# Ateneo Physics faculty Adler Santos gets his paper on polarized light via geometric algebra published in JOSA A

by Quirino Sugon Jr.

Dr. Quirino Sugon Jr., Adler Santos, and Dr. Nofel Lagrosas of the Department of Physics of Ateneo de Manila University

Ateneo Physics faculty Adler Santos wrote a paper entitled, Polarization ellipse and Stokes parameters in geometric algebra,” which was published this January 2012 in Journal of Optical Society of America:

In this paper, we use geometric algebra to describe the polarization ellipse and Stokes parameters. We show that a solution to Maxwell’s equation is a product of a complex basis vector in Jackson and a linear combination of plane wave functions. We convert both the amplitudes and the wave function arguments from complex scalars to complex vectors. This conversion allows us to separate the electric field vector and the imaginary magnetic field vector, because exponentials of imaginary scalars convert vectors to imaginary vectors and vice versa, while exponentials of imaginary vectors only rotate the vector or imaginary vector they are multiplied to. We convert this expression for polarized light into two other representations: the Cartesian representation and the rotated ellipse representation. We compute the conversion relations among the representation parameters and their corresponding Stokes parameters. And finally, we propose a set of geometric relations between the electric and magnetic fields that satisfy an equation similar to the Poincaré sphere equation.

Adler’s co-authors were his mentors Dr. Quirino Sugon Jr. and Fr. Daniel J. McNamara, SJ who are both based in the Department of Physics and Manila Observatory. Adler finished his BS and MS Physics in Ateneo de Manila University last 2008 and 2011, respectively.

Below is an interview of Adler Santos by Ateneo Physics News:

About polarized light. About polarized light formalism using geometric algebra. The main result of the paper was to translate different polarized light basis representations to one another using geometric algebra. Many of these transformations can be found in the literature, but we found some new ones, such as the relation for the electromagnetic field parameters which satisfy the poincare sphere equation.

Geometric algebra is a multivector algebra. This means that you have different elements and you can add them together: scalars, vectors, imaginary scalars, imaginary vectors,  and even higher things that are a bit strange for other mathematicians, since  the addition of elements that do not belong in the same space are not normally added. The way we handled rotations of electromagnetic fields using exponentials of imaginary vectors (or oriented planes) is strange since we don’t use matrices for rotations and that the exponential does not contain scalar or imaginary scalar arguments.  But it works.

2.  How long did it take you to finish your paper?

I was already starting my research on geometric algebra last July 2010, but I had to change multiple topics. I worked with Lorentz transforms in Special Relativity. I got stuck. I worked with Snell’s law  derivation in the law of refraction using electromagnetic boundary conditions. I got stuck. Finally, I was given by mentor a problem on Stokes parameters using geometric algebra. At last, we got results. I started the Stokes parameters paper on December 2010 and finished the first complete draft on the same month. I was able to do this because I worked full time in Manila Observatory–8 hours a day for 6 days. Part of my job is to finish my thesis.

3.  How did you get published?

My advice to those who wish to get published is this: Follow your adviser. You may write a good paper by yourself, but there will always be things that are not really clarified until your adviser checks your ideas, sentences, and equations.

We submitted the manuscript to American Journal of Physics (AJP), but the  the editors replied that paper was did not meet the domain of their journal, because they wish a more didactic paper for physics teachers; our paper was rejected because it was too mathematical and specialized. Two of the reviewers suggested that we send the manuscript to the Journal of Optical Society of America (JOSA). So we submitted to JOSA A: Optics, Image Science, and Vision. The referees suggested a lot of revisions, but many of these are just expansions or further clarifications of what we already did. After the first revisions, the referees made another round of comments to polish the paper.  After two days, we sent the second revised manuscript.  Finally, the peer reviewers and the topical editor agreed to publish our manuscript.

4.  What are you teaching this semester?

I teach part-time.  I teach PS 31.  My class is comprised of second year chemistry and math majors. They are a very curious batch. They asked about Dark Matter,  time travel. Einstein’s General Relativity, and Maxwell’s equations.  They even asked me about the hydrogen wave function. They wanted to know what  quantum Mechanics has to say about atoms. For chemistry majors, they only know so far are the bonds and chemical equations. They don’t yet know that there there are deeper principles. I showed them the hydrogen wave functions and they were intimidated. I showed them the graphs. Even if you have this joint volumes of space, the electron can be in either one of those even without travelling in between. These are the icing on the cake when I teach.

5.  What are your career plans?

Adler Santos's Mac computers with Python codes

Right now, just last night, I got the second interview. I got a job offer from a French web startup company that specializes in Python programming language and using Python to create Google apps. And then they gave me an exercise: they asked me to code a web app using Python, Html, CSS, and Javascript. They gave me two weeks to do it. I finished this weekend after  10 days. What is really fun about the exercise is that they did not guide me. I was only asked to Google everything I need. I have no guide. Being a developer, you know that all references are already in the internet: you just need to find them.  After submission. I have the second interview for code review. They asked me to explain my codes. Finally, they told me, that I am fit for the job. They sent me the contract. I have seven days to sign.  I can already start this February. I am still thinking about it.

I was originally planning to appy for Phd, to work with planetary magnetic fields. But I figured out that being a developer is more fun. There is instant gratification: you write a code , compile it, and see the results.