# Research Journal

### April 17th, 2015

Today I calculated the equations for a sphere, parabola, and debatably an ellpise that each satisfy the depth and outer most points along the edge of the bubble. If I am to assume that the surface of the bubble is a sphereand if I state that the orgin is the point of greatest depth within the bubble, the sagitta method used previously found the radius to be 47 inches and from this information an equation of a sphere with radius 47 centered at (0,47) can be made.

As for the parabola, using the known points along the outer edge of the bubble in the equation y=ax^2, the parameter a can be found very simply. The equation is given by y=0.014993x^2. As for the focus of this parabola, I found the directrix using the distance formula to find the distance from the focal point to the outer point and from the directrix to the out point and solved the equation for the focal point. This point was calculated to be at (0,21.5)

The ellipse is a little more tricky when it comes to pinning down an exact equation. I assumed that one focus would be found at a relatively far distance away to the bubble comparably similar to where the door is located. Using the constraints of depth and outer points, I used a computer program to calculate an ellipse. Below is all three equations on the same plot. The sphere is the outer-most curve, the middle curve is the ellipse, and the inner-most curve is the parabola.

### April 16th, 2015

Today I had the chance to meet with Emeritus Professor Peter B. Kahn to ask some questions regarding the history of the math tower basement bubble. Professor Kahn is an ex-chair of the Physics Department at Stony Brook who I met through my mother. With his green tea in hand, Professor Kahn told me several snipits of history regarding his time at the University ranging from building challenges to old professor interactions.

According to Professor Kahn, the construction of the math and physics buildings began in the early 70s where the math tower bubble was included within the original plans. The math tower bubble was constructed using a plastic mold, however beyond this information Professor Kahn did not have any other additional information regarding the bubble itself. There were several challenges with the construction of the basement where flooding had occurred on two separate occasions during the first few years of completion. Professor Kahn had told me that the office of Professor Metcalf took on some water damage as a result however even after the incident Professor Metcalf was reluctant to move his office to locations other than the basement.

### April 13th, 2015

Today Dr. Noé, Max, and I walked over to the Math Tower to take measurement of the basement bubble and observe some of its characteristics. Using a tape measure, I measured the diameter of the bubble to be 83 inches. With the help of Max and Dr. Noé we used this tape measure as a chord to measure the sagitta of the bubble with a meter stick. Measured from the center of this chord to the maximum depth of the curve, the sagitta was calculated to be 25 inches. Using half of this chord length and the sagitta, the radius of curvature could be calculated using the sagitta method.

Whereas if we are to assume that the bubble is spherical, the radius can be calculated using the following equation:

The calculated radius of curvature was determined to be 46.945 inches, however in all practicality this can be rounded to 47 inches. If the bubble is indeed a portion of a sphere, an omnidirectional sound source would indeed reflect sound waves off of the surface of the bubble directly back to the sound source. When speaking into the bubble at this radius of curvature, my voice appeared to be loudest.

Next, we looked at how the characteristics of the bubble behaved at a distance d away from the bubble. From the opposite facing doors of the math tower basement to the bubble was measured to be a distance of 266.5 inches which includes the recessed wall lip where the bubble is set into the wall.

From here Max and I were able to test some characteristics. The first of which involved Max speaking into the bubble near its presummed focal length to observe how sound is heard at the door. If we are to assume that the bubble is a sphere, the focal length is approximated to be located half of a radius distance away from the bubble. If the calculated radius of curvature is 47 inches, the focal length according to this value is approximately 23.5 inches. Rays that emanate from the focal point of a perfectly spherical concave mirror/surface reflect off of the surface and travel parallel to the central axis line of the surface. At the door, Max’ whispers appeared to be comparably louder when he spoke into the bubble at a distance near the focal point. Although Max’ projected voice is not omnidirectional in nature, this finding provides some insight into the characteristics of the bubble. From the door we also observed how the light from a candle behaved when held at various distances. It is important to note of the fact that images from this distance appear to be inverted in the bubble i.e, the ceiling reflection appears to form the lower half of the image.

When held at a distance of approximately less than 24 inches away, there appeared to be a ring formation along the surface of the curvature. Note how the candle image appears to be upright and only slightly magnified.

When held at a distance of 25 inches away, the surface of the bubble reflected the largest lit surface area.

At approximately 28 inches, the image of the candle appeared inverted and greatly magnified.

### April 6th, 2015

Today we further discussed our projects and goals for the upcoming days. Max finalized his project idea with studying far-field diffraction pattern transitions from the Fresnel regime to the Fraunhofer regime. The entire experiment seems to an interesting endeavor where distance, aperture size, and wavelength are all factors in determining where the transition in diffraction patterns will occur. As for Jay, Dr. Noe suggested creating a comparing model for the GRIN tank to describe why the path of light bends however it is uncertain whether this idea will be entirely pursued.

For my project, I finalized some of the initial goals to be described in the abstract of the project. It is important to emphasize the presentation of the math tower basement bubble as a selling point for other students on campus to be interested in seeing for themselves other than to just plainly describe the surface and be done with it. Dr. Noé suggested that I make a rig for the parabolic pig mirror that he has in the LTC to use as a prop to visually represent the filling of light across the surface of the mirror when light is placed at its focal point. I think this would be an intriguing demonstration that would certainly catch some eyes when presenting. It would be smart to take advantage of the tangible aspect of the rig demonstration for several other projects do not have the same ability to involve the readers of their posters in quite the same way.

As a fun side note, Jay brought to the LTC a silvered towel rack that he had removed from his dorm bathroom. When shined with laser light at an angle along the length of the rack and interesting reflection pattern formed where the near flat surface of the rack reflected a section of a circle whose radius of curvature increased with decreased angle of incidence with respect to the normal.

### March 30, 2015

Today, Max had shown to the group the derivation of light intensity for Young’s double slit experiment by using Euler’s equation to describe the intensity within complex form. I then moved onto describing the derivation for calculating the radius of curvature using the sagitta Method, which I include below:

The diagram above is a geometric representation of a section of a circle’s curve. The radius of the circle is indicated by R. L indicates half of a chord that perpendicularly bisects a radial line. The distance from the point at which the chord bisects the radial line to the outer curve is the sagitta, represented as a distance S. A right triangle is formed with sides R, R-S, and L. By using the pythagorean theorem, the sides can be used in the following formula:

With rearrangement of this formula, we can represent the radius as:

The equation above only requires the length of the half chord and and Sagitta to calculate the radius of curvature.

After this derivation, we began discussing our projects more in depth by first deciding to create clear goals to form a plan of action for the experiment. By focusing on main goals rather than getting hung up on the specifics of an experiment, an idea or theory can truly be addressed without becoming too narrow in its scope. For example, I had discussed with the group a certain method that I was willing to try out for my experiment with the dish mirror to observe its ability to focus rays. For my devised method I proposed to use a microphone and oscilliscope to measure the intensity of a stationary frequency of sound at various distances away from the dish mirror. Although my method in theory would work, Dr. Noé emphasized the fact that my experiment is in its essence to observe the focusing ability of the dish mirror of any ray, including light. By using light, the experiment can become far less complicated to demonstrate the same effect of focusing that an experiment using sound would, however by using less equipment there are less parameters to worry about. Another important notion that was addressed in the meeting was the importance in questioning whether the dish mirror is truly a portion of a sphere or rather a different curve. It is important to keep an open mind and let experimental values of the curvature lead the way to the true characteristics of the dish mirror rather than just to assusme. Dr. Noé did a fantastic demonstration to show how light behaves with curved surfaces. By using a candle , Dr. Noé placed the flame at various distances to demonstrate certain effects. When near the focal point of his small dish mirror, the candle light reflected off of nearly the entire surface of the dish mirror and was focused into our faces. When at a distance equal to the radius of the dish mirror, the light was reflected off of the surface and back to the candle. As a result, the silvered surface was not lit with candle light from our perspective. After this demonstration, I really began to understand how light can be a very powerful tool as a demonstration of focusing ability for experiment and I am excited to see how the demonstration will look like on a larger scale. As for Jay’s ideas, he introduced to Max and I the Tablot effect which features light interference from grating that produces fascinating patterns that repeat themselves according to the grating width and relative distance away. Dr. Noé showed us past experiments which involved the Tablot effect. Jay was suggested to create an excel representation of the GRIN tank by modeling the path that light travels along. This seems like an endeavor that will lead to results that shall be fascinating to each of us.

### March 23, 2015

Today Dr. Noé noted at an analysis of the dish mirror in the math tower as a possible research idea. To elaborate, Dr. Noé drew the geometric model of the dish mirror on the board and showed to the group how the radius of curvature could be found by using a chord across two points along the curvature and the depth distance from the center of the chord to the curve know as the “sagitta” (latin for arrow). As previously stated in earlier journals, according to the thin lens equation the concentration of rays (light or sound) after reflecting off the surface of the dish mirror will converge to a point according to the object distance from the dish mirror. Dr. Noé had mentioned that a measure of sound intensity using a point source of sound, a microphone, and an oscilloscope could be used to arrive at the dish mirrors focal point experimentally. By using this method, one could investigate the accuracy of the sagitta method model for the dish mirror against the collected sound intensity data. I had learned that the sagitta method was actually a very important tool in nuclear physics in past decades as a method to calculate the energy of a decay particle using the radius of curvature that is generated within a bubble chamber along the path of the energized particle. This experiment, with its large size and hands-on nature, is something that I am excited to hopefully perform!

As for other topics, Dr. Noé had showed to the group a myriad of physics demonstrations involving the polarization of light. We dabbled into the description of light as a circularly polarized electromagnetic wave that when reflected between two media may change polarization. Very fascinating demonstrations were done showing this property using mirrors and polaroid film to show the diminishing of the light intensity of reflections as the film is rotated. The Brewster angle (named after the Scottish physicist David Brewster) was mentioned by Dr. Noé as the angle at which light is perfectly polarized after reflecting off of a surface. This angle was demonstrated by looking at the reflection of a person off of the white board through the polaroid lens and rotating the lens until the reflection of the person was completely out of sight. Fascinating to say the very least!

### March 9, 2015

Today, Max, Jay, and I received our passwords and user names for access to the LTC server. While Jay and Max have windows computers, I on the other hand have a Mac which features slightly different commands for navigation in the SSH client. Mac computers use a program called Terminal which combines certain operating Mac commands directly into the program intended for ease of use. After comparing each windows command, I recorded a list of mac commands for my own reference.

We had briefly mentioned a few topics including frequency cones and whispering galleries. The whispering galleries featured interesting effects with transferring sound around the circumference of a circle. This acted as transition into a brief description of parabolic surfaces and enclosed elliptic surfaces. Within these ellipses light traveling from one foci will always reflect off of the enclosed elliptic surface and travel through the opposite foci. Sadly my time within this group meeting session was cut short due my work shift.

### March 2, 2015

Today was a group meeting with Dr. Noé, Jay, Max, and I. We were building upon some of the other topics that we had gone over last meeting such as the Law of refraction and Fourier transformations, leading to several different topics that we ran into as the conversation progressed. Dr. Noé brought geometric optics into the conversation when Jay mentioned the concave semi-circle structure located in the basement of the math tower. Sound waves and light rays within this system can be represented as rays when drawing the diagram for this concave system. By using geometry, Fermat’s principle, and the law of reflection, we were able to show how within the concave system, if a point light source was fixed at a distance equal to the radius of the concave semi-circle, all rays of light would reflect from the surface of the semi-circle back to the light source. If the point source was moved into the semi-circle at a distance equal to half the radius, the emanating rays from the point source would reflect off of the semi-circle’s surface and travel out of the semi-circle perfectly parallel to one another. These rays are traveling “together” in a sense that if they were to hit a perfectly perpendicular surface to the semi-circle, each ray would be the same in length, satisfying Fermat’s principle. This is point is defined to be the focal length of the circle, where there is an interesting way of showing this distance in an auditory fashion by having one person stand several feet away from the semi-circle directly behind someone standing in front of the semi-circle with his or her mouth at a distance between the radius and the focal length. If the closest person whispers into the semi circle, all of the sound waves are reflected off of the surface of the semi-circle and are focused onto the listener, giving the auditory perception that the whisper is coming from a closer distance.

This cool geometric description led the conversation into the thin lens equation, which describes the relationship between source object distance, image distance, and focal distance from the plane/lens in question. Many times when drawing diagrams, an ideal thin lens is used for simplification. Dr. Noé had mentioned the relationship that a small angle approximation had when calculating the angle of reflection from the incident ray of an ideal Plano convex lens when taking the tangent of the distance from the incident ray to the central axis and the focal length of the lens. Dr. Noé mentioned that using vector matrices could represent ray transfers such as these.

When discussing Fourier Transforms, Dr. Noé move on to speaking about delta functions and how they relate to signal processes. He had mentioned that if you were to decompose a signal into its constituents using a Fourier transform, you could represent the information using a delta function. Stemming from this, we ended the conversation with a review of complex numbers.

### February 27, 2015

During this personal meeting between Dr. Noé and I, I was introduced to the linux environment of the LTC server. Dr. Noé showed how the website was organized by operating through the SSH interface to access the server. Dr. Noé went over my biography and offered his advice to improve my writing. Dr. Noé created my LTC site using a template and uploaded my bio to the page. He had mentioned the process in which passwords and user names are generated and it appears that I will have access to the site in just a few days.

### February 23, 2015

The first group meeting with Dr. Noé and Dr. Cohen started off on the right foot with a discussion of project ideas between Jay Rutledge, Max Stanley, and I. Max first brought up some of the recent information that he had been gathering about Fourier Transforms and their applications. A Fourier transformation can be a powerful tool for decomposing several types of waves into their smaller constituent parts. The applications are endless in signal decomposition and have been used by pervious LTC research students. Fourier Transforms stem from Fourier series which describe how periodic functions can be represented as a summation of other periodic functions i.e., sine and cosine functions. Dr. Noé gave Max a book on Fourier transforms and told him to search Jon Wu’s research on Fourier Transform spectroscopy to see how he had used Fourier Transformations to his advantage within his research project. Fourier transforms are very interesting and I am looking forward to see how Max shall integrate them into his project.

Jay Rutledge raised the topic of Optical Vortices into the discussion, which he had described as light twisted around a central axis where light cancels within the enclosed region. He briefly mentioned how there are some scientists that believe that optical vortices may one day be helpful in quantum data processing and computing. Dr. Noé mentioned that running a circular plate through water could show an interesting visual representation of a vortex. One can observe the way light behaves in this example by looking from the side of the plate directly into the vortex and noting that it appears much darker than the surrounding water.

As for my own ideas, I discussed the techniques of ellipsometry and its use in investigating the dielectric properties of thin films. By using ellipsometry, the characteristics of thin films can be calculated, gathering information regarding the composition, roughness, depth, crystalline nature, electrical conductivity, and many other characteristics. Using polarized light to reflect off of a film surface and detecting the difference in polarization, this powerful tool is used throughout industry for contact free determination of the thickness and optical constants of several kinds of films. The problem with ellipsometry however is the vast challenge of modeling the data, where it takes professionals and experienced student’s extensive training over a significant time to gather the aptitude to perform ellipsometry on a thin film. Sadly, Dr. Cohen and Dr. Noé do not have much experience with the practice of ellipsometry. It appears that an avenue down that road for a research project might be a far stretch.

Dr. Noé went on to discuss Fermat’s principle, which describes how light travels from one point to another along the path that is the shortest in time to complete. It was mentioned that to create a more complete description of how light travels, complex wave functions are necessary. From this insight, we moved onto proving the law of reflection by using geometry and discussed Snell’s law of refraction by using a boy scout analogy which highlighted the strange behavior of light almost "knowing" where to go. With Snell’s law including the refractive indices of varying mediums, Dr. Noé mentioned a rather fascinating experiment done by a previous student within the Laser Teaching Center, which showed the path that laser light followed when moving through a tank of enclosed water and corn syrup. The mixture found within the tank has varying indices of refraction that are related to the depth of the tank. When laser light enters the tank from an upward angle, it proceeds to curve downwards!

This bewildered the three of us, as we all attempted to explain the phenomenon that perplexed our minds. With all that occurred this afternoon, this group meeting was an excellent introduction to new concepts and ideas.