## Journal

Thursday, July 30, 2009

Over the past couple of days I have been investigating my new idea. The reason that I have not posted earlier is that I wanted to post good news that my theory worked. It has, but not as well as I had wanted. I thought that these new equations would allow for accurate predictions about the waist of the laser. I took a large number of pictures of waists with a 750 mm lens and 1000 mm lens at the predicted distances from the front of the laser and where I had previously fitted the minimum waist of the laser to be. The difference between these two distances is 3.15 cm. If my idea was a good measure of waist size, then the measured value would vary slightly from the predicted value as the lens is moved around. The waist sizes that I found from 36 microns to 56 microns in both locations! This means that this method can have changes of 40% with just slight variations of the lens location. Even thought this aspect came to nothing, at least this is a simplified version of the Peatross and requires one less known parameter.

Aside from this I have been working on my abstract and paper for the REU program as well as working on the webreport for the LTC. I also am setting up the two lens setup again to get some more data points and to use a different size lens for the interferometer lens.

Monday, July 27, 2009

Today I spent almost all of the day doing fits of my data with the theory that I had transfered over to Excel. But the exciting part did not occur until 4 o'clock today. I finally got around to modeling what determines the size of the waist in the two lens setup and found that it was a linear function of:

Here f0 is the focal length of the first lens and B is the diameter of the laser.

When I was talking to Dr. Noe, he mentioned that this setup only worked within the Rayleigh range of the laser, so that got me thinking. Why doesn't it work outside the Rayleigh range? Of course that seems obvious because the wavefront is no longer approximately collimated, but what happens? The beam diverges to so diverge angle &theta. Then an idea struck me. The first lens is providing a focus to the further lens and that beam now has a constant divergence angle. Why not just use the divergence angle of the laser to act in stead of the first lens. So outside of the Rayleigh range a lens can be placed such that it is a focal length away from the minimum waist of the laser, and the ghost beam should form in a collimated reference beam. To test this i grabbed a meter stick and my 1m focal length lens and put them in front of the laser. After magnifiying the image, I beheld a beautiful ring pattern that inverted as I passed its waist.

I spent the last hour and a few hours at home thinking about this and doing some math. I came up with one equation and one statement about this.

From this equation I am confindent in saying that for every laser, there can be only one waist size that it can produce. This means that this setup has an application of being able to determine the diverenge angle of any laser and that it can therefore be used to determine the waist of the laser. Tomorrow I hope to take some pictures of from this setup and see if my prediction of the waist formed by the laser I am using is 39.17 microns!

Thursday, July 23, 2009

The past two days were very exciting for me. The reason is that I final figured out themath model for the Gouy phase! It has taken a couple weeks of stop and go work, but it is finally done. Also I have spent some time decoding my pictures. This involes taking the .TIFF files that I took with the camera and then cropping them down to some 60x1 pixel file and saving them as a .pgm in a linux program called xv. After doing this I wanted to get some numbers out of the picture so I used a command that a past student here figured out ( Jon Fuchs). cat file.pgm | tr -s " " "\n" > file.dat . What this does is that it takes every grayscale pixel and converts it to a number from 0 to 255. After getting these files I put them into excel and made slice graphs of the ring patterns. Here you can see a theory plot on the left and a slice graph on the right.

As can be seen, these two graphs have the same general shape. Tomorrow I will be working on getting the fit graph on the same graph as my data and fitting them.

Tuesday, July 20, 2009

Yesterday I got the Peatross experiment setup. This procedure was extremely simple and much easier than the ordeal that was the Mach-Zehnder interferometer. The reason that this setup was simple was the once the focal lengths of the lens were figured out, it was a matter of placing them down at the sum of their focal lengths apart, which is just down with a tape measure. After doing this I had to attenuate the beam and, because of table size, I had to use a mirror to steer the beam sideways.

I spent the rest of the day and today looking at the images that I got from it. I noticed one interesting thing about it. Depending on where I had the second lens and the angle of the mirror, strange results. This is that the location of the waist would shift in a straight line when I would the camera back and forth, also, in serve cases, the rings would look like ovals. This is due to the beam not hitting the center of the plano-convex lens and being perpendicular, so the ghost beam is being steered.

Tomorrow I will work on my math model and, hopefully, get it working and on here for you to see!

Friday, July 17, 2009

Today I wanted to be finished with my interferometer setup, so I started doing calculations to see if the results I measured make sense. The most important thing to look at is if the waist appeared where it was supposed to. In the pictures that I took, the waist would appear at around 26cm, with a little variation. This distance is larger than the 20cm focal length of the lens that I was using. At first I was very confused as to why this would happen and then realized that this probably occurred due to the divergence of the laser beam. Knowing this I then proceeded to calculate the placement of the waist. To do this I used matrix methods and found that the waist should be at 15cm! This made me extremely confused. I stared at it for a while and then realized that I had forgotten a sign in my final results. Quickly changing my mistake I got the waist to appear at 30cm. This was still not what I was looking for, so I looked to see if there was any way that I had the wrong divergence angle. I saw that the divergence angle formula was only valid for extremely far distances. To combat this I took the derivative of the waist formula and saw that this agreed with the angle formula within 0.25%. Using this I got the waist to be at 29cm, but this is not close enough. So I looked to see what would need to change to get the desired 26cm. The divergence angle would have to change from 0.00076 radians to 0.00056 radians, the height of the beam above the lens axis would have to change from 0.484 mm to 0.431 mm, or some combination of the two. After seeing this I was satisfied that the waist was in the right spot, because just slight misalignments of my setup could lead to these small changes.

The other thing that I accomplished today was the calculating of the lens combination that I would need to accomplish the Peatross setup. The way I did this was to just take the equations that he had in his paper and use trial and error with different lens combinations. After a while I finally found one that worked. Using a 250 mm and 1000 mm lens pairing results in the ghost beam having a waist of 33.6 microns, a Rayleigh length of 5.6 mm, and an intensity ratio at the ghost beams waist of 1.4. Hopefully these will give me good results when I get the setup constructed.

And before I forget, here are the final pictures that I got from my setup. As can be seen, the waist picture (on the left) has no interference lines, so both arms then must have the same phase. Also the other pictures show that the center spot does invert as the waist is passed. The pictures are in order before the waist to after the waist starting from the bottom left, going up and over, and then down.

Tuesday, June 14, 2009

Sorry about the lack of posting but I have had a busy week.This week I finally got pictures representing the Gouy phase. Yesterday I recalibrated the Mach-Zehnder interferometer that I had set up and was able to get pretty good pictures. The interferometer was not as well setup as I could get it, but I wanted to see some results. I focused one arm of the interferometer and was able to see ring patterns form in the reference beam. Half of the image was blurry, but this was due to the improper aligning of the interferometer. The images did as I expected. As you can see, on one side of the focused beams waist the center is bright and on the other side the center is dark. This is representative of the Gouy phase, which is an Arctan function, going from positive to negative values, which cause the constructive and destructive interference respectively. After I had gotten my pictures onto my computer Hal came gave us a talk on liquid crystals, to prepare for a talk he gave today. We spent the rest of the day discussing liquid crystals and trying to figure out how they work. The way the pictures are setup is that the top left picture is the one that is located near where the waist should be and then the left column is further out and the right column is closer to the focusing lens.

Today I made a little more progress with the Gouy phase experiment. When I got in this morning I went to adjust the interferometer make the two arms better aligned, with some success. The images that I then took do show a bit more of the rings, though about a fourth of them are still blurry, which I will try to fix over the course of the week. There was a slight problem with these pictures though. As can be seen the center remains bright throughout. This happened due to some construction going on outside and the vibrations from thier machines were shaking my table and causing the phase of the beams to change. The last picture is one that I took once the construction workers stopped making such a racket. The pcitures came out okay, but there is something strange about them. Normally we should expect to see that the center should never switch to dark or bright again once the waist is passed, but this can be seen in the bottom right picture. This can be explained by there being some initial phase, probably caused by my attenuator. Tomorrow I will try to resolve this phase difference and get pictures that are as nice as the first batch, but have less blur.

Monday, July 6,2009

Today I spent the day setting up a Mach-Zehnder interferometer to try and view the Gouy phase. This process was difficult due to the ammount of precision that is necessary to get the beam to meet up again with the correct amount of angle and height in the final beam splitter. This took most of the day, but this was a nice change from trying to get the razor blade experiment to work. With that the readings never came out the way they should, but perhaps one of the high school students can get that setup to work correctly.

Once the Mach-Zehnder was fully working I put a 1m focal length lens into one arm of the beam to view the interference pattern of one arm being focused and the other collimated. What I saw was that the interference pattern did not change when the lens was inserted. I found this to be very odd. It was at this point that Dr. Noe pointed out that the focusing power of the lens was being canceled out by the diverging power of the laser beam. To see the interference pattern that I was expecting, I got a smaller focal length lens, this time 200 mm, and saw that the interference pattern now had the circular rings that I wanted to see. Tomorrow I will take a look at the interference pattern with the CCD camera to see if the Gouy phase is apparent.

Thursday, July 2, 2009

Sorry for the delay in posting another journal entry, its been a good week. I think I have a good direction for my project. I am going to look more deeply into the Gouy (or Guoy, I'm not really sure) phase of the Gaussian beam. To start off with this I have spent the week getting the profile for the laser beam that I am using. I have finished taking pinhole measurements of the beam and then found the waist of the beam for each of the slices of the beam. After that I fit the waist equation for the Gaussian to my waist measurements and got the fit rather well. From this I found out that the initial size of my beam is 0.25 mm. This will be important to know when I want to expand it. Hopefully tomorrow I will be able to setup a razor blade experiment to measure the profile in the Rayleigh range, in this case around 32 cm, and compare these results against the pinhole. Then I plan on collecting Gouy phase data from a Mach-Zehnder interferometer setup, after which I will hopefully be able to reproduce the Peatross experiment. If I don't run into too much trouble and have enough time Marty said he would be willing to help me setup a Bessel beam. Perhaps I can measure the change of curvature of a Bessel beam to see if it matchs up with that of a Gaussian beam in the Rayleigh range.

Friday, June 26, 2009

Today I got the experiment fully setup, with a slight tweak. Dr. Noe suggested that since the pinhole was so small that it was causing Airy patterns to form that the first lens was unecessary, so it was removed. We then tried view the expected interference pattern shown by the paper, but this was unsuccessfull. On Monday I will try to find the correct values given by the experiments formulas and see if I can try to find the interference patterns again. If not I will remove the pinhole and add the 200 mm lens back in to see if the pattern is there.

Thursday, June 25, 2009

We went to a lunch meeting yesterday where there were two talks. The first one was a made by a grad student who talked about a diffraction project that he did in his optics rotation course. He talked about the difference between Fresnel (near-field) and Fraunhofer (far-field) diffraction and then about his construction of a setup to look at Fresnel diffraction. Then he gave an overview of the Mathematica code that he used to compare his experimental intensity profiles to the theoretical results. Then another sutdent came in and gave a presentation about laser stabilization. This is the process of controlling the length of the resonating cavity of a laser so that the frequency of the laser does not change has much. Since laser light is of such a short wavelength, even minute changes in the cavity length can cause a large change in the frequency. He described the process that he used to control the temperature of the cavity and then showed the results that his got.

Today I finally got around to measuring the focal lengths of the 200 mm and 1000 mm focal length lenses. To do this I used a method called auto collimation. This involves taking a point source of light, such as light coming out of a fiber, and shining it through the desired lens and onto a plano mirror. If the lens is a focal length away from the point source the light going onto the mirror will be collimated (parallel to each other). When it goes back into the lens it will come to a focus on the point source again. To see if the returned image is a point the mirror should be angled slightly so that the image appears near the point source. Then I began to setup the experiment that was described in the Peatross paper. The difference in my experimental setup is that the pinhole used is a 250 μm. This causes an Airy pattern to form. This makes me wonder if the 200mm fl lens is necessary anymore, or if this will still create a "ghost beam?"

Tuesday, June 23, 2009

Today I was very excited! I think I have a projects in the works. I was talking to Dr. Noe about the Peatross paper (soon to be on a links/reference page) and we decided to set up this experiment. Tomorrow I will measure the correct focal lengths of our lens (I will use a 200mm fl lens and a 1000mm fl lens), and hopefully get the beam working correctly. I will also try to see if this setup can be used with other applications and work on get some sort of plot up that shows what sort of lens ratios would work for this experiment.

Friday, June 19, 2009

Today we did more reading, so that we can come up with a project idea. Dr. Noe showed us a demonstration using polarizing materials, and then inserting other things, such as cellophane. We saw that depending upon the orientation of the polarizers the material in between would change color, or in the case of plastic objects, like a fork, it would show the stress patterns in the material. Then we learned how this worked and saw that different parts of different materials may retard (this means to slightly phase shift) certain wavelengths of light depending on their orientation with the polarizers. Also Dr. Metcalf came in a showed me that a glass tube gives a similar type of ray trace to the axicon. Maybe this can be used to create a different type of Bessel beam? Also I was thinking about what would happen if different HG modes were used to create the beam? I'll have to bring this up to Dr. Noe next week.

Thursday, June 18, 2009

Today we started the day with a discussion on the use of ray matrices to calculate the position and angle of a ray after going through a system of thin lens. Yesterday Dr. Noe posed a question to us to find out why the focus through a magnifying glass outside is not a point but has a size. Today we derived that reason. The light that comes out of the magnifying glass is actually an image of the sky, and, since the sun does not appear to be a point source, has a minimum size at the focus. The data that we took correlated to the expectation that the ratio of the image size to the focal length is equal to the angular size of the sun, which is roughly 1/100. Also yesterday Dr. Noe sent me an article about using a single lens to act as an interferometer for a Gaussian beam. This was of interest to me, since one possible topic I am interested in is Bessel beams. After reading this I realized that there were many terms that I did not remember from my optics course and others that I had never seen before. I then spent some of the day looking up these terms: Fraunhofer limit, Fresnel-Kirchhoff integral, Fresnel approximation, Cornu spiral, Mach-Zehnder type interferometer, and coherence length. Two terms that I did not find definitions for were Gaussian aperture and the Gouy experiment. Hopefully Dr. Noe can help me understand these. After lunch I stopped by Dr. Metcalf's office and asked him about Bessel beams. During this I found out that they can be simply made with an axicon lens. I have never heard of this, so I looked it up and made a simplistic ray model for it. This lens has a conical surface with a plane back. I think that my project could have to do with the properties of Bessel beams, I just need to refresh myself on some Gaussian optics!

Tuesday, June 16, 2008

Today was my first day on the REU program. I got to meet up with all of the other students at the morning breakfast and they all seem like a great bunch of people. I am looking forward to working with them over the summer. Also I got to meet up to Dr. Noe and he showed Mara A., Max T., and me around the Laser Teaching Center. He showed us the different projects that current students are working on around the lab and told us to start thinking of subjects that we would be interested in studying.

Thomas Videbaek
June 2009
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