# Research Journal

### Friday April 3, 2015

I was playing with my laser pointer as usual, shining onto and into anything cool I can think of. I first shot a few drops of my roommate's oil based facewash into water to try to make a temporary GRIN medium. The beam seemed to curve slgihtly, though this may have been deceiving due to the curvature of the water-container.

That is not the part I found particularly cool, though. Below is a picture I took of my shining the laser pointer onto a metal towel rack in my dorm room.

The rack is a long, rectangular hollow bar with square faces,(seemingly) flat sides, and rounded yet distinct edges. When I point the bar at my wall, whether I direct the beam at an angle onto a flat side or cut the beam along an edge, a circle of (red) light is projected onto the wall. Along this circle is a rather bright spot which seems to be the point of expected relfection, but I am surprised by the formation of the circle. Only a small arc is present when I shine onto only a flat side, but when I shine onto an edge almost an entire circle is present--I guess this is because the beam width spans a little bit of both adjacent sides.I detached the bar to see how big of a circle I could make and was able to fill my entire room with its diameter.

My first thought was that this is a result of very slight curvature of the "flat" bar, but I wondered then why the circle (its position, diameter) remained exactly the same even as I shifted the same angle slightly over onto the edge of the bar. Even as I rotated the bar (about its length), the circle still remained fixed and the "bright dot" simply rotated about the circumference with the bar.

I am going to bring the towel rack with me to our meeting on Monday.

### Monday March 30, 2015

Today's meeting began with Max at the whiteboard, working through a derivation of the difference in path-length between two light rays from separate slits to some point of observation a distance away. This involved a good bit of work with complex numbers, which I to had try to recall (from last semester...). He seemed to have a pretty good handle on most of it, though reached a point where all that's needed is for him to sit down again to work out the remaining algebra clearly and carefully. As Max worked at the board, I recalled some articles I had just been reading earlier in which the difference in path-length was a simple one-step find. This, however, was under the assumption that the rays are traveling very far and near the grating can be considered parallel. This simplies the math a great deal, but "does not sit well" with Dr. Noe. What Max was deriving was the path-difference which is valid without the need of this assumption (and as result is a bit lengthier). The signifance of this difference in path-length is that it indicates the difference in phase of the light [of a single wavelength], where the in-phase, out-of-phase characteristics determine the interference patterns immediately associated with light and slits.

Following Max, Brandon took over at the whiteboard to demonstrate the "sagitta method" in finding the radius of curvature of a sphere, given only a small portion of it (less than half--that'd be too easy). Discussion of this method up was brought after talk of the "dish mirror", or "Math Tower Basement Bubble" as we like to call it. With Brandon considering an analysis of this mirror for his project, one idea for starters was to measure the dimensions of the partial-sphere and determine its radius of curvature, and from this evenf further its focal length. We already knew that by standing at a certain point near the mirror the sound was much louder. Considering the geometry of any such type of mirror, we were able to see that at this point all rays reflect directly back through their origin. This was easily observed by finding the spot and just standing there, but to calculate it this sagitta method was applied. Turns out, and which is news to me, a 'sagitta' is the length drawn from a chord to the curve of a circle/sphere/(ellipse?). With some drawing, geometry, and algebra, the radius of curvature of the (assumed) sphere can be found. Brandon was working it out very well until he stumbled at the point where he was to reorganize the terms into a recognizable form, and simply solve a quadratic. He actually worked this out correctly in this notebook previously, or had it down in his head at least, and merely had a forgetful moment up at the board. After this, we brainstormed as a group some other ideas for a project involving this. This time, however, Dr. Noe suggested we begin to organize our thoughts as a "planner" or outline-- main points, sub points, etc. He emphasized the importance of not stating the goals of a research project two narrowly, but to have a more open and general statement of goals. For example, Brandon suggested using a microphone and oscilloscope to map the intensity of sound at different points from the mirror. Dr. Noe reformed this, while still providing the same essence, instead as "demonstrate the focusing ability of the mirror." In doing this, of course, what Brandon had said could be included also. We then came to question of testing whether the surface was even a portion of a true sphere or not, which was a real curveball.

On the topic of surfaces, radiuses, focal lenths, Dr. Noe carried out a first-ever Laser Teaching Center demonstration. A week or two ago we had found the focal length of a mirror by "shaking our own hand", which of course was just the image. This time, however, Dr. Noe would find the focal length but with a lit candle. We reasoned prior that the image of the flame would span the entire mirror when placed at the focal point, which we were immediately able to verify with the very bright and sudden reflection of light. Brandon was always one to get excited in the physics lab, but this time his face really lit up.

I learned of the Talbot effect for the first time in an article, or maybe blog rather,that Dr. Noe had shared with me. The Tablot effect was immediately capturing with its elaborite and deep patterns. When light passes through a diffraction grating--a thin screen with regular, cut-out slits--it, well, diffracts, and certain wavelengths diffract differently. The diffracting light from each slit interferes with light from others. As a result, interference of both construction or deconstruction occurs. It turns out that in the region near the grating, the diffraction and interference pattern of light will repeat at regular intervals such that the image of the initial grating is reformed. This was first observed by Talbot, and so the effect took name after him. The distance from the grating at which the grating image is reformed [nearly] exactly is called the Talbot distance, and is dependent only on the spatial period of the slits and the wavelength of the light incident on them. Even more interestingly, the image of grating is formed also in the region less than one Talbot length. The period of the grating is not necessarily maintained, but instead many new versions of the gradient appear in fractional form, producing a fractal. For example, while half the distance of the Talbot length produces the grating image shifted half a period, a third of that distance produces the grating image with triple the frequency. I find it fascinating how the image of the gradient shrinks and regrows and repeats this at regular intervals, at each and any point along the way forming some different version of the orginial grating. I wonder at what distance is too great thst this near-field effect is no longer valid or well-defined, and am curious what it is about the waves that would dilute the otherwise repeaint and clear Talbot images. Past projects at the LTC have created Talbot images only with laser light, however the original observation was performed with white light. Dr.Noe suggested I attempt this in some way with white light, but then as the time constraint was relealized we agreed this may not be reasonable to attempt in the remaining time. It is at this point that Dr. Noe suggested to me the Excel model of the path of light in a GRIN tank, and I look very much forward to exploring this.

[Insert some pictures of Talbot effect]

### Monday March 23, 2015

Today was the first day back from spring break and usual group meeting. Dr. Noe asked to see our individual lab notebook, which I had been filling out along the way. The notebook is a very useful tool to reinforce what we've been discussing and also a place to include some of my own research and findings. Dr. Noe provided some tips on maintaining and making most of the notebook.

Afterward, we continued our development of project ideas. Brandon is on the possible track of doing a project on the "giant-bubble", or "dish mirror" in the basement of the math tower, following Dr. Noe's suggestion. Some ideas right now are a detailed analysis and ray-tracing of the mirror geometry, including measurements of the dimensions of the mirror as well as possibly even mapping out the 'sound field'.

I mentioned my continued curiosity about the GRIN Tank, particularly the point where the laser beam reverses direction. I attributed this to total internal reflection where the gradually curving light reaches one of the many media-boundaries at too large an angle to continue refracting upward, and so the beam reflects and continues to refract and curve but now downward. Dr. Noe mentioned how the beam of light will bend downward even when incident to the tank perpendicularly; I figure this could be due to an unevenly paced mixing of the water into corn syrup. I am not sure if this will be a project I can pursue, but I've wondered if I could perhaps create a model of the index gradient to show that the beam would in fact undergo total internal refraction. Dr. Noe said he is going to talk to Professor Graf who had some clever ideas on modeling the index gradient and has done GRIN related work in the past with a student(s) of his.

Max at the moment is still on the track of studying Fourier, but is not sure just yet of how he can find application in a project. We turned the discussion to optical vortices when Brandon had mentioned them just before he had to leave, which I then mentioned again I have been interested also. He raised the question how optical vortices would interact with each other given their spiraling nature. Turns out, when of the same wavelength, optical vortices will add depending on their topological charge (magnitude indicates number of twists per wavelength; sign indicates direction of twist). So, one of +1 and +2 would simply add for an optical vortex +3, whereas one of +2 and -1 would form an optical vortex of +1. This perfect construction or deconstruction, however, is difficult to produce, Dr. Noe mentioned, as the two vortices would have to be lined up exactly and the waves in phase. With the very name given by the dark spot created in the center of this light, I naturally wonder if the 'vortex' has as much use as the spiraling portion of the light.

### Friday March 6, 2015

Today, only Max and I were able to meet with Dr. Noe given the Friday-turned-Monday schedule. We used this time to familiarize ourselves with the Linux environment that is home to the Laser Teaching Center server. We do not have access to our own accounts just yet, but Dr. Noe walked us through most of the general commands we must know for operation and navigation of our side of the website. He showed us where to find helpful information regarding this and how to eventually go about doing it ourselves from our personal computers. One of the most important takeaways, though, is that you CANNOT UNDO.

### Monday February 23, 2015

Today, Brandon, Max, and I each shared with Dr. Noe some ideas we have researched over the past week, and were fortunate to have Dr. Marty Cohen sit in with us also. The meeting opened with a continued talk of Fourier Transforms, which Max had first introduced earlier and decided to look more into. We learn that these transforms are present throughout all of physics and considered some cases where they are helpful, though we do not yet have a handle on mathematics and so our understanding is not great. Dr. Noe lent Max a book titled "Who Was Fourrier?" so he can get a bit more of a basic introduction and background of the theory. This seems like a possible direction Max can lead his project, and I am interested also in learning more about these Fourrier Transforms

Afterward, I mentioned that I had noticed quite a few appearances of the term "Optical Vortex" in various projects on the LTC website. I was curious what these were so I decided to look it up after passing over the words several times. Turns out, the 'vortex' portion is just that-a spiraling form of light about a central axis causes the optical field in the center to be zero, and so when projected onto a surface a circular ring with a dark center is observed, i.e. a 'vortex'. I learned of some very meaningful and potential uses of optical vortices, such as optical trapping, tweezing and quantum computing. The rotating nature of the light gives rise to an angular momentum from which a torque can be applied to electric dipoles, which is helpful for in the careful manipulation required for trapping and optical tweezing. Also, the number of twists per one wavelength of the light, called the topological charge, is significant; this number can be positive or negative depending on direction of twist, and the greater the number the more rapidly the light is twisting about the axis. There is no theoretical limit to topological charge and so therefore is an infinite number of states for the optical vortex, which is of particular interest to quantum computing. Dr. Noe continued with some examples of vortices in nature; whirlpools in a bathtub drain, ones created by oars or plates in water, etc. I am curious how these optical vortices are produced; what kick starts the twisting motion? How does this twisting light interact with other objects? Is it only the twisting portion that is useful, or does the dark-center also play an important role?

Then, Brandon brought up "ellipsometry" which seemed to surprise Dr. Noe a little bit, who hasn't dealt with the process much or at all during his career. Brandon explained ellipsometry as a way method of 'mapping a surface' using laser light, measuring and determining the characteristics of a thin-film against a background material of known index of refraction. From this, properties of the film such as thickness, index of refraction, etc can be calculated. However, it turns out that making use of the data collected by this method is very complicated, and there are even entire math courses devoted to this. This appears beyond the scope of PHY287, and so likely will not be the direction Brandon takes his project.

After each our personal offerings, we turned the discussion to some basic optics talk. We began with Fermat's principle, which states that a ray light will travel between two points along the path that requires the least time. He introduced an analogy in which boy scouts want to get from one camp to another in the least time, with the camps separated by a river and one downstream a bit. Of course, moving through water takes longer time than on land, as we assume humans can walk or run faster than swimming. After considering several paths, we arrived at one that would be quickest: an angled path to reach the river but still shy of opposite camp, followed by a more sharply angled path through the river (carried downstream), and then a continued path at less an angle toward the camp. This of course wqs soon clear to demonstrate refraction, where the speed of travel differed between two media. Dr. Noe amended this definition later to include that light will also 'choose' the path of stationary optical length. This whole idea of light 'choosing' anything is strange, as it seems natural that the path taken is a result of the refraction, and not that the refraction was a result of a desired path.