BEAM ACTIVE λ = 632.8nm MODE: TEM₀₀ LOC: 77 MASS AVE
OPTICAL BENCH / OPERATOR 01

jocabe

Inside the mind of Jose Carvajal-Beltran — MIT undergraduate, materials scientist, and crazy laser enthusiast.

Portrait of Jose Carvajal-Beltran
SUBJECT / 01.001

Welcome. I'm an undergraduate at MIT studying Materials Science & Engineering.

My current research focuses on optimizing light-dose orientation algorithms for Volumetric Additive Manufacturing — a printing technique where collimated beams converge inside a rotating vat of photoresin to sculpt geometry out of light. I enjoy the slow satisfaction of rigorous problem-solving.

Off the bench, you'll find me on a soccer pitch, hanging out on campus with friends, or overbuilding custom hardware for the fun of it.

Institution MIT · Course 3
Focus Lasers and VAM
Location Cambridge, MA
Status Active

Résumé

DOCUMENT / 02.001

If the document below does not load, view or download the PDF.

Log

TRANSMISSION / 03

Understanding the Fourier Slice Theorem

2026.03.05 // ENTRY_001

This is a sample blog post.

The Fourier Slice Theorem (also known as the Projection-Slice Theorem) is a fundamental principle underpinning medical imaging modalities like CT scans. In simple terms, it states that the 1D Fourier transform of a parallel projection of a 2D object is exactly equal to a 1D slice of the 2D Fourier transform of that object through its origin.

Mathematically, let's represent an object's density as a 2D function $ f(x, y) $. If we take a projection of this object along the y-axis, we integrate over $ y $ to get a 1D function $ p(x) $:

$$ p(x) = \int_{-\infty}^{\infty} f(x, y) \, dy $$

Next, we take the 1D Fourier transform of this projection:

$$ P(u) = \int_{-\infty}^{\infty} p(x) e^{-i 2\pi u x} \, dx $$

Now, consider the 2D Fourier transform of the original object $ f(x, y) $:

$$ F(u, v) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x, y) e^{-i 2\pi (ux + vy)} \, dx \, dy $$

If we examine the slice of this 2D transform exactly along the x-axis (where $ v = 0 $), the equation simplifies perfectly to match our 1D transform of the projection:

$$ F(u, 0) = \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty} f(x, y) \, dy \right) e^{-i 2\pi u x} \, dx = P(u) $$

Because of this relationship, we can take physical projections of an object from multiple angles, compute their 1D Fourier transforms to build out a 2D frequency domain plane, and then apply an inverse 2D Fourier transform to reconstruct the original object's internal structure.

Projects

APPARATUS / 04
APPARATUS 04.001

Custom Air Hockey Table

Edgerton · MIT · Interphase Edge
Watch on YouTube

In this project, Diego Salcedo and I were both in the same Edgerton (Makerspace at MIT) section for Interphase Edge. We decided that it would be a great idea to build an air hockey table with unique V shape. After going through a million failures, we found that it would be virtually impossible to create what we wanted to with a very short timespan and while taking many difficult courses.

Instead, we created our own air hockey table from scratch, with Diego working on the machining aspects and myself working on the electronics components. I managed to 'safely' configure a 120V AC OEM blower to the table and hook it up to power and an e-stop to blow air through the many holes Diego drilled. After this, I used a protoboard and infrared sensors to create a beam-break sensor and a scoreboard to track who is winning.

All-in-all, this is definitely the most fun I have had creating a project!

APPARATUS 04.002

8.012 Final Project

Classical Mechanics · Ring & Beads Experiment
Download Report (PDF)

This was my final project for 8.012, it was revolving a problem from a Problem Set that felt very counterintuitive even if the math checked out. We made a physical model and coded a simulation (with a BIT of exaggeration) to see if the phenomenon was real and if it could be explained better. The problem was of two beads on this ring, it is said that if the beads were a certain mass or greater, then the ring would 'jump' up at a certain angle. We tested this in real life (after hours of making models that didn't work and using ancient scales) and found that the phenomenon did actually happen, seeing it made it feel more intuitive. We submitted our report and ended up getting selected to present in front of the entire class (Shoutout to Kaku & Ishan)!

Academics

COURSEWORK & RESEARCH / 05
COURSE 05.001

8.012 Classical Mechanics

MIT Physics · Advanced Track

Definitely the hardest and most time consuming endeavor so far in my journey (albeit the journey has just started). This class is jokingly called 'physics for masochists', and it tracks. People would ask me all the time "why are you taking that class", and I genuinely had no response other than for curiosity. This class focused very heavily on some heavy calculus as well as vector algebra (in VERY ugly ways).

COURSE 05.002

6.2020 Electronics Project Laboratory

MIT EECS · Prof. Jim Bales

One of the most interactive and interesting classes I've taken. Taught by the great Jim Bales, this class was an introduction to circuits and basic laws about electricity. We built many projects with protoboard and learned how to use new electrical components every week. My second favorite moment was laughing for a solid 30 minutes with the two other people in my group because absolutely nothing was working and the software kept crashing.

RESEARCH 05.003 · ACTIVE

UROP: Optimal Orientation for VAM

Volumetric Additive Manufacturing · Light Dose Optimization
FIG. 05.003 — How Volumetric Additive Manufacturing works, visualized.

This is a current research project that I am undertaking, it involves a really cool 3D manufacturing technique that is known as Volumetric Additive Manufacturing (VAM). The most basic way to explain it is to say that light from a projector 'shoots' a bunch of 2D photos at a vat of resin that hardens with light. The vat spins and for each degree, a new photo is projected. All of these projections form a 3D object. What I am setting out to determine is whether the orientation ($\phi$ and $\theta$) have any effect on the resulting quality of the print due to dosage of light. I will update this section more and I learn and undertake this challenge more!

RESEARCH 05.004 · PENDING

MIT MISTI Mexico

Univ. Anahuac Mérida · Carbon Nanostructures · Summer 2026

This is the project I will be up against over the summer as I travel to Yucatan, Mexico to do research with a team in the University of Anahuac Merida. I have been dipping my toes into the existing research on this and will update this as I explore over the summer.

Docs

STANDARD OPERATING PROCEDURES / 06

Documentation and standard operating procedures for various laboratory processes.