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It's my first day here. I arrived about 1:15, and listened to Dr. Noe explain the "negative resistance" from a HeNe laser tube. After that, I listened to a student from another lab speak about her experimental apparatus for studying quantum chaos with a microwave cavity. After that, I set up my UNIX account and webpage. I began reading "Experimental Observation of Wave Chaos in a Conventional Optical Resonator" (J. Dingjan, E. Altewischer, M.P. van Exter, and J.P. Woerdman, from Physical Review Letters).
I got a copy of Chaos and Nonlinear Dynamics by Robert C. Hilborn from the library upstairs, and have started reading it. One example was about an ac circuit with a diode in it, which prompted me to go and spend some time studying ac circuits, their resonance, semiconductor, diodes, and transistors in the Hecht physics textbook. I listened to a seminar on Atomic Nanofabrication. Brendan showed me his apparatus for detecting chaos in the flickering of a HeNe laser tube without enough voltage to maintain a steady beam.
I did some further reading in Chaos and Nonlinear Dynamics, and then spent a lot of time writing a computer program as an exercise from the book. It simulted the logistic growth map, and found the periodic behaviors, and where the bifurcations of its behavior occur. My hand calculations of Feigenbaum's delta based on the data were fairly accurate, but the automated delta-calculator still isn't working. But the program can very nicely and quickly generate data, and analyze the periodicity. I also did some more reading on applications of nonlinear dynamics to lasers, specifically in the competition between transverse modes. It has been shown that this competition does lead to chaotic behavior.
I finished, debugged, improved, and re-debugged my logistic growth map simultar. The most important part of the program is its ability to calculate the Feigenbaum delta from the simulation data, and results are interesting. For the bifurcations 1->2, 2->4, and 4->8 (Delta-sub-1), I got Delta = 4.74737, a 1.67% error. For 2->4, 4->8, and 8->16 (Delta-sub-2), I got 4.52381, a 3.11% error. For 4->8, 8->16, and 16->32 (Delta-sub-3), I got 5.25000, an ugly 12.44% error. This is interesting, because the definition of the Feigenbaum delta is the limit as n goes to infinity of Delta-sub-n, or 4.66920161..., and my data seems to match that better at lower n's, as opposed to the expected higher n's. The improvements still to be made are increased resolution in searching for higher-order bifurcations (with self-similarity), and the calculation of the Feigenbaum alpha scaling constant. Here are the files:
logistic map simulator simulator binary (Linux)
source code (C++)
apvector class source code
apvector class header file
The apvector class files are graciously stolen from ETS (Educational Testing Service), from their AP Computer Science curriculum.
I also continued research on transverse intensity distributions of lasers, and nonlinear dynamics state spaces (1D and 2D).
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I started off by doing some web research on fiber optics and hollow laser beams, and assembling about 15 links on my weblinks page. I'm thinking of working with the transverse modes that come out of multimode cable, specifically the strange hollow beam that results from shining a Gaussian TEM00 beam incident at an angle to the coherent fiber cable we have here. It may be interesting to see if it acts as a Laguerre-Gaussian (LG) mode when put through an astigmatic mode converter. If it does, it would be expected to emerge as a TEM01 Hermite-Gaussian (HG) mode, rotated at 45 degrees.
I did a little experimentation with the coherent cable today. First, I verified that the "circle effect" occurred for laser light, but not with incident images, such as text. The text tilted and blurred, as expected. I plan to check if a flashlight-type source will exhibit this behavior. If it doesn't, that means that this effect is truly unique (or almost unique) to laser light, which means that it is probably linked to some special property of laser light: coherence, monochromaticity, divergance, etc. Then, I verified what I had heard, which is that linearly polarized light coming in will come out unpolarized. I believe this is because the light bounces around so much inside the cable, that the independant beams are no longer "rotated" the same way with respect to each other. I have set up a small apparatus to get the HeNe light through the cable and out onto a breadboard. On the other end of the cable, I am setting up an interferometer with a beam splitter and two plane mirrors, so I can test coherence.
Further playing with the coherent cable defeated the idea of some link between laser-specific properties and the circle behavior. This was because I found two other light sources that are able to produce the effect: a Mini-Maglite flashlight, and the reflection of that light off of plain white paper. Now the question is why this occurs with these light sources, and not with an image such as text. The next thing I tried was a complex pattern from a light source. For this, I used a laser pointer pen through the lens from "3D glasses", which is like a kind of diffraction lens. The incident image on one end of the cable was rectangular grid array of small filled-circular dots. Coming out of the cable when whe incident image was parallel to the normal of the plane, were a few overlapping empty circles. As the angle was increased from 0, the circles increased in radius, which means it exhibits the "circle behavior". As well as these things, I also tested the effect of intensity on the behavior. I set up a laser pointer to shine through two polarizers, then into the cable. As one polarizer rotates, the component of the light though it that is parallel to the polarization direction of the other polarizer is changed. The effect is that this is a simple intensity dimmer. I found intensity to not effect the shape of the circle pattern. There are pictures of all of these things on my pictures page.
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The day has just started, and I already have something to report. As I was leaving last Friday, Professor Metcalf handed me a box of fiberoptic cable to play with. This morning, I experimented with three of them. Two are incoherent fiberoptical bundles, one of which is bifurcated. The third is a very small coherent bundle. The coherent bundle, as expected, exhibited the circle behavior from a TEM00 beam. But the interesting thing was that the two incoherent bundles exhibited the same kind of multiple circle behavior with a TEM00 beam that the coherent cable had with a complex incident beam. So the circle behavior of a TEM00 beam through the coherent bundle can be seen as a specific and trivial example of a more generalized kind of transformation. So from now on, I will refer to all instances of the circle behavior as the circle transformation, since it transforms an image. I still do not understand why reading text at an angle through the coherent bundle does not cause this behavior. Also, I was able to see the circle transformation occur in multimode fiber, which is not a bundle, but a single strand. I still haven't gotten it with single mode fiber, because I don't have it coupled to the HeNe laser yet (the diode laser I used for most of the other experiments can't be mounted securely enough to couple it to single mode fiber). I spent the rest of the day showing things in the lab to and talking to the new people here, from the Simons Fellowship program
Today, Professor Metcalf gave his second talk on quantum mechanics, focusing on explaining the nature of wavefunctions: they are complex, and the "recipes" for removing the complexity correspond to measureable properties of the object. The wavefunction itself is unmeasureable, but it can be calculated. Later, REU students came in and looked at what we're doing here, and I explained my fiber optics setup.
Early in the day, Dr. Noe had us all discuss possible methods for Jennifer to take data on her project. She has a fishtank filled with liquid that has a gradient of index of refraction, with the higher indices towards the bottom. We were trying to find a way to measure the index as a function of the height above the bottom of the fishtank, without seriously disturbing the system. We came up with a lot of ideas, but non-disturbance is difficult. After that, we listened to Professor Metcalf give another talk on quantum mechanics. It was mostly about electrons around an atom, and their properties, specifically: mass, magnetic moment, and charge. Also important is their indistinguishability. I spent the afternoon trying to get the CCD array running so I can take data on the transverse intensity distribution of the circle transforms. With Owen's and Jose's help, I got it working a little, but it still needs a lot more adjustment to make it really useable. Tomorrow and Friday are off, so Monday's mission: get the CCD array reliably working, and start taking transverse mode data on the circle transform patterns.
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Within 20 minutes, I was able to adjust the CCD array and start taking nice pictures. This was mostly achieved with a 10cm focal length lens, intensity cutting with paper and a polarizer, and manipulation of the software parameters, gain and bias. The pictures are posted. In working with the CCD array and its software, EDC-1000N IS, I discovered two annoying bugs. Firstly, if you save a file to a location where there is not enough room, the program will just save an empty file, and not even tell you. This caused me to re-setup my apparatus and take my data all over again for most of my readings. And even more annoyingly, the program has serious problems with saving files in .BMP format. The pictures come out distorted, as you can see in my two side-by-side comparisons on the pictures page. This does not occur for the default file format, .TIF, and I was not able to see if it happens for the two other options, .FTS and .RAW. After that, I took these pictures and analyzed them with Scion Image, an advanced image processing program. The results of this are still being put together on my fiberoptics page.
I got better acquainted with Scion, and cleaned up a lot of my images. Then I continued work on my fiberoptics page, and did a best-fit graph for the transverse intensity plot of the red diode laser. As I expected, it fit well with a Gaussian curve.
I spent some more time today with the fiberoptics page. I also generated approximate best-fits of superimposed Gaussian and superimposed Lorentzian curves. Since there is a problem mounting floppy disks, I haven't been able to match them up agaisnt the transverse data for the Schott cable beam, but from eyeing it it looks like the Lorentzian is more like the data. This may seem unlikely, but there is no real reason to favor Gaussian over any other curve, since what is coming out of the cable is not a solution to the same set of boundary conditions and paraxial wave equation as a laser cavity. Lorentzian curves are quite common in physics, most notably as the shape of certain spectral lines. So there is no reason not to believer Lorentzian is the better model.
Since the floppy drive was not fixed yet, and once it was the computers were all taken, I spent the morning reviewing and learning more about classical EM theory, spherical and cylindrical coordinate systems, and related topics. I did this by going through Chapters 2 and 3 of Optics by Hecht, which is an excellent book. After lunch, I finally was able to generate and superimpose best-fit curves for the transverse data from the hollow beam coming out of the Schott coherent bundle. Both plots are now on my fiberoptics page. It turns out that the two superimposed Gaussian curves were way off, and that it is very close to two superimposed Lorentzian curves. This is not surprising, since there is no real reason to favor any Gaussian or other "normal laser mode", since the beam from the coherent bundle is not a solution to the same wave equation and set of boundary conditions. The fact that the Lorentzian curves fit indicates that some other physical process is at work. Now I have to figure out what.
Unfortunately, I made a mistake. My data showing that the red diode laser had a Gaussian beam was absolutely wrong. Diode lasers can't be Gaussian, because they have rectangular apertures. So I commented out that data from the fiberoptics page, and I am working on re-taking the data with the open-cavity HeNe laser that Owen got working yesterday. I helped him couple the CCD array to it, and took some nice pictures that I will post tomorrow. I am going to try to best-fit the HG function to what looks like a very nice TEM02 mode. The surface plot looks quite accurate. One thing I thought of investigating, that I will probably get to on Monday, is seeing if the beam coming out of the coherent cable when the incident angle is zero degrees is actually Gaussian, as it appears. It seems to me that logically it should be Lorentzian, analogous with its behavior when the angle is greater than zero degrees. If it is, that means we are dealing with what might not be a perfect image translator. I got some data sheets on the Schott coherent cable, but couldn't get any from Dolan-Jenner, the company that makes the two non-coherent bundles I have. Contrary to my hunch, the two Dolan-Jenner cables turned out to be the exact same model, B424.
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At the suggestion of David McGloin from University of St. Andrews, I interfered a Gaussian beam from a HeNe with the hollow beam resulting from the Gaussian beam going through the coherent fiber bundle. Due to its phase singularity, an LG mode interfering with a Gaussian beam would make a spiral pattern. This is what I was looking for, even though I was quite sure the beam was not LG. Although I got a pattern appearing to be spiral, but upon closer examination, this was just a side-effect of the beam expander I was using on the Gaussian beam to try and match its size up with the hollow beam. Actually, I don't even know if the coherent bundle output is coherent, and I suspect that it is not. If it is not, then an interference pattern would be unnoticeable anyway. After that, I went to a talk on neuroscience. Specifically, the speaker was analyzing the effect of synchronous synapse firing, and how it is affected by "top-down" learning. When I came back, I discussed project ideas with Dr. Noe. I like the idea of investigating bessel beams, which have extremely little divergence. This makes them especially useful in atom-trapping applications. One way of making them is by shining an LG mode into an axicon lens, which is a special and expensive conical lens that makes hollow beams.
I did some more reading on hollow beams, and specifically hollow beams. I also spoke with Seung Hyun Lee, a grad student who worked with the astigmatic mode convertor, and he said he will get me a copy of his paper on the topic. I did some reading in Optics by Hecht and Lectures, Vol. II by Feynman on two topics I have been meaning to learn about: ray tracing and matrix geometrical optics in Hecht, and tensors in Feynman. Laser summer, when I was working at the Cooper Union, the mechanical engineering professor I was working with unsuccessfully tried to explain tensors to my group. Reading Feynman is much better than listening to him.
I got the papers from Seung Hyun Lee, which I will read later today and tomorrow. I also got two of the refernces from papers by the St. Andrews group, but most were so old, I couldn't get them online in full text (80s-early 90s). Professor Sam Goldwasser, of Sam's Laser FAQ fame, came today. He gave a talk on a diode laser he is building with a team at Drexel, and talked to us about laser safety and other laser topics. Then he came back to the lab and looked around at what we are doing. I also spent some time on a very interesting computer simulation problem that Owen was having. I haven't quite got it yet, though.
Most of today was spent doing background research about the astigmatic mode convertor, and hollow beams. I read a few journal articles on the topic, and I plan to put together the astigmatic mode convertor that is mentioned in "Astigmatic laser mode converters and transfer of orbital angular momentum" (M.W. Beijersbergen, L. Allen, H.E.L.O. van der Veen, J.P. Woerdman). The article discusses the mathematical relationship between HG and LG laser modes, and the Gouy phase shift involved in the experimental conversion. Another fascinating article I read was "Second-harmonic generation and the orbital angular momentum of light" (K. Dholakia, N.B. Simpson, M.J. Padgett, L. Allen). In that experiment, an HGnm mode with m=0 is converted into an LG mode with l=n-m and p=0 using an astigmatic mode converter. This LG beam is then frequency-doubled, using a nonlinear crystal. Since p=0, a simplified expression for the LG beam is used. When this is squared (as is done in the power expansion of the electric field for the second harmonic), the l index becomes 2l. So now, the LG beam is p=0 and l=2(n-m). When the frequency-doubled LG beam is put through another astigmatic mode converter, the HG beam the comes out is n=2n and m=0. So by doubling the frequency of the beam in a different mode, the order of the mode is doubled as well. When p>0, however, the resultant frequency-doubled mode is too complex to analyze.
I started the day with more journal-reading. I got the open-cavity HeNe to give off a nice TEM01, which appears to be 45 degrees tilted. But when I put human hair across the cavity, I didn't get a TEM01 and TEM10 as expected. Monday I will try and get cylindrical lenses so that I can make the mode converter.
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I was able to obtain two cylindrical lenses today, from Professor Metcalf's group. They appear to have the same focal length, about 15cm, though they were not labeled. One seems to have some type of coating, probabl antireflective, that the other does not. I separated them by f*sqrt(2), facing opposite directions, and shone HeNe light through. I played with lens combinations, but was unable to get a mode conversion. Tomorrow I will remember to bring the paper I was reading on the construction of an astigmatic mode converter, so I will hopefully get it to work then. I also attended a seminar on francium trapping. I didn't understand most of it, but I got the cool fact that at any given time, only about a teaspoon of francium exists on earth.
I spent some time trying to adjust the lenses on the astigmatic mode converter today, without much luck. I went back to the paper to see if I could get anything out of them. It seems that my main problem is now trying to achieve mode-matching. This required that the Rayleigh range of the beam hitting the first cylindrical lens is a certain amount. This is required so that the rays can be properly manipulated so that there is astigmatism on just one axis between the cylindrical lenses.
I tried to calculate specifically what I need to do to make the converter work. The required Rayleigh range is 25.6cm, and I believe the one out of the HeNe is about 80cm. I know HeNes can have ranges that are a few meters, but I think this one is shorter. Therefore, I need a way to drastically decrease the Rayleigh range. Alternatively, I could increase the focal length of the cylindrical lenses. This is what doesn't make sense to me. The people who did this experiment already used focal lengths of 19mm, implying a Rayleigh range of only 3.2cm. This seems very small, although they did use a f=16cm lens to mode-match. Maybe I would just need to match up my lens correctly. I also listened to the undergraduates (Doug, Brendam, Jose, Jill, and Jennfier) give practice talks, since they have their REU presentations tomorrow.
Today we listened to the REU students give their final presentations. We saw the ones that worked here at the LTC, and others that did optical and laser projects. They were all really good, especially one about laser seeding, which is using a "master" laser to pump a "slave" laser at the same wavelength as the master. It allows much greater power than would otherwise be possible. Pete took us to Wendy's. Wendy's rules.
I spent most of today teaching myself more optics. I started with matrix ray optics, because I find it so elegant. It's the kind of thing that should've definately been taught in AP physics, since it's not very difficult. After that, I moved on to Gaussian beam optics, which is very cool. It's the extension of geometrical optics made by taking into account diffraction effects, so that the width of a beam at a focal point is non-zero, unlike in geometrical ray optics. As a Gaussian beam is focused, the shape of the boundary of the beam (defined as the 1/e2 intensity point) is hyperbolic. The equations and concepts of Gaussian optics are necessary to understand the mode-matching difficulties that I'm having with the astigmatic mode converter. Unfortunately, I don't know how to treat non-zero order HG beams in the language of Gaussian optics, or even if it is possible. I need to look into that.
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I found out today that we are hosting a brown-bag lunch for the Simons Fellows tomorrow, and that we all need to prepare short presentations. Since none of my equations look very good, and it currently takes a long time to create the images of them, I taught myself some basic LaTeX, and then extensively covered its math mode. This took almost the entire day, but it ended with some very nice results that can be seen on my transverse modes research page. The equations were ripped out of pages generated by LaTeX2HTML. My Gaussian optics studies last Friday have already come in handy; I discussed with Waldo and Pete how to better focus and collimate the near-IR diode laser they're using for their optical tweezers. They need an extremely tiny spot size.
Our presentations and demonstrations for the Simons Fellows were today. They ran long, but at least they went well. I got to show off the open HeNe and its modes, and got excited at one point when I accidentally generated what looked like a hollow beam, right out of the laser! After the Simons Fellows left, and then after lunch, I went back and analyzed it, and it turned out to just be a very complex array of HG modes. I could tell it was because it had very small but symmetrical "wings" at opposite sides of the beam, and the circular-appearing part had symmetrical intensity differences going around the circumference.
I spent the day trying in vain to get the astigmatic mode converter to work. I calculated, using beam optics, the relationship between the distances, the lenses, and the Rayleigh range of the HeNe. I think that my mistake is the assumption that the waist is at the aperture of the laser. In fact, I think it should be at the plane mirror. The cavity has a 600mm focal length spherical mirror on one end, and a partially transmitting plane mirror at the other. Tonight, I will re-calculate, and tomorrow, I will re-experiment.
Finally, IT WORKS!!! Well, kind of. I got what looks like like both an LG01 and the input HG01 mode at once. Based on a rough analysis in Scion Image, I think it's 30% LG01 and 70% HG01. Not bad, considering I had nothing until now. The only problem with the calculations from yesterday was what I had expected: I took the waist to be in the wrong place. On my transverse modes page, I added a section on the astigmatic mode converter. It explains how it works, shows pictures of it and the output beam, has diagrams of it, and shows the calculations leading up to the parameters (distances and focal lengths) I chose.
I woke up at 6am, so I could get here on time to go on the trip to Brookhaven National Lab. I went with Pete, Waldo, and Evan for some good food at Country Kitchen, and we hurried to get to the SAC Circle on time. We were the ONLY ones on time, including the bus and the head of the Simons program. Anyway, the trip itself was really cool. It's such an amazing place, and the STAR detector is a massive piece of super-sexy machinery.
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To start off the day, I re-read the paper about the efficiency of
astigmatic mode converters, and noticed that it mentions that improperly
aligned AMCs often produce beams with the two intensity blobs across from
each other on the hollow beam. So my AMC isn't insane, it's just somewhat
misaligned. Tinkering only made it worse, so I decided to re-construct it
with a larger focal length mode-matching lens, which means that
absolute changes in the distance (on the order of centimeters) will cause
a smaller percentage change of Rayleigh range, which should hopefully
allow better fine-tuning of the AMC. I was unable to get an exact focal
length, unfortunately, but it is between 510 and 550mm. I double-checked
my calculations against the ones in the efficiency-related journal
article. We arrived at the same results, but just expressed
differently. Their results were expressed as desired beam widths, and
mine as desired Rayleigh ranges. They are simply related, but upon
reflection, I think it may be effective to directly measure spot size
directly. Since the new AMC isn't working yet, I'll probably use that
method to align it, as soon as I get it to reliably work. I will have to
use the CCD camera, since working "by eye" is too inaccurate. Scion
should be able to easily find the 1/e2 points with its profile
plot, as long as I avoid saturation. So that's tomorrow's goal. By the
way, I momentarily had the open-cavity HeNe produce an LG mode directly,
and got one poor but unsaturated CCD image before it misaligned somehow
(it was probably super-unstable, and some small vibration did
it). Tomorrow I will analyze it with Scion and make sure it is really
LG.
I made a list of topics and sub-topic for my eventual report, and then a list of contacts for the astigmatic mode converter, in case I need some help. I also measured the far-field divergance angle for the open-cavity HeNe, and therefore got the Rayleigh range. I did some calculations, and tomorrow, I will re-make the astigmatic mode converter.
I re-made the astigmatic mode converter with my re-calculated settings, and I got something resembling an elliptical LG beam. Then Tina Shih visited us, and gave a talk, and we hung out with her and showed her our projects. I also started a re-measuring of the open-cavity HeNe's divergance, since yesterday's measurements weren't too good.
I set up my new apparatus to measure the divergence of the open-cavity HeNe, which in its third form consists of a fan in the path of the laser beam, which is later focused by a lens into a photodetector, which is connected to an oscilloscope. This is because the digital multimeter gave heavy fluxuations which were too large to average out, and get a measurement from. The periodic cutting off of the laser beam by each of the five fan blades allows calculation of the angular velocity of the fan blades, the tangential velocity at the point where the laser beam is incident, and most importantly, the diameter of the beam. I also spent a long time helping Hilary take some nice CCD pictures of vortices in her soap film apparatus, and listened to a short talk on cleaning coated optical parts.
At Dr. Noe's suggestion, I investigated the source of the great instability in the laser's power, which was so quickly fluxuating that taking the divergence readings from the oscilloscope was nearly impossible. The difficulty came from the fact that the length of time is measured as the time from the 10% to the 90% intensity points, but with fluxuating power, these points were not well defined. I manipulated the mirros until the greatest overall power was achieved, which caused a 4.68V potential difference in the oscilloscope, as opposed to the usually 300mV-1V that a diagonal HG01 mode gives. I'm not sure what power this corresponds to, but I had a 10KOhm resistor across the leads. Therefore the current was around 0.468mA. The translation to power depends on a characteristic curve for the particular photodetector I used, which I don't have at this time. This higher power operation, which was a massive multimode superposition, also had STABLE POWER! Even more importantly, so does the Gaussian mode. Another unstable mode, besides the diagonal HG01 is the direct LG beam I have gotten again. The direct LG is so unstable, it occasionally momentarily decays into the annular profile with two opposite intensity peaks, strikingly similar to the pattern from a misaligned astigmatic mode converter. Both of these unstable modes, which are related by well-documented mathematical relations, are probably unstable because of mode competition. The solution to the instability problem eventually came from Jennifer's head. Specifically, HG modes were stably produced by putting her very thin hair across a multi-mode beam. But being the idiot I am, while making intra-cavity modifications to aid in the process of adjusting the hairs (I had two), I gave myself a SHOCK!!!. Very odd feeling, and I let out a loud scream. So from now on, the laser will be OFF and ELECTRICALLY NEUTRALIZED before I make any modifications.
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In preparation for tomorrow's presentations, I spent a good part of the day re-doing the astigmatic mode converter section of my transverse modes page, so that it reflects the work with AMC 0.4, not AMC 0.2. I also did some work with analyzing the LG beam. The (n) dark blobs in the center of the LG beam are the results of the bright lines across the center of the original HG beam, which isn't suppressed fully. The LG from the HG02 is about 60% of the composition, with the original HG composing about 40%. This is significant, since it's my first >50% LG beam. I also started thinking about performing some interferometry between the input and output beams, which I'll probably try to do tomorrow, after the talks, if I have time.
The presentations were this morning, but more importantly, I had 5 slices of pizza. Owen's presentation reminded me of something: moving the plane mirror closer to the curved mirror, thereby reducing the cavity length, allows a more powerful and varied multi-mode operation. So in search of HG beams with two non-zero indices, which had been difficult to get previously, I tried a more powerful multi-mode. This involved moving Jennifer's hair and its rotation stage dangerously close to forward of the two electrical leads, about the distance I was at when I got shocked on Friday. This time the laser was OFF. I got an HG12 mode from the laser, and put that through the AMC. WOW!!! I got an extremely super-cool double-concentric-ring pattern, which is the LG12 pattern. At first it seemed to be an HG01 inside a ring, but that was because I was viewing it after too much unstable propagation. I moved my viewing screen closer to the AMC, and took a CCD image. After analyzing both the inner and outer rings, I found the inner ring to have 93.2% LG composition, and the outer to have 83.2% LG composition. That means overall I can safely say it has 88-90% LG composition, MUCH better than yesterday's 60%.
I did some reading in Laser Fundamentals by William T. Silfvast, which is a really good book. I improved my background on lasers in general, and on certain quantum mechanical ideas. Because Professor Metcalf had to leave in the middle of presentations yesterday, some of us re-presented for him today. I made a failed attempt to determine the beam waist in the region of the cylindrical lenses. I'm back on the analog oscilloscope, since Oleg took back the digital one, and I think that's the main problem. I'm having trouble locking in on the signal and reading it properly. I want to determine it so that I can precisely fix the alignment of the astigmatic mode converter.
Today was the last day for the Simons students, and Pete as well. Pete was nice and brought us a dozen donuts and a Box o' Joe (Dunkin Donuts meal #6), and then we ordered some pizza from Luigi's for lunch. Not much work got done. Ken probably won't be around much, and Pete is gone. Hilary, we're not sure about. Evan's going to come less frequently, and Waldo and I will pretty much be coming the same as before.
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The lab is really empty now. It was Evan, Waldo, and I, and Evan won't be around again until next Monday. I got an email reply from Dr. Johannes Courtial, who was the correspondance author of an article of the performance of AMCs. He gave several good recommendations, and a reference to an earlier article from his group. I read it, and it has some beautiful pictures of the interferometry between an LG beam and an approximate plane wave (they used an expanded and collimated intensity spot from the input HG mode). They used a Mach-Zehnderr interferometer, which I am going to try to make a replica of. Dr. Noe pointed out my idiocy in my idea to get a plane wave from another HeNe to interfere with. I was completely forgetting my own research, which is based on modes (in this case, longitudinal). For just a 100mm cavity length difference, that means a mode spacing difference of c/2L = 1.5 GHz, which would mean that there would be 1.5 BILLION beats per second, which is quite impossible to see. So, I'll have to replicate their setup, which means a whole new round of Gaussian optical calculations (I'm excited. Feel the sarcasm.) to prepare a proper beam expander with our given lenses. That's tomorrow's project.
Not much got done today. I ran a few calculations, checking my mode-matching results with an article, and found a discrepancy. Tomorrow, I will email Dr. Courtial from St. Andrews, asking if he knows what's wrong with my approach. Also, I checked out cylindrical lens prices from Edmund Scientific and Thor Labs, for f=25mm.
I got a copy of "Laser Beams and Resonators" (H. Kogelnik and T. Li, from Applied Optics Vol. 5 No. 10 pg. 1550-1567), which is an excellent summary of Gaussian optics and ABCD matrices of various types. I used it to re-calculate things from my experiment. It seems that I had the waist size of the beam off by a factor of two. My equations gave very closely the same results as the equations from the article, although the article's equations are easier to work with. I tried making a new mode converter with the values from the article, and unfortunately, the beam is not as good as it was before. I have a feeling that I had the waist size correct to begin with. But with these equations, eventually putting together the Mach-Zehnder interferometer should be much easier.
I did some more adjustments on the astigmatic mode converter today, since it's been acting up. I can't get any HG modes with two non-zero indices anymore, I'll have to work on that some more. I started taking pictures of the mode conversions, and I intend to get all of the modes and document them. Also, I've managed to get almost completely circular beams within a small range, by increasing the distance from the mode-matching lens to the center of the cylindrical lens assembly.
Alex is sick. Alex isn't at the lab.
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Today, I took pictures of every HG mode I have achieved, and their corresponding converted LG beams. I also took data on multimodes, and how their output power affects HG mode selection, specifically for the case of the astigmatic mode converter. All of this went on the transverse modes page. I am still trying in vain to interfere the LG beams and produce spiral patterns.
Today, I worked on getting the interferometer working. It
doesn't. Professor Metcalf pointed out my mistake in that I
wasn't matching the radii of curvature of the beams, which would
disallow any noticeable interference pattern. So to start, I am
bolting down all of my optics, which were previously loose, to
allow for easy adjustment. My apparatus is now satisfyingly
complicated-looking.