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BS, Pontifical Catholic University
PhD, University of Notre Dame

* = Student co-author
Article links to the file of the publication  
 

"Electronics with Discrete Components" E.J. Galvez (Wiley, Hoboken, 2025). 
This new edition incorporates the latest technological advances in the digital industry as they apply to making electronic circuits with individual integrated circuits. It adds many more exercises and problems plus labs that use Arduino and Teensy boards.

•  "" V. Rodriguez-Fajardo, T. Nguyen* and E.J. Galvez, Journal of Optics, 27, 045609 (2025).
  We studied preparing optical beams in modes of the beam that mimicked the states of the quantum pendulum (Imagine a pendulum of atomic size). We prepared superpositions of modes (states) in such a way that the propagation represented time. Thus as we moved the camera along the beam we saw the quantum state changing. The image shown is of the beam in the "Fourier plane." It is a remarkable optical analog graph of the wavefunctions of the states involved in the superposition, where the angular position from the center is the angle of the pendulum and the distance from the center is the energy of the state. It is an analog energy-level diagram. The top and bottom halves of the image mirror each other.  
• “,” B. Luo,* L. Francis,* V. Rodriguez-Fajardo, F. Khoshnoud and E.J. Galvez, American Journal of Physics, 92, 308-316 (2024). .
  This work reports on a method to implement a demonstration of Young's double-slit experiment with single photons. It modifies the standard setup to do table-top experiments with correlated photons. In our work we resolved some of the challenges associated with this demonstration: spatial coherence and resolution.
• “Nonlocal Mueller Polarimetry,” C.-J. You,* V. Rodriguez-Fajardo, L. Francis,* and E.J. Galvez, Proceedings of SPIE 12845, 1284509 (2024). We note that tables 1 and 3 contain typos. Will submit an erratum.
  &�Բ�����;“Einstein Beams and the Diffractive Aspect of Gravitationally-lensed Light,” V. Rodriguez-Fajardo,  T.P. Nguyen,* K.S. Hocek,* J.M. Freedman,* and E.J. Galvez, New Journal of Physics, 25, 0833033 (2023).
  We developed a new form of laboratory astrophysics: simulating gravitational lensing. With it we were able to image the diffraction of gravitationally lensed light, as predicted by general relativity but never observed in astrophysical measurements. We also found a new type of optical beam that is a hybrid between non-diffracting beams and Gaussian beams, called Einstein beams.
  &�Բ�����;“,” N.S. DiBrita* and E.J. Galvez, American Journal of Physics 91, 307-315 (2023).
  &�Բ�����;“Einstein beams carrying orbital angular momentum,” V. Rodríguez-Fajardo, T.P. Nguyen,* K.S. Hocek,* J.M. Freedman,* and E.J. Galvez, Proceedings of SPIE 12436, 124360C (2023).
• “Direct measurement of the density matrix of a two-photon polarization qubit,” E.J. Galvez, A.D. Goldstein,* C.J. You,* V. Rodríguez-Fajardo, Li. Shi, and R.R. Alfano, Proceedings of SPIE 12373, 123730C (2023).
• “A curriculum of table-top quantum optics experiments to teach quantum physics,” E.J. Galvez, Journal of Physics: Conference Series 2448, 012006 (2023). 

Quantum Optics Experiments To Teach Quantum Mechanics
Over the past 20+ years we have developed table-top experiments to teach the fundamentals of quantum mechanics: superposition,indistinguishability, entanglement. This article is a review of the experiments that we have designed and implemented.

• “,” E. J. Galvez, B. Sharma,* F. K. Williams,*  C.-J. You,* B. Khajavi, J. Castrillon, L. Shi, S. Mamani L.A. Sordillo, L. Zhang, and R. R. Alfano, Biomedical Optics Express 13, 6621-6630 (2022). 

Quantum Decoherence as a Diagnosis Tool
Over the past seven years, we have been working on a technique that transmits entangled photons through tissue. We characterize the tissue by what it does to the entangled state of the photons. In this article, we found that entangled photons going through brain tissue with Alzheimer's disease experienced less decoherence than healthy samples. The figure shows the measured density matrix of the sample with the disease (left) and the healthy (right). 
 

• “(2D+1) Pendulum Beams: Nin-diffracting Optical Spatial Wavepackets that Simulate Quantum Pendulum Dynamics,” T.P. Nguyen,* V. Rodriguez-Fajardo, and E.J. Galvez, Proceedings of SPIE 12017, 1201704 (2022).
• "Einstein Beams: Optical Beams Following Gravitationally Lensed Trajectories," E.J. Galvez and J.M. Freedman,* Proceedings of SPIE 11701, 117010U (2021).
• "Remote Quantum Optics Labs," E.J. Galvez, Proceedings of SPIE 11701, 1170106 (2021).
• “,” E.J. Galvez, F.J. Auccapuclla, Y. Qin,* K.L. Wittler,* and J.M. Freedman,* Journal of Optics 23, 024001 (2021).
   

Non-diffracting Pendulum Beams Reproduce the Quantum Pendulum
In this research, we create optical beams in a specific type of optical mode where the wave equation has the same form as the quantum equation for the single pendulum. As a consequence, the images that appear on a screen have an intensity proportional to the quantum probability.

“,” A. Bista,* B. Sharma,* and E.J. Galvez, American Journal of Physics 89, 111-120 (2021).

Demonstrating how Quantum Superposition can Uncover the Eavesdropper in Quantum Communications
In the quantum world, a measuring device gives outcomes of a measurement. To each outcome corresponds a state. So after a measurement, the system is left (or projected) in the state corresponding to the measured value. A quantum system  can be in a superposition of states of the measuring device, so that a measurement modifies the state of the system. These quantum principles can be harnessed to detect an eavesdropper in a communication with photons. The measurement device detects the photon in one of 2 orthogonal states or possibilities. If the photons are sent in a superposition of these states and the eavesdropper makes a measurement, it would change the state of the photon so to reveal the action. We implement the generation and detection of the single photons using quantum entanglement, and simulate the eavesdropper with a dephasing optical element.

• “Transmission quantum state tomography of biological tissue,” E.J. Galvez, B. Sharma,* F.K. Williams,* B. Khajavi, and L. Shi, Asian Journal of Physics 29, 379-386 (2020).
• “Experiments with Correlated Photons” E.J. Galvez,  in Experimental Physics: Principles and Practice for the Laboratory, W. Smith, Editor (Taylor & Francis, Boca Raton, 2020), pp 377-402.
• , B. M. Holmes* and E.J. Galvez, Journal of Optics 21, 104001 1-7 (2019).
• , J. Castrillon, E.J. Galvez, B. Rodriguez, and O. Calderon-Losada, European Journal of Physics 40, 055401 (2019).
• , E.J. Galvez, Proceedings of SPIE 11143, 111431A 1-6 (2019).
• Pendulum Beams: A Window into the Quantum Pendulum, E.J. Galvez, F.J. Auccapuclla, K.L. Wittler,* and Y. Qin,* Proceedings of SPIE 10935, 1093509 (2019).

B. Khajavi, J.R. Gonzales Ureta, and E.J. Galvez, Photonics 5, 5030016 (2018).

Determining Optical Vortices by their Wave-Ripples
Optical beams can carry optical vortices. These are wave tornados in the light. Should we combine two beams with vortices of distinct vorticicity, a striking phenomenon occurs. The resulting beam rearranges its vorticity so that it consists of a central vortex surrounded by an array of singly-charged vortices. The pattern of vortices reveals the vortex content of the original beams. We studied this phenomenon ten years ago, in work led by then Sean Baumann '08 (see story below Optics Express from 2009). In this work we use a "shear interferometer" to unravel this pattern in a simple and direct way. Behzad Khajavi, recent Ph.D. from Florida Atlantic University, did his dissertation research at �󷢲�Ʊ and led the discovery of using shear interferometry to determine the vorticity of optical beams (Optics Letters 2017 below). Using this method, and with important contributions from Junior Gonzales Ureta, graduate student visitor from Peru, we developed a method to determine the vortex content of superpositions of vortex beams. These can be used to store information in optical beams. The figure below shows the wave ripples in the light beam, color coded to enhance the peaks and troughs of light waves (radians), featuring dislocations in the ripples caused by the vortices, and revealing that it was made by two beams with vorticity of +1 and -2.

Laser Imaging Polarimetry of Nacre, J.A. Jones, R.A. Metzler, A.J. D'Addario,* C. Burgess,* B. Regan,* S.Spano,* B.A. Cvarch,* and E.J. Galvez, Journal of Biophotonics (in press, March 28, 2018, DOI: 10.1002/jbio.201800026). Preprint.

Nacre Through Polarizing Glasses
In this work, we investigate the structure of nacre by passing polarized light through it. The way nacre scrambles the polarization of the light tells us about the type of shell that produced the nacre. It can be bivalve (such as pearl oyster), gastropod (such as abalone- figure) or cephalopod (such as nautilus). Six �󷢲�Ʊ students participated in the research. B. Holmes has done more recent studies.

, E.J. Galvez, I. Dutta,* K. Beach,* J.J. Zeosky,* J.A. Jones, and B. Khajavi, Scientific Reports 7, 13653 1-9 (2017). 

Möbius Twists in Light Fields 
This publication reports on a stunning realization: light fields in generic conditions twist in 3 dimensions. We went to investigate on a simple situation: two optical beams crossing in free space. Our aim was to measure it the simplest way possible: with polarizers. It was not easy; it took us 5 years to solve this problem. In the article we report that knowing our particular geometry we can extract the 3-dimensional fields, and show that they indeed twist, forming either Möbius strips or twisted ribbons along a closed path in space. We show an example of experimentally extracted of the polarization orientation at the crossing of two beams- more details can be found in the article.
 

, B. Cvarch,* B. Khajavi, J.A. Jones, B. Piccirillo, L. Marrucci, and E.J. Galvez, Optics Express 25, 14937 (2017).

Generating Monstars using Liquid-Crystal Q-plates
Monstars are asymmetric topological dislocations. We have created them previously using superpositions of optical beams. A new type of engineered device invented by our colleagues at the University of Naples (B.P. and L.M.) known as the "q-plate" can use the birefringence of liquid crystals to impart dislocations onto optical beams. This has been shown by them previously with symmetric q-plates. As part of his summer (2016) and senior thesis research, Ben Cvarch '17 studied q-plates that were elliptically symmetric, and showed that they produce monstar disclinations of index zero (see images below). Behzad Khajavi continued Ben's work and measured monstars of index 1/2 with the same q-plates. Kudos to both. The article was selected as the "Editor's Pick" from the journal as an excellent research publication.

• , B. Khajavi and E.J. Galvez, Optics Letters 42, 1516-1519 (2017).

Determining the Orbital Angular Momentum of a Light Beam
In doing the previous work on monstars we stumbled onto a new method to determine the topological charge of a beam carrying orbital angular momentum. The orbital angular momentum per photon is the topological charge times Planck's constant over two-pi. In many instances it is important to know this parameter and we found a method to determine it visually using a minimum number of optical elements. All we need is a pair of lenses to make the wavefront planar, and a wedged optic. If the beam carries an optical vortex, and hence orbital angular momentum, then by reflecting it off a wedged optic we find a pattern of conjoined forks as shown in the figure. The topological charge is given by the number of tines minus 1.
 

• , Enrique J. Galvez and Behzad Khajavi, Journal of the Optical Society of America A 34, 568-575 (2017).

Optical Monstars
This work is the culmination of a study of topological patterns that we can embed in the polarization of the light. In presents a study of all the monstars that can be produced with three modes of light. Monstars are disclinations or disruptions in a rotational order. Why do we care? They are present in many physical systems: crystal dislocations, magnetism, and topologies of surfaces and materials, to name a few. However, in most situations we cannot control the pattern that appears, or is very difficult to generate a pattern deliberately. Here we use light to explore all the possibilities. We have found many new patterns that have not been seen before. We find them surprisingly variable, often showing wild and crazy shapes. The experiments reproduce the theory very well. The figure below shows three types of monstars (predicted shapes are in first row, and measured ones are in the second row). They are characterized by their index: the number of turns that the lines (polarization) makes per turn around the center. Their indices are:  +3/2 for the left pattern, 0 for the middle pattern, and -1/2 for the right pattern.

• Deducing 3-Dimensional Polarization Fields from Projective Measurements (Revised), Enrique J. Galvez and Ishir Dutta, Proceedings of SPIE 10120, 101200B (2017).
   
• Searching for the Helical-Gradient Force on Chiral Molecules Joshua A. Jones, Brian Regan,* Joshua Mills,* Behzad Khajavi, Jackson Painter* and Enrique J. Galvez, Proceedings of SPIE, 10120, 101200M (2017).
   
• Polarimetry of Pinctada Fucata Nacre Indicates Myostracal Layer Interrupts Nacre Structure, Rebecca A. Metzler, Joshua A. Jones, Anthony J. D'Addario* and Enrique J.  Galvez, Royal Society Open Science 4, 160893 (2017).

Using Light's Polarization to Investigate Nacre
Nacre or Mother of Pearl is the colorful iridescent surface of shells. It has an amazing structure: a brick and mortar pattern of micron-sized crystals (bricks) embedded in organic matter (mortar). The crystals, made of aragonite, modify the polarization of the light as it passes through them. In this study we study the structure of nacre by examining what it does to the polarization of the light after it passes through a thin section, edge on. A stage of turbulence or disruption in the growth of the shell leaves a mark in the layers of nacre, similar to the way major geological events in Earth's history leave a mark in the geological layers of the soil. In our case the layer is known as the myostracal layer. In this work we were able to show a difference in the structure of nacre before and after this event using light. It is a unique experiment! The photo below shows imaging photos of the sample and a glimpse of our analysis. For more you must read the paper!

• , Lingyan Shi, Enrique J. Galvez, and Robert R. Alfano, Scientific Reports 6, 3774 (2016).

Characterizing Brain Tissue with Entangled Photons
In this work we investigated how the entanglement in polarization between two photons of light survived when one photon passes through a thin slice of brain tissue. Before we did the experiment we thought that traveling through a messy medium of neurons and axons would wipe out the entanglement. What we found was surprising. The entanglement survived passage through tissue regardless of thickness! What mattered was not the thickness, but the amount of water and the type of tissue. Gray matter was quite transparent, but white matter was not. Other tissues, like kidney were not as transparent. The results that we present leaves us with one question: could we diagnose disease with this method? The question is still open. Stay tuned!

• Complex Light Beams, Enrique J. Galvez, Chapter article to appear in Deep Imaging in Tissue and Tissue-like Media with Linear and Nonlinear Optics, R. Alfano and L. Shi Eds. (Pan Stanford Publishing) forthcoming. Preprint
• The Poincare'-Sphere Approach to Polarization: Formalism and New Labs with Poincare' Beams, Joshua A. Jones, Anthony J. D'Addario,* Brett L. Rojec,* Giovanni Milione, and Enrique J. Galvez, American Journal of Physics 84, 822-835 (2016).
   

A New Approach and Lab for Teaching the States of Polarization of Light
In this article we propose incorporating the Poincare' sphere in introductory treatments of optics as a way to explain states of polarization. In particular, this approach makes an easy and intuitive connection between the polarization ellipse and its analytical description. We also present a new advanced laboratory where students explore all states of polarization in an interesting way: creating "holes" in laser beam and moving them within the beam. The holes are in reality the nulling of a certain state of polarization within a beam where the polarization varies across its transverse dimension. The figure below shows an example of the image of the beam with a particular state filtered out (left) and the extracted pattern after measuring it via polarimetry.

• High-Order Disclinations in Space-Variant Polarization, Behzad Khajavi and Enrique J. Galvez, Journal of Optics 18, 084003 (2016). 
   

Exploring High-Order Disclinations in Polarization of Light Fields
In this work we investigated encoding topological features, known as disclinations, on the polarization of the light. Disclinations are dislocations of the rotational order of vectors, directions or lines. Think of your fingerprint. The skin ridges contort and at some points form patterns that are known as lemons and stars in topology, but loops and deltas in fingerprinting lingo. A next high-order pattern of a lemon is  one where the lines are radial or form concentric circles (whorls in fingerprints). Since we deliberately encode these patterns in light, we can do so with both types in all orders, as shown in the figure. In this article we present the technique to generate these patterns. In the figure we show two data sets. False color encodes the orientation relative to the radial direction (radial is yellow; orthogonal is blue).

• High-Order Disclinations in the Polarization of Light, E.J. Galvez and B. Khajavi, Proceedings of SPIE 9764, 97640R (2016). Preprint
• Preparation of Poincare' beams with a same-path polarization/spatial-mode interferometer, Behzad Khajavi and Enrique J. Galvez, Optical Engineering 54, 111305 1-6 (2015).  
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Investigating Polarization Singularities with a more Efficient Interferometer
The work of the previous article uses a same-path interferometer using two spatial light modulators to study further polarization singularities that are not centered on the beam of light. The particular singularities that we study are called C-points, which are points where the orientation of the polarization ellipse is undefined, but around which the orientation rotates. This is seen in the figure below, where false color denotes ellipse orientation. The structure of the singularities has a triangular shape, where the vertices of the triangle and its center contain the C-points (lemon/monstars at the vertices and star at the center). Simulation is on the left and measurements are on the right. Small ellipses placed in random locations specify the state of polarization at that point.

• Physics from Planet Earth, J.C. Amato and E.J. Galvez (Taylor and Francis, 2015). New general physics textbook that offers a fresh new approach to teaching calculus-level mechanics.
• "Light Beams with Spatially-Variable Polarization," E.J. Galvez, in , D.L. Andrews, Editor (Wiley-Wise, 2015)
• "Space-Variant Polarization Patterns of Non-Collinear Poincare Superpositions," E.J. Galvez, K. Beach,* J.J. Zeosky, and B. Khajavi, Proceedings of SPIE 9379, 93790A 1-8 (2015).
  
  "Encoding and Decoding Non-separable States of Polarization and Spatial Mode of Single Photons," B. Khajavi, X. Cheng,* K. Kebede,* and E.J. Galvez, Proceedings of SPIE 9379, 93790G 1-8 (2015)
• "Resource Letter SPE-1: Single-Photon Experiments in the Undergraduate Laboratory," E.J. Galvez, American Journal of Physics 82, 1018-1028 (2014). 

Quantum Optics Laboratories to Teach Quantum Physics
The previously listed article is a review of experiments with single photons adapted to teach quantum physics. It also lists the original artic