the fibre beam splitter, I found that for the for an input power of 50mW in the 1064nm fibre, there was a total of 22mW coming out of the output ports. On checking the 1064m fibre, we found that the mode shape coming out of the fibre wasn't clean. Inspection through a fibre microscope showed no damage to the fibre ends. We have changed the fibre, but lost the 75% mode-matching in the process. I'm getting a maximum of 50% with the new fibre.

TODO -

Noise analysis of photodetectors in the ATF Lab (I have collected data from the FFT network analyser. I will plot it and put it up on the elog)

Make the Homodyne Setup.

See Squeezing.

Measure Squeezing.

Calculate Squeezing Losses.

Convince people that these losses can be reduced.

Successfully finish SURF project

Attachment 1: 532_MM.pdf
Attachment 2: 532_f1.pdf
Attachment 3: 532_f2.pdf
Attachment 4: sm1064att.pdf
Attachment 5: sm1064out.pdf
2183   Sun Nov 5 15:12:06 2017 ranaSummaryComputingdiaggui on Mac laptops

Its now possible to install DTT (aka diaggui) ont your personal laptop using macports. This would allow you to access data from the LIGO sites, the 40m NDS, as well as the W. Bridge sub-basement labs (once Jon and I set up a way to move the data to the CIT cluster and NDS it).

as usual with MacPorts, YMMV, but this way works for me on more than one machine:

1. sudo port install gds (this starts installing a huge pile of dependencies and may take a long while - you may need a charger)
2. It will install root5 and then while getting the gds package, it displays the error:
3. Error: Failed to archivefetch gds: root5 must be installed with +python27.
4. This is because the MacPorts package for GDS is misconfigured. So at this point you do:
5. sudo port uninstall root5
6. sudo port install root5 +cocoa+gcc6+graphviz+gsl+http+minuit2+opengl+roofit+soversion+ssl+tmva+xml+python27
7. sudo port install gds

This works on my old laptop (El Capitan), but so far not on Sierra.

After that you can run diaggui to open up your old .xml files, look back at old data, etc.

Draft   Wed Jan 17 10:41:45 2018 awadeSummaryWOPOMaking a transformer coupled photodiodes

[DRAFT]

These are some some summary notes on bandwidth and noise budget of transformer coupled photodiode detectors and and example made for the WOPO experiment.

## PD transformer coupling: what and why?

A lesser known strategy in photodetector design is to transformer couple the photocurrent generated by a photodiode before applying amplification.  It offers a potential benefit in terms of improved input current signal-to-noise at the amplifier input. Monolithic style RF amplifiers only come in a couple of fixed input impedances, determined mostly by mass market commercial imperatives. Transformer coupling their inputs allow us to adapt common 50 Ω, low-noise figure, wide bandwidth RF amplifiers for use in TransImpedance Amplifiers (TIAs), boosting their current signal to noise at a tradeoff of bandwidth.  For the addition of this simple passive element we get an AC coupled output with added overall gain that adds (almost) no additional noise.

We don't, however, get a free lunch. There are tradeoffs in bandwidth and dirt effects of transformers that need to be carefully considered against the requirements. We also miss out on a DC coupled output that is useful for diagnostics and initial alignment.

A schematic of the basic circuit is shown below:

The photodiode is reverse biased and connected to the primary coil of a low resistance RF transformer of coil ratio* N:1, where N is the integer ratio of extra turns on the primary coil.  With less turns on the output secondary coil the voltage is stepped down but the current is scaled by a factor of N.  This means that for a fixed current noise at the input of a given amplifier we are able to increase the relative size of the input current: this means better signal to noise.

### Shot noise

Shot nosie is given by

$i_\textrm{sn} = \sqrt{2eI_p}$

where e is the charge per electron (1.602•10-19 C) and Ip is the photocurrent. Ip is given by

$\inline I_p = \eta\frac{eP}{h\nu} = \eta\frac{eP\lambda}{hc}$

where eta is the quantum efficiency of the diode (from 0 to 1.0), P is power incident, h is plank constant (6.626•10-34 J.s) and c is the speed of light. At some choice of incident laser power shot noise will become the dominate noise source.

Vendors tend to quote responsivity of diodes which is current per unit laser power

$\inline \mathcal{R} = \eta\frac{e\lambda}{hc}$

for 1064 nm light the maximum is 0.858 A/W and responsivity is scaled by the quantum efficiency of the diode.

### Input noise of the amplifier

Op amp first stage amplifiers have quoted voltage and current noise. Typically we choose FET input op amps for their lower current noise: these can be order 0.8 fA/rtHz and 5 nV/rtHz. For monolythic RF amplifiers there is a terminating impedance at the input. Thermal noise of this termination sets the limiting signal to noise.

Thermal noise (Johnson–Nyquist) associated with a 50 Ω termination resistance at the input of the RF amplifier given by

$i_\textrm{th} = \sqrt{\frac{4k_B T_n}{R}}$

where kB is Boltzmann's constant (1.38•10-23 J/K), Tn is the amplifier noise temperature (Ideal ~ 293K) and R (=50Ω) is the input resistance of the amplifier. For 50 Ω termination at room temperature, current noise is 18 pA/rtHz. Referenced to the input at the photodiode, the transformer reduces this apparent noise by a factor of N (the number of coils on the primary side).

$i_\textrm{th, in ref.} = \frac{1}{N}\sqrt{\frac{4k_B T_n}{R}}$

Also, there is a voltage induced by thermal noise at the input termination.  This goes as $\inline \sqrt{4k_B T_n R}$ and these fluctations in voltage accross the diode capactitace induce a $\inline \dpi{100} C\mathrm dV/\mathrm dt$  convertion to current noise, where C is the inherent capacitance of the PD.  At the diode the voltage is stepped up going from the amplifier to the diode. In the frequency domain this means the input refered current noise induced from amplifier voltage noise is given by

$\inline \dpi{100} i_\textrm{cv} = 2\pi fCN\sqrt{4k_B T_n R}$.

Thus, there is some tradeoff between current and voltage noise. Ignoring dark current of the PD, the total noise is

$\inline i_n = \sqrt{i^2_\textrm{sn}+(i_\textrm{th}/N)^2+ (i_\textrm{cv}N)^2}$.

There is another bonus here: reduction in the equivalent capacitance of the PD as seen by the TIA. Voltage noise at the input pins of the first stage op amp can be converted to equivalent current noise by the intrinsic capacitance of the photodiode.  Applying a voltage across a capacitive junction induces a current

$\inline \dpi{100} i_\textrm{in} = C_\textrm{in} \frac{\textrm d V_\textrm{in}}{\textrm d t}$

that, in the Fourier domain, is

$\dpi{100} \tilde{i}_\textrm{input}(f) = 2\pi e_n C_\textrm{in} f$

where $\inline \dpi{100} e_n$ is the amplifier voltage noise density [V/sqrtHz]. The effective step up of the amplifier voltage noise going back to the PD input g is N for a N:1 transformer the source impedance apprent at the amplifier input stage for a N:1 is N

$\tilde{i}_\textrm{input}(f) = 2\pi e_n NC_\textrm{in} f$

The migration of charge carriers to the two sides of a photodiode N-P region means that natural l capacitor is formed across the active region of the photodiode

## Noise Budget

• Effective input referred noise from apparent transformed impedance
• Insertion loss of typical and best in class transformers
• Hysteresis of ferromagnetic core devices
• Inherent thermal noise of transformer coils
• Bandwidth limitations of transformers themselves
• Bandwidth relating to impedance ratio of transformer configuation (N:1 turns)
• Amplifier stage noise

Optimization given desired noise profile and bandwidth

Dynamic range estimate

## WOPO Requirements

[xxx] Chaoyong Chen, Shaoping Shi, and Yaohui Zheng, "Low-noise, transformer-coupled resonant photodetector for squeezed state generation", Review of Scientific Instruments,  Volume 88, Issue 10 (2017). https://doi.org/10.1063/1.5004418

[xxx] Shreyas Potnis and Amar C. Vutha, "Broadband low-noise photodetector for Pound-Drever-Hall laser stabilization", Review of Scientific Instruments 87, 076104 (2016) https://doi.org/10.1063/1.4960088 https://arxiv.org/pdf/1607.01816.pdf

[xxx] Malcolm B. Gray, Daniel A. Shaddock, Charles C. Harb and Hans-A. Bachor, "Photodetector designs for low-noise, broadband, and high-power applications", Review of Scientific Instruments 69, 3755 (1998); https://doi.org/10.1063/1.1149175

Some typical 'chip' sized RF transformers: https://www.coilcraft.com/pdfs/pwb.pdf Range is limited, you might want to wind your own but this is challenging

* Note: in this post I refer to the primary coil as the one with more turns connected to the input side.  Some of the literature makes a habit of defining the secondary coil as the larger number of terms (the assumption is that you always want to step up voltage) and refer to hooking up the transformer with secondary on the input.

Attachment 1: BasicSchematic_TranCoupledPD.pdf
2199   Wed May 23 20:38:10 2018 ranaSummaryWOPOTransformer coupled photodiodes

What is going on with this fossilized squeezing experiment? I reckon that there is a loss diagram out there somewhere. Le'ts see some ellipses and arches:

https://youtu.be/WyF8RHM1OCg

2202   Tue Jun 5 13:59:43 2018 EricSummary LabJack Summary

We used the setup specified in the previous eLog post. The design inputs a digital signal which is stored as a control signal to compare against. Next, the digital signal is passed through the DAC and comes out as an Analog signal through terminal DAC0. Since only the DAC or the ADC can run at one time, the DAC is then paused until the ADC converts the signal back to digital, at which point the ADC is paused and the DAC resumes functioning. Theoretically, this conversion should be happening at 100 Hz, and in practice, this number will be very close to 100 Hz. See Figure 1 for the setup diagram.

We passed two signals through the LabJack – 1 Hz sine wave and 10 Hz sine wave. These signals were converted to an analog signal through the DAC and then converted back to a digital signal using the ADC and the return signal was saved. We then ran Rayleigh statistic tests on this obtained data, which measured the continuous probability of a circular distribution for random variables. In GWpy, the Rayleigh statistic is a calculation of the coefficient of variation of the power spectral density (PSD) of a given set of data. It is used to measure the ‘Gaussianity’ of those data, where a value of 1 indicates Gaussian behaviour, less than 1 indicates coherent variations, and greater than 1 indicates incoherent variation. It is essentially a p-value for a frequency range where a Rayleigh value closer to zero means that we can reject the null hypothesis that the signal is not of the corresponding frequency. These methods were obtained from GWpy (https://gwpy.github.io/docs/stable/examples/spectrogram/rayleigh.html  and https://gwpy.github.io/docs/stable/examples/frequencyseries/rayleigh.html#gwpy-example-frequencyseries-rayleigh). This produced both Rayleigh spectrograms and Rayleigh spectrums. We also produced Rayleigh spectrograms and spectrums for the digital signal inputted into the LabJack to compare against the data read back from the LabJack. The attached 8 plots are these spectrograms and spectrums – 4 for each frequency containing 2 expected and 2 observed spectrograms and spectrums. We can see that the LabJack is mostly consistent with converting the data. We see clear signals in the observed plots that correspond to 1 Hz and 10 Hz. We can see in the spectrograms that the LabJack occasionally drops data, but very rarely – only around 3 times for sampling sessions of around an hour. The data drops no more than a few milliseconds each time (usually only one packet).

Attachment 1: Figure1.pdf
Attachment 2: 1hz_spectrograph_observed.pdf
Attachment 3: 1hz_spectrograph_expected.pdf
Attachment 4: 1hz_spectrogram_observed.png
Attachment 5: 1hz_spectrogram_expected.png
Attachment 6: 10hz_spectrograph_observed.pdf
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2211   Wed Jul 11 16:30:40 2018 Vinny W.Summary2micronLasersAcoustic and Thermal Sensitivity

I found some relevant work done on the discussion of acoustic and thermal sensitivity within a optic fiber. For reference, it's important to have an idea of the geometrics of the fiber. The core, cladding, and coating are 11 +/- 1 um , 125 +/- 1 um, and 245 +/- 10um in diameter, respectively. Though the cladding is pure silica, the coating is Ge-doped. ( http://www.thorlabs.com/drawings/65f0b20de1051938-7503B425-91B4-7335-B52672A2FD1F6447/SM2000-SpecSheet.pdf ). Additionally, the fiber's sensitivity to some pressure depends on its characteristic elastic coefficients ( Young's Modulus, $E$, and Poisson's Ratio, $\sigma$) and Pockels coefficients, $P_{12}$ and $P_{44}$ [1]. The elastic coefficients for fused silica can be found online but are also referenced below.

Acoustic Sensitivity:
To have an idea of the approximate sensitivity we could expect, we can consider some preform silica cylinder that is undergoing a uniform pressure. Many of the references I found measured that pressure from variations of the relative phase change between two interferometer arms- one static and the other undergoing a pressure difference relative to it. The optical phase retardation per unit of pressure (in dyne/cm^2) can be expressed as:

$\frac{\Delta\phi}{\phi} = \frac{(1-2\sigma)}{E}[\frac{n^2}{2}(3P_{12}+2P_{44})-1]$

This comes out to be approximately  $-1.23*10^{-14}{dyn}/{cm^2}$ .

Obviously there's more to the picture than just that. We need to consider the differences in refractives indices and and elastic coefficients between the different materials present in our optic fiber, as well as account for the radial and axial displacements in the "cylinder" caused by some pressure. The approach in reference [2] consideres a two layer cylinder, and assumes that the center is of homogenous material much like the example above. We can place the optic fiber in a cylindrical coordinate system as shown below:

(Insert crudely made cylinder coordinate system here)

Following reference [2], we'll assume that the axial stress at the very ends of the fiber is zero. ( $\sigma _{22} = 0$ at $z = \pm L$, where $2L$ is the full length of the fiber). This is referred to as the radial model, a form of boundary condition. Since axial symmetry is also assumed, the stresses and strains will be functions of $r$ and $z$. For a single material solid cylinder, the solutions for the differential equations for the axial/radial displacements are expressed as products the trigonometric and modified Bessel functions. When we consider a multilayered cylinder, the general solution will be the the series expansion of those products, and their coefficients are determined by the boundary conditions.

It was shown both experimentally and theoretically that a 2D model of the above scenario (i.e. a plane strain ) gives nearly identical results as the 3D model, and involves much less calculation power. The reference goes in-depth at how the following equation was is derived, but to keep this concise, the average induced fractional phase change can be expressed as:

$\frac{\Delta \phi}{\phi}= e_z-\frac{1}{2}n^2[2e_r(p_{11}-p_{44})+e_z(p_{11}-2p_{44})]$

where $e_z,e_r$ are the axial and radial strain, respectively. When the strains are a function of position along the axis of the fiber, we would need to average of its length:

$\frac{\Delta \phi}{\phi} = \frac{1}{2L}[\int_{-L}^{+L}e_zdz-\frac{n^2}{2}[2(p_{11}-p_{44})\int_{-L}^{+L}e_rdz+(p_{11}-2p_{44}\int_{-L}^{+L}e_zdz)]]$

[Work in progress. -Vinny (7/19/18)]

2264   Wed Nov 28 17:14:11 2018 awadeSummaryTMTFNotes on optimal length mismatch for fiber delay line MZ

I just came across some notes I made about optimal fiber length for the TMTF fiber stabilization scheme.  I'm putting them here for future reference. The idea is to construct a fiber Mach-Zehnder interferometer with one arm much long than the other.  The arm length mismatch gives us a frequency discriminator to which we can lock our laser to.  The question is what length is best, given expected losses, for maximum signal-to-noise?

## Choosing the right arm length mismatch

Larger path length mismatch always increases the slope of the ideal MZ interferometer in frequency space.  However, at some point the propagation losses in real fiber reduce the fringe visibility, degrading the signal slope and, therefore, signal-to-noise.

Vinny made some initial calculations that indicated that optimal path length mismatch for SM2000 fiber MZ interferometer was 117 m and in other estimates on order of kilometers. I redid these calculations myself to see if there was an nice analytic expression for the long arm length given a known loss.  It turns out there is:

$\Delta L = \frac{1}{\alpha}$

Where alpha is the loss per unit length that goes as $\dpi{100} \textrm{Loss} = \exp(-\alpha L)$.  For SM2000 fiber, which is the standard type that Thorlabs ship, the losses are estimated at 37.5 dB/km @ 2000 nm.  This makes an alpha of 8.6e-3/m and the estimated optimal fiber mismatch length of 116 m. This agrees with the last number Vinny publish on the elog (ATF:2207).

I've attached a Mathematica notebook (as a pdf and .nb file) so that we have a common point of reference for future questions of design.  This way we don't have to redo it every time. In the book there is an estimate of the expected frequency equivalent shot noise if we operate the 2 µm MZ with 100 µW of power.  In there I find that, at the shot noise limit, we are looking at order 50 mHz/rtHz noise floor.  This is an order of magnitude worse that the next best standard 1064 nm fiber.

These calculations were only considering shot noise and not fiber acoustic noise or PD dark noise.  If we up the input power to 1 mW then we should expect a frequency equivalent shot noise of 15 mHz/rtHz.

## Detection at the output

During construction of this experiment for the SURF project Vinny built a single TransImpedance Amplifier (TIA) for each MZ: in loop and out of loop.  We then subtracted a DC offset with a stable voltage reference to place ourselves roughly at the half fringe.  This was ok as a quick dirty first pass. However, the ANU people pointed me to a better way.  We make a subtraction measurement between the two outputs of the MZ.  This doubles the signal slope and to first order is immune to intensity fluctuations of the laser.

We had trouble locking because when we feedback to the laser current to control frequency the power is also affected.  By making a ballanced measurement this way would make us immune to power changes in the laser and always keep us at the half fringe.

At this stage the experiment is turned off and unmanned.  When we get another undergraduate we should get them onto make a pair of matched low noise TIA to make a ballanced readout scheme from the MZ.  We need to also look into getting some more PD to be able to also do the out of loop detection at the same time.

Attachment 1: 2umMissmatchWorkings_v1.pdf
Attachment 2: 2umMissmatchWorkings_v1.nb.zip
2265   Thu Nov 29 16:52:50 2018 RahulSummary Cryo-vacuum chamber

I would like to give a brief update on the ongoing effort for the assembly of the cry-vacuum chamber at the QIL. Gabriele has been kind enough to spare some time to help me with safe crane operation, considering the chamber and plates are 70 kg each.

At first, I prepared an aluminum spacer (made out of bosch extrusion) which is 90 mm in height. The idea is to lift the bottom plate (which has 16 through holes and counterbore for bolting purpose) such that the bolts can be easily inserted from the bottom. The spacer is strong enough to take the load of the entire assembly (i.e. around 200 kg, bottom and top plate + collar). The assembly will be resting on this spacer, even during the in-situ baking procedure. After baking and pump down testing is complete, the spacer will be removed and the vacuum chamber will be resting flat on the optical bench.

Next, with the help of a crane we tried lifting the collar using eye bolts and nylon slings, however the suspension point was way too high giving us no room for the crane to lift. This required shortening the length of the nylon sling. Firstly, I got shorter length eye bolts (2.5 inch against 4 inch) and secondly I used carabiners to tie up the sling. Using this set up we were able to lift the collar successfully.

The collar has groove for single dovetail O-ring (pictured attached) at the top and bottom surface.

The specifications for the viton O-ring was provided by Nor-Cal. I bought the viton O-ring (2-474V, 24.94’’ ID) from Kurt Lesker, the dimensions of which is given below,

O-RING,FKM,24.940"ID X .275" W(ACT),25"ID X 1/4"W (NOM),ISO 630

The challenge is to keep the O-ring held in the groove at the bottom of the collar (against gravity) during it’s mating with bottom plate. However, the O-ring always falls off. At this point I am not inserting the O-ring completely inside the groove (and I will come to this point next) and just slightly pushing it in. I want the O-ring to move in naturally from the pressure created by the plate and collar. I contacted Zach since he has faced similar issues with his tank (Chris gave me this information) and he used vacuum grease to keep the O-ring sticking on to the groove. However, outgassing from the grease could be a point of concern.

Next, I tried to insert the O-ring completely inside the grove, strangely in this case I find it to be longer by at least an inch. The inner diameter of the groove is 25 inches.

I contacted Nor-Cal to cross-check the size they have recommended. They realized that they have made a mistake with the size and have agreed to ship one size smaller O-ring (i.e. 2-473 having an ID of 23.94) which should give a stretch of 4%. I will use this once we receive it.

I found a Youtube video of the O-ring installation in a half or full dovetail groove. Given below is the link, since this is different from the usual method, hence important. The advantage of using dovetail groove is that the O-ring will stay inside during the assembly without falling off.

I asked Nor-Cal about in-situ baking to reduce outgassing and they recommended that with viton O-ring seals 200C is max and 150C or lower is safe. I will go with 100 C. Steve doesn’t agree and doesn’t like the idea of baking it during pump down. As per him, once a turbo pump was destroyed (I guess at 40m) after some parts of the O-ring got damaged and got sucked inside. However, I think keeping things at 100 C should be safe as other folks at the cryolab have done the same, successfully.

Attachment 1: bottom_plate_on_spacer.jpg
Attachment 2: Crane_dovetail.jpg
2299   Mon Feb 25 20:42:32 2019 KojiSummaryScatterometerSi Block stored in Cryo Lab

On the Friday cleaning, we vacated the east optical table. The Si scatterometer was disassembled and the Si block was moved and stored to the cryo lab.

Attachment 1: P_20190221_115501.jpg
2306   Wed Mar 13 17:49:14 2019 awadeSummaryWOPODamage threshold estimates for optical fibers

This is just a note about damage tolerances for fibers so we have a reference of the amount of power that can be used.

Thorlabs gives the maximum theoretical CW power as 1 MW/cm² and a 'practical' safe power level of 250 kW/cm² (or a quarter of the max).  They don't seem to provide information about wavelength dependence and assume its the same for all wavelengths.  We don't expect to be affected by any of the exotic ultra high power effects like bend loss induced damage and photodarkening.  The air/glass interface is where the damage will occur.  This can either be because of heating of the ferrule/connector (causing epoxies etc to break down and damage the interface by depositing on the optical surface) or regular mechanism that are the same as bulk optics (dielectric break down and thermal effects).

The intensity profile of light confined in the fiber is defined by the Mode Field Diameter (MFD) -- the cross-sectional diameter of the light that includes the core of the fiber and a region just beyond the cladding the mode occupies. MFD of 1064 PM fiber (PM980-XP) is 6.6 ± 0.5 μm @ 980 nm and for the 532 nm PM fiber (PM460-HP) is 3.3 ± 0.5 µm @ 515 nm.

Fiber effective area is

$\dpi{100} A = \pi (\textrm{MDF}/2)^2$

which is 8.5x10^-8 cm^2 for PM460-HP and 3.4x10^-7 cm^2 for PM980-XP.  Taking the conservative 'practical' damage threshold this indicates a maximum power of 21.4 mW into the 532 nm fiber and 85.5 mW into the 1064 nm PM fiber. The absolute maximum is just a factor of four more than this: 85.6 mW into 532 nm PM fiber and 342 mW into 1064 nm fiber. If the fiber ends are kept clean then we should be fine if the power level is kept below 85 mW.

## Fiber beam splitter

Of particular concern is the power handling capability of the fiber beam splitter (PN1064R5A2).  There will be some waste 532 nm light coming out of the WOPO and I don't want these potentially multi-mode components to exit the cladding at the point of the coupler and damage the surrounding material. The 1064 nm maximum power rating of this 50:50 PM beam splitter is listed as 1 W (for the connectorized fiber), so we should be well clear of that threshold for the LO light.  For 532 nm its less clear. The equivalent 532 nm PM 50:50 beam splitter (PN530R5A2) has a rated power of 100 mW @ 530 nm for bare or connectorized fibers. As the MFD of the 1064 nm version of this PM beam splitter has a much larger MFD and the exiting 532 nm light will already be expanded in the cable patching the WOPO to the BS, we should be well clear of this damage threshold point.

So bottom line is that we need to keep power below 85 mW going into the WOPO device and keep all the end connectors super clean and it will be fine.

2327   Mon Apr 22 19:35:16 2019 anchal and awadeSummaryWOPOShot Noise Intercept Current measurement of Photodiodes

Andrew measured the output voltage noise of photodiodes through an AC coupled pre-amplifier (G=200, 0.7 nV/rtHz) while varying the intensity of white light falling on it. The setup uses a regular 2xAA flashlight (torch) with an incandescent bulb to illuminate the photodiode. The flashlight focus is adjusted to give a non-expanding beam and a 25.4 mm (1") lens is then used to focus the collimated beam down. The total power of light was controlled by trimming it with an iris.

### Data Analysis:

Below is the fitting of data to estimate the shotnoise intercept current. Here the fitted function is the quadrature sum of input referred shot noise and dark noise modeled as an equivalent shot noise quanitiy i_snint (called intercept current):

$i_\textrm{in} = \sqrt{2e^-(i_\textrm{pc}+i_\textrm{snint})}$

I have included zip file with data and notebook in it as well. With measured shotnoise intercept current of 48 ± 7 uA and 31 ± 7 uA, we are good to go to use these for measuring shot noise with light power in the order of 5.0 mW falling on the photodiodes. With 5.0 mW (~3.9 mA) on each PD, this gives us a clearance of 10dB from the dark noise floor on average for each detector.  Plots with fits are attached below.

Attachment 2: WOPO_BHD_Photodiodes_SNInterceptAnalysis.pdf
Attachment 3: 20190419_MeasuredInterceptCurve.zip
2366   Wed Jun 26 11:45:09 2019 KojiSummary2micronLasers2um PD transimpedance

The transimpedance amps for the 2um (unamped) InGaAs detectors were made and evaluated.

Attachment 1: The circuit diagram

The usual transimpedance configuration. The detector (Thorlabs DET-10D) is an extended InGaAs which is sensitive up to 2.2um. I believe the detector is biased to 1.8V although it is not obvious and the 12V battery is used. The feedback resistor was chosen to be 5kOhm so that the circuit can handle up to ~2mA (~1.7mW). The feedback capacitance pf 100pF for compensation was chosen kind of arbitrary to keep the circuit stable and also the RC cut off to be more than 100kHz. The output resistance is 100Ohm. The selection of the opamp is described below.

Attachment 2: The amplifier noise Part I

The amplifier noise (the first unit called Amp #2)  was evaluated with the opamp swapped with OP27 (BJT), LT1128 (BJT), OPA604 (FET), and LT1792 (FET), chosen from the 40m stock. For the given environment, the FET amps exhibited better performance while the BJT amps suffered from more line noise coupling and the larger 1/f noise. Particularly, LT1792 reached at the level of ~2pA/rtHz, with lower line noises. This looks the best among them. Note that the 5kOhm feedback resistor gives 1.8pA/rtHz current noise.

Attachment 3: The amplifier noise Part II

Then the second unit (called Amp #1) was made. This unit has more high-frequency noise. It turned out that the noise was coming from the power supply which was the +/-12V from the rear panel of an SR560 which was connected to the AC power. The noise dramatically went away with the battery mode operation of SR560 (by disconnecting the AC power). The floor level was 2.2pA/rtHz and it was slightly higher than the quadratic sum of Johnson noise of 5kOhm and the voltage noise of the amp (4nV/rtHz). This noise level was just sufficient for the purpose of the 2um detector.

Attachment 4: The detector noise levels

Now the detector #1 and #2 were paired with the amp #1 and #2, respectively. In fact the detector 1/f noise was way too large compared to the amplifier noise. There is no hope to detect shot noise level of the mA photocurrent.

Attachment 5: The detector response

The detector response of each PD+AMP pair was measured using Jenne's laser and Thorlabs PD10A (~150MHz). There was some systematic error of the absolute level calibration, therefore the transfer functions were adjusted so that they have 5kOhm transimpedance at ~1kHz. The phase delay is ~30deg at 100kHz. This partially comes from the combination of 100pF//5kOhm and the ~4MHz bandwidth gain of the opamp. If we want faster response we need to modify these.

Attachment 1: PD_model.pdf
Attachment 2: AMP2_Noise.pdf
Attachment 3: AMP_Noises.pdf
Attachment 4: PD_noise.pdf
Attachment 5: PD_TF.pdf
2368   Mon Jul 1 21:20:55 2019 KojiSummary2micronLasersThe PDs delivered to the lab

The amplifier sets for the thorlabs 2um PDs were delivered to the lab.

- PD1 and Amp1, PD2 and Amp2 are the proper combination. If a high quality power supply is used, it is not an issue.
- The cables for the external bench supply or the 9V batteries have been made.

2392   Mon Aug 12 12:09:00 2019 Nathan HollandSummary2micronLasersAdjusting the Voltage/Frequency Offset on the Brimrose AOM Driver.

The 80 MHz brimrose AOM driver, which came with the AOM, can actually drive between 75 MHz and 85 MHz. It has an input port, which accepts between 0V and 10V, for altering the frequency.

Previously, before Friday 2019-08-09, I had set the offset to + 5V, using a signal generator because that was what was available in the QIL lab. The signal generator, for some unclear reason, had then moved this offset to +4.4 V, when plugged into the AOM driver. Attachment 1 shows a power spectrum, from the BB PD, One notices a large peak negatively detuned from the 80 MHz signal frequency. Attachement 2 shows a zoom in, around 70 MHz, revealing this peak to be two peaks, one near 78 MHz, and another smaller one near 79 MHz.

I decided to adjust the voltage, input to the frequency port, to minimise the RF sidebands I observed. The results of this adjustment can be seen in attachments 3, and 4. Notice that there no observable sidebands. This was achieved with an offset of 4.52V.

Attachement 5 is the manual for the AOM, and its driver. (Removed by KA)

Attachment 1: 2um_MachZehnder_wide_PowerSpectrum_Screenshot.png
Attachment 2: 2um_MachZehnder_narrow_PowerSpectrum_Screenshot.png
2400   Thu Aug 15 02:30:29 2019 KojiSummary2micronLasersInAsSb PDs accomodated in PD housings / PD cables

InAsSb PDs were housed in the PD cages. The cages were engraved to indicate the batch (Sb3512 or 3513) and the serials (A1, A2, ...).

The PD legs does not have an indicator for the pin1. So, the tab of the PD case is directed "UP". Also the direction of the tab is marked on the cage. The tab of the short plug was also aligned to Pin1. However, the PD case is too thin and the PDs can rotate in the cases.
So the face photo was also taken so that it indicates how Pin 1 looks like from the PD face. (Attachment 4)

Also made the cable for the LaserComponents PD and the InAsSb PD. Pin n shows up as Pin n of DB9 Male connector.

Once we have the PD test is the bias circuit (with a monitor) and some patch panel kind of preparation, we can start working on the PD test.

Attachment 1: P_20190814_214159.jpg
Attachment 2: P_20190814_214537.jpg
Attachment 3: P_20190814_215119.jpg
Attachment 4: P_20190814_215158.jpg
Attachment 5: P_20190814_220035.jpg
Attachment 6: P_20190814_220553_003.jpg
Attachment 7: P_20190814_220832.jpg
Attachment 8: P_20190814_220841.jpg
Attachment 9: P_20190814_235137.jpg
2404   Fri Aug 16 01:28:28 2019 KojiSummary2micronLasersSwitchable breakout bok

DB9 switchable breakout box is ready. We are ready to do some PD test now.

Attachment 1: P_20190815_210355.jpg
Attachment 2: P_20190815_210419.jpg
2405   Sun Aug 18 14:16:59 2019 DuoSummary Measuring the dark current of PD

Koji set up an experiment measuring the dark current of the photodiodes. A bias voltage is given and the current is converted to voltage via a TIA, where it is measured. Also note that in order to provide a high quality bias voltage, we LP the output of the device with a second order sallen key filter cutoff at 1Hz.

Attachment 1: exp.pdf
Attachment 2: IMG_2921.png
2423   Mon Sep 23 10:49:27 2019 KojiSummaryPD QEQE and dark current of InAsSb sensors

The QE and dark current of all the InAsSb sensors were measured. All the measurements were done in room temperature.
- The incident beam power of the 2004nm beam was 0.95mW.
- The beam was focused down to 50um gaussian radius, which was confirmed by DataRay BeamR.
- The angle of incidence was ~0deg.

- The element side (nominally Pin 2, 3, or 6) were connected to the vias boltage (negative) and the common ground was connected to the transimpedance amplifier (Shalika OP140 R=5100Ohm)
- The dark current was highly dependent on the reverse bias voltage. The QE was also bias dependent.
- Sb3512 A2 have different behavior compared to others. Alex mentioned that Sb3512 is the test batch. We can exclude this sensor from the test.
- The best QE was ~0.7 for Sb3513 A3 P2 (Pink) and Sb3513 A2 P6 (Purple). Both have the area of 500um^2. These two particular elements have low dark current of <1mA. The dark noise of this specific sensor should be measured.

Some issues of the measurements

- The transimpedance amp (TIA) has suspicious behavior. The saturation voltage was ~17V rather than <-15V. This indicates that the voltage regulators possibly have leakage of the input voltage (+/-18V) to the output line. This needs to be checked, particularly before the dark noise test.

- TIA saturation: The bias voltages could not be raised to ~1V for some PDs because of the dark noise and the saturation of the TIA. The transimpedance should be lowered by a factor of ~5.
- Because of the low bias voltages of these saturated cases, the max QEs were not reached. This also prevented from checking if there was any clipping loss. This should be checked again with the lower transimpedance.

- TBD: The angular dependence and the reflectivity of the sensor should be checked. It is difficult to carry out these tests without a sensor card.

Attachment 1: InAsSb_QE.pdf
Attachment 2: InAsSb_DarkCurrent.pdf
Attachment 3: 190921_SbPD_QE.zip
2425   Wed Sep 25 01:05:30 2019 KojiSummaryPD QEQE and dark current of InAsSb sensors

The lenses were arranged so that the spot on the PD can become smaller. A quick measurement on a (500um)^2 element showed the QE of ~80%

With the strong focusing lens of f=40mm, the beam was once expanded to a few mm. Then f=75mm lens focuses the beam to ~30um (radius). (See Attachments 1&2)

With this new beam, the QE was quickly checked. The new measurement is indicated as "Sb3513 A2P6new" in the plot. It showed the QE of ~80%.
The AOI was scanned to find any maximum, but the AOI of 0deg was the best at least with the given beam. I'm not sure yet why 500umx500um requires such small beam radius like 30um. Awesome

Attachment 1: P_20190924_233507_vHDR_On.jpg
Attachment 2: P_20190925_003614_vHDR_On.jpg
Attachment 3: InAsSb_QE.pdf
Attachment 4: InAsSb_DarkCurrent.pdf
2446   Fri Nov 15 20:10:00 2019 KojiSummary2micronLasersInAsSb PD Mounts designed

LIGO-Number Title Author(s) Topic(s) Last Updated
E1900369-v1 InAsSb Photodiode Mount (Short) Assembly Koji Arai Assembly
Basic R&D
15 Nov 2019
E1900368-v1 InAsSb Photodiode Mount Assembly Koji Arai Assembly
Basic R&D
15 Nov 2019
D1900490-v1 InAsSb Photodiode Mount Face Plate Koji Arai Basic R&D 15 Nov 2019
D1900489-v1 InAsSb Photodiode Mount (short) Koji Arai Basic R&D 15 Nov 2019
D1900488-v1 InAsSb Photodiode Mount Koji Arai Basic R&D 15 Nov 2019

Draft   Mon Nov 18 17:17:29 2019 DuoSummary Restart model in QIL lab

Here are the commands.

ssh fb4 sudo /sbin/rmmod c4tst c4iop && cd /opt/rtcds/caltech/c4/target/c4iop/scripts/

./startupC4rt && cd ../../c4tst/scripts/ && ./startupC4rt && systemctl start rts-awgtpman@c4iop.service && systemctl start rts-awgtpman@c4tst.service && systemctl restart daqd@standiop.service

2460   Mon Nov 25 21:46:56 2019 KojiSummaryPD QESystem Diagram

System diagram of the PD QE test with the IRLabs cryostat.

PT-SE (MS/PT-SE) connector data sheets

Connector/receptacles/tools https://www.peigenesis.com/images/content/pei_tabs/amphenol/pt-ptse-series/new-thumbs/123-146_pt_series.pdf
Amphenol catalog http://www.amphenol-industrial.com/images/catalogs/PT.pdf

Detoronics Hermeic Sealed Connectors (DT02H-18-*PN) http://www.hselectronics.com/pdf/Detoronics-Hermetic-Connectors.pdf

TEC spec: Mounted TEC type: 2MD04-022-08/1 https://www.tec-microsystems.com/products/thermoelectric-coolers/2md04-series-thermoelectric-coolers.html
2MD04-022-08/1 dTmax = 96, Qmax = 0.4W, Imax = 0.7A,  Umax = 2.0, ACR = 2.29 Ohm

Attachment 1: cryo_pd_test.pdf
Attachment 2: InAsSb_PD_mount_short.PDF
Attachment 3: PD_pin.pdf
2463   Wed Nov 27 20:38:57 2019 KojiSummaryPD QESystem Diagram

The external Dsub cable is ready except for the 32pin connector to be plugged-in to the chamber. See QIL ELOG 2460 for the pin assignment.

Attachment 1: P_20191127_203612_vHDR_On.jpg
2464   Sun Dec 1 01:32:19 2019 KojiSummaryPD QEPD TEC cooling test

While I'm still waiting for the proper connector for the vacuum feedthru of the IRLabs cryostat, I have connected to the Dsub9/15 split cable to another Dsub9 connector so that I can test the cooling of the InAsSb sensor in air. Also, the 2004nm laser, a fiber-coupled faraday isolator, and 90:10 beam splitter was moved to the cryostat table and fixed on a black al breadboard. [Attachment 1]

The InAsSb TEC was controlled by the TEC controller of ITC-50. I didn't change the PID parameters of the controller but the temperature nicely setteled to the setpoint. The sensor has a 2.2kOhm thermister. And the max current for the TEC was unknown. The TEC driver had the current limiter of 0.3A and it was not changed for now. With this current limit, the thermistor resistance of 10Kohm was realized. This corresponds to the temperature of about -20degC. According to the data sheet given by Alex, the resistance/temperature conversion is given by the formula

1/T = 7.755e-4 + 3.425e-4*log(R)+1.611e-13*log(R)^3

Questions:

• What is the max current for the TEC?
• What is the calibration formula of the thermister (TB04-222) at cryogenic temperature?

To satisfy the curiosity, the dark current of a (500um)^2 element was measured between -250K and -300K. At -254K, the dark current went down to the level of 40uA (1/15 of the one at the room temp). For the measurement, the bias voltage was set to be 0.5 and 0.6V. However, it was dependent on the diode current. (Probably the bias circuit has the output impedance). This should be replaced by something else.

To Do

• 2um laser beam setup (w=100um beam)
• Bias circuit
• Quick check of the QE and dark noise at -20degC

Attachment 1: 20191129191814_IMG_9146.JPG
Attachment 2: cooling_dark_current.pdf
2466   Tue Dec 3 15:32:39 2019 KojiSummaryPD QEPD TEC cooling test

The quantities we want to measure as a function of the temperature:

- Temperature: 2.2k thermister resistance / 100ohm platinum RTD

- QE (Illuminating output / Dark output / Reference voltage / Reference dark output)

- Dark current (vs V_bias) -> Manual measurement or use a source meter

- Dark noise (PSD) 100kHz, 12.8k, 1.6kHz, 100Hz

2468   Thu Dec 5 13:50:59 2019 KojiSummaryPD QEDark current measurement with the sourcemeter

I borrowed KEITHLEY 2450 source meter from Rich. The unit comes with special coaxial cables and banana clips. Most of the peripherals are evacuated in the OMC lab.

The dark current of A2P2, A2P3, A2P6 were measure with different temperatures (300K, 270K, 254K). The plot combined with the previous measurement ELOG QIL 2425.

== How to use the source meter ==

- Two-wire mode: Connect the wires to the diode

- Over voltage protection: [MENU] button -> SOURCE / SETTINGS->Over Voltage Protectiuon 2V

- Sweep setting: [MENU] button -> SOURCE / SWEEP -> e.g. Start -750mV, Stop +500mV, Step 10mV, Source Limit 1mA -> Select Generate

- Graph View: [MENU] button -> VIEWS / GRAPH

- Start measurement: Note: The response of [TRIGGER] button is not good. You need to push hard
This starts the sweep, or a menu shows up if your push is too long -> Select "Initiate ..."

- Data Saving: [MENU] button -> MEASURE / READING BUFFERS -> Save to a USB stick

Attachment 1: InAsSb_DarkCurrent_markedup.pdf
2475   Wed Dec 11 01:29:26 2019 KojiSummaryPD QESb3513 A2P6 Dark Current / QE / Dark Noise measurement @77K

[Raymond, Aidan, Chris, Koji]

P6 element (500um)^2

- We looked at the current amp (FEMTO) output. The amplifier saturated at the gain of 10^3 V/A. Looking at the output with a scope, we found that there is a huge 1.2MHz oscillation. Initially, we thought it is the amplifier oscillation. However, this oscillation is independent of the amplifier bandwidth when we tried the our-own made transimpedance amp.

- Shorting the cryostat chamber to the optical table made the 1.2MHz significantly reduced. Also, connecting the shield of the TEC/Laser controller made the oscillation almost invisible. This improvement allowed us to increase the amp-gain up to 10^7.

- Then the dominant RMS was 60Hz line. This was reduced by more grounding of the cable shields. The output was still dominated by the 60Hz line, but the gain could be increased to 10^8. This was sufficient for us to proceed to the careful measurements.

----

- The dark current was measured by the source meter, while the photocurrent (together with the dark current) was measured under the illumination of the ~1mW light on the PD.

- Attachment 1 shows the dependence of the dark current against the swept bias voltage. We had ~mA dark current at the room temp. So, this is ~10^5 improvement.

- Attachment 2 shows the dependence of the apparent QE against the swept bias voltage. The dark current was subtracted from the total current, to estimate the contribution of the photocurrent in the measurement.

- Attachment 3 shows the dark noise measurement at the reverse bias of ~0.6V. Up to 1kHz, the noise level was below the equivalent shotnoise level of 1mA photocurrent.

---------

All the data and python notebook in the attached zip file.

Attachment 1: Sb3513_A2P6_DarkCurrent_77K.pdf
Attachment 2: Sb3513_A2P6_QE_77K.pdf
Attachment 3: Sb3513_A2P6_DarkNoise_77K.pdf
Attachment 4: 191210_3513A2P6.zip
2481   Fri Dec 20 13:20:53 2019 AidanSummaryPD QEQE results from A2P6 (500um) and A2P2 (1mm)

The QE measurements from the first couple of photodiodes are attached below.

• plot_JPL_diode_results.m - A2P6 analysis
• plot_JPL_A2P2_diode_results.m - A2P2 analysis

QE = [I_photocurrent]/[P_PD] * h *nu/e

P_PD = Power incident on photodetector = 0.966*power_incident on cryo window

Power incident on cryo window = F(voltage on reference PD)

Attachment 1: PC_DC_v_T.pdf
Attachment 2: A2P2_001_test.pdf
Attachment 3: PC_DC_v_T.pdf
Attachment 4: A2P6_001_test.pdf
Attachment 5: plot_JPL_diode_results.m
% load JPL data
f0 = dir('*dark*.txt');
f1 = dir('*photo*.txt');
f2 = dir('*cond*.txt');

% get temperature vs time
tempList = [];
pList = [];
for ii = 1:numel(f2)-1

... 102 more lines ...
Attachment 6: plot_JPL_A2P2_diode_results.m
close all
clear all
f0 = dir('*dark');
f1 = dir('*bright*');

% get temperature vs time
tempList = [];
refPDList = [];
for ii = 1:numel(f1)

... 113 more lines ...
2488   Thu Feb 27 14:26:52 2020 KojiSummaryLab MonitoringItem lending: Particle Counter from OMC Lab to QIL

Item lending as per Ian's request: Particle Counter from OMC Lab to QIL

The current particle class of the room was measured to be 800.

The particle counter went back to the OMC lab on Aug 10, 2020.

Attachment 1: P_20200227_134755_vHDR_On.jpg
2489   Fri Feb 28 13:36:32 2020 Ian MacMillanSummaryLab MonitoringItem lending: Particle Counter from OMC Lab to QIL

Still trying to figure out how to set up the particle counter remotely. The current particle count is 576.

 Feb. 28 at 12:30pm 576 Feb. 28 at 5:00pm 594 Mar. 2 at 8:30am 393 Mar. 2 at 11:30am 650

Note: the particle count is the number of particles detected over 0.3um size.

2491   Mon Apr 6 18:35:48 2020 KojiSummaryGeneralWest Bridge flooding Apr 6th

West Bridge flooding Apr 6th due to rain in the night

Looks like the first responder was Calum. The attached photos were sent from him.

Attachment 1: image2.jpeg
Attachment 2: image1.jpeg
2509   Thu Oct 22 11:19:44 2020 anchalSummaryECDLProspects of using ECDL for Auxiliary laser

We can use Thorlabs SAF1450S2 gain chip to generate 1418 nm light using an ECDL design similar to the one described in Kapasi et al. Optics Express Vol. 28, Issue 3, pp. 3280-3288 (2020) (ANU 2um ECDL design).

### PZT Transfer function

I have contacted Disha and Johannes to get the actual measured data for the PZT transfer function of this ECDL design. Fig.5b in their paper plots the transfer function of the PZT. Since, in ECDL PZT directly changes the cavity length, it has a more powerful actuation strength (2 orders of magnitude more) with actuation of 560 MHz.V upto 1 kHz. It however had a very low pole at 1 kHz and two mechanical resonance-antiresonance pairs near 1 kHz and 2 kHz. I modeled a transfer function by eye using Fig.5b of the paper. Page 1 in the attached pdf shows this modelled transfer function.

### AUX PDH Loop

Next, we need to change the PDH loop for the auxiliary laser lock with the 40m cavity since the PZT has changed. I modelled one from scratch. This simple analog loop's performance is shown in orange in pages 2-5. This loop seemed stable from all the metrics I know, viz: phase margin of about 55 degrees (Page 2), no strong peak in close loop transfer function (page 3), and no remanant oscillations in time domain response (page 4).

I also modeled a similar loop but with digital compensation of the resonance-antiresonance features. This loop is plotted in green on pages 2-5. Both these loops have 300 kHz of bandwidth just by using PZT. I beleive this could be increased but I have not taken into account any saturation of PZT.

### ECDL Frequency Noise

From Fig.4. of the paper gives a frequency noise estimate for free running ECDL. They mentioned that a roll-off below 10 Hz was due to their thermal feedback to remain in linear range of their frequency noise emasruement method. I modeled the noise of ECDL hence by
$\frac{10^4}{f} + 15\, Hz/\sqrt{Hz}$
where the flicker noise contribution is similar to NPRO noise but ECDL has a white noise of 15 Hz/rtHz due to natural linewidth of spontaneous emission or Schawlow-Townes linewidth (with several broadening factors). I think this is an inherent limitation of ECDLs.

Page 5 shows both unsuppressed and suppressed frequency noise estimate for ECDL with the loops mentioned above and current values of NPRO noise are also plotted for comparison.

Code

Attachment 1: AUX_Loop_Study_With_ECDL.pdf
2510   Fri Oct 23 12:19:19 2020 anchalSummaryECDLEffects of chosen AUX finesse and source on Calibration requirements

Following up on the last post, here I presented a near back of the envelope calculation of how different choices of AUX cavity finesse and laser source for mariner would affect the prospects of calibration scheme.

### Laser sources considered:

As mentioned in the last elog post, here I considered using an NPRO seeded auxiliary laser source (converted to 1418nm by whatever method), ECDL based on ANU design with a modified PDH loop and same ECDl with a digital compensation of PZT resonances. I have taken the residual frequnecy noise of these lasers as the dominant noise source in the calibration scheme. Craig and Gautam in their proposal for SoCal wanted the AUX laser to be locked to the arm cavity in a PDH shot noise limited way. That would be necessary for 4km interferometers and would be easier to achieve there with higher laser powers and higher cavity finesse.

### Finesse of 40m Arm Cavity for 1418nm:

Here I considered three cases. First assumes about 3% transmittance of 1418nm in ITM and ETM HR coatings for mariner. This gives a finesse of about 100 and a cavity pole of 18.9 kHz. I believe this is the existing case at 40m. Next we consider transmittance of 0.5% and 0.05% (500 ppm) of 1418nm in ITM and ETM HR coatings for mairner. These cases give finesse of 625 and 6.28k respectively with cavity poles at 3 kHz and 299 Hz respectively.

Page 1: Consideres the case of finesse of 100. The green dashed line shows the amount of drive strength (in m) required at different frequencies if we use ECDL with PZT resonance compensation, to get an SNR of 1000 in 100s of integration time.

Page 2: Same as above but for Finesse of 625.

Page 3: Same as bove but for Finesse of 6280.

Page 4: Comparison of different finesse cases for the ECDL with PZT compensation option. Dashed curves represent requried drive strength (in m) for different cases.

Page 5: Same as above but for NPRO seeded auxiliary laser.

Note: For the NPRO seeded auxiliary laser, we have assumed that the noise of conversion to 1418 nm is similar to noise due to SHG process which is not dominant. There would be an effect of multiplying with a factor ranging form 1-1.5 due to frequency conversion but I have ignored it here for simiplicity. Also, NPRO case is limited in bandwidth due to PZT resonances. We might be able to get away with them using digital compensation like the case study for ECDL. But I haven't attempted that here as we do not know our NPRO PZT's resonance features yet.

Attachment 1: AUX_Finesse_and_Source_Study.pdf
2511   Wed Oct 28 14:05:19 2020 ranaSummaryECDLEffects of chosen AUX finesse and source on Calibration requirements
 Quote: would be easier to achieve there with higher laser powers and higher cavity finesse. But I haven't attempted that here as we do not know our NPRO PZT's resonance features yet.

I don't know why it would be easier to have higher finesse with longer arms. Something about beam size???

The NPRO PZT TF's are all in the 40m elog - there are many measurements of TF made over the past 10 years. Its like Raiders of the Lost Ark - you have to believe its there while searching.

2512   Thu Nov 5 11:20:45 2020 anchalSummaryECDLTrue PDH Error signal TF and including FSR effects in approximated models

If we use ECDL for auxiliary frequency in 40m and hope to stabilize it up to 1 MHz with digital compensation of PZT, it is important to take into account any phase effect of the nearby FSR at 3.97 MHz. This should ideally be included in the Input Mode Cleaner loop considerations as well. These effects would be more prominent in longer cavities like aLIGO and LISA where FSR is very low and should we attempt to stabilize a laser lock beyond cavity's FSR.

I did a no assumptions calculation for getting a general transfer function fo PDH error signal in units of [W/Hz] assuming 1 W of incident power. This calculation would soon be uploaded here. I'll put down here primary results.

For incident field on a Fabry-Perot cavity (with fsr of $\nu_{fsr}$), reflected electric field transfer function (unitless) is given by:

$\dpi{300} \tiny R(\omega) = \frac{-r_1 + (r_2^2 + t_2^2) r_2 e^{-i \omega/\nu_{FSR}}} {1 - r_1 r_2 e^{-i \omega/\nu_{FSR}}}$

Then, PDH error signal for a modulation frequency of $\Omega$ at a modulation index of $\Gamma$, in units of [W/Hz] (i.e. error signal power per Hz of error in laser frequency from cavity resonance) is given by:

$\dpi{300} \tiny H_{\nu 2P}(\omega) = -\frac{i \pi P_0 J_0(\Gamma)J_1(\Gamma)}{\omega} \Bigl(R(\Omega)R(\omega) - R(0)R(\Omega + \omega) + R^*(\Omega)R(\omega) - R(0)R^*(\Omega - \omega)\Bigr)$

after demodulation and low pass filtering. Note this transfer function is a complex quantity as it carries phase information of the transfer function too. The real signal is obtained by multiplying this signal at $\omega$ with $e^{i \omega t}$ and taking the real value of the product.

Having done this, we can see how in the real PDH error signal, there is a low pass at cavity pole, given by $- \frac{\nu_{fsr}}{2\pi}log(r_1 r_2)$ and a notch every fsr. The notch creates a zig-zag in the phase of the tranfer function and has a HWHM same as cavity pole. After this point, I just fitted a ZPK model to the transfer function to obtain a empirically derived model for PDH error signal transfer function. Apart from the cavity pole, this model needs to have resonance and antiresonance features present at each FSR with resonance having a linewidth of cavity pole while anti-resonance having a linewdth of $\pi/\nu_{fsr}$. Here's how the ZPK model would look like:

\dpi{200} \begin{aligned} z:&\, \pi/\nu_{fsr} \pm n \nu_{fsr} j \\ p:&\, f_p \pm n \nu_{fsr} j, f_p \end{aligned}

I've attached my notebook where I did the fitting analysis and the overlap plot of real PDH error signal TF and the modelled approximation.

Attachment 1: PDHErrorTFModel.pdf
Attachment 2: PDHTFforACavity.ipynb.zip
2515   Mon Nov 9 10:08:38 2020 anchalSummaryECDLAUX wavelength finesee requirements in mariner with 1418 nm ECDL (Preliminary)

I have a preliminary calculation to post here. This does not include noise sources from cavity fluctuations and main frequency noise. But it gives some idea about shot noise and frequency noise of AUX laser conttribution to the noise in calibration.

### What's included?

• I have put in measured PZT transfer function of ECDL at ANU upto 25 kHz. Above this point, they did not measure it so I couldn't make it artificially dirty. I just assumed 1/f roll off above (which is definitely incomplete picture).
• I have included phase effects of cavity FSR in the loop by adding the resonance features as mentioned here.
• I have added attempted resonance compensation for 1kHz and 2kHz features in the PZT after fitting the data with poles and zeros and iverting them.
• 10mW of incident AUX light is assumed on the arm cavity.
• Total 10 mW of combined power at 709 nm is assumed to fall on the beatnote. So AUX light would be frequency doubled for this beatnote.

### What's left?

• Need to add seismic noise and other measured excess noise that come from the cavity motion.
• Need to add laser frequency noise of main laser, however, it must be small since it is locked to mode cleaner.
• Need to add digital delay of Red Pitaya or whatever filter would be used for PZT resonance compensation.
• Need to model PZT transfer function of ECDL above kHz properly. ANU replied that they can't measure it for higher frequencies due to lack of time.
• Need to do time domain stability analysis. This I haven't been able to do as I have just been using python-controls package as black box to compute impulse and step responses of state space systems. When simply adding easured transfer function data, I couldn't create the state stpace representation for the system. I tried to fit multiple resonances above 2 kHz but couldn't really capture the magnitude of the response well. Maybe I can just assume higher harmonics of the 1kHz and 2kHz resonances?

### Attachments:

• Page 1 is the measured PZT transfer function fo 1900 nm ECDL from ANU along with the modeled 1/f roll off after 25 kHz.
• Page 2 is the open loop transfer function of the AUX PDH loop for the three different finesse cases studies. Note that the blue curve is hidden beneate the other two curves. Before objections came, I know this is unreal and incomplete, but I have to start somewhere.
• Page 3 is the calibration noise budget with different colors showing the three finesse cases.
• This is also incomplete but we can takeaway what the shot noise contribution would look like and initiate a dialogue about the integration time chosen (which is 100s here), the SNR aim chosen (1000 here) and what drive strength would be good enough.
• From notes of Craig and Gautum, I think we can drive the mirrors at 0.1pm ampltude in the calibration band. From my first and only calibration measurement in 40m, I could drive upto a pm without loosing lock in the cavities.
• But that was a simple single arm lock and full ifo lock must be more sensitive. So is 0.1 pm drive strength good enough or do we want to aim lower?
Attachment 1: AUX_Finesse_Study_With_ECDL.pdf
2518   Wed Nov 11 16:09:22 2020 anchalSummaryECDLAUX wavelength finesee requirements in mariner - Added Excess Scatter Noise

An issue was raised with last calculation about the fact that our sensing of PDH signal isn't ideal and in the real world there is scattering, clipping extra adding excess noise in the PDH loop. This noise primarily comes by the intensity noise imparted on promptly reflected light from the cavity via various shaking optics etc on the table before it goes to the PDH reflection RF photodiode.

This noise's coupling to the PDH loop is identical to how shot noise of light couples into the PDH loop i.e.:

• Intensity noise of light is converted into voltage oise by the PDH photo diode.
• This is compared against the cavity finesse amplified real PDH error siganl at this stage.
• Therefore, in frequency noise, the affect of this intensity noise is smaller for higher finesse cavities since cavity finesse only amplifies the PDH signal anf not the scattering noise.

### Excess noise estimate

• I used this measurement taken in 40m with Koji to estimate this noise.
• This measurement contained a beatnote between IR coupled AUX light and the main laser IR pick-off when X-arm is locked to the main laser and AUX laser is locked to X-arm.
• So this noise measurement is an upper bound on the total noise in AUX laser frequency when it is locked to the X-arm.
• I compared this against the noise budget model for AUX PDH loop I have which uses the same control loop as the uPDH box used here.
• I found a bulge of excess noise below 100 Hz and it seemed to go done as 1/f^2 there. I was reminded by a chat I had with Rana and another professor sometime last year when Rana mentioned scattering noise showing up as "Scatter shelf" looking something like this.
• So I modeled excess noise as the difference between the noise budget and the measured noise with it extending after 100 Hz with the same roll off as in 10-100 Hz.

### Calibration noise budget

• I took the excess noise measured, converted it to W/rtHz by using current AUX PDH discriminant and photodiode gain, and normalized by the power (9.6mW) to get this noise in RIN/rtHz.
• Then I assumed that the same RIN would be imparted in the Mariner AUX loop and calculated excess intensity noise at the PDH loop by multiplying the above number with assumed 10mW of incident power to get it in W/rtHz.
• From here, I fed it to the same input as I feed the shot noise in the loop and calculated the effect in the overall noise budget.
• For high bandwidth and gain PDH loops required for calibration, this kind of noise would dominate up to a kHz before getting taken over by the residual laser frequency noise.
• I have again plotted cases for three choices of finesse/mirror transmission. If we used 99.95% reflectivity (1000 ppm transmission, finesse of 3140) we would be fine for most calibration lines except the one around 40 Hz. (assuming drive strength of 1e-13 m everywhere).
• Otherwise, if possible, we should go for higher finesse. Case (c) plotted here (Page 3), shows that for 99.995% reflectivity (100 ppm transmission, finesse of 31415), we will be fine in all over the range with cavity pole dropping to 63 Hz. This would be really nice of course, if it is possible.
• So we recommend HR coatings for 1418 nm in the Mariner to be 99.995% reflective (giving total power transmission of 100 ppm).
Attachment 1: AUX_Finesse_Study_With_ECDL.pdf
2521   Fri Nov 20 18:47:42 2020 KojiSummaryGeneralPermenant exchange of TED200C(QIL) and TED200C(2umECDL)

I moved the brand new TED200C on the workbench to Crackle for 2um ECDL (permanently)
The TED200C temp controller used in the 2um PD test setup will stay there (permanently)

http://nodus.ligo.caltech.edu:8080/SUS_Lab/1851

2522   Fri Nov 20 18:49:43 2020 KojiSummaryGeneralInstrument loan

FEMTO DLPCA200 low noise preamp (brand new)

Keithley Source Meter 2450 (brand new) => Returned 11/23/2020

were brought to the OMC lab for temporary use.

2525   Tue Dec 22 10:42:17 2020 ranaSummaryGeneralInstrument loan

I was thinking about getting this new current pre-amp from NF:

http://www.nfcorp.co.jp/english/pro/mi/loc/pre/ca5351/index.html

It seems to have a good noise performance and has a built in low pass filter and also a remote interface.

The FEMTO seems less fancy, but their noise performance is actually 2-3x better.

 Quote: FEMTO DLPCA200 low noise preamp (brand new) Keithley Source Meter 2450 (brand new) => Returned 11/23/2020 were brought to the OMC lab for temporary use.

2526   Tue Dec 22 15:20:14 2020 KojiSummaryGeneralInstrument loan

Is the reverse bias programmable? FEMTO has a bias trimmer on it. It's useful in the usual application, but for automation, the configuration of the input becomes cumbersome.

2527   Tue Dec 29 17:53:21 2020 ranaSummaryGeneralInstrument loan

doesn't seem so, but they sell this one:

http://www.nfcorp.co.jp/english/pro/mi/loc/m_lp/p_lp/lp_6016_01/index.html

which has a USB interface and pretty good voltage noise spectrum

2551   Mon Apr 5 18:50:54 2021 RadhikaSummary Current PD testing schematic

I'm attaching my rough first draft of the QIL photodiode testing schematic. Please provide comments for fixes/improvement!

Attachment 1: QIL_PD_testing.jpg
Attachment 2: QIL_PD_testing.graffle
ELOG V3.1.3-


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Frank Gutch Jr: Charlottesville Revisited (Revisited),

Posted in Opinion, Review with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , on May 3, 2016 by segarini

The reason Charlottesville is being (Revisited) is because I do believe that somewhere out there in the ether Charlottesville has already been Revisited and, hell, I am sure I confuse people often enough without reusing headers.  It would be akin to writing a book and naming all the chapters “Chapter One,” which, now that I think about it, is a pretty good idea.  But the second Revisited, placed in parentheses, separates the first from the second, does it not?

Frank Gutch Jr: Bubbling Under in the Swinging Sixties; My Father and the War; Plus Notes

Posted in Opinion with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , on May 26, 2015 by segarini

Rock ‘n’ roll radio.  There was nothing like it.  It was fresh and exciting and exposed teens to a whole new world beyond their doorsteps.  If the fifties had cracked the egg, the sixties had spilled the eggs guts.  All over the place.  Humpty Dumpty magnified.  Omelet City.

format: plain


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# Question 392bf

Sep 24, 2015

$8$ tablets.

#### Explanation:

The idea here is that you know how much tetracycline the patient must take every six hours, and that you have some stock tablets that contain $\text{500 mg}$ of tetracycline each.

The first thing that you need to determine is how many tablets will you need to provide $\text{1.0 g}$ of tetracycline.

To do that, first convert the amount of tetracycline each tablet contains from miligrams to grams

500color(red)(cancel(color(black)("mg"))) * (10^(-3)"g")/(1color(red)(cancel(color(black)("mg")))) = "0.5 g"

This means that you need to give your patient

1.0color(red)(cancel(color(black)("g tetracycline"))) * "1 tablet"/(0.5color(red)(cancel(color(black)("g tetracycline")))) = "2 tablets"

every six hours in order to get the correct dosage per day.

If you take into account the fact that one day has 24 hours, the total number of tablets you must have on hand is

24color(red)(cancel(color(black)("hr"))) * "2 tablets"/(6color(red)(cancel(color(black)("hr")))) = color(green)("8 tablets")#

Give your patient a total of eight $\text{500-mg}$ tablets every day.


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# Can we prove nowhere differentiability of Brownian path via Karhunen–Loève coefficient?

This post is partly inspired by Fourier Coefficients and Hölder Continuity.

Typical proofs of the nowhere differentiability of Brownian paths is by contradiction using binary expansion from real analysis. For example, Prop 11. of S.Lally's notes.

But usually one can prove a function $f$ is nowhere differentiable by examining their Fourier coefficients $\hat{f}$. For example Simple Proofs of Nowhere-differentiability Weierstrass's Function and Cases of Slow Growth

Can we prove the nowhere differentiability (w.p.1) of Brownian paths by examining its Karhunen–Loève coefficients?

Moreover, what is the relationship between Karhunen–Loève coefficients $\hat{X_t}$ and the smoothness of the path of an $L^2$ stochastic process $X_t$?

If there is such a relationship, is this relationship similar to the relationship between Fourier coefficients and original function?

In response to Christian's comment,

The Fourier coefficients don't look like the right thing to look at to me if you want to establish that Brownian paths are nowhere differentiable, since the FT responds to lack of smoothness somewhere by lack of decay.

A natural thing to ask is that what object characterizes/detects global non-smoothness in in situation?

• For a start, I think the Kolmogorov 0-1 law together with the fact that all the coefficients are smooth should get you that nowhere differentiability has probability 0 or 1 (because tweaking any finite number of your iid random variables will not change the differentiability). – Nate Eldredge Apr 18 '17 at 5:39
• @NateEldredge Yes, we can observe that convergence of KL series is a tail event but do you mean that differentiability can be determined by the convergence rate of KL series itself and hence it's a tail event too? – Henry.L Apr 18 '17 at 10:55
• The Fourier coefficients don't look like the right thing to look at to me if you want to establish that Brownian paths are nowhere differentiable, since the FT responds to lack of smoothness somewhere by lack of decay. – Christian Remling Apr 22 '17 at 18:22
• @ChristianRemling So what is the correct thing to look at here? And can't we discover nowhere differentiability by showing lack of decay with high probability? – Henry.L Apr 25 '17 at 17:29
• No, not that easily because lack of decay of the FT just implies that the function is non-smooth somewhere. For me, the natural intuitive explanation of the fact is that typically $B_{t+h}\simeq h^{1/2}$. – Christian Remling Apr 25 '17 at 19:27


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Forum moved here!

Home / Find Function Not visibly working

ChristianDieudonne

When I click on ‘Find’ under ‘Go To’, no search box or other search mechanism appears.

Using CTRL F does not produce anything either.

I have the toolbar enabled (though it’s not showing either).

GitHubRulesOK

Find is active when the Layout is in Fixed User Interface Mode so it sounds like you are in ebook mode. newer versions make it easy to switch usually using the “Debug” option then pressR to reload/refresh display.
If you need to switch ChmUI or are using an older version then you need to edit Advanced Options to switch EbookUI[
UseFixedPageUI = true

ChristianDieudonne

Hello.

I updated Sumatra to the latest version and now there is now not even a ‘Find’ function at all under ‘Go To’.

I don’t understand how to switch to Fixed User Interface Mode. In the newer version I no longer find ‘Debug’ as an option either.

In ‘Default Layout View’ under ‘Options’, Fixed User Interface Mode is not listed.

So, even though I’m now presumably using the newest version after updating, do I still need to edit Advanced Options? If so, how?

Thank you

GitHubRulesOK

Sorry that 3.2 debug option was removed again in 3.3x
So back to the old way

in settings advanced it should open in notepad then scroll down to this section

EbookUI [
FontName = Georgia
FontSize = 12.5
TextColor = #5f4b32
BackgroundColor = #fbf0d9
UseFixedPageUI = false
]

and change that false to true

ChristianDieudonne

HI, Thank you! I had somehow already figured that out myself in the meantime though, even though instructions I found online elsewhere were different - I just reasoned it out as far as the command information. So, problem solved. Now, with the ‘False’ direction changed to ‘True’ the ‘Find’ function appears readily and prominently.


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Math Help - Logarithmic Equation

1. Logarithmic Equation

Solve the equation, log4 ^(m + 2) - log4^ (m - 5) = log4^ 8

Is there a solution to this at all?

2. Originally Posted by BlueStar
Solve the equation, log4 ^(m + 2) - log4^ (m - 5) = log4^ 8

Is there a solution to this at all?
no there isn't

i assume the 4 is not the base. in that case

$\log 4^{m + 2} - \log 4^{m - 5} = \log 4^8$

$\Rightarrow \log \left( \frac {4^{m + 2}}{4^{m - 5}} \right) = \log 4^8$

$\Rightarrow \log 4^7 = \log 4^8$

which is not true

3. Originally Posted by BlueStar
Solve the equation, log4 ^(m + 2) - log4^ (m - 5) = log4^ 8

Is there a solution to this at all?
Here's another way to see that something is off with this equation

$\log 4^{m + 2} - \log 4^{m - 5} = \log 4^8$

$\Rightarrow (m + 2) \log 4 - (m - 5) \log 4 = 8 \log 4$

$\Rightarrow \left[ (m + 2) - (m - 5) \right] \log 4 = 8 \log 4$

equating coefficients we find

$\Rightarrow (m + 2) - (m - 5) = 8$

$\Rightarrow 7 = 8$ ---------> False

4. Hello, BlueStar!

I bet those 4's are bases . . .

Solve the equation: . $\log_4(m + 2) - \log_4(m - 5) \;= \;\log_4(8)$

We have: . $\log_4\left(\frac{m+2}{m-5}\right) \;=\;\log_4(8)$

Then: . $\frac{m+2}{m-5} \;=\;8\quad\Rightarrow\quad m + 2 \;=\;8m - 40\quad\Rightarrow\quad -7m \;=\;-42$

. . Therefore: . $m \,=\,6$

5. Originally Posted by Soroban
Hello, BlueStar!

I bet those 4's are bases . . .

We have: . $\log_4\left(\frac{m+2}{m-5}\right) \;=\;\log_4(8)$

Then: . $\frac{m+2}{m-5} \;=\;8\quad\Rightarrow\quad m + 2 \;=\;8m - 40\quad\Rightarrow\quad -7m \;=\;-42$

. . Therefore: . $m \,=\,6$

yes, i was thinking that as well. don't you just hate those typos?

i made the statement that i assumed 4 was not the base, but BlueStar said nothing about it, unless he/she doesn't know what that means ...or it really wasn't the base, let's give BlueStar the benefit of the doubt


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# Show that $11^{10} \equiv 1 (\mod 100)$

$11^{10} \equiv 1 \pmod{100}$

I tried to solve by using euler's theorem, But I got stuck.

$\gcd(11, 100) = 1$

$11^{φ(100)} \equiv 1 \pmod{100}$

$11^{40} \equiv 1 \pmod{100}$

I don't know how to go on as $11^{40}$ is bigger than $11^{10}$

• Binomial theorem? – Angina Seng Aug 28 '17 at 18:22
• I would suggest multiplying by hand $11 \times 11$ reducing modulo $100$ as you go, and see what transpires. Repeated squaring can be quicker for computing large powers. You could also try a binomial expansion of $(1+10)^{10}$. But try something and look out for helpful patterns. – Mark Bennet Aug 28 '17 at 18:23
• @MarkBennet I don't know this binomial theorem, this example can't be solved by euler's theorem ? – Goun2 Aug 28 '17 at 18:24
• Even if you do not use it for this question, the binomial theorem is a tremendously valuable thing to know. I recommend learning it. – David K Aug 28 '17 at 18:34

Hint:

$$(1+10)^n\equiv1+\binom n110^1\pmod{10^2}$$ for positive integer $n$

• I didn't understand :/, what do you mean with this ? – Goun2 Aug 28 '17 at 18:30
• Why doesn't the euler's theorem work for this case ? (11,100) are relatevely prime, it should work. – Goun2 Aug 28 '17 at 18:31
• @hjx He used the binomial theorem. – user236182 Aug 28 '17 at 18:33
• The theorem does work, bit it just isn't useful for this particular problem. – Batominovski Aug 28 '17 at 18:33
• @hjx Not knowing the Binomial Theorem is not an excuse for not trying to search for it, understand what it says, and find out about the proof, especially, when this theorem is one of the most fundamental theorems in mathematics that everybody should have learned at some point in their life. – Batominovski Aug 28 '17 at 20:09

Well..., you got $11^{40} \equiv 1 \pmod{100}\Rightarrow (11^{10}-1)(11^{10}+1)(11^{20}+1)\equiv 0 \pmod{100}$

Now it is easy to see that $(11^{10}+1)$ and $(11^{20}+1)$ end with $2$. Then $(11^{10}-1)\equiv 0 \pmod{25}$

Again $11^{2} \equiv 1 \pmod{4}\Rightarrow 11^{10} \equiv 1 \pmod{4}$

Since $\gcd(25,4)=1$, we will have $11^{10} \equiv 1 \pmod{100}\space\space\space\space\space\blacksquare$

$11^{10}-1$

$=(1+10)^{10}-1$

$=(1+\binom{10}{1}10+\binom{10}{2}10^2+...+10^{10})-1$

$=\binom{10}{1}10+\binom{10}{2}10^2+...+10^{10}$

is divisible by $100$.

Alternatively: $$11^{10}-1=(11^5-1)(11^5+1)=(11-1)(11^4+11^3+11^2+11+1)(11^5+1)=$$ $$10\cdot (...5)(...2)=...00.$$

Here is an indirect proof

Note that $\varphi (4) = 2$ and $\varphi (25) = 20$ so that if $r$ has no factor in common with $100$ we have $r^{20}\equiv 1 \bmod 4, \text{ and } \bmod 25 \text { hence also } \bmod 100$.

Note also that $6^2=36 \equiv 11 \bmod 25$

We have $$11^{10}=(11^2)^5 \equiv 1\bmod 4$$ and also $$11^{10}\equiv 6^{20}\equiv 1 \bmod 25$$ so that $$11^{10}\equiv 1 \bmod 100$$

Note that $11$ is not a square modulo $4$ or $100$, but the reduction of exponent by a factor of $2$ (which doesn't apply to the factor $4$ - with $\varphi (4)=2$ needing no reduction - $2$ is still a factor of the target exponent $10$) suggests looking for a square root modulo $25$.

You can use the binomial theorem if you want, but here it is a silly solution:

$11^2=21\operatorname{mod}100$, thus $11^{10}=(11^2)^5=21^5\operatorname{mod}100$.

If you compute $21^5=4.084.101=1\operatorname{mod}100$.

$$\qquad{11^2\equiv 121\equiv 21\mod 100\\\\ (11^2)\times 11\equiv 231\equiv 31\mod 100\\ (11^4)\equiv 31\times 11\equiv 341\equiv 41\mod 100\\ (11^5)\equiv 41\times 11\equiv 451\equiv 51\mod 100\\ (11^6)\equiv 51\times 11\equiv 61\mod 100\\ (11^7)\equiv 61\times 11\equiv 71\mod 100\\ (11^8)\equiv 71\times 11\equiv 81\mod 100\\ (11^9)\equiv 81\times 11\equiv 91\mod 100\\ (11^{10})\equiv 91\times 11\equiv 1001\equiv 1\mod 100\\}$$


text:
## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Dissipative property for non local evolution equations

#### Abstract

In this work we consider the non local evolution problem $\begin{cases} \partial_t u(x,t)=-u(x,t)+g(\beta K(f\circ u)(x,t)+\beta h), ~x \in\Omega, ~t\in[0,\infty[;\\ u(x,t)=0, ~x\in\mathbb{R}^N\setminus\Omega, ~t\in[0,\infty[;\\ u(x,0)=u_0(x),~x\in\mathbb{R}^N, \end{cases}$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N; ~g,f: \mathbb{R}\to\mathbb{R}$ satisfying\linebreak certain growing condition and $K$ is an integral operator with symmetric kernel, $Kv(x)=\int_{\mathbb{R}^{N}}J(x,y)v(y)dy.$ We prove that Cauchy problem above is well posed, the solutions are smooth with respect to initial conditions, and we show the existence of a global attractor. Furthermore, we exhibit a Lyapunov's functional, concluding that the flow generated by this equation has the gradient property.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 26, Number 1 (2019), 91-117.

Dates
First available in Project Euclid: 20 March 2019

https://projecteuclid.org/euclid.bbms/1553047231

Digital Object Identifier
doi:10.36045/bbms/1553047231

Mathematical Reviews number (MathSciNet)
MR3934083

Zentralblatt MATH identifier
07060318

#### Citation

da Silva, Severino H.; Garcia, Antonio R. G.; Lucena, Bruna E. P. Dissipative property for non local evolution equations. Bull. Belg. Math. Soc. Simon Stevin 26 (2019), no. 1, 91--117. doi:10.36045/bbms/1553047231. https://projecteuclid.org/euclid.bbms/1553047231


text:
# Homework Help: Limits and derivatives

1. May 20, 2016

### Ujjwal28

1. The problem statement, all variables and given/known data
What will be lim[2sin(x-1)/(x-1)], where x tends to 1?
[ ] denotes greatest integer function.
2. Relevant equations
Can I directly solve it using the formula sinx/x =1 when x tends to 0

3. The attempt at a solution
Okay so the quantity inside [ ] can be written as ——>>2 sin(x-1)/(x-1).. And sin(x-1)/(x-1) should equAte to 1 and only 2 should be left inside the greAtest integer function and therefore the answer should be 2... But the answer is 1 what am I doing wrong please explain...

2. May 20, 2016

### Tom MS

You really should be right. Can you make sure you wrote it down correctly?

3. May 20, 2016

### Ujjwal28

Lol yes it the same question I too was shell shocked to see the answer

4. May 20, 2016

### Samy_A

What you claim is that $\displaystyle \lim_{x \rightarrow 1} [2\sin(x-1)/(x-1)] = [\lim_{x \rightarrow 1} (2\sin(x-1)/(x-1))]$.

But is that claim correct? Is [] a continous function?

5. May 20, 2016

### SammyS

Staff Emeritus
This limit is not 2 .

The given answer, that the limit is 1, is indeed correct.

6. May 20, 2016

### Ujjwal28

How did you get to that? Could you please explain?

7. May 20, 2016

### Samy_A

Can you explain why you think that $\displaystyle \lim_{x \rightarrow 1} [2\sin(x-1)/(x-1)] = [\lim_{x \rightarrow 1} (2\sin(x-1)/(x-1))]$?

8. May 20, 2016

### Ujjwal28

I don't know what effect does it really have? So please explain it to me in detail

9. May 20, 2016

### Samy_A

Let's start with the RHS: $\displaystyle \ [\lim_{x \rightarrow 1} (2\sin(x-1)/(x-1))]$.
As you stated, $\displaystyle \lim_{x \rightarrow 1} (2\sin(x-1)/(x-1))=2$, so that $\displaystyle \ [\lim_{x \rightarrow 1} (2\sin(x-1)/(x-1))]=[2]=2$.

For the LHS, $\displaystyle \lim_{x \rightarrow 1} [2\sin(x-1)/(x-1)]$, you can't simply reverse the order of operations, swapping the limit with the []. That only works for continuous functions.

[] is not continuous at x=1. If that is not clear to you, try to prove this first.

To evaluate $\displaystyle \lim_{x \rightarrow 1} [2\sin(x-1)/(x-1)]$, you first have to evaluate $[2\sin(x-1)/(x-1)]$ for x close to 1, but not equal to 1. Try it, and see what you get.

10. May 20, 2016

### Ujjwal28

I have no idea what you're telling me

11. May 20, 2016

### Samy_A

That was quick.

Anyway, then we are back at post #7. Why did you think that the limit should be 2? Give your detailled reasoning, then someone will hopefully be able to help you find the error(s).
Because, as @SammyS wrote, the correct limit is 1.

12. May 20, 2016

### Ujjwal28

Someone please explain it to me

13. May 20, 2016

### SammyS

Staff Emeritus
Under what conditions is $\ \lim_{x\to a} f(g(x)) = f(\lim_{x\to a} g(x)) \ ?$

14. May 20, 2016

### Ujjwal28

15. May 20, 2016

### SammyS

Staff Emeritus
It's dangerous to use a theorem without paying attention to the conditions under which the theorem is valid.

We can get back to that later. You might continue to ponder that.

As an alternate way to understand this:
Graph the function $\displaystyle \ \left[ \frac{\sin(x-1)}{x-1} \right ] \ .$

16. May 20, 2016

### Staff: Mentor

Just to be clear, the function in question is $\lfloor \frac{\sin(x-1)}{x-1}\rfloor$, the greatest integer function -- the largest integer less than or equal to $\frac{\sin(x - 1)}{x - 1}$

The LaTeX for this is $\lfloor$\dots$\rfloor$

17. May 20, 2016

### SammyS

Staff Emeritus
Right. Since OP was using [ ] , I continued using that.

With the standard floor symbols: $\displaystyle \ \left\lfloor \frac{\sin(x-1)}{x-1} \right\rfloor \$

Last edited: May 20, 2016
18. May 20, 2016

### Ray Vickson

Although the function $f(h) = \sin(h)/h$ has a limit of 1 as $h \to 0$ (where $h = x-1$), the question is: how does $f(h)$ behave for $h \neq 0$ but near 0? Is it a little bit bigger than 1? Is it a little bit smaller than 1? You cannot answer these questions just from the limit alone; you need a bit more information.

If $f(h) < 1$ for small $|h| \neq 0$ then $1 < 2 f(h) < 2$ for small, nonzero values of $h$, and that will mean that $\lfloor 2f(h) \rfloor = 1$ for small nonzero values of $h$.

19. May 20, 2016

### ehild

Evaluate $\lfloor 2 \frac {\sin(x-1)}{x-1}\rfloor$ for x=0.9, 0.99, 0.999, and x= 1.1, 1.01, and 1.001. What do you get? So what is the limit if x-->1?
(Set your calculator to SCi digit 9)

Last edited: May 21, 2016
20. May 20, 2016

### Ujjwal28

I don't know how to procede

21. May 20, 2016

### Ray Vickson

What is stopping you from taking the advice offered in post #19?

22. May 21, 2016

### ehild

What is the greatest integer of 1.999999?


text:
# Consider the basic strength of amines in aqueous solution,which one has the smallest $pK_b$ value?
$\begin{array}{1 1}(A)\;(CH_3)_3N\\(B)\;C_6H_5NH_2\\(C)\;(CH_3)_2NH\\(D)\;CH_3NH_2\end{array}$


text:
Australia's war on encryption: the sweeping new powers rushed into law
theguardian.com/technology/201

With all the vague "end of the internet!" or "why is everyone so upset at such good idea" stories elsewhere on the new Australian backdoor-your-apps law, it's good to see a piece that explains what it actually does.

The moral seems to be: don't allow or agree to any automatic updates while in Australia. Australians, if you want security from govt snooping and from security flaws, tough luck.

A Mastodon instance for maths people. The kind of people who make $\pi z^2 \times a$ jokes.
Use $ and $ for inline LaTeX, and $ and $ for display mode.


text:
# Toughest ACT Curves of the Past Decade

We sifted through almost every single released ACT curve from the past decade to show you the worst and the best.

A tough curve means that even a few wrong answers will make your score plummet.

A forgiving curve means that a few wrong answers won’t hurt your score too much.

If you don’t know what a curve is or how you’re scored based on one using your raw score, I highly recommend you start here for some background information.

First, a note about curves: some curves are harsh at the top yet forgiving in the middle (1-2 wrong will make your score plummet, but once you’re up to 9-10 wrong, your score doesn’t drop as much as you get more wrong). Other curves are forgiving at the top and harsher in the middle (1-2 wrong won’t make your score drop too much, but once you’re up to 9-10 wrong, your score drops a bit more as you get more wrong).

Because most of the students reading this blog are aiming for the top, we focused on how harsh the top of each curve was (scores of 30+).

Read further below for how we determined curve harshness.

#### ACT English Curves

ToughestMost Forgiving
Raw ScoreJun 2012Jun 2015
753636
743536
733435
723335
713235
703134
693033
682932
672831
662731
652730
642629
632628
622528
612527
602427
592426
582426
572325
562325
552325
542224
532224
522124
512123
502123
492123
482022
472022
462022
451921
441921
431920
421820
411820
401719
391719
381618
371618
361617
351517
341516
331516
321416
311415
301315
291315
281214
271214
261114
251113
241113
231012
221012
21911
20911

#### ACT Math Curves

ToughestMost Forgiving
Raw ScoreApr 2005Dec 2008
603636
593436
583336
573235
563135
553134
543034
533033
522933
512932
502831
492830
482729
472729
462728
452628
442627
432627
422526
412526
402425
392425
382424
372324
362323
352223
342222
332121
322121
312020
302019
291919
281918
271818
261818
251817
241717
231717
221716
211616
201616

ToughestMost Forgiving
Raw ScoreDec 2009Red Book 3*
403636
393436
383336
373135
363034
352933
342832
332731
322630
312529
302428
292428
282327
272226
262225
252124
242124
232023
222022
211921
201820
191719
181718
171617
161616
151515
141415
131414
121313
111313
101212
91112
81111
71011

* Red Book 3 stands for the third test in The Real ACT, 3rd Edition, which contains five actual past exams.

#### ACT Science Curves

ToughestMost Forgiving
Raw ScoreJan 2006Dec 2013
403636
393436
383236
373035
362935
352834
342732
332631
322630
312529
302528
292427
282326
272325
262225
252224
242123
232123
222022
212022
201921
191920
181920
171819
161819
151718
141717
131616
121515
111414
101413
91312
81211
71110
6109

#### Methodology

Warning: For ACT Geeks Only

We graded each curve on several factors. First, we looked at the minimum raw score (minimum # correct) needed to get a 30 for each section. The higher the minimum, the tougher the curve.

Red – the toughest curves
Green – the most forgiving curves

Out of 75Out of 60Out of 40Out of 60
Red Book 169523337
Red Book 267523235
Red Book 367533234
Apr 200569533336
Jun 200568513536
Dec 200568523435
Jan 200667503337
Apr 200668533335
Jun 200668533435
Apr 200766523436
Apr 200867503533
Dec 200868493432
Apr 200967503634
Jun 200967503535
Dec 200968533635
Apr 201069533436
Apr 201169523536
Apr 201269513534
Jun 201269513434
Dec 201267513435
Apr 201367523433
Dec 201366523432
Apr 201467513434
Jun 201466503335
Dec 201467523434
Apr 201566523234
Jun 201565513434

* Red Book is short for The Real ACT, 3rd Edition. Red Book 1 stands for the first test in that book.

To get the toughest curves, we looked for those with the highest minimums and filtered out the rest. Because there are multiple curves with the same minimum, we took each of the remaining curves and averaged the scaled scores above the 30 mark. The lower the average, the tougher the curve. The higher the average, the more forgiving the curve.

For example, the June 2015 English section:

Finding the average of the scaled scores above the red line for the English section, we get

$\dfrac{30 + 31 + 31 + 32 + 33 + 34 + 35 + 35 + 35 + 36 + 36}{11} = 33.4545$

Here are the overall results that we used for the basis of comparison:

#### ACT English Curves

TOUGH:

 Test Date: Red Book 1 Apr 2005 Apr 2010 Apr 2011 Apr 2012 Jun 2012* Average: 33 33 33 33 33.2857 33
* Because of the ties, we averaged out a few more scores further down along each curve. The Jun 2012 test won out as the toughest curve.

FORGIVING:

 Test Date: Jun 2015 Average: 33.4545

#### ACT Math Curves

TOUGH:

 Test Date: Red Book 3 Apr 2005 Apr 2006 Jun 2006 Dec 2009 Apr 2010 Average: 32.625 32.125 32.75 32.75 33 33.125

FORGIVING:

 Test Date: Dec 2008 Average: 33.75

TOUGH:

 Test Date: Apr 2009 Dec 2009 Average: 33 32.8

FORGIVING:

 Test Date: Red Book 2 Red Book 3 Apr 2015 Average: 33.1111 33.6666 33.2222

#### ACT Science Curves

TOUGH:

 Test Date: Red Book 1 Jan 2006 Average: 33.75 33

FORGIVING:

 Test Date: Dec 2008 Dec 2013 Average: 33.6666 33.8888

It turned out to be pretty accurate in quantifying how tough or forgiving each curve was.

Enjoy!


text:
# tiOPF/fr

Deutsch (de) | English (en) | français (fr) | italiano (it) | 日本語 (ja)

## Contents

The TechInsite Object Persistence Framework (tiOPF) is an Open Source framework of Delphi/Object Pascal code that simplifies the mapping of an object oriented business model into a relational database. The framework is mature and robust. It has been in use on production sites for over seven years. It is free, open source, and available for immediate download with full source code.

Some of the key features of the tiOPF include:

• Lets you build an object oriented application that can swap databases with the flick of a command like a command line parameter or a change of a compiler directive. Currently, there are persistence layers for Interbase-IBX, Oracle-DOA, XML-MSDOM, XML-XMLLite, Paradox-BDE, MSAccess-ADO, MSSQLServer-ADO, FireBird-FBLib, FireBird-SqlDB, HTTP Proxy-Remote, CSV files and TAB files.
• Family of abstract base classes for building a complex object model
• 27 Persistent object aware components for building complex GUIs (Delphi only).
• Model-GUI-Mediators for enabling any standard GUI component to become Object Aware.
• 1400+ DUnit/fpcUnit tests to guarantee stability
• 160 pages of documentation to get you started
• News groups for support
• Automated, daily builds and unit testing
• Lots of demos focusing on specific parts of the framework for easy learning.
• Cross platform. Currently tested on Windows and Linux.

### Authors

Peter Hinrichsen - Original Developer.
Graeme Geldenhuys - Ported to Free Pascal.

The latest stable release can be found on the tiOPF Download page.
The latest development code can be found on the SubVersion server as follows

svn co https://tiopf.svn.sourceforge.net/svnroot/tiopf/tiOPF2/Trunk Source

An easy way to get the code, is to use the command line svn client as follows. The commands below will checkout the Source, Demos and Docs directories of tiOPF version 2.

 mkdir tiOPF2
cd tiOPF2
svn co https://tiopf.svn.sourceforge.net/svnroot/tiopf/tiOPF2/Trunk Source
svn co https://tiopf.svn.sourceforge.net/svnroot/tiopf/tiOPF2_Demos Demos
svn co https://tiopf.svn.sourceforge.net/svnroot/tiopf/tiOPF2_Docs Docs

Try and keep the above directory layout. It is much easier to support and the documentation build scripts will work without modification. The Source checkout will download some external links. Most of them are only related to Delphi, but there is one directory called FBLib which is required for the FBLib (Firebird RDMS) support. tiOPF requires a slightly modified version of FBLib 0.85 that adds some extra feature which will hopefully make it into the next FBLib release.

### Dependencies / System Requirements

• Compiler: Free Pascal 2.0.4 (FPC 2.1.1 though more risky, also works. It is used for the nighly builds under Linux).
• Components for your required persistence layer, if it is not included with the compiler. eg: FBLib for FireBird RDMS.

Status: Stable (Tested on Windows and Linux.)

Issues: None

### Installation

#### The Packages

Inside the Source\Compilers\FPC directory there are four packages.

tiOPF
Core units (run-time only package)
tiOPFGUI
GUI related units and components (run-time only package).
tiOPFGUIDsgn
Registers/Installs the components into the Lazarus component palette (design-time only package). The GUI components used under Lazarus are still experimental and under heavy development. It is recommended to get frequent updates from SubVersion if they are going to be used.
tiOPFHelpIntegration
Integrates the fpdoc generated help files into Lazarus's help system (design-time only package)

#### The Setup

• Unzip the zip file or check out the source from SubVersion to a location of your choice.
• Open Lazarus
• Open the package tiOPF.lpk with Component/Open package file (.lpk) located in the Source\Compilers\FPC directory.
• Click on Compile
• Open the tiOPFGUI.lpk package and click Compile

Optional

• Open the tiOPFGUIDsgn.lpk package and click Compile and then Install (Lazarus should rebuild and restart).
• Open the tiOPFHelpIntegration.lpk package and click Compile and the Install (Lazarus should rebuild and restart).

NOTE #1
I included the FBLib Firebird Library components as the default persistence layer for Free Pascal in the tiOPF.lpk package. That is the only one I use under Free Pascal/Lazarus. So make sure you have FBLib installed, or you need to remove it from the tiOPF package.

Persistence layers are controlled by a Compiler Directive under Compiler Options -> Other -> Custom Options. eg: The LINK_FBL directive relates to the FBLib components. For all the available options see the end of the tiOPFManager.pas unit.

NOTE #2
For the Integrated Help to work, Lazarus needs to know how to find the html help files. Please read the tiOPFHelpIntegration.txt file located in \Source\Compilers\FPC for further instructions.

### Usage

In Lazarus, open your project and add tiOPF as a Required Package (Project -> Project Inspector -> Add). Include tiObject in your uses clause. You are now ready to create objects descending from TtiObject or TtiObjectList.

See the example projects in the Demos directory for additional examples.