Research Journal
Wednesday, August 5th, 2014
Dr. Noe e-mailed Miles Padgett and Grover Swartzlander asking for advice on my project- which was very neat. Both seemed to indicate that my experiment would work better in two dimensions, which makes sense. I tried following this advice and made sample cells today with just a microscope slide and coverslip. This was not particularly successful, but I can try again tomorrow. Dr. Padgett also suggested that I use silica spheres, but unfortunately these would need to be rather small. If the spheres are larger than the radius of the focused vortex, they will spin (instead of orbiting) and it is not possible to see the rotation of a sphere with a microscope.
Tuesday, August 5th, 2014
I think I resolved some major issues today. First was the realization that copper oxide cannot be trapped at the focus of a Gaussian beam. Just as the ray-optics model is only applicable to particles of a certain size, it is also only applicable to particles of a certain shape. Since the copper oxide particles are irregularly shaped, light transmitted through the particles will be refracted irregularly, resulting in a net force on the particle in a random direction. This explains why I have been observing that the particles are pushed away by the beam with great force. The one thing I am still unclear about is why the repulsive force only occurs when the beam intensity is above a certain threshold. I also realized that the focused laser beam is likely creating convection currents in the water. This might explain why small particles are attracted in the vicinity of the beam. Continuing with this train of thought, I believe that the bubbling effect I observed yesterday was in fact a manifestation of absorptive heating of the copper oxide particles. The effect occurred when the beam was brought over large particles- which of course absorb more light and dissipate more heat to the surrounding water.
In the afternoon, I returned to trying to rotate the copper oxide particles. Although the particles cannot be trapped at the focus of a Gaussian beam, I believe that they can be contained within the low-intensity region of a vortex. Don't quote me on that though. When I did try trapping the particles with a slightly defocused vortex beam, however, I think I finally saw rotation. I am not convinced that this was a direct consequence of angular momentum transfer, but it is possible. I captured the rotation on video, and when I removed the spiral phase plate the rotation ceased (even though the beam remained in the same place)- which is good. I just really need to pick up the pace for tomorrow.
Monday, August 4th, 2014
More strange results. I have determined the the repulsive force I have been observing is only an issue with the copper oxide particles. The polystrene spheres can be (sort of) trapped with the current arrangement and they are certainly not repelled by the beam...
Friday, August 1st, 2014
I reconstructed the entire optical tweezers setup twice today. It was misaligned the first time- but it is as good as it is going to get now. By the end of the day, I had time to try to trapping the copper oxide particles again with the vortex beam. I thought I may have observed rotational motion, but the particle in question was knocked out of the trap by another particle before I could document anything. Strangely, the copper oxide particles seemed to be repelled by the beam almost all of the time. This couldn't be a result of the scattering force, because that should displacement the particles in the direction of beam propagation. There's no mention of a repulsive gradient force in the literature I've read either. The only possible explanations I have right now are that copper oxide is diamagnetic or that the particle topography is causing light to refract at strange angles. Neither theory is very convincing.
Thursday, July 31th, 2014
I wrote my abstract this morning. I wish I had more conclusive results to talk about, but my experiment is not yet finished. Dr. Noe also taught me the importance of careful word choice.
The afternoon was occupied with alignment and a talk by graduate student from the University of Vienna. I am being extra-meticulous with the placement of the optical components because I no longer have time to go back and fix things. I started by positioning the two dielectric mirrors and the dichroic mirror. In hindsight, I believe some of my alignment issues from before may have come from a disregard for the degrees of freedom of the optical components. I am now more conscious of making sure that all mirrors and lenses are aligned perpindicular to the axis of propagation.
I am really down to the wire with my project now. In theory, if I am very careful about aligning everything, I should be able to observe the transfer of orbtial angular momentum because others have done so with the same equipment. In practice, I'm not sure if things are going to work and I may need a new direction for my final week. Tomorrow will tell.
Wednesday, July 30th, 2014
Forward and backwards progress today. I started in the morning doing intensity profiles of vortex beams with different topological charges. I have read different things about the radius of the intensity ring varies with l, and I wanted to experimentally derive the relationship myself. I used a 150 micron aperture and a photodiode on a translation stage to take the measurements. I measured the profile where the beam was expanded for greater precision. Some of the results are below.
After this, I tried to repeat my results from yesterday and once again observed the vortex repulsion. I also tried trapping the copper oxide particles with a Gaussian beam, and found that the particles exhibited the same quasi-rotational motion. Thus, the particle rotation I observed with the vortex beam cannot be attributed to any sort of beam angular momentum.
The backwards progress came later in the day, when Marty recommended that I examine what the beam looked like after passing through the objective. I had tried to this before, but hadn't noticed anything unusual. This time, however, it was clear that the vortex beam was not centered coming through the objective, nor was it travelling perpendicular to the optical bench. Further adjustments revealed that the whole system was not as well aligned as I would have liked to imagine. In addition, it was clear that a sizeable portion of the vortex beam was being truncated while passing through the objective aperture. The only solution to this litany of problems seemed to be to take things apart and start over. This is not an ideal thing to be doing so late in the research, but I am not going to observe any sort of orbital motion unless the beam is properly aligned. I've got a headache waiting for me when I return.
Tuesday, July 29th, 2014
Today was eventful. I started off trying to trap Copper (II) Oxide particles because these are commonly used in the demonstration of orbital angular momentum transfer. I thought that the scattering force would be somewhat of an issue with these particles (because they are absorbing) but this did not appear to be the case. The particles were successfully trapped with a Gaussian beam in three dimensions. It is helpful to get many copper oxide particles in one drop of water, because a vast majority of the particles end up on the circumference of the drop and are difficult to see (like the coffee ring effect).
After succesfully trapping the particles with a Gaussian beam, I moved on to using an l=2 vortex. The result was interesting. I was successfully able to trap the copper oxide particles on the region of doughnut-shaped high-intensity region of the vortex, but orbital motion was not patently observed. The particles did appear to rotate, but I am not convinced that this had anything to do with the properties of the vortex beam. The interesting part, however, was that particles trapped in the beam would be periodically repulsed in the transverse direction. This could not be due to the scattering force, as this would push the particles in the axial direction. The forcefulness of the repulsion was also curious. The trapped particles were thrown of the vortex faster than the frame-rate of the CCD camera could detect. It almost appeared like teleportation.
Monday, July 28th, 2014
This morning, we reviewed the Fourier Transform, Fourier Series and the basics of Fourier optics. Afterwards, I returned to the optical tweezers setup and tried trapping 5 micrometer polystyrene spheres. I had to remove the voltage box for the light source from the table, because it was vibrating and causing the drop of microspheres to shake. The microspheres were drawn into the beam more easily than the yeast cells, but were also more quickly pushed away by the scattering force.
Thursday, July 24th, 2014
This morning I taught the rest of the double slit derivation.
Wednesday, July 23rd, 2014
There was a lot of painstaking alignment today. I spent most of the morning trying to get the third lens back to its correct position. Though tedious, this process did yield several interesting observations. First, I noticed that a sharp image of the laser appears in planes both above and below the specimen. Presumably, one of these is an image of a beam reflection at the air-glass interface, and the other is the beam reflection at the water-air interface. Although there should theoretically be a reflection at the glass-water interface as well, I did not see this. Using the Fresnel equation, I determined that this interface should have the lowest percent reflection (0.6%)- which might explain why I cannot see an image of the beam. This focused image of the laser becomes an issue when I am looking at cells near these interfaces, because it prevents me from seeing what I am trapping. This issue could probably be circumvented with the placement of a red filter in front of the camera.
Secondly, both Marty and I noticed that the third lens does not actually focus the laser beam very well. This is not good for the strength of the trap, because it means that the objective will not focus the laser precisely to a point either. The third issue I ran into with trapping was the aggregation of the yeast cells. The yeast cells seem inclined to form colonies, and once in a colony, they are nearly impossible to trap.
In the afternoon, we listened to a talk from Dr. Figueroa about his work with quantum information processing. When we got back to the lab, I returned to aligning things, and experimented with using microscope slides, instead of coverslips, to hold the sample. Although the coverslips work terrifically because they are so thin, the microscope stage cannot be moved in the x-direction unless a standard size slide is used. However, it was clear that the microscope slides on hand were thicker than the working distance of the objective, so I’ll have to figure out something later.
Despite all of these issues, the tweezers did end up working in the afternoon. I was careful to bolt everything down so that I wouldn’t have to go through the process of moving the third lens again. I trapped multiple cells throughout the day- which is a good sign. The tweezers were also able to manipulate the trapped particles in the z-direction (perpendicular to the stage), which is an indication that the gradient force is definitely stronger than the scattering force.
Tuesday, July 22nd, 2014
Good progress today. This morning we derived the equation for the position of dark fringes in the single-slit experiment. Afterwards, I returned to the optical tweezers and tried to get an image of the laser beam to appear on the TV monitor. My first thought was to move Lens 3 back and forth- but this clearly did not work. I then realized that the camera image might not be on the same region of the sample as the laser. Thus, I adjusted the position of the gold mirror beneath the microscope until I finally got a diffracted image of the laser on the screen. This image was of the reflection of the laser off of the bottom of the microscope coverslip (something I did not realize at the time).
The next step was to position the third lens so that the waist of the laser beam would lie on the image plane. The only real way to know if these two intersect is to see if you can trap a particle. I started out trying to trap the dirt on a microscope coverslip, but this failed presumably because the particles were not sufficiently transparent. Therefore, I decided to look at yeast cells, which Dr. Noe had recommended before. I twice diluted a sample of aggregated yeast cells, and pipetted a drop of this solution onto a microscope coverslip. At first, I was not sure how to differentiate yeast cells from dirt; I knew only that the yeast were small, irregularly sized particles- which is basically how everything looks under a microscope. Dr. Noe cleared up this confusion and showed me that the yeast cells are generally in a different plane of focus. Once I could definitively identify the yeast, I began adjusting the position of the third lens. After a long period of adjusting things ever so slightly, I positioned the beam over a yeast cell and moved the stage. The cell didn’t move. It was about time something started working.
For the remainder of the day, I experimented with moving the particle around. At some point, I moved the microscope stage too fast, and the original trapped cell broke free. Near the end of the day, I decided to test the trapping efficiency of a lower magnification objective, but I accidentally knocked the camera while switching lenses. This means I will have to realign the 3rd lens again. Luckily, I recorded its position before anything was moved. Lastly, I did notice (to my surprise) that even one cell near the focus of the laser will scatter light all over the room.
Monday, July 21st, 2014
Today was an eventful day. We started off by discussing where the 1.22 in the Rayleigh criterion comes from. Then, after doing some more reading on orbital angular momentum, I returned to the alignment of the optical tweezers. I started off by readjusting the dichroic mirror beneath the inverted microscope- when I realized that it was a rather dim-witted idea to align the components at the end of the beam path first. Thus, I turned my attention towards putting the first dielectric mirror in the correct place. I fortunately did not have to tinker with the beam expander very much because it was still well-aligned from Kathy’s setup. That saved me some time.
Aligning the dichroic mirror and the inverted microscope proved to be a challenge. I ended up removing the third lens in the setup so that I could work with an expanded beam instead a focused one. I then took out the objective of the microscope, and put a mirror on the microscope stage. When I had adjusted things so that the beam was reflected back along the path of incidence, I bolted everything down. Afterwards, Melia showed me how to clean the lenses and mirrors- which were covered in chalk dust. Lastly, I returned to repositioning the gold mirror and the CCD camera. Interestingly, I discovered that when the CCD camera is pointed at the television monitor (which displays the camera image), the monitor starts flashing like a strobe light. I think the television has a CRT screen, and the CCD camera is picking up the flicker (refresh rate). It’s still the most interesting thing I’ve seen on television in four weeks.
The third lens is the final element I have to position. The lens is there to direct the laser beam so that it will focus on the image plane after passing through the objective. For this to be achieved, the laser beam must come to a focus at the image distance. This way, the light rays will enter the objective at the same angles the image rays exit the objective. Thus, the laser beam will be focused to the origin of the image rays. I mounted the lens on an optical rail and will try to find this position tomorrow.
Friday, July 18th, 2014
I started to fiddle with the inverted microscope today. The first step was to get a focused image on the camera, which was difficult because a lot of parts were misaligned (and covered with chalk dust). To start, I removed the angled dichroic mirror from underneath the microscope, and repositioned the gold mirror. Then, I placed the camera on an adjustable stand, and moved it around until some semblance of an image appeared on the monitor. In this process of tinkering, I by chance discovered that standing at a certain position relative to the camera greatly enhanced the contrast of the image. This led me to believe that the room-light was washing out much of the image contrast- a hypothesis that was confirmed when I turned the lights off. At this point, I was able to rework the dichroic mirror into the setup without sacrificing the integrity of the image.
Another inadvertent observation led to an interesting point. While positioning the camera, I noticed that the image largely stayed in focus even as I moved the camera back and forth. This seemed curious at first, because I thought that the image should come to focus almost at a discrete point. Dr. Noe suggested I plot the image distance vs. object distance (related by the thin lens equation) in Excel. I calculated the focal length of the objective using the image distance and magnification factor. Plotting the thin lens equation revealed that the camera could be moved 5 cm without having to adjust the stage height more than 1 mm. That explained the positional ambiguity.
Before we left for lunch, Dr. Noe did a derivation of intensity as a function of position in the double-slit experiment. The phase of a wave can be represented in complex notation as eikx where k is the wavenumber. The principle of superposition stipulates that for two-wave interference, amplitude can be rewritten as the sum of the phases: Ap = eikx1 +eikx2. The intensity at a point can be derived by multiplying by the complex conjugate: (eikx1 + eikx2 )( (e-ikx2 +e-ikx2). We ended here, but Dr. Noe said we could finish the derivation some other time.
After lunch, I returned to working with the inverted microscope, and mounted the CMOS camera on the optical bench. Tweaking some other parameters, I realized that there were issues with the specimen illumination. Even after Dr. Noe helped me adjust the vertical position of the halogen bulb, I could still see that the light was neither uniform nor centered. Curiously, I could also see an image of the filament in the projection- which should not be the case in a Kohler illumination system (or so I thought). Nonetheless, I eventually figured out that by nearly closing the illumination aperture, I could see exactly where the light was centered. This made alignment much easier. Lastly, I took some measurements and determined that rotating the focus knob by one tick changes the stage height by around a micrometer.
Thursday, July 17th, 2014
We began this morning by reviewing the topics we had covered in previous lessons. This dovetailed into two small experiments: determining the wavelength of a laser using the Rayleigh criterion, and determining the diffraction patterns resulting from different shaped apertures. Probably by luck, we first calculated the wavelength of the Helium-Neon (632.8nm) laser as 615nm. However, when we repeated the experiment, but measured the angular resolution (sin-1 Radius of 1st Ring / Distance from laser) much farther from the laser, our estimated wavelength had a much higher margin or error. This was puzzling. The second part of the experiment went a little smoother- after we figured out how to work the camera. Our results affirmed the axiom that diffraction patterns mirror aperture geometry.
Later in the day, I began mulling over some ideas related to angular momentum transfer and topological charge. Although the angular momentum per photon in a vortex goes as lħ, where l is the topological charge, the intensity distribution of a vortex also changes with the charge. Thus, I am not sure that higher order vortices necessarily impart more angular momentum to trapped particles. According to at least one paper,
I was curious to see what these intensity plots looked like for different charges, so I plotted a few in Wolfgram Alpha.
If I am interpreting everything correctly, these graphs demonstrate that peak intensity decreases with topological charge, while the radius of the bright ring increases. Thus, for the same integrated power, much less light would be absorbed by a particle in a high order vortex. Looking at these graphs, I also saw that the radius appeared to increase linearly with charge. I confirmed this observation by taking the first derivative of the intensity function. When a=1, the first derivative is equal to 0 when l = x.
Wednesday, July 16th, 2014
Today began with brief presentations on our proposed experiments. Dr. Noe and Marty made it clear that just establishing a working optical tweezers setup would involve a lot of work (more than I anticipated). Nevertheless, the tweezers setup is mostly there, even if the alignment needs fixing. But before I even worry about trapping particles, I have a lot to learn about the apparatus.
The middle part of the day was taken up by the weekly pizza lunch. The pizza was good. Professor Schneble gave a talk on his work on ultracold atomic physics, which he (thankfully) simplified for us. After the talk, he showed his extremely complicated apparatus. The thought of aligning so much equipment…
I started the afternoon by reviewing the basic configurations and ray-diagrams of simple and compound optical microscopes. Next, I read up on the principal components. The objective, which seems to be heavily discussed in optical tweezers literature, collects the illumination light and produces a magnified image of the sample. It basically functions like a magnifying glass with a very short focal length. The numerical aperture of the objective is a measure of the range of angles over which the aperture will accept light from a point. It is defined as
Where n is the refractive index of the medium that the lens is in, and θ is half of the maximum angle. Refractive index is included in the formula as to make the numerical aperture of a lens a constant for a given distance. The numerical aperture of the objective is a determinant of the resolution, which is diffraction limited in most cases. As we have experimentally demonstrated, light passing through a circular aperture will produce an Airy pattern. The image resolution of an optical microscope is compromised by this interference pattern. The angular resolution is given Rayleigh’s Criterion, θ = 1.22λ / D, where D is lens diameter. Since presumably numerical aperture is a function of lens diameter, it makes sense that lenses with higher numerical apertures have better resolution.
The optical tweezers setup also uses a Kohler system, which is designed to uniformly illuminate the specimen. This is achieved by using lenses to defocus the image of illumination source in the specimen image plane, and all of the conjugate image planes (other planes in the microscope where an image of the specimen is formed). Other methods, like critical illumination, produce an image of the illumination source in the specimen image plane as well as in the eyepiece. The diagrams on Wikipedia illustrate this clearly.
Tuesday, July 15th, 2014
Today was a Wikipedia day. I found an interesting paper that discussed a lot of the finer points about the transfer of orbital angular momentum. The paper defined the rate of absorption of angular momentum by the equation:
where τ is torque (time derivative of angular momentum), S is the Poynting vector, σ is the polarization state (±1 for left or right handed circular polarization), k is the wavenumber, and dA is a small area on the surface of a particle. I assume that this equation holds true if topological charge is substituted for σ. The rate of absorption for the particle as a whole can be found by integrating this equation over the entire illuminated surface of the particle. My only issue with this equation was that I was not able to reconcile the dimensions at all…
Another important equation I came across was the formula for the orbital velocity of a small particle in a viscous medium. The force of friction acting on these particles is given by Stokes Law:
When the optical force (Fo) on a particle is equal to the drag force, the particle will reach a constant velocity.
This equation could be useful in figuring out certain parameters of the system. I also read a paper discussing the transfer of orbital angular momentum in fractional order vortices. This ends up not being a great idea because the low-intensity gap disrupts the rotation.
Monday, July 14th, 2014
This morning we began with another estimation problem, this time about angular resolution. Using the diffraction grating formula, we determined that two headlights on a truck could be resolved as separate objects if the truck was within 6 km. After this, we all talked with Dr. Noe about final project ideas. I am most interested in characterizing the mechanical effects of optical vortices on small particles. In particular, I would like to characterize angular velocities of absorptive particle entrained in optical vortices as a function of the topological charge and intensity of the vortex.
Previous experiments have demonstrated the transfer of orbital angular momentum between an optical vortex and small (10µm radius) particles. My current understanding is that this transfer is only possible if the particles are at least partially absorbing. Particles with a small diameter are not trapped in the center of the optical vortex, but on the ring of light intensity. Transfer comes from scattering force. It cannot come from the gradient force because there is no azimuthal intensity gradient.
Friday, July 11th, 2014
This morning we listened to a talk by a sales representative from SAES Group. He was pitching a product, called a getter, which lowers pressures in a vacuum by adsorbing select gases . Having worked with electron microscopes last summer, I know what fun it can be to achieve a very low-pressure vacuum.
After the talk, I returned to reading papers about the use of optical vortices in optical tweezers. Optical tweezers take advantage of the fact that light has momentum and thus can impart momentum to objects. In an optical tweezers setup, focused light from a laser beam refracts through a small particle, and endows the particle with momentum. For particles with a diameter much larger than the wavelength of the laser light, a gradient force pushes the particle towards the region of the beam with the highest intensity. The gradient force applies in both the axial and transverse directions, and can be illustrated by a simple ray-diagram. It is what traps a particle.
A ‘scattering force’ also acts on trapped particles. Reflected light transfers momentum to the particle in the direction of propagation of the beam. The magnitude of the scattering force is proportional to the intensity of the focused light. In driving the particle away from the region of highest intensity, the scattering force decreases trap efficiency. An ideal beam in an optical tweezers setup has a low intensity, but a pronounced intensity gradient. Furthermore, it was established by Dr. Arthur Ashkin that rays with higher convergence angles contribute more to the gradient force. This is intuitive and can be proved mathematically. The doughnut shaped profile of the optical vortex therefore lends itself well to this application.
After enough reading, I still had one major unresolved question: what if the refractive index of the particle is lower than that of the surrounding medium?
Monday, July 7th, 2014
Today we started out with doing a derivation of golden ratio constant. Having learned from the previous derivation, we were quicker to recognize shortcuts this time. Afterwards, I decided I should look up Euler’s Formula, because it has been coming up a fair amount in the papers I have been reading. There are about half a dozen proofs listed on Wikipedia, of which the calculus-based one is easiest to follow.
Afterwards, I began to do more research into optical vortices (which I had started looking into on the weekend). This yielded several interesting insights. First was the idea that light has angular momentum. Angular momentum is classically defined as a function of a body’s moment of inertia and its angular velocity. Since light clearly does not have a moment of inertia in the conventional sense, a quantum mechanical definition must be applied. Nevertheless, angular momentum in light is still a conserved quantity just as it is in the classical sense.
Angular momentum in light is apparently broken up into two components- spin angular momentum and orbital angular momentum. Spin angular momentum is tied to circular polarization. Orbital angular momentum, on the other hand, arises when dealing with helical wavefronts (unrelated to polarization). Vortices are also assigned a topographical charge based on the number of 'phase- twists' in one wavelength. Thus, orbital angular momentum goes as a function of topographical charge.
Thursday, July 3rd, 2014
In the afternoon we had a lecture on research ethics and conduct. That was a little too much excitement for one day. I reviewed Maxwell’s equations and the Lorentz Force Law on the train-ride back to New Jersey. Here is my Sparknotes version.
Gauss’ Law of Magnetism- Magnetic fields have a divergence of 0 (No
magnetic monopoles)
Gauss’ Law for Electric Fields – The divergence of an electric field is
proportional to the total charge density
Faraday’s Law- Voltage in a circuit is equal to the negative rate of
change of magnetic flux
Ampere’s Law- A flowing current and changing electric flux density both
create circling magnetic fields
I continued looking into frequency combs as well. A natural concern related to frequency combs is the frequency stabilization of the comb itself. Although the frequency spacing between modes can vary, most literature seems to discuss the stabilization of the absolute frequency of these modes (carrier envelope offset frequency). To achieve stabilization, a comb must span more than an octave of frequencies. A self-referencing technique is employed. One frequency, nfrep + f0, is doubled by means of a nonlinear optical process, and compared to another non-doubled frequency, 2nfrep + f0. The beating between these two frequencies yields the carrier envelope offset frequency.
Once f 0 is known, it can be controlled using a feedback loop which varies the pump power of the laser.
Tuesday, July 1st, 2014
Day 2 in the LTC began with rehearsal of the Wednesday presentations. I am going to talk about the most recent project I did for my science research class, “Characterizing IR Transmission of Materials Using a Novel Thermo-Mechanical Detector”. The neat part about this experiment was that I was able to force a bimetallic strip into mechanical resonance by heating it at its natural frequency. In the meantime, I calculated the smallest detectable amplitude of oscillation of the bimetallic strip using the small angle approximation and some trigonometry. It turned out to be 18.5 micrometers.
After lunch I decided to look into some potential topics for a long-term project. Dr. Noe had suggested that I might be interested in laser frequency stabilization, and so I had looked at a paper written by Victor Zhao (a Simons fellow from several years back) before I got here. He utilized a technique that took advantage of the orthogonal polarization of adjacent modes in a laser. The length of his laser only allowed for only two possible modes within the Doppler broadened gain curve. The principle of operation was that as the laser cavity expanded (due to heating), the two modes would ‘sweep’ across the gain curve, because modes must be half integer multiples of the cavity length. By separating out the modes with a polarizer, he was able to monitor the intensity of each as it moved across the curve. One mode would increase in intensity as it moved to a position with higher gain, while the other decreased. A feedback circuit was used to stabilize these intensities by adjusting the cavity length with a heater. I wasn't sure if using a heating element was the quickest way to adjust cavity length, but then again I’m not sure of a better way to do it.
Since I read this paper while I was still back in New Jersey, I was able to talk it over with my grandfather, who figured out how to explain the hanging question at the end of Victor's paper. To test the stability of the system without his feedback loop, Victor altered the length of the laser cavity by means of a sinusoidal variation in heater power. He then examined the intensity of one of the modes using a photodetector. In every trial, superposition of the intensity vs. time and heating power vs. time graphs revealed a π/2 phase offset. The cause of this offset was unresolved in the paper. My grandfather realized (in hindsight it does not seem difficult, but I probably would not have figured this out) that the heating power graph must be integrated over time to get the total amount of heat added to the system. The heating power is ultimately the rate at which heat is being added to the system, and thus is not equivalent to the temperature of the system.
Monday, June 30, 2014
Today was day one in the LTC. It was an experience. We started off the day with a discussion on the optics behind the pig-toy, which segued into conversation on conic sections. Dr. Noe introduced the geometric concept of the sagitta, which is a useful parameter in determining curvature. We then moved on to the binomial approximation, (1 + E)n = 1 + nE, for values of E close to 0. This can be justified with a simple linear approximation:
F’(E) = n(1 + E)n-1
F’(0) = n
F(E) ≈ F(a) + F’(a)(E - a)
F(E) ≈ F(0) + n(E – 0)
F(E) ≈ 1 + nE
We discussed the binomial approximation in the context of modeling diffraction patterns. After lunch, we went outside to burn holes in paper with magnifying glasses. Although the standard magnifying glass was able to intensely focus light and burn the paper, a sample convex lens with a longer focal length was not. Dr. Noe explained that because the sun has a non-negligible angular diameter, not all of the rays from the sun are exactly parallel. The angular differences of incident rays are exaggerated in lenses with longer focal lengths, resulting in a larger (and less intense) image of the sun. An ideal lens for burning paper would have a low F-number. We finished the day discussing the Taylor series and working on some derivations.