Selected Publications

Jonathan Tran, Ted Yoo, Shane Stahlheber, Alex Small, Journal of Statistical Mechanics: Theory and ExperimentP05014 (2013)

This paper describes a project on determining the percolation threshold of 3D lattices in which each atom only forms 3 bonds.  Most of the crystals that we learn about in solid-state physics have 4 or more bonds per atom.  However, lattices with just 3 bonds per atom are possible, and we computed their percolation thresholds.  One motive for this work was simply that the lattices are pretty, the algorithms are fun, and we got to fill in a gap in the literature.  Another motive is that I want to learn something about percolation theory to use some of the ideas in other projects.  Of course, now that I've delved into this a bit, I've come up with additional projects.
Alex Small, Nature Methods, 9, pg. 655-656 (July 2012)

This is an invited article discussing two papers in Nature Methods.  The papers introduce better image analysis tools for super-resolution localization microscopy (PALM and STORM).
Ertan Salik, Alex Small, Optics and Photonics News, pg. 10-11 (June 2012)

We were invited to submit an article describing our efforts to grow the number of physics majors at Cal Poly Pomona.  In the past 3 years we've gone from averaging 10 or fewer per year to a graduating class of 21 in 2012!  I am one of the coordinators of the group of faculty that informally steers our recruitment efforts, and it was gratifying to see this article come out describing our accomplishments in the same month that our first growth cohort (we began our intensive efforts with the group that were sophomores in 2010) graduates.
Rebecca Starr, Shane Stahlheber, Alex Small, Optics Letters, 37, pg. 413-415 (2012)

Here we describe an algorithm for determining the coordinates of a fluorescent molecule by looking at a picture of it.  Essentially this amounts to fitting a 2-dimensional function to the image data (i.e. an array of photon counter per pixel).  While many algorithms are available for this task, ours has the advantage that the number of steps in most time-consuming part is only proportional to the square root of the number of pixels, rather than the number of pixels.  Since our results converge in 2-3 iterations, and we do it twice (once for the x-coordinate, once for the y-coordinate) the number of steps is about 6*the square root of the number of pixels, multiplied by the number of function evaluations per step.  There are single-step algorithms out there, but they still require a number of operations proportional to 2x the number of pixels.  Our algorithm wins if the number of pixels is about 9x9 or more.  Also, our algorithm is based on Maximum Likelihood Estimation (MLE) and thus should achieve the theoretical minimum variance in the position estimates (or, at least, something very close to it, given that we make a few assumptions and thus don't have a perfect probability model).

Alex Small, Biomedical Optics Express, 2, 934-949 (2011)

This paper is a follow-up to the Optics Letters paper below.  Now we consider what happens if you have only one laser in the experiment instead of 2.  The laser that bleaches molecules is also the one that activates them, so controlling error rates, activation, and bleaching is actually more complicated rather than less complicated.  Sure, you have only one variable to play with, but now that means that everything changes together; you can't separate activation and bleaching.  The lesson is that in this more complicated case you need much more detailed knowledge of the photophysics of your system.
Alex Small, Kai S. Lam, American Journal of Physics, 79, 678-681 (2011)

This paper describes a simple way of deriving two classic equations of optics and mechanics.  I worked with my colleague Kai Lam on this after teaching upper division classical mechanics and deciding that I want an easier to way to show the students Hamiltonian mechanics.  The usual approach from Goldstein is way too tedious for undergraduates; this approach is just based on the action integral, and makes for nice analogies with optics.
Theoretical investigation of optical patterning of monolayers with subwavelength resolution
Triet Nguyen, Michael Mansell, Alex Small, Physics Letters A, 374, 2681-2687 (2010)
    We decided to do some simulations of nanolithography controlled with STimulated Emission Depletion (STED), to see what the effects of diffusion and chemical kinetics would be if you try to use lasers to attach soluble excited molecules to a surface.  Surprisingly, diffusion hardly matters at all, and repeated laser pulses can compensate nicely for slow kinetics.

    Due to an oversight on our part, the following acknowledgment was not included in the article:  "This work was supported by the Research Corporation, the Keith and Jean Kellogg Honors College, and California State Polytechnic University."  We regret the error and apologize to our sponsors.
    This is a continuation of the work in the paper below, showing that the algorithms used in microscopy techniques like PALM and STORM have implications for resolution. When a student finishes some simulations we will combine those calculations with the theoretical results here and send them to a journal.
  • Theoretical Limits on Errors and Acquisition Rates in Localizing Switchable Fluorophores (pdf)

    Alexander R. Small, Biophysical Journal, 96, L16-L18 (2009)

    This manuscript describes the theoretical limit to speed and error rates when using microscopy techniques that involve randomly activating a handful of molecules, imaging their positions, then randomly activating a different handful of molecules, and continuing this process until all of them have been imaged. I got the idea of working on this problem from attending a talk at Biophysical Society in 2007. In summer of 2008 I asked a student to work on an image analysis algorithm. The algorithm didn't work so great, but I asked myself "Well, what is the maximum performance of any algorithm?" and came up with this work sort of on the fly. Since then I've found an extension that introduces a whole new concept of resolution limits (more details coming soon).

  • Spatial Distribution of VEGF Isoforms and Chemotactic Signals in the Vicinity of a Tumor (pdf)

    Alexander R. Small, Adrian Neagu, Franck Amyot, Dan Sackett, Victor Chernomordik, Amir Gandjbakhche

    Journal of Theoretical Biology, 252, 593-607 (2008)

    This manuscript describes a reaction-diffusion model of Vascular Endothelial Growth factor (a molecule responsible for the growth of blood vessels) being released from a tumor and interacting with the tissue. We explain a number of significant experimental results with our model, and rule out a common hypothesis. This model was first developed on the back of a napkin over a lunch break during the angiogenesis workshop at the UCLA Institute for Pure and Applied Mathematics (May 2006). It was a fun workshop and a fun project, and we're exploring extensions and applications of the model right now.

  • Enhancing Diffraction-Limited Images Using Properties of the Point Spread Function (pdf)
    Alexander R. Small, Ilko Ilev, Victor Chernomordik, and Amir Gandjbakhche
    Optics Express, 14, 3193 (2006)

    This paper won me an NIH Fellow's Award for Research Excellence. It describes a really simple way to get information on fluorescent probes separated by less than the diffraction limit. I'm working right now on applying this algorithm to speeding up a super-resolution method that others have developed. Aside from the fact that it can do some neat tricks (alone or in combo with other methods), the key insight is that it shows which parts of the diffraction-limited image carry the most information.

  • Delocalization of Classical Waves in Highly Anisotropic Random Media (pdf)
    Alexander R. Small and David Pine
    Physical Review E, 75, 016617 (2007)
    *Selected for inclusion in the Virtual Journal of Biological Physics, February 1, 2007

    This paper describes a third of my thesis work. Of the three projects that went into my thesis, this one was most uniquely mine, conceived entirely on my own without any suggestions. Of the papers that I've been an author on thus far, this is the one that I'm proudest of. It may not have turned the world upside down, but it proved to me that I am an independent scientist. This one is my baby. It describes interesting phenomena related to Anderson localization of light in layered systems. More specifically, it describes ways that Anderson localization can be thwarted, resulting in diffusive transport in systems that exhibit 1D characteristics.

    I don't know how it got selected for the Virtual Journal of Biological Physics, because it has nothing to do with biology. I guess it must have been because it's a Physical Review E article with an author at NIH. The study of waves in layered systems is indeed important for tissue optics, but Anderson localization is not known to occur in tissue optics. Well, if it ever is observed, maybe everybody will cite my paper. So that's something.

  • Scattering Properties of Core-Shell Particles in Plastic Matrices (pdf)
    Alexander R. Small, Sheng Hong, and David Pine
    Journal of Polymer Science, Part B-Polymer Physics, 43, 3534 (2005)

    This paper describes a project done in collaboration with industrial scientists at Arkema. We were working on finding ways to reduce the amount of scattering from small rubber particles embedded in plastic. The rubber particles make the plastic tougher, but they scatter light. We came up with some design rules. It was a fun project because I learned that I could do things with theoretical physics that people might actually find useful.
  • Patterned Polymer Photonic Crystals Using Holographic Lithography and Soft Lithography (pdf)
    Jun Hyuk Moon, Alexander R. Small, Gi-Ra Yi, Seung-Kon Lee, Won-Seok Chang, David J. Pine, Seung-Man Yang
    Synthetic Metals, 148, 99 (2005)

    This paper describes the part of grad school where I learned how to make lemonade from lemons. I was getting nowhere with my efforts to make good colloidal crystals in the channels of a PDMS stamp (soft lithography). Then I noticed that another student in the lab (Jun Hyuk Moon) was making polymer photonic crystals by holographic lithography. And I thought to myself "Hey, why not pattern the polymer with the stamps as well as the light? Then we could get wave guide geometries!" So I talked to Jun Hyuk and we decided to try it. And it worked.