Jose Carvajal-Beltran
Inside the mind of Jose Carvajal-Beltran — MIT undergraduate, materials scientist, and crazy laser enthusiast.
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
Log
Understanding the Fourier Slice Theorem
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
ISEF: Potential Shift in Velocity as Temperature Changes
International Science & Engineering Fair · Physics · PHYS005 ↗
Motivated by the growing interest in drone delivery systems and rotor-based space capsules, this project used maple samara seeds as a physical analogue to study how temperature affects aerodynamic descent. I dropped 30 seeds twice each across three controlled-temperature rooms (180 total trials), recording fall times from 2.3 meters and computing terminal velocity, energy equilibrium, and Reynolds numbers for each condition.
The results showed a clear shift: at 27 °C terminal velocity exceeded 1 m/s, while at 20 °C it stayed below that threshold. The higher-temperature runs also yielded lower Reynolds numbers, indicating a more laminar flow regime. The data point to a positive correlation between ambient temperature and descent velocity, and suggest that future work with more temperature points could sharpen these findings considerably. This project won a CIA award
ISEF: Dynamics of Space Capsule with Rotors
International Science & Engineering Fair · Physics · PHYS002 ↗
Despite decades of NASA interest in rotor-equipped re-entry capsules, the literature rarely drills down to the rotor geometry itself. This project set out to fill that gap, testing the hypothesis that a larger chord length produces superior autorotation performance. I ran a CFD simulation of the NACA 4412 airfoil to map pressure distributions across the blade, then applied Momentum Theory to derive closed-form autorotation equations and estimate Reynolds numbers under idealized assumptions.
A Blade Element Momentum (BEM) simulation revealed a counterintuitive result: the efficiency curve is actually wider — meaning more forgiving across a range of conditions — for shorter chord lengths than for longer ones. Supplementary heat-flux analyses of the capsule outer surface produced consistent results across two independent runs. The overall conclusion is that a rotor-based space capsule is plausible, but optimal performance likely favors thinner, high-strength blades and a coaxial rotor arrangement to manage the highly turbulent Reynolds regime predicted during descent. This project won a full tuition scholarship to F.I.T.
Custom Air Hockey Table
Edgerton · MIT · Interphase Edge
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 a unique V shape. After going through many iterations, we found that it would be virtually impossible to create what we wanted to within 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 configured a 120V AC OEM blower to the table and hooked 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!
8.012 Final Project
Classical Mechanics · Ring & Beads Experiment
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
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 informally called 'physics for masochists' — and for good reason. 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 heavily on advanced calculus and vector algebra.
6.2020 Electronics Project Laboratory
MIT EECS · Prof. Jim Bales
One of the most interactive and interesting classes I've taken. Taught by 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. A memorable lab session involved our entire group troubleshooting a stubborn circuit that simply refused to cooperate — a rite of passage for any engineer.
UROP: Optimal Orientation for VAM
Volumetric Additive Manufacturing · Light Dose Optimization
FIG. 05.003 — How Volumetric Additive Manufacturing works, visualized.
This is a research project that i took part in, 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 set 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 found that there was in fact a variation in the print quality, sometimes it was obvious, but othertimes it was not. I utilized a technique involving many iterations at a small scale (using less voxels) to see which angles looked promising. This almost always translated to the actual print having a higher quality as well, where quality is determined by measures of a histogram involving in-part and out-of-part voxels.
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.