Research Journal

8 May 2006

Nick Bigelow gave a talk titled "Cold Atoms, Cold Molecules and Optics on Chips!" He is my first invited speaker! A lot of great physics, and many nice meals shared together. I will miss Nick and his group this summer so very much.

Maaneli gave a talk about Bohmian quantum mechanics on 4 May! I'd love to learn more about this, but I'm glad Maaneli isn't in my quantum course - we'd never get anywhere.

1 May 2006

SK Gayen of CUNY City College gave a seminar titled "Optical Imaging of Targets in Turbid Media." Ballistic light is coherently forward-scattered. Snake like is paraxially forward scattered. Diffusive light is multiply scattered. Photon sorting gates can restore some information. In a time gate, ballistic light is the only light captured by a kerr cell shutter. A Fourier space gate selects lower spatial frequency photons from paraxial scattering. In a polarization gate, perpendicularly-polarized light is subtracted from parallely-polarized light. Absorption preferentially reduces diffusive light. The image can be restored!

20 April 2006

Frank Wilczek of MIT spoke about "The Origin of Mass and the Feebleness of Gravity." Wilczek is one of this year's Simons Lecturers. Maaneli was of course elated. "Why is gravity so feble?" is apparently equivalent to "why is the proton so light?" And I really haven't kept up on my cosmology, as I'm not too familiar with axions.

4 April 2006

A colloquium by Angela Olinto of Chigcao about Ultrahigh Energy Cosmic Rays reveals that animals adore detectors: cows keep messing up their 1,600 detectors in Argentina! [Recall long-tailed rabbits in Chile.]

3 April 2006

Jenny Sebby-Strabley of NIST gave a talk titled "A Double Well Lattice for Dynamically Manipulating Pairs of Cold Atoms." A lattice can have in-plane or out-of-plane polarization. The former is seperable into x and y components, and the latter gives a deeper trap. A double well lattice combines these to have very stable phase in a configuration new to me. The lattice can be made state-dependent by a "ficiticious magnetic field" somehow. And what is a Brillouian zone?? And how can it map bands? Phase winding is as fun as the name implies. A diffraction pattern of the atoms washes out and revives with frequency h/U (where U is the potential) for a deep trap (J/U << 1, where J is the tunneling). ...I look forward to reading her papers and learning more about this fun stuff!

Both Jenny and Gretchen were once SURF students at NIST!

27 March 2006

Gretchen Campbell of MIT gave a talk today about the deep 3D lattices their BEC is playing in. By adjusting the depth of the lattice, the BEC can be changed from a Mott insulator (deep) to a superfluid (shallow) state. The lattice depth changes the tunneling between lattice sites. About 1-5 atoms occupy 150,000 lattice sites! One of the many mysteries of the talk is Kapitza-Dirac diffraction. Gretchen is finishing her PhD with Ketterle and Prichard. Dominik worked with the group as a postdoc until joining Stony Brook a year ago.

22 March 2006

A talk by P. Vanden Bout from NRAO blew us all away: the Atacama Large Millimeter Array being constructed in Chile consists of 50 12m detectors. Running around the installment is the vizcacha, a long-tailed rabbit!

We've made progress on the cellophane waveplate idea. Rotation about either of the cellophane's axes will change the retardance. See our brief report here. This seems simple once you get it, but to see it you have to get past the ingrained notion that a birefringent material has two axes. It actually has one axis, that of the polymer molecules. The way a birefringent material is cut defines the second axis.

The OR presentations were today as well. Rebekah has been working on a polarization reflection problem Hal brought up. As always, it all comes back to spirals: σ+ andσ- have orthogonal helicities, not orthogonal polarizations. Atoms in a MOT always see a certain helicitiy; the polarization need not be defined by k.

15 March 2006

I've been accepted into NIST's SURF program for the summer! I'll be working in Bill Phillip's Laser Cooling group with Kris Helmerson's sodium BEC and vortices!!

28 February 2006

Hideo Mabuchi of Caltech spoke about Quantum Metrology with Cold Atoms at today's colloquium. He cited Nick Bigelow's spin squeezing! Spin squeezing implies entanglement. Quantum filtering is a gradual collapse of the wavefunction. On one end of the spectrum is the coherent state, and on the other is Dike state. The Dike state is unattainable due to decoherence issues. Many of the concepts in the talk are familiar now, but I'm not intimate with them yet.

28 February 2006

Vladimir Shalaev gave a colloquium on Negative-Index Mematerials. Ironically, we spoke about this in EM this week!

Dr. Noé and I have been working on a cellophane. We hope to make a waveplate for locking a laser by turning the cellophane. More later...

We've been interviewing so many potential Simons Fellows. Most of them have been extremely disappointing! Long Island's "Science Research Course" has ruined a generation of potential scientists.

13 February 2006

On Friday, Johnny Huckens of NIST visited Stony Brook to speak to Hal. We showed him the LTC and the Nuclear Structure Lab. He's finishing up a graduate degree at Maryland after working in acoustics for a time. Today, Dr. Noé and I traveled to Adelphi University to hear Johnny's talk on Ultracold Atoms in Lattices: Making Pancakes and Playing the Accordian. (Lattices are so hot these days!!) Amazingly, his current results parallel a 1994 paper by Janicke and Wilkens (PRA 50 3265) on channeling diffraction. The theorists must have been thrilled to hear about Johnny's results!

While at Adelphi, we met Eugene Hecht! He was carrying around his latest Optics book, and is working on his next edition. He plans to include something about singular optics!!

9 February 2006

Louise Kindt and Sidsel Damkgaer spoke today about laser cooling magnesium at the Neils Bohr Institute in Copenhagen (yay, Denmark!). They're having some weirdness, and are on a physics road trip across the US to all the exciting LCAT labs!

31 January 2006

Rob Schoelkopf from Yale spoke today about quantum optics on a superconducting chip. I really want to learn more about superconductivity!! He compared atomic/optical and solid state systems in a very clear way. His Cooper pair box is something else I'd like to read more about.

On the 30th, Erich Ippen from MIT gave an excellent talk titled "Femtosecond Optics." The ultrafast jargon is still rather mysterious to me: chirping and FROGs and attoseconds

This semester should be fun; I'm taking EM2, quantum physics, lasers, complex analysis and music theory. As for my LTC research, I hope to continue to work on the SLM-making idea, and/or study the OAM sorter.

December 6th, 2005

Grover's optical vortex coronagraph appeared on Slashdot yesterday. This is soon after I wrote a Wikipedia article on optical vortices. So thousands of geeks all over the world turned to my wiki article when they wanted to learn more about the spirals behind Grover's wonderful new results. Yay.

See my new SLM-specific journal.

November 4th, 2005

Less than a month ago I went to the OSA Annual Meeting in Tucson, AZ. This year felt like a reunion, whereas 2004's meeting was more of an introduction. I love physics for the physics itself, but I also love the physics community! I got many new ideas for research, made wonderful contacts, and attended some wonderful talks (Tom Bergeman's was perhaps the most incomprehensible!). Read more here. Infinite thanks to everyone that made this adventure possible!!

As for my summer at Rochester, it was incredibly amazing. I had a wonderful time learning a ridicuolus amount of physics and engineering and had fun in Room 5, with vacuum grease, soldering fumes, tweaking little knobs in total darkness, etc. And now I'm in love with quantumy optics.

The next year or so will be filled with exceedingly difficult physics-related decisions, including where to spend the summer, what field to pursue, where to go to graduate school... Advice is appreciated. :)

How I love physics.

June 1st, 2005

I am spending the summer in Nick Bigelow's Cooling And Trapping (CAT) lab at the University of Rochester! I'm going to try to *gasp* make vorts to shoot at a BEC!!!!!!!!!!!!!!!!!

April 5th, 2005

Ben Oppenheimer of the American Museum of Natural History (apparently there are tenured research positions there!!) gave a colloquium about exoplanet detection!! I had lunch with him and some graduate students, and showed him my vorts. He had never heard about Grover's Peering into Darkness idea, but seems excited by it! Ben and I also taught some very un-interested high school WISE girls about planets, stars and galaxies. Perhaps most exciting, Ben also dislikes the term "exoplanets!" Perhaps we can get "xenoplanets" into mainstream astronomy. I look foward to visiting his lab in NYC in the future.

March 22nd, 2005

Göran Tranströmer of Sweden's Royal Institute of Technology visited the LTC for a few days during his stay in New York. His university has an undergraduate laser lab and is interested in teaching methods. Göran also works at CERN, and his group is learning about using high-power lasers in the accelerators. He was kind enough to teach Dr. Noé and me some Swedish. I really really want to re-visit Scandinavia now!!

Jag älskar fysik!

March 8th, 2005

A dozen very bright students from The Experimental High School in Beijing enjoyed a few hours of optics demonstrations. They also toured the Nuclear Structure Lab, but unfortunately could not visit biology and engineering labs due to a snow storm.

Last week Michal Simon gave an Astronomy Open Night lecture about exoplanets. Afterwards, I saw Saturn through the 0.35 m telescope on the roof.

February 18th, 2005

Sambasivan Ramaswami, a physicist from India, visited the lab today. He has a great interest in optics and now writes popular articles.

I WAS IN THE VAN DE GRAAFF!!!!!!!!! The stripper foils needed to be changed and Rich Lefferts was kind enough to give me a personal tour. When I originally toured Stony Brook in 2002, I was shown the Nuclear Structure Lab...never did I think that a few short years later I would see the beautiful 60's-style lime green on the inside of the accelerator.

February 11th, 2005


The 2005 Simons Foundation Lecturer gave four amazing talks over his two weeks at Stony Brook, including a colloquium entitled Optical Vorticulture (which he defined as "the study and tender loving care of optical vortices and singularities"). I had the honor of joining Berry for dinners and learning about some of his many interests, including the Severn bore, the music of the primes, astronomy, Newton's phase eels, reflecting spheres and the interconnectivity of things. Apparently, the term "topological charge" makes sense in two dimensions only! Berry came to the LTC and saw my vorts created by Kiko's CGH and my Mach-Zehnder interferometer. I created forked linear and spiral interferograms of a Gaussian beam and charge one vort, and Berry modeled the interference on Mathematica, in a way much simpler and more elegant than my previous attempts. The whole two weeks were AMAZING. I never thought I would get to meet the man who coauthored pages 165-90 of Proc.Roy.Soc.Lond, A336 in 1974 and inspired the field of singular optics! Thank you to all responsible!

A spiral shockingly similar to my own.

December 18th, 2004

My Waves & Optics term paper was titled "Twisting Light with Twisty Molecules" (keeping with the spiral and singularity themes of my life) and described various spatial light modulators. Hamamatsu and Holoeye make two very different models, employing optically-addressed refelctive parallel aligned nematic liquid crystal and electrically-addressed transmissive twisted nematic liquid crystal technologies, respectively. Holoeye is the much more affordable SLM that Miles Padgett told us about at the OSA meeting. When the LTC purchases a SLM, we'll be able to make dynamic high-intensity vorts for tweezers and more. It seems that the topological charge of a vortex created by an SLM is limited only if the SLM is pixel-based (electrically addressed).

November 30th, 2004

Azi Genach of Queens College gave a stimulating colloquium talk entitled "Photon Localization in the Time Domain." He was José's freshman physics teacher and visited the LTC before his talk. He was very interested in my vortices since he is encountering similar singularities in his own work.

November 11th, 2004

I met Mary Dillon, Yaagnik's very cool (for lack of a better word) science research teacher!

Tom Weinacht's lasers class came by the LTC to see some HG modes made by my hair and the open cavity laser.

Jan and Matt are working on some great temporal coherence stuff!

November 2nd, 2004

Jon, Dr. Noé and I visited Yingtian Pan's OCT lab in the Stony Brook hospital. It was fun to see Jon meeting one of his gods because I know exactly how it feels. :)

James is working at Photonics Industries International! He's in charge of the "spinny thing" project!

October 15th, 2004

Dr. Noé and I returned last night from a dream-like week at the Optical Society of America's Annual Meeting in Rochester, New York. We visited Kiko's lab at Colgate, met Grover Swartzlander and Miles Padgett, heard talks by Taco Visser, Greg Gbur, Sabino Chavez-Cerda, Grover, Petras Zerom, Isaac Freund, Mikhail Vasnetov, Tom Brown and Miles Padgett. I also got to surprise my sister since she didn't expect to see me until late November! It was an amazing experience. Thanks to all who helped make it happen!!

September 18th, 2004

Apparently, "I am Debian Linux. People have difficulty getting to know me. Once I finally open my shell they're apt to love me."

Lidiya was here today to work on her presentation for the OSA meeting. Oh, how I've missed Fourier.

Dr. Noé generously took Jon, Ani, Yaagnik (who became offical physicists Thursday night), Danielle, me and our newest LTC family member Rajesh out to the Curry Club on Friday. I love dal! Raj spent two days here, and is already graphing bessel functions and playing with Airy patterns. The rest of the kids are making wonderful progress on their projects (that Siemen's deadline is approaching!).

The Laser Teaching Center now has a laundry service!

I'm attempting to make interference patterns with Mathematica. Four books, two papers, my notebook, two computers...

The grin tank has molded!

I'm an Inter-Library Loan criminal! It's proof of how much I love Yaagnik, Ani and optical vortices.

September 8th, 2004

I met another LTC legend, Tina Shih!

September 2nd, 2004

Melissa gave a talk today at the offical last pizza lunch ever (garlic and broccoli!). She worked on, of all things, vortices in BECs.

Pretzels are just Lissajous with a frequency ratio of 4:3 and a phase difference of π.

September 1st, 2004

I am back in Stony Brook! The past three weeks went crazy fast and were packed with NYC, Toronto, kayaking, catsitting, Star Trek, landscaping, computer restoration, canoeing, Dance Dance Revolution, reading about imaginary numbers (thanks to Dr. Noé's perfect REU gift...Rita recieved a Feynman book, Lidiya a math book and Yiyi a CD), seeing my old friends, lawnmower repair, scaring people with physics-related remarks, visiting my high school physics teacher, scouting Rochester... Anyhoo, it was good to be home and it's good to be back.

August 6th, 2004

My home. Buy it.

My REU presentation went okay. Rita's was excellent, and not just because she included The Lion King and Smurfs. The lunch they provided had a horrible vegetarian option, which Rita and I griped about for a long time.

I was running around all day escorting Courtney to meetings with various biochemistry and chemistry people. She really likes Stony Brook! :)

After checking out of Douglass, touring the Wang Center and speaking with Professor Metcalf, we headed to the keg. Rita, James, José and I want to start a mariachi band in the spirit of Feynman.

I said to goodbye to everyone and headed to NJ. I had an amazing summer, and I'll write more tear-inducing stuff later. Just know that everyone helped me fall in love with physics more than I thought possible.

August 5th, 2004

Pizza lunch, complete with garlic broccoli, was a rehersal for our REU talks tomorrow. Professor Metcalf provided helpful comments, especially about the order of my slides. I was getting mixed messages from Dr. Noé and Professor Metcalf about how serious my talk should be...

Kira and Courtney arrived on the train today. Dr. Noé and I showed them some of the classic LTC demonstrations. We played with the multimode fiber, sucked helium and poured liquid nitrogen on ourselves into the night.

August 3rd, 2004

I took many interesting photos of various vortex-related things today. I put one of the Mach-Zender mirrors onto a translator and Yaagnik immediately aligned an amazing interference pattern.

Today I noticed that my vitamin E capsule has interesting optical was magnifying my fingerprint, and that part of my finger looked like a fork.

August 2nd, 2004

Dr. Noé and I spent a lot of time writing my abstract. The Talbot effect in the fork grating was discovered by accident. The Mach-Zender interferometer is working to interfere two LG's of opposite topological charge. The fringes create a nice fork!

Optical vortices and optical fibers.

July 31st, 2004

Today is a fork-related holiday! (I did not make this up!)

July 30th, 2004

Laser Sam fixed the Zeeman effect by putting some voltage on the piezo attached to one of the cavity mirrors. The slight change was enough and produced a very stable signal. We went to Jillian's lab to see her "real lasers" (which means you wear goggles!), including femtosecond and dye. Sam then set up the Fabry-Pirot he made. He and Matt played with the open cavity laser and made some Hermite-Gaussian modes. We all enjoyed Sam's visit.

Aurora borealis and neutrinos.

Ani and I spoke to Professor Metcalf about angular momentum. We are walking a finer line between classical and quantum than we thought!

Today I look like this.

Electron orbitals look a little too much like LG & HG intensity and phase distributions...

"Optical Sprinklers are peculiar light field distributions with one or several lines of darkness. Optical sprinklers rotate in time and can be used for the trapping and distribution of cold atoms. Slow moving atoms remain trapped at the centre of the beam while hot atoms escape through the sprinkler?s arms." from Glasgow.

July 29th, 2004

Laser Sam was here today. After garlic broccoli pizza (a superposition of three of my favorite foods), we summerized our projects, and he showed us his pumped green laser and Zeeman effect. Sam is so passionate about his work, always enjoyable to see in someone.

While observing the Zeeman Effect, Professor Metcalf put another magnet in the field to observe its effect on the oscilloscope readings. He got a bit too close to the uncased tube and it was knocked over onto the metal table, which no one happened to be leaning on at the time. I was centimeters away from the table, so as everyone was screaming, I turned off the power supply. We all almost died. But it was worth it for Zeeman! We tried to observe the beats on a 0.2 mW laser of our own, but Zeeman mysteriously decided not to work.

I met Oleg, the LTC legend who built the amazing device in the back room to measure a magnetic field. Circularly polarized light was sent into a cesium cell to align the magnetic moments of the atoms. When another beam of circularly polarized light is sent into the cell orthogonally, the magnitude of an applied magnetic field may be determined.

The Metcalf group had its photo taken today!

Making a Power Point on Windows really made me appreciate HTML and Linux...who wants to click an extra six times per Greek letter? My laptop is going to need an upgrade...

Professor Metcalf and Dr. Noé took Sam, Rita and me out to Carnival. I had spiral phase ramps with garlic, parmesan, escarole & cannelini beans, with a side of garlic broccoli rabe. The topological charge of the pasta was not conserved though! Dr. Noé had the exact same thing, and Rita had the same thing but with nanotubes. The resident vegetarians (Rita & I) convinced Professor Metcalf not to get veal!!

July 28th, 2004

Now I understand complex numbers.

July 27th, 2004

Professor Hemmick gave a lecture on the search for quark gluon plasma at Brookhaven's Phenix detector. Jet quenching is when a medium inhibits the production of the usual number of particles in a collision. Colored glass condesate is what the nuclei become at the relavistic conditions in the accelerator. The nucleus reaches its maximum gluon saturation from a stationary reference frame, and any additional gluons merge. String fragmentation (unrelated to string theory) is what prohibits quarks from existing alone. Field lines between quarks and antiquarks do not exist because their force carrier, the gluon, also carries a color charge (photons, the electric force carriers, do not carry a charge themselves, so the field lines themselves go off to infinity). When two quarks are seperated, another quark is produced using the energy it takes to tear the string. Overall, an increase in energy density creates quark gluon plasma, and the current results from Phenix may provide sufficent proof of this new state of matter within a year. It was a thrilling lecture, as particle physics, early-universe conditions and the like are some of my many turn-ons.

Lab life has been crazy lately. Ani figured out what lenses to add to Yiyi's tweezer setup so the field of view on the CCD would take up the whole telescreen. Danielle set up a Fresnel zone plate on the carriage and calculated where the Poission spots should be. Matt used a multimeter and DOS to graph the intensity profile of a HeNe. Lidiya is getting great results with her spatial filtering (and running around using three computers because Windows 95 Paint can't save files as jpgs!). I'm having a cleaning and organization fit, and helping everyone with various setup, theoretical and computer stuff. Yiyi's been at Bell Labs.

A long overdue addition to the LTC library: Siegman's Lasers.

"I don't know which is more fun: hardware or software problems!" Me, on mouse debugging.


  • Induced gluon radiation
  • Primary colors include green if it's an addition and yellow if it's subtraction
  • Jet suppression
  • Virtual & real photons/gluons
  • Unitarity

July 23th, 2004

Happy 17th birthday Jon! To celebrate, Danielle bought LG00 doughnuts!

I love people who use cardioids for the closing of an email!

Working on planet & star simulation...

Physics books make good pillows.

José recreated his "which-way" experiment. Quantum physics is fun!

The kegger was inside today. Jon and Ani became obsessed with mango peach salsa.



  • Spiral zone-plate-style patterns result when there is no angle between the interferring Laguerre-Gaussian wave and the Gaussian wave. Forked interference patterns are due to an angle between the two waves.
  • A fork grating produces a diffraction pattern with vortices of increasing charge since a binary grating can be decomposed into a superposition of harmonic components. From OV book, page 49, chapter 2 by Smith, et al, "Each component can be recognized as a grating capable of producing a higher order singularity with correspondingly closer spacing leading to a larger diffraction angle."
  • Dove prisms flip the transverse cross-section of the transmitted beam.


  • The best vortices are formed when the grating is placed a certain range of distances away from the source (for my fork the best vortices are at ~1.52m). Perhaps this is due not only to beam radius, but the curvature of the wavefronts. As z→∞ in a Gaussian beam, the wavefront becomes planar.


  • Beijersbergen, Allen, van der Veen, Woerdman, "Astigmatic laser mode converteers and transfer of OAM," Optics Commun, 96, 123 (1993)
  • Padgett, Arlt, Simpson, Allen, "An experiment to observe the intensity and phase structure of LG laser modes," Am J PHys, 64, 77 (1996)
  • Photocopy page 46 of OV book - shows graphical transition from spiral to fork interference pattern!
  • He, Heckenberg, Rubinsztein-Dunlop, "Optical particle trapping with higher order doughnut beams produced using high efficiency computer generated holograms," J Mod Opt, 42, 217 (1995) - order-specific CGH
  • Photocopy graph on page 56 of OV book - graphical Re(E)=0 and Im(E)=0 behavior in fractional topological charges!

July 22th, 2004

Happy Pi Approximation Day! (22/7≈π)

Dr. Noé left for the Suzuki Institute in Ithaca this morning. I inherited the office :)

Hawking shook astrophysics by retracting his claim that all the information that enters a black hole is irreparably lost.

Today I'm in love with the University at Buffalo. Inter-library loan finally got me the elusive Horizons in World Physics Volume 228: Optical Vortices book!! That's right, a whole book devoted to my favorite singularites with an introduction by Sir Berry and an entire chapter by Swartzlander.

Prior to our pizza lunch, Professor Metcalf spoke with us a bit about angular momentum (the name is better in French). Ani and I felt better knowing that he doesn't really understand spin either! It's confusing because it's such a mathematical construct, and can be better interpreted using the magnetic moment of the particle rather than thinking of it as actually spinning. Photons have spin 1/2 because they only have two states; a value of one would allow three (-1, 0, 1), as was discussed in a previous MOT lecture. This makes sense, whereas the explanation that spin 1/2 is a 720° revolution (as I once read in Hawking) is impossible to comprehend. Spin 0 isn't really linear polarization, but a superposition (yes, cat again) of spin +1 and -1 (circular polarization). The superposition describes the magnitude of the magnetic moment, and its direction is given by exp(iφ). We also wondered why zebras aren't green and white striped, and why there aren't striped bears. Cicadas' period of emergence is only prime numbers of years (13 or 17)!! Anyway, during GARLIC pizza we had a serious discussion about our projects. Jon is interested in spectroscopy and fireflies. Danielle's project is heading towards the Talbot effect in a Fresnel zone plate. Matt's heat feedback loop project seems sound - aside from the challenging electronics aspect. Lidiya's filtering and Fourier setup is going nicely. Ani's understanding of LG modes in optical trapping is amazing, and our thought process on the setup is progressing. Yaagnik had a great idea to do something with adaptive optics (not Keck-style!). He could deform mirrors and fix a signal less complicated than those distorted by the atmosphere. Of course, we were also discussing Euros, presidents and other classic pizza day randomness.

José gave a great talk on diode lasers, which contain two semiconductors. One has an excess of electrons (often Germanium) and the other an excess of holes (silicon). Holes are places that electrons could possibly go. The emission of light comes from electrons crossing the semiconductor interface to fill the holes. The output beam is not Gaussian, but eliptical (in Yiyi's tweezers, the laser was a diode so she had to correct the beam shape with lenses) due to the shape of the interface. Since the electron transitions are between energy bands (instead of descrete levels), the diode laser emits a spectrum instead of monochromatic light. Increasing the voltage on the diode ensures that the emitted light is coherent by increasing the potential difference between the electrons and the holes; too little voltage will give you only a "fancy LED" (light-emiting diodes also work via semiconductor interfaces). The number of holes and electrons increases with current, so more light is emitted. Diode lasers have a high gain; few free photons are needed for amplification of the signal. The advantage of diode lasers is their tiny size and high power, but the beam is elliptical, divergences rapidly, and has a random polarization. José also explained the energy level configuration of conductors, semiconductors and insulators. The energy bands associated with energy level splitting may create an energy gap between the valence level (ground) and conduction level. A large gap, as in an insulator, requires too much energy for electrons to transverse. The gap is noexistent in a conductor, allowing for the easy flow of electrons. A semiconductor's energy band configuration is somewhere in-between.

After a berry & Berry overload, I decided to make a random quote show up on my home page. Not wanting to use Switch again, I cranked out a compact little array.


  • Modulated vortices!
  • Polarized beam splitter - depends on Fresnel coefficient. What would it do to an LG0l≠0?
  • In studying the effect spacial coherence has on the vortex core, Swartzlander suggested plotting the intensity ratio of the core to the total verses the distance between the light traveled (which is therefore proportional to the amount of spatial incoherence). Professor Metcalf pointed out that the aforementioned ratio may only be relevent in astronomy, and that I should use something other than that ratio for my analysis. However, if the vortices are being used in other applications - optical tweezers, for example - the light is guaranteed to be coherent. You can control your laser, but not your starlight. Am I missing something in Professor Metcalf's comment?
  • I want to make crazy CGH's of my own with multiple tines & forks, and - most exciting - channel most of the intensity to a specific order!

July 21th, 2004

July is National Blueberry Month.

This morning Ani, Yaagnik, Yiyi, Matt and I decided to join Danielle and Jon on their Simons tour of the Center for Molecular Medicine. We were asking so many questions that we were thirty minutes late for lunch and never got to the fifth floor neurology stuff! On the structural biology floor, we saw the nuclear magnetic resonance machines, which analyze proteins using a modified Zeeman effect. Another protein had to be studied in the dark because light would change its structure. The Center for Infectious Diseases was studying lyme disease and urinary tract infections. Ani previously worked on E. Coli, so he spoke with the (very informative and enthusiastic) graduate student about the living conditions of the spirochetes. The precious little bacteria move by spiraling (like a certain Laguerre-Gaussian mode I know and love), which we saw live via an amazing charge coupling device camera that made Yiyi drool. The E. Coli study found that the infection has a virus-like behavior, in that they enter the cell itself and replicate (outside the nucleus) until it burts. Only the E. Coli with cicilla can cause infection, as the flow of urine helps rinse others away. The Developmental Genetics group is working with fruit flies to study deafness in humans, although their analysis is far from having direct implications. By hooking electrodes up to the fly's brain and antenna, you can see whether or not the fly is deaf by the electrical responsivity of the synapses. Usually a computer-generated fly mating call is used, but we got to talk directly to the fly and see its synaptic reaction!! I had the best mating call... We also learned that flies don't actually speak, just beat their wings right in front of their potential lover's face. The researchers also learned something about how physical stimiuli of the antenna leads to a chemical reaction. The same mechanism is employed in various human systems. In all the labs, the professors and graduate students figured out we were physics people by our questions! I asked whether parity was a consideration in a protein study; Yiyi was intrigued by the feedback system on an incubator; Yiyi and I wondered about electric noise in the fruit fly experiment (ironic, as the experiment was testing the deafness of the flies!), which was corrected by a Faraday cage and some grounds; we were all facinated by the application of the Zeeman effect in protein experiment. After all this intense learning, we met up with the other Simons kids for lunch (oatmeal raisin cookies!). Jon and Danielle also enjoyed the tour, and the seven of us started screaming about the amazing specifications of various lasers, confocal lenses and CCD's we'd seen...the other Simons kids were very afraid.

Afterwards, we went to play virtual reality games in the SAC plaza. Ani and Jon raced in little VR cars that they were way too big for. Danielle and I had a virtual tour through Ali Baba's Cave (there was nothing Ali Baba about it). There were crystals everywhere and we hoped that they were birefringent...

Lidiya gave a ninety minute lecture on her best friend Fourier!! Basically, any periodic function can be described by a sine functions and a multiple of λ/n, n being an integer less than or equal to one. The order m of the function is in the trigometric argument, and accuracy increases with m. A periodic function is described by the normalized axis A0 and the Fourier coefficents (the amplitudes of the constutient cosine and sine functions), Am and Bm. The Fourier series is f(x)=A0/2 + ∑m=1Amcos(mkx) + ∑m=1Bmsin(mkx) where k is the wavenumber. Since 0λsin(akx)cos(bks)dx=0, A0=2/λ∫0λf(x)dx, Am=2/λ∫0λf(x)cos(mkx)dx, and Bm=2/λ∫0λf(x)sin(mkx)dx. That all made sense when Lidiya explained it, too! Fourier integrals are used in evaluating nonperiodic waves, where the period and wavelength are infinity. As λ→∞,  m→1/∞, and all non-integer orders are represented (therefore the Fourier function is very accurate). The Fourier integral is f(x)=1/π[∫0A(k)cos(kx)dx + ∫0B(k)sin(kx)dx]. A(k) and B(k) are the Fourier transforms in terms of inverse space, which is the spacial frequency and distance between the minima, u=1/x. An/2=na/Dsinc(πna/D) and u=sin(θ)/λ=n/D. The intensity is the square of the Fourier transform: IN=I0(0)[sinc(πma)]2[sin(NπuD)/sin(πuD)]2, where N is the number of slits. Threfore, as N→∞ IN goes to zero. A grating with ideally narrow slits have ampliudes of infinity. The even-odd effect observed in both the Ronchi gratings and my fork grating is due to the suppression of some maxima due to the interaction of the single-slit envelope and the interference from multiple slits. Mathematically, the even coefficents of the Fourier transform goes to zero. Lidiya didn't even get to image formation or her setup!

July 20th, 2004

Happy 35th Anniversary Apollo 11!!

The REU kids came for a tour of the LTC today...except only a few of them showed up. Honestly. Anyway, Rita discussed her saturation spectroscopy, Lidiya Fourier diffraction and transforms, Yiyi optical tweezers, and I vortices.

Professor Metcalf gave another joyous MOT lecture, focusing on the position-dependent magnetic force. The energy between states is 100hB (in Tesla). The point where the magnetic field is zero (due to cancellation) is where the energy states split up since linearly (π), right circularly (σ+) and left circularly (σ-) polarized light interacts differently with the magnetic field. The frequency of light is chosen so it will not initially resonate with the magnetic moment of the atom. As the distance along the z-axis (chosen to suit the problem) increases, the energy of the energy level slopes downward (proportional to the magnetic field gradient) and meets the energy of the light: resonance. The atom is forced towards the center, where the magnetic field is zero.

In case you're wondering, my sister plays the didgeridoo.

July 19th, 2004

Today we went to BROOKHAVEN NATIONAL LAB!!!! The thrills were RHIC (touching the beam lines!) and STAR (wanting to hug the whole three-storey mess of wires).

José set up a Michelson interferrometer for the kids to play with today. It's really fun to see the fringes form and dissapear with the slightest disturbence.

We invented new physics this afternoon. Lidiya set up her spatial filter, and we put the small square Ronchi (say "ron-key") grating in front of it, expecting to see a nice dotty diffraction pattern. Well, about a half meter from the iris we were getting LG01 modes (bull's eyes). Another quarter meter out we saw LG10 -like modes (semi-vortices), although the zero order had a singularity!! AND in the far field we were getting a good old Ronchi diffraction grating (solid). The current theories are that the grating is dirty or damaged, and/or Lidiya's cockeyed way of setting things up at random angles affected things.

July 17th, 2004

Happy 18th Birthday Yiyi!!!!

July 16th, 2004

Today was the best kegger ever - strawberries, grapes, pineapple, broccoli, cherry tomatoes, cheese, carrots, 'loupe, the works! (I hear the beer was bad though...) Yiyi and I realized we have a misconception about salsa. We think we like it, but we don't. It's simply too hot. Anyhoo, Dr. Noé and I were talking with Professor Sprouse about cameras (I have very little confidence in our CCD being able to distinguish between minute intensity differences in the core due to incoherence). He pulled out his really spiffy digital one, making us realize that we don't have to buy something industrial. Perhaps the LTC's use of floppy disks will soon cease...

The high schoolers, Dr. Noé and I had a nice long discussion about potential projects and current interests this morning. Topics included GRIN (gradient index of refraction, and its applications in gravitational lensing), fractals, optical coherence tomography (using scattering to image tissue), edible optics, Jell-O lasers, optical vortices in trapping, holography, evanescence.

"What are you smiling about?" "Linux."

Swartzlander's proposition is to vary the spatial coherence by varying the distance between the source and the grating. We need a pinhole right before the grating, or else the beam size is too large and no singularites are created.

From a Swartzlander paper (yet to be published!), "...a vortex of infinite charge m' may be expressed as an infinite series of vortices having integer charge m:  f(θ) = exp(im'θ) = ∑ +∞m=-∞ Cmexp(imθ)  where  Cm = (-1)msinc(π(m'-m))exp(iπm')  assuming  f(θ) = f(θ+2πl)  for  l = ±1, ±2,..." In a vortex of integer charge, the amplitudes of the infinite number of constituent vortices is zero, as required by the sinc function (a function with a removable singularity).


  • The sinc function is sin(x)/x
  • In linear media, a beam with multiple vortices will stablize in the near field (a distance smaller than the beams' diffraction length). In nonlinear media the vortices interact for longer periods.
  • Vortices in a linear medium have diffraction rings.
  • In a multi-vortex beam, the larger vortices interact slower, and the closer neighboring vortices are, the sooner they'll interact.
  • Hermite polynomials are Hn(x)=(-1)nexp(x2)(d n/dn)exp(-x2)
  • Laguerre polynomials are Ln(x)=(n! exp(-x))-1(d n/dx n)(xnexp(-x))
  • This Vienna group made a forked CGH that distributed most of the incident Gaussian beam's intensity to the vortex of charge one!! The rest of their work on orbital angular momentum entanglement is great too.
  • From an HGm n beam created by affecting the gain distribution in the laser cavity an LGlp mode can be made using an an astigmatic mode converter (as Alex did) and the resulting LG beam will have indicies l = m-n  and  p = minimum(m, n). Why? I don't know.
  • LG beams with p ≠ 0 can be produced with a CGH fork grating (Arlt, et al.)
  • Astigmatic mode converters make use of the Guoy phase shift. This is analagous to the fork grating in that the difference between the top and bottom of the fork grating is π, as is the change in phase in the Guoy shift (pronounced goo-ey, much to my suprise).
  • Gegenbauer-Gaussian modes are LG modes with nonzero p indicies, and may also be described as a superposition of LG modes. Why haven't I ever heard that term before?!
  • Bottle beams occur in combined GG beams (LGlp≠0). Their radial intensity pattern changes dramatically (from a bright to dark center and back again) along the axis of propegation because each constutient mode is affected differently by the Guoy shift. Bottle beams are unstable, as the is different on both sides.


  • How does the superposition of charge one vortices described in Swartzlander and Schmit's equations relate to the migration of the vortex train away from the z-axis in the far field described by Berry?
  • Kalidescope lasers (Danielle's current interest)
  • Swartzlander says that a charge three vortex is simply three charge one vortices. This not only conflicts with my understanding of LG modes, but he wrote (in "Optical Vortex Filaments" from Volume 228 in Horizons in World Physics), which differentiates between "large-core" vortices and vortex filaments. And let us not forget the enigmatic point vortices and vortex solitons...I have no idea what a vortex is.

July 15th, 2004

Dr. Noé took Jon and Danielle to the Nuclear Structure lab for the first time, and Yaagnik and I tagged along because each trip there is an adventure. We saw the target room, which was overrun by radiation only yesterday. I found a Scientific American in the control room with a cover article on string theory, so I opened it up and found an article on vortices (citing Harwit and Swartzlander) that noted that kids wear shirts with Maxwell's equations on them. Woah.

The point of describing waves with exp(ikx) is apparently to make the math easier...but why must a beam with a helical phase ramp have an exp(-ilφ) component? What does that translate into for the non-easy math? I just don't understand how the complex plane relates to real life...

The Melles Griot lasers we use come from Carlsbad, California - the town where I learned how to surf. Busy, busy, busy.

Professor Metcalf said (during double GARLIC pizza) that the summer will be succesful if we leave feeling that we know less than when we arrived.

The time on that jpg is currently accurate...I'm making use of my assembly skills acquired through my multi-thousand dollar Lego investment. There are some CCD issues though; the vortices are too bright, but just about the right size (an iris was used to get rid of most of the diffraction orders), so I sadly don't actually have any data from 01:00.


  • A stable phase is a predictable phase.
  • A vortex beam can also be made directly in the laser cavity (Lim, et al.)!
  • Temporal coherence is used in Michelson interferrometers and is how far two waves can travel before they destructively interfere.
  • Spatial coherence is the area over which the light is coherent - the resulting pattern in Young's double slit experiment has low spatial coherence because there are many cases of destructive interference.
  • What's trying to happen experimentally (per Swartzlander's suggestions) will vary primarily the temporal coherence of the light.


  • How is a vortex filament different from an LG beam?!
  • I don't understand why Swartzlander says that a vortex of topological charge one is optimal - in his photos/simulations from the Peering into Darkness paper, the most dramatic intensity patterns indicitive of a second source are those with higher charges. The dimmer the source, the greater you want the effect of the fork to be. The incident light from both astronomical sources is temporally incoherent. If a charge one phase mask is used, one of the vortex cores is centered on the axis of propegation, so if the planet were very very dim, you wouldn't notice a difference in the star and planet pattern verses a lone star system. Using a charge four vortex (in a fork grating the fourth order is suppressed...unless you're using a higher pronged fork, in which case the order immediately next to the center LG00 spot would have charge tines-1), though, the axis of propegation is not in the center of a singularity, so it'd be obvious that there is a second dimmer source, whether your CCD picks it up or not. Swartzlander sais that the non-charge one vortices are unstable...but I can't imagine a good old charge three decaying in the near field, which is where you would photograph the incoming starlight. I think this has to do with the superposition of charge one vortices and filament thing again. [Remind me to fix this horribly written paragraph - it's 3am and apparently my English brain centers shut down before my physics ones.]

July 14th, 2004

Ani, Rita, Yaagnik, Matt and I toured to the Van der Graff and the Francium  lab.

José gave a nice long talk about lasers - coupling, population inversion, gain, stable and unstable mirror configurations for the cavity, Raleigh range, waist, radius of curvature, modes.

I learned some JavaScript so I could make a photo change randomly on every page load. It's basically the same as C++, except that the type of variable (int, float, bool, etc.) isn't mandatory to define. Strange. I suddenly remembered Switch statements and gladly deleted my ten billion If-Else statements. I still don't know what was wrong with my initial For loop. (My code is Copylefted.)

I'm reading through a special issue on singular optics...unfortunately I can't take it out of the library so it's going slow (and I'm not about to print the whole issue).

In preparation is an actual setup with tangible components.

"...Le Mans? Isn't that the pregnancy class?" No, me, "Le Mans" is a race (not a car) and "lamaze" is the pregnancy class!

July 13th, 2004

Professor Metcalf continued his MOT talk today. The total angular momentum J=jħ where j=±(1/2, 1, 3/2, 2). The magnetic moment μ=gμgJ where g is the fudge factor and μg is the Bohr magneton. Since J is an integer, magnetism is quantized. The change in the component of J must be ±1, so the maximum number of orientations of the magnetic moment m is 2j+1. The magnetic moment of the electron can only be +1/2 or -1/2. The magnitude of the angular momentum must be ħ and the direction depends on the polarization of the incident photon (linear along magnetic field if m=Δ0, +(-)circular if m=Δ+(-)1 [for m=ħ2 there must be two seperate excitations, two must be absorbed at once, or the orbital angular momentum in addition to the angular momentum of the photon can move the electron up two states]).

Professor Allen gave a talk to the REU kids about solid state physics, which is basically the study of crystals. He is trying to figure out why a 3x3x3 semiconducting nanocrystal spontaneously decays into dipoles. In the Earth, an increase in pressure can lead to a new metastable crystal structure. Professor Allen also discussed the shape of galaxies - why aren't they more ergotic? Using Newtonian mechanics and conservation of energy and angular momentum, planets should take a precessing orbit (like a Spirograph!) around a star (or a starsystem around a black hole). The Henon-Heiles Hamiltonian equation destroys a conservation law. A triangular potential well binds the planets and breaks the cylindrical symmetry if r3/3sin(3φ)+r2/2 < 1/6. A BASIC program demonstrated how sensitive the system is (chaos).

Professor Sprouse took Yiyi, Yaagnik and me to his Francium lab and the accelerator. We all climbed up and gazed at the trap itself! Yiyi had a wonderful time talking to the graduate student.

Dr. Noé, Yiyi and I toured Professor Metcalf's lab as well and were amazed by the azure blue laser! We also tried to steal an optical table.


  • Surface of section (from Professor Allen's talk, where vy vs. y is smooth if the planet trajectory isn't too chaotic)
  • Anapole moment (it violates parity [from Professor Sprouse's tour])

July 12th, 2004

Yaagnik arrived today, so the other kids gave him a summary of what we've been up to for the past two weeks.

The laser cooling group and its honorary member went to the Curry Club for the buffet. There was an amazing dal (lentil) dish, an interesting cheese cube dish, some eggplant stuff, and of course mango lassi, Kingfisher, naan and orange slices. Yiyi went crazy with cantalope. Memorable quotes include "Glass is vegetarian, but I don't eat that!" and "My future dust hurts!"

At the lab, I optimized the single mode fiber. I stuck in the mulitmode for no reason and the pattern wasn't as fine...Jon started seeing things in them and it led to this. Hysterical.

Isn't it amazing that 1/0 is illegal but 1/(0!) is okay?

We played with the single mode fiber, a lens, various pinholes, the two alleged Ronchi gratings and the fork grating. As expected, we were getting Talbot effect on the Ronchi grating (and we verified that the little Ronchi is not even close to being binary!). Talbot was showing up for the fork too! The lens was producing a crazy edge diffraction pattern that destroyed the vortices because the incident beam was not Gaussian. Removing the lens, we placed various sizes of pinholes before the fork grating (if the incident beam - laser or otherwise - is too big, vortices will not be created). The pinholes were holes made by actual pins, so they produced crazy diffraction patterns, including this inexplicable monster.

July 10th, 2004

What do physicists do on Saturday nights? This.

July 9th, 2004

Professor Metcalf gave another MOT lecture today, focusing on magetism and that everything is a magnet, right down to quarks. Optical traps are position dependant; optical molasses is velocity dependent. Therefore, six-dimensional phase space is used to describe the particles' position and velocity.

Yiyi, Lidiya and I went to a luncheon about applying to graduate school. We learned that we shouldn't get a Master's before going to graduate school.

Ani, Jon, Danielle and I made actual vortices with my fork, and I helped them figure out how scattered incident light would alter the vortex intensity pattern. Earlier we discussed the joy of style sheets.

At the kegger, Yiyi and I wore our semimatching Euler/Heisenberg shirts!! We also brought thirty wraps we "stole" from the aforementioned luncheon ("Oops I dropped some diced pistashios into this chicken wrap I'm making.").

Besides reading Tuva or Bust and billions of singular optics papers, I recently read The Physics of Star Trek. It was good, although it has more physics background than the typical Trekkie needs. In Voyager, the ship was falling into a black hole but escaped because there was a "crack" in the singularity!! The singularity is purely mathematical, so how can math crack? My favorite mistake is from Deep Space Nine. Quark (name of a Ferengi, not a particle) bought a machine to create an excess of left-handed neutrinos so the laws of probability in his casino would be altered in his favor. Apparently parity violation does not apply to Star Trek particles! Such mistakes are great opportunities to teach my sister about physics.

I have a message on my phone from Tuesday saying I have mail waiting for me in the dorm office, which is open from 10:00 to 22:00 every day. I haven't yet picked it up because I'm doing physics all day. My non-physics friends are shocked that I "work" all day - whether it's teaching the kids html or LG modes, reading journals (art majors think scientific journals are physicists' personal diaries), attending seminars, reading popular math-free physics books, or just eating garlic, my life revolves around physics. And that's the way I want the rest of my life to be. As Professor Metcalf said, "Why go on vacation? That's what I do every day."

Today light is a wave.

REU is half over...which means the summer is half over. Crazy.

Helium vortices in Helsinki!

Break out your blue and red glasses for this!


  • The orbital angular momentum of a vortex is always quantized. The orbital angular momentum of a photon can be fractional.
  • The coherence length of a beam is λ2/2Δλ or c/2Δν
  • "The modulus of the complex degree of coherence represents the visibility of the interference fringes, the argument of the complex degree of coherence is related to the location of the interference fringes. The complex degree of coherence describes the spatial correlation of the quasi-monochromatic wavefields in the space-time domain." (Arimoto)

Experiment ideas:

  • Interfere two LG10 modes - I would like to do this both physically and on Maple/Mathematica/etc.
  • Scattering


  • Mutual coherence function
  • Debye vectorial theory
  • The mathematical meaning of the | and > and < in Schrödinger's equation
  • Why do I sneeze when I look at bright lights??

July 8, 2004

Kiko Galvez was here today! Yiyi showed him her tweezers, he answered my questions, we discussed quantum entanglement computing, there was GARLIC pizza, and he gave a talk based on the new geometrical phase work he did.

I also got to meet the famed Alex Ellis, master of mode conversion. I was shocked that all he used to make his HG modes was a hair! I gave him the Michael Berry paper on fractional topological charges and we decided to learn about confocal ellipsoidal coordinates.

José, Alex and I had a long heart-to-heart with Jon, Ani and Danielle about circular polarization and modes. They soaked it up like happy little sponges. Explaining stuff to other people really helps clarify it in one's mind. Next week they have to explain what we taught them to Matt and Yaagnik.

Then Professor Metcalf, Professor Koch and Dr. Noé took Professor Galvez, Alex, Yiyi and me out to Carnival, an Italian restaurant in Port Jefferson Station. I had spaghettini with escarole and cannelini beans in GARLIC sauce and broccoli rabe. I wish there was garlic fusilli - it resembles a spiral wavefront!!! Yiyi had perciatelli, commonly known as nanotubes. Everyone around us must have known we were a bunch of physicists by our strange conversations.

July 7, 2004

Rita gave a talk on the research she did last summer on Generalized Synchronization of Spatio-Temporal Chaos. A feedback loop using a liquid crystal display created the inital chaotic drive pattern. When the loop was closed, the response patterns eventually evolved into the same chaotic patterns due to coupling, regardless of the initial conditions of the system. This shows that chaos can be controlled. An application of this is in encrypting digital messages. Rita's work will be published in Physical Review Letters.

Yiyi and I went to a BBQ today and had picture-perfect (veggie) burgers while sitting with a swarm of wasps. Then we stood in line forever to get airbrushed shirts. We wanted Maxwell's equations on them, but those wouldn't fit, so we decided to get Schrödinger's equation with a cat! Unfortunately the airbrush guy was afraid of the Ψ. Yiyi got ei&pi+1=0 and I got Heisenberg's uncertainty; both are surrounded by blue and hot green ("That's so 1992!") swirls. We shall wear them to the keg.


  • Autocorrelation is the overlap when a function is shifted to the right and left
  • Crosscorrelation is the overlap of two functions


July 6, 2004

Cassini meets Titan.

I had this in my house growing up. That explains a lot.

Scientific American has an article this month on topology and the yet-to-be-disproved proof of the Poincaré Conjecture. Topology is really fun, and applies directy to my interest in cosmology and string theory (although I still don't know why the word was chosen to describe the phase dislocation of an optical vortex). Poincaré's conjecture is that the 3-sphere is the simplest compact 3-manifold...the 3-sphere is not just a sphere, but the surface of a 4-dimensional ball. Crazy. The guy who proved the Conjecture is Perelman, who spent some post-doc time at Stony Brook and gave a week of lectures on his proof both here and at Princeton during spring 2003!! If his proof survives a period of scrutiny from other mathematicians, Perelman will get $1 Million for solving one of the Millenium Problems. The proof adds something to the Ricci flow equation, which is the morphing of a manifold to smooth out its hills and valleys. This smoothing can sometimes create a singularity, which is bad. Perelman's "surgery" cuts off the part of the manifold about to create a singularity and glues a little half sphere on the wound so the Ricci function can continue to smooth the surface out, eventually creating 3-sphere(s). Perelman's Ricci addition has implications in studying the spacetime foam (crazy dynamical stuff smaller than strings at the Planck Length due to gravity-quantum interactions).

An entire Optics & Photonics News issue from 2002 is devoted to solitons. Solitons do not decay as they propegate because they create a potential well and then capture themselves in it. They have particle-like behavior and can interact with each other.

We're listening to Mozart's Clarinet Quintet right now, a piece I've played a billion times. It's fun to hear how different people interpret the cadenza and little parts (Alfred Prinz just did something crazy).


  • Topological dispersion is when the incident beam contains multiple wavelengths, each recieves slightly different phase changes (topological charge) by the step-phase mask, so you end up with a beam where the net charge may not be an integer. This is also called "frequency-dependent topological charge" and it may destabilze a temporal correlation vortex (whatever that is).
  • From Swartzlander & Schmit: "The transition from coherent to incoherent behavior [of vortices] may link the correspondence between the wave (e.g. quantum) and classical realms." Woah.
  • Catastrophe theory is when a small change in a dynamical system's control parameter yields a huge change in behavior. (Chaos theory depends on the system's initial conditions.)
  • "Partially coherent" can refer to temporal, spatial or both
  • Temporal coherence is the phase correlation along the axis of propegation and describes how monochromatic the wavefront is.
  • Spatial coherence is the phase correlation transverse to the axis of propegation and describes how uniform the wavefront is.
  • When an LGl0 beam is blocked at its waist, the pattern will continue to spiral, so at the Rayleigh range the intensity pattern will have rotated by &pi/4 regardless of the topological charge of the beam. The direction of rotation depends on the sign of the topological charge (positive clockwise and negative counterclockwise) (Arlt).
  • The even-odd diffraction pattern of the Ronchi grating is due to coefficents in the Fourier series...
  • Nonlinear effect is when the electric field polarizes the medium unproportionally to the amplitude.


  • Twisted Gaussian Schell-model beams
  • Seperable (coherent LG) vs. inseperable (partially coherent) phases (Ponomarenko, J Opt Soc Am A 18, 150)
  • Cross-spectral density of beams (Ponomarenko)
  • Phase dislocation in detail (in Berry & Nye...)
  • LG mode diffraction length
  • Circular edge dislocation (in Berry & Nye)
  • Spectral degree of coherence
  • Spherical phase front

Garlic naan at midnight is delish.

I live in the Finger Lakes region of New York, halfway between Irondequoit and Sodus bays, 25 miles east of Rochester (the 3rd largest city in the state) and 410 miles from Stony Brook.

July 2, 2004

There was no kegger today!!! Instead we raided a chemistry luau. My trying to blend in: "If anyone asks, hydrogen has one electron, helium has two, Francium is really rare, and the two types of bonding are covalent"(Lidiya came up with "ionic" ten minutes later.) They had delicious boca veggie burgers and pretzel windows (we'll have to make preztel penguins, but I would never eat a pretzel apple), but the only drink left was beer. Woohoo for chemistry! However, Yiyi missed it because she fell into an optical vortex or got stuck in some optical molasses.

Professor Metcalf gave another talk in his laser cooling series. Motion-dependent forces were most interesting; apparently all textbooks are afraid of them.

José gave a talk on how to use the CCD (charge coupling device) camera.

Jon's journal sums both talks up nicely.

Swartzlander replied to our questions. Apparently, a vortex of charge other than one is unstable and "breaks into" vortices of charge one. I knew that any vortex (fractional or integer charge) can be created by superposition of fundamental (charge one) vortices, but I didn't know that they decay. Does this happen in the far field? Also, a mode one's core is described (seemingly only by Swartzlander) as a filament, which agrees my previous definition of filament.

Correlation functions are coming up in other papers (Swartzlander and Schmit to be published in PRL), so I'm going to work more on comprending them. The cross correlation function relates what's happening to the phase/wave at one point to what's happening at another, but I don't understand how increasing the coherence increases the strength of the cross-correlation ring charateristic of a vortex.


  • Starburst pattern in Sacks et al. paper and its elimination by using an aperature
  • Pillbox beam
  • Topological dispersion (frequency dependant)
  • Temporal correlation vortex
  • Complex degree of coherence (I'm assuming complex means it has an imaginary component, not that it's hard)
  • Mutual coherence function

July 1, 2004

We saw the copy of Newton's "Optiks" from the 1700s in the department office today!

Dr. Noé, Professor Metcalf and the rest of us had a GARLIC pizza and strawberry luncheon and talked about random nothingness, including this (although I like this and this better).

I read in a Sacks et al. paper about vortex filaments, which are vortices whose core is much smaller than its beam size (although the core is still large with respect to the wavelength). Due to diffraction, the filaments create rings (I don't understand why). The angular momentum is strongest closest to the axis of propagation.

Cross correlation function is a lost cause for now...

Dr. Noé taught me how to spy on who's looking at my site. How devious.

Look what I learned about today!!! This too. (Not related to optics at all.)

Real physicists brush their teeth in the lab.

This Schouten, Visser and Wolf paper entitled "New effects in Young's interference experiment with partially coherent light" will be interesting to read in the future uncluttered with vortices.

Something for the particle physicists to ponder by J.R.R. Tolkien: "He that breaks a thing to find out what it is has left the path of wisdom."


  • A Cornu spiral is the resulting curve when the two Fresnel integrals, S(x) = &int0xsin(t2)dt and C(x) = &int0xcos(t2)dt are parametrically plotted in the complex plane (C has an i component). And most scary, Berry published a paper with Shelankov called "The Aharonov-Bohm wave and the Cornu spiral"!!!!! The AB effect is somehow analogous to fractional topological charge. Everything is so interconnected.
  • Bessel beams are cylindrical beams whose radius does not expand with distance from the source.
  • Bessel functions are any function that solve Bessel's differential equation. They have cylindrical and spherical coordinate properties and therefore applicaitons in a lot of crazy things. Friedrich Wilhelm Bessel calculated various stars' parallaxes!
  • Hankle functions are solutions of the cylindrical wave equation that express inward and outward propagating cylindrical wave.


  • Correlation functions - auto, cross
  • Is the vortex core size related to the Bessel zeros (like the Airy disk of a circular aperture)? (Although I don't really know what that means.)

Yiyi and I were in the lab until 12:30am. We're officially physicists now.

June 30, 2004

"Monday, Wednesday, Friday light is a wave; Tuesday, Thursday, Saturday light is a particle; Sunday, take your pick." - Professor Metcalf

This morning about twenty to-be eighth graders in a summer science camp from South Huntington came for a tour - instead we brought the LTC to them. They had fun burning paper and seeing various demonstrations, including diffraction glasses, fiber optics and the speaker laser light show.

Dr. Noé generously took us to the Curry Club, where we had mango lassi (drink), yellow (vellow) dal (yellow lentils), zeera alu (spiced potatoes), navrattan korma (coconut sauce vegetables), naan (bread) and peas pilaf. I'm really enjoying the interesting foods and I deduced all by myself that alu means potato.

Professor Metcalf gave a talk on the background of laser cooling - light has momentum, is a particle and is a wave. I asked the unanswerable question: how is conservation of energy time reversal invariance?

We poured over Swartzlander's most recent email - he suggests sending halogen light through a fiber light pipe through a pinhole to vary the coherence area of the light (incoherent light will not produce a vortex) and plotting the intensity of the core divided by the total intensity vs. the distance between the fork grating and the pinhole; compare this to a plot without the fork grating. The greater the distance between the pinhole and the grating, the more scattering. That the light is only partially coherent means that there will be some light in the vortex core: "scattering may superimpose other modes into the dark vortex core region, shifting or brightening the core if the light is, repsectively, coherent or incoherent." For some reason the planet light is not coherent, so it would not be affected by the grating. Everything about viewing the planets depends on their light being scattered. We're going to order some stuff and get things cracking.

Feynman's "Optics: The Principle of Least Time" lecture describes how the path light takes is the sum of probability phasors (not Star Trek...why are Kirk's weapons called that?) whose angles are proportional to the time it would take to get from one point to another. "Almost all of [the total] accumulated probability occurs in the region where all the arrows are in the same direction (or in the same phase). All the contributions from the paths which have very different times as we change the path, cancel themselves out by pointing in different directions." Light does take every (violation of conservation of energy!) possible path (sounds like quantum...) but it interferes so we only see the real path. (This can actually be the longest path, as light takes the one where the phase/time doesn't change much if the path is slightly changed.) The relation to my vortices is that the phase in the core is so spiraly that they all cancel each other out (when the source is coherent). The exact boundary conditions for the edge of the core depend on how much the phase spirals in one azimuthal revolution (the topological charge!); higher degree of phase step in the beam, the greater the interference in the core and the larger the core size (and smaller the vortex radius).

Rich Lefferts said that science is questions, so here they come:

  • Why is the planet light scattered?
  • Spatial correlation functions
  • Phase dislocation vs. vortex
  • Coherent scattered light shifts the core (Swartzlander, Phys. Rev. Lett. 88, 103902); is the core off-center then? Also, how can scattered light be perfectly coherent (or does this refer to relatively coherent scattered light?)
  • Speckle patterns and temporal (temporal meaning phase?) fluctuations (in this case Star Trek has hurt me, as all I can think is time travel).

Questions for Swartzlander:

  • Advantages and disadvantages of using a CGH or a phase plate in ESP observation
  • Why is the planet light scattered? And why must the planet and star wavenumber be the same?

June 29, 2004

We verified experimentally that the dimmer higher modes of diffraction ARE supposed to be there, so they're dimmer because they're being suppressed somehow (Kiko knows why!!). We thought the even-odd thing may be related to the Ronchi grating having equal amounts of black and white, but on the muliple diffraction grating plate, the gratings with a 3:1 ratio of black:white also displayed dimmer higher order modes. The 3:1 gratings were not every other spot, though, but every two spots. Therefore the even-odd thing should be related to the spacing, but one of the supposed Ronchi gratings does not exhibit the expected effect! In addition, the low intensity first order seems to maintain its intensity through the odd orders of diffraction, while the even orders decrease in brightness.

Lidiya found a graph about halfway down this York site, which corresponds to the intensity pattern we observed.

Dr. Noé called Grover Swartzlander today. The higher the order of diffraction, the higher the topological charge! So my fork grating does produce m = 0, 1, 2, 3... while a phase mask is engineered to only produce a certain charge. Swartzlander also wanted to know if I was "serious" so we could crank out a paper in August. Phone conversation with my sister/friends/cats: "I'm not coming home. Ever. I'm going to Arizona, the optical vortex capital of the world." *sound of jaw dropping, then deathly silence*

June 28, 2004

Gravitational lensing and a non-optical singularity.

The Colgate paper discusses geometric and dynamical phases, and they use the word eigenstate instead of mode (from linear algebra). "In general, the phase of the final state differs from that of the initial state by &phi = &phid + &phig where &phid and &phig are the dynamical and geometric phases, respectively. If only the topology of the path is altered, then only &phig varies." The dynamical phase is kept constant in the experiment. All the non-Colgate information I've found on &phid and &phig relates to quantum (Berry's phase). They also use a Poincaré sphere, which represents all the possible polarizations of light. One pole is right circularly polarized, the other left circularly polarized. The equator is linearly polarized light at varying angles. Between the poles and equator the light is elliptical. In Colgate's paper, they used a modified sphere, the Orbital Poincaré. Instead of circularly polarized light, the poles represent LG+10 and LG-10. The equator is instead HG modes, with 0° being HG01 (two horizontal lobes), and 180° HG10 (two vertical lobes). The various transformations (using a &pi/2 converter or Dove prism) can be described by moving along the sphere's lattitudes and longitudes. The Colgate people transformed only the topological phase, and they moved from the south pole (LG charge -1) to the equator (HG01), along the equator 20°, and back to the south pole by using two &pi/2 converters at -&pi/4, two dove prisms, and two &pi/2 converters at &pi/4 respectively.

We went to a seminar today by Tim Chupp of the University of Michigan entitled "A Proposed Electric Dipole Moment Measurement with Radon." Professor Metcalf talked to us (including the newly arrived high schoolers) before the lecture, stressing that physicists like to measure zeros and find symmetry in things. That the electron has an electric dipole moment is strange enough, but if it's found to be closer to zero than the standard model permits, it'll be another symmetry violation (CP being the other) and we'll need a new/revised model.

My prime question is currently why do the vortices change shape in higher orders of diffraction? Secondly, why is every other order dimmer?

We tried to figure out the relationship between every other diffraction order being dim and the grating's black and white spacing and ratio. Numbers and conclusions to come...

June 25, 2004

Kerry vows to lift ban of governmental funding for stem cell research! Woohoo!

String theorists are coming to Stony Brook!

Dr. Noé took us out to the Curry Club today - Indian! I had wonderful garbanzo & lentil, spinach and mixed vegetables dishes, and a mango drink. Delish.

The kegger was entertaining and full of Smart Food. Yiyi and I were dissapointed in the poster, though.

The lab got a makeover today in preparation for the storm coming Monday...

The Colgate site says, in reference to the varying shapes of the vortices in higher orders of diffraction, "Since [the computer generated fork] is a binary grating with equal fringe to space ratio, the second order doughnut is suppressed. You will see the third order (LG03) doughnuts much better." Sadly there is no further explaination!

I must reread a Colgate paper by the same people as the site.


  • What happens when a vortex is passed through another fork?
  • How do the vortices change when the fork's colors are inverted?
  • Qualitative and quantitative description of why higher orders of diffraction in a non-fork grating decrease in intensity
  • Are the differences in near and far field intensity patterns due to Fraunhofer and Fresnel?
  • Why is every other order dim, as seen here?

June 24, 2004

Quantum entanglement and transporters in Science News!!

The fractional charge paper by Basistiy (sometimes misspelled as Basisty) revealed that what I gathered from the Berry paper yesterday is basically right, with the exception of the difference in intensity distribution at the near and far fields. In the near field, there is one vortex of charge S&alpha (I think) and the vortex chain relating to charge &alpha. In the far-field the chain dissolves (it moves away from the z-axis) and the core breaks up into the closest integer to S&alpha + &alpha vortices. Why this happens, I don't know.

I spent an insane amount of time searching for a SPIE paper referenced in the Basistiy, but it does not exist!


  • When reflected in a mirror, the sign of the topological charge will change.
  • A semi-answer from further down this page! "...any vortex beam maintains its charge as it propagates, and will always have a null at the beam center, although its radial irradiance pattern will genearlly change due to propagation related phase shifts along the different [radial] modes." (Smith and Armstrong.)


  • Far-field pattern changes (multiple vortices and non-integer charge)
  • OAM and Poynting vector behavior of fractional charge vortices
  • Integer charge = tines - 1?? (Then why the increase in core size with distance from z-axis? And why solid beam on x-axis??)
  • Multiple vortices created with a fork grating (with multiple Y's)?
  • Relationship between LG's azimuthal and radial modes, and how that ties into the core size change on the x-axis.
  • Propagation-related phase shift (radial mode).

Experiment ideas:

  • Study the far-field of OV (fractional charge if possible!)
  • Destruct vortices by interferring those of opposite charge (this can also test that the charge of the many OVs in the fork diffraction pattern are of like charge)
  • Effect of non-Gaussian incident beam on vortex pattern
  • Investigation of "propagation-related phase shift along radial mode" by examining very near field, mid field and far field.

A search for singular optics books in the library returned "Singultus - See: Hiccups."  ...Busy, busy, busy.

Stardate 41474

The Advanced Accelerator Concepts Workshop is at Stony Brook, so the campus is swarming with more physicists than usual!

The Lee, Yuan and Dholakia paper is mainly an experimental confermation of a Berry paper on fractional topological charges published earlier this year. When the charge is at a half-integer or above, a new vortex is "born." This is what is circled in all the photos of the fork interference patterns, although I can't tell the difference between a pre- and postnatal vortex. A non-integer charge is "a superposition of serveral axially symmetric optical beams of different integer charge" that would require an infinite weighting for each constituent LG mode.

While the NEW Berry paper about fractional phase steps is saturated in rather scary math, I now (somewhat) understand where the C shape of non-integral topological charge comes from. There is an infinite number of zeros in an equation describing the wavefunction oscillation (equation 28). As &xi and &eta (which are like helps to pretend they're just x and y) approach zero and as &alpha (the phase step) increases, a chain of little vortices are created. The chain always lies near the &xi axis and the vortices' signs alternate for &alpha = m + 0.5 + v (m is the nearest integer to &alpha using 2nd grade rounding and v is an arbitrary constant whose absolute value is « 1) as the far-field is approached. The chain vortices annhilate each other in pairs (of +v and -v) and the chain gets shorter until only one vortex remains with an integer charge, and S&alpha = m + 1. The chain oringinates near the core center, and spreads out along the x-axis, which is what makes the C shape. I love when pure math proves something, even if I can't understand it. There is no vortex on the z axis (that of propagation) and there are edge-diffracted waves for fractional &alpha.

"Vortices with fractional strength [topological charge] cannot propagate in free space; instead...[the beam] break up into a collection of strength ± 1 vortices...with interactions [manifested as the chain of vortices] as &alpha increases through half-integer values (Berry 260)." As the phase step (&alpha in Berry's paper) is increased beyond an integer charge S&alpha the central core branches into S&alpha vortices seperated by regions of slightly higher intensity, and a chain of mini vortices extends outward along the positive x-axis. The S&alpha vortices all have the same sign, and form an S&alpha-gon (for example, if &alpha = 6.2, then there will be a chain and six vortices in the core making a hexagon). The chain increases for 0 < &alpha < 0.5, reaches its maximum at &alpha = 0.5 (creating the nice C shape), and decreases for 0.5 < &alpha < 1.0. At &alpha = 0.5 a new vortex is "born" in the far-field due to destructive interference of the chain vortices, although I don't understand where that comes from yet.

Dr. Noé believes it's my "life's work" to understand the Berry fractional charge paper. Very true.

Also from the new Berry, I learned that the topological charge is the vortex strength S&alpha while the &alpha of the 2&pi&alpha azimuthal mode is the phase step. These terms are usually interchangable, but when dealing with fractional &alpha, the distinction should be made, as S&alpha=int(&alpha + 0.5).

My many papers have a new home in a heavy duty industrial binder!


  • The light passing through the Y of the fork gets diffracted by different orders, hense the phase ramp.
  • The Zeeman Effect is when spectral lines split into their components when exposed to a magnetic field. Several electron configurations exist in most atoms that have the same energy. The magnetic field interacts differently with electrons of varying quantum numbers (principal, azimuthal, magnetic and spin) and their energies slightly change. This is useful in studying the magnetic field of stars.
  • Evanescent waves have an imaginary wave number (?) and are formed when a sinusoidal wave undergoes total internal reflection. Their intensity decreases exponentially instead of sinusoidally, and their modes do not have a phase shift. The evanescent zone is the region where the intensity rapidly dimishes.
  • Topological charge is also called a phase step (sometimes!)
  • Quantum entanglement is when two particles are so closely related that one must have a certain property and the other another property. Before you observe one of them, both have both properties (think cat), and as soon as you observe one particle, both of their wavefunctions collapse. Einstein didn't like the idea and called it "spooky action at a distance" because if the two particles are infinitly seperated and one of their states collapses, it has to tell the other one instantly what it should be doing. I have no idea how this relates to fractional phase steps, but quantum mechanics is fun.
  • Glory scattering produces a little circular rainbow caused by diffraction that can be seen from mountains and planes in the direction opposite the sun. The plane or mountain's shadow appears in the center of the circle. Photos from a German meterological site.
  • The Aharonov-Bohm effect "is a quantum mechanical phenomenon by which a charged particle is affected by electromagnetic fields in regions from which the particle is excluded, first proposed by Aharonov and Bohm in 1959. Such effects are predicted to arise from both magnetic fields and electric fields, but the magnetic version has been easier to observe. In general, the profound consequence of Aharonov-Bohm effects is that knowledge of the classical electromagnetic field acting locally on a particle is not sufficient to predict the quantum-mechanical behavior." (From Berry relates the increase in charge to this effect, which I'll ponder further in the future.


  • Edge dislocations vs. screw dislocations (the answer lies somewhere in Nye/Berry)
  • Guoy phase mismatch
  • Core width proportional to &lambda?
  • Weighting of beam superposition
  • Evolution of the intensity pattern in far-field of multiple vortex beams and fractional phase step
  • Evanescent zone/wave
  • Aharonov-Bohm effect (analogy between topological charge and it's quantum flux)
  • Plane waves vs. beams
  • Glory scattering
  • Angular momentum of fractional charge vortices.

June 22, 2004

Dr. Noé just handed me an article in today's New York Times Science News called "Grasping For Light." It's about the search for ESPs using the first coronograph of its kind (at Haleakala, no less). The telescope is the Advanced Electro-Optical System and uses adaptive optics. There is also an infrared camera named Kermit! Coronagraphs, like optical vortices, reduce the glare from the sun so you can see the nearby planets. However, it merely uses a disk instead of interferometry.

There's also an article in the Times Science News about Mars' hematite blueberries, with a quote from McLennan, a geologist here at Stony Brook!

QuNit: "The connection between quantum entanglement and the exchange symmetry of the states of N identical particles is made explicit using the duality between the permutation group and the simple unitary group. Each particle has n-levels and spans the n-dimensional Hilbert space. We shall call the general state of the particle as a qunit. The direct product of the N qunit space is given a decomposition in terms of states with definite permutation symmetry." (From this Suranjana abstract.) Vortices apparently have some application in the study of quantum mechanics, but I think I'll save that for later.

In Brian Greene's PBS miniseries "The Elegant Universe" about the quest for a grand unified theory, he goes to the Quantum Cafe to explain how everything is just probability - there's a superposition on what drink he recieves until he observes it and the wave function collapses. They have a Quantum Cafe at Starfleet Headquarters too! (Species 8472 recreated it when they were supposedly training to invade [it was actually reconnaissance] non-fluidic space.)

The REU students had a tour of the Nuclear Structure lab today.

H-bar is &#295!!

I did a lot of reading today, out of Hecht, Mansuripur, Nye/Berry and others.

I guess I'll be missing school for Monday - Thursday for the OSA meeting (and hopefully I'll be leaving on Friday afternoon!), which I think is worth it. How many opportunities do I have to hear a whole string of lectures on singular optics? And by Grover?! Midterms seem to be the first week in October, so hopefully things will work out!!

Swartzlander's "Peering into darkness with a vortex spatial filter" requires that the light from the planet and the star have the same wave number. Light is, of course, only reflected by the planet, so this would always be the case (unless you factor in Doppler). Why, though, is this necessary? The vortex is only a "window" and it doesn't care what the planet's light is doing. If the wavenumber matters, does the phase matter? When light moves to a medium of higher optical density it is inverted. The atmosphere of a planet certainly has a higher index of refraction than space, so the planet light will always be out of phase with the sun's. But is that relevant?

This Lee, Yuan and Dholakia paper uses a fork grating to create vortices with non-integer topological charges!! The gratings are nonsymmetrical in that on one side of the fork, the lines have a break. If you start out at m=0, you just have vertical stripes. If you cut the lines in half on half of the grating and gradually slide the bottom away from the others, the topological charge increases. Eventually the bottom lines will line up with the top lines again, and after bending them a little to account for the extra half line, you get a Y with a line nested in its top. (Sorry that every other word in that explanation was either "line" or "half".) That paper also debunks my "topological charge = number of tines - 1" theory, since it goes from m=0 to m=1 without making a two-pronged fork (also known as a Y). The images of the fractional topological charge vortices look like O's turning into C's and back again as m increases from 0 to 1. This is similar to the Leiden paper, as their graphics showed three m=1 vortices and one C-looking m=0.5 vortex. Gladly, the Lee paper has images of vortices of charges 0 - 2 and the 1.5 vortex is not multiple vortices contained in the same beam (I still have to figure out how to make those, as described in the Mansuripur book). I'll read the Lee paper and take notes tonight until my sister calls so we can watch Voyager over the phone together.


  • Phase mask (and phase plate) vs. hologram in astrometric applications.
  • Increase in core size as distance from z axis increases
  • Solitons
  • Planet-star relative wavenumber and phase
  • Non-integral topological charge!

June 21, 2004

Happy solstice!

On half-integral topological charge, the Leiden paper explains that it can only be made using a spiral phase plate, which is like one turn of a smoothed out spiral staircase. A fork grating makes a beam with topological charge = the number of tines in the fork - 1. So my fork creates only vortices of topological charge ± one (if I'm interpreting that right...the gratings are made by using pure mathematics or interferring a plane wave with a pre-existing vortex of desired charge (and making a hologram). This m = prongs - 1 theory can be tested by splitting the post-fork beam and recombining it with any vortex on the other side of m = 0. If a two pronged fork creates only m = ± 1, then where does the m = 0 center come from?). Making a spiral phase plate with any value of topological charge is easy, using m = hs(n-n0)/&lambda and h = hs&theta/2&pi + h0 where h = phase plate height, hs = step height, &theta = azimuthal angle and h0 = base height, n = index of refraction of phase plate and n0 = index of refraction of medium. The paper provides 3D intensity representations of a theoretical and actual 3.5 topological charge beam. There seems to be three and a half seperate vortices instead of a vortex of intensity and core width between those of m = 3 and 4. The difference between integer and half-integer vortices in the graphic is that the integer ones are completely surrounded by ridges of opposite intensity, while the m = 0.5 vortex is only half surrounded by a ridge. This paper (as always) brings up more problems than it solves, as a beam with N vortices each of charge m does not have a topological charge of Nm, according to the Mansuripur book (if the distance between the vortices is less than the radius of the beam at its waist, they will combine and possibly anhilate each other). So, the Leiden paper easily addressed how to make beams of non-integer topological charge, but led to more confusion as to what it looks like (m = 3.5 is not the same as three m = 1 vortices and an m = 0.5!). I can't find any other references to a non integral topological charge...

I attempted to read the Nye and Berry paper yesterday, but only got through a third of it.

In the December 2001 Optics & Photonics News, there's an interesting article called "Critical Foliations and Berry's Paradox." Berry says that "there is no such thing as a 3D line of maximum intensity in a [HeNe] or any other optical beam." When you obstruct the beam, the location of the maximum depends on how the obstruction is oriented. A tilt of a mere half degree can turn a rectangular vortex into three small oval vortices. Because of these critical foliations, related to Berry's paradox, observers can get drastically different results, while both are correct.

The helical phase fronts come from the phase term exp(il&phi), l being the topological charge and i being imaginary numbers. I have no idea how &radic -1 relates to physics...

Answers / revelations:

  • Rayleigh range zR is the distance along the axis of propagation z where the cross sectional area is twice as great as it was at the beam's waist w0 (or, zR is where the beam radius is 21/2 times larger than the radius at the waist)
  • Beam waist is the radius of the beam at the point where the phase is uniform. According to Mansuripur (57), "there is a symmetry between the locations before and after the waist...the intensity profiles on opposite sides of the wiast are identical, while the phase profiles differ by a minus sign. The beam is convergent before and divergent after the waist." There is only one waist in a Gaussian beam moving through free space.
  • The beam radius is the radius at the point where the amplitude and intensity are e-1 and e-2 respectively.
  • Guoy phase shift is the 180° phase shift when light goes through a focal point (when the waist is crossed).
  • Order is the same as topological charge is the same as vortex charge.
  • The Poynting vector spirals around the axis of propagation in a vortex, which is why only spiral phase fronts can have OAM. OAM = ± mh-bar (LG only; independant of polarization) and SAM = ± h-bar (circularly polarized; independant of phase). A beam can have both OAM and SAM. The total angular momemtum is the sum of SAM and OAM.
  • Spanner is the cool British way to say wrench.
  • A screw phase dislocation is an optical vortex, so it always has a zero intensity core.
  • Vortex streets are simply bunches of vortices in a row. In water and clouds, they can arrise when something zips through the medium.
  • Poynting vector walk-off is when the directions of energy flow/momentum and wave propagation diverge.
  • In LGlp, the l is the topological charge and azimuthal mode, and the p is the radial mode. If two LG modes from TEM01 are superimposed and one is rotated 90°, if p = 0 and l &ne 0, you get a vortex; if l = 0 and p &ne 0, you get a bull's eye. Modes come from the laser's boundary conditions and describe the intensity distributions.
  • Lasers, like plane waves, have no electric or magnetic field component in the direction of propagation (TEMpl ).
  • The wave equation requires that the core be dark.
  • The cavity in the laser where the ruby is stimulated can only have standing waves when there is an integer number of waves between the mirrors. That is where the longitudinal cavity mode m comes from in TEMmn. The transverse mode n describes the y component of the beam.

More questions:

  • Astigmatic (which carry no angular momentum) and pseudostigmatic (which carry angular momentum) beams
  • Vortex streets
  • Why in "Topological defects on moiré fringes of spiral zone plates" by Huguenin, et. al, don't m = +1 and m = -1 destroy each other?
  • Poynting vector walkoff
  • Multiple vortices with opposing charges: pattern changes as distance from source increases.
  • quNit "...the photon can be viewd as a N-dimensional quantum system..." (From Oemrawsingh, Eliel, Woerdman, et. al)
  • If a two pronged fork creates only m = ± 1, then where does the m = 0 center come from?
  • At Rayleigh range, "phase singularity is mixed with diverging wavefront curvature." (From Mansuripur)
  • Why isn't m directly related to OAM, as Padgett, Courtial, Allen claim? After all, OAM is explicitly equal to ± mh-bar.
  • H-bar in HTML!
  • Imaginary numbers and physics
  • How TEM modes are physically modified.
  • The wave equation.

SpaceShipOne launched today in a history-making flight reminscent of Cochrane. It is one of many entries in the race for the Ansari X Prize.

June 18, 2004

I have the taste of green tea ice cream in my mouth...which is amusing because I don't know what real green tea tastes like. And as tea is only tea after you steep the leaves, is the ice cream made from actual tea, or just the leaves? There's also some intriguing orange/carrot/sweet potato, zucchini and potato tempura, miso, and lots of edamame floating around in me from Sushi Ichi. The Dragon Roll, a foot long eel (e-e-eel) atop sushi, provided visual trauma. Many thanks to Dr. Noé!

Later was the weekly physics kegger, where Yiyi had her first marshmallow in like eight years and I met my roommate for the first time.

Sir Michael Berry is coming here in February!

I have a book (Mansuripur) now with a whole chapter on OV! It also details LG (and HG) modes. I've been reading that and writing down some tangible notes. I think topological charge is finally defined in my mind, and vortex charge is exactly the same thing (although some papers reverse the indices of the LG mode just to confuse me). That book has some nice pictures of how the intensity of the Poynting vector varies with time as the phase of the vortex spirals, again stressing that the photons themselves are not vortex-ing. Topological charge is directly related to orbital angular momentum, where OAM = topological charge * h-bar. The topological charge, l is the number of cycles the phase goes through when you rotate the beam once. So, assuming l is quantized, this should come only in multiples of 2&pi. But what's stopping it from going through 2.5829(2&pi) phase cycles?? When I comprehend how exactly l is varied, that question might be answered (Alex's site will help).

Dr. Noé just handed me a wonderful article "Light's Orbital Angular Momentum" by Padgett, et. al in May 2004's Physics Today, among a sea of other pertainent stuff.

It's not only Swartzlander who thinks optical vortices could be applied in astronomy: Harwit of Cornell.

About the term "extrasolar planet"...every planet is extrasolar, since there's not going to be one inside a star. Our sun and solar system really need names! ("The United States of America" isn't really a name either, as more than one country on the America continents have states that are united.) Maybe xenoplanets?

Questions / Further ponderings:

  • Screw phase dislocation always create zero intensity? (Phase dislocation in general)
  • Undefined phase must mean zero intensity?
  • Half integer topological charges physically means what? (It's strange that topological charge is always described as being an integer, including in the aforementioned May 2004 Physics Today article. That article also says that the OAM is quantized, meaning the topological charge is an integer.)
  • Spin angular momentum.
  • Angular Doppler shift.
  • Phase conjunction.
  • Guoy phase.
  • Rayleigh range.

Experiment ideas:

  • Making vortices with half integral topological charge (ha).

I found a mulberry tree on campus. I think I'll name it Michael.

June 17, 2004

From "Generation of vortex beams by an image-rotating optical parametric oscillator":

"Vortex beams can be conveniently expressed as unique linear combinations of Laguerre-Gauss modes. These are cylindrically symmetric modes characterized by two integer indices, p and m, which are unaltered by propagation. A line integral of the phase gradient around the dark beam center is equal to (m2&pi), where m is the vortex charge. This charge gives [the vortices] an orbital angular momentum of mh-bar per photon along the propagation axis in addition to the usual spin angular momentum associated with polarization. They have p + 1 radial nodes. Any vortex beam of charge m can be decomposed into a linear combination of Laguerre-Gauss modes, each of charge m but possibly having different p's. These constituent modes each propagate without changing character, so their dark center is preserved. Hence any vortex beam maintains its charge as it propagates, and will always have a null at the beam center, although its radial irradiance pattern will genearlly change due to propagation related phase shifts along the different p modes."

In the Brand paper, the LG mode is described by [l][p], l being the azimuthal mode and topological charge, p the radial mode. In the above quoted Smith and Armstrong paper, the LG is [p][m], where p is the radial mode and m is the vortex charge!!! Both describe the loop integral to be 2&pil or 2&pim, implying that m, the vortex charge, is also the azimuthal mode, and is the exact same thing as the topological charge. However, our good friend Grover and just about every vortex paper uses "topological charge" a lot more often than "vortex charge." From this Vortex Filaments paper of his, he calls the topological charge the "harmonic index" and says that the fundamental units of the TC/HI, M, are &sigma, where &sigma + ± 1. "In genearl, any vortex having |M| &ne 1 may be represented as the product of 'fundamental vortices'...Thus, the propagation dynamics of optical vortices may be described in terms of fundamental vortices [those with &sigma = ± 1]." I really think it's time to ask Grover to clarify these definitions.

This and this describe how to create optical vortices.

From Heckenberg, et. al, "The signature of a phase singularity is the defect [intensity=0 and phase is undefined] in the fringes structure where an interference fringe starts at the location of the singularity." A topological charge of 2 = 2(2&pi) phase integral, 3 = 3(2&pi), etc. By using a Mach-Zehnder interferometer, they "misaligned and misdirected doughnut [vortex] modes", and created some fairly complex (but nicely symmetric) patterns. They also created an "optical leopard" by combining Gaussian Hermite TEM20 and TEM02 with a &pi/2 phase shift (TEM means Transverse ElectroMagnetic and TEM00is the normal output of the laser).

A very familar picture, 2/3 of the way down this Colgate page. They have some really good graphics there too. "These Hermite-Gauss solutions (or modes) are categorized by their indices (n,m) and `order' N=n+m...It is important to understand that the light is not following a helical path. The phase of the light is changing in such a way that it describes a helix..." The second order LG mode (LG20...but wouldn't it also be LG11 and LG-810, etc.?) has a phase change that spirals twice as fast as the first order. HG modes do not carry angular momentum, and its indices describe the number of intensity minima in the magnetic (n) and electric (m) fields.

June 16, 2004

This island weather is vile and it's only going to get worse. Sadly, there are no cicadas about, but there are plenty of lightening bugs.

In attempt to answer yesterday's question, what does "topological" mean in this context? (The "charge" refers only to the sign of which way the vortex is spinning [the vortices on both sides of charge 0 are symmetric] and vortices of opposite charge will cancel each other out.) In math, topology involves the connectedness of things - a solid sphere can be transformed into a solid square, but a sphere can't become a circle because dimension is a topological property. The sphere and the cube are topologically the same, since they can be changed into each other by squishing or bending or cutting (if they are glued back together along the exact same seams). I remember going to the science museum in Baltimore or Toronto and seeing this crazy shape that could be made by cutting and warping and glueing a hollow doughnut - the doughnut and the crazy shape are topologically equivilant. Great, but how does that relate to the "order of the dislolation" of the phase? Maybe that's like asking how a quark can be flavored...

Now I'm looking at "Phase Singularities in Beams" by GF Brand, Am. J. Phys. 67 (1), January 1999.

"...The beam is linearly polarized...where Llp is an associated Laguerre polynomial, and l and p are azimuthal and radial mode numbers. Of special interest to us is the last term exp(il&phi). Consider the case where l &ne 0. Imagine a loop encircling the axis of the beam. Around this loop the phase changes by l times 2&pi. Now imagine the loope becoming infinitesimally small. The loop reduces to a point on the axis where the phase can take on any value between 0 and l times 2&pi. Ambiguity is avoided only if the amplitude of the field on the axis is exactly zero. It follows that the beam must be hollow. The axis of the beam is described as a phase singularity. The integer l is known as the topological charge."

The two modes l and p describe the power distributions of the beam. So here, the topological charge is described as the azimuthal mode number. As l increases, the hole in the vortex becomes bigger (more interference).

According to this Mamaev paper, which deals with optical vortices in photorefractive media, "excitation of the medium with a circular vortex leads to charge-dependent rotation, stretching, and eventual decay of the localized structure. Another unique manifestation of the dominant role of anisotropy in photorefractive self-focusing is the recent experimental observation of the decay of a high-charge vortex. The very existence of this decay is due to the anisotropy-driven stretching of the vortex core." While my vortices do not deacy, this also suggests that the charge describes the motion. The post-fork laser light is circularly polarized (and therefore carries angular momentum proportional to the the topological charge multiplied by how many photons are in the beam). So does a higher vortex-charge mean it's spinning faster? Can it be either positive or negative? If so, it wouldn't indicate which way the vortex is spinning, as that is what the topological charge's sign does.

Revelation, courtesy of Mamaev, et. al: "The direction of rotation is uniquely determined by the vortex charge: changing its sign changes the sign of the roation." Woohoo. So the sign of the topological charge is exactly the same as the sign of vortex charge. But then what does the absolute value of the vortex charge mean? It can be half-integral (topological charge can be too), as this Leiden paper states. That paper also seems to indicate that vortex charge is a measure of vorticity (circulation per unit area).

Experiment ideas

  • Destroying vortices by combining two of opposite topological charge.
  • Verifying that the vortex is circularly polarized by using a linear polarizer (there should be no change in intensity).

June 15, 2004

I'm back in Stony Brook after being home for the shortest three weeks ever. Yesterday I woke up at 3:30am and I can still feel it. I watched the Magic School Bus on the plane (it was the one about sound). Right now I'm reading about various vortex-related things, including this and this.

"Optical vortices occur naturally in cylindrical beams and in scattered light" (from "Observed Scattering into a Dark Optical Vortex Core" Palacios, Rozas, Swartzlander, 2002). Cylindrical beams satisfy the Bessel function; the beam's radius remains constant regardless of the distance from the source. The Bessel beam is a "natural" OV, even though it cannot exist because it would require infinite energy, but approximations still create vortices. "

Orbital angular momentum is best described by this: "The mechanical effects are visible in optically trapped specimens: Spin rotates trapped items on their own access axes, similar to the rotation of the Earth, where orbital angular momentum rotates them around the beam's axis, similar to the path of a satellite." Helical waves carry OAM.

In ESP news, NASA found one by detecting a gap in the dust disk surrounding a young star. It is the youngest planet ever found, and supports a theory on how gas giants form.

In other news, parity violation has been observed in low(ish)-energy electron-electron scattering; the top quark is 2% heavier than previously thought.


  • Vortex charge = topological charge?
    • "...for a point vortex on a Gaussian background field, we find the vortex charge is a constant of motion (as expected from the principle of conservation of orbital angular momentum)." (From here)
    • "...vortices are spiral phase ramps around a singularity, where the phase of the wave is undefined and its amplitude vanishes. The order of the dislocation multiplied by its sign is referred to as the vortex's topological charge...The sense of spiraling is dictated by the sign of the vortex's topological charge..." (from here)

April 16, 2004

This week Dr. Noé and I took many photos of the fork grating, and the optical vortices that it creates. We also studied the diffraction pattern of a Ronchi grating, which is a grating without the Y that produces a series of solid dots. The laser beam was diffracted by the Ronchi grating and the image projected on the wall a distance R away was traced. The most intense spot was in the middle, as expected. The angle &theta is the distance between the spots, x, divided by R. Since &theta is very small, the small angle approximation may be used where sin(&theta) = &theta. The wavelength, &lambda, equals the distance between scattering centers, d multiplied by &theta. It was assumed that d equals 254 &mum from a previous WISE experiment. Next, the Ronchi grating was replaced with the fork grating and x' measured. Knowing λ and θ, d' was calculated. As expected, the grating period is larger for the Ronchi grating, which results in more dots produced. The space between the lines on the fork grating is about three-fourths of that on the Ronchi grating. This leads to the effect we observed when the light is diffracted by both gratings at 90o out of phase. The horizontal spacing is less than the vertical spacing between the dots. However, we later realized that the distance between the lines on the fork grating changes, so the value d' is not well defined. We decided to measure the grating period near the top and bottom of the Y. The grating had to be moved as close as possible to the laser, since at the previous distance the beam had spread out so much that vortices were unavoidably produced. This was undesirable because we were trying to study the difference in line spacing at the very top and very bottom of the grating. I determined the distance between the lines at the two positions using d=R&lambda/x, and Dr. Noé counted the number of lines at the top (26) and bottom (25) of the grating using a printout of this photo. There was a 3.8% difference in the grating period, which was confirmed experimentally. We translated the grating up and down as far from the singularity of the fork as possible, and could observe the diffraction pattern expanding and contracting as the grating period changed.

The discovery of the first extrasolar planet via microlensing was announced yesterday. The gravity from the star bends light, creating a giant lens. The location of the Earth with respect to the extrasolar system must be perfect, which is probably why a planet has not been discovered using this long-theorized method until now. This chart needs to be updated now! The microlensing also permitted a red dwarf star to be seen for the first time.

Azure Hansen
April 2004
azure.hansen (a)
Laser Teaching Center