August 16, 2008
Well, it's been quite a while since I last updated the journal, and the Simons Program is officially over. For the last two weeks, I've been working on my project, in which I've been trying to create a circularly polarized vortex tweezer wtih Hamsa. My part in the whole thing was to get the laser circularly polarized, and that was harder than I expected. The easiest way to do this, obviously, would have been to use a 1/4 wave-plate, but I couldn't actually get my hands on one designed for the 632.8 nm He-Ne laser that we were using. I tried tilting a number of wave-plates designed for wavelengths in the seven and eight hundred nanometer range, but getting circularly polarized light with those ended up meaning that I'd have to tilt the plates so much that we'd lose way too much intensity. The circular polarizer that Hamsa and I borrowed from another lab had the same problem: we lost 50% intensity while using it.
I also tried using cellophane and tape again, which are two other birefringent materials, but both those materials had some, shall we say, quality, issues.
By the end, I had switched to using two right triangular prisms with total internal reflection; information about this method can be found on Kristine Horvat's page. I had to adjust the prisms in 3 dimensions, because the incident beam has to hit at a 45 degree angle, and there was no way Hamsa and I were going to tilt the laser itself. The setup, with everything tilted all over, looked pretty funny, or as Nityan would say, "bootlegged." I actually managed to get pretty good circularly polarized light rather quickly with this method, but then Hamsa said that the setup was positioned too low for her to insert a mirror to redirect the beam, so I had to take everything to the opposite end of the table and do it over again, higher up. For some reason, that took me a lot longer. By the end of Thursday, I had managed to get close to, but not quite perfect, CP light.
ps - A copy of my abstract can be found here.
July 28, 2008
There's been mostly more experimenting this week. First off, here's an updated version of the charts from last week.
Purple is the curve predicted by Malus' Law, blue is the actual experimental data.
Next, I did a very similar experiment, this time with two polarizers. Doesn't it seems weird that once you get extinction with one polarizer, you can bring light back by inserting another polarizer into the setup? Here's a very basic explanation: When you get extinction, that means that there is no component of the light travelling in the direction of the polarizer's orientation. By inserting a second polarizre into the setup, you give the light a component in the original polarizer's direction, and hence you get some light back.
For this experiment, the resultant light should have an intensity I = I0cos2θsin2θ an equation that is proven through either geometry or the Jones matrices. Here is my graph of my actual experimental results vs the predicted intensities:
On Wednesday, we attended a lecture on sonoluminescence, which is the production of light through sound. The emission of light also occurs at temperatures high enough to be interesting to the study of thermonuclear fusion.
July 18, 2008
I've started doing actual experiments this week, involving polarized light. This first step was, as usual, to read up a little more on the topic and its subtopics. It turns out, there's several different ways to polarize light: reflection, scattering, the use of birefringent materials, or the use of polaroid sheets. I started off using reflection to determine the direction of the laser's polarization - when the incident light hit the surface of the glass at Brewster's Angle, the reflected light is linearly polarized, and the parallel component of the light is zero.
Next, I polarized light using a polaroid sheet, of which there are several types. The first type, the J-sheet, was invented in the late 1920s by Edwin Herbert Land, and used aligned iodiquinine sulfate crystals. The crystals absorbed any light travelling perpendicular to their alignment, and transmitted the light travelling parallel. A decade later, the H-sheet was developed, this time transmitting perpendicularly travelling light and absorbing parallel travelling light. The new H-sheet reduced some of the scattering problems that had resulted in a "milky" color in the old J-sheet.
In my experiment, I attempted to measure the intensity of light as a function of the degree of rotation of the polarizer. I basically ran a laser beam through a polarizer, then used a photodetector to measure the amount of light after being polarized. Hopefully, as I changed the angle of the polarizer, I would notice that the intensity of the light was changing periodically as well, according to Malus' Law, where I = I0cos2θ -- I0 being the original intensity and θ being the difference between the angle of the polarizer and the orientation of the laser beam.
Here's a graph of my results (top) vs. the results predicted according to Malus' Law (bottom):
Not bad, but looking through my data tables (not shown), I noticed that most of my measurements were a tad below the predicted intensities. A possible reason for this is because there is a bit of reflection going on: during the experiment, I noticed two reflections in addition to the original laser; one reflection was off the surface of the polarizer, the other was off the surface of the photodetector itself.
In other news, here's a pretty short and to-the-point summary of "Lasers
and Quantum Optics".
July 11, 2008: More Matrix Mechanics
Well, it's been a pretty informative week. In a somewhat unorthodox move, Dr. Metcalf has been introducing us to quantum mechanics from Heisenberg's matrix mechanics approach first, rather than the Schrödinger, equation-happy approach. We started off by reviewing the solution to last week's homework assignment, regarding the method for finding the eigenvectors for a given matrix. Finding the eigenvalue was pretty straightforward: if you have a matrix A, you can find the eigenvalue λ with the formula det(A-λI) = 0. I is the identity matrix. Once you know the eigenvalue, though, there can be any number of possible eigenvectors. In quantum mechanics, you must choose the eigenvector whose elements x1 and x2 satisfy the equation x12 + x22 = 1.
After we had gotten that concept down, we started using it in the next lecture, to find the eigenvector of the Hamiltonian. By arbitrarily (and somewhat incorrectly, since we're ignoring any time dependence) defining Ω as Ω = Ωr + iΩi, we were able to write the Hamiltonian as a (E, ħΩ, ħΩ*, -E) matrix. The eigenvalue we then calculated was ±√(E2 + ħ2|Ω|2); after a bit of algebra, the eigenvector was a sin θ, cos θ vector. This conclusion is very significant because it indicates that a particle can occupy both states simultaneously, for certain values of θ. That small fact is one of the most important facets of quantum mechanics.
Later, we attempted to correct the "swindle" of ignoring Ω's time dependence by showing that, at least approximately, time dependence really doesn't affect anything. First we proved that in some instances, matrices can commute. If AΨ = aΨ and BΨ = bΨ then ABΨ = BAΨ, i.e., they commute. Then, we applied this concept, as well as a lot of algebra, to the equation ℋΨ = iħΨ(dot). After a rotating wave approximation, we were able to get a result that did not include any time calculations, even though we defined Ω with the appropriate time dependency, unlike the previous "swindle."
July 3, 2008
Note: I don't know how to display math (specifically, matrices) on this page right now, so I've linked to pictures of the matrices I'm talking about where appropriate.
There's actually a lot to talk about today, despite (or because of?) Dr. Noé's pegging of today as a cleanup day. We started off the day with a lesson on polarized light. Nityan and I both initially defined polarized light as light that only oscillates in one direction. As Dr. Noé ended up explaining to us, what we were talking about was specifically linearly polarized light, which was actually a special case of polarized light. The electrical vector of the EM wave can be described in terms of components, just like forces. When the x and y components are exactly in phase and have the same amplitude, you get linearly polarized light. There are other cases, though: when the components are 90 degrees out of phase, you get circularly polarized light, and when the components are anything else, you get elliptically polarized light. Here are some pretty good diagrams of the linear, circular, and elliptical polarizations.
After lunch, we sat down for Part Deux of Dr. Metcalf's "elegant" (read: Schrödinger Equation-less) quantum mechanics presentation. We started by naming some of the matrices we found for homework last night. Quite a few matrices were provided, including a very clever one by Hamsa, but Dr. Metcalf only accepted four of them.
These four matrices, aka the Pauli matrices plus the 2x2 identity matrix, were four very special Hermitian (transposed = complex conjugate) operator matrices. More on Pauli's stuff as we go on.
Dr. Metcalf described a seeming conflict: Heisenberg's assertion that "any observable has a Hermitian operator matrix," vs. the observed two level Bohr atom with two states: excited (|e>) and ground (|g>). The |e> atom is represented by a (0, 1) two-by-one matrix, and the |g>, a (1, 0) matrix of the same dimensions. So, mathematically, to excite an atom means using an operator on the (0,1) matrix that will transform it into a (1,0) matrix. So, a 0-1-0-0 two-by-two matrix should be a valid operator. But, that operator is not the Hermitian operator that Heisenberg insisted upon. Uh oh...
Einstein, in 1917, solved this conflict by introducing a new criteria: the correct operator must change |g> to |e>, but must also be able to change |e> back to |g>. Under this new standard, the 0-1-0-0 matrix that Heisenberg couldn't stand was indeed wrong. The correct operator, which satifisfied Eistein's criteria? This one. Hermitian operator? Check. Coincidentally, it's a check for Pauli matrices as well.
As it turns out, the Pauli matrices plus the identity matrix, put together, "span the space" of 2x2 matrices. That is, any 4-dimensional particle can be described using a combination of these four matrices (much in the same way that any point in the Cartesian plane can be described using a combination of x and y).
After Dr. Metcalf's talk, our group headed back to the lab to clean it up a little bit. I ended up sorting through the extension cords and AC adaptors, testing batteries, and, as Dr. Noé quipped, "fixing" my first laser. While we were cleaning, Dr. Noé was showing me some of the equipment around the lab: various lasers, diffraction gratings, oscilliscopes, etc. He ended up teaching me a bit about Fourier analysis. Here are some examples: 1, 2, and 3, along with a decent applet. I plan on reading up on Fourier analysis a bit more over the long weekend.
Homework: Figure out how to determine the eigenvectors of any 2x2 matrix.
July 2, 2008
Today was lecture day at the LTC. Dr. Metcalf gave the first presentation about complex numbers and matrices (thank you, Hamsa!). He began the talk by demonstrating that some operations, like addition and multiplication, are commutative, i.e. a + b = b + a, while other operations, such as rotations, were not. Matrix multiplication, he iterated, fell into the latter, non-commutative category.
In the next lecture, Dr. Schneble gave us what I think was two presentations mixed into one. He talked about his work with Bose-Einstein Condensates, which are extremely cold forms of atomic gas. Normally, if you take a container of gas, separate it into two halves by, say, inserting a wall in the middle, and take the wall back out, you'd observe the gas atoms mixing evenly back together, right? As it turns out, that's not always true. At ultracold temperatures, the two samples of the gas would actually demonstrate interference effects. Yep, wave-particle duality has struck again.
As it turns out, the gases have to be at temperatures so close to absolute zero that you need more than one method of cooling them. The first method Dr. Schneble described was laser cooling, which first seemed counterintuitive to me because you'd think that the last thing you want to do to get something close to absolute 0 (no atomic motion) is to introduce a high energy beam of light. What actually happens, though, as I understand it, is that the photons from the laser bombard the gas atoms with opposite-direction momentum. The gas atoms absorb the photons and their momentum, while emitting the photons and momentum in random directions. The overall effect is that the momentum of the gas atom is reduced, and the Doppler Effect ensures that the photons will only affect the gas atoms moving at a certain velocity.
The other step involved in cooling the (rubidium, btw) gas to ultracold temperatures is evaporative cooling. On of the professors (Dr. Metcalf, I think) made a funny little analogy. Think of a class where the teacher scales the grades based on class average. Wouldn't it be nice if you could just get rid of the top 10 scoring kids? Well, that's kinda like what evaporative cooling does: it removes the highest energy atoms from the atom cloud. Dr. Schneble spent the rest of the presentation describing work on optical lattices, which are very similiar, but more "coherent," than solid, crystalline lattices. After the presentation, we were given a tour of Dr. Schneble's lab. Among other things, we were shown a hole burned into the heavy-duty curtain in the doorway, reminding us (or me, at least) just how dangerous these high energy buggers can be.
Homework: Find all the two by two matrices whose square are the identity matrix. Include complex solutions.
A couple of side notes:
July 1, 2008
The second day at the Laser Teaching Center has just passed, in a much different fashion than the first. Today, Nityan and I spent the majority of the day going through various physics texts, publications, and websites (hello, Wikipedia!) I spent most of my time looking into two topics: optical tweezers and optical vortices.
In layman's terms (aka the only terms I somewhat understand right now), optical tweezers work by focusing laser beams onto dielectric particles. The light carries a momentum proportional to its energy and direction, and that momentum is changed when the beam hits the dielectric particle. Because of the good ol' Law of Conservation of Momentum, the momentum of the particle therefore also has to change. Thus, the particle experiences a trapping force (remember, force is the derivative of momentum). One popular application of the tweezers is to trap and manipulate microscopic (in the nm range) particles such as viruses, bacteria, organelles, and DNA.
Optical vortices are created by twisting light - in other words, introducing an orbital angular momentum to the beam. The light around the axis of rotation cancels itself out, leading to the phase singularity that you see (or rather, don't see) at the center. Optical vortices have a very interesting potential use in quantum computing. Computers currently store information in binary terms: 0 and 1. Optical vortices, on the other hand, have many states (theoretically, an infinite number), and can thus store information much more efficiently. Moreover, using vortices to send information is naturally secure: anyone but the intended reciever of the information would obviously have to be viewing the beam from a different location, so they would be observing a different (distorted) beam from the intended reciever. As my old physics teacher, Dr. Watt, would say, "Ooh, look! Fascinating interdisciplinary topic of the day!"
Oh right, I also had to go to a mandatory bio and chem lab safety lecture today. I went to sleep within the first couple of minutes, but I'll be sure to refer to my handout the next time we deal with pathogens in a laser lab.
June 30, 2008: First Day
Today was my first day in the Laser Teaching Center, and boy am I glad to finally be somewhere that actually has A/C. My mom was with me for Dr. Noé's tour of the Center as well as the opening reception; we were both struck with how busy and bustling the lab seemed to be. There are three other folks here in the lab already: Hamsa, Will, and Victor; Nityan and I are the newbies to the Center.
After the parents left, we heard the expected lab safety talk, got some really nice lab notebooks, and went outside for some interesting hands-on lens and mirror demonstrations. During this time, some of the questions Dr. Noé asked me about the physics behind what we were witnessing made me realize that I really needed to brush up on my optics. Example: Dr. Noé: "So why doesn't this produce an image on the paper?" Me: "Um, something to do with real and virtual images...?" Ah, well, with an answer that unspecific, there was no way that I could technically be wrong.
After lunch at the Student Activities Center ("SAC"), Hamsa put Nityan and me to work by having us derive/prove the Law of Reflection and Snell's Law. The day pretty much ended with an impromptu lesson on complex math and matrices, courtesy of Hamsa, and I headed back to the dorm knowing that I had to brush up on my precalc in addition to the optics.
Here's an interesting question to end the day: What is i (the square root of negative one) raised to the i?